Lawrence M. Widrow, Pascal J. Elahi, Robert J. Thacker, Mark Richardson, Evan Scannapieco, Power spectrum for the small-scale Universe, Monthly Notices of the Royal Astronomical Society, Volume 397, Issue 3, August 2009, Pages 1275–1285, https://doi.org/10.1111/j.1365-2966.2009.15075.x
Navbar Search Filter Mobile Enter search term Search Navbar Search Filter Enter search term SearchThe first objects to arise in a cold dark matter (CDM) universe present a daunting challenge for models of structure formation. In the ultra small-scale limit, CDM structures form nearly simultaneously across a wide range of scales. Hierarchical clustering no longer provides a guiding principle for theoretical analyses and the computation time required to carry out credible simulations becomes prohibitively high. To gain insight into this problem, we perform high-resolution (N= 720 3 –1584 3 ) simulations of an Einstein–de Sitter cosmology where the initial power spectrum is P(k) ∝k n , with −2.5 ≤n≤− 1. Self-similar scaling is established for n=−1 and −2 more convincingly than in previous, lower resolution simulations and for the first time, self-similar scaling is established for an n=−2.25 simulation. However, finite box-size effects induce departures from self-similar scaling in our n=−2.5 simulation. We compare our results with the predictions for the power spectrum from (one-loop) perturbation theory and demonstrate that the renormalization group approach suggested by McDonald improves perturbation theory's ability to predict the power spectrum in the quasi-linear regime. In the non-linear regime, our power spectra differ significantly from the widely used fitting formulae of Peacock & Dodds and Smith et al. and a new fitting formula is presented. Implications of our results for the stable clustering hypothesis versus halo model debate are discussed. Our power spectra are inconsistent with predictions of the stable clustering hypothesis in the high-k limit and lend credence to the halo model. Nevertheless, the fitting formula advocated in this paper is purely empirical and not derived from a specific formulation of the halo model.
In the standard cosmological paradigm, present-day structures arise from small-amplitude density perturbations which have their origin in the very early Universe. These primordial perturbations are presumed to form a Gaussian random field whose statistical properties are described entirely by the power spectrum, P(k). While higher order statistics are required to describe the density field once non-linearities develop, the power spectrum remains central to our understanding of structure formation.
During the radiation-dominated phase of the standard cold dark matter (CDM) scenario, the power spectrum evolves from its primordial, approximately power-law form, P(k) ∝k to one in which its logarithmic slope, neff≡ d ln P(k)/d ln k, decreases from neff≃ 1 on large scales to neff≃−3 on small scales. 1 At the start of the matter-dominated phase, which signals the beginning of structure formation, the dimensionless power spectrum, Δ 2 (k) ∝k 3 P(k), decreases monotonically with scale. The implication is that structure forms from the bottom up. Hierarchical clustering, as this process has come to be known, is the central idea in our understanding of structure formation. Hierarchical clustering also explains why cosmological N-body simulations are able to provide a reasonable facsimile of true cosmological evolution; with enough dynamic range, simulations are able to follow the development of virialized, highly non-linear systems on small scales while properly modelling the large-scale tidal fields that shape them. However, the dynamic-range requirement becomes increasingly difficult to achieve as neff→−3 since, in this limit, Δ 2 becomes independent of k and structures collapse nearly simultaneously across a wide range of scales. Put another way, as neff→−3, the infrared divergence of the power spectrum becomes increasingly problematic for numerical (as well as theoretical) studies.
Interest in the small-scale limit of the CDM hierarchy was prompted by the realization that dark matter haloes have a wealth of substructure ( Klypin 1999; Moore et al. 1999a) and that this substructure may have important implications for both direct and indirect dark matter detection experiments (see, for example, Stiff, Widrow & Frieman 2001; Diemand, Kuhlen & Madau 2007; Kamionkowski & Koushiappas 2008; Kuhlen, Diemand & Madau 2008). High-resolution simulations suggest that the subhalo mass function extends down to the dark matter free-streaming scale with approximately constant mass in substructure per logarithmic mass interval. These simulations probe structures which form from an initial power spectrum with −3 < neff < −2. For example, in the simulation of the first CDM objects by Diemand et al. (2007), neff≃−2.8. With such extreme spectra, care must be taken in order to insure that the results are not corrupted by finite-volume effects. This issue, as it relates to the halo and subhalo mass functions, is discussed in Bagla & Prasad (2006) and Power & Knebe (2006) as well as in the companion to this paper, Elahi et al. (2009).
In this paper, we provide insight into the small-scale limit of CDM by focusing on scale-free cosmologies, that is Einstein–de Sitter cosmologies where the initial power spectrum is a power-law function of k, P(k) ∝k n . The guiding principle for understanding structure formation in these models is self-similar scaling which implies that the functional form of the dimensionless power spectrum is time independent, up to a rescaling of the wavenumber k (see below). Self-similar scaling provides a diagnostic test of whether a simulation has sufficient dynamic range ( Jain & Bertschinger 1998). Contact with the standard ΛCDM cosmology is made by treating parameter n as a proxy for scale: n≃−1.8 corresponds to cluster scales and n≃−2.2 to galactic scales. The limit n→−3 corresponds to the bottom of the CDM hierarchy.
We perform N-body simulations with n=−1, − 2, −2.25 and −2.5, and compare our results directly with theoretical models. The computation costs to conduct credible simulations with n < −2.5 are prohibitively high (see below) and even our n=−2.5 results must be considered suspect because of finite box-size effects. We compare our results for the power spectra in the quasi-linear regime with predictions from one-loop perturbation theory (PT) and demonstrate that for n=−2 and −2.25, the agreement is vastly improved if one implements the renormalization group (RG) approach suggested by McDonald (2007).
To predict the full non-linear power spectrum, one must resort to semi-analytic models such as the ones described in Hamilton et al. (1991) and Peacock & Dodds (1996). These models are based on the stable clustering hypothesis ( Peebles 1974; Davis & Peebles 1977) which holds that gravitationally bound systems decouple from the rest of the Universe once they collapse. An alternative, known as the ‘halo model’ ( Ma & Fry 2000; Peacock & Smith 2000; Seljak 2000), allows for the continual accretion of mass on to existing haloes. The density field is treated as a distribution of mass concentrations, each characterized by a density profile. The power spectrum then involves the convolution of this density profile with the halo mass function. Simulations by Smith et al. (2003) of structure formation in a number of scale-free cosmologies demonstrate a clear departure from the stable clustering hypothesis and appear to support the halo model, at least qualitatively.
Peacock & Dodds (1996) and Smith et al. (2003) provide fitting formulae for non-linear power spectra. Formally, these formulae apply to all initial power spectra with n > −3 though they are calibrated using simulations with n≥−2. One goal of this paper is to provide an alternative fitting formula which applies when n≤−2.
The overall layout of this paper is as follows: in Section 2, we present background material including a discussion of self-similar scaling, PT and semi-analytic models. We describe our simulations in Section 3 and our results in Section 4. We also provide a new and improved fitting formula for the non-linear power spectra. In Section 5, we discuss the implications of our results for the halo model and stable clustering hypothesis. We conclude, in Section 6, with a summary and a discussion of directions for future investigations.
In keeping with standard definitions (e.g. Peacock 1999), we express real-space density perturbations as deviations from the mean background density, ρbg(t), and then construct the k-space representation as follows: