Using Dark Matter Haloes to Learn about Cosmic Acceleration:A New Proposal for a Universal Mass Function

Chanda Prescod-Weinstein1,2

NASA Postdoctoral Program Fellow, Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD,20770, USA

Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON, N2L 2Y5 Canada

Department of Physics and Astronomy, 200 University Ave. W, University of Waterloo, Waterloo, ON,N2L 3G1 Canada

and

Niayesh Afshordi3

Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON, N2L 2Y5 Canada

Department of Physics and Astronomy, 200 University Ave. W, University of Waterloo, Waterloo, ON,N2L 3G1 Canada

ABSTRACT

Structure formation provides a strong test of any cosmic acceleration model because a suc-cessful dark energy model must not inhibit or overpredict the development of observed large-scalestructures. Traditional approaches to studies of structure formation in the presence of dark energyor a modified gravity implement a modified Press-Schechter formalism, which relates the linearoverdensities to the abundance of dark matter haloes at the same time. We critically examine theuniversality of the Press-Schechter formalism for different cosmologies, and show that the haloabundance is best correlated with spherical linear overdensity at 94% of collapse (or observation)time. We then extend this argument to ellipsoidal collapse (which decreases the fractional time ofbest correlation for small haloes), and show that our results agree with deviations from modifiedPress-Schechter formalism seen in simulated mass functions. This provides a novel universal pre-scription to measure linear density evolution, based on current and future observations of cluster(or dark matter) halo mass function. In particular, even observations of cluster abundance in asingle epoch will constrain the entire history of linear growth of cosmological of perturbations.

Subject headings: cosmology: large-scale structure of universe

1. Introduction

An important part of the effort to explain cos-mic acceleration and the cosmological constantproblem is testing proposed models in the contextof what are, at this stage, better-established phys-ical pictures. Structure formation could prove to

1Current address: Goddard Space Flight Center, Green-belt, MD, 20770, USA

2chanda.prescod-weinstein@nasa.gov3nafshordi@perimeterinstitute.ca

be an incredibly useful phenomenological methodfor distinguishing models of cosmic acceleration.

It is currently believed that large-scale struc-ture formation has its seeds in small quantumfluctuations in the early universe (e.g., Mukhanovet al. (1992)). The current model for structureformation is elegant in its fundamental simplicity.Random inhomogeneities, artifacts of cosmic in-flation, create a runaway effect in which overdenseregions attract more matter, thus becoming moredense and leading to galaxies, stars, and planets.

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https://ntrs.nasa.gov/search.jsp?R=20110008448 2020-03-30T04:20:46+00:00Z

Better understanding of this process is inde-pendently an intriguing enterprize in the field ofcosmology. However, in this work we focus onthe relationship between the cosmic accelerationand structure formation. More specifically, differ-ent cosmological pictures (cosmologies with differ-ing causes of acceleration, such as a cosmologi-cal constant, dark energy, or modifications of Ein-stein gravity) might have expansion histories thatare similar to one another but leave different im-prints on large-scale structures, and in particularon galaxy clusters. Therefore, structure forma-tion provides a unique testing ground for modelsof cosmic acceleration (e.g., Ishak et al. (2006);Acquaviva et al. (2008); Vikhlinin et al. (2009)).Here we critically examine some of the assump-tions in this program, and develop a framework toenhance the accuracy of this kind of work.

The first step in this direction is to revisit howthe Press-Schechter formalism (Press & Schechter1974) (PSF) is used to predict the cluster massfunction. Press & Schechter (1974) have arguedthat the number density of dark matter haloes (orgalaxy clusters) of mass M is given by:

dn(M, z)

dM= f [σ(M, z)]

ρm(z)

M

∂ lnσ−1(M, z)

∂M,

(1)where σ2(M, z) is the variance of linear overden-sity in spherical regions of mass M at redshift z,while ρm(z) is the mean matter density of the uni-verse. For random gaussian initial conditions, f [σ]is given by:

fPS [σ] =

√2

π

δscσ

exp

[− δ

2sc

2σ2

], (2)

where δsc (' 1.68 for most ΛCDM cosmologies) isthe spherical collapse threshold for linear overden-sities (Gunn & Gott 1972).

While the PSF successfully predicts the broadfeatures of the simulated cluster mass functions,it proves too simplistic for detailed model com-parisons required for precision cosmology. Conse-quently, several authors including Sheth & Tormen(1999), Jenkins et al. (2001), Evrard et al. (2002),Warren et al. (2006), and Tinker et al. (2008) haverefined the function f(σ) to better match the massfunctions in N-body simulations. We shall refer tothese approaches as modified PSF. For example,Warren et al. (2006) and Tinker et al. (2008) pro-

pose a fitting formula of the form:

f(σ) = A

[(σb

)−a+ 1

]e−c/σ

2

, (3)

where (A, a, b, c) = (0.186, 1.47, 2.57, 1.19) give agood fit to simulated haloes of overdensity ∆ =200, in a concordance ΛCDM cosmology at z = 0(Tinker et al. 2008) (see Fig. 1). While most ofthis work is based on fitting formulae to simulatedmass functions, Sheth et al. (2001) argue that anapproximate implementation of ellipsoidal collapsecan account for most of the deviations from thePSF.

Fig. 1.— A comparison of Press-Schechter prediction forthe function f(σ) (dotted line; Eq. 2), with a parameterizedfit to the numerical simulations (solid line; Eq. 3).

However, a more pressing question for cosmo-logical applications is whether the function f(σ) isuniversal, or rather can vary for different cosmolo-gies or cosmic acceleration models. In other words,could the same modified PSF be used to accuratelypredict halo abundance in cosmologies with differ-ent cosmological parameters? While earlier stud-ies failed to find such dependence, Tinker et al.(2008) first noticed a systematic evolution of f(σ)with redshift, implying a breakdown of universal-ity at the 10 − 30% level (also see Reed et al.(2007); Crocce et al. (2010); Bhattacharya et al.(2010)). Courtin et al. (2010) note that universal-ity is limited by the nonlinearity of structure for-mation, and the cluster mass function shows some

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redshift dependence at higher redshifts that can be(partially) understood in the context of sphericalcollapse. However, the spherical collapse modelfalls short of explaining the 10 − 30% deviationsfrom universality in all but the most massive clus-ters (see Fig. 7).

In this work, we contribute to the effort to bet-ter understand the role and limits of universal-ity in the cluster mass function by introducing anew parameter that appears to be universal acrosscosmological models.1 In particular, the modi-fied PSF relies on σ(M), the root-mean-squareof linear density fluctuations at the time of ob-servation, when in reality, observed clusters arevery non-linear objects with overdensities exceed-ing 200. We thus seek to find a universal timein the past when we could make a connection be-tween the nonlinear structures that we observe inthe present and the linear structures that existedat that time, since all structures go through a lin-ear phase. Our basic strategy is to find the time inthe past at which the linear density of the struc-tures that collapse today show minimum disper-sion as we vary cosmologies. This is illustrated inFig. (2).

In § 2, we present the complete nonlinear differ-ential equation that governs the growth of matterperturbations in spherical overdense regions in thepresence of a cosmological constant.

In § 3, we describe a numerical method whichwe developed in order to solve both the nonlin-ear and linear structure formation equations inthe presence of a cosmological constant, or darkenergy with a constant equation of state.

In § 4, we discuss the implications of our nu-merical study, i.e. we find a universal fraction ofcollapse time, where linear density is the same in-dependent of cosmology.

In § 5 we reexamine the idea of universality ofmass functions in light of the results of the previ-ous section, including the effect of ellipsoidal col-lapse on the formalism, and propose a new massfunction for general dark energy models.

1Here we define cosmological models to mean different val-ues of the cosmological constant density ΩΛ at the timeof observation, or different values of the dark energy equa-tion of state parameter, w. For now, we do not considerquintessence models where w is dynamical in nature, orpossible modifications of Einstein gravity. Future work willextend our study to such possibilities.

Finally, in § 6, we conclude with an overview ofour results and a discussion of future prospects.

2. Λ & Non-linear Structure Formation

In Appendix A, we derive the differential equa-tion that governs the growth of linear matter per-turbations. We used a familiar fluid dynamics pic-ture to do so. Here we derive the full non-linearequation for spherical overdensities using only cos-mological considerations. It should be noted thatthis particular form of the non-linear equation isonly strictly valid for ΛCDM cosmologies, wherethe inside of a spherical top-hat overdensity can beconsidered as a separate universe. More complexmodels such as dark energy models with differentvalues of w require additional considerations whichwill be discussed below.

We consider a physical picture in which struc-ture formation arises due to a uniformly positivespherical perturbation away from an average mat-ter density, i.e. a top-hat matter overdensity. Thisscenario is similar to considering two cosmologieswith two distinct scale factors: one for the outeruniverse and another for the overdense region. Ofcourse, we are interested in a scenario where adark energy component similar to a cosmologicalconstant is at play, so we will assume the presenceof one as part of our base model.

For the external universe, we write the Fried-mann equation with zero curvature:(

aoao

)2

=8πGN

3(ρ+ ρΛ) = H2. (4)

We can assume ρ ∝ a−3 for ordinary matter, whileρΛ = const. denotes the cosmological constantdensity. We also note that the value of the cosmo-logical constant will be the same inside the over-dense region and the background.

In a general scenario with dynamical dark en-ergy models we cannot assume that curvature, of-ten denoted by k, will be a constant inside theoverdense region due to the presence of pressuregradients. Therefore, Wang & Steinhardt (1998)point out that we are compelled to use the time-time component of Einstein’s equations, as thesedo not explicitly involve the curvature term. How-ever, in the presence of a cosmological constant, orw = −1, we can ignore these considerations andbegin with the first order Friedmann equation. We

3

tobs tobs

Standard Universal Mass Functions Current Proposal

Cosmic Time Cosmic Time

Fig. 2.— A schematic picture of our proposal for a universal mass function: While standard universal mass functions relatethe observed cluster abundance to linear growth at the same time, we propose that cluster abundance is best predicted by lineargrowth at an earlier time. The top charts show the abundance for clusters of different mass (nominally, masses of 1013, 1014

and 1015h−1M, from top to bottom) as a function of cosmic time, while the bottom charts show the linear growth. As weargue in § 5, the abundance of less massive clusters depends on the linear growth at an earlier time, which explains why thearrows point to different stages in the history of linear growth.

then calculate k, which can be seen as an integra-tion constant that arises in going from second tofirst order Friedmann formulations, using initialconditions.

The scale factor in the overdense region is gov-erned by:(

aiai

)2

+k

a2i

=8πGN

3(ρi + ρΛ). (5)

Again, we define ρi = ρ(1 + δ). A little bit of al-gebra yields the following full differential equationfor δ:

[ρ(1 + δ)]− 2

3

[−8πGN

3δρ− 2

3

Hδ

1 + δ+

1

9

δ2

(1 + δ)2

]+k = 0

(6)

It is important to reiterate the importance ofhaving access to both linear and nonlinear solu-tions. As noted by Pace et al. (2010) amongstothers, although initially both the complete solu-tion and its linear approximation will track, even-tually the nonlinear solution will grow much fasterrelative to the scale factor.

Following Liddle & Lyth (2000), we can showthat knowing nonlinear theory is necessary. A crit-ical point in the evolution of a structure’s collapseis the turnaround event in which universal expan-sion’s dominance over the perturbation is eclipsedby gravitational collapse. In other words, at theturnaround point, a potential structure has de-tached from background expansion, but completegravitationally-bound structure formation has notyet begun. This might be thought as the true birthof a structure within the void. Knowing the non-linear solution allows us to find out the value of thescale factor and the overdensity at this so-called“turn-around point.”

Numerically, at the point of complete collapsethe nonlinear solution will “blow up” and ap-proach infinity (see Fig. 3). The linear density atthis point, δsc is the quantity that enters the origi-nal Press-Schechter formalism (see § 1) and is usedas proxy between linear and non-linear structures.Physically, we do not expect an actual singularity.This “blow up” point is considered to be the be-ginning of virialization, a process whereby energy

4

in the bulk infall of matter is redistributed intorandom motion of dark matter particles, leadingto a system in virial equilibrium, where kinetic en-ergy T and the potential energy U are related bythe virial theorem:

Tvir =1

2(R ∂U/∂R)vir, (7)

(see Maor & Lahav (2005)). Another point of viewon the spherical collapse picture, including a dis-cussion of virialization, can be found in Somerville& Primack (1999). However, for the purpose ofthis study, more details on this process are notnecessary.

We can generalize to cases where w 6= −1,i.e. non-cosmological constant scenarios. Abramoet al. (2007) provides a comprehensive derivationof the full non-linear equation, which we refer theinterested reader to for complete details. For ourpurposes, it will suffice to show the final result,which is Equation 7 in Abramo et al. (2007):

δj +

(2H − wj

1 + wj

)δj −

4 + 3wj3(1 + wj)

δj2

1 + δj

−4πG∑k

ρk(1 + wk)(1 + 3wk)δk(1 + δk) = 0. (8)

The subscripts j and k refer to different fluids inthe system, e.g. matter and dark energy. In ascenario where the dark energy can clump, thisbecomes a system of equations. As we discuss be-low, we do not allow this possibility. Therefore,noting that w = 0 for matter, we get the follow-ing non-linear equation governing the behavior ofmatter perturbations:

δm + 2Hδm −4

3

δ2m

1 + δm− 4πGρmδm(1 + δm) = 0.

(9)But, why exactly are we allowed to ignore theclumping in dark energy?

The discussion about dark energy perturbationsis often cast in terms of the effective speed of soundfor dark energy, e.g., Bean & Dore (2004). Typ-ically one may expect that for an adiabatic fluidwith a constant equation state parameter w, thespeed of sound is given by c2 = δP/δρ = w. How-ever, when wde < 0, clearly this becomes imagi-nary, suggesting a catastrophic instability, e.g., seeAfshordi et al. (2005), and thus we must use a

more general definition of the c. In other words,dark energy with constant w cannot be an adi-abatic fluid, implying w = P

ρ 6=δpδρ . While the

equation of state parameter can still give us infor-mation about the background evolution, the fullaction of the fluid is necessary to compute its effec-tive speed of sound: c2eff ≡

δpδρ . It turns out that

for the simplest quintessence models ceff = 1, al-though for more general actions ceff could essen-tially take any value, e.g., Armendariz-Picon et al.(2001).

As to the question of dark energy clumping,we know that pressure fluctuations propagate withthe speed of sound ceff . Therefore, dark energyshould be smooth on scales smaller than its soundhorizon ∼ ceff/H. As long as ceff ∼ 1, all thecollapsed structures at late times are much smallerthan the sound horizon, implying that dark energyperturbations δDE should be negligible for theirformation.

3. Numerical Techniques

Although Gunn & Gott (1972) showed thatthe perturbation equation with the assumption ofspherical symmetry (also known as the sphericaltop-hat problem) can be solved analytically forthe case of an Einstein-de Sitter (EdS) universe(Λ = 0), we are interested in cases where thecosmological constant/dark energy are non-zero.Therefore, a numerical solution is necessary.

We built a code in C++ that utilizes the Runge-Kutta method for numerical solutions of ordinarydifferential equations. The full code can be foundin Prescod-Weinstein (2010). The code runs aninstance of a loop for value of ΩΛ,present in therange of 0 to 0.7 (the currently measured valueof dark energy density), which solves the second-order linear differential equation, Equation A7, aswell as the full non-linear equation, Equation 6.In order to solve both equations, the solution tothe differential equation for the background scalefactor, Equation 4, is found. The curvature con-stant is calculated for the initial overdensity be-fore proceeding to a solution for both the linearand nonlinear differential equations.

Collapse time is formally defined to be the timeat which nonlinear overdensity, δNL, goes to infin-ity. However, we define collapse numerically byrequiring a large value of nonlinear overdensity,

5

δNL = 200. The initial conditions are calibratedsuch that they provide the same results as the ana-lytical EdS model, namely that δL(tcollapse) ≈ 1.7.This is essentially done by assuming that at earlytimes δ scales linearly with the scale factor a,as expected for the growing solution to Equation(A7) in the matter era 2.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

100

105

1010

1015

δL

δN

L

Fig. 3.— This plot shows nonlinear vs. linear overdensi-ties in the Einstein-de Sitter Universe. The code produces,as expected, a δNL that diverges when δL ≈ 1.68.

4. Results

As stated in the Introduction, we wish to dis-cover the time (as a fraction of the collapse time)such that the variance of δL(t) is at a minimum,as we vary cosmologies. In doing so, we find thetime in cosmological history where linear theoryis most likely to accurately predict gravitationalcollapse, independent of cosmology.

For simplicity, we chose to plot the relative dif-ference of δL(t) (in each cosmology) with respectto the fraction of time to the collapse time in theEinstein-de Sitter universe. However, it shouldbe noted that the results are independent of thischoice, and one could easily calibrate with respectto a universe with a non-zero Λ instead. Havingfound a common ground, all the data was searchedfor a single point in time (in units of collapse

time) where the variance of δL(Q)δL(Λ=0) − 1 was at

a minimum, where Q stands for different dark en-ergy models under consideration, whether a gen-eral fluid or a cosmological constant.

The interpolations and variance computationswere carried out using a script in MatLab, which

2It is important to remember that δ is a ratio of two numberswhose units are that of density. Therefore, δ is dimension-less.

can be found in Prescod-Weinstein (2010). In thecase of the simple cosmological constant models,Fig. 4 shows the result that at t

tcollapse' 0.94, the

variance in δL(t/tcollapse) for different cosmologi-cal constant models hits a minimum of 1.80×10−9.Values of δL at range from 1.602 for Einstein-deSitter 3 to 1.599 in the Λ = 0.7 universe.

Computing the solution to Equation (9) re-quires some modifications to the code. Instead ofscanning over different cosmological constant val-ues, this version of the code varies between con-stant values of w. Moreover, as current observa-tions, e.g., Komatsu et al. (2010) set a (very con-servative) upper limit of −1/3 for the value of w ,we studied cases where w was smaller than −1/3.

Because the cosmological constant is a specialcase of this scenario with w = −1, we are ableto check the self-consistency of our methods (Eq.6 vs. Eq. 9) finding that both versions of thecode produce the same results for a universe withΩΛ = 0.7 (approximately the universe that we livein).

0.2 0.4 0.6 0.8 1 1.2-0.06

-0.055

-0.05

-0.045

-0.04

-0.035

t/tcollapse

(δL/δ

L,

Λ=

0)-

1

Fig. 4.— Relative change in δL(t/tcollapse) in ΛCDMcosmologies (ΩΛ = 0.1, 0.2, ..., 0.7), with respect to theEinstein-de Sitter Universe. tcollapse is calculated forspherical overdensities. The curves seem to intersect att/tcollapse ' 0.94, and a calculation of the point of mini-mum variance between the lines confirms this.

Fig. (5) shows a similar comparison to that ofFig. (4), but with different values of equation ofstate, w, with ΩDE = 0.7. Interestingly, we canagain clearly see a point of minimum variance att/tcollapse ' 0.94, suggesting that this result mightbe quite independent of the dark energy model (atleast within the spherical collapse approximation).

3This is below the predicted analytic value of 1.68 at collapsebecause while δL = 1.68 is expected as δNL → ∞, wehave set the collapse to occur at δNL = 200, resulting in alowered collapse δL.

6

0.2 0.4 0.6 0.8 1 1.2-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t/tcollapse

(δL/δ

L,

Λ=

0)-

1

Fig. 5.— Same as Fig. (4), but with varying valuesof dark energy equation of state (w = −0.3,−0.4, ...,−1),at ΩΛ(today) = 0.7. Again, curves seem to intersect att/tcollapse ' 0.94, and, again, a calculation of the pointof minimum variance between the lines confirms this. Webegin at the value -0.3 because current data constrains theparameter to be smaller than this value.

The variance at this minimum is 7.5×10−7. In thenext section, we discuss the physical significanceof this result.

5. What does the Cluster Mass Functionteach us about cosmology?

At first, one might be puzzled by the fact thatδL happens to have almost the same value at 94%of the collapse time, independent of w or Λ, eventhough the linear approximation breaks down longbefore this point. In other words, why should non-linear collapse show such strong correlation withthe linear evolution? This can be understood asthe near cancelation of two different effects withopposite signs:

1. With the exact same initial conditions, thepresence of dark energy weakens the gravita-tional attraction near the turn around point,which in turn stretches the collapse time.

2. The linear growth factor D(t), which is thegrowing solution to Equation (A7), slowsdown as dark energy starts to dominate,since the Hubble friction 2Hδ stops decaying(H → const.), while matter density decaysexponentially ρ ∝ a−3 ∝ exp(−3Ht).

It turns out that for near ΛCDM cosmologies,these two effects nearly cancel each other, i.e. thelinear growth is slowed down, but tcollapse is alsolonger, resulting in nearly the same value of δLclose to (i.e. at 94% of) the collapse time.

Fig. 6.— The expected time of minimum variance inlinear overdensity, in units of ellipsoidal collapse time. Ob-serving the cluster number counts at tcol should tell us thelinear overdensity of the collapsing region at t∗, indepen-dent of the dark energy model.

However, we should note that this result isspecific to the spherical collapse scenario. Onthe other hand, real collapsing regions are farfrom spherical. Even though very rare peaks ofa random gaussian field can be approximated asspheres, more abundant haloes could significantlydeviate from sphericity, e.g., Bardeen et al. (1986).In addition, in a study of halo assembly histories,Angulo & White (2010) find that extending thePSF to account for ellipsoidal effects could sup-port a more accurate picture. For Einstein-de Sit-ter universe, Sheth et al. (2001) give a simple nu-merical fit for the impact of ellipticity on the linearcollapse threshold, δec:

δec ' δsc[1 + 0.31× σ(M)1.23

], (10)

where δsc ' 1.686 is the spherical collapse thresh-old (but see Robertson et al. (2009) for the limita-tions of the ellipsoidal collapse formalism). SinceδL ∝ t2/3 in an Einstein-de Sitter universe, thisimplies that, for the same initial overdensity, thecollapse time of an elliptical region is longer thanthat of a spherical region by a factor of:

tcollapse,ellipticaltcollapse,spherical

'[1 + 0.31× σ(M)1.23

]3/2.

(11)

7

In other words, we need to extrapolate the lineartheory predictions in Figs. (4-5) farther beyondthe point of intersection to actually hit gravita-tional collapse. Therefore, assuming that Equa-tion (11) is approximately independent of the darkenergy model, it can be combined with the resultsof previous sections to show that the point of min-imum variance in δL is shifted to smaller values oft/tcollapse, i.e.:

t∗tcollapse

=0.94

[1 + 0.31× σ(M)1.23]3/2

, (12)

if we include the impact of ellipsoidal collapse.This result is shown in Fig. (6), and demonstrateshow measuring the mass function of clusters at agiven era may tell us about the entire history oflinear growth, and not just a snapshot at the timeof observation (as is in the traditional universalmass function hypothesis).

Inspired by the fitting formula Eq. (3), Equa-tion (12) leads us to propose a new universal clus-ter mass function:

f(σ;Q) ' g(σ)e−h(σ)/σ2∗ , σ∗ = σD

[0.94× to

(1 + 0.31× σ1.23)3/2

],

(13)where the actual mass function is related tof(σ;Q) through equation (1), and to is the time ofobservation at which D(t) is normalized to unity.In other words, Equation (13) suggests that theexponential cut-off in the cluster number countsat any time t is set by the linear density fluctu-ations σ∗ at an earlier time t∗, which is set byEquation (12). As suggested by several recent nu-merical studies (see Introduction) f(σ;Q) dependson cosmology (denoted by Q), but we propose thefunctions g and h to be universal, while the de-pendence on cosmology (or dark energy models)only enters through D(t∗). The explicit depen-dence of g and h on σ at the time of observation isjustified, as the value of σ acts as a proxy for theasphericity of the collapsing region (Sheth et al.2001).

Comparing to Eq. (3), we fix g and h as:

g(σ) = A

[(σb

)−a+ 1

],

h(σ) = cD2

[0.94× to

(1 + 0.31× σ1.23)3/2

]ΩΛ=0.7

, (14)

where (A, a, b, c) = (0.186, 1.47, 2.57, 1.19) 4.

Fig. 7.— The relative change in f(σ) at high redshifts,compared to z = 0. The colored regions show the simu-lated results from Tinker et al. (2008). Curves in top panelshow our analytic prediction without ellipticity corrections,while curves in the bottom panel include the ellipticity cor-rections (Eq. 13). The solid, dotted, short-dashed, andlong-dashed curves refer to z = 0, 0.5, 1.25, and 2.5 respec-tively.

Fig. (7) compares the evolution in the func-tion f(σ;Q) in N-body simulations of Tinker et al.

4We recognize that these choices for g and h are not unique,but as Fig. (7) demonstrates, these are sufficient to fitcurrent simulations

8

(2008), with our prediction from Eqs. (13-14). Wesee that while the result from spherical collapse,t∗ = 0.94 × tcollpase, underpredicts the evolution(top panel), Eq. (13), which includes the ellip-soidal correction to the collapse time (Eq. 12),can successfully reproduce the evolution in f(σ;Q)(bottom panel). Notice that, while we use thesame σ-dependence as Tinker et al. (2008) at z = 0to fix functions h(σ) and g(σ), our predicted red-shift evolution is based on Eq. (13). Therefore,comparison with their simulated evolution pro-vides an independent test of our proposal.

In the end, we should point out a subtlety incomparing our predictions with simulated clustermass functions. The most widely used algorithmto identify haloes in N-body simulations, known asfriends-of-friends (FoF), is based on linking neigh-boring particles. However, this method has thetendency to bridge between two close-by haloes,and thus counting them as one. An alternativemethod, which is known as the Spherical Over-density method, (SO) identifies haloes by search-ing for spheres of large overdensity (200 or more).The SO haloes are more closely related to the ob-servational notion of clusters, as large overdensecompact regions of the Universe. Consequently,we expect our predictions based on spherical (orellipsoidal) collapse to be relevant for SO clustermass functions, such as in Tinker et al. (2008).Indeed, the deviations from universality show op-posite trend with mass for FoF clusters, with more(less) massive clusters having closer to (fartherfrom) universal behavior (Reed et al. 2007; Crocceet al. 2010; Bhattacharya et al. 2010). We notethat predictions from FoF and SO methods aredivergent, leading to substantive challenges to ourability to compare models with cluster observa-tions. Further discussion of this topic, however, isbeyond the scope of this article.

6. Conclusions and Future Prospects

We have presented a study of non-linear grav-itational collapse in different models of dark en-ergy/cosmic acceleration. In particular, we crit-ically examined the correlation between the lin-ear growth of fluctuations and the emergence andstatistics of collapsed objects (such as dark mat-ter haloes or galaxy clusters). First, we focusedon the collapse of spherical overdensities, and dis-

covered that they all have the same linear ovder-density (' 1.50), at ' 94% of the time of col-lapse/virialization, independent of the density orequation of state of dark energy. We then used asimple prescription to include the impact of el-lipsoidal collapse in our finding, and then usedthis result to propose a new universal mass func-tion for galaxy clusters/dark matter haloes. Oursemi-analytic predictions are consistent with theobserved evolution in mass function of haloes inN-body simulations.

Future work will include the adaptation of thisprescription to study models with dynamical equa-tions of state, such as quintessence or modifiedgravity models. A particularly challenging appli-cation would be to the gravitational aether/blackhole model (Prescod-Weinstein et al. 2009). Be-cause of the way dark energy is sourced in thegravitational aether model, there are unique tech-nical challenges associated with properly describ-ing structure formation in its presence. EarlyDark Energy (Grossi & Springel 2009) cosmolo-gies also provide an interesting example of aquintessence model where dark energy signa-tures in structure formation would be particularlystrong.)

Finally, we recognize that a more systematic ap-proach to the question of universal mass functionsshould be possible, and given the level of theoret-ical and observational activities in this field, it isunlikely that the present work provides the lastword on this subject. For example, the result willbe sensitive to the shape of the primordial powerspectrum (Bagla et al. 2009), and can be extendedto the conditional mass function (e.g., Neisteinet al. 2010). However, the novel (albeit trivial) ob-servation of this work is that measuring the clus-ter mass function will teach us about the entirehistory of linear growth. This is in contrast tomany previous cosmological applications of clustermass functions, which assume a one-to-one corre-spondence with the linear σ(M) at the time ofobservation. Similar to a multi-level archeologi-cal excavation, dark matter haloes can be thoughtof as artifacts of linear growth. As Fig. (6) andEq. (13) suggest, the number of smaller haloes(with larger σ(M)) can teach us about the earlyevolution of linear growth, while the bigger haloes(with smaller σ(M)) tell us about its more recenthistory.

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We thus speculate that this perspective caneventually lead to yet untapped information aboutthe nature and history of cosmic acceleration, es-pecially as the releases of several large scale clus-ter surveys such as Atacama Cosmology Tele-scope (ACT) (Menanteau et al. 2010), South PoleTelescope (SPT) (Andersson et al. 2010), Planck(Geisbusch & Hobson 2007), and Red SequenceCluster Sequence 2 (RCS2) (Yee et al. 2007) arenow looming on the horizon.

Acknowledgements

We would like to acknowledge helpful commentsand discussions with Robert Brandenberger, GilHolder, Brian McNamara and Ryan Morris. Dur-ing the time that this work was conducted, CPand NA were supported by Perimeter Institute forTheoretical Physics. Research at Perimeter Insti-tute is supported by the Government of Canadathrough Industry Canada and by the Province ofOntario through the Ministry of Research & In-novation. CP is supported by an appointment tothe NASA Postdoctoral Program at the GoddardSpace Flight Center, administered by Oak RidgeAssociated Universities through a contract withNASA.

10

A. Linear Perturbations

The linear perturbation theory that is used to describe structure formation can be derived via a fluidpicture. We use a Newtonian treatment because when density perturbations are small, the gravitationalpotential will be nonrelativistic (Peebles 1993). The standard equations of fluid dynamics are reviewed here.First, the continuity equation:

∂ρ

∂t+∇ · (ρ~v) = 0, (A1)

where ρ is the matter density and ~v is the fluid velocity. The Euler equation is:

∂~v

∂t+ (~v · ∇)~v = −∇P

ρ−∇φ, (A2)

where φ is the gravitational potential. Finally, the Poisson equation is:

∇2φ = 4πGNρ. (A3)

Introducing the comoving coordinates ~x = ~ra(t) , we can write the velocity in terms of the comoving coordinates

and the scale factor a(t):

~v =d~r

dt= a~x+ ~xa =

a

a~r + ~u(

~r

a, t) (A4)

where ~u = ~xa(t) is the peculiar velocity. Note that at constant ~r, d~rdt = 0 so a~x+a~x = 0, giving us aa~x = −∂~x∂t .

This leads us to the following relation:(∂ρ(~r, t)

∂t

)r

=

(∂ρ(~r, t)

∂t

)x

+∂~x

∂t· ∂ρ∂~x

=

(∂ρ(~r, t)

∂t

)x

− a

a~x · ∇xρ(~x, t). (A5)

We now wish to rewrite Equation A1 in terms of the comoving coordinates, which essentially meansreplacing the partial differential with the modified one from Equation A5. Moreover, what we are reallyinterested in is the development of relative deviations from the mean density, or the density contrast: δ =ρρ − 1. Thus, we write ρ = ρ(1 + δ) and assuming that ρ is the mean density of regular matter, we expect

ρ ∝ a−3. This gives:

0 =

(∂δ

∂t

)~x

+1

a∇ · (1 + δ)~u. (A6)

We make similar transformations for the Poisson and Euler equations. Next, we drop higher order terms (e.g.O(u2) or uδ). We differentiate the linearized continuity equation and take the divergence of the linearizedEuler equation. This gives us a differential equation that depends entirely on δ and not on ~u:

δ + 2a

aδ = 4πGNρδ. (A7)

The linear growth factor D(t) is defined as the growing solution for δ in this equation.

11

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