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-scale structures. Traditional approaches to studies of structure formation in the presence of dark energy or a modified gravity implement a modified Press-Schechter formalism, which relates the linear overdensities to the abundance of dark matter haloes at the same time. We critically examine the universality of the Press-Schechter formalism for different cosmologies, and show that the halo abundance 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 of best correlation for small haloes), and show that our results agree with deviations from modified Press-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 a single epoch will constrain the entire history of linear growth of cosmological of perturbations. Subject headings: cosmology: large-scale structure of universe 1. Introduction be an incredibly useful phenomenological method for distinguishing models of cosmic acceleration. An important part of the effort to explain cos- It is currently believed that large-scale struc- mic acceleration and the cosmological constant ture formation has its seeds in small quantum problem is testing proposed models in the context fluctuations in the early universe (e.g., Mukhanov of what are, at this stage, better-established phys- et al. (1992)). The current model for structure ical pictures. Structure formation could prove to formation is elegant in its fundamental simplicity. Random inhomogeneities, artifacts of cosmic in- 1Current address: Goddard Space Flight Center, Green- belt, MD, 20770, USA flation, create a runaway effect in which overdense [email protected] regions attract more matter, thus becoming more [email protected] dense and leading to galaxies, stars, and planets. 1 Better understanding of this process is inde- pose a fitting formula of the form: pendently an intriguing enterprize in the field of σ −a 2 cosmology. However, in this work we focus on f(σ) = A + 1 e−c/σ ; (3) b the relationship between the cosmic acceleration and structure formation. More specifically, differ- where (A; a; b; c) = (0:186; 1:47; 2:57; 1:19) give a ent cosmological pictures (cosmologies with differ- good fit to simulated haloes of overdensity ∆ = ing causes of acceleration, such as a cosmologi- 200, in a concordance ΛCDM cosmology at z = 0 cal constant, dark energy, or modifications of Ein- (Tinker et al. 2008) (see Fig. 1). While most of stein gravity) might have expansion histories that this work is based on fitting formulae to simulated are similar to one another but leave different im- mass functions, Sheth et al. (2001) argue that an prints on large-scale structures, and in particular approximate implementation of ellipsoidal collapse on galaxy clusters. Therefore, structure forma- can account for most of the deviations from the tion provides a unique testing ground for models PSF. of 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 to enhance the accuracy of this kind of work. The first step in this direction is to revisit how the Press-Schechter formalism (Press & Schechter 1974) (PSF) is used to predict the cluster mass function. Press & Schechter (1974) have argued that the number density of dark matter haloes (or galaxy clusters) of mass M is given by: dn(M; z) ρ¯ (z) @ ln σ−1(M; z) = f[σ(M; z)] m ; dM M @M (1) where σ2(M; z) is the variance of linear overden- sity in spherical regions of mass M at redshift z, whileρ ¯m(z) is the mean matter density of the uni- verse. For random gaussian initial conditions, f[σ] is given by: Fig. 1.| A comparison of Press-Schechter prediction for r the function f(σ) (dotted line; Eq. 2), with a parameterized 2 δ δ2 fit to the numerical simulations (solid line; Eq. 3). f [σ] = sc exp − sc ; (2) PS π σ 2σ2 However, a more pressing question for cosmo- where δsc (' 1:68 for most ΛCDM cosmologies) is logical applications is whether the function f(σ) is the spherical collapse threshold for linear overden- universal, or rather can vary for different cosmolo- sities (Gunn & Gott 1972). gies or cosmic acceleration models. In other words, While the PSF successfully predicts the broad could the same modified PSF be used to accurately features of the simulated cluster mass functions, predict halo abundance in cosmologies with differ- it proves too simplistic for detailed model com- ent cosmological parameters? While earlier stud- parisons required for precision cosmology. Conse- ies failed to find such dependence, Tinker et al. quently, several authors including Sheth & Tormen (2008) first noticed a systematic evolution of f(σ) (1999), Jenkins et al. (2001), Evrard et al. (2002), with redshift, implying a breakdown of universal- Warren et al. (2006), and Tinker et al. (2008) have ity at the 10 − 30% level (also see Reed et al. refined the function f(σ) to better match the mass (2007); Crocce et al. (2010); Bhattacharya et al. functions in N-body simulations. We shall refer to (2010)). Courtin et al. (2010) note that universal- these approaches as modified PSF. For example, ity is limited by the nonlinearity of structure for- Warren et al. (2006) and Tinker et al. (2008) pro- mation, and the cluster mass function shows some 2 redshift dependence at higher redshifts that can be Finally, in x 6, we conclude with an overview of (partially) understood in the context of spherical our results and a discussion of future prospects. collapse. However, the spherical collapse model falls short of explaining the 10 − 30% deviations 2. Λ & Non-linear Structure Formation from universality in all but the most massive clus- ters (see Fig. 7). In Appendix A, we derive the differential equa- tion that governs the growth of linear matter per- In this work, we contribute to the effort to bet- turbations. We used a familiar fluid dynamics pic- ter understand the role and limits of universal- ture to do so. Here we derive the full non-linear ity in the cluster mass function by introducing a equation for spherical overdensities using only cos- new parameter that appears to be universal across mological considerations. It should be noted that cosmological models.1 In particular, the modi- this particular form of the non-linear equation is fied PSF relies on σ(M), the root-mean-square only strictly valid for ΛCDM cosmologies, where of linear density fluctuations at the time of ob- the inside of a spherical top-hat overdensity can be servation, when in reality, observed clusters are considered as a separate universe. More complex very non-linear objects with overdensities exceed- models such as dark energy models with different ing 200. We thus seek to find a universal time values of w require additional considerations which in the past when we could make a connection be- will be discussed below. tween the nonlinear structures that we observe in the present and the linear structures that existed We consider a physical picture in which struc- at that time, since all structures go through a lin- ture formation arises due to a uniformly positive ear phase. Our basic strategy is to find the time in spherical perturbation away from an average mat- the past at which the linear density of the struc- ter density, i.e. a top-hat matter overdensity. This tures that collapse today show minimum disper- scenario is similar to considering two cosmologies sion as we vary cosmologies. This is illustrated in with two distinct scale factors: one for the outer Fig. (2). universe and another for the overdense region. Of course, we are interested in a scenario where a In x 2, we present the complete nonlinear differ- dark energy component similar to a cosmological ential equation that governs the growth of matter constant is at play, so we will assume the presence perturbations in spherical overdense regions in the of one as part of our base model. presence of a cosmological constant. For the external universe, we write the Fried- In x 3, we describe a numerical method which mann equation with zero curvature: we developed in order to solve both the nonlin- ear and linear structure formation equations in a_ 2 8πG o = N (ρ + ρ ) = H2: (4) the presence of a cosmological constant, or dark a 3 Λ energy with a constant equation of state. o In x 4, we discuss the implications of our nu- We can assume ρ / a−3 for ordinary matter, while merical study, i.e. we find a universal fraction of ρΛ = const.