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Universidade de Sao PauloInstituto de Fısica
Melhorias na Predicao da Estrutura de LargaEscala do Universo por meio de Teorias Efetivas
de Campo
Henrique Rubira
Orientador: Prof. Dr. Marcos V. B. T. Lima
Dissertacao de mestrado apresentada ao Instituto deFısica da Universidade de Sao Paulo, como requisitoparcial para a obtencao do tıtulo de Mestre em Ciencias.
Banca Examinadora:Prof. Dr. Marcos V. B. T. Lima - Orientador (USP)Prof. Dr. Rogrio Rosenfeld (ICTP/IFT/UNESP)Prof. Dr. Miguel Boavista Quartin (UFRJ)
Sao Paulo2018
University of Sao PauloPhysics Institute
Towards Precise Large Scale StructurePredictions with Effective Field Theories
Henrique Rubira
Supervisor: Prof. Dr. Marcos V. B. T. Lima
Dissertation submitted to the Physics Institute of theUniversity of Sao Paulo in partial fulfillment of therequirements for the degree of Master of Science.
Examining Committee:Prof. Dr. Marcos V. B. T. Lima - Supervisor (USP)Prof. Dr. Rogrio Rosenfeld (ICTP/IFT/UNESP)Prof. Dr. Miguel Boavista Quartin (UFRJ)
Sao Paulo2018
La speranza è nell’opera.
Io sono un cinico a cui rimane
per la sua fede questo al di là.
Io sono un cinico che ha fede in quel che fa.
Vincenzo Cardarelli
Dedico primeiro ao meu irmão Fernando e meus primos, que serão companheiros da vida toda.
A minha mãe Nadya e meu pai Wagner que sempre priorizaram a educação dos filhos mesmo
nos momentos mais difíceis. E a todos meus amigos que estiveram por perto nesses anos.
Agradeço Prof. Dr. Marcos V. B. T. Lima, pelos cinco anos de orientação. Ao Prof. Dr. Rafael
L. Porto, pela ajuda gigante. Ao DESY e ao Prof. Thomas Konstandin, que me abrigaram no
mestrado e onde estarei no Doutorado.
Agradeço pelo auxílio financeiro a FAPESP/CAPES. Processo nº 2016/08700-0 e
2017/09951-9, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).
Agradeço também a Rodrigo Voivodic, Renato Costa, Vinicius Busti, Hugo Camacho e Michel
Aguena por todas as discussões e ajudas com a dissertação.
vii
Resumo
Com os próximos grandes projetos the observação do Universo, a cosmologia entrará
em uma era de alta precisão de medidas. Novos dados trarão um novo entendimento
da evolução do Universo, seus principais componentes e do comportamento da gravi-
dade. Sendo assim, é fundamental também ter uma boa predição teórica para a
formação de estrutura de larga escala em regime não-linear.
A melhor maneira de resolver as equações hidrodinâmicas que descrevem o nosso
universo é por meio de simulações cosmológicas na rede. Entretando, estas contém
desafios, como a correta inclusão de física bariônica e a diminuição do alto tempo
computacional. Uma outra abordagem muito usada é o cálculo das funções de cor-
relação por meio de métodos perturbativos (em inglês, Standard Perturbation Theory,
ou SPT). Entretanto, esta contém problemas variados: pode não convergir para algu-
mas cosmologias e, caso convirja, não há certeza de convergência para o resultado
correto. Além disso, há uma escala privilegiada nos limites integrais que envolvem o
método perturbativo. Nós calculamos o resultado por esse método até terceira ordem
e mostramos que o termo de terceira ordem é ainda maior que o de 2-loops e 3-loops.
Isso evidencia alguns problemas descritos com o método perturbativo.
O método de Teorias Efetivas de Campo aplicado ao estudo de LSS busca corrigir
os problemas da SPT e, desta forma, complementar os resultados de simulações na
rede. Em outras áreas da física, como a Cromodinâmica Quântica de baixas energias,
EFTs também são usadas como um complemento a essas simulações na rede. EFTs
melhoram a predição do espectro de potência da matéria por meio da inclusão dos
ix
chamados contra-termos, que precisam ser fitados em simulações. Estes contratermos,
que são parâmetros livres, contém importante informação sobre como a física em
pequenas escalas afeta a física nas escalas de interesse. Explicaremos os resultados
para a predição em 3-loops de EFT, trabalho inédito.
É possível usar as EFTs também no problema de conectar a campo de matéria
com outros traçadores, como os halos e as galáxias, chamado de bias. Com as EFTs
podemos construir uma base completa de operadores para parametrizar o bias. Será
explicado como utilizar esses operadores para melhorar a predição do bias em escalas
não-lineares. Serão calculados esses termos de EFT em simulações. Também será
mostrado como renormalizar o bias em coordenadas de Lagrange.
Por fim, será explicada outra importante aplicação das EFTs em cosmologia, mais
especificamente em teorias de inflação. EFTs parametrizam desvios nas teorias de um
campo único no chamado regime de slow-roll.
Palavras Chaves: Cosmologia, Estrutura de Larga Escala, Teorias Efetivas de Campo
Áreas do conhecimento: Cosmologia
x
Abstract
With future cosmological surveys, cosmology will enter in the precision era. New
data will improve the constraints on the standard cosmological model enhancing our
knowledge about the universe history, its components and the behavior of gravity. In
this context, it is vital to come up with precise theoretical predictions for the formation
of large-scale structure beyond the linear regime.
The best way of solving the fluid equations that describe the large-scale universe
is through lattice simulations, which faces difficulties in the inclusion of accurate
baryonic physics and is very computationally costly. Another approach is the theoreti-
cal calculation of the correlation statistics through the perturbative approach, called
Standard Perturbation Theory (SPT). However, SPT has several problems: for some
cosmologies, it may not converge and even when it converges, we cannot be sure it
converges to the right result. Also, it contains a special scale that is the loop momenta
upper-bound in the integral. In this work, we show results for the 3-loop calculation.
The term of third order is larger than the terms of 2-loops and 3-loops, making explicit
SPT problems.
In this work, we describe the recent usage of Effective Field Theories (EFTs) on
Large Scale Structure problems to correct SPT issues and complement cosmological
simulations. EFTs are used in other areas of physics, such as low energy QCD, serving
as a complement to lattice calculations. EFT improves the predictions for the matter
power spectrum and bispectrum by adding counterterms that need to be fitted. The
free parameters, instead of being a problem, bring relevant information about how
xi
the small-scale physics affects the scales for which we are trying to make statistical
predictions. We show the calculation of the 3-loop EFT counterterms.
EFTs are also used to explain main points connecting the matter density field with
tracers like galaxies and halos. EFTs highlighted how to construct a complete basis of
operators that parametrize the bias. We explain how we can use EFT to improve the
bias prediction to non-linear scales. We compute the non-linear halo-bias by fitting the
bias parameters in simulations. We also show the EFT renormalization in Lagrangian
coordinates.
Finally, we explain another critical EFT application to cosmology: in primordial
physics. It can be used to parametrize deviations to the slow-roll theory within the
inflationary paradigm.
Keywords: Cosmology; Large Scale Structure; Effective Field Theories
Knowledge Areas: Cosmology
xii
List of Figures
1.1. Self-portrait of Vincent van Gogh (1887) [1], which would have been a
great fan of EFTs if he had lived enough. We can see that the artist uses
thick brush strokes in his painting. Even though, we can still recognize
the men in the portrait. It happens because even with a painting that
does not contain the complete description of small scales, it contains
enough information in large scales that allow our brain to reconstruct
Vincent van Gogh’s image. That is the basis of EFT idea: we don’t need
the full UV complete theory, which may be very complicated. We only
need to access the relevant information to our portrait of the reality. . . 16
1.2. Representation of one of the main EFT concept in particle physics. In the left
we see the most complete theory, which works well at the UV. In this case
we have the 2 fermions and one vector field vertex. But at low energies we
can integrate out the vector field and consider an interaction of four fermions.
Extracted from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1. The nth-order kernel can be represented by a vertex with n internal legs
in the diagram. Extracted from [3]. . . . . . . . . . . . . . . . . . . . . . 23
xiii
List of Figures
2.2. Diagrammatic representation of the 1 and 2 loop calculations of SPT.
There are two diagrams in the 1 loop expansion. For the two loop there
are four diagrams expanding each one of the two δ of the power spec-
trum. The vertices represent the kernels. The internal lines represent
linear power spectra. Diagrams from [4]. . . . . . . . . . . . . . . . . . 25
2.3. Size of each SPT term separately. We can see that the 3Loop (pink) con-
tribution is larger than both 2Loop (blue) and 1Loop (red). This shows
the failure of SPT. The continuous line means a positive contribution,
while a dotted line means negative sign. . . . . . . . . . . . . . . . . . 26
2.4. Total SPT power spectrum in 1,2 and 3 loops. The results are compared
with the cosmic emulator [5] (PNLIN ). The gray lines show the 1% and
2% error limits. The 3Loops SPT fails even before the 1Loop and 2Loops. 27
2.5. Results for the 1Loop SPT Bispectrum calculated for two different trian-
gles configuration. To be compared with Figure 4 of [6]. We can see that
the result is very similar to what was expected and the small differences
can be attributed to different cosmologies. . . . . . . . . . . . . . . . . . 31
3.1. Diagrammatic representation of the 1loop matter power spectrum. We
need to add a counterterm, here represented by a crossed vertex inside
the parenthesis, that cancels out the UV cutoff (Λ) dependence of the
P13 diagram. When Λ k, we have the leading scaling of F3 being
k2/Λ2 such that P13 ∝ k2P0. That is parametrized in EFTofLSS by
adding an effective term in the stress tensor like ∂j∂iσij = c2s∂
2δ, where
c2s is an effective parameter that needs to be measured in small-scale
simulations. Figure from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . 34
xiv
List of Figures
3.2. Comparison between the non-linear emulator spectrum and differ-
ent EFT calculations up to 2 loops. In all of them we used the IR-
resummation of 1606.03633. For the 1Loop EFT (red) we fitted the spec-
trum in 0.15hMpc−1 < k < 0.25hMpc−1. The theory fails around k ∼
0.3hMpc−1 and we found c2s1 = 0.96(0.75)(1/hMpc−1)2. For the 2 Loops
EFT we compare two different counterterm structures (from [7] and [8]).
In both we fitted c2s1 also in 0.15hMpc−1 < k < 0.25hMpc−1 and set the
renormalization scale kren = 0.005hMpc−1 to determine c2s2 as described
in the text. For the 2LoopsEFT described in 1310.0464 (yellow) we found
c2s1 = 0.64(0.53)(1/hMpc−1)2. The theory has an 1% agreement until
k ∼ 0.27hMpc−1. For the 2LoopsEFT described in 1507.05326 (blue) we
found c2s1 = 0.44(0.48)(1/hMpc−1)2, c1 = 0.48(−1.6)(1/hMpc−1)2, c4 =
−1.12(−7.0)(1/hMpc−1)4. The theory has a 1% agreement until k ∼
0.58hMpc−1. The values in (red) reflect the results of [8], which we plot
in dashed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3. Test of how the choice of kren changes the spectrum. P 2LoopEFT and P 3Loop
EFT
divided by the respective standard spectrum, where by standard we
mean the one with kren1 = 0.005hMpc−1 and kren2 = 0.3hMpc−1. For
instance, the blue lines show the 2 loops EFT prediction Eq.3.17 using
kren1 = 0.008 divided by the same theory using kren1 = 0.005. We can
see that both choices do not affect the spectrum up to k ∼ 0.7hMpc−1. . 43
xv
List of Figures
3.4. 3LoopEFT result (pink with 5 free parameters using renormalization
conditions and black with all 7 parameters) compared with 1LoopEFT
(red) and 2LoopEFT (blue). Considering the 1% goal, the reach of
the theory is improved to k = 0.74hMpc−1, a sensible improvement
compared with k = 0.58hMpc−1 (k = 0.30hMpc−1) of the 2LoopEFT
(1LoopEFT). As the number of counterterms is increased from three to
five, we must be worried about overfitting and the physical predictabil-
ity of the theory. Comparing the 3LoopEFT with either five or seven
parameters, we see that the use of renormalization conditions does not
affect where the theory is 1% accurate. . . . . . . . . . . . . . . . . . . . 44
3.5. Same as Figure 3.4, now also showing the version without performing
the IR-Resummation in dashed lines with the respective colors. While
for 1LoopEFT and 2LoopEFT the wiggles amplitude cause a difference
of at most 2%, in the case of the 3LoopEFT the amplitude leads to a
difference of more than 100% (the flutuations are outside the picture
scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
xvi
List of Figures
3.6. Estimative of how much new information the 2Loop (blue) and 3Loop
(pink) terms bring to the spectrum. In the top panel we show the total
contribution and in the bottom we divided by the respective spectrum
to have a dimensionless quantity. For this, we fitted the counter-terms
of PZero in three different ranges, which we plot here with continuous,
dashed and dotted lines. The continuous lines represent the respective
fit range of each EFT order (0.15hMpc−1 < k < 0.6hMpc−1 for the
2LoopEFT and 0.15hMpc−1 < k < 0.75hMpc−1 for the 3LoopEFT). In
dotted line, we fitted in low k regime and in dashed we fitted in high k
regime. We can see by the bottom plot that the 2Loop term brings more
information on all scales. The 3Loop contribution was around zero in
almost all scales. Fitting in high k (dashed) the same happened. . . . . 47
3.7. Now we calculate PZero as defined by Eqs 3.21, 3.22 and 3.23 for the
best-fit value of the counter-terms that fit the emulator spectrum. In
top panel we show the absolute value and in bottom panel the ratio to
Pi, also defined in Eqs 3.21, 3.22 and 3.23. Solid lines represent the 1, 2
and 3Loops spectrum calculation, but using the best-fit sound speeds
from the 3LoopEFT. We see that the best-fits are not cancelling any of
the terms. For the dashed lines we used the best-fit of the respective fits
of the 1LoopEFT and 2LoopEFT. . . . . . . . . . . . . . . . . . . . . . . 48
3.8. How the parameters of the 3LoopsEFT change with the kmax used in the
fitting. The cs parameters do not change because we fit them at low k
regime. The shaded region is the 1σ error estimated. The regions where
the parameters are nearly independent of k (k ≈ 0.75hMpc−1) indicate
we are avoiding overfitting. . . . . . . . . . . . . . . . . . . . . . . . . . 49
xvii
List of Figures
3.9. 1Loop term of SPT calculated for Λ = 100hMpc−1 (yellow), Λ = 60hMpc−1
(red) and for Λ = 0.7hMpc−1 (black). The difference is minimal between
them. The red and yellow curves are nearly identical and appear over-
lapped. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.10. 2Loop term of SPT term calculated for Λ = 100hMpc−1 (yellow), Λ =
60hMpc−1 (blue) and for Λ = 0.7hMpc−1 (black). . . . . . . . . . . . . . 50
3.11. 3Loop term of SPT term calculated for Λ = 100hMpc−1 (yellow), Λ =
60hMpc−1 (purple) and for Λ = 0.7hMpc−1 (black). While the pink and
yellow curves are nearly indistinguishable, the difference relative to the
black line is huge all over the spectrum, and is particularly large close
to the lowest cutoff (kmax = 0.7hMpc−1). . . . . . . . . . . . . . . . . . 51
3.12. Checking the UV consistency of the theory for the 1LoopEFT (red),
2LoopEFT (blue), 3LoopEFT (pink). We try to address the following
question with this plot: by adding counterterms, can we cancel out
the UV dependence of the SPT terms? For P1, P2 and P3 the answer is
yes on all scales. For P3 we found a small disagreement at low k, but
if fails severely only in the region that was not used in the fits of the
counterterms (k > 0.75hMpc−1). Compare with Figure 4 of [8], where
they find a similar result. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1. Scheme of the connection of the universe initial conditions, the most
basic building block, to galaxy observations. While in chapter 3 the aim
was connecting primordial universe to the underlying matter statistics,
in this chapter we address the problem of connecting dark matter to
observables through the bias. Extracted from [3]. . . . . . . . . . . . . . 58
xviii
List of Figures
4.2. Computation of the single (left plot) and double (right) operators that renor-
malize the halo-halo and halo-matter power spectrum. We can see that our
results agree perfectly with Figures 3 and 4 of Assassi [3]. . . . . . . . . . . . 68
4.3. Halo mass function of the three DEUS simulation boxes. Each box
works well in the mass range it is expected to work (for instance, we
expect the large simulation to have less low mass halos). We compare
the halo abundance found with the prediction from Sheth and Tormen
(ST) [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4. Best fit of the halo-matter power spectrum for three different mass
bins (a,b,c). In each plot we show the residue between the theory and
data for three different counterterms structures: in red fitting only b1,
in blue fitting b1 and b2 and in green the full set of counterterms of
Equation 4.30. We also can see that the reduced χ2 is best for the EFT
prediction. In shaded lines we show the estimated error, which is larger
for the smaller mass bins (small box sizes) due to cosmic variance. . . . 71
4.5. Same as Figure 4.4 but now for the halo-halo power spectrum. As
expected, the cosmic variance effects for the halo-halo spectrum are
larger. The EFT prediction (green) and its reduced χ2 are still better. . . 72
4.6. Linear bias dependence with the mass. In bottom we divide by the
theory [10]. We did the fit using only the halo-matter spectrum (red),
only the halo-halo (blue) and both (green). The theory is consistent with
the data, especially for low masses. The black continue line show the
excursion set prediction for the bias [11]. . . . . . . . . . . . . . . . . . 73
4.7. Second order bias prediction for two theories: in the top, we fit only b1
and b2; in bottom we show the bδ2 we got fitting all EFT counterterms.
The black continue line show the excursion set prediction for the bias [11]. 74
xix
List of Figures
4.8. Mass dependence of the two EFT counterterms (bF and bI). We fit them
using the halo-matter spectrum (red), the halo-halo (blue) and both of
them together (green). We can see that the larger the mass of the halo is,
more important the counterterms are to fit the power spectrum. . . . . 75
5.1. Slow-roll potential. This potential is needed to generate an accelerated ex-
pansion of the universe after the Big Bang, in the Inflation period. Extracted
from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2. Planck 1σ and 2σ constraints on the tensor-to-scalar ratio r and on the spectral
index ns. The constraints are compared to the values expected by the main
inflationary models. We can see that Starobinsky Inflation [13], a simple R2
gravitation, is still inside all the confidence regions. Also the φ2 model is still
close to the contours. We can see that the scalar field theories with different
polynomial potentials are pretty acceptable, justifying the EFT parametrization
around this model. Result from [14]. . . . . . . . . . . . . . . . . . . . . . . 98
5.3. WMAP5 1σ, 2σ and 3σ constraints on the EFT of Inflation parameters c3 and
cs. Figure from [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.1. Description of time evolution of the modes. The interaction of two linear
modes gives rise to the second order expansion. And the interaction of
this with another linear makes the third order expansion. This coupling
can happen at any point of time, being non-local. That way we need to
integrate over time. Diagrams from [16]. . . . . . . . . . . . . . . . . . 113
C.1. Expansion parameters of the perturbation theory of Lagrangian and
Euler frameworks. We can see that εs,<k. grows faster than the others
and start being order higher than one in nonlinear scales. . . . . . . . . 132
xx
Contents
Resumo ix
Abstract xi
List of Figures xiii
1. Introduction 1
1.1. Standard cosmological model for physics . . . . . . . . . . . . . . . . . 5
1.1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2. Perturbing the Background . . . . . . . . . . . . . . . . . . . . . 6
1.1.3. The linear solution . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2. Trying to go beyond the linear regime . . . . . . . . . . . . . . . . . . . 10
1.3. Statistical measurements in physics . . . . . . . . . . . . . . . . . . . . . 14
1.4. The philosophy behind Effective Field Theories and the most famous
example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1. 4-Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2. SU(2)× U(1) Electroweak Theory . . . . . . . . . . . . . . . . . 18
1.5. Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6. Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
xxi
Contents
2. Standard Perturbation Theory 21
2.1. Euler Standard Perturbation Theory (SPT) . . . . . . . . . . . . . . . . . 21
2.2. Lagrangian Standard Perturbation Theory (LPT) . . . . . . . . . . . . . 28
2.2.1. Connecting Lagrange to Euler . . . . . . . . . . . . . . . . . . . 29
2.3. The 1Loop SPT Bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . 30
3. Effective Field Theory 33
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum . . . . . 33
3.1.1. The data and fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2. The Most General Counterterms Structure . . . . . . . . . . . . 35
3.1.3. The 1 and 2 Loops Calculation . . . . . . . . . . . . . . . . . . . 38
3.1.4. The 3 Loops Result . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.5. The IR Resummation . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.6. Avoiding Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.7. UV completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.8. Conclusion of the 3loops EFT . . . . . . . . . . . . . . . . . . . . 52
3.2. Lagrangian space EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4. The Bias Problem with EFT 57
4.1. Bias tracers, bias renormalization and its time evolution in Euler space 58
4.1.1. Euler bias: time evolution . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2. Euler bias: renormalization . . . . . . . . . . . . . . . . . . . . . 63
4.1.3. Euler bias: basis of operators and correlations . . . . . . . . . . 66
4.2. Halo Bias Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1. The data used and the fitting procedure . . . . . . . . . . . . . . 68
4.2.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
xxii
Contents
4.3. Lagrangian space renormalization . . . . . . . . . . . . . . . . . . . . . 75
4.3.1. Lagrangian Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2. Lagrangian Expansion . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.3. Renormalization of M2 . . . . . . . . . . . . . . . . . . . . . . . . 78
5. Inflation 85
5.1. Big Bang problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2. The Main Inflation Models . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3. Perturbations on the Inflaton . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4. Predictions of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5. Most General Action in the Unitary Gauge . . . . . . . . . . . . . . . . 91
5.5.1. The Slow-Roll in this Parametrization . . . . . . . . . . . . . . . 94
5.6. Action in the NGB Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.7. Constraints on Inflation and the EFT parametrization . . . . . . . . . . 97
6. Conclusion 101
A. The Fluid Equations 105
A.1. From Vlasov to Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.2. The Fourier transform of equations . . . . . . . . . . . . . . . . . . . . . 107
A.3. Perturbation theory in Einstein-de Sitter cosmology . . . . . . . . . . . 109
A.4. Matrix notation and the non-local in time propagator . . . . . . . . . . 112
A.5. Lagrangian Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B. Statistics Calculations complements and Diagrams 117
B.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.2. Perturbation to Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 118
xxiii
Contents
B.3. Halo bias counter terms derivation . . . . . . . . . . . . . . . . . . . . . 121
B.3.1. Halo-Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B.3.2. Halo-Halo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.3.3. F terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.3.4. I terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.3.5. Double I terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C. IR Resummation 127
C.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
C.2. Lagrangian Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
C.3. Euler Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
C.4. The resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Bibliography 134
xxiv
1Introduction
The concept of the western physical cosmology has drastically changed in the
twentieth century: going through Einstein spacetime, Hubble expansion, Penzias and
Wilson homogeneous and isotropic universe with the Cosmic Microwave Background
(CMB) [17] and the accelerated expansion of the universe obtained by the Supernova
groups [18, 19]. After Planck 2015 results [20] we know that the universe is very
well described by General Relativity (GR) with a dark energy component containing
almost 70% of the total energy density fraction, and the remaining 30% split between
non-interaction cold dark matter (CDM)(25%) and baryons (4.6%). This theory, called
ΛCDM, has essential predictions that are consistent with CMB experiments results:
1
1. Introduction
the Big Bang theory and the origin of the main elements of the universe1.
While the CMB improved our knowledge of the universe in the last 30 years [20,
21], in the next decades there will be large galaxies surveys [22, 23] that will allow
for vital improvements in cosmological constraints. Modern surveys in large-scale
structure (LSS) [24, 25] will probe cosmology with impressive accuracy and reach even
deeper regions of the space. With the improvements in experiments, which will bring
cosmology to the precision era, we also need to improve theoretical prediction of our
models. This dissertation focus on how we can get a better prediction for large-scale
structure formation via Effective Field Theories (EFTs).
The main observables of LSS surveys, which will be the main focus of this disser-
tation, are the N-point functions of the detected objects. The two-point function is
a measurement of the excess probability of finding two objects at a given distance
compared with a randomly generated sample. Its Fourier transform (FT) is called the
power spectrum. The three-point function represents the excess probability of finding
a triangular configuration, and its FT is called the bispectrum [26]. The importance
of statistical objects is that they allow us to reconstruct the evolution of the density
perturbations. The matter power spectrum, for instance, gives us information about
the baryon fraction, the dark energy equation of state, the dark energy fraction, the
primordial fluctuations amplitude and the spectral index. The matter bispectrum
gives one more important information: it can detect eventual non-gaussianities in the
universe evolution, which may be traced back to check eventually present primordial
non Gaussianities2.
This dissertation has the following structure: this introduction contains the backbone
of this work. In 1.1 we explain the standard cosmological model, the context in which
1Of course there still incomplete parts in the Big Bang theory that are very studied nowadays likebaryogenesis, which tries to explain the asymmetry between matter and anti-matter.
2Primordial non Gaussianities in the initial matter density field are one of the main predictions ofinflationary models [27].
2
this work is inserted. We also define main quantities, make clear the background
evolution and consider perturbations on this background. In 1.3 we further consider
the power power spectrum and the usage of statistics in cosmology. In 1.2 we compare
different methods for improving theoretical prediction for LSS. We also introduce
the Effective Field Theory approach as a method that fixes the problems of the other
techniques and serves as a complement to cosmological simulations. Finally, in 1.4 we
provide a more general panorama of EFTs, explaining how it works in other areas of
physics.
In Chapter 2 we explain the Standard Perturbation Theory (SPT) [26], a method
that attempts to improve the prediction of the power describing perturbations on
the matter density field. The correlation of this perturbative modes will give rise to
loop integrals, very common in Quantum Field Theory (QFT). However, even though
in QFT these loops represent virtual particles, here they are nothing more than a
statistical description of different realizations of initial conditions [16]. We calculate
this expansion until the 3loops order and show the main problems of this theory that
EFT will try to fix. We also show how to do the perturbation in the Lagrangian frame,
which instead of using the fluid density and velocities field (Euler frame), takes the
fluid lines as a referential frame and calculate their displacements.
In Chapter 3 we describe the Effective Field Theory approach to the Large Scale
Structure problem. We show how EFT will fix SPT problems through the right descrip-
tion of small-scale physics3. Up until now, the EFT was calculated only until 2loops
order [7]. We calculate the 3loops Effective Field Theory and show its results.
In Chapter 4 we address the issue of how to relate the statistics of the surveys
observables (galaxies, quasars, etc.) with the underlying dark matter density field.
While the formation of these objects is dependent on baryonic physics and challenging
3In this dissertation, we call the small-scale physics as UV. UV in Effective Field Theory is every modek that is above an energy cutoff limit Λ, which is integrated out. Integrating out these modes willbring the EFT counterterms explained in the body of this dissertation.
3
1. Introduction
to model, the evolution of the dark matter non-interacting particles is predictable on
large scales. The relation between the tracers and the dark matter field is called bias.
We expect the tracers to follow the gravitational potential in large scales, which are
dominated by the dark matter component. We call the bias in this large-scale regime
the linear bias. In not so large scales we expect non-linear physics to contribute. We
explain how to use Effective Field Theories to improve the predictions of the halo-bias.
We explicitly show measurements made on simulations for the EFT halo-bias terms
and how they change with the halo mass. And finally, we show how to extend the EFT
halo bias approach to the Lagrangian frame. We now briefly explain the big picture of
how to apply EFTs to the halo problem, which will be better explained later.
The so-called Eulerian bias tries to find out the final time best description of the
relation between both halo and matter overdensities δh(x,τ) = F [δ(x,τ)]4. Basically
Eulerian picture parametrizes in relevant bias parameters the unknown physics [28,29].
The usual procedure to deal with the halo bias is to assume that we can use a Taylor
series [30] in the local matter density5:
δh(x,τ) = F [δ(x,τ)] =∞∑i=0
bii! δ
i(x,τ). (1.1)
But considering the above expansion, the terms of order higher than one receive
contributions from all scales 6, and we can no longer assume that they are small and
our series would converge. One way to deal with this problem is to renormalize our
theory [28, 32, 33]. Splitting δ = δL + δS , such now we have a short-wavelenght and a
4We will use δ = δm to refer to the matter overdensity field. We also refer to the matter overdensity,sometimes, simply as matter density. We will not work with the local density field by itself in thisdissertation.
5 [31] uses the nomenclature Local-in-Matter-Density (LIMD) as a nice disambiguation, because theremight be a lot of other local operators like (∂i∂j∂kΦ)2.
6Taking the FT, the product is a convolution and this convolution is integrated over all scales. InQuantum Field Theory (QFT), two operators evaluated at the same point are called a compositeoperatorand they are very sensitive to small distance UV physics.
4
1.1. Standard cosmological model for physics
long-wavelenght term7 it is possible now to Taylor expand only in the long-wavelenght
overdensity. But of course this division is arbitrary and sets a cutoff scale Λ. To cancel
out this dependence and to parametrize the hard modes integrated out we must add
counterterms [3]:
δh(x,τ) = b1δL(x,τ)︸ ︷︷ ︸Linear Bias
+ b2
2 δ2L(x,τ)︸ ︷︷ ︸
Second Order Bias
+∑
bOO(x,τ)︸ ︷︷ ︸Counterterms
+ . . . (1.2)
Finally, we leave to the following section of this chapter a discussion of Effective
Field Theories in general. In Chapter 5 we talk about another successful application of
EFTs, this time to parametrize different models of primordial physics. We conclude in
Chapter 6.
We stress that in order to make this dissertation more readable, we placed most of
the explicit calculations on Appendices. We preferred to leave in the main text mostly
the physical meaning and central concepts, which in the end are the most exciting part.
Also – if the reader hasn’t noticed yet – the author has an inclination for footnotes8.
Since many jargons of EFT-of-LSS are a heritage from particle physics, which may get
complicated fast, the author will use footnotes to explain them.
1.1. Standard cosmological model for physics
1.1.1. Background
The metric that satisfies the Cosmological Principle, which determines the isotropic
and homogeneous universe, is known as Friedmann-Robertson-Walker metric (FRW).
7We define δL as a convolution over a filter WΛ at a scale Λ: δL =∫d3yWΛ(x− y)δ(y). In the Fourier
space this convolution become a simple multiplication.8Which sometimes may be very pedantic, but we hope they contain useful complements.
5
1. Introduction
With a being the scale factor and K the universe curvature, we have:
ds2 = −dt2 + a(t)2[ dr2
1−Kr2 + r2(dθ2 + sin2 θdφ2)] . (1.3)
The time evolution of the scale factor is given by the Friedmann equations, which are
just the Einstein equations solution for the FRW space-time:
H2 ≡ a2
a2 = 13M2
PL
ρ− K
a2 , (1.4)
H +H2 = −16(ρ+ 3p) . (1.5)
The evolution of the scale factor in an Einstein-de Sitter universe (just non-relativistic
matter) is a ∼ t2/3 and for the deSitter universe (only cosmological constant) is a ∼ eHt.
These predictions lead to the Big Bang picture, a singularity in space-time in the past.
The Big Bang scenario was confirmed with the Cosmic Microwave Background (CMB)
detections.
To work with Inflation theory, it is convenient to define the conformal time τ , such
that the metric is now conform:
τ =∫ τ
0
dt
a(t) , such that ds2 = a(t)2(−dτ 2 + dr2
1−Kr2 + r2(dθ2 + sin2 θdφ2)).
(1.6)
1.1.2. Perturbing the Background
Now that we have described the simplest universe, isotropic and homogeneous,
we will describe the nature of perturbations in the FRW metric 1.3. We follow [34]
and [26].
There are three possible ways to perturb the FRW metric: scalar, vectorial and
6
1.1. Standard cosmological model for physics
tensorial perturbations. Tensorial perturbations are essential for Inflation theory,
where they produce CMB polarization, and also in the gravitational waves study.
Vector perturbations decay as 1/a2, and are not relevant for this study. The scalar
perturbations will be the most important in this dissertation 9.
Another important note is that we can choose an appropriate gauge, called Newtonian
conformal gauge , in which scalar perturbations are described by the following fields
Ψ(~x,t) and Φ(~x,t):
ds2 = (−1− 2Ψ)dt2 + δija2(1 + 2Φ)dxidxj , (1.7)
where we used Equation 1.3 in cartesian coordinates and considered a universe without
curvature (K = 0). Notice that the metric now depends also on the spatial coordinates
~x.
We can calculate the zeroth order perturbation in Einstein equations10 in Fourier
space as:
δG00 = −6HΦ,0 + 6ΨH2 − 2k
2Φa2 , (1.8)
(traceless)δGij = 2
3a2k2(Φ + Ψ). (1.9)
Where by traceless, we mean applying the traceless operator to take the non-degenerate
longitudinal part of Einstein equations11. With the geometrical quantities in hands we
still need to calculate the energy-momentum tensor, the RHS of Einstein equations for
each component. Here we focus only on matter field perturbations, since these are the
9It is important to note that by the decomposition theorem we can study scalar and tensorial, perturbationsseparately, because they decouple in linear regime.
10Neglecting non-linear perturbations contribution.11For completeness, this operator is simply kik
j − 13δijk
ikj , whose trace is zero.
7
1. Introduction
most important in the universe recent evolution12.
A consequence of neglecting radiation component is that there is no anisotropic
stress and Equation 1.9 gives Ψ = −Φ. The interpretation of Equation 1.8 is also pretty
clear: it is a simple Poisson term −k2 × Grav. Potential with a relativistic correction.
In the limit of scales smaller than the Horizon scale (k H) we recover Newtonian
physics.
Since for Baryons and Dark Matter we define the density contrast δ as:
T 00 = −ρ[1 + δ], (1.10)
where bar means background quantities. We have by Einstein equation 1.8 in the
Newtonian limit:
k2Φ = 32ΩmH2δ. (1.11)
WithH = Ha. Now that we have an equation describing the gravitational potential
changes with the matter density, we need an equation that describes how matter
will change with the gravitational potential. The reader will find in A.2 a complete
calculation through Boltzmann equations, that now we summarize as :
∂τδ + ∂i[(1 + δ)ui] = 0, (1.12)
and:
∂τui +Hui + ui∂ju
j = −∂iΦ−1ρ∂j(ρσij), (1.13)
which as we can see, is a non-linear solution. By u we mean the velocity field and σ is
12Of course radiation is important in the beginning of the universe evolution and for generating initialconditions for matter. But since matter perturbation evolves mostly in a matter dominated universeand we are dealing only with late evolution time (z < 100) we can neglect this component. Baryoniceffects here are also neglected and treated as dark matter. For very large scales, whis is a goodapproximation, but for scales k > 0.5hMpc−1 we need to start considering them [35]
8
1.1. Standard cosmological model for physics
the energy-momentum tensor.
1.1.3. The linear solution
We now go ahead and linearize the equations and solve them in the regime where
field perturbations are small. We consider the fields to be irrotational and in the regime
of zero stress tensor, valid on large scales. The full discussion of this approximation
and an attempt to solve the full non-linear equation is done in Chapter 2. The linear
solution reads (already using 1.11):
∂τδ + θ = 0,
∂τθ +Hθ = −32ΩmH2δ, (1.14)
where θ = ∂iui. Solving the system of equations (differentiate in time the first equation,
replace the second in this new equation and change θ = −∂τδ again):
d
dτ 2 δ +H∂τδ = 32ΩmH2δ, (1.15)
which is a second order equation. Writing δ(x,τ) = D(τ)δ(x,τ = 0) gives us two
solutions, the growing and decaying modes (D+ and D−). D+ is also called growth
factor. The solution, of course, will depend on the universe contents, but for Einstein-
de-Sitter universe (only matter) it will give the known growth function that scales
with the scale factor: D+ = a [26].
The following section will briefly discuss attempts to solve the non-linear equation,
which is the main topic of this dissertation.
9
1. Introduction
1.2. Trying to go beyond the linear regime
The behavior of the dark matter field in the linear regime is a problem that can
be solved by an Einstein-Boltzmann integrator code [36, 37], which also provides
appropriate initial conditions and evolution during radiation dominated era. However,
differently from the CMB, which is very well described by the linear physics since
decoupling at the beginning of the universe, the matter field is not. The matter
overdensity evolved substantially by the gravitational influence, which is non-linear.
Also, the range of scales probed by future surveys and the BAO peak are outside the
linear scale (k < 0.1hMpc−1). That justifies the need for accurate non-linear predictions
for the matter evolution.
In past years, several methods of reaching non-linear scales for the matter over-
density statistics predictions arose. Since it is an essential point for the motivation of
Effective Field Theories (EFT), we will explain some of these methods, their advantages,
and failure points.
• Standard Perturbation Theory (SPT): consists in perturbing the matter over-
density field assuming small perturbations for large enough scales and solving
the fluid equations perturbatively. The hope is that by going to higher orders
in perturbation will improve the calculation. Each term in the perturbation is
associated with a diagram. The first problem with this method is that it does not
converge and going to higher orders in perturbation (loops) does not necessarily
improve the prediction. Another problem is that the integrals that appear in
the loops require in practice a UV cutoff (Λ), which has no physical meaning.
The last problem is that the meaning of the loops is the interaction between
modes of different scales. But we do not know how small-scale physics behave
accurately enough [16], leading us to conclude that even if SPT converges, it is
not converging to the right result. We explore much more about SPT in Chapter 2
10
1.2. Trying to go beyond the linear regime
and we will see how EFT fixes all these problems.
• Resummation Methods13: it corresponds to a set of methods that try to fix SPT
by rearranging the order of its terms and constructing a non-linear propagator
[38–43]. Despite sometimes being called renormalization, this class of methods
does not incorporate the UV physics in their calculation. They fix IR problems
of the fluid approximation, but it is not reasonable to expect improvements in
the UV prediction correcting long-wave modes physics. Also, the problem of the
non-physical cutoff Λ in the SPT integrals persists, breaking Galilean symmetry.
Even though correcting the BAO peak, the corrections of the two-loop RegPT [43]
do not beat the percent precision of EFT (even for z = 1) and do not reach scales
k > 0.2hMpc−1.
• Vlasov solvers: we cite here for completeness the Vlasov solvers [44], which
despite having singularities problems and not presenting virialization (they fail
in the shell crossing limit), reproduce the moments of Boltzmann distribution
very well.
• Schrodinger method: we would like to cite also this method that is gaining a lot of
attention after the Fuzzy Dark scalar particle models [45, 46]14. The Schrodinger
method is based on the fact that the system of a Schrodinger equation with a
Poisson potential coincides with the Vlasov set of equations, and a map between
both can be done through a Wigner transformation [47]. Here the parameter
~/mscalar15 is treated as a resolution parameter. It was shown that this method
does not suffer from singularities coming from shell crossing, reproduces the
moments of Boltzmann distribution [48] and has structure formation [49]. The13They were also dubbed Γ expansion, multi-point propagator, eikonal approximation or even renor-
malization methods.14Despite the fact that it is not necessarily motivated by Fuzzy Dark Matter models but in a coincidence
of Schrodinger equation with a Poisson potential and Vlasov equations.15Where mscalar is the mass of the scalar particle
11
1. Introduction
problem is that only recently it was solved in two dimensions [50] and solving
for D = 3 is still a problem.
• Halo Model: the Halo Model [51] is one of the most beautiful theories without any
free parameter to predict the matter power spectrum. Based on the assumption
that all matter is inside halos, the power spectrum calculation naturally is split
into a 1-Halo term (correlation of the matter particles inside a single halo) and
a 2-Halo term (correlation of a particle from one halo with another one from a
different halo). The main ingredients of the Halo Model are the halo density
profile, its mass function, and the halo bias. The pros of the Halo Model are that
its predictions are right (usually 5% agreement) for very large scales and also
very small (halo scales). The problems are in the transition between these two
scales, where neither the 1-Halo term nor the 2-Halo (nor their sum) can capture
the right physics.
• Fits to simulations(Halofit, Coyote): there are several attempts to fit formulas for
the non-linear power spectrum [5, 52, 53]. The last Halofit formula [53] contains
35 free parameters fitted to simulations around the WMAP cosmology [21]. Their
range of precision is of 5% for k < 1hMpc−1(for z < 10) and 10% on small scales
(5% for k < 1hMpc−1 for z < 3). Of course, this kind of fit is good for its purposes,
but the huge number of free parameters can overshadow the physics. Also, there
is no guarantee that these fit values are valid for higher order statistics and for
cosmologies far from where they were calibrated.
• Numerical simulations: they use brute force to time evolve initial conditions16
calculating the particle interactions. They are the most trusted method, but
the very fast scaling in computational time with the number of particles make
them impossible to use for constraining cosmology through a Markov Chain16Generated displacing particles according to a linear power spectrum [54].
12
1.2. Trying to go beyond the linear regime
Monte Carlo (MCMC). Two widely used simulation codes are RAMSES [55]
and GADGET2 [56]. We refer the reader to [57] for a comparison of methods.
Another difficulty of working with simulations is to include the baryonic physics
interaction with Cold Dark Matter [58, 59], which start becoming important for
k > 0.5hMpc−1 [57]. Here we don’t want to dismiss the usage of simulations
since they are a fantastic and well-tested way to overkill the problem. But we
suggest using lattice as a partner of Effective Field Theories, as it is usually done
in other areas of physics, such as Quantum Chromodynamics.
• EFT: The primary goal of using EFT is correcting the problems of the methods
described above. The Effective Field Theory (EFT) approach is by construction
the right way to understand how small scales affect the scale of interest and to
parametrize our ignorance about these unknowns. The price to pay in the EFT
approach is the inclusion of counterterms to be fitted in small-scale simulations
to account for UV effects. The number of counterterms can grow fast with the
loop order of calculation17. While in one loop level it was shown that one free
sound speed parameter is enough, in two loops it is necessary to include three
free parameters [8]18. EFT corrects SPT the right way: we cancel out the UV-
cutoff reliance, guarantee the convergence of the series and insert the proper
small-scale modes interaction. It is also the unique method that ensures a 1%
precision within its regime of validity, something not achieved by neither one of
the above-described methods. We describe more about the EFT calculation for
the matter-matter power spectrum in Chapter 3. There we present the 3LoopEFT
calculation that will be submitted for publication soon. We also show the one
loop bispectrum calculation in SPT.
17One vital point here is that having free EFT parameters, even if many, is not trouble but a solution.Each one of the counterterms has a clear physical meaning in a series expansion.
18In [7] they use only one free parameter, but later on in [8] they conclude that to remove the UVdependence of the theory properly, we need three free parameters.
13
1. Introduction
1.3. Statistical measurements in physics
Statistics are fundamental to cosmological analysis. Differently from other areas,
in which an experiment can be done numerous times, we live in a single observable
universe. We have only one realization for our experiment and its initial conditions19
are stochastic. The best, and often the only, way to extract information from the matter
perturbations is through statistical analysis.
The probability functions used to put cosmological constraints in large scale struc-
ture are the N-point functions. The 2-point correlation function, for instance, measures
the excess probability, relative to a random distribution, of finding two objects at a
specific distance r. In fluid coordinates, the 2-point correlation function ξ(r) is defined
as:
ξ(r) = 〈δ(x)δ(x + r)〉 (1.16)
such that its Fourier transform, the matter power spectrum, can be calculated as:
〈δ(k)δ(k′)〉 = δD(k + k′)P (k), (1.17)
with:
P (k) =∫ d3r
(2π)3 e−ik.rξ(|r|) . (1.18)
In Equation 1.17 we highlight what we call a statistical homogeneous cosmic field
through momentum conservation: the Dirac’s delta function guarantees the invariance
under translation. What we call a statistically isotropic field is evidenced by the non-
dependence of P (k) with the direction of k vector.
Likewise, we can calculate the probability of finding a triangular configuration (the
3-point function) and its Fourier transform, the bispectrum. The bispectrum may
19Considering the inflationary paradigm that relates the initial condition with fluctuations in a primor-dial Inflation field. This initial conditions may be generated by stochastic quantum fluctuations.
14
1.4. The philosophy behind Effective Field Theories and the most famous example
depend on three coordinates usually parametrized as two triangle sides and one angle
or the three triangle sides. The excess probability of a quadrilateral configuration is
the 4-point function and its Fourier transform is the trispectrum. The bispectrum is
defined as :
〈δ(k1)δ(k2)δ(k3〉 = δD(k1 + k2 + k3)B(k1,k2,k3). (1.19)
Of course, measuring higher points statistics becomes harder as the number of points
increases. But all N-point statistics contain essential information that together allows
us to probe different aspects of cosmology. While the 2-point function measures
the variance in the density field on different scales, the 3-point function measures
deviations from a Gaussian field.
For a perfect Gaussian field, Wick’s Theorem relates higher order N-point functions
to the 2-point correlation function. For odd N, the N-point function is zero. For even N
it can be calculated as sums of products of correlations of lower odd N-point functions.
1.4. The philosophy behind Effective Field Theories
and the most famous example
The purpose of this section is to discuss ideas behind Effective Field Theories (EFTs)
with an example of a very famous artist: Vincent van Gogh. The artist is known for
his instantly recognizable self-portraits 1.1. But the impressionist artist also is also
well known for using thick lines with his brush, a feature present in the work of other
painters of the same epoch such as Claude Monet. One question that comes is how
our brain can process the image such that even with the thick brush stroke we can
still recognize the man in the painting? The answer is that we do not need the best
resolution possible but only the relevant information that allows us to link the traces
we know about Van Gogh, the man, with the picture. We recognize large-scale patterns
15
1. Introduction
that may emerge from the small-scale description as the most relevant information.
The idea of EFTs is precisely the same.
Figure 1.1.: Self-portrait of Vincent van Gogh (1887) [1], which would have been agreat fan of EFTs if he had lived enough. We can see that the artist usesthick brush strokes in his painting. Even though, we can still recognizethe men in the portrait. It happens because even with a painting that doesnot contain the complete description of small scales, it contains enoughinformation in large scales that allow our brain to reconstruct Vincent vanGogh’s image. That is the basis of EFT idea: we don’t need the full UVcomplete theory, which may be very complicated. We only need to accessthe relevant information to our portrait of the reality.
Our second example will be about one of the most basic physics equation F = ma
[60]. Newton stated the proportionality between force and acceleration but never
told us how to measure the mass as a sum of the most fundamental constituents of
particle physics. He suggested, otherwise, different experiments that would allow us
to measure this effective parameter.
In the last 40 years, the Effective Field Theories (EFTs) are at the center of particle
physics discussions. The point is that to describe a phenomenon that happens at a
certain energy scale we do not need to know what is going on higher energy scales.
We can encode the effects of this physics in the coefficients of the higher dimension
16
1.4. The philosophy behind Effective Field Theories and the most famous example
operators of the theory. This way, EFTs are a hierarchical construction of physics, where
it is clear which are the relevant operators at a given energy scale. The emergence
of EFTs appeared first in condensed matter physics and later particle physics. For
examples of EFT applications in Nuclear physics, see [61]. For a broad review of EFTs
with a large set of examples and discussion see [62]. For applications of EFT in the
gravitational waves problem see [60]. For EFTs in Inflation see [63] and 5 . For the EFT
applications of quantum gravity (and the emergence of modified gravities as EFTs)
see [64].
Here we will present a basic example of EFT in the Standard Model (SM) of particle
physics, namely the 4-Fermi Theory. This section is based on [2].
1.4.1. 4-Fermi Theory
At the early 1930s, it was well known the β decay, in which a proton can decay to a
neutrino, a positron and a neutron. The correspondent Lagrangian written by Enrico
Fermi to describe this interaction was [2]:
LFermi = GFermiψpψnψeψν . (1.20)
If we count the dimensions, since each fermion ψ has energy dimension 3/2, we
have that GFermi goes with −2. Our theory is therefore non-renormalizable in loops
calculation, since we need infinity counter-terms δF , δ1, δ2, . . . in the bare Lagrangian
for the theory to became measurable:
L = ZFGF ψψψψ + Z1a1G2F ψψψψ + Z2a2G
3F ψ∂ψψ∂ψ + . . . , (1.21)
with Zi = 1 + δi.
17
1. Introduction
The GFermi constant measured has the value of [2]:
GFermi =( 1
292.9 GeV
)2. (1.22)
What makes 4-Fermi theory predictable is that it has fermions with mψ G−1/2Fermi.
The consequence is that in low energy experiments we should have higher powers of
GFermi terms in the Lagrangian becoming negligible. But when the experiment energy
raises to 292.9 GeV our perturbation theory is no longer valid and we should consider
a UV theory. The UV theory, in this case, is the SM electroweak Lagrangian, which we
will describe later.
This is Effective Field Theory in front of our eyes! At low energy limit, we don’t
need to care about what is going on at the UV. The degrees of freedom of high energy
physics are integrated out in low energy.
1.4.2. SU(2)× U(1) Electroweak Theory
The Electroweak model is a symmetry breaking model of SU(2)×U(1)Y → U(1)EM
L = −14(W a
µν)2 − 14B
2µν + (DµH)2 +m2H†H − λ(H†H)2. (1.23)
The first two terms are the kinetic term for the vector fields of the SU(2) × U(1)Y
groups local symmetry. Dµ is the covariant derivative, and H is the Higgs field. When
the Higgs field assumes a vacuum expectation value (vev), we have the symmetry
break. When it happens, we can rewrite our W aµν and Bµν as other 4 fields: Zµ, W±
µ and
Aµ. The first three are massive bosons, and the last one is the massless photon field.
The purpose here is not to explain entirely the SM of particle physics, but to illustrate
its EFT mechanism. This shows how a sophisticated model of high energy physics has
its degrees of freedom integrated out in low energy.
18
1.5. Notations and Conventions
The measured masses of the fields in the electroweak theory above are:
mZ = 91.2 GeV mW = 80.40 GeV mh = 126 GeV. (1.24)
With this values we can understand why we do not see the influence of these fields
at experiments with energy around a few MeV . We can consider an effective action
schematically as:
∫ΠDFermionsΠDEWFields exp (iSEW ) =
∫ΠDFermions exp (iSEff ). (1.25)
The Figure 1.2 shows how, at low energy, we can integrate out the vectorial degrees
of freedom of our theory and model it as a four fermions effective interaction.
Figure 1.2.: Representation of one of the main EFT concept in particle physics. In the left wesee the most complete theory, which works well at the UV. In this case we have the2 fermions and one vector field vertex. But at low energies we can integrate outthe vector field and consider an interaction of four fermions. Extracted from [2].
1.5. Notations and Conventions
For completeness, in this section we define the notations and conventions used
throughout this work.
• For∫ d3p
(2π)2 we use the short notation∫
p;
• We use boldfont and arrows above to describe vector quantities;
19
1. Introduction
• We use ′ to omit the momentum conservation in the power spectrum, for instance:
〈δ(k)δ(k′)〉′ = P (k) instead of 〈δ(k)δ(k′)〉 = δD(k + k′)P (k).
1.6. Acronyms
Acronym Definition
CMB Cosmic Microwave Background
LSS Large Scale Structure
FT Fourier Transform
EFT Effective Field Theory
(S)PT (Standard) Perturbation Theory
LPT Lagrangian Perturbation Theory
MCMC Monte Carlo Markov Chain
LIMD Local In Matter Density field
QFT Quantum Field Theory
SC Spherical Collapse
GR General Relativity
CDM Cold Dark Matter
FRW Friedmann Robertson Walker
EdS Einstein-de-Sitter
BAO Baryonic Acoustic Oscillation
20
2Standard Perturbation Theory
2.1. Euler Standard Perturbation Theory (SPT)
In this section we describe the Standard Perturbation Theory (SPT) approach for the
calculation of matter 2-point statistics [26]. The dynamics of the matter overdensity δ
and the velocity u can be derived directly from Boltzmann equation A.1 or through
21
2. Standard Perturbation Theory
the Einstein energy conservation to be:
∂τδ + ∂i[(1 + δ)ui] = 0, (2.1)
∂τui +Hui + ui∂ju
j = −∂iΦ−1ρ∂j(ρσij) ,
∂2Φ = 32ΩmH2δ .
The first equation is the continuity equation for matter density sourced by the velocity.
The second equation states the local conservation of momentum sourced by the stress
tensor σ and gravity. The last equation is the Poisson equation for the gravitational
potential Φ in the matter dominated era. Ωm is the present matter density fraction
relative to critical and H = Ha. Considering u to be irrotational and the fluid to be
collisionless, it can be described by its divergence θ = ∂iui1. In Fourier space, Eqs. 2.1
become (see A.2):
∂τδ(k,τ) + θ(k,τ) = −∫dk3
1dk32 δD(k− k1 − k2)α(k1,k2)θ(k,τ)δ(k,τ) , (2.2)
∂τθ(k,τ) +Hθ(k,τ) + 32ΩmH2δ(k,τ) = −
∫dk3
1dk32 δD(k− k1 − k2)β(k1,k2)θ(k,τ)δ(k,τ) ,
1Here we discuss the validity of these two approximations: the irrotational flow and the collisionlessapproximation (null stress tensor σ). The validity of the non-rotational approximation is based onthe fact that we can always decompose a vectorial field in a scalar part θ defined by the divergentand a curl part W. But the curl part decays with the universe expansion as W ∝ a−1 [26]. Alsothe only source of W is the stress tensor, which is being neglected here. The point that underlinesneglecting the stress tensor is considering the perfect fluid approximation without shell crossingof the fluid lines. Of course this approximation, as time evolves, should start failing in even largerscales. But we expect it to be a good approximation around linear scales. One of the things EFT willcorrect in SPT is parametrize the effective stress tensor.
22
2.1. Euler Standard Perturbation Theory (SPT)
with α and β the well known coupling functions that generate nonlinearities:
α(~k1,~k2) = (1 +~k1 ·~k2
k22
),
β = |~k1 + ~k2|2(~k1 · ~k2)
2k21k
22
. (2.3)
We can also see that neglecting the RHS we have the usual linear solution2. As 2.2 is a
highly non-linear equation we use perturbation theory separating time (through the
growth function D) and space coordinates like:
δ(k,τ) =∑
Dn(τ)δ(n)(k),
θ(k,τ) =∑
Dn(τ)θ(n)(k). (2.4)
That way, we can put 2.4 in 2.2 such that we find (with kT = ∑ni=1 ki):
δ(n)(~k) =∫d3k1 . . . d
3knδD(~k − ~kT )Fn(~k1, . . . ,~kn)δ(1)(~k1) . . . δ(1)(~kn),
θ(n)(~k) =∫d3k1 . . . d
3knδD(~k − ~kT )Gn(~k1, . . . ,~kn)δ(1)(~k1) . . . δ(1)(~kn), (2.5)
...
Figure 2.1.: The nth-order kernel can be represented by a vertex with n internal legs inthe diagram. Extracted from [3].
The kernels F and G in Equation 2.5 are represented by the diagram in Figure 2.1
2After setting the RHS to zero, differentiating the first equation of 2.2 and replacing the second inthe first we find a second order equation. The solutions are the growing mode D and the decayingmode A.4.
23
2. Standard Perturbation Theory
with:
Fn(k1, . . . ,kn) =n−1∑i=1
Gi(k1, . . . ,ki)(2n+ 3)(n− 1)×
[(2n+ 1)α(k1,k2)Fn−i(ki+1, . . . ,kn) + 2β(k1,k2)Gn−i(ki+1, . . . ,kn)] ,
Gn(k1, . . . ,kn) =n−1∑i=1
Gi(k1, . . . ,ki)(2n+ 3)(n− 1)×
[3α(k1,k2)Fn−i(ki+1, . . . ,kn) + 2nβ(k1,k2)Gn−i(ki+1, . . . ,kn)] . (2.6)
Also we have F1 = G1 = 1. A derivation of Eqs. 2.6 is shown in A.3. Them we solve
for δ and θ order by order using 2.63. In Appendix A.4 we show the same equations
in the matrix notation usually found in literature4. The respective terms in each loop
order are:
P 1LoopSPT = P0 + 2P13 + P22︸ ︷︷ ︸
P1
;
P 2LoopSPT = P 1Loop
SPT + 2P15 + 2P24 + P33I + P33II︸ ︷︷ ︸P2
;
P 3LoopSPT = P 2Loop
SPT + 2P17 + 2P26 + 2P35I + 2P35II + P44I + P44II︸ ︷︷ ︸P3
, (2.7)
and their diagrammatic pictures are shown in Figure 2.2. We highlight the notation:
for the complete spectrum in i loop order we label P iLoopSPT , for the sum of terms that
belong only to one order i of loop we call Pi and for each diagram we call Pij , with i
and j being the number of order expansion of each one of the two vertices. For the
3We underline here one important approximation made. The separation of time and space in thesolution of perturbation theory is only valid in an Einstein-deSitter cosmology. The usage of thegrowth function D is an approximation for a general cosmology known to be correct within a 1%error [26].
4There is also another important point treated in A.4 that is more developed in a few resummationmethods attempts [38]. When we expand as Eq. 2.4 we are considering the interaction to be local intime. However this is only approximately true on sub- horizon scales, since the typical time scale isHubble H−1. A more complete solution should include a time integral in the coupling of the modes.
24
2.1. Euler Standard Perturbation Theory (SPT)
linear matter power spectrum we use P0, P11 or even PL. The difference between each
term in Eqs. 2.7 is how to contract each leg in the expanded δ(n) during the Wick’s
Theorem usage.
Plin
Plin
P22
F2 F2
Plin
Plin
P13
F3 Plin
Plin
2
Plin
Plin
P22
F2 F2
Plin
Plin
P13
F3 Plin
Plin
2
(a)
P11
Plin
P22
F2 F2
Plin
Plin
P13
F3 Plin
Plin
P15
F5 Plin
Plin
Plin
P24
F4 F2
Plin
Plin
Plin
P33I
F3 F3Plin
Plin
PlinP33II
F3 F3Plin
Plin Plin
2
P11
Plin
P22
F2 F2
Plin
Plin
P13
F3 Plin
Plin
P15
F5 Plin
Plin
Plin
P24
F4 F2
Plin
Plin
Plin
P33I
F3 F3Plin
Plin
PlinP33II
F3 F3Plin
Plin Plin
2
(b)
Figure 2.2.: Diagrammatic representation of the 1 and 2 loop calculations of SPT. Thereare two diagrams in the 1 loop expansion. For the two loop there are fourdiagrams expanding each one of the two δ of the power spectrum. Thevertices represent the kernels. The internal lines represent linear powerspectra. Diagrams from [4].
For clarity, we leave all the explicit calculation involving loops in B with the diagram-
matic interpretation. Also, the reader will find in [16] the formulation of perturbation
theory with the path integral formalism, which we consider to be the most complete
and elegant. We should also mention that [4] connects SPT with QFT.
Recently, [65] provided the first 3loops SPT solution. We reproduce their results
below in Figures 2.3 and 2.4 for a WMAP5 cosmology [21]. We used for this the CUBA
SUAVE Monte-Carlo integrator [66] to do the 8 dimension integral of the 3loop terms.
We implemented the IR-safe version of [67] in our calculations to avoid the numerical
instability of each term separately.
25
2. Standard Perturbation Theory
10-3 10-2 10-1 100
k [h/Mpc]
10-1
100
101
102
103
104
105
P(k
)
P0
P1
P2
P3
Figure 2.3.: Size of each SPT term separately. We can see that the 3Loop (pink) contri-bution is larger than both 2Loop (blue) and 1Loop (red). This shows thefailure of SPT. The continuous line means a positive contribution, while adotted line means negative sign.
We can see in Figure 2.3 the third order term being more relevant than the other
terms for this scales. It basically shows the failure of the perturbative solution since
the higher order terms are more important than lower order terms. In Figure 2.4 we
compare with the emulator [5] and it is clear that the 3 loop theory fails even before
the 1 loop and 2 loop SPT.
26
2.1. Euler Standard Perturbation Theory (SPT)
0.0 0.2 0.4 0.6 0.8 1.0
k [h/Mpc]
0.10
0.05
0.00
0.05
0.10
P(k
)
PNLIN
(k)−
1
P0
P 1LoopSPT
P 2LoopSPT
P 3LoopSPT
Figure 2.4.: Total SPT power spectrum in 1,2 and 3 loops. The results are comparedwith the cosmic emulator [5] (PNLIN ). The gray lines show the 1% and 2%error limits. The 3Loops SPT fails even before the 1Loop and 2Loops.
But more than this: even if the series converges, the SPT approach still has problems
that we will discuss now. First, the integrals in 2.7 do not converge for some cosmolo-
gies. It means we must put by hand an artificial cutoff Λ in the integrals, which gives
us a theory that depends on an arbitrary choice. Second, the way that small scale
modes interact with large scale modes is something far from being known, leading
to the conclusion that SPT incorporates the small-scale effects incorrectly. It means
that even if the series is convergent (see [68] for the discussion of convergence in 1
dimension, which has an exact result), it is not converging to the right result.
All this together motivated the formulation of the Effective Field Theory of LSS, a
theory that by construction:
• is independent of the arbitrary cutoff Λ;
• parametrize our ignorance about small-scale interactions and gives a description
27
2. Standard Perturbation Theory
of the size of these non-linearities;
• has a hierarchical construction that is made to assure convergence order-by-order
in perturbation.
In the following chapters, we explain with details the EFTofLSS.
2.2. Lagrangian Standard Perturbation Theory (LPT)
The Eulerian picture considers the Navier-Stokes conservation equations for the mat-
ter overdensity field δ(x,t) and for the velocity divergence field θ(x,t). The Standard
Perturbation Theory (SPT) approach is to perturb the fields δ = ∑i δ
(i) and solve itera-
tively. The Lagrangian approach (LPT), on the other hand, considers the trajectories of
the fluid particles from the initial condition q:
z(q,t) = q + s(q, t), (2.8)
where the position z is a function of the displacement s. A map between the two
frames is given by the Jacobian measure J = det[∂z/∂q] 5
1 + δ(x,t) = J−1(q,t) . (2.11)
The dynamics of the trajectories will be given by a simple Newton law under a
5 Since:
d3q = ρ(z,t)ρ(t) d3z = (1 + δ(z,t))d3z, (2.9)
multiplying both sides by δ3D(x− z(q,t)) and integrate both sides∫
d3~qδ3D(x− z(q,t)) =
∫(1 + δ(z,t))δ3
D(x− z(q,η))d3~z = 1 + δ(x,t). (2.10)
28
2.2. Lagrangian Standard Perturbation Theory (LPT)
universe comoving expansion rateH = Ha:
d2
dη2 zi(q,η) +H d
dηzi(q,η) = −∂xi
Φ[z(q,η)], (2.12)
where Φ is the newtonian comoving potential and η is the conformal time dη = dt/a(t).
The evolution of the gravitational potential is driven by the Poisson equation:
∂2xΦL(x,η) = 3
2H2Ωmδ(x,η). (2.13)
The perturbative expansion in LPT is done in the displacement [69, 70]: s = ∑i s(i).
Each order of the displacement is given by:
s(n)i (k) = i
n!
∫p1. . .∫pn
(2π)3δ3D
(n∑i
ki − k)L
(n)i (k1, . . . ,kn)δ0(k1) . . . δ0(kn). (2.14)
The coupling of the non-linear modes with the linear ones is given by the kernels L(n)i ,
described in the Appendix A.56. We can write the spectra C for each perturbation, in
Fourier space as:
〈s(a)i (k1)s(b)
j (k2)〉′ = −C(ab)ij (k1,− k1)
〈s(a)i (k1)s(b)
j (k2)s(c)l (k3)〉′ = +iC(abc)
ijl (k1,k2,− k1 − k2) (2.15)
2.2.1. Connecting Lagrange to Euler
As the observables are the density and velocity fields we want to connect the
Lagrange frame to Euler observables. As mentioned before, we do it using the Jacobian
6Note this is very similar to the expansion of δ in Euler 2.5. One of the disadvantages of the Lagrangeframe is that the recursive relations for L are much more complicated [71–73]
29
2. Standard Perturbation Theory
2.11:
δ(k,η) =∫d3xe−ik · xδ(x,η) =
∫d3xe−ik · x(
∫d3qδ3(x− z(q,η))− 1)
=∫d3qe−ik · z(q,η) −
∫d3xe−ik · x =
∫d3qeik · q(eik · s(q)) − 1), (2.16)
and calculating the power spectrum we find7:
P (k) =∫d3qeik · q(〈eik · ∆s〉 − 1) . (2.17)
with ∆s = ~s(~q)− ~s(0). The reader can find more about the connection of both frames
in [69], where the C’s found in Eqs. 2.15 are used to calculate the expected value in the
RHS of Equation 2.17.
2.3. The 1Loop SPT Bispectrum
In this short section we show the preliminary tests that we developed for the
bispectrum (the Fourier transform of the 3-Point function) theoretical prediction. The
3-Point statistics is very important to constrain primordial non-gaussianities generated
by different inflationary scenarios [27] (see also the chapter on Inflation).
Up to this level, within SPT we have the bispectrum calculated as [74]:
BSPT1Loop = B0 +B222 +BI
321 +BII321 +B411︸ ︷︷ ︸
B1
(2.18)
with the tree level term being defined as:
B0 = PL(k1)PL(k2)F2(k1,k2), (2.19)
7 Since 〈δ(k,η)〉 = 0, then the result of the first term in expansion is 〈eik1 · s(q1)〉 = 1.
30
2.3. The 1Loop SPT Bispectrum
and the 1Loop terms as:
B222 = 8∫qF2(−q,q + k1)F2(q + k1,− q + k2)F2(k2 − q,q)PL(q)PL(q + k1)PL(q − k2),
BI321 = 6PL(k3)
∫qF3(−q,q − k2,− k3)F2(q,k2 − q)PL(q)PL(q − k2),
BII321 = 6PL(k2)PL(k3)F2(k3,k2)
∫qF3(−q,q,k3)PL(q), (2.20)
B411 = 12PL(k2)PL(k3)∫qF4(−q,q,− k3,− k2)PL(q).
For the same cosmology described above, we reproduced the calculation made in [6].
Each term of the bispectrum composition is described in Figure 2.5 calculated for two
different triangle configurations (equilateral and isosceles).
10-2 10-1
k [h/Mpc]
10-1
100
101
102
103
104
10−
6B
(k,k,k
)
Bispectrum - Equilateral Configuration
B TreeB 1Loop|411|
321I|321II|
222
(a)
10-2 10-1
k [h/Mpc]
10-1
100
101
102
103
104
10−
6B
(k,2k,2k)
Bispectrum - Isosceles Configuration
B TreeB 1Loop|411|
321I|321II|
222
(b)
Figure 2.5.: Results for the 1Loop SPT Bispectrum calculated for two different trianglesconfiguration. To be compared with Figure 4 of [6]. We can see that theresult is very similar to what was expected and the small differences canbe attributed to different cosmologies.
We can see that the 1Loop bispectrum in Standard Perturbation Theory was correctly
reproduced compared with [6]. In a future work, we plan to calculate the bispectrum
at the two-loop level and compare our results with measurements from simulated
31
2. Standard Perturbation Theory
data. For that we will run N-body simulations and develop unbiased estimators for
3-point statistics.
32
3Effective Field Theory
3.1. The 3Loops Effective Field Theory of the LSS
Power Spectrum
The Effective Field Theory approach, as said before, is the ideal framework to
understand the size of the non-linearities. As in SPT, we still integrate the loops until a
UV cutoff Λ, but now we add the proper counterterms to cancel out the Λ dependence.
The counterterms will fix the integral problem and simultaneously provide information
about the small-scale physics. It is similar to a fluid, in which you parametrize the
33
3. Effective Field Theory
small-scale physics with a few parameters that embrace the phenomenology of UV
physics.
Each counterterm is associated with a free parameter usually called sound speed,
which is associated with the speed of propagation of the non-linear modes, similarly to
an effective fluid, for which a sound speed is associated with small-scale physics that
we are integrating out, generally defined as how the pressure perturbation changes
with density perturbation: c2s = δp/δρ. But the sound speed can also be read as a
non-linear scale lNL : cs = H × lNL. See Figure 3.1 for a diagrammatic formulation
for the matter power spectrum, in which we add a counterterm to cancel out the UV
dependence of the P13 diagram.
P 1LoopEFT (k) = + 2×
(+
)+ , (3.1)
Figure 3.1.: Diagrammatic representation of the 1loop matter power spectrum. Weneed to add a counterterm, here represented by a crossed vertex insidethe parenthesis, that cancels out the UV cutoff (Λ) dependence of the P13diagram. When Λ k, we have the leading scaling of F3 being k2/Λ2 suchthat P13 ∝ k2P0. That is parametrized in EFTofLSS by adding an effectiveterm in the stress tensor like ∂j∂iσij = c2
s∂2δ, where c2
s is an effectiveparameter that needs to be measured in small-scale simulations. Figurefrom [3].
In QFT, the usual approach to include the counterterms is at the level of the La-
grangian. In our approach, however, we will introduce the counterterms directly
in the equations of motion, and it will play the role of an effective stress-tensor. In
3.1.2 we discuss the most general counterterms structure. In 3.1.3 we summarize the
current work on the 2LoopsEFT. In 3.1.4 we show our result for 3LoopsEFTofLSS. In
the following subsections, we discuss the risk of overfitting and the UV-completion of
the theory.
34
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
3.1.1. The data and fitting
In order to compare the theoretical results we have in this dissertation with the
non-linear matter power spectrum from N-Body simulations, we used the Franken
Emulator1 [75]. This emulator works with an interpolation of different simulations,
for different cosmologies and redshifts. Its error is estimated to be around 3%. For our
calculations, this level of precision was enough, following [7]. Notice that [7] used the
Coyote emulator that consists of a previous version of the Franken Emulator.
For generating the linear matter power spectrum we used the CLASS code [36].
To fit all the parameters described here, we used a simple χ2 minimization, with the
errors being given by the estimated cosmic variance.
3.1.2. The Most General Counterterms Structure
In this section, we construct the most general structure the counterterms should
have and justify the origin of each one, somewhat based on Section 3.3 of [8] and
in [74].
We underline that in EFTofLSS the counterterms are added at the level of the
equations of motion through the stress tensor (∂τ)i, which is neglected by SPT. Of
course, the EFT counterterms must follow the symmetries of the fluid equations
[3, 76, 77]. It means that the theory must contain only scalar operators composed by:
∂i∂iΦ, with Φ = Φg,Φv, (3.2)
and recall that Φg is the gravitational potential defined above, but without labels in
1Available at http://www.hep.anl.gov/cosmology/CosmicEmu/emu.html
35
3. Effective Field Theory
order not to overload the notation and Φv is the velocity potential defined as:
Φv = θ
∂2 . (3.3)
We should also remember that the matter density field is included in one of the
contractions of the gravitational potential through the Poisson equation: δ ∝ δij∂i∂jΦg2.
However, we have degeneracies in both potentials because Φ(1)g = Φ(1)
v and at second
order we can include these differences in the quadratic counterterms [3]. This allows
us to consider only operators of Φg.
1. Linear counterterms:
The only linear term allowed is:
(∂τ)i ⊃ ∂iδ. (3.4)
Here we use the ’containing’ symbol to emphasize that this is only one of the
infinity allowed terms in the stress tensor, which can be ranked by their relevance.
In this case this is the leading order term. It will lead to counterterms like:
Pc.t.−lin(k) = −2(2π)(c2s1 + c2
s2 + c2s3)k2P0
−2(2π)(c2s1 + c2
s2)k2P1 − 2(2π)(c2s1)k2P2. (3.5)
Here the careful reader may ask why we need three counterterms and not only
one. At one loop level, it is true that only c2s1 would work. But at two loops we
need a new counterterm c2s2 to cancel out the k2P0 scaling of the first couterterm
added, and it has the same relevance of other two loop counterterms. The same
arguments takes place for c2s3 in three loop order.
2Just a warning for not misinterpreting the Kronecker delta which will always appear with indices,the matter overdensity δ and the Dirac delta, which will appear as δD.
36
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
2. Quadratic counterterms:
When we think about renormalizing the 3point-function, it is clear that we
must also include quadratic operators to the counterterms. The set of quadratic
operators to be included is [8]:
(∂τ)i ⊃ (1 + δ)× ∂i(∂jv
j
−Hf− δ
), ∂i[∂2φ]2, ∂i[(∂j∂kφ)2], ∂i∂jφ∂j∂2φ], (3.6)
which are degenerated and can be summarized as a single term like:
Pc.t.−quad(k) = −2(2π)c1k2∫qP (k − q)P (q)F2(q,k − q)︸ ︷︷ ︸
Pquad
. (3.7)
3. Higher derivative terms:
Operators with more derivatives of the gravitational potential should start con-
tributing as we go further down to small scales:
(∂τ)i ⊃ ∂2∂iδ, (3.8)
with:
Pc.t.−HD(k) = 2(2π)2(c4 + c4,2)k4P0 + 2(2π)2c4k4P1. (3.9)
4. Stochastic term:
The stochastic term – which we assume to be uncorrelated with any perturbation
δ(n) – is often modeled as a term proportional to k4 (see Section 5 of [78] for a
37
3. Effective Field Theory
broad discussion):
(∂τ)i ⊃ ∂i4τ, (3.10)
and:
Pstoch(k) = (2π)2cstochk4. (3.11)
5. Cubic and Quartic:
At this level of expansion in counterterms the cubic and quartic operators should
be sub-leading or degenerated:
(∂τ)i ⊃ ∂iδ3, . . . . (3.12)
3.1.3. The 1 and 2 Loops Calculation
This section is designed to summarize the current results in EFTofLSS, which up to
now has been calculated up to 2Loops level. At one loop, the results of [79, 80] use the
following theoretical prediction for the matter power spectrum:
P 1LoopEFT = P 1Loop
SPT − 2(2π)c2s1k
2P0 . (3.13)
They fitted the counterterm c2s1 in N-body simulation (or used an emulator, in the case
of [80]) in the range 0.1hMpc−1 < k < 0.3hMpc−1, being inside the 1% error limit in
all this range.
In two loops, we implemented two different approaches. First, the result of [7]
38
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
consists in only one free parameter c2s1:
P 2LoopEFT = P 2Loop
SPT − 2(2π)(c2s1 + c2
s2)k2P0 − 2(2π)c2s1k
2P1 + (2π)2(c2s1)2k4P0. (3.14)
The other c2s2 is constrained by:
P 2LoopEFT (kren) = P 1Loop
EFT (kren) at kren ∼ 0.2hMpc−1, (3.15)
which implies:
c2s2(c2
s1,kren) = P2(kren)2(2π)k2
renP0(kren) − c2s1P1(kren)P0(kren) + π(c2
s(1))2k2ren. (3.16)
However, as shown in [8], this number of counterterms is not enough to cancel out the
UV dependence of the theory. They suggest including two more counterterms, leading
to three free parameters. Also they argue that the renormalization scale chosen by [7]
was too high, which leaves the theory very dependent on this choice. The final form of
counterterms suggested by [8] to cancel out the UV is3
P 2LoopEFT = P 2Loop
SPT − 2(2π)(c2s1 + c2
s2)k2P0 − 2(2π)c2s1k
2P1 + (2π)2(c2s1)2k4P0
−2(2π)c1k2∫qP (k − q)P (q)F2 + 2(2π)2c4k
4P0. (3.17)
with a much lower kren:
P 2LoopEFT (kren) = P 1Loop
EFT (kren) at kren = 0.005hMpc−1. (3.18)
This choice of renormalization scale reduces the reach in k of [7] substantially. To
fit the spectrum we used for [7] the range: 0.1hMpc−1 < k < 0.3hMpc−1. For [8]
3In [8], they show that the stochastic counterterm is not needed to cancel out the UV dependence ofthe theory, dropping this term from the EFT theoretical prediction.
39
3. Effective Field Theory
we fitted the c2s counterterms in the same range and the other counterterms in the
range 0.15hMpc−1 < k < 0.6hMpc−1. The values of the counterterms we found using
Eq. 3.13 and Eq. 3.17 are described in Table 3.1. We show that the theory does not
change with the choice of kren in Figure 3.3.
In Figure 3.2 we show the results found for our fits (continuous lines): in yellow
we use only c2s1, as did by [7]; in blue we use Eq. 3.17, as did by [8] . In dashed lines,
we show the results obtained with counterterms values described in the respective
references. The one-loop approach works in the 1% range up to k < 0.3hMpc−1. The
2Loops EFT described in [7] (in yellow, but here with the different renormalization
scale kren described above) does not improve sensibly with only one free parameter4.
The 2Loops EFT described in [8], in blue, reaches an impressive improvement with 1%
precision until k < 0.58hMpc−1. The differences of counterterms values obtained are
attributed to the different cosmologies used5.
4We highlight that if we set kren = 0.2hMpc−1 we found the same precision described in the article, butthe results are very sensible to the choice of kren, which suggest we must use a different approach.
5Here we highlight that we checked the values obtained for the counterterms for their cosmology, butopted to use the WMAP5 cosmology to be consistent with [65] results.
40
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
0.0 0.2 0.4 0.6 0.8 1.0
k [h/Mpc]
0.10
0.05
0.00
0.05
0.10
P(k
)
PNLIN
(k)−
1
P0
P 1LoopEFT
P 2LoopEFT - 1310.0464, fitting cs
P 2LoopEFT - 1310.0464, using cs of article
P 2LoopEFT - 1507.05326, fitting cs
P 2LoopEFT - 1507.05326, using cs of article
Figure 3.2.: Comparison between the non-linear emulator spectrum and different EFTcalculations up to 2 loops. In all of them we used the IR-resummationof 1606.03633. For the 1Loop EFT (red) we fitted the spectrum in0.15hMpc−1 < k < 0.25hMpc−1. The theory fails around k ∼ 0.3hMpc−1
and we found c2s1 = 0.96(0.75)(1/hMpc−1)2. For the 2 Loops EFT we
compare two different counterterm structures (from [7] and [8]). In bothwe fitted c2
s1 also in 0.15hMpc−1 < k < 0.25hMpc−1 and set the renor-malization scale kren = 0.005hMpc−1 to determine c2
s2 as described inthe text. For the 2LoopsEFT described in 1310.0464 (yellow) we foundc2s1 = 0.64(0.53)(1/hMpc−1)2. The theory has an 1% agreement untilk ∼ 0.27hMpc−1. For the 2LoopsEFT described in 1507.05326 (blue) wefound c2
s1 = 0.44(0.48)(1/hMpc−1)2, c1 = 0.48(−1.6)(1/hMpc−1)2, c4 =−1.12(−7.0)(1/hMpc−1)4. The theory has a 1% agreement until k ∼0.58hMpc−1. The values in (red) reflect the results of [8], which we plot indashed.
41
3. Effective Field Theory
3.1.4. The 3 Loops Result
For the 3Loops EFT, the structure of the seven counterterms that contribute is:
P 3loopEFT = P 3loop
SPT − 2(2π)(c2s1 + c2
s2 + c2s3)k2P0
−2(2π)(c2s1 + c2
s2)k2P1 − 2(2π)(c2s1)k2P2+
(2π)2[(c2s1)2 + 2c2
s2c2s1]k4P0 + (2π)2(c2
s1)2k4P1−
2(2π)c1k2∫qP (k − q)P (q)F2 + (2π)2cstochk
4+
2(2π)2(c4 + c4,2)k4P0 + 2(2π)2c4k4P1. (3.19)
The strategy to fit the theory with the non-linear data was fitting the c2s parameters
in low k (0.15hMpc−1 < k < 0.3hMpc−1) and the others in the range that we expected
the validity of the theory (0.15hMpc−1 < k < 0.75hMpc−1). This is because the c2s
parameters are constructed to cancel out the divergences that go with k2 [81].
We tested different structures of counterterms with five and all seven free parameters.
For the case with five free parameters, we fix c2s2(c2
s1,kren1) and c2s3(c2
s1,kren2) now for
two different renormalization scales:
P 2LoopEFT (kren1) = P 1Loop
EFT (kren1) at kren1 = 0.005hMpc−1,
P 3LoopEFT (kren2) = P 2Loop
EFT (kren2) at kren2 = 0.3hMpc−1. (3.20)
The choice of two different renormalization scales is made such that at low scales
most of the contribution comes from the 1LoopEFT term and the 3LoopEFT start
contributing only after the failure of the 2LoopEFT. Also, it is important to discuss if the
five-parameter theory depends on the choice of these two scales, which we address in
Figure 3.3. We can see that in the 2Loop case, the difference in the spectrum with kren =
0.005hMpc−1 and kren = 0.008hMpc−1 was insignificant. The same happened for the
42
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
determination of kren1 in the 3LoopsEFT. The change in kren2 affects the spectrum only
in the region where the theory is not valid anymore.
0.2 0.4 0.6 0.8 1.0
k [h/Mpc]
0.04
0.03
0.02
0.01
0.00
0.01
0.02
0.03
0.04P(k
)
Pstd(k
)−
1P 2LoopEFT : kren1 = 0. 008hMpc−1
P 3LoopEFT : kren1 = 0. 008hMpc−1, kren2 = 0. 3hMpc−1
P 3LoopEFT : kren1 = 0. 005hMpc−1, kren2 = 0. 4hMpc−1
Pstd : kren1 = 0. 005hMpc−1, kren2 = 0. 3hMpc−1
Figure 3.3.: Test of how the choice of kren changes the spectrum. P 2LoopEFT and P 3Loop
EFT
divided by the respective standard spectrum, where by standard we meanthe one with kren1 = 0.005hMpc−1 and kren2 = 0.3hMpc−1. For instance,the blue lines show the 2 loops EFT prediction Eq.3.17 using kren1 = 0.008divided by the same theory using kren1 = 0.005. We can see that bothchoices do not affect the spectrum up to k ∼ 0.7hMpc−1.
The final result of the 3LoopsEFT is described in Figure 3.4. We can see that with
five free parameters we reached the same precision as with seven free parameters.
Considering the 1% precision threshold, we reach a scale of k = 0.74hMpc−1, which is a
sensible improvement if compared with the 2LoopsEFT. The values of the counterterms
found are described in Table 3.1.
In the next section, we discuss one crucial result, which is the resummation of
the IR modes. With this procedure, we eliminated the oscillations of SPT due to the
displacements in the perturbative expansion.
43
3. Effective Field Theory
0.0 0.2 0.4 0.6 0.8 1.0
k [h/Mpc]
0.10
0.05
0.00
0.05
0.10
P(k
)
PNLIN
(k)−
1
P0
P 1LoopEFT
P 2LoopEFT
P 3LoopEFT − 5param
P 3LoopEFT − 7param
Figure 3.4.: 3LoopEFT result (pink with 5 free parameters using renormalizationconditions and black with all 7 parameters) compared with 1LoopEFT(red) and 2LoopEFT (blue). Considering the 1% goal, the reach of thetheory is improved to k = 0.74hMpc−1, a sensible improvement comparedwith k = 0.58hMpc−1 (k = 0.30hMpc−1) of the 2LoopEFT (1LoopEFT). Asthe number of counterterms is increased from three to five, we must beworried about overfitting and the physical predictability of the theory.Comparing the 3LoopEFT with either five or seven parameters, we seethat the use of renormalization conditions does not affect where the theoryis 1% accurate.
c2s1 c2
s2 c2s3 c1 c4 c4,2 cstoch
1LoopEFT 0.96 - - - - - -
2LoopEFT 0.44 -1.56 - 0.48 -1.12 - -
3LoopEFT 1.84 -1.6 -9.64 0,28 -3.04 20.0 5414.4
Table 3.1.: Values of the counter-terms in each EFT order:c2s1, c2
s2, c2s3, c2
1 are in unitsof (1/hMpc−1)2, c4 and c4,2 are in units of (1/hMpc−1)4 and cstoch is in(1/hMpc−1)7. The counterterms size for the 3LoopEFT increase comparedto the 2LoopEFT.
44
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
3.1.5. The IR Resummation
Here we describe one important result of this work, which is how the resummation
of the IR modes affects the 3Loops spectrum. This is an original contribution of this
work. The physics behind this is described in [82, 83] and summarized in C. It is
mainly associated with the fact that the SPT expansion is implicitly expanding in the
IR displacement fields, which for the current expected ΛCDM cosmology ( [20]) are
quite large and not order one. It leads to wiggles in the spectrum as we can see in
Figure 3.5. But we can resum these expansion terms within the Lagrange description,
which makes the wiggles vanish.
In Figure 3.5, where we plot both the non-resummed theory (dashed lines) and the
resummed theory (continuous lines). We can see that the effect of resummation is even
more critical as we go to higher loops. For the resummation, we modified the code
of [84] to include the new terms of the spectrum.
0.0 0.2 0.4 0.6 0.8 1.0
k [h/Mpc]
0.10
0.05
0.00
0.05
0.10
P(k
)
PNLIN
(k)−
1
P0
P 1LoopEFT
P 2LoopEFT
P 3LoopEFT
Figure 3.5.: Same as Figure 3.4, now also showing the version without performingthe IR-Resummation in dashed lines with the respective colors. While for1LoopEFT and 2LoopEFT the wiggles amplitude cause a difference of atmost 2%, in the case of the 3LoopEFT the amplitude leads to a differenceof more than 100% (the flutuations are outside the picture scale).
45
3. Effective Field Theory
3.1.6. Avoiding Overfitting
With a large number of parameters, it is a must to discuss if we should measure
physical parameters or if we should focus on best fitting simulated data. In order to
avoid overfitting, in this section we aim to answer two questions:
1. Do the 2nd, and 3rd order terms of the SPT bring any new information (can we
fit them with lower order terms?)?
2. How stable are the counter-terms found?
Beginning with the first question, we define the following quantities:
P 1LoopZero = P1︸︷︷︸
1st term only
−2(2π)(c2s1)k2P0, (3.21)
P 2LoopZero = P2︸︷︷︸
2nd term only
−2(2π)(c2s1 + c2
s2)k2P0 − 2(2π)(c2s1)k2P1 + (2π)2(c2
s1)2k4P0
−2(2π)c1k2∫qP (k − q)P (q)F2 + 2(2π)2(c4)k4P0, (3.22)
P 3LoopZero = P3︸︷︷︸
3rd term only
−2(2π)(c2s1 + c2
s2 + c2s3)k2P0 − 2(2π)(c2
s1 + c2s2)k2P1 − 2(2π)(c2
s1)k2P2+
(2π)2[(c2s1)2 + 2c2
s2c2s1]k4P0 + (2π)2(c2
s1)2k4P1 − 2(2π)c1k2∫qP (k − q)P (q)F2 + (2π)2cstochk
4
+2(2π)2(c4 + c4,2)k4P0 + 2(2π)2c4k4P1. (3.23)
We will then try to find – via χ2 minimization – if there are values for the counterterms
that fit each term of SPT. Basically, this will show the amount of information that
each term of SPT brings. In Figure 3.6 we can see that while the two loop term brings
46
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
relevant information in the low and mid k regime, the three loop term can be fitted
entirely by the counterterms. This increases the risk of overfitting the data with the
five counterterms.
400
200
0
200
400
Pi ZERO
P 2LoopZero
P 3LoopZero
0.2 0.4 0.6 0.8 1.0
k [h/Mpc]
0.4
0.2
0.0
0.2
0.4
Pi ZERO/P
i
Figure 3.6.: Estimative of how much new information the 2Loop (blue) and 3Loop(pink) terms bring to the spectrum. In the top panel we show the totalcontribution and in the bottom we divided by the respective spectrumto have a dimensionless quantity. For this, we fitted the counter-termsof PZero in three different ranges, which we plot here with continuous,dashed and dotted lines. The continuous lines represent the respective fitrange of each EFT order (0.15hMpc−1 < k < 0.6hMpc−1 for the 2LoopEFTand 0.15hMpc−1 < k < 0.75hMpc−1 for the 3LoopEFT). In dotted line, wefitted in low k regime and in dashed we fitted in high k regime. We cansee by the bottom plot that the 2Loop term brings more information onall scales. The 3Loop contribution was around zero in almost all scales.Fitting in high k (dashed) the same happened.
Also, we can ask the inverse question and ask if we use the values of the coun-
terterms found by the fitting the data, the SPT term is canceled. In Figure 3.7 we
show the values of PZero fitted in different ranges. Even though we can cancel out the
dependence on the three loop, as was shown in Figure 3.6, in practice the 3Loop term
47
3. Effective Field Theory
seems to be contributing.
600
400
200
0
200
400
600
Pi ZERO(cbest) P 1Loop
Zero (c3LoopEFTbest )
P 1LoopZero (c1LoopEFT
best )
P 2LoopZero (c3LoopEFT
best )
P 2LoopZero (c2LoopEFT
best )
P 3LoopZero (c3LoopEFT
best )
0.2 0.4 0.6 0.8 1.0
k [h/Mpc]
4
2
0
2
4
Pi ZERO(cbest)/Pi
Figure 3.7.: Now we calculate PZero as defined by Eqs 3.21, 3.22 and 3.23 for the best-fitvalue of the counter-terms that fit the emulator spectrum. In top panel weshow the absolute value and in bottom panel the ratio to Pi, also defined inEqs 3.21, 3.22 and 3.23. Solid lines represent the 1, 2 and 3Loops spectrumcalculation, but using the best-fit sound speeds from the 3LoopEFT. We seethat the best-fits are not cancelling any of the terms. For the dashed lineswe used the best-fit of the respective fits of the 1LoopEFT and 2LoopEFT.
For the second question considered above, about the stability of the counterterms,
consider Figure 3.8. We fitted the theoretical predictions for different values of kmax.
For the maximum k that we used in the final fits (k = 0.75hMpc−1) all counterterms
but c4 and cstoch are quite stable, indicating that this is the right region to fit.
48
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0864202468
c s(1
) 5 parameters7 parameters
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0864202468
c s(2
)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01210
86420
c s(3
)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.005
101520
c 1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01412108
6420
c 4
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.001020304050607080
c 42
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
kmax [h/Mpc]
100000100002000030000400005000060000
c stoch
Figure 3.8.: How the parameters of the 3LoopsEFT change with the kmax used in thefitting. The cs parameters do not change because we fit them at low kregime. The shaded region is the 1σ error estimated. The regions wherethe parameters are nearly independent of k (k ≈ 0.75hMpc−1) indicate weare avoiding overfitting.
3.1.7. UV completion
One of the main goals of EFT, differently from SPT, is to be independent of the cutoff
used, once the cutoff is chosen to be in a length scale smaller than those we want our
theory to be valid. In Figures 3.9,3.10 and 3.11 we show each term of SPT for different
cutoffs Λ: 0.7hMpc−1, 60hMpc−1, 100hMpc−1. We used as the standard cutoff of this
work Λ = 60hMpc−1 to compare with [8]. We can see that while the two very high
cutoffs have a small difference between them, the difference relative to a small cutoff
Λ = 0.7hMpc−1 is huge, and the difference is driven mainly by the 3Loop term.
49
3. Effective Field Theory
10-3 10-2 10-1 100
k [h/Mpc]
100
101
102
103
P(k
)
P1
PΛ = 0. 71
PΛ = 1001
Figure 3.9.: 1Loop term of SPT calculated for Λ = 100hMpc−1 (yellow), Λ = 60hMpc−1
(red) and for Λ = 0.7hMpc−1 (black). The difference is minimal betweenthem. The red and yellow curves are nearly identical and appear over-lapped.
10-3 10-2 10-1 100
k [h/Mpc]
10-1
100
101
102
103
104
P(k
)
P2
PΛ = 0. 72
PΛ = 1002
Figure 3.10.: 2Loop term of SPT term calculated for Λ = 100hMpc−1 (yellow), Λ =60hMpc−1 (blue) and for Λ = 0.7hMpc−1 (black).
50
3.1. The 3Loops Effective Field Theory of the LSS Power Spectrum
10-3 10-2 10-1 100
k [h/Mpc]
10-1
100
101
102
103
104
P(k
)
P3
PΛ = 0. 73
PΛ = 1003
Figure 3.11.: 3Loop term of SPT term calculated for Λ = 100hMpc−1 (yellow), Λ =60hMpc−1 (purple) and for Λ = 0.7hMpc−1 (black). While the pink andyellow curves are nearly indistinguishable, the difference relative to theblack line is huge all over the spectrum, and is particularly large close tothe lowest cutoff (kmax = 0.7hMpc−1).
In order to check the UV completion of the theory we tried to find if there are
counterterms that, when summed to the SPT term of each order, will result in the same
SPT term calculated for Λ = 0.7hMpc−1. This answers if, by adding counterterms, we
can make the difference between SPT terms derived for different cutoffs vanish. This
result is plotted in Figure 3.12 for the three different SPT loop orders. We can see that
in all the terms the UV is well completed by the counterterms.
51
3. Effective Field Theory
0.2 0.4 0.6 0.8 1.0
0.4
0.2
0.0
0.2
0.4
(PΛ
=∞
i+c.t.
)/P
Λ=
0.7
i−
1
P1
P2
P3
Figure 3.12.: Checking the UV consistency of the theory for the 1LoopEFT (red),2LoopEFT (blue), 3LoopEFT (pink). We try to address the followingquestion with this plot: by adding counterterms, can we cancel outthe UV dependence of the SPT terms? For P1, P2 and P3 the answer isyes on all scales. For P3 we found a small disagreement at low k, butif fails severely only in the region that was not used in the fits of thecounterterms (k > 0.75hMpc−1). Compare with Figure 4 of [8], wherethey find a similar result.
3.1.8. Conclusion of the 3loops EFT
We have shown that the 3LoopsEFT complements the 2LoopsEFT improving the
reach of the theory from k = 0.58hMpc−1 to k = 0.74hMpc−1. Also, the IR resumma-
tion process becomes even more critical at this loop order. The counterterms structure
seems enough to correct the different cutoffs Λ that can be used in the loop integrals.
We also show that the risk of overfitting the data is very high since we have only one
simple function as the power spectrum is, to constrain a few free parameters. However,
we notice that this is a preliminary work, and in future articles, we plan to improve
these parameter constraints by also using the matter bispectrum. The bispectrum – the
52
3.2. Lagrangian space EFT
Fourier transform of the 3 point function – brings a lot of extra information, as it can
be calculated for different triangles configurations.
3.2. Lagrangian space EFT
All EFT formulation described above is made for the Euler fluid framework. Here
we present how to extend it to Lagrange space, work done by [85]. We reproduce their
results explaining the main steps. We use the same notation of Section 2.2.
The idea of the LEFT is smoothing all fields in the scale we want to make predictions.
As consequence, now we have the equations 2.12 and 2.13 evaluated for the long-
wavelenght fields:
d2
dη2 zL(q,η) +H d
dηzL(q,η) =
−∂x
ΦL[z(q,η)]︸ ︷︷ ︸Poisson Term
+12Q
ij(q,η) ∂i∂jΦL[z(q,η)︸ ︷︷ ︸Tidal Term
] + . . .
+ aS(q,η)︸ ︷︷ ︸Overlaped Regions
Acceleration
, (3.24)
and:
∂2xΦL = 3
2H2Ωm
[δL(x,η) + 1
2∂i∂jQij + . . .
]. (3.25)
One crucial fact is that now he must add new terms that will dictate how the small-
scale physics interfere in the dynamics at the scale of interest. For example, we must
add an acceleration aS due to an overlap of different regions along time evolution,
and at first order, we also need to include the tidal forces that will set how these fields
behave due to the finiteness of the objects. This tidal force will be proportional to the
53
3. Effective Field Theory
quadrupole Qij of the region:
δL(x,η) =∫d3qδ3
D(x− zL(q,η))− 1,
Qi...l(x,η) =∫d3qQi...lδ3
D(x− zL(q,η)). (3.26)
Another important aspect of convoluting our fields at a scale 1/Λ is that physics
must not depend on this arbitrary choice of scale. That is the main constraint that will
appear in the parametrization of the multipoles and in the acceleration. Integrating
out the small scale physics we should find out how they interfere in the large modes.
With the introduction of counterterms we cancel out these arbitrary scale such that the
physics is still independent of Λ. For example, for the quadrupole, we can choose the
following form:
Qij(q,η) = 〈Qij〉S +QRij +QSij = l2S(η)δij3︸ ︷︷ ︸
Short Modes
(3.27)
− δij3 l2T (η)∂ksk(q,η)− l2TF (η)(
12(∂isj(q,η) + ∂jsi(q,η))− δij
3 ∂ksk
)︸ ︷︷ ︸
Response Modes
+ QSij(q,η)︸ ︷︷ ︸Stochastic Term
,
(3.28)
and for the acceleration:
aS(zL(q,η)) = aS(q,η) + 32HΩml
2ΦS
(η)∂q (∂q · sL(q,η)) . (3.29)
Now we have the set of time-dependent parameters l2S,l2T , lTF ,l2ΦS that describe
how the UV cutoff affects the modes of interest. They shall be fitted using small-scale
simulations. Notice that we have separated out the contributions of the quadrupole
into three pieces: the first coming from the expectation value of the small-scale modes;
54
3.2. Lagrangian space EFT
the second being a response of the long modes and the third is a stochastic piece. The
cutoff independence states that the following quantities must not depend on Λ:
〈siL(q,η)sjL(q,η) +Qij(q,η)〉 , and also 〈smL (q2,η)(siL(q,η)sjL(q,η) +Qij(q,η)
)〉.
(3.30)
The first is just the expectation value of small scales6 while the second term is a
contraction with a point far away and will give rise to the response term.
Calculating the divergences at the expectation value we have:
〈si(q,η)sj(q,η)〉 = 13δijl
2Λ(η), with l2Λ(η) ≡
∫ Λ
0
dp
2π2PL(p), (3.31)
which will be canceled out by counterterm of the expetation value of quadrupole
(background level S):
l2S(Λ,η) = 〈Qii,S〉 = −l2Λ(η). (3.32)
Now the divergences coming from correlation with a distant point:
〈sl(q2)si(q)sj(q)〉 F.T.→ 〈sl(k)∫psi(p− k)sj(−p)〉 = 2i
∫pC
(112)ij (k,− p,p− k)
p→∞→ ik2PL(k) 435 l
2Λ(−kikjkl
k6 + 2δijklk4 ), (3.33)
which will be canceled analysing the correlation of the quadrupole response part with
a distant point:
〈Qij,Rsl〉 = ik2PL(k)(klk4
13(l2T − l2TF )δij + l2TF
kikjklk6
). (3.34)
6Notice it is a composite operator
55
3. Effective Field Theory
Comparing this to 3.33 we find:
l2TF = 435 l
2Λ l2T = −4
7 l2Λ. (3.35)
Also we need to renormalize the displacement field s(n)i (q,η) itself. This is done
defining a source termF (n) such that s(n)i (q,η) is the result of the linear time propagator
G in this source [60]:
s(n)i (q,η) =
∫dη′G(η,η′)F (n)(q,η′), (3.36)
It is possible to show that the correlation of this source term expanded in second and
third orders with one external leg will give rise to the divergences that will be canceled
out by the acceleration. The coefficient l2ΦSis:
l2ΦS= 121
105 l2Λ. (3.37)
Of course, this part treats how to cancel out the UV dependence. We still need to
parametrize the UV physics, fitting its free parameters to N-body simulations.
56
4The Bias Problem with EFT
As said in the introduction, the bias is defined as the relation between the matter
density field and the number density of observable tracers (halos, galaxies, etc.).
Figure 4.1 is a scheme that connects all the steps from initial conditions to observations.
57
4. The Bias Problem with EFT
Initial Conditions Large-Scale StructureSPT, EFT-of-LSSN-body simulations
dark matter
halos
galaxies
halo biasing
galaxy biasing
RSD
Figure 4.1.: Scheme of the connection of the universe initial conditions, the most basicbuilding block, to galaxy observations. While in chapter 3 the aim wasconnecting primordial universe to the underlying matter statistics, in thischapter we address the problem of connecting dark matter to observablesthrough the bias. Extracted from [3].
We start this chapter with the more theoretical aspects in Section 4.1. After this, in
Section 4.2 we show our main results measuring the bias parameters. Next in 4.3 we
discuss our work on renormalizing the bias in Lagrange space.
About the notation of this chapter, we stress that there are different representations
of higher order operators (K2, δ3, δK2, K3, O(3)td or G2, δ
3, δG2,G3,Γ3). We use both
notations in different parts, according to which notation is found in the literature we
used as basis for each topic. For a map between the operators set, it is a simple linear
combination of terms. See Appendix C of [31] for a table explaining this change of
basis.
4.1. Bias tracers, bias renormalization and its time
evolution in Euler space
This section aims to discuss different achievements on studies of halo bias, such as
the renormalization of the Euler bias [3] and the systematic construction of a basis
of operators [29]. These will be useful tools for the renormalization procedure in
Lagrange space introduced in Section 4.3. Most of the conventions follow [31].
58
4.1. Bias tracers, bias renormalization and its time evolution in Euler space
The usual parametrization to deal with the halo bias is to use a Taylor series [30] on
the local matter overdensity:
δh(x,τ) = F [δ(x,τ)] =∞∑i=0
bii! δ
im(x,τ) . (4.1)
The reason why this is called Euler bias expansion is because it sets a general relation
between the two fields δh and δm for a particular time slice. Here we will emphasize two
crucial aspects of this parametrization: first that if we consider this evolution to be local
in the matter overdensity1 as in 4.1 for a time slice, then time evolution will naturally
generate other new operators showing that this basis is incomplete [86]. Second, we
show that the Taylor expansion is not well defined in the UV and must be renormalized.
This renormalization will also introduce new (non-local in δ) operators [3].
4.1.1. Euler bias: time evolution
We start assuming that the tracer formation– in this case the halos, though the same
remains valid for galaxies – occurred at a specific time τ∗2. At this time we must set
the initial conditions for the Euler bias parameters. The most complete set of operators
of the gravitational potential in Euler description up to third order is:
δ∗h = δh(xfl(τ∗),τ∗) =3∑
n=1
b∗n(δ∗)nn! + b∗K2(K∗)2
2! + b∗K3(K∗)3
3! + 16b∗δK2δ∗(K∗)2 + (stochastic), (4.2)
1We highlight that by local in matter density, we call all operators like δ,δ2,δ3 and other powers of δ.In some references the reader may find the term ’local’ to describe this terms. See Section 1.3 of [31]for a broad discussion about terminology.
2The label ∗ in coordinates and operators will always refer to this particular time. We comment belowon a generalization of this single time trace formation.
59
4. The Bias Problem with EFT
where from Equation 2.13 δ = Tr[∂i∂jΦ] and K is the traceless component: Kij = Dijδ
with Dij = ∂i∂j − 13δij
3, and Kn ≡ Tr[Kn].
The time evolution for the δh and δ fields is driven by the continuity equation,
coupled to the Euler equation for the velocity divergence θ:
D
Dτδh = −θ(1 + δh),
D
Dτδ = −θ(1 + δ),
D
Dτθ = −Hθ − 3
2ΩmH2δ − (∂ixvj)2,
θh = θ. (4.3)
The last equation reflects the assumption of no velocity bias. For scales much larger
than the halo scales this is valid because halos comove with the matter flow. For
a deeper discussion see Section 2.7 of [31]. A matrix representation for this set of
equations is:
D
DτΨ = −σ · Ψ + S, (4.4)
with:
Ψ(x,τ) =
δh(x,τ)
δ(x,τ)
θ(x,τ)
, σ(τ) =
0 0 1
0 0 1
0 32ΩmH
2 H
, S(x,τ) =
δhθ
δθ
−(∂ixvj)2
.
(4.5)
We solve this system of equations by the convective SPT approach described in [29].
The idea is that the integration of the system is much easier if done along the fluid flow,
3For simplicity we will use from now on the gravitational potential and the rescaled potential Φ =2
3H2ΩmΦ interchangeably, as it will be obvious what they stand for in different places.
60
4.1. Bias tracers, bias renormalization and its time evolution in Euler space
basically due to the fact that the convective derivatives become simple derivatives.
One immediate problem is the Eulerian derivative in the third component of the
source S. In order to change to Lagrangian derivative we use the Jacobian |1 + M|
with Mij = ∂isj4 and the relation between the displacement and the velocity fields:
s(q,τ0) =∫ τ0
0v(xfl(q,τ),τ)dτ, (4.6)
such that:
∂ixvj =
(1
[1 + M]
)ik
∂kq vj = ∂iq s′j + (∂iq sk)∂kq s
′j. (4.7)
Squaring and expanding order by order, we find for the third term of the source [31]:
S(1)3 = −[a′(τ)]2
[(K(1))2 + (1
3δ(1))2
],
S(2)3 = 2a(τ)[a′(τ)]2
[−2
3δ(1)σ[2] − 2K [1]
ij Dijσ[2] + (1
9δ(1))3 + δ(1)(K [1])2 + (k(1))3
]. (4.8)
We first find a Green solution for the homogeneous equation:
∂τA(τ,τi) + σ(τ)A(τ,τi) = 0; with A(τ,τ) = 1, (4.9)
and use it to construct the Green function solution for the full equation including the
source term:
Ψ(x,τ) = A(τ,τ∗)Ψ(x∗,τ∗) +∫ τ
τ∗dτ ′A(τ,τ ′)S(xfl(τ ′),τ ′), (4.10)
with Ψ(x∗,τ∗) = (δ∗h,δ∗,θ∗). In order to look for a solution for A we use the well-known
approximation of considering an Einstein-deSitter universe in σ, and find the solution.
4The same dynamics described in Section2.2
61
4. The Bias Problem with EFT
We then insert the corresponding growth function D.
The linear and the second order solution are:
Ψ[1](x,τ) = D(τ)
bE1 (τ)
1
−H(τ)
δ1(q) +
ε∗0
0
0
, (4.11)
Ψ[2](x,τ) =
bE1 (τ)δ[2](q) + bE2 (τ)(δ[1](q))2/2 + bEK2(τ)K2
[1](q)/2 + εEδ (τ)δ[1](q)
D2(τ)[
1721(δ[1](q))2 + 2
7K2[1](q)
]−D(τ)D(τ)
[1321(δ[1](q))2 + 4
7K2[1](q)
]
,
(4.12)
with:
bE1 (τ) = 1 + (b∗ − 1)D(τ ∗)D(τ) , (4.13)
bE2 (τ) = b∗2
(D(τ ∗)D(τ)
)2
− 821(b∗ − 1)D(τ ∗)
D(τ)
(D(τ ∗)D(τ) − 1
), (4.14)
bEK2(τ) = b∗K2
(D(τ ∗)D(τ)
)2
+ 47(b∗ − 1)D(τ ∗)
D(τ)
(D(τ ∗)D(τ) − 1
), (4.15)
while the third order solution can be found in Appendix B of [29].
Now we want to shift back the fields to Euler coordinates:
ΨE(x) = Ψ(x− s(q,τ)), (4.16)
and the third order Taylor term will be:
ΨE(x) = Ψ[3] − si[1]∂iΨ[2] −(si[2] − s
j[1](∂js
i[1]))∂iΨ[1] + 1
2si[1]s
j[1]∂i∂jΨ
[1]. (4.17)
62
4.1. Bias tracers, bias renormalization and its time evolution in Euler space
We have now a few considerations to highlight for this remarkable result:
• The first thing to notice is that the factors with a single D(τ ∗)/D(τ) in b1 and b2
correspond to the results that come from the spherical collapse (SC) solution [87].
As D(τ∗)D(τ) < 1 we have the other terms being corrections around SC.
• We can find the same solutions going through other approaches. We used here
the one that we consider to be the most elegant. An even simpler way (see section
2.3 of [31]) is to evolve fluid equations in fluid trajectories directly:
11 + δh
D
Dτδh = 1
1 + δ
D
Dτδ = −θ ⇒
1 + δh|τ = 1 + δ|τ1 + δ|τ∗
(1 + δh|τ∗), (4.18)
which can be expanded in δ. After collecting the terms of each order and moving
to Euler coordinates we find the same set of bias solutions.
• From the solution 4.15 we can see the emergence of non-locality in matter over-
density terms from time evolution. Supposing we start only with LIMD terms
(b∗K2 = 0) after a while bEK2(τ) will increase.
4.1.2. Euler bias: renormalization
As said before, as the composite operators are highly sensible to UV physics, they
must be renormalized:
[δi](x,τ) ≡ δi(x,τ) +∑OZOO(x,τ), (4.19)
where the square brackets mean a renormalized operator. For each order i there are
different operators O allowed by the symmetries that cancel out the cutoff Λ depen-
dence and encode the small-scale physics. As an example, the following calculation
63
4. The Bias Problem with EFT
will focus on how to find out these operators for [δ2].
As we have the freedom to choose our renormalization conditions, in [3] this choice is
made such that O cancels out loop divergences of the operator and at low momentum
the renormalized operator is equal to the non-renormalized. For instance, if we
contract δ2 of Equation 4.1 with m external legs δ(1)qi
5 and neglecting the loop terms
coming from O:
〈[δ2]δ(1)q1 . . . δ(1)
qm〉 = 〈(δ2)δ(1)
q1 . . . δ(1)qm〉tree + 〈(δ2)δ(1)
q1 . . . δ(1)qm〉loop +
∑OZO〈Oqδ1
q1 . . . δ(1)qm〉tree.
(4.20)
As we want the counterterms to cancel out loop divergences we must find O such that:
∑OZO〈Oqδ(1)
q1 . . . δ(1)qm〉tree = −〈(δ2)δ(1)
q1 . . . δ(1)qm〉loop. (4.21)
For no external legs, we must add the variance σ:
⟨(δ2)x
⟩ 1 Loop=⟨(δ(1)δ(1))x
⟩F.T→⟨∫
pδ
(1)p−kδ
(1)−p
⟩=∫ Λ d3p
(2π)3PL(p) ≡ σ2(Λ). (4.22)
For one external leg:
⟨δ(1)x1 (δ2)x
⟩ 1 Loop= 2⟨(δ(2)δ(1)δ(1)
x1 )x⟩
F.T→
2⟨δ
(1)k1
∫pδ
(2)p−k1δ
(1)−p
⟩=∫ Λ d3p
(2π)3F(2)(p,− k1)PL(p)PL(k1) = 68
21σ2(Λ)PL(k1), (4.23)
where F (2) is the second order kernel of SPT. The counterterm that fix it is: 6821σ
2(Λ)δ.
That happens because contracting with δ(q1) at the tree level:
6821σ
2(Λ)〈δ(1)(q)δ(1)(q1)〉tree F.T→ 6821σ
2(Λ)PL(k1). (4.24)
5Contracting with the linear part of δqiwe can separate the divergent part coming from the composite
operator from the ones that come from the non-linear part of the external legs. The last will berenormalized by the EFT of LSS.
64
4.1. Bias tracers, bias renormalization and its time evolution in Euler space
And for two external legs, the calculation is a bit longer and neglecting NLO diver-
gences it leads to:
⟨[δ2]xδ(1)
x2 δ(1)x1
⟩→ σ2(Λ)PL(k1)PL(k2)
[5248735 + 508
2205k1.k2
k21k
22
]. (4.25)
About the counterterms:
1. The first term will be canceled by 2624735 σ
2(Λ)δ2;
2. The second term, though, will be canceled out only by a term that is not propor-
tional to δ. Here the term that renormalize it is: 2542205σ
2(Λ)(∂i∂jΦg)2. We can see
that this term still preserves boost invariance, as it has two derivatives of the
rescaled gravitational potential.
The final renormalization until m = 2 is given by:
[δ2] = δ2 − σ2(Λ)[1 + 68
21δ + 81262205δ
2 + 2542205G2(Φg)
], (4.26)
such that with Φi = Φg,Φv6 we have:
G2(Φi) ≡ (∇i∇jΦi)2 − (∇2Φi)2. (4.27)
A last addendum is that if we consider the contraction with three external legs, a last
operator will contribute at 1loop level for the halo-matter and halo-halo correlations,
where the velocity potential starts being considered:
Γ3(Φv,Φg) = G2(Φg)− G2(Φv). (4.28)
6∇2Φv = θ, where θ is the velocity field.
65
4. The Bias Problem with EFT
4.1.3. Euler bias: basis of operators and correlations
We have finally constructed a set of operators that work as a basis for renormalizing
δ2 in the Euler expansion for the halo overdensity field O = δ,δ2,G,Γ3:
δh = b1δ + b2
2 δ2 + bG2G2 + bΓ3Γ3 . (4.29)
Thus, now we can calculate a functional form for the correlations [3]:
Phm = 〈δhδ〉 = b1PEFTmm (k) + b2
2 I[δ2](k) + bG2(F [G2](k) + I [G2](k)) + bΓ3F [Γ3](k),
Phh = 〈δhδh〉 = b1[b1PEFTmm (k) + b2I [δ2](k) + 2bG2(F [G2](k) + I [G2](k)) + 2bΓ3F [Γ3](k)]+
b2b2
4 I[δ2,δ2](k) + bG2b2I [δ2,G2](k) + bG2bG2I [G2,G2](k),
(4.30)
with the functions I [δ2],F [G2], I [G2],F [Γ3], I [δ2,δ2],I [δ2,G2] and I [G2,G2] being different com-
66
4.1. Bias tracers, bias renormalization and its time evolution in Euler space
binations of integrals of PL [3] deduced in B.3, which give 7:
I [δ2] ≡ 〈(δ(1))2δ(2)〉 = 2∫pF2(q − p,p)PL(p)PL(|q − p|),
F [G2] ≡ 〈G2δ(1)〉 = 4PL(q)
∫pσ2p,q−pF2(q,− p)PL(p),
I [G2] ≡ 〈G2δ(2)〉 = 2
∫pσ2p,q−pF2(q − p,p)PL(p)PL(|q − p|),
F [Γ3] ≡ 〈Γ3δ(1)〉 = −8
7P (q)∫pσ2p,q−pσ
2p,qP (p), (4.31)
I [δ2,δ2] ≡ 〈(δ(1))2(δ(1))2〉 = 2∫pPL(k)PL(k − p),
I [G2,G2] ≡ 〈(G2)(G2) = 2∫p(σ2
p,q−p)2PL(k)PL(k − p)〉,
I [δ2,G2] ≡ 〈G2(δ(1))2〉 = 2∫pσ2p,q−pPL(k)PL(k − p),
and the degeneracies are:
F [G2] = 52F
[Γ3],
I [δ2] ≈ −54I
[G2],
I [δ2,G2] ≈ −75I
[G2,G2]. (4.32)
We plot the shape of the single operators on the left of Figure 4.2, showing degen-
eracies between them. The double operators are plotted on the right of Figure 4.2.
7We emphasize that the number of free parameters can be reduced to 3 in the case of Phm – the linear,bi and bf – using the degeneracy between the integrals. In the case of Phh it can be reduced to 4.
67
4. The Bias Problem with EFT
10-2 10-1
k[hMpc−1 ]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Bias Renormalization - Single Terms
Iδ2
/P
IG2 /P
−5/4IG2 /P
−1/4 ∗F/P
(a)
10-2 10-1
k[hMpc−1]
20
15
10
5
0Bias Renormalization - Double Terms
−7/5 ∗ IG2G2/P
I δ2G2/P
−I δ 2δ 2
/P
(b)
Figure 4.2.: Computation of the single (left plot) and double (right) operators that renormalizethe halo-halo and halo-matter power spectrum. We can see that our results agreeperfectly with Figures 3 and 4 of Assassi [3].
4.2. Halo Bias Measurements
This section describes one of the works that is being developed by the student. The
idea was to fit the counterterms of the EFT-bias described above in this chapter, using
N-body simulations and find the mass dependence of these parameters. A similar
work was published in [88] while this work was being developed. We are planning to
extend our work, and here we present our results for the halo bias counterterms fit.
This work was done in collaboration with the Ph.D. student Rodrigo Voivodic.
4.2.1. The data used and the fitting procedure
For this work we used the simulated data from Dark Energy Universe Simula-
tion project (DEUS)8 with WMAP5 cosmology. We used three different box sizes:
162Mpch−1, 648Mpch−1 and 2592Mpch−1; with 10243 particles each. From the sim-
ulated data (dark matter particles), halos were found using a Friends-of-Friends
8http://www.deus-consortium.org/deus-data/
68
4.2. Halo Bias Measurements
algorithm9. We checked that the halo mass function is comparable with a prediction
by Sheth and Tormen [9] in Figure 4.3. For the following computations, we divided
each halo catalog into six mass bins.
Figure 4.3.: Halo mass function of the three DEUS simulation boxes. Each box workswell in the mass range it is expected to work (for instance, we expectthe large simulation to have less low mass halos). We compare the haloabundance found with the prediction from Sheth and Tormen (ST) [9].
We fit the theory to the N-body simulation data using a simple χ2 minimization.
The theoretical prediction come from the set of b in 4.3010. We first fitted b1 using
data in linear theory11, using k in the range [kmin, 0.2Mpch−1] and setting the other
coefficients equal to zero, as they are not expected to contribute in this range. We then
fix b1 to this fitted value and fitted the other coefficients using the data in the range
[kmin, 0.2Mpch−1]. By kmin we mean the respective minimum k of each one of the three
box. In the following sections, we present the main results.
9See [89] for a description and comparison of this halo finder method.10We also tested a different approach, where we minimize the ratio between halo-matter cross spectrum
and the matter power spectrum in order to reduce cosmic variance, but it leads to similar results.11Since the fail of linear theory is different for different halo masses, we used as kmax to fit b1, respec-
tively for the smaller, mid-size and larger box: 0.06Mpch−1,0.03Mpch−1 and 0.015Mpch−1.
69
4. The Bias Problem with EFT
4.2.2. Results
We begin showing the results for the halo-matter cross-spectrum in Figure 4.4 for
three different mass bins. In this plot we show the relative difference between the
theory and data for three different theory predictions: considering only b1, i.e. the
linear theory (red), considering only b1 and b2, i.e. a theory with second order bias
(in blue) and fitting the full set of counterterms in Equation 4.30 (in green). The EFT
prediction provided the best fit. The full EFT prediction works even better for the
largest mass bins, compared to the linear and second-order predictions. This happens
because linear theory is expected to fail earlier and the counterterms will be more
critical in this case. Notice that for low mass halos, which were found in the smallest
boxes, cosmic variance is much higher at these scales.
Of course we expect the theory with more free parameters, in this case the EFT
prediction, to give the best fit. Therefore we also show for this case we show also in
the plots the reduced χ2 that includes the number of degrees of freedom:
χ2 =∑(theory− data)2
len(data)−#D.O.F − 1 . (4.33)
Again the best result was provided by the EFT estimative. We can see that, for the larger
halo mass and for all theories, χ2 1, suggesting a failure of the analysis. Or, even
more than this, for this halo mass the 1Loop calculation fails before k = 0.2Mpch−1.
The solution for this is using different kmax for the fits or going one loop further in the
expansion.
70
4.2. Halo Bias Measurements
0.020.040.060.080.100.120.140.160.180.20k [h/Mpc]
0.4
0.2
0.0
0.2
0.4
P(T
heo
ry)
hm
/P(S
im)
hm
−1
M= 1. 55e+ 11M¯/h
Linear: χ2 = 9. 31
2nd Order: χ2 = 2. 01
2nd + c.t: χ2 = 1. 39
(a)
0.00 0.05 0.10 0.15 0.20k [h/Mpc]
0.4
0.2
0.0
0.2
0.4
P(T
heo
ry)
hm
/P(S
im)
hm
−1
M= 4. 17e+ 12M¯/hLinear: χ2 = 69. 36
2nd Order: χ2 = 2. 73
2nd + c.t: χ2 = 2. 38
(b)
0.00 0.05 0.10 0.15 0.20k [h/Mpc]
0.4
0.2
0.0
0.2
0.4
P(T
heo
ry)
hm
/P(S
im)
hm
−1
M= 4. 03e+ 14M¯/hLinear: χ2 = 16901. 74
2nd Order: χ2 = 472. 52
2nd + c.t: χ2 = 29. 27
(c)
Figure 4.4.: Best fit of the halo-matter power spectrum for three different mass bins(a,b,c). In each plot we show the residue between the theory and data forthree different counterterms structures: in red fitting only b1, in blue fittingb1 and b2 and in green the full set of counterterms of Equation 4.30. Wealso can see that the reduced χ2 is best for the EFT prediction. In shadedlines we show the estimated error, which is larger for the smaller massbins (small box sizes) due to cosmic variance.
We did the same calculation for the halo-halo power spectrum, which we show
in Figure 4.5. Cosmic variance is larger for the halo-halo spectrum compared to the
halo-matter cross-spectrum since the number of dark matter halos is much smaller
than dark matter particles. The results of the EFT prediction are still better than for the
71
4. The Bias Problem with EFT
other two. Its reduced χ2 is still better.
0.05 0.10 0.15 0.20k [h/Mpc]
0.4
0.2
0.0
0.2
0.4
P(T
heo
ry)
hh
/P(S
im)
hh
−1
M= 1. 55e+ 11M¯/h
Linear: χ2 = 2. 86
2nd Order: χ2 = 2. 12
2nd + c.t: χ2 = 1. 52
(a)
0.05 0.10 0.15 0.20k [h/Mpc]
0.4
0.2
0.0
0.2
0.4
P(T
heo
ry)
hh
/P(S
im)
hh
−1
M= 4. 17e+ 12M¯/hLinear: χ2 = 7. 61
2nd Order: χ2 = 8. 77
2nd + c.t: χ2 = 2. 34
(b)
0.05 0.10 0.15 0.20k [h/Mpc]
0.4
0.2
0.0
0.2
0.4
P(T
heo
ry)
hh
/P(S
im)
hh
−1
M= 4. 03e+ 14M¯/hLinear: χ2 = 64. 23
2nd Order: χ2 = 44. 85
2nd + c.t: χ2 = 20. 60
(c)
Figure 4.5.: Same as Figure 4.4 but now for the halo-halo power spectrum. As expected,the cosmic variance effects for the halo-halo spectrum are larger. The EFTprediction (green) and its reduced χ2 are still better.
Now we want to see how the bias changes with the mass bin and what is the mass
dependence of the new EFT counterterms. We also compare the measured bias with
the bias theoretical prediction from excursion set [10] (see also [90] for an elegant
formalism using path integrals).
The linear bias results are shown in Figure 4.6, where we plot the linear halo bias as
a function of halo mass. All measurements of the linear bias b1 were performed in low
72
4.2. Halo Bias Measurements
k regime without fitting more counterterms, such for all theories b1 found were the
same, changing only with mass. In the bottom, we show the relative difference to the
theory prediction. We show the fits also using only the halo-matter spectrum (red),
only the halo-halo (blue) and both (green). We can see that the values we found are
pretty close to theoretical prediction.
1011 1012 1013 1014 10150
5
10
15
20
b Lin
ear
Theory
HM
HH
HM+HH
1011 1012 1013 1014 1015
M¯/h
1.0
0.5
0.0
0.5
1.0
bdata
Lin
ear/btheory
Lin
ear−
1
Figure 4.6.: Linear bias dependence with the mass. In bottom we divide by thetheory [10]. We did the fit using only the halo-matter spectrum (red),only the halo-halo (blue) and both (green). The theory is consistent withthe data, especially for low masses. The black continue line show theexcursion set prediction for the bias [11].
The second order bias is plotted in Figure 4.7. In the top plot we show the values for
the prediction fitting only b1 and b2 (which was dubbed as bδ2 = b2/2, and represents
the coefficient appearing in front of δ2 in bias expansion). In the bottom, we show the
values found for the EFT prediction of bδ2 , which means we fitted all counterterms
together. Both plots are similar, and in the bottom, we chose to zoom in to show the
73
4. The Bias Problem with EFT
error bars compared with the theoretical prediction from excursion set.
1011 1012 1013 1014 10150
10
20
30
40
50
b(2nd)
δ2Second Order Halo Bias
TheoryHMHHHM+HH
1011 1012 1013 1014 1015
Mass [M¯/h]
2.01.51.00.50.00.51.01.52.0
b(EFT)
δ2
Figure 4.7.: Second order bias prediction for two theories: in the top, we fit only b1and b2; in bottom we show the bδ2 we got fitting all EFT counterterms. Theblack continue line show the excursion set prediction for the bias [11].
Finally in Figure 4.8 we show the mass dependence of the counterterms fitted. We
can see that for low masses their values are close to zero. This is consistent with the
first plots of 4.4 and 4.5, where linear, second order and EFT predictions seem close to
each other. For larger masses, the importance of the counterterms is greater since the
deviation from linear theory is also larger. That is also consistent with the last plots of
4.4 and 4.5, where EFT improves the fits even more.
74
4.3. Lagrangian space renormalization
1011 1012 1013 1014 10156
4
2
0
2
4
6
8
10
12
b I
Renormalized Halo BiasHM
HH
HM+HH
1011 1012 1013 1014 1015
Mass [M¯/h]
30
25
20
15
10
5
0
5
b F
Figure 4.8.: Mass dependence of the two EFT counterterms (bF and bI). We fit themusing the halo-matter spectrum (red), the halo-halo (blue) and both ofthem together (green). We can see that the larger the mass of the halo is,more important the counterterms are to fit the power spectrum.
In further works, the student is planning to extend this approach to voids and
consider improvements in the Halo Model using this non-linear bias prediction.
4.3. Lagrangian space renormalization
There are two approaches to dealing with the bias. First, the Euler formulation,
parameterizing the most general operators formed by the gravitational potential in
the final time-slice. This approach was described in the past sections of this chapter. In
the second approach, called Lagrangian space formulation, linear theory can predict
the formation of objects12 assuming conservation of the number of these objects with12For instance in the excursion set formalism [91].
75
4. The Bias Problem with EFT
respect to the nonlinear calculation. As the large-scale universe evolution is determin-
istic, this is the same as considering the initial time slice that leads to object formation
in the final time slice through linear growth [33, 70, 92, 93].
In this section, we further discuss the Lagrangian bias and it renormalization, work
that is currently being developed in collaboration with Prof. Rafael Porto. Recently, [94]
considered a renormalization scheme in Lagrange space. They do the perturbation in
the Fourier space, and figure out something similar to the resummation techniques
described in Section 1.2. Here we will consider a much simpler calculation: we describe
the creation of a Lagrangian basis of operators in Section 4.3.1. In Section 4.3.2 we
explain two different ways to do the expansion. In 4.3.3 we renormalize M2, similarly
to what was done above for δ2.
4.3.1. Lagrangian Basis
Previously, we found the most complete basis in Euler space can be written as
all possible contractions and powers of the gravitational potential derivatives ∂i∂jΦ.
Likewise, now we seek for an analogue basis composed by the distortion tensor
Mij = ∂qi sj (recall the derivative is taken with respect to the Lagrangian coordinate).
Following [29] we can write from si +Hsi = −∂iΦ:
∂i∂jΦ = εkmnεjpl2J (δpn +Mpn)(δlm +Mlm)(Mik +HMik), (4.34)
where J = |1+M |. Now we have the basic building block — the gravitational potential
derivatives — as a function of M and its derivatives. But since from perturbation
theory we can write any operator O that is composed by dO factors of ∂i ∂jΦ(1) as:
O = DdOO(dO) +DdO+1O(dO+1) + . . . , (4.35)
76
4.3. Lagrangian space renormalization
its time derivatives will be simply derivatives of the growth factor D and we can write
this as functions of M only (see Section 2.5 of [31] for a complete discussion).
Thus, writing M as a perturbation series:
M = DM (1) +D2M (2) +D3M (3) + . . . , (4.36)
with M (n) = ∂qi s(n)j , we can form a complete basis:
1st Order: Tr[M (1)]
2nd Order: Tr[M (1)]2,Tr[M (1)M (1)],Tr[M (2)]
3rd Order: Tr[M (1)]3,Tr[M (1)M (1)] Tr[M (1)], (4.37)
Tr[M (1)M (1)M (1)],Tr[M (2)] Tr[M (1)],Tr[M (2)M (1)],Tr[M (3)]
4th Order: Tr[M (2)] Tr[M (2)],Tr[M (3)] Tr[M (1)] . . .
Here, the red terms can be expressed as lower order terms by the equations of motion
and can be dropped from the basis [72].
4.3.2. Lagrangian Expansion
We now make clear how to relate the main blocks of our theory. There are two ways
of doing this. The first, easier, is expanding directly as all possible functions of M . Just
like we wrote δh(x,τ) = F [δ(x,τ)] we can expand δh(x,τ) = F [M (1),M (2),M (3), . . . ], as
(see 2.42 of [31]):
δh = −1 + |1 +Mh|−1 =
−1 + bTr[M ] Tr[M ] + bTr[MM ] Tr[MM ] + bTr[M ] Tr[M ] Tr[M ] Tr[M ] + . . . . (4.38)
77
4. The Bias Problem with EFT
This can be directly mapped into the Euler bias parameters, but the contribution of
each bias term is spread over each bM term, making it hard to figure out the physical
observables in this expansion.
Another way is through the Lagrangian-Euler relation:
δ(x,t) = J−1(q,t)− 1 = det[1 +Mij]−1|x=q+s − 1, (4.39)
with Mij = ∂isj . Expanding M in Lagrange coordinates:
Mij(x,τ) =∑n
Dn(τ)M (n)ij (q) =
∑n
Dn(τ)∂is(n)j , (4.40)
and s(n)j being defined in momentum space by Equation 2.14.
Using the relation of the determinant and the trace of M 13, we find the following
expansion through Equation 4.38 evaluated in the Lagrange coordinates [29]:
δ = −D(τ) Tr[M (1)] + 12D
2(τ)[Tr[M (1)]2 + Tr[M (1)M (1)]− 2 Tr[M (2)]
]+
D3(τ)[−Tr[M (1)]3
6 − Tr[M (1)M (1)] Tr[M (1)]2 + Tr[M (1)] Tr[M (2)]−
Tr[M (1)M (1)M (1)]3 + Tr[M (1)M (2)]− Tr[M (3)]
]+ . . . . (4.41)
We can than substitute this back into the Euler basis. We will take the first more
straightforward approach and renormalize the M operators.
4.3.3. Renormalization of M 2
We proceed now to renormalize the M2 operator, similarly to what was done for δ2
in Section 4.1.2. Here we contract the number of external legs m with each 2nd order
operator. We want to show the closure of the M basis 4.37 under the renormalization
13det(1 +M) = 1 + Tr[M ]− 12 (Tr[M2]− Tr[M ]2) + 1
6 (2 Tr[M3]− 3 Tr[M2] Tr[M ] + Tr[M ]3).
78
4.3. Lagrangian space renormalization
procedure. We did this calculation using the Mathematica software [95]. As the calcula-
tion is very long and boring we omit it here, reproducing only the results. The arrows
in calculations mean the Fourier Transform (FT):
• No external leg: m = 0:
There are two allowed contractions at this order
〈(Tr[M (1)]Tr[M (1)]
)q〉′ = 〈(∂is(1)
i ∂js(1)j )q〉′ F.T→
i2∫p(−pi)L(1)
i (−p)pjL(1)j (p)PL(p) = −
∫pPL(p) = −
∫ Λ dp
2π2p2PL(p) ≡ −σ2(Λ) ,
(4.42)
and:
〈(Tr[M (1)M (1)]
)q〉′ = 〈(∂is(1)
j ∂js(1)i )q〉′ F.T→
i2∫p(−pi)L(1)
j (−p)pjL(1)i (p)PL(p) = −
∫pPL(p) = −
∫ Λ dp
2π2p2PL(p) ≡ −σ2(Λ) .
(4.43)
See Section 2.2 and Appendix A.5 for a definition of the Lagrange kernel L.
Conclusion: at m = 0 we have the same result as Euler. It is a simple number
that can be absorbed in the counterterm in front of the operator.
• One external leg: m = 1:
Again the two allowed contractions with one external leg s(1)m (q2) are:
〈(Tr[M (2)]Tr[M (1)])q s(1)m (q2)〉′ = 〈(∂is(2)
i ∂js(1)j )q s(1)
m (q2)〉′ →
i2〈(∫
p(pi − ki1)s(2)
i (p− k1)(−pj)s(1)j (−p)
)s(1)m (k1)〉′ =
−PL(k1)27km1k2
1σ2(Λ) , (4.44)
79
4. The Bias Problem with EFT
and:
〈(Tr[M (2)M (1)])q s(1)m (q2)〉′ = 〈(∂is(2)
j ∂js(1)i )q s(1)
m (q2)〉′ →
i2〈(∫
p(pi − ki1)s(2)
j (p− k1)(−pj)s(1)i (−p)
)s(1)m (k1)〉′ =
−PL(k1)27km1k2
1σ2(Λ) , (4.45)
as we have two different contractions :
−PL(k)47kmk2 σ
2(Λ) . (4.46)
The counterterm we need to absorb this term already belongs to the basis:
47σ
2(Λ)∂isi = 47σ
2(Λ)Tr[M ] ⊂Mbasis, (4.47)
because if we operate with s(1)m :
47σ
2(Λ)〈(∂isi)s(1)m 〉 →
i47σ
2(Λ)〈kisis(1)m 〉 = 4
7σ2(Λ)km
k2 PL(k) . (4.48)
• Two external legs: m = 2
Expanding until this order, we have now two kinds of terms expanding M : as a
31 expansion and as a 22.
80
4.3. Lagrangian space renormalization
The 22 terms are:
〈(Tr[M (2)]Tr[M (2)])q s(1)m (q1)s(1)
l (q2)〉′ = 〈(∂is(2)i ∂js
(2)j )q s(1)
m (q1)s(1)l (q2)〉′ →
i2〈(∫
p(pi − ki1)s(2)
i (p− k1)(−pj − kj2)s(2)j (−p− k2)
)s(1)m (k1)s(1)
l (k2)〉 =
− 18245PL(k1)PL(k2)k
l2k2
2
km1k2
1
(1 + 1
3(k2 · k1
k2k1)2)σ2(Λ) , (4.49)
and:
〈(Tr[M (2)M (2)])q s(1)m (q1)s(1)
l (q2)〉′ = 〈(∂is(2)j ∂js
(2)i )q s(1)
m (q1)s(1)l (q2)〉′ →
i2〈(∫
p(pi − ki1)s(2)
j (p− k1)(−pj − kj2)s(2)i (−p− k2)
)s(1)m (k1)s(1)
l (k2)〉 =
− 18245PL(k1)PL(k2)k
l2k2
2
km1k2
1
(1 + 1
3(k2 · k1
k2k1)2)σ2(Λ) . (4.50)
The 31 terms are:
〈(Tr[M (3)]Tr[M (1)])q s(1)m (q1)s(1)
l (q2)〉′ = 〈(∂is(3)i ∂js
(1)j )q s(1)
m (q1)s(1)l (q2)〉′ →
i2〈(∫
p(pi − ki1 − ki2)s(3)
i (p− k1 − k2)(−pj)s(1)j (−p)
)s(1)m (k1)s(1)
l (k2)〉 =
−PL(k1)PL(k2)kl2k2
2
kmk2
1
∫pPL(p)(pi − ki1 − ki2)L(3[A+B+C+D])
i (p,− k1,− k2) (4.51)
We did each term of L(3)i (p,− k1,− k2) separately (A,B,C,D). Summing them:
(A) + (B) + (C) + (D) = −PL(k1)PL(k2)kl2k2
2
km1k2
1
163
[15 + (k2 · k1
k2k1)2]σ2(Λ) .
(4.52)
81
4. The Bias Problem with EFT
The next contraction term is:
〈(Tr[M (3)M (1)])q s(1)m (q1)s(1)
l (q2)〉′ = 〈(∂is(3)j ∂js
(1)i )q s(1)
m (q1)s(1)l (q2)〉′ →
i2〈(∫
p(pi − ki1 − ki2)s(3)
j (p− k1 − k2)(−pj)s(1)i (−p)
)s(1)m (k1)s(1)
l (k2)〉 =
−PL(k1)PL(k2)kl2k2
2
kmk2
1
∫pPL(p)−p
2 + k1px+ k2 · pp2 (−pj ·L(3[A+B+C+D])
j (p,− k1,− k2)).
(4.53)
Doing each term of L(3)j (p,− k1,− k2) separately (A,B,C,D) using Mathematica we
found the same result:
(A) = 1063
(1− (k1 · k2)2
k21k
22
)σ2(Λ) (4.54)
(B) = 263
[3 + (k2 · k1
k2k1)2]σ2(Λ) (4.55)
(C) = 221
[1 + 1
3(k2 · k1
k2k1)2]σ2(Λ) (4.56)
(D) = 19
(1− (k2 · k1
k2k1)2)σ2(Λ) (4.57)
Again:
(A) + (B) + (C) + (D) = −PL(k1)PL(k2)kl2k2
2
km1k2
1
163
[15 + (k2 · k1
k2k1)2]σ2(Λ)
(4.58)
82
4.3. Lagrangian space renormalization
Summing up, the final result with two external legs is:
(22) + 2× (31) = PL(k1)PL(k2)kl2k2
2
km1k2
1
(−1212
2205 + 1242205(k2 · k1
k2k1)2)σ2(Λ) . (4.59)
The counterterms we need to cancel out these terms are:
1. The first term will need a counterterm like σ2(Λ)(∂s)2:
σ2(Λ)〈(∂isi∂jsj)qsm(q1)sl(q2)〉 →
i2σ2(Λ)〈ki1si(−k1)kj2sj(−k2)sm(k1)sl(k2)〉 = −kim
k21
kl2k2
2σ2(Λ)PL(k1)PL(k2) .
(4.60)
2. The second term will need a counterterm like σ2(Λ)(∂isj∂jsi):
σ2(Λ)〈(∂isj∂jsi)qsm(q1)sl(q2)〉 →
σ2(Λ)〈ki1sj(−k1)kj2si(−k2)sm(k1)sl(k2)〉 =
kimk2
1
kl2k2
2
(k1 · k2
k21k
22
)σ2(Λ)PL(k1)PL(k2) . (4.61)
All counterterms needed to renormalize M2 are, therefore, also elements of the
basis. This shows we have a complete set of operators.
We have presented the Lagrangian bias renormalization procedure, in which we
can similarly expand the bias as Equation 4.38. Despite the relation of these bias
parameters bM with observables being not so clear as in Euler, the Lagrangian
basis is already complete under renormalization.
We have described the procedure of renormalizing M2 at one loop level, point
out the interesting fact that do not need to include any new operator to the basis,
83
4. The Bias Problem with EFT
differently from what we saw in Euler’s base in Sec. 4.1.2.
84
5Inflation
This is an independent part of the dissertation in which we describe how EFTs have
been used in Inflation. We first motivate inflation, describe its most important model
based on [12, 96] and then see how EFT intends to parametrize these models.
85
5. Inflation
5.1. Big Bang problems
The first problem in the Big Bang scenario is called the Horizon Problem. The
particle horizon grows with conformal time (recall dτ = dt/a):
τ − τi =∫ a
ai
da
Ha2 ∼ a(1+3ω)/2. (5.1)
Since a grows with time, regions, regions that have never been in causal contact are now
inside inside each other’s horizon. This means that we would expect inhomogeneities
in the CMB. However, the inhomogeneities in the CMB temperature are tiny, being
around 10−3%. The conclusion is that somehow these points must have been in causal
contact in the past.
The other problem is called Flatness problem. Multiple measurements indicate
that the curvature of the universe is very small (Ωk < 10−3). A Universe with zero
curvature is an unstable point, not an attractor. Since:
∂Ωk
∂ log a = Ωk(1 + 3ω), (5.2)
for a matter dominated universe (ω = 0):
∂Ωk
∂ log a = Ωk. (5.3)
As time goes by, any small curvature grows. For the curvature to be small as it is
today, at the time of Nucleosynthesis (BBN) we should have Ωk < 10−18. This is a very
strict and fine-tuned constraint, but it still need to say that it can be allowed by any
symmetry.
Alan Guth proposed a solution solution to both paradoxes in the 1980s [97] with
Inflation. Inflation simultaneously puts all points in causal contact and converts the
86
5.2. The Main Inflation Models
Ωk = 0 into an attractor.
One important consideration done by [12] is that this severe constraints on the
initial conditions (very homogeneous and flat universe) can be assumed a priori by
the standard Big Bang model. What Inflation has as an advantage is that this initial
condition can be generated and not simply imposed.
5.2. The Main Inflation Models
The Big Bang problems arise when we consider a universe dominated in the past
by a component with equation of state ω > −1/3. Both problems are solved if we
consider that in the past some other component with ω < −1/3 dominated, something
like dark energy [97].
This extra component provided an accelerated expansion in the early universe. This
acceleration allows for the CMB photons to be in causal contact in the past, such that
they could reach the thermal equilibrium. Moreover, an accelerating universe has
Ωk = 0 as an attractor.
Ok, so now we have an Inflation theory that solves the Big Bang problems. But
Inflation generates another set of questions: how did Inflation end? Inflation must
end, otherwise we would observe a deSitter like universe expansion at all epochs. So
now we have two requirements: the universe must have had an accelerated expansion
in the past, but this acceleration must end somehow. And its end should provide the
right initial conditions for BBN. Let us now take a look at the simplest inflationary
model, generated by a single scalar field undergoing slow-roll [98], usually called
inflaton field. One should keep in mind that there are much more complicated models,
even considering supersymmetry [99].
87
5. Inflation
The action of a scalar field minimally coupled to gravity and its stress tensor are:
S =∫d4x√−g
[M2
PL
2 R + 12(∂φ)2 − V (φ)
], (5.4)
Tµν = ∂µφ∂νφ− gµν[12(∂φ)2 + V (φ)
]. (5.5)
We want to generate an accelerated expansion (something like ω < 1/3). If the scalar
field has potential energy much larger than its kinetic energy, this condition can be
satisfied:
ω = p/ρ =12 φ
2 − V (φ)12 φ
2 + V (φ)≈ −1, (5.6)
The second equality above holds if we identify the density and pressure terms in the
stress tensor. This solution is called slow-roll solution, as described in Figure 5.1.
Figure 5.1.: Slow-roll potential. This potential is needed to generate an accelerated expansionof the universe after the Big Bang, in the Inflation period. Extracted from [12].
From the Euler-Lagrange equation, we can obtain the equation of motion for this
field to be:
φ+ 3H + ∂φV = 0. (5.7)
The equalities above determine the two Inflation conditions. We define two slow-
roll parameters of Inflation: ε is the time rate of change of the Hubble length and η
88
5.3. Perturbations on the Inflaton
guarantees the slow-roll condition with a large friction term, such that Inflation holds
for a sufficiently long time [96]:
ε = − H
H2 = 12φ2
H2 1, (5.8)
η = − φ
Hφ 1, (5.9)
ε is also related to the numberN of e-folds of the universe expansion during Inflation:
ε = −d logHdN
. (5.10)
In order to solve the BB problems we should have N ∼ 60, i.e. the universe should
increase in size by a factor e60.
5.3. Perturbations on the Inflaton
For a time-dependent background solution φ0(t), a perturbative expansion around
this solution is:
φ(x,t) = φ0(t) + δφ(x,t). (5.11)
Under the coordinate transformation:
xµ → x′µ = xµ + ξµ therefore by first order Taylor δφ→ δφ(x′) + φ0ξ0. (5.12)
Taylor expanding again and using the fact that√−g = e3Ht for a deSitter universe, the
action now reads [96]:
S = S0 +∫d4xe3Ht[−gµν∂µδφ∂νδφ], (5.13)
89
5. Inflation
in Fourier Space this Lagrangian is a simple quadratic Lagrangian such that we can
consider a harmonic oscillator frequency ω(t) = k/a(t) and write the action as:
S =∫dtd3ka3[δφkδφ−k −
k2
a2 δφkδφ−k]. (5.14)
We want to calculate the correlation statistics of δφ, mainly the two-point function:
〈δφδφ〉 = 〈δφ2k〉. Just like for the harmonic oscillator now we have:
〈δφ2k〉 ∼
1a3ω
. (5.15)
If we want to calculate this quantity for when ω ∼ H (freeze-out), as will be explained
in following section, we have a ∼ k/H and:
〈δφ2k〉 ∼
H2
k3 . (5.16)
This is a scale invariant two-point function, because perturbations are generated with
equal power on all scales (integrating in volume all scales contribute equally).
5.4. Predictions of Inflation
Let us first consider a classical universe. The universe is all filled up with φ at the
top of its potential. At each point of the universe, the field starts rolling down this
potential the same way. We expect a constant surface of φ = φend at t = tend (see
Figure 5.1), with all points reaching this value at the same time.
But if we consider quantum fluctuations in this field, we no longer expect this equal-
time surface. The duration of inflation, and therefore the expansion of each point, will
be different. At the end of the day, what have a φ = φend surface, but a locally different
90
5.5. Most General Action in the Unitary Gauge
scale factor:
aeff = a(1 + ζ) ∼ δa
a, (5.17)
where ζ is the local metric or curvature fluctuation. The reason for defining ζ is that,
for modes outside the horizon (k aH), ζ is conserved (for a proof see [27]). The
main point is then trying to constrain Inflation by these curvature fluctuations because
in all after-Inflation universe evolution ζkaH is "untouched" 1. The way we probe ζ is
by the coupling with the CMB and with LSS (its primordial non-gaussianities).
Here we intend to give just a small introduction to Inflation. For a more complete
study, the references in which we tightly based this section are [12, 63, 96].
Until now we concluded that Inflation solves the main problems of the Big Bang
Theory with a quasi deSitter universe. It is a quasi deSitter universe because it must
end somehow generating the right initial conditions for the subsequent evolution of
the universe.
Since Inflation must end at some time, there must be a special clock setting its
end. It is a time symmetry breaking event. Time diffeomorphisms are no longer
valid. In Quantum Field Theory any broken symmetry generates the so-called Nambu-
Goldstone bosons (NGBs), which live in the coset space of the broken generators of
the group symmetry. Likewise, Inflation will have NGBs associated to time translation
symmetry breaking.
5.5. Most General Action in the Unitary Gauge
Consider a single scalar field φ model, inside a FRW universe (FRW metric field).
We suppose this field can be expanded as a background plus a perturbation, repeating
1It is outside the horizon! This is the freeze-out described by Section 5.3.
91
5. Inflation
what was done above:
φ = φ0(t) + δφ(x,t). (5.18)
As a scalar, φ is invariant under diffeomorphisms (diffs). However, the perturbation
is not scalar under time diffs (though it is under space diffs) :
t→ t+ ξ0(t,x) then by Taylor δφ→ δφ+ φ0(t)ξ0. (5.19)
Let’s remember QFT classes: in a symmetry breaking process, we have two types
of gauges. One in which we see the NGBs and another in which we absorb this field
in the gauge field components2. The former is usually called π − gauge and the latter
unitary gauge. When the massless vector field eats the NGB it gains mass, and now it
has one more degree of freedom3.
By GR we know that we can go to a frame where there are no perturbations φ = φ0(t).
We say that in this frame the perturbation was eaten by the graviton, that now has 3
d.o.f (scalar mode + 2 helicities).
In order to respect all the spacetime diffs symmetries, the only terms allowed in
the action are those proportional to the Riemann tensor Rµνρλ. Now we have more
freedom because we can break time diffs. From Appendix A of [63] we know that the
most general action in the unitary gauge can have only dependence on the following
terms:
S =∫d4x√−g F (Rµναβ, g
00,Kµν ,∇µ,t), (5.20)
with terms proportional to time slices extrinsic curvature Kµν = (gσµ + nσµ)∇σnν , where
nµ is the vector orthogonal to this slices.
2We say that the NGBs were eaten by the vector field.3Remember a photon has two polarization, but a massive boson has three. A photon can find no
rest! [100]
92
5.5. Most General Action in the Unitary Gauge
Considering the Lagrangian dependence above, we can write the following action:
S =∫d4x√−g [12M
2PLR +M2
PLHg00 −M2
PL(3H2 + H)+
M2(t)4
2! (g00 + 1)2 + M3(t)4
3! (g00 + 1)3 − M2(t)2
2! δKµµ
2 + . . . ]. (5.21)
This action seems somewhat complicated, but it is just the most generic Lagrangian
that preserves spatial symmetries but breaks time diffs. Let’s enumerate the main
points about this action, to make it easy:
1. It describes perturbations around a flat FRW with expansion H .
2. We are in the unitary gauge. That is the reason we do not have φ terms but
δg00 = g00 + 1.
3. δKµν = Kµν − a2(gµν + nµnν) is the differential with respect to the FRW metric.
The most general Lagrangian might have a dependence on the extrinsic curvature
of the constant time slices.
4. We have time-dependent operators, breaking time diffs. But we do not have
space dependent operators.
5. The first term is the GR Einstein-Hilbert term. The following two terms are set
by the background unperturbed FRW solution.
6. The higher order terms parametrize all different Inflation theories in the M
coefficients.
7. This is a clearly hierarchical approach. We can see order by order the deviation
from the slow-roll model (the simplest one, as we will see next). This treatment
is similar to what is done in beyond Standard Model physics.
93
5. Inflation
5.5.1. The Slow-Roll in this Parametrization
Slow-roll field action is:
S =∫d4x√−g
[−1
2g00φ2 − 1
2(∂iφ)2 − V (φ)]
=∫d4x√−g
[−1
2g00φ2
0 − V (φ0)]. (5.22)
The equality holds for the unitary gauge. From the Friedmann equation we have:
φ20 = −2M2
PLH and V (φ) = M2PL(3H2 + H). (5.23)
so that:
S =∫d4x√−g
[g00M2
PLH −M2PL(3H2 + H)
]. (5.24)
Thus for slow-roll Inflation, we have all high-order terms set to zero. That is why the
M operators are the deviation from the slow-roll model.
5.6. Action in the NGB Gauge
In order to understand how from the action 5.21 we can insert the NGB again, let us
repeat the same procedure for a SU(N) theory. In the unitary gauge:
S =∫d4x
[−1
4 TrF 2 − m2
2 TrA2]. (5.25)
Under a gauge transformation U , defining Dµ = ∂µ − igAµ:
Aµ →i
gUDµU
† such that S =∫d4x
[−1
4TrF2 − m2
2g2TrDµU†DµU
]. (5.26)
The last term, a mass term, is not gauge invariant. We want the gauge invariance back.
It can be restored parametrizing U = exp itaπa(x,t), where π is our NGB scalar. Then
94
5.6. Action in the NGB Gauge
we normalize π such that we have a canonical kinetic term πc = mgπ:
S =∫d4x
[−1
4TrF2 − m2
2g2Tr(−ita∂µπ
a)(ita∂µπa)]
(5.27)
=∫d4x
[−1
4TrF2 − 1
2(∂µπac )2]. (5.28)
It is important to notice that in the first equality we have neglected interaction terms
like mAaµ∂µπac . This is true in the limit of large kinetic term, for E m. Both fields
decouple there. We will make the same approximation for the gravity coupling with
the NGBs!
Let’s repeat the most general action. To make it easy we will consider here only the
linear order operators:
S =∫d4x√−g [12M
2PLR +M2
PLHg00 −M2
PL(3H2 + H) . . . ] (5.29)
=∫d4x√−g [A(t) +B(t)g00]. (5.30)
if we consider time diffs:
t→ t+ ξ0(x), x→ x = x then g00 → g00 = ∂x0
∂xµ∂x0
∂xνgµν . (5.31)
so that in the new coordinates:
S =∫d4x
√−g(x)
∣∣∣∣∣∂x∂x∣∣∣∣∣ [A(t) +B(t)∂x
0
∂xµ∂x0
∂xνgµν(x)] (5.32)
=∫d4x
√−g(x) [A(t− ξ0) +B(t− ξ0)∂(t− ξ0)
∂xµ∂(t− ξ0)∂xν
gµν(x)]. (5.33)
For simplicity we drop the tilde hereafter. In the SU(N) case, we just parametrized
our gauge transformation with the NGBs. Here we do the same, substituting the time
95
5. Inflation
diffs by our π:
ξ0(x)→ −π(x) which implies: (5.34)
S =∫d4x
√−g(x) [A(t+ π(x)) +B(t+ π(x))∂(t+ π(x))
∂xµ∂(t+ π(x))
∂xνgµν(x)]. (5.35)
Here is our NGB back again! The full action 5.21 is now written as:
S =∫d4x√−g [ 1
2M2PLR−M2
PL(3H2(t+ π) + H(t+ π))
+M2PLH(t+ π) ((1 + π)2g00 + 2(1 + π)∂iπg0i + gij∂iπ∂jπ)+
M2(t+ π)4
2! ((1 + π)2g00 + 2(1 + π)∂iπg0i + gij∂iπ∂jπ + 1)2
+M3(t+ π)4
3! ((1 + π)2g00 + 2(1 + π)∂iπg0i + gij∂iπ∂jπ + 1)3 + . . . ].
(5.36)
But what do we gain reintroducing π? The main point is that: just like in the past
for E m we neglected the interaction of π with the gauge boson, now we can do
the same. Locally, at very short distances (analogous to E m) there is no metric
perturbation affecting π. No interaction between the fields! We can neglect terms with
δg in the Lagrangian for high-energy limits.
Let‘s first consider the slow-roll case. The mixing terms are like: M2PLHπδg
00 =
H1/2πcδg00c , so that the mixing energy:
Emix = H1/2 = ε1/2H → 0 when ε→ 0. (5.37)
The action, in the regime E Emix and after neglecting low derivative terms4 is:
S =∫d4x√−g [12M
2PLR−M2
PLH(π2 − (∂iπ)2/a2
)]. (5.38)
4remember ∂t ∼ E
96
5.7. Constraints on Inflation and the EFT parametrization
We can do the same for the full action. The only difference is redefining what is our
limit E Emix5. The full action reads:
S =∫d4x√−g [ 1
2M2PLR−M2
PLH(π2 − (∂iπ)2/a2
)+2M4
2
(π2 + π3 − π(∂iπ)2/a2
)− 4
3M43 π
3 + . . . ]. (5.39)
It is interesting to notice the hierarchical structure of the π operators here. One
question that arises is: if in cosmology we consider IR scales (size of the universe, etc.),
in which the decoupling limit above is no longer valid, why should we consider this
limit? This question was answered at the end of Section 5.4. Do you remember there
is a conserved quantity ζ that is constant outside the horizon? If we have H > Emix,
then 5.39 is still valid! Even though it is strange to think H as a UV limit, that is
what happens! It is the UV of greater scales outside the horizon. The horizon is a UV
compared with what is farther. Our expansion parameter, surprisingly, is Emix/H .
5.7. Constraints on Inflation and the EFT
parametrization
In this short section we expose the main constraints that were imposed over infla-
tionary models by [14] and how the CMB experiments were used to put bound in the
EFT parameters.
In Figure 5.2 we show the constraints that the Planck 2015 results imposed on the
main Inflation theories. For this, they show the plot of the contours on the spectral
index ns and the scalar-to-tensor ration r, defined as:
ns − 1 ≡ d ln k3PR2π2d ln k , (5.40)
5e.g for the action with terms proportional to M2, we should have Emix ∼M2/MP L
97
5. Inflation
rs = PTPR
, (5.41)
and PR and PT are, respectively, the power spectrum of the primordial scalar and
tensorial fluctuations. While the spectral index measures the deviation from a scale-
invariant spectrum for the scalar perturbations, the scalar-to-tensor ratio measures
the difference between tensorial and scalar perturbations. They are very consistent
predictions of Inflation that can be related to other physical observables.
Figure 5.2.: Planck 1σ and 2σ constraints on the tensor-to-scalar ratio r and on the spectralindex ns. The constraints are compared to the values expected by the maininflationary models. We can see that Starobinsky Inflation [13], a simple R2
gravitation, is still inside all the confidence regions. Also the φ2 model is still closeto the contours. We can see that the scalar field theories with different polynomialpotentials are pretty acceptable, justifying the EFT parametrization around thismodel. Result from [14].
We can see in Figure 5.2, taken from [14], that most Inflationary theories stay close
to the confidence regions. Specially the Starobinsky Inflation [13], which is a f(R)
model with R2 that makes GR singularities vanish. Also scalar field models with
a polynomial potential are still acceptable, suggesting that a expansion around this
theory is a good deal. This fact justifies the usage of the EFT approach described above.
In Figure 5.3 we show the constraints on the EFT parameters obtained by [15]. There
98
5.7. Constraints on Inflation and the EFT parametrization
they plot the c3 and cs parameters, which are just combinations of the M parameters
of Eq. 5.21.
Figure 5.3.: WMAP5 1σ, 2σ and 3σ constraints on the EFT of Inflation parameters c3 and cs.Figure from [15].
We can see that instead of analyzing the whole zoo of Inflation theories, it is easy
to map them in the EFT parameters that can be constrained by themselves. Also EFT
parameters bring important information, since each term in Eq. 5.21 is associated with
a different relevant operator.
That way, we finish the discussion of Inflation. It was given a short explanation of
vital discussions around the other EFT usages in cosmology.
99
6Conclusion
In this dissertation, the focus was on improving the theoretical prediction of the
statistics for matter perturbations and for matter tracers, like halos. We described
the Standard Perturbation Theory approach, which perturbs the matter field and
computes the correlations order by order. These correlations are usually associated
with diagrams with loop shapes and each new loop order is expected to improve the
previous one. We have seen that this did not happen, suggesting several problems
in theory. The first evident problem is that the approach contains a special scale Λ,
without physical interpretation, which is the cutoff in the loop integrals. Another
problem is that for some cosmologies the series may not converge. Also, the severe
101
6. Conclusion
problem is that SPT parametrizes the interaction with small-scales physics incorrectly,
since the fluid equations considered are not valid on small scales. We have also shown
that the three-loop term, numerically calculated by a 9-dimensional integral, is larger
than the one-loop and two-loop contributions. It shows the failure of this approach
explicitly.
We presented the effective field theory approach as a solution to all SPT problems:
it correctly parametrizes the small-scale physics influence on large scales through
counterterms. The counterterms cancel out the cutoff dependence of the theory and
improve the power spectrum predictions. The counterterms are inserted directly in the
equations of motion, parameterizing the stress tensor as an effective stress tensor that
is a function of the density field derivatives. We calculated the 3-loop EFT, expanding
previous calculations that were done only in the 2-loop level.
Next, we have shown the applications of the Effective Field Theory ideas to the
bias problem. The relation between tracers and the matter density field is usually
calculated as a Taylor expansion in the matter overdensity field. We presented the
primary results in the literature that have shown that this expansion is not a complete
basis under the effects of time evolution. Time evolution naturally generates other
terms that are not proportional to the matter overdensity. We also explained a second
reason why this basis is not complete: the operators with powers higher than one
needed are very sensitive to small-scale physics and must be renormalized. This
renormalization brings operators non-local in density. We determined the halo bias
non-local parameters by fits to simulation data and showed their mass dependence.
The EFT parameters provide a much better prediction for the halo-bias than the usual
non-linear bias expansion.
We also computed the halo-bias renormalization in Lagrange space. Lagrange
bias has several advantages compared to the Euler bias because their operators are a
102
complete basis of operators.
Finally, we described other application of EFTs in cosmology, more specifically to
the inflationary scenario. Inflation sets a special clock and breaks time shift symmetry.
This symmetry breaking generates Nambu-Goldstone bosons that can be eaten by
the metric. EFT constructs the most general theory allowed by spatial translation
symmetries, which will enable the construction of a basis that parametrizes deviations
from slow-roll single field theory.
103
AThe Fluid Equations
Following [26] we aim here to arrive in the fluid equations for the dynamics of matter.
For this, we used Boltzmann equation (Vlasov with conservation) and Poisson to
describe gravity. I thank Renato Costa for sharing and cross-checking the calculations
in this appendix.
A.1. From Vlasov to Fluid
From the Vlasov equation which is basic the conservation of particle number in a
phase space, f(τ,~x,~p), say:
105
A. The Fluid Equations
df(τ,x,p)dτ
= 0, (A.1)
we obtain
∂f
∂τ+ ∂xi
∂τ
∂f
∂xi+ ∂pi
∂τ
∂f
∂pi= 0, (A.2)
which can be rewritten as
∂f
∂τ+ pi
ma
∂f
∂xi−ma∂iφ ∂f
∂pi= 0, (A.3)
where we use pi = ma∂xi
∂τ, and ∂pi
∂τ= −ma∂iφ. If we integrate out the momentum
dependence of the above equation, i.e, applying the operator∫d3p, on it we get
∂
∂τ
∫d3pf +
∫d3p
pi
ma
∂f
∂xi+∫d3p
∂pi
∂τ
∂f
∂pi=
∂
∂τ
∫d3pf + ∂
∂xi
∫d3p
pi
maf −ma∂iφ ∂
∂pi
∫d3pf. (A.4)
Now, defining
∫d3pf(x,p,τ) = ρ(x,τ),∫
d3ppi
maf(x,p,τ) = ρ(x,τ)ui(x,τ), (A.5)∫
d3ppi
ma
pj
maf(x,p,τ) = ρ(x,τ)ui(x,τ)uj(x,τ) + σij, (A.6)
substituting the first two equations of (A.6), into (A.4), we get the conservation energy
equation
∂τρ+ ∂xi(ρui) = 0, (A.7)
106
A.2. The Fourier transform of equations
which can be rewritten by using the density contrast definition δ(x,τ) = ρ(x,τ)ρ(τ) − 1
arriving in:
∂τδ + ∂i[(1 + δ)ui] = 0 (A.8)
The same way for the momentum, we can multiply Equation A.3 by the operator∫d3p pi
ma:
∂
∂τ
∫d3p
pi
maf + ∂
∂xj
∫d3p
pi
ma
pj
maf −ma∂jφ ∂
∂pj
∫d3p
pi
maf (A.9)
leading to :
∂τ (ρui) + ∂xj(ρuiuj + σij) + ρ∂xi
Φ = 0 (A.10)
which substituting the continuity equation times ui:
∂τui +Hui + ui∂ju
j = −∂iΦ−1ρ∂j(ρσij) (A.11)
A.2. The Fourier transform of equations
The continuity equation in real space, for θ = ∂iui is given by
∂δ(~x,τ)∂τ
+ θ(~x,τ) = −∂i(δ(~x,τ)ui(~x,τ)). (A.12)
Using the fact tha ~u = ∇γ +∇× ~w, where∇× (∇γ) = 0, and∇ · (∇× ~w) = 0. We also
assume from now on that ~w ∼ 0, which leads to ~u = ∇γ. Since θ = ∇ · ~u, we get
θ = ∇2γ,
107
A. The Fluid Equations
or
∇−2θ = γ.
Back in the formula for ~u, we get
~u = ∇−2(∇θ).
Now, applying the operator∫ d3x
(2π)3 e−i~k · ~x on equation (A.12) and using the fact that
A(~k,τ) =∫ d3x
(2π)3 e−i~k · ~xA(~x,τ),
we get
∂δ(~k,τ)∂τ
+ θ(~x,τ) = −∫ d3x
(2π)3 e−i~k · ~x∇
[ ∫d3k1e
i ~k1 · ~xδ(~k1,τ)∇−2∇(∫
d3k1ei ~k2 · ~xθ(~k2,τ)
)].
= −∫ d3x
(2π)3 e−i~k · ~x
[ ∫d3k1(i~k1)ei ~k1 · ~xδ(~k1,τ)(−i)
~k2
k22
(∫d3k1e
i ~k2 · ~xθ(~k2,τ))
+∫d3k1e
i ~k1 · ~xδ(~k1,τ)(∫
d3k1ei ~k2 · ~xθ(~k2,τ)
)]
= −∫d3k1
∫d3k2
∫ d3x
(2π)3 e−i(~k− ~k1+ ~k2) · ~xδ(~k1,τ)θ(~k2,τ)
)
−∫d3k1
∫d3k2
∫ d3x
(2π)3 e−i(~k− ~k1+ ~k2) · ~xδ(~k1,τ)
~k1 ·~k2
k22θ(~k2,τ)
= −∫d3k1
∫d3k2δ
D(~k − ~k12)(1 +~k1 ·~k2
k22
)δ(~k1,τ)θ(~k2,τ)
= −∫d3k1
∫d3k2δ
D(~k − ~k12)α(~k1,~k2)δ(~k1,τ)θ(~k2,τ),
where α(~k1,~k2) = (1 + ~k1 ·~k2k2
2) and δD(~k − ~k12) =
∫ d3x(2π)3 e
−i(~k− ~k1+ ~k2) · ~x.
108
A.3. Perturbation theory in Einstein-de Sitter cosmology
The same way we can find that:
∂θ′(~k,τ)∂τ
−H2(τ)θ′(~k,τ) + 32ΩmH2(τ)δ(~k,τ) =
−H(τ)2∫d3k1d
3k2δD(~k − ~k12)β(~k1,~k2)θ′(~k1,τ)θ′(~k2,τ) (A.13)
with:
β = |~k1 + ~k2|2(~k1 · ~k2)
2k21k
22
. (A.14)
A.3. Perturbation theory in Einstein-de Sitter
cosmology
Assuming a flat universe with only matter (Ωm = 1 and ΩΛ = 0) one can solve the
Friedman equations and find (using conformal time) thatH(τ) = 2/τ and a(τ) ∼ τ 2.
∂δ(~k,τ)∂τ
−H(τ)θ′(~k,τ) = H(τ)∫d3k1d
3k2δD(~k − ~k12)α(~k1,~k2)θ′(~k1,τ)δ(~k2,τ)
∂θ′(~k,τ)∂τ
−H2(τ)θ′(~k,τ) + 32ΩmH2(τ)δ(~k,τ) =
−H(τ)2∫d3k1d
3k2δD(~k − ~k12)β(~k1,~k2)θ′(~k1,τ)θ′(~k2,τ) (A.15)
and from what we can see that from the solution for the scale factor and for the Hubble
parameter the separation of variables works out since all the terms will be of the same
order in τ , which makes the above equations to depend only on ~k. Note that this is
very particular for a Einstein-de Sitter universe.
So, using the ansatz
δ(~k,τ) =∑n
an(τ)δ(n)(~k), θ(~k,τ) = −H(τ)∑n
an(τ)θ(n)(~k) (A.16)
109
A. The Fluid Equations
plugging it back to equation (A.15) and assuming small values for the scale factor
(which means we can neglect the nonlinear terms of the RHS), gives
δ(1)(~k)∂τa(τ)−H(τ)a(τ)θ(1)(~k) = 0. (A.17)
Using the values for a(τ) andH(τ) in a Einstein-dS universe gives δ(1)(~k) = θ(1)(~k).
Now, to go to the next step we consider the next order in perturbation. This means,
the equation for θ(2)(τ) is given by
1a2(τ)H(τ)
∂(δ(2)(~k)a2(τ))∂τ
− θ(2)(~k) =∫d3k1d
3k2δD(~k − ~k12)α(~k1,~k2)δ(1)(~k1)δ(1)(~k2)
− 1a2(τ)H2(τ)
∂(H(τ)a2(τ)θ(2)(~k))∂τ
− θ(2)(~k) + 32Ωmδ
(2)(~k) =
−∫d3k1d
3k2δD(~k − ~k12)β(~k1,~k2)δ(1)(~k1)δ(1)(~k2)
(A.18)
where we have used the fact that θ1(τ) = δ1(τ). Now usingH(τ) = 2/τ , a(τ) ∼ τ 2 and
Ωm = 1, the above equations become
2δ(2)(~k)− θ(2)(~k) =∫d3k1d
3k2δD(~k − ~k12)α(~k1,~k2)δ(1)(~k1)δ(1)(~k2)52 θ
(2)(~k)− 32 δ
(2)(~k) =∫d3k1d
3k2δD(~k − ~k12)β(~k1,~k2)δ(1)(~k1)δ(1)(~k2)
(A.19)
Multiplying the first equation by 5/2 and adding it to the second equation above, we
get
72 δ
(2)(~k) =∫d3k1d
3k2δD(~k − ~k12)(
52α(~k1,~k2) + β(~k1,~k2)
)δ(1)(~k1)δ(1)(~k2) (A.20)
110
A.3. Perturbation theory in Einstein-de Sitter cosmology
Since α(~k1,~k2) = (1 + ~k1 ·~k2k2
2) and β(~k1,~k2) ≡ k2
12(~k1 ·~k2)2k2
1k22
, we have
δ(2)(~k) =∫d3k1d
3k2δD(~k − ~k12)(
57α(~k1,~k2) + 2
7β(~k1,~k2))δ(1)(~k1)δ(1)(~k2)
=∫d3k1d
3k2δD(~k − ~k12)(
57 + 5
7~k1 ·~k2
k22
+ 27
[(~k1 ·~k2)
2k22
+ (~k1 ·~k2)2k2
1+ (~k1 ·~k2)2
k21k
22
])δ(1)(~k1)δ(1)(~k2)
=∫d3k1d
3k2δD(~k − ~k12)(
57 + (~k1 ·~k2)
2k22
+ (~k1 ·~k2)2k2
1+ 2
7(~k1 ·~k2)2
k21k
22
)δ(1)(~k1)δ(1)(~k2)
=∫d3k1d
3k2δD(~k − ~k12)F2(~k1,~k2)δ(1)(~k1)δ(1)(~k2) (A.21)
where
F2(~k1,~k2) = 57 + (~k1 ·~k2)
2k22
+ (~k1 ·~k2)2k2
1+ 2
7(~k1 ·~k2)2
k21k
22
,
and we have used the fact that (~k1 ·~k2)2k2
2= 1
2(~k1 ·~k2)
2k21
+ 12
(~k1 ·~k2)2k2
2inside the integral.
The same way we can write with kT = ∑ni=1 ki:
δ(n)(~k) =∫d3k1 . . . d
3knδD(~k − ~kT )Fn(~k1, . . . ,~kn)δ(1)(~k1) . . . δ(1)(~kn)
θ(n)(~k) =∫d3k1 . . . d
3knδD(~k − ~kT )Gn(~k1, . . . ,~kn)δ(1)(~k1) . . . δ(1)(~kn) (A.22)
with:
Fn(k1, . . . ,kn) =n−1∑i=1
Gi(k1, . . . ,ki)(2n+ 3)(n− 1)×
[(2n+ 1)α(k1,k2)Fn−i(ki+1, . . . ,kn) + 2β(k1,k2)Gn−i(ki+1, . . . ,kn)] (A.23)
and
Gn(k1, . . . ,kn) =n−1∑i=1
Gi(k1, . . . ,ki)(2n+ 3)(n− 1)×
[3α(k1,k2)Fn−i(ki+1, . . . ,kn) + 2nβ(k1,k2)Gn−i(ki+1, . . . ,kn)] (A.24)
111
A. The Fluid Equations
A.4. Matrix notation and the non-local in time
propagator
Based on [26, 38] we set here the often used matrix notation. We also show how to
consider a non-local interaction in time for the modes (see Figure A.1). Up to now we
considered the evolution of the modes (even the perturbed modes generated after the
interaction of two linear modes) following the linear growth factor D1. A more unified
way of doing it is by defining the vector
Ψi(k,z) ≡(δ(k,z),−θ(k,z)
H
), (A.25)
where i = 1,2 and z = ln a (here we use EdS universe, Ωm = 1). The equations of
motion can be rewritten in terms of this vector as
∂zΨi(k,z) + ΩijΨj(k,z) = γijk(k,k1,k2)Ψj(k1,z)Ψk(k2,z) (A.26)
where repeated Fourier modes are supposed to be integrated over and the only non-
zero components of the γ matrix are γ121(k,k1,k2) = δ(k − k12)α(k,k1) and γ222 =
δ(k− k12)β(k1,k2). Also
Ωij ≡
0 −1
−3/2 1/2
. (A.27)
112
A.4. Matrix notation and the non-local in time propagator
τ
τin
=φi(1)(τ) =φi
(2)(τ) =φi(3)(τ)
τ
τin
τ
τin
Figure A.1.: Description of time evolution of the modes. The interaction of two linearmodes gives rise to the second order expansion. And the interaction ofthis with another linear makes the third order expansion. This couplingcan happen at any point of time, being non-local. That way we need tointegrate over time. Diagrams from [16].
A way to solve the above equation formally is by first taking the Laplace transform
of it, say
σ−1ij (ω)Ψj(k,ω) = φi(k) + γijk(k,k1,k2)
∮ dω1
2πiΨj(k1,ω1)Ψk(k2,ω − ω1), (A.28)
where σ−1ij (ω)Ψj(k,ω) = L∂zΨi(k,z) + ΩijΨj(k,z))(ω) and φi(k) = Ψi(k,z = 0) is the
linear solution. Remembering that the Laplace transform of a product of two functions
is
La(z)b(z)(ω) =∮ dω1
2πiA(ω1)B(ω − ω1). (A.29)
Now, multiplying equation (A.28) by σij(ω) it becomes
Ψi(k,ω) = σij(ω)φj(k) + σil(ω)γljk(k,k1,k2)∮ dω1
2πiΨj(k1,ω1)Ψk(k2,ω − ω1). (A.30)
Calling
f(z) = γijk(~k,~k1,~k2)Ψj(~k1,z)Ψk(~k2,z) (A.31)
113
A. The Fluid Equations
and
hl(ω) ≡ Lf(z)(ω) = γljk(~k,~k1,~k2)∮ dω1
2πiΨj(~k1,ω1)Ψk(~k2,ω − ω1) (A.32)
the equation becomes
Ψi(~k,ω) = σij(ω)φj(~k) + σil(ω)hl(ω). (A.33)
Using the mathematical result that the inverse Laplace transform of one function is
L−1F (ω)(z) =∮ dω
2πif(ω)eωz (A.34)
and for a product of two functions is given by
L−1F (ω)G(ω) =∫ z
0f(z′)g(z − z′)dz′ (A.35)
Applying the inverse Laplace transform on (A.33), we finally get
Ψi(~k,ω) = gij(z)φj(~k) + (ω)γljk(~k,~k1,~k2)∫ z
0dz′gil(z − z′)Ψj(~k1,z
′)Ψk(~k2,z′). (A.36)
Here
gij(z) =∮ dω
2πiσij(ω)eωz = ez
5
3 2
3 2
− e−3z/2
5
−2 2
3 −3
(A.37)
and we can see that both, increasing and decreasing modes are present. We highlight
a few aspects of this result: SPT normally collapse the time interaction (make it local)
to z = 0 such we have gij = δij . [38] tried to model this non-local in time interaction
with a non-linear propagator that depends also on k. Even though this method is well
114
A.5. Lagrangian Kernels
succeeded to estimate how much information about the initial conditions φ remains in
the final vector Ψ it doesn’t correct the main problem of SPT that is the lack of a right
UV theory that couples the small scale physics with large scale.
A.5. Lagrangian Kernels
It is important to make explicitly is the Lagrangian kernels1. The n-th order displace-
ment field:
s(n)(p,τ) = −iD2(τ)∫
p(2π)3δ3
D(p12...n − p)Ln(p1,...,pn)δ0(p1)...δ0(pn). (A.38)
where, Ln(pi1 ,...pin) ≡ 1p2
i1,...in
[pi1...inSn(pi1 ,...,pin)+pi1...in×Tn(pi1 ,...,pin)
]and p12...n ≡
p1 + p2 + ... + pn. We split the L kernel in a scalar S part and a tensorial part T. At
linear order we consider a irrotational fluid,
S1(p1) = 1, T1(p1) = 0. (A.39)
Using the recursion relations for the Lagrangian kernels [72, 73] one can show that the
above initial conditions imply the following relations for the second order kernels:
S2(p1,p2) = 37κ2(p1,p2), T2(p1,p2) = 0. (A.40)
where
κ2(p1,p2) = 1−(
p1 · p2p1p2
)2
. (A.41)
1I would like thank Renato Costa for the contributions to this part of the text.
115
A. The Fluid Equations
For the third order we have,
S3(p1,p2,p3) = 53κ2(p12,p3)S2(p1,p2)− 1
3κ3(p1,p2,p3) (A.42)
T3(p1,p2,p3) = w2(p1,p23)S2(p2,p3) (A.43)
where
w2(p1,p23) = (p1 × p2)(p1 · p2)(p2
1p22) , (A.44)
κ3(p1,p2,p3) = 1−(
p1 · p2p1p2
)2
−(
p1 · p3p1p3
)2
−(
p2 · p3p2p3
)2
+ 2(p1 · p2)(p1 · p3)(p2 · p3)p2
1p22p
23
.
(A.45)
The scalar and vector part of the fourth order kernel is
S4(p1,p2,p3,p4) = 2811
(κ2(p1,p234)S3(p1,p2,p3)−w2(p1,p234) · T3(p2,p3,p4)
)
+1711κ2(p12,p34)S2(p1,p2)S2(p3,p4)− 26
11κ3(p1,p2,p34)S2(p3,p4). (A.46)
T4(p1,p2,p3,p4) = 2(
w2(p1,p234)S3(p2,p3,p4) + p1 × p234p2
1p2234
(p1 × p234) · T3(p2,p3,p4)).
(A.47)
The last expressions were first obtained in [101].
116
BStatistics Calculations complements
and Diagrams
B.1. Definitions
The correlation function of the matter is defined to be:
ξ(r) = 〈δ(x)δ(x + r)〉 (B.1)
117
B. Statistics Calculations complements and Diagrams
such its Fourier transform, the matter power spectrum can be calculated as:
〈δ(k)δ(k′)〉 = δD(k + k′)P (k) (B.2)
with:
P (k) =∫ dr
(2π)3 e−ik.rξ(r) (B.3)
B.2. Perturbation to Diagrams
We illustrate here schematically how from the perturbation theory defined in Equa-
tion ?? we find the loop expansion at the same time τ :
δD(k + k′)P (k,τ) = 〈δ(k,τ)δ(k′,τ)〉 = 〈∑n
Dn(τ)δ(n)∑m
Dm(τ)δ(m)〉
∑n,m
Dn+m(τ)〈δ(n)δ(m)〉 = D2(τ)〈δ(1)δ(1)〉+D4(τ)(〈δ(2)δ(2)〉︸ ︷︷ ︸P22
+2 〈δ(1)δ(3)〉︸ ︷︷ ︸P13
) + . . . (B.4)
where we use the Wick theorem, such that for Gaussian fields – as we expect the linear
density field to be in simple inflationary models – the contraction of an odd number of
δ(1) is zero1.
Following Equation A.22 we have:
〈δ(1)δ(1)〉 = P0(k) (B.5)
P21 = 〈δ(2)δ(1)〉 =∫d3k1F2(~k1,~k − ~k1)〈δ(1)(~k1)δ(1)(~k − ~k1)δ(1)(−~k)〉 = 0 (B.6)
1Remembering that following Equation A.22 δ(n) contains n orders of δ(1)
118
B.2. Perturbation to Diagrams
because the field is gaussian and any odd number of contractions will lead to this.
P22 = 〈δ(2)δ(2)〉 =∫d3k1
∫d3k1′F2(~k1,~k − ~k1)F2(−~k1′ ,− ~k + ~k1′)×
〈δ(1)(~k1)δ(1)(~k − ~k1)δ(1)(−~k1′)δ(1)(−~k + ~k1′)〉 (B.7)
The term insede brackets, by the Wick’s theorem can be written as the sum of other 3
terms:
〈δ(1)(~k1)δ(1)(~k − ~k1)δ(1)(−~k1′)δ(1)(−~k + ~k1′)〉 =
〈δ(1)(~k1)δ(1)(~k − ~k1)〉〈δ(1)(−~k1′)δ(1)(−~k + ~k1′)〉+
〈δ(1)(~k1)δ(1)(−~k1′)〉〈δ(1)(~k − ~k1)δ(1)(−~k + ~k1′)〉+
〈δ(1)(~k1)δ(1)(−~k + ~k1′)〉〈δ(1)(−~k1′)δ(1)(~k − ~k1)〉 =
δD(k)P0(k1)δD(−k)P0(k1′)+
δD(k1 − k1′)P0(k1)δD(k1 − k1′)P0(|k − k1′|)+
δD(k1 + k1′ − k)P0(k1)δD(k1 + k1′ − k)P0(k1′) (B.8)
The first term will be zero due the dirac delta will lead to a term like F2(k,− k) = 0.
The other two will be equal when integrated over k1′ :
P22 =∫k1′ ,k1
F2(~k1,~k − ~k1)F2(−~k1′ ,− ~k + ~k1′)
2δD(k1 − k1′)P0(k1)δD(k1 − k1′)P0(|k − k1′|) =
2∫k1
(F2(~k1,~k − ~k1))2P0(k1)P0(|k − k1|) (B.9)
The same procedure can be done for P13 and the other terms of the two and three
119
B. Statistics Calculations complements and Diagrams
loops diagrams [65, 102]:
P13 = 3P0(k)∫k1F3(~k1,− ~k1,k)P0(k1) (B.10)
The two loop diagrams are:
P15 = 15P0(k)∫k2,k1
F5(~k1,− ~k1,~k2,− ~k2,k)P0(k1)P0(k2)
P24 = 12∫k2,k1
F4(~k1,− ~k1,~k2,k − ~k2)F2(−~k2,− k + ~k2)P0(k2)P0(|k − k2|)P0(k1)
P33A = 6∫k2,k1
F3(~k1,~k2,k − ~k1 − ~k2)2P0(k2)P0(|k − k2 − k1|)P0(k1)
P33B = 9P0(k)∫k2,k1
F3(~k1,− ~k1,k)F3(~k2,− ~k2,k)P0(k2)P0(k1) (B.11)
and the three loops:
P17 = 105P0(k)∫k3,k2,k1
F7(~k1,− ~k1,~k2,− ~k2,~k3,− ~k3,k)P0(k1)P0(k2)P0(k3)
P26 = 90∫k3,k2,k1
F6(~k1,− ~k1,~k2,− ~k2,~k − ~k3,~k3)F2(−~k + ~k3,− ~k3)P0(|k − k3|)P0(k1)P0(k2)P0(k3)
P35A = 60∫k3,k2,k1
F5(~k1,− ~k1,~k2,~k − ~k3 − ~k2,~k3)F3(−~k + ~k2 + ~k3,− ~k2,− ~k3)
P0(|k − k2 − k3|)P0(k1)P0(k2)P0(k3)
P35B = 60P0(k)∫k3,k2,k1
F5(~k1,− ~k1,~k2,− ~k2,~k)F3(−~k,− ~k3,~k3)
P0(k1)P0(k2)P0(k3)
P44A = 24∫k3,k2,k1
(F4(~k1,~k2,~k3,~k − ~k1 − ~k2 − ~k3))2
P0(k1)P0(k2)P0(k3)P0(|k − k1 − k2 − k3|)
P44B = 72∫k3,k2,k1
F4(~k1,− ~k1,~k − ~k3,~k3)F4(~k2,− ~k2,− ~k + ~k3,− ~k3)
P0(k1)P0(k2)P0(k3)P0(|k − k3|) (B.12)
120
B.3. Halo bias counter terms derivation
B.3. Halo bias counter terms derivation
In this section we show explicitly each counter-term statistics of the halo bias,
described in Section 4.1.3.
In Euler usually space we expand:
δh = b1δ + b2
2 δ2 + bG2G2 + bΓ3Γ3 + b∇2δ∇2δ =
b1′δ + b2
2 δ2 + bG2G2 + bΓ3Γ3. (B.13)
With the shorthand notation b1′ = b1 + b∇2δ∇2 or in Fourier space b1′ = b1 + b∇2δk2.
B.3.1. Halo-Matter
Here we reproduce the halo-matter correlation:
〈δhδ〉 = 〈[b1(δ(1) + δ(2) + δ(3)) + b2
2 [(δ(1))2 + (δ(2))2 + 2δ(2)δ(1)] + b3
6 (δ(1))3 + bG2G2
+bΓ3Γ3 + b∇2δ∇2(δ(1) + δ(2) + δ(3))]×
(δ(1) + δ(2) + δ(3))〉 =
b1′(PL(k) + P22(k) + 2P13(k)) + bδ2 [I [δ2] + F [δ2]] + bG2F [G2] + bG2I [G2] + bΓ3F [Γ3] .
(B.14)
121
B. Statistics Calculations complements and Diagrams
with:
〈(δ(1))3δ(1)〉 = 0,
F [δ2] ≡ 2〈δ(2)q δ(1)
q δ(1)−q1〉,
I [δ2] ≡ 〈(δ(1))2qδ
(2)−q1〉,
F [G2] = 〈G2δ(1)〉,
I [G2] = 〈G2δ(2)〉, (B.15)
F [Γ3] = 〈Γ3δ(1)〉,
I [Γ3] = 〈Γ3δ(2)〉 = 0.
The formulas for each term calculated are just below.
B.3.2. Halo-Halo
Here we reproduce the halo-halo correlation:
〈δhδh〉 =
〈[b1(δ(1) + δ(2) + δ(3)) + b2
2 [(δ(1))2 + (δ(2))2 + 2δ(2)δ(1)] + b3
6 (δ(1))3 + bG2G2 + bΓ3Γ3 + b∇2δ∇2δ]
×[b1(δ(1) + δ(2) + δ(3)) + b2
2 [(δ(1))2 + (δ(2))2 + 2δ(2)δ(1)] + b3
6 (δ(1))3 + bG2G2 + bΓ3Γ3 + b∇2δ∇2δ]〉 =
b1′ [b1′(PL + P22 + 2P31) + 2bδ2(F [δ2] + I [δ2])+
2bG2(F [G2] + I [G2]) + 2bΓ3(F [Γ3] + I [Γ3])] + bδ2bδ2I [δ2,δ2]
+2bG2bδ2I [δ2,G2] + bG2bG2I [G2,G2] (B.16)
122
B.3. Halo bias counter terms derivation
, with
I [δ2,δ2] ≡ 〈(δ(1))2(δ(1))2〉,
I [G2,G2] ≡ 〈(G2)(G2)〉,
I [δ2,G2] ≡ 〈G2(δ(1))2〉. (B.17)
which we calculate explicitly below.
B.3.3. F terms
Here we show explicitly the calculation of the F terms.
F [δ2] = 〈δ2(δ(1))〉 F.T.→
2× 〈∫p[F (2)(p,− k)δ(1)(p)δ(1)(−k)δ(1)(−p)][δ(1)(k)]〉 =
4PL(k)∫pF (2)(p,− k)PL(p) =
4PL(k)∫pF (2)(p,− k)PL(p), (B.18)
which is a number times PK and is degenerated with b1.
123
B. Statistics Calculations complements and Diagrams
F [G2] = 〈G2(δ(1))〉 = 〈[(∇i∇jΦ)2 − (∇2Φ)2](δ(1))〉 F.T.→
2× 〈∫p[(pi − ki)(pj − kj)
F (2)(p,− k)δ(1)(p)δ(1)(−k)(~p− ~k)2
pipjδ(1)(−p)
p2 −
F (2)(p,− k)δ(1)(p)δ(1)(−k)δ(1)(−p)][δ(1)(k)]〉 =
4PL(k)∫p[ (p
2 − ~p ·~k)2
(~p− ~k)2p2− 1]F (2)(p,− k)PL(p) =
4PL(k)∫pσ2p,q−pF
(2)(p,− k)PL(p). (B.19)
F [Γ3] = 〈Γ3(δ(1))〉 = 〈(G2(Φg)− G2(Φv))(δ(1))〉 =
〈[(∇i∇jΦg)2 − (∇2Φg)2](δ(1))〉 − 〈[(∇i∇jΦv)2 − (∇2Φv)2](δ(1))〉 F.T.→
4PL(k)∫pσ2p,q−p(F (2)(p,− k)−G(2)(p,− k))PL(p). (B.20)
B.3.4. I terms
Here we show explicitly the calculation of the I terms.
I [δ2] ≡ 〈(δ(1))2qδ
(2)−q1〉
F.T.→ 〈∫pδ(1)(−p)δ(1)(p− k)F (2)(k − p,p)δ(1)(k − p)δ(1)(p)〉 =
2∫pF (2)(k − p,p)PL(|k − p|)PL(p). (B.21)
124
B.3. Halo bias counter terms derivation
I [G2] = 〈G2δ(2)〉 F.T.→ 2× 〈
∫p[(pipj
δ(1)(−p)p2 )((pi − ki)(pj − kj)
δ(1)(|p− k|)(p− k)2 )− δ(1)(−p)δ(1)(p− k)]
F (2)(k − p,p)δ(1)(k − p)δ(1)(p)〉 =
2∫pσ2p,q−pF
(2)(k − p,p)PL(|k − p|)PL(p). (B.22)
B.3.5. Double I terms
Here we explicitly show the calculation of the double I terms.
I [δ2,δ2] ≡ 〈(δ(1))2(δ(1))2〉 F.T.→
〈∫pδ(1)(p)δ(1)(k − p)δ(1)(−p)δ(1)(−k + p)〉 = 2
∫pPL(k)PL(k − p). (B.23)
I [G2,G2] ≡ 〈(G2)(G2)〉 = 〈[(∇i∇jΦg)2 − (∇2Φg)2]2〉 F.T.→
〈(∫p[(pipj
δ(1)(−p)p2 )((pi − ki)(pj − kj)
δ(1)(|p− k|)(p− k)2 )− δ(1)(−p)δ(1)(p− k)])2〉 =
2∫q(σ2
p,q−p)2PL(p)PL(|p− k|). (B.24)
I [δ2,G2] ≡ 〈G2(δ(1))2〉
〈∫p[(pipj
δ(1)(−p)p2 )((pi − ki)(pj − kj)
δ(1)(|p− k|)(p− k)2 )− δ(1)(−p)δ(1)(p− k)]×∫
p′δ(1)(−p′)δ(1)(|p′ − k|)〉 =
2∫qσ2p,q−pPL(p)PL(|p− k|). (B.25)
125
CIR Resummation
Here we show how appears the implicit expansion parameters of the SPT expansion
and open the calculation showing that the expansion in Euler coordinates leads to an
expansion in the large-scale displacements, which are large and can be resummed. It
is different of what happen in Lagrange coordinates, where the expansion parameters
are well behaved.
127
C. IR Resummation
C.1. Parameters
We define the following parameters that will be contained in the asymptotic behavior
of the loops in SPT and Lagrangian Perturbation Theory.
εδ,<k =∫ k
0
dq
2π2 q2PL(q), is responsable for tidal effects of modes larger than k;
εs,<k =∫ k
0
dq
2π2PL(q), coordinates the displacement generated by modes larger than k ;
εs,>k =∫ ∞k
dq
2π2PL(q), coordinates the displacement generated by modes smaller than k ;
η =∫ ∞k
dq
2π2P 2L(q)q2 , which are subleading stochastic contributions.
C.2. Lagrangian Calculation
The Lagrangian correlation between the displacements are calculated as [70, 85]:
C(11)ij (k) = −kikj
k4 PL(k), (C.1)
C(13)ij (k) = − 5
21kikjk4
k3
4π2PL(k)∫ ∞
0drPL(kr)
∫ 1
−1dx
r2(1− x2)2
1 + r2 − 2rx,≡ −521kikjk4 R1(k)
(C.2)
C(22)ij (k) = − 9
98kikjk4
k3
4π2
∫ ∞0
drPL(kr)∫ 1
−1dxPL[k(1 + r2 − 2rx)1/2] r2(1− x2)2
(1 + r2 − 2rx)2
≡ − 998kikjk4 Q1(k). (C.3)
We can split the integrals into two asymptotic behaviors. Let’s analyse the (13)
128
C.2. Lagrangian Calculation
integral divergence:
C(13)ij (k) '
− 521kikjk4
k3
4π2PL(k)[∫ k
0drPL(kr)
∫ 1
−1dx
r2(1− x2)2
1 + r2 − 2rx︸ ︷︷ ︸r → 0
+∫ ∞k
drPL(kr)∫ 1
−1dx
r2(1− x2)2
1 + r2 − 2rx︸ ︷︷ ︸r →∞
] =
−1663kikjk4
k3
4π2PL(k)[∫ k
0drPL(kr)r2 +
∫ ∞k
drPL(kr)] =
− 863kikjk4 PL(k)[εδ,<k + k2εs,>k], (C.4)
and now the (22) integral divergence:
C(22)ij (k) '
− 998kikjk4
k3
4π2 [∫ k
0drPL(kr)
∫ 1
−1dxPL[k(1 + r2 − 2rx)1/2] r2(1− x2)2
(1 + r2 − 2rx)2︸ ︷︷ ︸r → 0
+
∫ ∞k
drPL(kr)∫ 1
−1dxPL[k(1 + r2 − 2rx)1/2] r2(1− x2)2
(1 + r2 − 2rx)2︸ ︷︷ ︸r →∞
] =
− 12245
kikjk4
k3
4π2 [2PL(k)∫ k
0drr2PL(kr) +
∫ ∞k
drP 2L(kr)r2 ] =
− 6245
kikjk4 [2PL(k)εδ,<k + k4η]. (C.5)
We can see that there is no expansion in the IR displacements εs,<k, which is the
problematic parameter as we will see.
129
C. IR Resummation
C.3. Euler Calculation
The same calculation of the correlations of the matter densities fields (parameters of
an Eulerian fluid) can be done [67]. By the first expansion terms:
P13(k) = 6P (k)∫qP (q)F (s)
3 (k,q,− q), (C.6)
P22(k) = 2∫qP (q)P (|k− q|)(F (s)
2 (q,k− q))2, (C.7)
we can expand like:
P22(k) = 2∫qP (q)P (|k− q|)(F (s)
2 (q,k− q))2 =∫ kdr(kr)2
4π2 P (kr)∫ 1
−1dxPL(k(1 + r2 − 2rx)1/2)(r (3− 10x2) + 7x)2
98r2 (r2 − 2rx+ 1)2 '∫ k
0
kdr(kr)2
4π2 PL(kr)∫ 1
−1dxPL(k(1 + r2 − 2rx)1/2)(r (3− 10x2) + 7x)2
98r2 (r2 − 2rx+ 1)2︸ ︷︷ ︸r → 0
+
∫ ∞k
kdr(kr)2
4π2 PL(kr)∫ 1
−1dxPL(k(1 + r2 − 2rx)1/2)(r (3− 10x2) + 7x)2
98r2 (r2 − 2rx+ 1)2︸ ︷︷ ︸r →∞
=
k2
6 PL(k)∫ k
0
dq
2π2PL(q) + 998k
4∫ ∞k
dq
2π2P 2L(q)q2
k2
6 PL(k)εs,<k + 998k
4η. (C.8)
130
C.4. The resummation
and:
P13(k) = 6P (k)∫qP (q)F (s)
3 (k,q,− q) =
6P (k)∫qP (q)×
[( 1k2 + q2 − 2kqx)
(5
126k2 − 11kqx
108 + 7q2x2
108 −k4x2
54q2 + 4k3x3
189q −23k3x
756q + 25k2x2
252 − 2kqx3
27
)+
( 1k2 + q2 + 2kqx)
(5
126k2 + 11kqx
108 − 7q2x2
108 −4k4x2
27q2 −53k3x3
189q + 23k3x
756q −121k2x2
756 − 5kqx3
27
)] '
−k2
3 P (k)∫ k
0
dq
2π2P (q) + 10k2
21 P (k)∫ ∞k
dq
2π2P (q) =
−k2
3 P (k)εs,<k + 10k2
21 P (k)εs,>k. (C.9)
which contains the expansion in εs,<k. In Figure C.1 we can see that in mid-nonlinear
scales, where we are trying to improve the spectrum, this is the most problematic
parameter that start to be order higher than 1. Each loop order will bring an intensifi-
cation of this problem.
C.4. The resummation
The calculation to the resummation is extensive, and here we only summarize the
idea behind this process and the recipe for resumming each term.
We start writing the matter density as a function of the displacement field s and the
initial coordinate of each fluid particle as:
δ(k,t) =∫dq exp(−ik.(q + s)) (C.10)
131
C. IR Resummation
0.01 0.1k [h/Mpc]
10-4
10-3
10-2
10-1
100
101
ε
k2 εs, >k
k2 εs, <k
εδ, <k
k4 η/PL (k)
Figure C.1.: Expansion parameters of the perturbation theory of Lagrangian and Eulerframeworks. We can see that εs,<k. grows faster than the others and startbeing order higher than one in nonlinear scales.
such the correlation is:
〈δ(k1,t1)δ(k2,t2)〉 =
(2π)3δD(k1 + k2)∫dq exp(−ik1.q)〈exp [ik1.(s(q,t1)− s(0,t2)]〉
(2π)3δD(k1 + k2)∫dq exp(−ik1.q) exp
[∑N
iN
N !〈X(k1,q; t1,t2)N〉c]. (C.11)
In the second line we used the cumulant theorem and:
X(k1,q; t1,t2) = k1. (s(q,t1)− s(0,t1)) , (C.12)
The procedure of [82] is, summarized, a Taylor expansion in the quantity
exp[∑
NiN
N !〈X(k1,q; t1,t2)N〉c]
in powers of only εδ< and εs>. We denote the sum up to
132
C.4. The resummation
order N in the εδ< and εs> by a bar like P|. The sum up to order N in all expansion
parameters (including εs<) is indicated by a double bar like P||. Here, as it is not the
goal, we leave the extensive calculation for the references [82, 84].
P (k,t)|N =N∑j=0
∫k′M||N−j(k,k′; t)Pj(k′,t). (C.13)
Where M is associated with the probability of a fluid particle that started in q (which
works also as a label) to finish the position r and the index j denotes the order of
expansion of each term. It is a kind convolution of each term of the usual power
spectrum with a filter M , described in simply way in [84]. The final calculation of the
matter power spectrum in three loops is:
P 3Loop,IRResumEFT (k)|2 = (2π)2cstochk
4 +∫k′M||3(k,k′)P0(k′)+
M||2(k,k′)[P1(k′)− 2(2π)c2
s1k′2P0(k′)
]+
M||1(k,k′)[P2(k′)− 2(2π)c2s2k′2P0(k′)− 2(2π)(c2
s1)k′2P1 + (2π)2[(c2s1)2]k′4P0
−2(2π)c1k′2∫qP (k′ − q)P (q)F2 + 2(2π)2(c4)k′4P0(k′)]+
M||0(k,k′)[P3(k′)− 2(2π)c2s3k′2P0(k′)− 2(2π)c2
s2k′2P1(k′)− 2(2π)(c2
s1)k′2P2(k′)
+(2π)22c2s2c
2s1k′4P0(k′) + (2π)2(c2
s1)2k′4P1(k′) + 2(2π)2(c4,2)k′4P0(k′) + 2(2π)2c4k′4P1(k′)..
(C.14)
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