Spherical collapse in νΛCDM

Marilena LoVerdeEnricoFermi Institute, Kavli Institute forCosmological Physics,Department of Astronomy andAstrophysics,

University of Chicago, Illinois 60637, USA(Received 11 June 2014; published 17 October 2014)

The abundance of massive dark matter halos hosting galaxy clusters provides an important test of themasses of relic neutrino species. The dominant effect of neutrino mass is to lower the typical amplitude ofdensity perturbations that eventually form halos, but for neutrino masses ≳0.4 eV the threshold for haloformation can be changed significantly as well. We study the spherical collapse model for halo formation incosmologies with neutrino masses in the range mνi ¼ 0.05–1 eV and find that halo formation is differentlysensitive to Ων and mν. That is, different neutrino hierarchies with a common Ων are in principledistinguishable. The added sensitivity to mν is small but potentially important for scenarios with heaviersterile neutrinos. Massive neutrinos cause the evolution of density perturbations to be scale dependent athigh redshift which complicates the usual mapping between the collapse threshold and halo abundance.We propose one way of handling this and compute the correction to the halo mass function within thisframework. For

Pmνi ≲ 0.3 eV, our prescription for the halo abundance is only ≲15% different than the

standard calculation. However for larger neutrino masses the differences approach 50–100% which, ifverified by simulations, could alter neutrino mass constraints from cluster abundance.

DOI: 10.1103/PhysRevD.90.083518 PACS numbers: 98.80.-k, 95.85.Ry, 98.65.Cw

I. INTRODUCTION

The exquisite measurements of temperature anisotropiesin the cosmic microwave background (CMB) by Planck[1], WMAP, [2], SPT [3] and ACT [4] reveal a Universethat, on large scales, can be remarkably well characterizedby just a few cosmological parameters. One parameter thatis not currently required is the energy density in massiverelic neutrinos. Cosmological evidence for massive neu-trinos is at present undetected, and as such is considered anextension to the ΛCDMmodel. Neutrino oscillation experi-ments require that massive neutrinos contribute at least afew tenths of a percent to the cosmic energy budget today,Ωνh2 ≳ 0.06 eV=94 eV [5].The presence of massive neutrinos changes the evolution

of matter perturbations. Matter perturbations with wave-lengths smaller than the neutrino free-streaming length aresuppressed and a detection of this suppression wouldprovide a measure of the energy density in relic neutrinos(for a review see Ref. [6]). The neutrino-induced suppres-sion to the matter power spectrum scales primarily withP

imνi and current bounds on the sum of the neutrinomasses from cosmological data sets are

Pimνi ≲ 0.2–1 eV

(see e.g. Refs. [1–4] for CMB constraints, Refs. [7–16] forconstraints from galaxy and Lyman-alpha forest surveys,and Refs. [17–22] for constraints from the abundance ofgalaxy clusters).The standard model of particle physics includes three

active neutrino species and three neutrino mass eigenstates.However, there are a number of anomalies in particlephysics, nuclear physics, and astrophysics data sets thatsuggest the presence of additional light neutrino species

with mass ∼1 eV (for a review see Refs. [23,24] and alsoRefs. [25–27]). The invisible decay width of the Z bosonlimits the number of weakly interacting neutrinos to three,so if an additional neutrino species exists it must be sterile[5]. Cosmological data sets bound the effective number ofall (active and sterile) neutrinos species (i.e. the number ofrelativistic fermionic degrees of freedom in the earlyUniverse). Current constraints from Planck are 2.72 <Neff < 4.04 at 95% confidence [1].As discussed in Ref. [28] neutrinos with masses mν ≳

0.2 eV (and the standard temperature of Tν ∼ 1.7×10−4 eV) cluster much more strongly around dark matterhalos so one might expect correspondingly larger effects onhalo formation and abundance. While the evidence for anadditional, more massive, sterile species is far from strong,one motivation for this work is to provide a frameworkfor understanding the signatures of neutrinos with massesas large as 1 eV on halo formation and abundance. In anycase, massive neutrinos exist and are one of two knowncomponents of dark matter in the Universe today. It istherefore important to understand how calculations ofcold dark matter (CDM) structure formation are alteredby the neutrino component.In this paper, we consider the effects of massive

neutrinos on the simplest, spherical collapse model ofhalo formation [29]. The effects of massive neutrinos onspherical collapse were first studied by Ichiki and Takada[30] and this paper largely follows their approach. Themain differences between this work and Ref. [30] are that(i) we start the spherical collapse calculations at later timeswhen the perturbations are subhorizon using the initialconditions from the publicly available CAMB code [31],

PHYSICAL REVIEW D 90, 083518 (2014)

1550-7998=2014=90(8)=083518(16) 083518-1 © 2014 American Physical Society

and (ii) we allow for multiple massive neutrino species andlarger individual neutrino masses than those consideredin Ref. [30]. Including multiple massive species allows usto separately study the effects of mν and Ων on sphericalcollapse. Larger neutrino masses also cause greater changesto the evolution of spherical overdensities which is helpfulto understand the different physical effects and ultimatelymay be important to model and constrain a sterile species.The semianalytic spherical collapse model we consider

here is, of course, not a precise description of cold darkmatter structure formation in the Universe. The currentcommunity standard for modeling structure formation ishigh-resolution N-body simulations (see e.g. Ref. [32] andreferences therein). Nevertheless, the spherical collapsemodel is a useful testing ground for developing an under-standing of how nonstandard cosmologies impact haloformation (see e.g. Refs. [33–39]). Furthermore, at presentthere are only a handful of N-body simulations that includethe effects of massive neutrinos [40–47] and even fewerthat include both cold dark matter and neutrino particles insimulations [40,41,43,44,46,48–50]. In the mixed-dark-matter simulations of Refs. [44–46] the Sheth-Tormenmodel for the halo mass function [51] was found tocontinue to describe the halo abundance in a cosmologywith massive neutrinos with mνi ≲ 0.2 eV provided onemakes the replacement Ωm → Ωc þΩb (i.e. the neutrinocontribution to Ωm is neglected). In Ref. [50] the samereplacement was found to give good agreement between theνCDMmass functions and the Tinker spherical-overdensitymass function [32] as long as the CDM power spectrumwas used to calculate the variance of mass fluctuations onthe scale M (rather than the total matter power spectrum).These descriptions of the change to the halo mass functionare in qualitative agreement with our calculations that showthat the collapse threshold is changed by ≲0.5% forP

imνi ≲ 0.3 eV and M ≤ 1015M⊙ (see also Ref. [30]).It would be interesting to compare our calculations whichinclude larger values of neutrino masses to mixed darkmatter N-body simulations.In the plots and numerical examples shown throughout

this paper we use Planck [1] values of the standard, flatΛCDM cosmological parameters: the Hubble parameterh ¼ 0.67, CDM density Ωch2 ¼ 0.1199 and baryon den-sity Ωbh2 ¼ 0.022. When solving for the evolution ofspherical overdensities we treat baryons and CDM as asingle fluid (for a discussion of baryon effects on sphericalcollapse calculations see Refs. [30,52]). We assume threespecies of massive neutrinos with variable massesmν1, mν2 and mν3. Massive neutrinos contribute a fractionΩνh2 ≈

Pmνi=ð94 eVÞ to the critical energy density so for

fixed CDM and baryon densities, changing the neutrinomasses leads to a different total matter (Ωm¼ΩcþΩbþΩν)density today. We adjustΩΛ to keep the universe flat, that isΩΛ ¼ 1 −Ωc −Ωb −Ων −Ωγ and these plots are referredto as being at fixed Ωc. In this paper, “inverted hierarchy”

means mν1 ¼ mν2 ¼ 0.05 eV and mν ¼ 0 eV, and “nomassive ν” means mν1 ¼ mν2 ¼ mν3 ¼ 0 eV. We alsomake comparisons between cosmologies with massiveneutrinos and a cosmology with massless neutrinos andthe same total matter density, Ωcomparison

c ¼ Ωc þ Ων; werefer to the difference between these two cases as changeswith fixed Ωm. In numerical examples, we consider severalrepresentative scenarios for the neutrino mass hierarchywhich are not all compatible with oscillation data, but theyallow us to compare scenarios with a fixed Ων but differentindividual neutrino masses. We solve for the backgroundcosmology and spherical halo collapse self-consistently foreach set of neutrino masses.The rest of this paper is organized as follows. In Sec. II

we outline our method for solving spherical collapse inνΛCDM and present numerical calculations of the haloevolution in this model. In Sec. III we relate sphericalcollapse results to halo abundance in νΛCDM cosmologies.Massive neutrinos cause the evolution of density perturba-tions to depend on the scale and this fact leads to somesubtleties in relating spherical collapse calculations tohalo abundance so we review the framework in detail.Numerical results for the neutrino-induced changes to thecollapse threshold and halo abundance are presented inthe same section. Conclusions are given in Sec. IV.Appendix A describes how we set up the initial conditionsusing the output of CAMB [31]. Appendix B presents testsof whether our results are sensitive to the approximationused to calculate the neutrino mass that clusters inside thehalo during collapse (they are not). Appendix C furtherexplores the meaning of the collapse threshold in νΛCDMand presents a comparison of several approaches todefining this quantity.

II. SPHERICAL COLLAPSE INA νΛCDM UNIVERSE

In this section we develop the spherical collapse modelfor halo formation in a universe with multiple componentsto the energy density. Our halo is a homogenous sphericaloverdensity in CDM and baryons that accumulates neutrinomass as the amplitude of the CDM and baryon perturbationgrows during gravitational collapse.The unperturbed background universe evolves according

to the Friedmann equation,

H2ðaÞ¼ 8πG3

ðρ̄cðaÞþ ρ̄bðaÞþ ρ̄νðaÞþ ρ̄γðaÞþ ρ̄ΛÞ ð2:1Þ

where a is the scale factor and we have included CDM ρ̄c,baryons ρ̄b, neutrinos ρ̄ν, photons ρ̄γ , and a cosmologicalconstant ρ̄Λ. The number density, energy density, andpressure of a single neutrino mass eigenstate with massmνi is given by

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-2

n̄νi ¼ 2

Zd3pð2πÞ3

1

ep=Tν þ 1;

ρ̄νi ¼ 2

Zd3pð2πÞ3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm2

νi

pep=Tν þ 1

;

P̄νi ¼ 2

Zd3pð2πÞ3

p2

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm2

νi

p 1

ep=Tν þ 1; ð2:2Þ

where the neutrino temperature is TνðaÞ ¼ 1.95491 K=aand the total neutrino energy density is ρ̄ν ¼

Piρ̄νi. The

other components evolve as ρ̄c ∝ 1=a3, ρ̄b ∝ 1=a3,ρ̄γ ∝ 1=a4, and ρ̄Λ ¼ const.

A. Equation of motion for R

We follow the evolution of the radius RðtÞ enclosing aconstant CDM and baryon mass M. We start our calcu-lations at zinit ¼ 200. At this redshift the density perturba-tions that form halos at late times have amplitudesδinit ∼Oð10−2Þ so we may set the initial conditions usinglinear theory (see Appendix A for further discussion).However, zinit ¼ 200 is late enough that the perturbations ofinterest are all well within the horizon. Furthermore,perturbations in the baryon density have nearly caughtup with perturbations of the CDM so we will treat them as asingle fluid with energy density ρ̄cb ≡ ρ̄c þ ρ̄b. The CDMand baryon density perturbations are given by

δcb ≡ ρ̄cδc þ ρ̄bδbρ̄c þ ρ̄b

: ð2:3Þ

The equation of motion for a mass shell of radius Renclosing constant (CDM þ baryon) mass M is

R̈ ¼ −GMR2

−4πG

RR0 drr2ðρrestðr; tÞ þ 3Prestðr; tÞÞ

R2;

ð2:4Þ

where “_” indicates a derivative with respect to time, andρrest and Prest are the energy density and pressure ofphotons, neutrinos, and the cosmological constant. For ahalo of mass M, we use the linear initial conditions for R

Rinit ¼ R̄init

�1 −

1

3δcb;init

�;

_Rinit ¼ HinitR̄init

�1 −

1

3δcb;init −

1

3H−1

init_δcb;init

�;

R̄init ¼�

3M4πρ̄cb

�1=3

ð2:5Þ

where “¯” indicates unperturbed quantities and we usethe subscript “init” to indicate quantities at the initialredshift zinit. In a cosmology with massive neutrinos, thegrowth of linear density perturbations is different for

perturbations with wavelengths above and below theneutrino free-streaming scale. In this paper we will useCAMB to find _δcb=δcb at the initial time but it is instructiveto study the analytic expressions for an Ωm ¼ 1 universe toget a sense of how massive neutrinos affect the initialconditions for RðtÞ. In an exactly matter-dominated phasethe linear growing mode with wave number k evolves as

δcbðk; aÞ ∝� a for k≲ kfree-streaming

a1−3=5fν for k≳ kfree-streaming

ð2:6Þ

where kfree-streaming ∼ aHðaÞmν=TνðaÞ and fν ≡Ων=ðΩcþΩb þ ΩνÞ. The perturbations that form halos at late timesare predominantly on scales smaller than the neutrino free-streaming scale. Therefore we expect that the presence ofmassive neutrinos will delay halo formation.The evolution of RðtÞ shown in Eq. (2.4) is sensitive to

perturbations in the non-CDM components as well. Byzinit ¼ 200, the linear perturbations in photons and neu-trinos are completely subdominant on the scales of interest(see Appendix A). However, at late times massive neutrinoscan cluster nonlinearly in the collapsing density perturba-tions so we allow for a contribution from the clustering ofmassive neutrinos given by

δMνð< R; tÞ ¼ZVR

d3rδρνðr; tÞ ð2:7Þ

where VR ¼ 43πR3.

The final expression that we use to solve for thesubhorizon, nonlinear evolution of RðtÞ is then,

R̈ ¼ −GMR2

−4πG3

ð2ρ̄rðtÞ þ ρ̄νðtÞ þ 3P̄ν − 2ρ̄ΛðtÞÞR

−GδMνð< R; tÞ

R2; ð2:8Þ

with the initial conditions in Eq. (2.5).Before proceeding it is helpful to rewrite Eq. (2.8) as a

nonlinear equation for the density perturbation δcb. UsingM ¼ 4

3πR3ρ̄cbð1þ δcbÞ ¼ const we have

δ̈cb þ 2H _δcb −4

3

_δ2cb1þ δcb

−3

2H2ðΩcbδcb þΩνδνÞð1þ δcbÞ ¼ 0 ð2:9Þ

where Ωcb ¼ ΩcðaÞ þ ΩbðaÞ and δν ¼ δMν=M̄ν. Collapseof a halo occurswhenR → 0 or equivalently δcb → ∞. In theabsence of neutrino perturbations δν (which depend on halomass M) the equation of motion for δcb [Eq. (2.9)] iscompletely independent of halo mass. Therefore, in theabsence of neutrino clustering a given set of initial con-ditions δcb;init, _δcb;init will collapse at the same time, regard-less of the halo mass. The value of _δcbðzinitÞ=δcbðzinitÞ,

SPHERICAL COLLAPSE IN νΛCDM PHYSICAL REVIEW D 90, 083518 (2014)

083518-3

however, is cosmology dependent and there are percent-level changes for the different choices of mν and Ωm(seeAppendixA). In this paperwe are particularly interestedin the importance of the final δMν term. Calculations thatignore the δMν term are referred to as neglecting neutrinoclustering while those that incorporate a nonzero δMν aresaid to include neutrino clustering. In the next section weoutline our methods for calculating δMν.

B. Expression for δMν(<R;t)

The neutrino mass perturbation interior to R, δMν, iscalculated by treating the halo as an external potential forthe neutrinos. We use the no-neutrino-clustering solution[that is, the solution to Eq. (2.8) with δMν ¼ 0] for RðtÞ todetermine the potential due to the CDMþ baryon densityperturbation. We then use this potential as input into thelinearized Boltzmann equation for the evolution of the

first-order perturbation to the neutrino distribution function(see Refs. [53–55]).The neutrino mass fluctuation interior to R at time t from

a single species of mass mν is given by

δMνð< R; tÞ ¼ mν

ZVc

d3rZ

d3qð2πÞ3 f1ðq; r; tÞ ð2:10Þ

where q ¼ ap and p is the particle momentum, f1 is alinear perturbation to the neutrino distribution function(f ¼ f0 þ f1 where f0 is a Fermi-Dirac distribution withtemperature Tν) and Vc ≡ 4

3πR3ðtÞ=a3ðtÞ.

For f1, we use an approximate solution to the Boltzmannequation for nonrelativistic particles in an external potential(for details see Ref. [28]). For neutrinos around a sphericaltop-hat halo, this reads

f1ðrcomoving;q; ηÞ ¼ 2mν

Tν

Zt

t0

dt0

aðt0Þeq=Tν

ðeq=Tν þ 1Þ2GδMðt0Þ

r2

�αqTν

− q̂ · r̂

�

×

�a3ðt0Þr3

R3Θðr2ð1þ q2=T2

να2 − 2q=Tναr̂ · q̂Þ < Rðt0Þ2=a2ðt0ÞÞ

þ Θðr2ð1þ q2=T2να

2 − 2q=Tναr̂ · q̂Þ ≥ Rðt0Þ2=a2ðt0ÞÞð1þ q2=T2

να2 − 2q=Tναr̂ · q̂Þ3=2

�ð2:11Þ

(a) (b)

FIG. 1 (color online). Left panel: The total neutrino mass fluctuation within radius R for neutrinos with mν ¼ 1 eV, calculated usingthe linearized Boltzmann equation [Eqs. (2.10) and (2.11)] for halos ofM ¼ 1015M⊙ that collapse at several different times. For the halothat collapses latest, we have also plotted δMν calculated using the exact calculation (see Appendix B for details). Right panel: Theneutrino density fluctuation interior to R, δMν=M̄ν (solid lines), compared with the CDMþ baryon density fluctuation (dashed lines),and the initial linear neutrino fluctuation from CAMB (dotted lines). The divergences in the density fluctuations correspond to the redshiftat which R → 0.

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-4

where δMðtÞ ¼ M − 43πρ̄cbR3, Θ is the Heaviside step

function, α≡ Tνðη − η0Þ=mνr, and η is a time variabledefined by a2dη ¼ dt. Figure 1 shows δMν along withthe fractional overdensity δMν=M̄ν for a single neutrinospecies with mν ¼ 1 eV in halos of M ¼ 1014M⊙ collaps-ing at several different times.The linearized solution in Eq. (2.11) underestimates the

neutrino clustering on scales interior to R [28,56]; this canbe seen explicitly in Fig. 1 where we have also plotted anexample calculation of δMν from a full nonlinear solutionto the Boltzmann Equation for mν ¼ 1 eV. The differencebetween the linear approximation for δMν and the exactsolution is large, particularly at late times. However, wefound in Ref. [28] that for mν ≲ 0.2 eV, significantdifferences between the linearized and exact calculationsof the neutrino mass in an external potential do not becomeimportant until after the halo has begun to collapse; at thispoint the cold dark matter overdensity is large and thedynamics of R are dominated by the CDM. As is inAppendix B, even for more massive neutrinos (mν ¼ 1 eV)the linearized solution is sufficient to determine theevolution of R and the collapse time to about a percent.The fractional differences between the values of tcollapse fora given δcb;i using the linearized Boltzmann solution andusing the exact solution for δMν remain about an order ofmagnitude smaller than the fractional differences betweentcollapseðδMνÞ and tcollapseðδMν ¼ 0Þ.In Fig. 1(b) we compare the fractional overdensity in

neutrinos accumulating in the halo (calculated using thelinearized Boltzmann solution), the fractional overdensityin CDM and baryons, and the initial linear density

fluctuation in neutrinos from the CAMB transfer functions.On halo scales, the initial linear fluctuation in the neutrinodensity has damped away and is irrelevant in comparison tothe neutrino mass that accumulates in the halo at later times.In Fig. 2(a) we plot solutions to Eq. (2.8) with

δMνð< RÞ ¼ 0 for halos with the same initial δcbðzinitÞ(and therefore the same Rinit) but in cosmologies withdifferent neutrino masses so the _δcbðzinitÞ and the back-ground evolution differ. Increasing the neutrino massdelays the evolution and subsequent collapse of the halos.In Fig. 2(b) we show the effect of neutrino clustering on theevolution of RðtÞ. As expected, neutrino clustering interiorto R causes a slight decrease in the collapse time.

III. FROM SPHERICAL COLLAPSE TOHALO ABUNDANCE

In this section we relate the spherical collapse calcu-lations from the previous section to the abundance of halosat late times. As we will discuss below, the scale-dependentevolution of density perturbations in νΛCDM introducessome subtleties that are not present in the CDM-only caseso we review the spherical collapse framework for haloabundance and our assumptions in detail.

A. The critical overdensity

In the spherical collapse model, one approximates thenonlinear evolution of the initial density field smoothed onthe scale R with the evolution of a spherical top-hat densityperturbation with the same initial amplitude δcbðzinitÞ andvelocity _δcbðzinitÞ. From the solution to the equation of

(a) (b)

FIG. 2 (color online). Left panel: Spherical collapse solutions for halos with common values of δinit but in cosmologies with differentmν; each curve above has mν1 ¼ mν2 ¼ mν3 and from bottom to top they are mνi ¼ 0 eV, 0.1 eV, 0.2 eV, and 0.4 eV. These curvesneglect neutrino clustering [δMνð< RÞ ¼ 0]. In cosmologies with massive neutrinos the linear growth is slower and collapse occurslater. Right: A comparison of the spherical collapse solutions including and neglecting neutrino clustering interior to R for a cosmologywith mνi ¼ 0.4 eV.

SPHERICAL COLLAPSE IN νΛCDM PHYSICAL REVIEW D 90, 083518 (2014)

083518-5

motion for RðtÞ one can determine how large δcbðzinitÞ and_δcbðzinitÞ need to be for a halo to have collapsed (R → 0) bya given redshift. The critical value of δcbðzinitÞ required for aspherical halo to collapse by redshift z is called the collapsethreshold or critical overdensity, δcritðzinitÞ, and is a keyingredient in analytic models of the halo abundance (see forinstance, Refs. [57,58]). In the next few paragraphs wediscuss our approach for determining δcritðzinitÞ.The initial amplitude of the smoothed density field

around a point x is given by

δcb;Rðx; zinitÞ ¼Z

d3kð2πÞ3 e

ik·xWðkRÞδcbðk; zinitÞ ð3:1Þ

and the initial velocity of the pertrubation is

_δcb;Rðx; zinitÞ ¼Z

d3kð2πÞ3 e

ik·xWðkRÞ d ln δcbdt

ðk; zinitÞ

× δcbðk; zinitÞ: ð3:2Þ

The initial perturbation amplitude δcb;Rðx; zinitÞ is aGaussian random variable with zero mean and variance

σ2ðM; zÞ≡ hδ2Ri ¼Z

dkkjWðkRÞj2 k

3Pcbðk; zÞ2π2

ð3:3Þ

evaluated at z ¼ zinit, where Pcbðk; zÞ is the power spec-trum of cold dark matter plus baryon perturbations and

Wðk; RÞ ¼ 3j1ðkRðMÞkRðMÞ ; ð3:4Þ

where j1ðxÞ is the spherical Bessel function andRðMÞ ¼ ð3M=ð4πρ̄cbÞÞ1=3. The variance of the velocityof the initial perturbations is

_σ2ðM; zinitÞ≡ h_δ2Ri

¼Z

dkkjWðkRÞj2

�d ln δcb

dtðk; zinitÞ

�2

×k3Pcbðk; zinitÞ

2π2: ð3:5Þ

where d ln δ=dtðk; zinitÞ is the linear evolution of thegrowing mode (as determined by CAMB, for instance).If the linear evolution is scale independent d ln δcb=dt

can be pulled out of the integrals in Eq. (3.2) so that_δcb;Rðx; zinitÞ is proportional to δcb;Rðx; zinitÞ regardless ofthe density profile around x. That is, for scale-independentevolution the value of δcb;Rðx; zinitÞ alone [rather than thefull profile δcbðk; zinitÞ] completely specifies _δcb;Rðx; zinitÞ.On the other hand, in a cosmology with scale-dependentgrowth (such as the case with massive neutrinos)d ln δcb=dtðkÞ cannot be factored out of Eq. (3.2) and

the value of δcb;Rðx; zinitÞ alone does not determine_δcb;Rðx; zinitÞ.To determine the collapse threshold for a halo of massM

we solve Eq. (2.8) for the evolution of R for a top-hatperturbation with some initial amplitude δcb;init and initialvelocity

_δcb;init ≡ _σðM; zinitÞσðM; zinitÞ

δcb;init: ð3:6Þ

We then determine how large δcbðzinitÞ needs to be for theperturbation to collapse by a given redshift z. This definesthe critical overdensity at the initial time δcrit;init. With thisdefinition the critical overdensity linearly extrapolated to adifferent redshift is

δcritðzÞ≡ σðM; zÞσðM; zinitÞ

δcrit;init: ð3:7Þ

We have chosen Eq. (3.6) and Eq. (3.7) to define thecollapse threshold because Eq. (3.6) is representative oftypical initial conditions for the quantity δcb;Rðx; zinitÞ.Moreover, from Eq. (3.7) we can see that the rarity ofperturbations large enough to collapse as characterizedby the ratio δcritðzÞ=σðM; zÞ is independent of redshift. Incontrast, the linear evolution of an initial density perturba-tion with an exact top-hat density profile, δtop-hatðk; zinitÞ ¼4π=ð2πÞ3k2WðkRÞ differs from the linear evolution ofσðM; zÞ if d ln δ=dtðkÞ is k dependent because δtop-hat

and σðM; zÞ depend differently on k. In Appendix C wediscuss this issue in detail and make comparisons betweenthe definition used here and some alternative approaches.In Fig. 3 we plot the collapse threshold linearly extrapo-

lated to the collapse redshift, δcritðzcollapseÞ, for a range ofneutrino mass hierarchies. Increasing the total amount ofneutrino mass increases the barrier for collapse, but in away that depends on the individual neutrino masses (asopposed to just Ων ∝

Pimνi). For instance, the scenario

mν1 ¼ 0.6 eV, mν2 ¼ mν3 ¼ 0 eV and the degenerate hier-archy mνi ¼ 0.2 eV have common values of

Pimνi (and

therefore Ων once the neutrinos are nonrelativistic) but theδcritðzÞ’s clearly differ. The nonlinear dependence on mν ispartially due to the nonlinear clustering of massive neu-trinos during collapse (roughly δMν ∼

Pim

5=2νi [28]),

which is shown in Fig. 3(b). However, even in the absenceof nonlinear neutrino clustering δcrit has some sensitivity tothe individual neutrino masses through the suppression inlinear growth: the net suppression in small-scale densityperturbations depends on both the fraction of mass inneutrinos and the redshift at which the neutrinos becomenonrelativistic (aNR ≈ 3Tν=mν) which gives an additionalsensitivity to the masses [59].In Fig. 4 we show the dependence of the change to the

collapse threshold δcritðzcollapseÞ on the abundance of relicneutrinos and their individual masses. For the range of

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-6

neutrino masses and halo masses we have considered(1014M⊙ < M < 1016M⊙ and 0.05 eV < mνi < 1 eV)increasing n̄ν causes a linear increase in δcrit. On the otherhand, δcritðzcollapseÞ depends nonlinearly on the masses ofthe individual neutrinos. This means that observablesdependent on δcritðzcollapseÞ such as the halo mass functionare in principle able to distinguish between differentscenarios for the neutrino mass hierarchy. In practice,

the nonlinear dependence of δcritðzcollapseÞ on the neutrinomasses is probably only important for scenarios with anadditional sterile species but it is nevertheless important tokeep in mind.

B. The halo abundance

The calculations of the collapse threshold from Sec. III Acan be used to estimate the effects of massive neutrinos on

FIG. 4 (color online). The change to δcritðzcollapseÞ due to massive neutrinos (including neutrino clustering and compared at fixed Ωc).Left: The dependence on the number density of massive neutrinos, plotted for several different neutrino masses in units of n̄1ν, i.e., therelic abundance of a single neutrino and antineutrino species given in Eq. (2.2). The change to the collapse threshold is roughly linear inn̄1ν. Right: The change to the linearly extrapolated collapse threshold plotted as a function of neutrino mass mν. The change toδcritðzcollapseÞ depends nonlinearly on the masses of the individual neutrinos.

(a) (b)

FIG. 3 (color online). Left: The linearly extrapolated value of the initial density perturbation δcbðzinitÞ, required to collapse at zcollapse[see Eq. (3.7)]. Here we have fixed Ωc and Ωb so curves with different Ων have different total matter density Ωm. Right: The fractionalchange to the collapse threshold when neutrino clustering interior to R is included. In both panels M ¼ 1014M⊙ and the order of thecurves matches the order of the legends.

SPHERICAL COLLAPSE IN νΛCDM PHYSICAL REVIEW D 90, 083518 (2014)

083518-7

the halo mass function. To convert the changes toδcritðzcollapseÞ and σðM; zÞ into predictions for changes tothe halo abundance we will need to make the assumptionthat the changes to the halo mass function are characterizedentirely by these two parameters, an assumption that willneed to be tested against N-body simulations with neutrinomasses across the ranges that we consider.To study neutrino mass effects on the halo mass function,

we use the fitting formula of Bhattacharya et al. [60] whichis calibrated off of high-resolution CDM-only N-bodysimulations. Their expression for the number density ofhalos with masses between M and M þ dM is

nBðM;zÞ¼−ρ̄cbM

d lnσðM;zÞdM

fB

�ν≡ δcritðzÞ

σðM;zÞ ;z�; ð3:8Þ

where

fB

�δcritσ

; z

�¼ A

ffiffiffi2

π

rexp

�−a2

δ2critσ2

�

×

�1þ

�σ2

aδ2crit

�p��

δcritffiffiffia

pσ

�q

; ð3:9Þ

with AðzÞ ¼ 0.333=ð1þ zÞ0.11, a ¼ 0.788=ð1þ zÞ0.01,p ¼ 0.807, q ¼ 1.795, and σ ¼ σðM; zÞ. This fitting func-tion was calibrated off of N-body simulations of ΛCDMusing a constant linearly extrapolated spherical collapsethreshold δcritðzÞ ¼ 1.686. To study the effects of massiveneutrinos on halo abundance we will therefore use

δcrit → 1.686δcritðzjmνÞ

δcritðzjmν ¼ 0Þ ð3:10Þ

in Eq. (3.8) where δcritðzjmνÞ and δcritðzjmν ¼ 0Þ are ourcalculated collapse thresholds from Eq. (3.7) [61]. We useσðM; zÞ from Eq. (3.3) including the effects of massiveneutrinos on the linear CDM and baryon power spectrum.For comparison, we will also consider the predicted

changes to the Sheth-Tormen (ST) mass function

nSTðM; zÞ ¼ −ffiffiffiffiffiffiffiffiffiffi2qSTπ

rAST

�1þ

�qSTδ2critσ2

�−pST�

×ρ̄cbM2

δcritσ

d ln σd lnM

exp

�−qSTδ2critσ2

�ð3:11Þ

where qST ¼ ffiffiffi2

p=2, AST ¼ 0.322184, and pST ¼ 0.3.

In Fig. 5, we show the suppression in variance of CDMand baryon density fluctuations σðMÞ due to massiveneutrinos [Eq. (3.3)]. The fractional change in σðMÞ dueto massive neutrinos is large compared to the shifts in δcritshown in Figs. 3 and 4. One can also see in Fig. 5(a) that thesuppression in σðMÞ is not entirely characterized by Ων:there are small differences between curves with commonΩν but different mνi and this causes nðMÞ to retain somesensitivity to individual neutrino masses even if δcrit isindependent of the neutrino mass hierarchy.In Fig. 5(b) we show the changes to the halo mass

function for cosmologies with massive neutrinos. We showthe fractional correction to both the Bhattacharya massfunction and the Sheth-Tormen mass function. The dom-inant changes to the mass function are from the shift in

(a) (b)

FIG. 5 (color online). Left: The neutrino-induced suppression in the amplitude of σðMÞ, i.e., the CDM and baryon density fluctuationssmoothed on the scaleM [Eq. (3.3)]. Right: The change to the halo abundance in a cosmology with massive neutrinos at z ¼ 0.15 usingthe corrections to δcrit for mν ≠ 0 calculated in this paper and the mass functions of Bhattacharya (solid lines) and Sheth-Tormen(dot-dashed lines). In both panels the order of the legend matches the order of the curves.

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-8

σðMÞ rather than δcrit shown in Fig. 11 and Fig. 4. Whilescenarios with common Ων but different mνi are notcompletely degenerate, the differences are small and itwould be very challenging to distinguish between them.In Fig. 6 we show the fractional difference between the

mass function using our neutrino mass-dependent values ofδcrit and the standard value of 1.686. The standard collapsethreshold δcrit ¼ 1.686 predicts more massive halos thanour calculated δcritðmνÞ. That is, our prescription for thehalo mass function predicts that massive neutrinos causegreater changes to nðMÞ than the standard prescriptionwhich leaves δcrit fixed to the Ωm ¼ 1 value. The fractionaldifference between our nðMÞ and the standard prescriptionusing δcrit ¼ 1.686 increases slightly with increasing red-shift. For

Pimν;i ≲ 0.3 eV, the error in using δcrit ¼ 1.686

remains ≲10% until M ¼ 1015h−1M⊙ for all redshiftsbetween z ∼ 0–1.Throughout this paper we have used the variance of

CDM and baryon fluctuations (σðMÞ defined in Eq. (3.3)in the spherical collapse model (this is also the prescriptionof Refs. [30,50]). A number of authors [44,45], includingthe analyses of Refs. [18–20], have used the variance ofthe total (CDMþ baryonþ neutrino) mass fluctuations tocalculate the mass function. That is, in the halo massfunction they have used σmðMÞ

σmðMÞ≡Z

dkkjWðkRÞj2 k

3Pmðk; zÞ2π2

ð3:12Þ

where Pm is the total matter power spectrum includingmassive neutrinos, rather than σðMÞ calculated from theCDM and baryons alone [defined in Eq. (3.3)]. In Fig. 7 we

compare the difference between the halo mass functionscalculated with our δcrit from Eq. (3.10) and σðMÞ with themass function calculated with δcrit ¼ 1.686 and σmðMÞ. Onhalo scales massive neutrinos suppress σmðMÞ more thanσðMÞ so the combination of using δcrit ¼ 1.686, which islower than δcritðmνÞ and σmðMÞ which is also lower thanσðMÞ yields a final mass function that is closer to ourprediction than the δcrit ¼ 1.686, σðMÞ shown in Fig. 6.

IV. CONCLUSIONS

In this paper we have extended the spherical collapsemodel to cosmologies with massive neutrinos withmν up to1 eV. In our calculations of spherical collapse we take theeffects of massive neutrinos into account in three ways:(i) in setting up the initial conditions for RðtÞ, (ii) in thebackground cosmology entering into the equation ofmotion for RðtÞ, and (iii) in that we allow for neutrinoclustering in the halo during collapse. As expected, massiveneutrinos delay the time of collapse (R → 0). Includingnonlinear clustering of neutrinos interior to the halo slightlydecreases the delay in the collapse time but the net changedue to massive neutrinos is still to delay collapse.In cosmologies with massive neutrinos the evolution of

density perturbations becomes scale dependent once theneutrinos become nonrelativistic. This scale-dependentevolution introduces some subtleties in mapping betweenspherical collapse solutions and halo abundance (see alsoRef. [62] and Refs. [63,64] for a discussion of scale-dependent evolution and halo bias). We have developed anapproach for studying spherical collapse in nonstandardcosmologies with scale-dependent growth at early timeswhich is outlined in Secs. II and III. We applied this tocalculate the effect of massive neutrinos on the linearlyextrapolated collapse threshold δcritðzcollapseÞ. We find thatfor cosmologies with

Pimνi ≲ 0.5 eV, the changes to δcrit

are≲1%, in agreement with previous works [30,44–46,50].For scenarios with larger neutrino masses the changes to

δcrit are considerably larger. The dominant effects ofneutrino mass on δcrit come from two competing effects:the suppression in the growth of density perturbations onhalos scales increases the collapse threshold, while non-linear clustering of massive neutrinos interior to the halodecreases the collapse threshold. The suppression in thegrowth of halo-scale perturbations dominates over theneutrino clustering in all the examples we considered.Interestingly, we found that the effects of massive neutrinoson δcrit [and σðMÞ] are not entirely characterized by Ων.That is, we find that the predicted changes to δcrit and nðMÞfor cosmologies with different neutrino mass hierarchies(values of mν1, mν2, mν3) but the same Ων are not exactlythe same (see Fig. 4). This difference was also seen inthe mixed dark matter simulations of Ref. [50]. In ourframework the different dependence on mνi and Ων ispredominantly due to the fact that the neutrino-inducedsuppression in the growth of structure depends on both the

FIG. 6 (color online). The fractional difference between thehalo abundance nðMÞ using δcrit ¼ 1.686 and nðMÞ calculatedusing our corrections to δcritðmνÞ. Massive neutrinos increaseδcrit relative to 1.686 so the abundance calculated assumingδcrit ¼ 1.686 is larger. The order of the legend corresponds to theorder of the curves.

SPHERICAL COLLAPSE IN νΛCDM PHYSICAL REVIEW D 90, 083518 (2014)

083518-9

energy density in neutrinos and the redshift at which theybecame nonrelativistic (see e.g. Refs. [6,59]). Clustering ofneutrinos interior to the halo during collapse also causeschanges to δcrit that depend nonlinearly on the neutrinomasses (Fig. 3) but the sense of this effect on δcrit isopposite to that of the scale-dependent growth and issubdominant in all cases we considered.In practice, distinguishing between neutrino mass hier-

archies with halo abundance should be extraordinarilychallenging. At the level of the spherical collapse model,the theoretical uncertainty in nðMÞ is easily as large as thedifference between the effects of different mass hierarchies(see e.g. Ref. [60]). It is nevertheless important to note thatthe halo mass function should not be entirely determined byΩν. We view the theoretical uncertainties here as additionalmotivation for N-body simulations with a range of neutrinomass hierarchies.An increasing number of authors have studied halo

abundance in N-body simulations with mixed dark matter(νΛCDM) cosmologies [44–46,50]. This literature includessimulations with three degenerate neutrino masses rangingfrom mνi ¼ 0.05–0.40 eV and measurements of haloabundance for masses M ≲ h−11015M⊙. For three degen-erate neutrinos with mνi ≲ 0.2 eV our predicted changes toδcrit are≲1% leading to a fractional change in nðMÞ of onlya few percent at M ¼ 1014M⊙ but they give a suppressionof ∼10% at M ¼ 1015M⊙ at z ¼ 0 and ∼20% at z ¼ 1.Our predictions for the changes to nðM; zÞ appear to beconsistent with the halo abundance measured from simu-lations shown in Fig. 2 in Ref. [50].On the other hand, for

Pimνi ¼ 1.2 eV, we predict that

δcrit is ∼2% larger at M ¼ 1015M⊙ which leads to asuppression in nðMÞ at this mass of ∼30% for fixedσðMÞ. The authors of Ref. [44] found that the Sheth-Tormen mass function with δcrit ¼ 1.686 and σðMÞ calcu-lated using the total matter power spectrum describes thehalo abundance at z ¼ 0 in their simulations to better than∼ 5–10%. The difference between nðMÞ predicted here[calculated with the larger value of δcrit and σðMÞ calcu-lated using the CDMþ baryon power spectrum] and theapproach of Ref. [44] [using δcrit ¼ 1.686 and σmðMÞcalculated using the total matter power spectrum] is10–15% for

Pimνi ¼ 1.2 eV at z ∼ 0 and larger at higher

redshift, which is marginally inconsistent with Ref. [44].It would be interesting to study the halo mass function inN-body simulations for the more extreme scenarios con-sidered here (e.g. mνi ∼ 1 eV) where we find more signifi-cant corrections to δcrit and nðMÞ (e.g. Fig. 5). Thedifference between different prescriptions for the massfunction is small for

Pimνi ≲ 0.3 eV, but for larger

neutrino masses the variation between predictions is largerand could therefore alter the constraints on heavy neutrinospecies from cluster abundance.In our discussion of the halo mass function we have

neglected the neutrino contribution to the mass of the halos.

For neutrino masses ≲0.2 eV the neutrino contribution tothe halo mass should be ≲10−3 so neglecting the neutrinosis justified [28]. On the other hand we found in Ref. [28]that for neutrino masses Oð1 eVÞ the neutrino massassociated with the halo could reach ∼10% (though theneutrino mass interior to the virial radius is ≲1% of theCDM mass). Even a few percent shift in the halo massescan have a substantial effect on nðMÞ so we caution thatFigs. 5, 6, and 7, which are written in terms of the CDMmass only, do not directly give the corrections to theobserved halo abundance. The density profile of neutrinosis different from the density profile of CDM [28,46,65] so itis not obvious how the neutrino contribution will changethe inferred halo mass and the answer will likely depend onthe observable. We leave this question to further study.

ACKNOWLEDGMENTS

M. L. thanks Brad Benson, George Fuller, Daniel Grin,Wayne Hu, Doug Rudd, and especially Matias Zaldarriagafor helpful discussions. M. L. is grateful for hospitality atthe Institute for Advanced Study while this work was beingcompleted. M. L. is supported by U.S. Department ofEnergy Contract No. DE-FG02-13ER41958.

APPENDIX A: INITIAL CONDITIONS FOR δcbðtÞThe CAMB code gives the values of δcðk; zÞ, δbðk; zÞ, and

δνðk; zÞ for a unit-magnitude initial curvature perturbationin the adiabatic mode. We modify the CAMB code so that thetime derivatives of the density perturbations in CDM and

FIG. 7 (color online). The fractional difference between thehalo abundance nðMÞ calculated using δcrit ¼ 1.686 andσ2mðM;mνÞ given in Eq. (3.12) (the variance of the total matterfluctuations including CDM, baryons, and neutrinos)—as issometimes done in the literature—compared to our prescriptionusing δcritðmνÞ and the variance of CDM and baryons only[σ2ðMÞ as in Eq. (3.3)]. The order of the legend corresponds tothe order of the curves.

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-10

baryons, _δcðz; kÞ and _δbðz; kÞ, are output as well. To set upthe initial conditions for R and _R we need d ln δcb=dtðkÞ ¼_δcbðk; zÞ=δcbðk; zÞ (which is independent of δcb for linearperturbations).For an initial density perturbation with a top-hat profile,

the initial velocity is given by

d ln δcbdt

¼Rd3kWðkRÞ_δcbðk; zinitÞ=δcbðk; zÞR

d3kWðkRÞ ; ðA1Þ

where WðkRÞ ¼ 3j1ðkRðMÞ=ðkRðMÞÞ and _δcbðk; zÞ=δcbðk; zÞ is the ratio of the amplitudes in each Fouriermode which can be obtained from the transfer functionsgiven by CAMB. Note that for δcb independent of k, theintegrals just return a constant. That is,

Zd3kð2πÞ3 WðkRÞ ¼

Zd3k

Z43πR3

d3xe−ik·x ¼ 1; ðA2Þ

but Eq. (A2) does not converge numerically. However, theratio of the two expressions in Eq. (A1) plotted as afunction of the maximum value k does reach an asymptote.In Fig. 8 we plot

d ln δcbdt

ðkÞ ¼Rk0 k

02dk0Wðk0RÞ_δcbðk0; zinitÞ=δcbðk0; zinitÞRk0 k

02dk0Wðk0RÞ ;

ðA3Þ

along with the ratio _δcbðk; zÞ=δcbðkÞ. In Fig. 8 we see that_δcbðzinitÞ as given in Eq. (A3) does not depend on halo massfor the range of masses considered but it does depend onthe neutrino masses; the difference at low k is due to the

different ΩmðzinitÞ’s but the larger suppression at high k isdue to mν > 0. We can also see that by k ∼ 1 h=Mpc thevalues of Eq. (A3) and _δcbðk; zÞ=δcbðkÞ are nearly identical.We therefore use d ln δcb=dt ¼ _δcbðk; zÞ=δcbðkÞ evaluatedat k ¼ 10 Mpc−1 to determine the initial conditions of Rfor a top-hat initial density perturbation. As discussed inSec. III the calculations of the critical overdensity in theplots of the body of this paper use the root-mean-squarevalue of the initial velocity given in Eq. (3.5), rather thanEq. (A3). We have used the top-hat initial conditions ofEq. (A3) in Fig. 2, Fig. 10, and Fig. 11.For adiabatic initial conditions there are perturbations

in the other components δν and δγ at the initial time as well.A top-hat initial density perturbation in CDM and baryonswith amplitude δcb;init and radius R has Fourier componentsgiven by

δcbðkÞ ¼ δcb;initWðkRÞ; ðA4Þwhich allows us to determine the amplitude of the pri-mordial curvature perturbation in each Fourier mode, andtherefore the perturbations of the other components

δcb;iðk; zinitÞ ¼ Tcbðk; zinitÞζðkÞ; ðA5Þ

so that

δνðk; zinitÞ ¼Tνðk; zinitÞTcb;iðk; zinitÞ

δcbðzinitÞWðkRÞ; ðA6Þ

where ζðkÞ is the primordial curvature. The linear evolu-tion of the initial density perturbation, δνðzÞ ¼Rd3kTνðk; zÞ=Tcbðk; zinitÞδcbðzinitÞWðkRÞ, for a top-hat

CDM and baryon perturbation is plotted in Fig. 8.

(a) (b)

FIG. 8 (color online). Left: The initial _δcbðzinitÞ from CAMB. The solid curves show Eq. (A3) as a function of k; the curves forM ¼ 1013M⊙; 1014M⊙, and 1015M⊙ are indistinguishable. The dotted curves are the ratio _δcbðk; zinitÞ=δcbðk; zinitÞ, which at high k isnearly identical to Eq. (A3). All curves are at z ¼ 200 in units of HðzÞ. Right: The linear neutrino perturbation,δνðzÞ ¼

Rd3kTνðk; zÞ=Tcbðk; zinitÞδcbðzinitÞWðkRÞ, plotted as a function of redshift in units of δcbðzinitÞ for a range of halo masses.

SPHERICAL COLLAPSE IN νΛCDM PHYSICAL REVIEW D 90, 083518 (2014)

083518-11

At our start time, zinit ¼ 200, the amplitude of densityperturbations that collapse by z≲ 1 is ∼10−2. The ampli-tude of nonlinear corrections to our initial conditions(which we have ignored) is Oðδ2initÞ ∼ 10−3–10−4 whichis not very different from the magnitude of the neutrinomass effects on δcrit for some of the smaller neutrinomasses. However, to make nonlinear corrections

completely subdominant we would need to start ourcalculation before decoupling and then follow the subhor-izon evolution of the baryons independently of the CDMuntil z ∼ 200. The total density perturbation would nolonger have a top-hat profile in this case and this seems likeoverkill for a model that is intended to be a crudeapproximation to halo formation. However, we have tested

(a) (b)

FIG. 9 (color online). Left: Several calculations of the neutrino mass interior to R, δMν, plotted for a cosmology with one massiveneutrino with mν ¼ 1 eV and halos of M ¼ 1014M⊙ and M ¼ 1015M⊙. Shown is the linearized Boltzmann solution, the exactcalculation, and a hybrid solution that uses the linearized Boltzmann solution at early times, where the exact calculation has notconverged and the exact calculation at late times once δMν;exact > δMν;linear. Right: The solutions to Eq. (2.8) using δMν ¼ 0, thelinearized Boltzmann solution for δMν, and the hybrid exact-linear solution. Despite the large differences in δMν in the right panel, thesolutions for RðtÞ are very similar; the collapse time changes by≲1% forM ¼ 1015M⊙ and< 0.5% forM ¼ 1014M⊙. These changes intcollapse are small compared to the change due to the effect we are interested in, i.e., a nonzero δMν.

(a) (b)

FIG. 10 (color online). A comparison of the neutrino-induced changes to the linearly extrapolated collapse threshold calculated in twoways: first, using the initial velocity _δcbðzinitÞ=δcbðzinitÞ≡ _σ=σ and the linear extrapolation given in Eq. (3.7) as in the body of this paper(solid lines); second, using the initial conditions and linear extrapolation of a top-hat perturbation at z ¼ 1000 (dashed lines). The twoapproaches are in agreement at the ≲20% level.

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-12

the sensitivity of our calculations to the initial conditions intwo ways: (i) by changing zinit to earlier times and repeatingthe spherical collapse calculation, and (ii) by shifting zinitbut keeping δinit fixed between calculations with differentvalues of the neutrino mass (so that the magnitude ofthe nonlinear corrections are similar). In both cases thechanges to ðδcritðmνÞ−δcritðmν¼ 0ÞÞ=δcrit are≲ðfewÞ 10−3.Changing the ratios of CDM to baryons has an effect that iscomparable in magnitude. We therefore consider ourcalculations of δcrit to be accurate (independent of zinitand Ωb) to about ∼0.5%.

APPENDIX B: HOW ACCURATE IS THEEXPRESSION FOR δMν? DOES ITMATTER FOR δlinearcb ðzcollapseÞ?

In Sec. II B we used a solution to the linearizedBoltzmann equation to calculate the neutrino mass interiorto R, δMνð< RÞ. The linearized Boltzmann solution isaccurate at early times before the CDM density fluctuationhas become nonlinear, but significantly underestimatesthe late-time nonlinear clustering of neutrinos (e.g.Refs. [28,56]). In this appendix we explore the sensitivityof our results to the approximation for δMνð< RÞ used inSec. II B. Our reference point for a more accurate calcu-lation of δMν numerically integrates neutrino trajectoriestraveling in the external potential of the collapsing halo.The initial conditions for the neutrino trajectories samplethe initial phase space and δMνð< R; tÞ is obtained bysumming elements of phase space with trajectories interiorto R at time t (for details see Ref. [28]). We refer to thiscalculation as the “exact” calculation because it is an exactsolution to the Boltzmann equation for neutrinos in anexternal potential. Our exact calculation, however, stillmakes the approximation that δMν can be calculated usinga halo potential described by the solution to RðtÞwith δMν ¼ 0.Figure 9 shows the difference between the linearized

Boltzmann solution for δMν and the exact calculation forhalos with M ¼ 1014M⊙, M ¼ 1015M⊙, and single mas-sive neutrino species with mν ¼ 1 eV. The differencebetween the exact calculation and the linear approximationis large. Plotted in the right panel of Fig. 9 are the solutionsfor RðtÞ from Eq. (2.8) with δMν ¼ 0, δMν from thelinearized Boltzmann equation, and δMν using the linear-ized Boltzmann equation at early times and the exactcalculation once δMν;exact > δMν;linear. As discussed inSec. II nonzero δMν has a large effect on the evolutionof RðtÞ and the collapse time. However, the differencebetween RðtÞ including the larger δMνexact and the linear-ized δMνlinear is negligible. For the examples we haveconsidered, halos with M ¼ 1014M⊙ and M ¼ 1015M⊙,the collapse times change by < 0.5%, < 1%. Apparently,the effect of δMν on the final collapse time of R is mostimportant at early times. Nonlinear clustering of neutrinosdepends strongly on the neutrino mass. Since the error in

tcollapse from using the linearized Boltzmann calculation forδMν is unimportant for mν ¼ 1 eV we conclude that it is asafe approximation for all neutrino mass hierarchiesconsidered in this paper.

APPENDIX C: CALCULATING δcritðzÞ IN ACOSMOLOGY WITH SCALE-DEPENDENT

EVOLUTION

There are a number of analytic models of halo abun-dance, from the original Press-Schechter (PS) ansatz [57] tomore sophisticated excursion set and peaks calculations[58,66]. These models make use of the fact that the linearevolution of the density field is well understood andidentify regions of the early-time, linear density field thatsatisfy certain criteria, with halos at late times. For instance,a halo of mass M forming at redshift z can be associatedwith a region in the early-time density field δcb;initsmoothed on the scale R ¼ ð3M=ð4πρcbÞÞ1=3 that exceedsthe threshold for collapse [δcbðzinitÞ > δcritðzinitÞ] and/or is apeak [∇δcbðx; zinitÞ ¼ 0 and detð∇i∇jδcbðx; zinitÞÞ < 0]. Ifthe statistics of the linear density field are known (in thestandard cosmology they are Gaussian) then the fraction ofthe initial volume, and therefore the mass, that satisfies thehalo criteria (peaks or thresholds) can be calculated.For instance, the excursion set model relates the abun-

dance of halos at z to the distribution of first crossings ofthe barrier δcrit;iðzÞ which (in the limit of a Fourier-spacetop-hat smoothing function) gives the usual Press-Schechter mass function

nPSðM; zÞ ¼ −2ρ̄cbM

ddM

�Z∞

δcbðzcollapse ;zinitÞσðM;zinitÞ

dνe−1=2ν

2

ffiffiffiffiffiffi2π

p�; ðC1Þ

where δcbðzcollapse; zinitÞ is the value of the density fluc-tuation at zinit required to collapse by redshift zcollapse.Typically, one works with the collapse threshold and thedensity field linearly extrapolated to the present time. This isa convenient choice because δcritðzcollapse; zcollapseÞ is nearlyindependent of redshift. And, for a cosmology with scale-independent growth a top-hat density perturbation (or adensity perturbation of any profile) δcbðzÞ evolves in thesameway as σðM; zÞ so the ratio δcbðzÞ=σðM; zÞ is constant.In cosmologies with massive neutrinos, or any scale-

dependent growth, the growth rates of density perturbationsdepend on their profile. To be completely explicit,

δcbðzÞ ¼Z

d3kð2πÞ3

Tcbðk; zÞTcbðk; zinitÞ

δcbðk; zinitÞ ðC2Þ

where Tcbðk; zÞ are the transfer functions. Furthermore, thescale-dependent growth will change the density profiles ofperturbations, e.g. an initially top-hat density perturbationcan evolve into a perturbation with a slightly differentprofile.

SPHERICAL COLLAPSE IN νΛCDM PHYSICAL REVIEW D 90, 083518 (2014)

083518-13

Quantities that describe the statistics of the density fieldwill evolve differently from individual density perturba-tions. For instance, the variance of density fluctuationsevolves as

σ2ðR; zÞ ¼Z

d3kð2πÞ3 jWðkRÞj2

Tcbðk; zÞTcbðk; zinitÞ

2

Pcbðk; zinitÞ

ðC3Þ

so the ratio δcbðzÞ=σðM; zÞ may depend on the redshift zbecause the numerator and denominator are weightedtowards different k ranges. On the other hand the ratioδcbðkÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik3PcbðkÞ

pis time independent. However one has

to be cautious in using the statistics of the initial densityfield smoothed on some scale δcb;RðzinitÞ to define halos atlate times so that the description of halo abundance doesnot depend on the initial time.The issue of scale-dependent growth and the definition

of the collapse threshold was also discussed in Ref. [62] inthe context of modified gravity theories. In the example ofRef. [62], the growth is scale independent at early timesso that δcb;RðzinitÞ is sufficient to specify _δcb;RðzinitÞ.Those authors noted that the relevant quantity for calculat-ing the halo abundance is δcb;RðzinitÞ=σðM; zinitÞ. Toevaluate δcrit at a different redshift while preserving theratio δcb;RðzinitÞ=σðM; zinitÞ one would use δcritðzÞ≡δcb;RðzinitÞσðM; zÞ=σðM; zinitÞ which is consistent withEq. (3.7) in this paper. However, in Ref. [62] the scale-dependent growth did not become important until late

times so the initial velocity of a top-hat perturbation andthe root-mean-square value of top-hat perturbations_σðM; zinitÞ=σðM; zinitÞ are the same and there is no ambi-guity in setting the initial conditions for RðtÞ.In this paper we have chosen to use _σðM; zinitÞ=

σðM; zinitÞ to set the initial conditions for RðtÞ for tworeasons: (i) this choice represents typical initial conditionsfor δcb;RðxÞ, and (ii) it allows Eq. (3.7), which shouldrepresent the rarity of initial conditions that collapse by z, tobe independent of zinit. Note that there still may be residualsensitivity to the initial time, due to the fact that we setinitial conditions using σðM; zÞ which does not evolveaccording to the same equation of motion as the linearperturbation δcb. In principle we could start our calculationsat earlier times before neutrinos become nonrelativistic(hence before they induce scale-dependent growth) so thatwe would avoid the need to make this choice for _δcb;R.Unfortunately this does not get around the issue because atsuch early times the linear evolution is scale dependentbecause of the presence of radiation and the fact thatbaryons are not exactly tracking CDM. However, as asanity check we can compare our predicted change to thecollapse threshold in the presence of massive neutrinosfrom Eq. (3.7) and Eq. (3.6) to an alternative definition ofδcrit using top-hat initial conditions at early times when thescale-dependent growth due to massive neutrinos is small.Specifically, we define

δearlycrit ðzcollapseÞ ¼δtop-hatcrit ðz ¼ 1000ÞσðM; z ¼ 1000Þ σðM; zcollapseÞ ðC4Þ

(a) (b)

FIG. 11 (color online). Left: The linearly extrapolated value of the initial amplitude of the density perturbation δcbðzinitÞ required tocollapse at zcollapse, where the initial _δcbðzinitÞ and the linear extrapolation are done for an exactly top-hat density profile at z ¼ 200 [seeEq. (C5) and surrounding text]. The solid lines include neutrino clustering, while the fainter dotted lines neglect it [δMνð< RÞ ¼ 0 inEq. (2.8)]. Here we have fixedΩc andΩb so curves with differentΩν have different total matter densityΩm. Right: The fractional changeto the collapse threshold when neutrino clustering interior to R is included. In both panels the order of the curves matches the order of thelegends.

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-14

where δtop-hatcrit ðz ¼ 1000Þ is the critical value of a top-hatdensity perturbation, with _δcb calculated for top-hat initialconditions as in Appendix A linearly extrapolated toz ¼ 1000. This definition is in the same spirit as theapproach of Ref. [62]. A comparison between the neutrinocorrections to δcrit calculated using Eq. (C4) and Eq. (3.7) isplotted in Fig. 10. The neutrino corrections from the twocalculations are in agreement at the ≲20% level.Another approach for treating the scale-dependent

growth would be to determine the correlated requirementson both δcb;RðzinitÞ and _δcb;RðzinitÞ for a halo to collapse bya given time. Then one could associate regions in theinitial density field that satisfy the joint criteria with halosforming at the collapse redshift. Developing such a modelwould be quite interesting but is beyond the scope ofthis paper.Finally, we have so far treated the threshold for collapse

as a criteria on the initial density field, δcbðzinitÞ withstatistics characterized by the initial power spectrumPcbðk; zinitÞ and looked for consistent ways of imposingthis criteria on the density field linearly extrapolated to latetimes. One could instead treat the linearly extrapolatedvalue of a top-hat perturbation δcbðzinitÞ as the parameter ofinterest, for instance as a quantity in a fitting function. Thatis, one could use the initial conditions of a true top-hat

density perturbation δcb;init, with _δcb;init as given inEq. (A3), for the initial conditions for RðtÞ and linearlyevolve δcb;init to zcollapse using the true top-hat evolution.This gives

δtop-hatcrit ðzÞ≡Rd3kTcbðk; zÞ=Tcbðk; zinitÞWðkRÞR

d3kWðkRÞ δcb;i:

ðC5Þ

This approach results in quite different values of δcritðzÞ forcosmologies with large scale-dependent growth near thehalo scale (for massive neutrinos this happens whenmν;i ≳ 0.3 eV). More importantly, in these cases δcritðzÞ=σðM; zÞ ≠ δcb;i=σðM; zinitÞ so the linearly extrapolated

δtop-hatcrit cannot be straightforwardly interpreted as a collapsethreshold. For comparison, we show these calculations inFig. 11. Interestingly, the sensitivity to neutrino mass inδtop-hatcrit ðzÞ is almost entirely due to the clustering ofneutrinos interior to R. If δMνð< RÞ ¼ 0, the “top-hat”collapse threshold for cosmologies with massive neutrinosin Eq. (C5) is ≲0.5% different from Eq. (C5) for cosmol-ogies with mνi ¼ 0. The change to δtop-hatcrit from δMν ≠ 0 isnearly identical to the change to δcrit using Eq. (3.7).

[1] P. Ade et al. (Planck Collaboration), arXiv:1303.5076[Astron Astrophys. (to be published)].

[2] G. Hinshaw et al. (WMAP Collaboration), Astrophys. J.Suppl. Ser. 208, 19 (2013).

[3] Z. Hou et al., Astrophys. J. 782, 74 (2014).[4] J. L. Sievers et al., J. Cosmol. Astropart. Phys. 10 (2013)

060.[5] J. Beringer et al., Phys. Rev. D 86, 010001 (2012).[6] J. Lesgourgues and S. Pastor, Phys. Rep. 429, 307 (2006).[7] U. Seljak, A. Slosar, and P. McDonald, J. Cosmol.

Astropart. Phys. 10 (2006) 014.[8] B. A. Reid, L. Verde, R. Jimenez, and O. Mena, J. Cosmol.

Astropart. Phys. 01 (2010) 003.[9] S. A. Thomas, F. B. Abdalla, and O. Lahav, Phys. Rev. Lett.

105, 031301 (2010).[10] S. Saito, M. Takada, and A. Taruya, Phys. Rev. D 83,

043529 (2011).[11] M. E. Swanson, W. J. Percival, and O. Lahav, Mon. Not. R.

Astron. Soc. 409, 1100 (2010).[12] J.-Q. Xia, B. R Granett, M. Viel, S. Bird, L. Guzzo, M. G.

Haehnelt, J. Coupon, H. Joy McCracken, and Y. Mellier,J. Cosmol. Astropart. Phys. 06 (2012) 010.

[13] S. Riemer-Sorensen et al., Phys. Rev. D 85, 081101 (2012).[14] G.-B. Zhao et al., Mon. Not. R. Astron. Soc. 436, 2038

(2013).[15] R. de Putter et al., Astrophys. J. 761, 12 (2012).

[16] F. Beutler et al., arXiv:1312.4611.[17] A. Vikhlinin et al., Astrophys. J. 692, 1060 (2009).[18] A. Mantz, S. W. Allen, and D. Rapetti, Mon. Not. R. Astron.

Soc. 406, 1805 (2010).[19] B. Benson et al., Astrophys. J. 763, 147 (2013).[20] C. Reichardt et al., Astrophys. J. 763, 127 (2013).[21] M. Hasselfield et al., J. Cosmol. Astropart. Phys. 07 (2013)

008.[22] P. Ade et al. (Planck Collaboration), arXiv:1303.5080

[Astron Astrophys. (to be published)].[23] K. Abazajian et al., arXiv:1204.5379.[24] J. M. Conrad, W. C. Louis, and M. H. Shaevitz, Annu. Rev.

Nucl. Part. Sci. 63, 45 (2013).[25] M. Wyman, D. H. Rudd, R. A. Vanderveld, and W. Hu,

Phys. Rev. Lett. 112, 051302 (2014).[26] J. Hamann and J. Hasenkamp, J. Cosmol. Astropart. Phys.

10 (2013) 044.[27] R. A. Battye and A. Moss, “Evidence for massive neutrinos

from CMB and lensing observations,” 2013.[28] M. LoVerde and M. Zaldarriaga, Phys. Rev. D 89, 063502

(2014).[29] J. E. Gunn and J. R. Gott, III, Astrophys. J. 176, 1 (1972).[30] K. Ichiki and M. Takada, Phys. Rev. D 85, 063521

(2012).[31] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538,

473 (2000).

SPHERICAL COLLAPSE IN νΛCDM PHYSICAL REVIEW D 90, 083518 (2014)

083518-15

[32] J. Tinker, A. V. Kravtsov, A. Klypin, K. Abazajian, M.Warren, G. Yepes, S. Gottlöber, and D. E. Holz, Astrophys.J. 688, 709 (2008).

[33] O. Lahav, P. B. Lilje, J. R. Primack, and M. J. Rees, Mon.Not. R. Astron. Soc. 251, 128 (1991).

[34] J. D. Barrow and P. Saich, Mon. Not. R. Astron. Soc. 262,717 (1993).

[35] L.-M. Wang and P. J. Steinhardt, Astrophys. J. 508, 483(1998).

[36] T. Basse, O. E. Bjaelde, and Y. Y. Wong, J. Cosmol.Astropart. Phys. 10 (2011) 038.

[37] P. Creminelli, G. D’Amico, J. Norena, L. Senatore, and F.Vernizzi, J. Cosmol. Astropart. Phys. 03 (2010) 027.

[38] F. Schmidt, W. Hu, and M. Lima, Phys. Rev. D 81, 063005(2010).

[39] M. LoVerde and K. M. Smith, J. Cosmol. Astropart. Phys.08 (2011) 003.

[40] P. Colin, O. Valenzuela, and V. Avila-Reese, Astrophys. J.673, 203 (2008).

[41] J. Brandbyge, S. Hannestad, T. Haugbolle, and B. Thomsen,J. Cosmol. Astropart. Phys. 08 (2008) 020.

[42] S. Agarwal and H. A. Feldman, Mon. Not. R. Astron. Soc.410, 1647 (2011).

[43] M. Viel, M. G. Haehnelt, and V. Springel, J. Cosmol.Astropart. Phys. 06 (2010) 015.

[44] J. Brandbyge, S. Hannestad, T. Haugboelle, and Y. Y. Wong,J. Cosmol. Astropart. Phys. 09 (2010) 014.

[45] F. Marulli, C. Carbone, M. Viel, L. Moscardini, and A.Cimatti, Mon. Not. R. Astron. Soc. 418, 346 (2011).

[46] F. Villaescusa-Navarro, S. Bird, C. Pena-Garay, and M. Viel,J. Cosmol. Astropart. Phys. 03 (2013) 019.

[47] A. Upadhye, R. Biswas, A. Pope, K. Heitmann, S. Habib, H.Finkel, and N. Frontiere, Phys. Rev. D 89, 103515 (2014).

[48] F. Villaescusa-Navarro, F. Marulli, M. Viel, E. Branchini, E.Castorina, E. Sefusatti, and S. Saito, J. Cosmol. Astropart.Phys. 03 (2014) 011.

[49] E. Castorina, E. Sefusatti, R. K. Sheth, F. Villaescusa-Navarro, and M. Viel, J. Cosmol. Astropart. Phys. 02(2014) 049.

[50] M. Costanzi, F. Villaescusa-Navarro, M. Viel, J.-Q. Xia,S. Borgani, E. Castorina, and E. Sefusatti, J. Cosmol.Astropart. Phys. 12 (2013) 012.

[51] R. K. Sheth and G. Tormen, Mon. Not. R. Astron. Soc. 308,119 (1999).

[52] S. Naoz and R. Barkana, Mon. Not. R. Astron. Soc. 377,667 (2007).

[53] I. H. Gilbert, Astrophys. J. 144, 233 (1966).[54] R. H. Brandenberger, N. Kaiser, and N. Turok, Phys. Rev. D

36, 2242 (1987).[55] S. Singh and C.-P. Ma, Phys. Rev. D 67, 023506 (2003).[56] A. Ringwald and Y. Y. Wong, J. Cosmol. Astropart. Phys.

12 (2004) 005.[57] W. H. Press and P. Schechter, Astrophys. J. 187, 425 (1974).[58] J. M. Bardeen, J. Bond, N. Kaiser, and A. Szalay,

Astrophys. J. 304, 15 (1986).[59] W. Hu, D. J. Eisenstein, and M. Tegmark, Phys. Rev. Lett.

80, 5255 (1998).[60] S. Bhattacharya, K. Heitmann, M. White, Z. Lukic, C.

Wagner, and S. Habib, Astrophys. J. 732, 122 (2011).[61] We have checked that using δcritðzjmνÞ directly, rather than

the replacement in Eq. (3.10), makes only a small differencein the predicted change to the mass function. Precisely, thedifference is ≲10% for M ≲ 1015h−1M⊙ and it is smallcompared to the total change in nðMÞ due to mν ≠ 0.

[62] K. Parfrey, L. Hui, and R. K. Sheth, Phys. Rev. D 83,063511 (2011).

[63] L. Hui and K. P. Parfrey, Phys. Rev. D 77, 043527 (2008).[64] M. LoVerde, arXiv:1405.4855.[65] L. Kofman, A. Klypin, D. Pogosian, and J. P. Henry,

Astrophys. J. 470, 102 (1996).[66] J. Bond, S. Cole, G. Efstathiou, and N. Kaiser, Astrophys. J.

379, 440 (1991).

MARILENA LOVERDE PHYSICAL REVIEW D 90, 083518 (2014)

083518-16