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Towards understanding the Large-Scale Structurein the Universe using perturbation theory
Zvonimir Vlah
Stanford University & SLAC
with:
Raul Angulo (CEFCA), Alejandro Aviles (ABACUS), Emanuele Castorina (Berkeley),Matteo Fasiello (Stanford), Yu Feng (Berkeley), Patrick McDonald (Berkeley),
Marcel Schmittfull (Berkeley), Uros Seljak (Berkeley), Leonardo Senatore (Stanford),Martin White (Berkeley),
Structure Formation and Evolution
CMB:āĻ/Ļ ā¼ 10ā6
LSS: āĻ/Ļ ā¼ 100
Galaxies: āĻ/Ļ ā¼ 106
z=1100
z=2
z=0
[Planck,2013]
[2dF, 2002]
[Parsons, 1845]
LSS using PT Introduction 2 / 38
LSS: motivations and observations
Theoretical motivations: Inflation - origin of structures Expansion history Composition of the universe Nature of dark energy and darkmatter
Neutrino mass and number ofspecies
Test of GR and modifications ofgravity
Current and future observations: SDSS and SDSS3/4: SloanDigital Sky Survey
BOSS: the Baryon OscillationSpectroscopic Survey
DES: the Dark Energy Survey LSST: the large synoptic surveytelescope.
Euclid: the ESA mission to mapthe geometry of the dark Universe
DESI: Dark EnergySpectroscopic Instrument
SPHEREx: An All-Sky SpectralSurvey
LSS using PT Introduction 3 / 38
Galaxy clustering
[SDSS] Measured 3D distributionā much more modes than projected quantities(shear from weak lensing, etc.)
Redshift surveys measure: Īø, Ļ, redshift zoverdensity: Ī“ = (nā n)/n,power spectrum: P(k) ā¼ āØĪ“(k)|Ī“(k)ā©LSS using PT Introduction 4 / 38
Galaxy clustering
[SDSS] Measured 3D distributionā much more modes than projected quantities(shear from weak lensing, etc.)
Redshift surveys measure: Īø, Ļ, redshift zGeneralization is the multi-spectra:
āØĪ“(k1) . . . Ī“(kN)ā©c ā¼ PN(k1, . . . , kN)LSS using PT Introduction 4 / 38
Galaxy clustering scheme
-1.0 -0.5 0.0 0.5 1.0-1.0
-0.5
0.0
0.5
1.0
galaxy clustering
dark matterdynamics
redshift spacedistortions
biasing,
stochasticity &
exclusioneffects
satellites
+ others: baryons, assembly bias, neutrinos, (clustering) dark energy,GR effects, multiple d.m. species ...
LSS using PT Introduction 5 / 38
Galaxies and biasing of dark matter halos
Galaxies form at high density peaks ofinitial matter density:rare peaks exhibit higher clustering!
0 2 4 6 8 10-3
-2
-1
0
1
2
3
Ī~1k
ov
er
de
ns
it
y
Tracer detriments the amplitude:Pg(k) = b2Pm(k) + . . .
Understanding bias is crucial forunderstanding the galaxyclustering
[Tegmark et al, 2006]
LSS using PT Galaxies and biasing of dark matter halos 6 / 38
Redshift space distortions (RSD)
Real space:
Kaiser Finger of God
Redshift space:
Object position in redshift-space:s = xā fuz(x)z, uz ā” āvz/(fH)
Density in redshift-space:Ī“s(k) =
ā«x eikĀ·xeāifkzuz(x)
(Ī“(x) + fāzuz(x)
), fāzuz(x) < 1.
LSS using PT Redshift space distortions (RSD) 7 / 38
Redshift space distortions (RSD)Fingers of God (FoG)
Object position in redshift-space:s = xā fuz(x)z, uz ā” āvz/(fH)
Density in redshift-space:Ī“s(k) =
ā«x eikĀ·xeāifkzuz(x)
(Ī“(x) + fāzuz(x)
), fāzuz(x) < 1.
LSS using PT Redshift space distortions (RSD) 7 / 38
Why perturbative approach? This problem is in principle amenable to direct simulation.
Though the combination of volume, mass and force resolution andnumerical accuracy is actually extremely demanding - especially for nextgen. surveys.
PT guides what range of k,Mh, etc. scales are necessary and whatstatistics need to be best converged.
N-body can be used to test PT for `fiducial' models. However PT can be used to search a large parameter space efficiently,and find what kinds of effects are most important.
Can be much more flexible/inclusive, especially for biasing schemes. It much easier to add new physics, especially if the effects are small (e.g.
neutrinos, clustering dark energy, non-Gaussianity) Hopefully we gain some insight, not just numbers! Our goal is to do highly precise computations at large scales, inpreparation for next gen. surveys, not to push to very small scales.
For complementarity; because we can, we should.LSS using PT Redshift space distortions (RSD) 8 / 38
Gravitational clustering of dark matterEvolution of collisionless particles - Vlasov equation:
dfdĻ
=āfāĻ
+1
mp Ā· āfā amāĻ Ā· āpf = 0,
and ā2Ļ = 3/2HĪ©mĪ“.
Integral moments of the distribution function:mass density field & mean streaming velocity field
Ļ(x) = maā3
ā«d3p f(x, p), vi(x) =
ā«d3p pi
am f(x, p)ā«d3p f(x, p)
,
LSS using PT Gravitational clustering of dark matter 9 / 38
Gravitational clustering of dark matterEvolution of collisionless particles - Vlasov equation:
dfdĻ
=āfāĻ
+1
mp Ā· āfā amāĻ Ā· āpf = 0,
and ā2Ļ = 3/2HĪ©mĪ“.
Eulerian framework - fluid approximation:
āĪ“
āĻ+ā Ā· [(1 + Ī“)v] = 0
āviāĻ
+Hvi + v Ā· āvi = āāiĻā 1
Ļāi(ĻĻij),
where Ļij is the velocity dispersion.
LSS using PT Gravitational clustering of dark matter 9 / 38
Gravitational clustering of dark matterEvolution of collisionless particles - Vlasov equation:
dfdĻ
=āfāĻ
+1
mp Ā· āfā amāĻ Ā· āpf = 0,
and ā2Ļ = 3/2HĪ©mĪ“.
Eulerian framework - pressureless perfect fluid approximation:
āĪ“
āĻ+ā Ā· [(1 + Ī“)v] = 0
āviāĻ
+Hvi + v Ā· āvi = āāiĻ.
Irrotational fluid: Īø = ā Ā· v.
LSS using PT Gravitational clustering of dark matter 9 / 38
Gravitational clustering of dark matterEvolution of collisionless particles - Vlasov equation:
dfdĻ
=āfāĻ
+1
mp Ā· āfā amāĻ Ā· āpf = 0,
and ā2Ļ = 3/2HĪ©mĪ“.
EFT approach introduces a tress tensor for the long-distance fluid:
āĪ“
āĻ+ā Ā· [(1 + Ī“)v] = 0
āviāĻ
+Hvi + v Ā· āvi = āāiĻā 1
Ļāj(Ļij),
with given as Ļij = p0Ī“ij + c2s Ī“ĻĪ“ij + O(ā2Ī“, . . .)-derived by smoothing the short scales in the fluid withthe smoothing filterW(Ī), where Ī ā 1/kNL.
[Baumann et al 2010, Carrasco et al 2012]
LSS using PT Gravitational clustering of dark matter 9 / 38
Lagrangian vs Eulerian frameworkEulerian: Lagrangian:
Coordinate of a (t)racer particle at a given moment in time r
r(q, Ļ) = q+ĪØ(q, Ļ),
is given in terms of Lagrangian displacement.Continuity equation:
(1 + Ī“(r)) d3r = d3q ā 1 + Ī“(r) =ā«qĪ“D (rā qāĪØ(q)) ,
Fourier space
(2Ļ)3Ī“D(k) + Ī“(k) =ā«qeikĀ·q exp (ik Ā·ĪØ),
LSS using PT Gravitational clustering of dark matter 10 / 38
Lagrangian dynamics and EFTFluid element at position q at time t0, moves due to gravity:Lagrangian displacement field; x(q, t) = q+ Ļ(q, t).Density field at any time is given by
1 + Ī“(x) =ā«q Ī“D
[xā qā Ļ(q)
]ā Ī“(k) =
ā«q eikĀ·q
(eikĀ·Ļ(q) ā 1
)The evolution of Ļ is governed by
ā2t Ļ + 2HātĻ = āāĻ(q+ Ļ).Integrating out short modes (using filterWR(q, qā²)) system is splitting tnto Lālongand Sāshort wavelength modes, e.g.
ĻL(q) =ā«q WR(q, qā²)Ļ(qā²), ĻS(q, qā²) = Ļ(qā²)ā ĻL(q).
This defines Ī“L as the long-scale component of the density perturbationcorresponding to ĻL and also Ī¦L as the gravitational potentialā2Ī¦L ā¼ Ī“L.E.o.m. for long displacement:
ĻL +HĻL = āāĪ¦L(q+ ĻL(q)) + aS(q, ĻL(q)
), [Vlah et al, '15]
and aS(q) = āāĪ¦S(q+ ĻL(q))ā 12Q
ijL(q)āāiājĪ¦L(q+ ĻL(q)) + . . .,
Similar formalism was also derived in [Porto et al, '14].
LSS using PT Gravitational clustering of dark matter 11 / 38
Lagrangian dynamics and EFT
The correlation function and power spectrum can now be defined through thecumulants of the displacement, e.g.
P(k) =ā«q eiqĀ·k
[āØeikĀ·ā(q)ā©ā 1
].
For one loop power spectrum results, keeping linear modes resumed:
P(k) =ā«qeikĀ·q exp
[ā1
2kikjāØāiāj
ā©c +
i6kikjkk
āØāiājāk
ā©c + Ā· Ā· Ā·
]Final results equivalent to the Eulerian scheme. [Sugiyama '14, Vlah et al, '14 & '15]
Allows for the insight in the counter term structure and IR resummationschemes (in particular one leads to the scheme in [Senatore&Zaldarriaga, '14]).Simple IR scheme was suggested also in [Baldauf et al, '15].
LSS using PT Gravitational clustering of dark matter 12 / 38
Lagrangian dynamics and EFT
The correlation function and power spectrum can now be defined through thecumulants of the displacement, e.g.
P(k) =ā«q eiqĀ·k
[āØeikĀ·ā
ā©ā 1].
For one loop power spectrum results, keeping linear modes resumed:
P(k) =ā«qeikĀ·qā(1/2)kikjAlin
ij
[1ā 1
2kikjAlpt+eft
ij +i6kikjkkWlpt+eft
ijk + Ā· Ā· Ā·]
where Aij(q) = 2āØĪØi(0)ĪØj(0)
ā©ā 2
āØĪØi(q1)ĪØj(q2)
ā©.
Final results equivalent to the Eulerian scheme. [Sugiyama '14, Vlah et al, '14 & '15]
Allows for the insight in the counter term structure and IR resummationschemes (in particular one leads to the scheme in [Senatore&Zaldarriaga, '14]).Simple IR scheme was suggested also in [Baldauf et al, '15].
LSS using PT Gravitational clustering of dark matter 13 / 38
Linear power spectrum, correlation function & BAOPEFT-1-loop = P11 + P1-loop ā 2(2Ļ)c2s(1)
k2k2NL
P11
PEFT-2-loop = P11 + P1-loop + P2-loop ā 2(2Ļ)c2s(1)k2k2NL
(P11 + P1-loop) + c.t.
[/]
/
= -------
[Carrasco et al, '12/'13, Senatore et al '14, Baldauf et al '15, Foreman et al '15, Vlah et al '15]
Well defined/convergent expansion in k/kNL (one parameter). IR resummation (Lagrangian approach) - BAO peak! [Vlah et al '15] Six c. t. for two-loop - approximate degeneracy! [Zaldarriaga et al, '15]
LSS using PT Gravitational clustering of dark matter 14 / 38
Clustering in 1D
1D case studied recently in: [McQuinn&White, '15, Vlah et al, '15]
-
-
-
[/]
(
-)
/
-
= ()=()=+Ī±
()=+(Ī±+Ī±)
()=+(Ī±+Ī±+Ī±
)
()
() () ()
LSS using PT Gravitational clustering of dark matter 15 / 38
Clustering in 1D
1D case studied recently in: [McQuinn&White, '15, Vlah et al, '15]
-
-
-
[/]
(
-)
/
-
= ()=()=+Ī±
()=+(Ī±+Ī±)
()=+(Ī±+Ī±+Ī±
)
()
() () ()
LSS using PT Gravitational clustering of dark matter 15 / 38
Linear power spectrum, correlation function & BAOLinear power spectrum PL: obtained form Boltzmann codes (CAMB, Class).Formally we can divide it into smooth part PL,nw and wiggle part PL,w sothat: PL = PL,nw + PL,w
[/]
āØÆ()[/] =
-
-
[/]
āØÆĪ¾()[/] =Ī¾
Ī¾
Wiggle power spectrum: PL,w ā Ļn =ā«q q
ānPL,w(q) = 0 for n = 0, 2.
[/]
()/ =
[Vlah et al, '14 & '15]LSS using PT Gravitational clustering of dark matter 16 / 38
Resummation of IR modes: simple schemeSeparating the wiggle and non-wiggle part AijL(q) = AijL,nw(q) + AijL,w(q);
P = Pnw +
ā«qeikĀ·qā(1/2)kikjAijL,nw
[ākikj2Aij
L,w + Ā· Ā· Ā·]ā Pnw + eāk2Ī£2
PL,w + . . .
[/]
Ī()[/] =
-
-
[/]āØÆĪ¾()[/] =Ī¾
IR-SPT resummation model:
Pdm(k) = Pnw,L(k) + Pnw,SPT,1āloop(k) + Ī±SPT,1āloop,IR(k)k2Pnw,L(k)
+ eāk2Ī£2(āPw,SPT,1āloop(k) +
(1 + (Ī±SPT,1āloop,IR +Ī£2)k2
)āPw,L(k)
).
Alternative derivation in: [Baldauf et al, 2015]LSS using PT Gravitational clustering of dark matter 17 / 38
Wiggle residuals in our schemes: BAO
[/]
/
=_
()
-
-
-
-
-
[/] =
-
-
-
-
[/]
= -
-
-
-
[/]
-
[/] = _-
-
-
-
[/]
-
[/] = -
LSS using PT Gravitational clustering of dark matter 18 / 38
BAO+: Monodromy
[/]
()
/() =_
-
-
-
[/] =_
-
-
-
[/]
=_
-
-
[/]
-
[/] =_ _-
-
-
[/]
-
[/] =_ -
[/]
()
/() =_
-
-
-
[/] =_
-
-
-
[/]
=_
-
-
[/]
-
[/] =_ _-
-
-
[/]
-
[/] =_ -
LSS using PT Gravitational clustering of dark matter 19 / 38
BAO++: Small scale wiggles
[/]
/
=_
-
-
-
-
[/]
=_ ()
-
-
-
-
[/]
=_
-
-
-
[/]
-
[/]
=_ _-
-
-
-
[/]
-
[/]
=_ -
LSS using PT Gravitational clustering of dark matter 20 / 38
Earlier approaches to halo biasingLocal biasing model: halo field is a function of just DM density field
Ī“h = cĪ“Ī“ + cĪ“2(Ī“2 ā
āØĪ“2ā©)
+ cĪ“3Ī“3 + . . .
[Fry & Gaztanaga, 1993]Quasi-local (in space) relation of the halo density field to the dark matter
[McDonald & Roy 2008, Assassi et al, 2014]
Ī“h(x) = cĪ“Ī“(x) + cĪ“2Ī“2(x) + cĪ“3Ī“3(x)+ cs2s2(x) + cĪ“s2Ī“(x)s2(x) + cĻĻ(x) + csts(x)t(x) + cs3s3(x)+ cĻµĻµ+ . . . ,
with effective ('Wilson') coefficients cl and variables:
sij(x) = āiājĻ(x)ā1
3Ī“Kij Ī“(x), tij(x) = āivj ā
1
3Ī“Kij Īø(x)ā sij(x),
Ļ(x) = [Īø(x)ā Ī“(x)]ā 2
7s(x)2 +
4
21Ī“(x)2,
where Ļ is the gravitational potential, and white noise (stochasticity) Ļµ.
LSS using PT Earlier modelling of halo bias 21 / 38
Effective field theory of biasingNon-local (time) and quasi-local (spece) relation of the halo density field tothe dark matter
[Senatore 2014, Mirbabayi et al, 2014]Ī“h(x, t) ā
ā« tdtā² H(tā²) [cĪ“(t, tā²) : Ī“(xfl, tā²) :
+ cĪ“2(t, tā²) : Ī“(xfl, tā²)2 : +cs2(t, tā²) : s2(xfl, tā²) :+ cĪ“3(t, tā²) : Ī“(xfl, tā²)3 : +cĪ“s2(t, tā²) : Ī“(xfl, tā²)s2(xfl, tā²) : + . . .
+ cĻµ(t, tā²) Ļµ(xfl, tā²) + cĻµĪ“(t, tā²) : Ļµ(xfl, tā²)Ī“(xfl, tā²) : + . . .
+cā2Ī“(t, tā²)ā2xfl
k2MĪ“(xfl, tā²) + . . .
]Novice consideration of non-local in time formation, which depends on fieldsevaluated on past history on past path:
xfl(x, Ļ, Ļ ā²) = xāā« Ļ
Ļ ā²dĻ ā²ā² v(Ļ ā²ā², xfl(x, Ļ, Ļ ā²ā²))
Alternative - all effects chaptered in Lagrangian approach.Note: Assembly bias effects captured in the scheme.
LSS using PT Effective field theory of biasing 22 / 38
Effective field theory of biasingNew physical scale kM ā¼ 2Ļ
(4Ļ3Ļ0M)1/3, which can be different then kNL.
Interesting case kNL ā« kM !We look at the correlations at k āŖ kM.Each order in perturbation theory we get new bias coefficients:
Ī“h(k, t) =ā«tcĪ“,1
[DtĪ“
(1)(k) + flow terms]+
ā«tcĪ“,2
[D2t Ī“
(2)(k) + flow terms]+ . . .
= cĪ“,1[Ī“(1)(k) + flow terms
]+ cĪ“,2
[Ī“(2)(k) + flow terms
]+ . . .
Emergence of degeneracy: choice of most convenient basisRenormalization! (takes care of short distance effects at long distances)In practice, cĪ“,1 is a bare parameter, the sum of a finite part and a counterterm:
cĪ“,1 = cĪ“,1, finite + cĪ“,1, counter,
After renormalization we end up with using 7 finite bias parameters bi.Observables: Phm, Phh, Bhmm, Bhhm, Bhhh
LSS using PT Effective field theory of biasing 23 / 38
Effective field theory of biasingConsistency with N-body simulations achieved up to the k < 0.3Mpc/h forthe Power Spectra, similar for the Bispectrum k < 0.15Mpc/h
()/
()
=- _ (Ī“=)
[/]
()/
()
=- _ (Ī“=)
[]
/
= ā¢=Ļ = =
Most of the constraint comesform the 3-pt function
If we had the simulations for the 4-pt function 2-pt function would be fully predicted.
LSS using PT Effective field theory of biasing 24 / 38
EFT of biased tracers: bias fits
Error bars of the theory are given bythe higher loop estimates:e.g. āPhm ā¼ (2Ļ) b1
(k
kNL
)3P11(k).
This determines the theory reach kmax.
bin0 bin1bĪ“,1 1.00Ā± 0.01 1.32Ā± 0.01bĪ“,2 0.23Ā± 0.01 0.52Ā± 0.01bĪ“,3 0.48Ā± 0.12 0.66Ā± 0.13bĪ“2 0.28Ā± 0.01 0.30Ā± 0.01bcs 0.72Ā± 0.16 0.27Ā± 0.17bĪ“Ļµ 0.31Ā± 0.08 0.76Ā± 0.17
ConstĻµ 5697Ā± 108 10821Ā± 169
Characteristic sharp drop in thep-value after themaximal Bispectrum scale kmax,B
-
[/]
-
_
_
_
Within these scales EFT results fit thedata well, and then fail after crossingthis scales.
LSS using PT Effective field theory of biasing 25 / 38
Adding baryonic effects- baryons at large distances described as additional fluid component (short distancephysics is encoded in an effective stress tensor)
Ī“h(x, t) āā« t
dtā² H(tā²)[cā2Ļ(t, t
ā²)ā2Ļ(xfl, tā²)H(tā²)2
+ cĪ“b (t, tā²)wb Ī“b(xflb)
+ cāivic (t, tā²) wc
āivic(xflc, tā²)H(tā²)
+ cāivib (t, tā²) wb
āivib(xflb, tā²)H(tā²)
+ cāiājĻāiājĻ(t, tā²)āiājĻ(xfl, tā²)
H(tā²)2āiājĻ(xfl, tā²)
H(tā²)2+ . . .
+ cĻµc (t, tā²) wc Ļµc(xflc, tā²) + cĻµb (t, tā²) wb Ļµb(xflb, tā²)
+cĻµcā2Ļ(t, tā²) wc Ļµc(xflc, tā²)
ā2Ļ(xfl, tā²)H(tā²)2
+ cĻµbā2Ļ(t, tā²) wb Ļµb(xflb, tā²)
ā2Ļ(xfl, tā²)H(tā²)2
. . .
]where xfl is defined by Poisson equation and:
xflb(x, Ļ, Ļ ā²) = xāā« Ļ
Ļ ā²dĻ ā²ā² vb(Ļ ā²ā², xfl(x, Ļ, Ļ ā²ā²)) , xflc(x, Ļ, Ļ ā²) = xā
ā« Ļ
Ļ ā²dĻ ā²ā² vc(Ļ ā²ā², xfl(x, Ļ, Ļ ā²ā²))
- similar expressions valid when including neutrinos, clustering dark energy ā¦LSS using PT Effective field theory of biasing 26 / 38
Adding Non-GaussianitiesWe assume that non-G. correlations are present only in the initial conditionsand effect can be described by the squeezed limit, kL āŖ kS of correlation functions.After horizon re-rentry, but still early enough to neglect all gravitationalnon-linearities, the primordial density fluctuation are given by
Ī“(1)(kS, tin) ā Ī“g(kS) + fNLĻ(kL, tin)Ī“g(kS ā kL, tin) ,
where Ļ(kL, tin) = 32
H20 Ī©m
D(tin)1
k2S T(k)
(kLkS
)Ī±Ī“g(kL, tin) and where T(k) is the transfer function.
In the presence of primordial non-Gaussianities, additional components:
Ī“h(x, t) ā fnl Ļ(xfl(t, tin), tin)ā« t
dtā² H(tā²)[c Ļ(t, tā²) + c Ļ
ā2Ļ(t, tā²)
ā2Ļ(xfl, tā²)H(tā²)2
+ . . .
]+ f 2nl Ļ(xfl(t, tin), tin)2
ā« tdtā² H(tā²)
[c Ļ
2
(t, tā²) + c Ļ2
ā2Ļ(t, tā²)
ā2Ļ(xfl, tā²)H(tā²)2
+ . . .
]+ . . .
Recently also studied in: [Assassi et al, 2015]
LSS using PT Non-Gaussianities 27 / 38
Bias in Lagrangian space- Eulerian bias: relation between the final mass density field and the final halodensity field- Lagrangian bias: relation between the initial mass density field and theinitial halo density field
[Matsubara, 2011]Lagrangian bias
nonlinear evolution& formation
nonlinearevolution
Eulerian bias
initial massdensity field
initial number density field
final number density field
final massdensity field
Ī“X(q) = cĪ“Ī“L(q) + cĪ“2Ī“2L(q) + cs2s2L(q) + cĪ“3Ī“3L(q) + cĪ“s2Ī“Ls2L(q) + cs3s3L(q)
+ cā2Ī“
ā2qk2LĪ“L(q) + āstochasticā + . . . ,
- Tracer defined in Lagrangian space need to be displaced to the final time.LSS using PT Redshift space distortions (RSD) 28 / 38
Bias in Lagrangian space in redshift spaceFinal and initial density in real space (Lagrangian mapping):
(1 + Ī“X(x, Ļ))d3x = (1 + Ī“X(q, Ļin))d3q,
Density Ī“s(s) can be obtained from Ī“(x) requiring that the redshift-spacemapping conserves mass:
(1 + Ī“(s))d3s = (1 + Ī“(x))d3x
Power spectrum in redshift space: Exact expression! [Vlah et al, '16]
Ps(k) =ā«reikĀ·r[1 + Ī¾(r)
]exp
(ikā„v
ā„12(r)ā
1
2k2ā„Ļ
ā„12(r) + . . .
),
- sometimes called as Gaussian streaming model (GSM) if cumulants beyondĻ12 are neglected,
Mass b1 b2 bs2 Ī±Ī¾ Ī±v Ī±Ļ12.5 < lgM < 13.0 0.68 -1.01 -0.92 -24 -52 -1813.0 < lgM < 13.5 1.28 -1.34 -0.14 -9 25 -3
LSS using PT Redshift space distortions (RSD) 29 / 38
Bias in Lagrangian space in redshift spaceResults for a sample of DM halo multipoles in configurations space
ā = 0, 13.0 < lgM < 13.5
ā = 0, 12.5 < lgM < 13.0
ā = 2, 13.0 < lgM < 13.5
ā = 2, 12.5 < lgM < 13.0
0
20
40
60
80
100
120
i`s
2Ī¾ `
(s)
[hā
2M
pc2
]
z= 0. 80
20 40 60 80 100 120
s [hā1Mpc]
0.90
0.95
1.00
1.05
1.10
N-b
ody/T
heory
LSS using PT Redshift space distortions (RSD) 30 / 38
Beyond the EdS-like approximationsstandard Eularian fluid solution: [Fasiello, Vlah 2016]
Ī“(k, a) =ānFn(q1..qn, a)Ī“L(q1, a) . . . Ī“L(qn, a)
Īø(k, a) =ānGn(q1..qn, a)Ī“L(q1, a) . . . Ī“L(qn, a)
where:
Fn(Ī·) =
ā« Ī·
āā
dĪ·C(Ī·)
e(nā1)(Ī·āĪ·) f+
f+ ā fā
[(h(n)Ī² ā fā
f+h(n)Ī±
)
+eĪ·āĪ·Dā(Ī·)
Dā(Ī·)
(h(n)Ī± ā h(n)Ī²
)]similar for Gn, D+ is linear growth rate and f+ logarithmic growth rate.- integral and differential formulation: [Bernardeau, 1994]
Fn(q1..qn, a) =āiIi(a)Fi(q1..qn).
[Schmittfull, Z.V., McDonald 2016]Fast! Both in time and momentum aspect! [McEwen et al. 2016]LSS using PT Beyond the EdS-like approximations 31 / 38
Beyond the EdS-like approximations
P1āloop = Plin + P22 + 2P13 + Pc.t. and P01 =dP00d ln a
[/]
/
=+-
[+- ]
[(+ĪĪ± ) +-][+- ]
[/]
/
=+-
[+- ]
[(+ĪĪ± ) +- ]
[+- ]
- important for RSD!- biasing models of galaxy clustering (brake some of the degeneracies?)- fast to evaluate - in differential form!
LSS using PT Beyond the EdS-like approximations 32 / 38
Clustering Quintessence System
system of clustering dark matter and Quintessence [Fasiello, Vlah 2016]
āĪ“m
āĻ+ āi[(1 + Ī“m)vim] = 0 ,
āĪ“Q
āĻā 3(wā c2s )HĪ“Q + āi[(1 + Ļ) + (1 + c2s )Ī“Q]viQ = 0,
āvimāĻ
+Hvim + vjmājvim = āāiĪ¦,
āviQāĻ
+H(1ā 3w)viQ + vjQājviQ = āāiĪ¦ā
c2s āiĪ“Q1 + w
,
ā2Ī¦ =3
2H2Ī©m
(Ī“m +
Ī©q
Ī©mĪ“Q
)ā”
3
2H2Ī©mĪ“T,
- consistency conditions, for cs = w, conserved outside the horizon butgenerically not inside- P.S. enhanced in the IR with respect to dark matter case (similar to thenon-equal time pure (DM) correlator)
LSS using PT Clustering-Quintessence System 33 / 38
Clustering Quintessence System
[/]
-
/-
Ī
=
- ()/ -Ī ()
-Ī ()/ -
Ī () =-
=-
=-
[/]
-
/-
Ī
=
- ()/ -Ī ()
(-Ī +#
Ī)()/ -Ī () =-
=-
=-
These effects may propagate all the way to biased tracers observables.
Ī“h(x, t) āā« t
H(tā²)[cĪ“T (t
ā²)Ī“T(xfl, tā²)H(tā²)2
+ cĪ“d.e. (tā²) Ī“d.e.(xfl)
+ cāvc (tā²)āivic(xfl, tā²)
H(tā²)+ cāvd.e. (t
ā²)āivid.e.(xfl, tā²)
H(tā²)+ cĻµc (t
ā²) Ļµc(xfl, tā²) + cĻµd.e. (tā²) Ļµd.e.(xfl, tā²)
+cā2Ī“T(tā²)
ā2xfl
k2M
Ī“T(xfl, tā²)H(tā²)2
+ . . .
].
- time evolution can brake degeneracies in bias operators (at third order)LSS using PT Clustering-Quintessence System 34 / 38
Efficient Evolution of LoopsP1āloop = Plin + P22 + 2P13 + Pc.t. where e.g.
P22 ā¼ā«qf(q)g(k ā q)Plin
q Plinkāq =
ā« ā
0r2j0(rk)
[ ā« ā
0q2f(q)Plin
q j0(qr)ā« ā
0p2g(p)Plin
p j0(rp)]
[/]
()[/]
=
----
-
-
-
-
[/]
=
- nonlinear corrections are products correlations of field derivatives- useful for variation of IC paremeters- very fast to evaluate - useful is FFTLog (public code) [Hamilton, 2000]
LSS using PT Clustering-Quintessence System 35 / 38
Efficient Evolution of LoopsP2āloop = Plin + P33 + 2P24 + 2P15 + Pc.t. where
I24(k, Ī±, Ī²) =ā«q1q2
eiĪ±Ā·q1eiĪ²Ā·q2
q2n11 |k+ q1|2nā²1q2n22 |k+ q2|2n
ā²2
PL(q1)PL(q2)PL(|k+ q2|)|q1 + q2|2n3 |k+ q1 + q2|2n
ā²3
[/]
āØÆ()
- P2āloop given by taking derivatives of generating function IN.- much more efficient for evaluate the using M.C. integration.- simpler way to obtained asymptotic solutions
LSS using PT Clustering-Quintessence System 36 / 38
Summary Large redshift surveys can be used for precision tests of the ĪCDM model.
Expansion history (BAO), Growth of structure (RSD), ā¦ Analytic models can shed light on the relevant physics and we hope they can be
made accurate enough to fit next-generation data (on large scales). Modeling BAO+RSD requires beyond-linear modeling. Lagrangian framework offers a nice physical insight in LSS, application is e.g.
IR resummation (BAO+) EFT gives a consistent expansion in (k/kNL)2, and for halos also in (k/kM)2,
nonlocal effect in time and space included EFT approach is well suited for galaxy clustering (one-loop power spectra
k ā¼ 0.25h/Mpc, tree level bispectra k ā¼ 0.1ā 0.15h/Mpc ) Consistent description of five different observables (Phm, Phh, Bhmm, Bhhm, Bhhh)
with seven bias parameters. Exact time evolution can be important! All integrals can be evaluated in a efficient way (FFTLog).
LSS using PT Summary 37 / 38
Summary
Outlook: Higher loops calculations in order to extend the kmax on one hand andimprove precision on large scales,
Higher statistics (e.g. 4-pt function - great potential), Calculation of observables taking into account baryons,non-Gaussianities ā¦,
Generalisation of the formalism in order include GR effects (becomeimportant as surveys grow).
How truly effective are effective approaches (degeneracies etc.)?
LSS using PT Summary 38 / 38