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THE NONLINEAR MATTER POWER SPECTRUM1 Zhaoming Ma Department of Astronomy and Astrophysics and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637; [email protected] Received 2006 October 6; accepted 2007 April 30

ABSTRACT We modify the public PM code developed by Anatoly Klypin and Jon Holtzman to simulate cosmologies with arbitrary initial power spectra and the equation of state of dark energy. With this tool in hand, we perform the following studies on the matter power spectrum. With an artificial sharp peak at k 0:2 h Mpc1 in the initial power spectrum, we find that the position of the peak is not shifted by nonlinear evolution. An upper limit of the shift at the level of 0.02% is achieved by fitting the power spectrum local to the peak using a power law plus a Gaussian. We also find that the existence of a peak in the linear power spectrum would boost the nonlinear power at all scales evenly. This is contrary to what the HKLM scaling relation predicts, but roughly consistent with that of the halo model. We construct dark energy models with the same linear power spectra today but different linear growth histories. We demonstrate that their nonlinear power spectra differ at the level of the maximum deviation of the corresponding linear power spectra in the past. Similarly, two constructed dark energy models with the same growth histories result in consistent nonlinear power spectra. This is hinting, not a proof, that the linear power spectrum together with linear growth history uniquely determine the nonlinear power spectrum. Based on these results, we propose that linear growth history be included in the next-generation fitting formulae of the nonlinear power spectrum. Finally, we comment on the precision of existing fitting formulae when applied to dark energy models parametrized by w0 and wa. Subject headings:g large-scale structure of universe Online material: color figuresONLINE MATERIALCOLOR FIGURES

1. INTRODUCTION theory matter power spectrum with high accuracy (e.g., Seljak For cosmologists, the clustering property of matter is a gold et al. 2003). However, we measure the nonlinear matter or galaxy power spectrum from large-scale structure surveys. Our under- mine. Although we are at the beginning stage of the operation, standing of the connection between linear and nonlinear power we have already been rewarded dearly: combined with WMAP spectrum is improving rapidly along with our ability to simulate (Spergel et al. 2003), SDSS data constrain cosmological param- the complex processes involved. eters to reasonably high precision (for example, the matter density of the universe and the amplitude of the matter perturbation m 1.1. Baryon Acoustic Oscillations 8 to 10%; Tegmark et al. 2004; Abazajian et al. 2005; Seljak et al. 2005). Fosalba et al. (2003) and Scranton et al. (2003) found The BAO features in the late-time matter clustering, character- direct evidence of the mysterious dark energy through the ISW ized by a single peak in the correlation function and oscillations effect using data from SDSS. Also using SDSS data, the baryon in the matter power spectrum, are imprints of the acoustic oscil- acoustic oscillation (BAO) feature is used to probe the expansion lations in the early universe that produce the peaks and troughs in history of the universe and cosmological distances (Eisenstein the cosmic microwave background (CMB) angular power spec- et al. 2005; Cole et al. 2005). Weak-lensing surveys (Jarvis et al. trum (Peebles & Yu1970; Sunyaev & Zel’dovich 1970). Although 2005) made the first attempt to constrain the time evolution of dark the BAO features in the matter power spectrum are much weaker energy. Moreover, numerous ambitious future surveys of large- compared to those of the CMB because the dominating dark scale structure are proposed, such as the Dark Energy Survey2 matter did not participate in the acoustic oscillations, these fea- (DES), PanSTARRS,3 Supernova/Acceleration Probe4 (SNAP; tures can be measured with high accuracy with sufficiently large Aldering et al. 2004), and Large Synoptic Survey Telescope5 surveys. Indeed, the first detection was made in the Sloan Digital (LSST). From these surveys, we want to answer questions like the Sky Survey (SDSS) in 2005 (Eisenstein et al. 2005; Cole et al. following: Is our concordance cosmology model correct? What is 2005). the nature of the mysterious dark matter and dark energy? Do we With the large galaxy surveys proposed, e.g., DES, PANSTARRS, need to modify gravity theory? Is the universe flat or curved? SNAP, and LSST, BAO is emerging as an important method to On the other hand, we have tremendous success in theoretical measure the expansion history of the universe and cosmological modeling of structure formation in our universe. Starting with a distances (Eisenstein et al. 1998; Cooray et al. 2001; Blake & handful of cosmological parameters, we can compute the linear Glazebrook 2003; Hu & Haiman 2003; Seo & Eisenstein 2003; Linder 2003, 2005; Matsubara et al. 2004; Amendola et al. 2005; Blake & Bridle 2005; Glazebrook & Blake 2005; Dolney et al. 1 Presented as a dissertation to the Department of Astronomy and Astrophysics, 2006). The main theoretical uncertainties, including galaxy bias, University of Chicago, in partial fulfillment of the requirements for the Ph.D. degree. nonlinear evolution, and redshift space distortions, are investi- 2 See http://www.darkenergysurvey.org. 3 See http://pan-starrs.ifa.hawaii.edu. gated in detail by White (2005), Eisenstein et al. (2006a, 2006b), 4 See http://snap.lbl.gov. and Huff et al. (2006). Nonlinear evolution smears BAO peaks due 5 See http://www.lsst.org. to mode-mode coupling. This effect makes BAO signal weaker 887 888 MA Vol. 665 and harder to detect. An even more worrying effect is that non- and the two-halo term is the contribution from two particles that linear evolution could potentially shift the peaks that would bias belong to two different halos, the distance measurement from which the cosmological parameters are inferred. Z 2 2h m Eisenstein et al. (2006b) made an order-of-magnitude estima- P ðÞ¼k dm nðÞ m bmðÞjukðÞjjm PLðÞk : ð7Þ tion of the amount of shift of BAO peaks due to nonlinear evolu- ¯ tion. They use a spherical collapse model to estimate how much the overdensity within 150 Mpc has grown if the center were Here m is the mass of the halo, ¯ is the mean density of the uni- overdense, and the change in radius is one-third of that in the verse, u(kjm) is the Fourier transform of the profile of a halo overdensity. Their conclusion is that the change in scale is on the (e.g., NFW profile; Navarro et al. 1997) with mass m, and n(m)is 4 order of O(10 ). In this work, we use N-body simulations to test the halo mass function, which has the following form (Press & how much the peaks are shifted due to nonlinear evolution. Schechter 1974; Jenkins et al. 2001; Sheth & Tormen 1999): 1.2. Fitting Formulae of the Nonlinear Matter Power Spectrum ¯ d nmðÞ¼ f ðÞ ; ð8Þ Calibrated by simulations, fitting formulae of the nonlinear m dm power spectrum have been developed (Peacock & Dodds 1996; Smith et al. 2003; McDonald et al. 2005). Their precisions, on the where ¼ c /(m), with (m) the rms of the linear density field order of 10%–20%, are sufficient for most of the current appli- smoothed by a spherical top-hat filter W(k; m), cations. Future surveys would have statistical uncertainties so low Z that fitting formulae with improved accuracy are highly desired 1 2 2 2 (Huterer & Takada 2005). As a step toward fulfilling this goal, we ðÞ¼m d ln kÁLðÞk W ðÞk; m : ð9Þ test the assumptions that go into the aforementioned fitting formu- 0 lae and propose the necessary ingredients to build a more accurate one. First, let us briefly review the existing fitting formulae. Note that adding a sharp spike in the matter power spectrum In a series of papers (Hamilton et al. 1991; Peacock & Dodds would only modify (m) below a mass threshold that is deter- 1994; Jain et al. 1995; Peacock & Dodds 1996), the first fitting mined by the location of the spike. The modification to the mass formula of the nonlinear power spectrum is worked out. The foun- function n(m) would also only occur below the same mass thresh- 2 dation of the fitting formula is the so-called HKLM relation, old. This could be easily shown by adding a -function to ÁL(k) which has two parts. The first part is to connect the nonlinear in equation (9). scale, r NL, to the linear scale, rL , by the volume-averaged corre- Finally, based on a fusion of the halo model and an HKLM lation function ¯(r), scaling, Smith et al. (2003) provided an empirical fitting formula with improved performance (rms error better than 7% is claimed). 1=3 rL ¼ 1 þ ¯NLðÞr NL r NL: ð1Þ In detail, they replaced the two-halo term with a Peacock & Dodds (1996)–type fitting form with an exponential cutoff at nonlinear The reasoning behind this is pair conservation (Peebles 1980, k to deal with the translinear regime better and a fitting form to p. 266). The second part of the HKLM procedure is to conjecture the one-halo term. Again, the fitting forms are calibrated using that the nonlinear correlation function is a universal function of ÃCDM simulation. the linear one, To summarize, fitting formulae of the nonlinear matter power spectrum have been developed for ÃCDM cosmology with ¯NLðÞ¼r NL f NL ¯LðÞrL : ð2Þ precisions on the order of 10%–20%. The foundations for these fitting formulae are the HKLM scaling relation and halo model. If one assumes that ¯(r) is equivalent to matter power spectrum A less obvious assumption behind all these fitting formulae is at an effective wavenumber k, then we have the power spectrum that the nonlinear power spectrum is determined by the linear version of the HKLM relation, power spectrum at the same epoch, i.e., the history of the linear power spectrum does not affect the nonlinear one. 2 1=3 kL ¼ 1 þ Á NLðÞk NL k NL; ð3Þ In order to achieve the 1% accuracy required by future surveys (Huterer & Takada 2005), and with more general parameteri- 2 2 zation of dark energy models, what approach shall we take to ÁNLðÞ¼k NL; t fNL ÁLðÞkL; t : ð4Þ build a new fitting formula of the nonlinear power spectrum? To It is commonly referred to as the Peacock & Dodds fitting for- address this issue, we test the assumptions that go into the exist- mula, and f NL is calibrated using N-body simulation assuming ing fitting formulae using carefully constructed numerical exper- ÃCDM cosmology. iments. In the end we will decide which assumptions should be Another approach of calculating the nonlinear power spec- abandoned, kept, and further included. trum is to use the halo model (for a review see Sheth & Cooray Although all the existing fitting formulae are calibrated for 2002), which splits up the contribution to the power spectrum ÃCDM cosmology, it has been a common practice to apply them into one-halo and two-halo terms, to dark energy models in general. We quantify how well this procedure works. 1h 2h PNLðÞ¼k P ðÞþk P ðÞk : ð5Þ The rest of the paper is organized as follows: In x 2 we briefly introduce the particle-mesh (PM) simulations. We demonstrate the The one-halo term P1h(k) is the contribution from two particles behavior of the peak in the initial power spectrum in x 3 and dis- that reside in the same halo, cuss what we could learn from it. We test basic building blocks of Z the fitting formulae in x 4. In x 5 we evaluate the performance of m 2 P1hðÞ¼k dm nðÞ m jukðÞjjm 2; ð6Þ the fitting formulae when applied to general dark energy models. ¯ Finally, we summarize our findings. No. 2, 2007 NONLINEAR MATTER POWER SPECTRUM 889

2. THE PM SIMULATION Equations (13) and (14), in terms of these dimensionless variables, become The public PM code developed by A. Klypin and J. Holtzman is used for this work. The source code and manual are available 3 m online.6 The code assumes ÃCDM cosmology. In order to sim- 9˜ 2˜ ¼ ðÞ˜ 1 ; ð17Þ 2 a ulate cosmology with an arbitrary equation of state of dark energy, w(z), and/or an arbitrary initial power spectrum, we have dp˜ dx˜ p˜ ¼F(a)9˜ ;˜ ¼ F(a) ; ð18Þ modiﬁed the code appropriately. In this section we lay out the da da a 2 basic formalism used in the simulation and the tests we have performed. where

FaðÞ¼H0=a˙ 2.1. Formalism pffiffiffi a We use r, t, and v as position, time, and velocity variables in ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÈÉR : ð19Þ 3 a 0 0 physical space and x, , and u as the corresponding variables in m þ Ãa exp 3 a¼1½1 þ waðÞd ln a comoving space. In physical space the Poisson equation is Here w(a) is the equation of state of dark energy that has been generalized to an arbitrary function of a. 9 2È ¼ 4G; ð10Þ r 2.2. Brief Summary of the PM Code where È is the gravitational potential, G is the gravitational con- The PM code solves equations (17) and (18) in four main steps: stant, and is the density of the universe including all constit- 1. Set up the initial positions and velocities of the dark matter uencies. If we define a new potential as particles. 2. Solve the Poisson equation using the density field esti- 1 a¨ È þ x 2; ð11Þ mated with current particle positions. 2 a 3. Advance velocities using the new potential. 4. Update particle positions using new velocities. where overdots represent a derivative with respect to and a is the scale factor, then, in comoving coordinates, the Poisson equa- The initial condition is set up using the Zel’dovich approxi- tion becomes mation (Zel’dovich 1970; Crocce et al. 2006). The density as- signment is done using cloud-in-cell (CIC) interpolation, which a¨ is the method that linearly interpolates particle mass to its eight 9 2 ¼ 4Ga 2 þ 3 : ð12Þ x a neighboring cells. The Poisson equation is solved using the fast Fourier transform (FFT), and the particle positions and veloci- Using the Fredmann equation to substitute the a¨ /a term, we have ties are solved using the leapfrog integration scheme. For further details of the implementation of these schemes, please check out the excellent write-up by A. Kravtsov online.7 9 2 2 2 x ¼ 4Ga ðÞ ¯ 4Ga ; ð13Þ 2.3. Code Testing where ¯ is the mean density of the universe. The corresponding We set up a few test runs to determine the parameters to use in equations of motion in comoving space are the simulations. These include the number of particles Np , number of grid cells Ng , the starting redshift zini, evolving time step da, and the box size L . p mu ¼ m˙x; p˙ ¼ mu˙ ¼m9x; ð14Þ box Ideally, the larger Np and Ng are, the better. In practice, they are limited by the computation resources. For a standard run used in with m the mass of the particle. 3 3 this work, Np ¼ 256 and Ng ¼ 512 , one time step including the The PM code solves the Poisson equation (13) and the equa- power spectrum evaluation takes about 2.5–6 minutes on an eas- tions of motion (eq. [14]). It is convenient to make the variables ily accessible Pentium box. Approximately 500 MB of memory dimensionless by choosing suitable units. A. Klypin’s PM code 3 3 is required. For a bigger run with Np ¼ 512 and Ng ¼ 1024 , defines the code variables (denoted by a tilde) as which is mainly used to check the uncertainties of the standard runs, one step takes about 8 times longer and memory usage is on p˜ ˜ x ¼ xx˜ ; p=m ¼ p ;¼ ˜ ;¼ ; ð15Þ the level of 4 GB. 0 a 0 0 a 3 0 Figure 1 shows the differences of the matter power spectrum at z ¼ 0 due to different initial redshift of the simulation, zini. with the units Simulations with lower zini underestimate the nonlinear power spectrum. Relative to zini ¼ 100, zini ¼ 50 and zini ¼ 30 have 1=3 power deficits of 1.5% and 3%, respectively. These are in good x0 ¼ Lbox=N ; p0 ¼ x 0H0; g agreement with the results from Crocce et al. (2006). For most of 2 our applications, we use the ratio of the power spectrum. Percent 2 3H0 0 ¼ ðÞx0H0 ;0 ¼ m: ð16Þ level power deficit is not a concern for us. So we choose zini ¼ 50 8G as our standard initial redshift unless stated otherwise. As to the

6 See http://charon.nmsu.edu/~aklypin/PM/pmcode/pmcode.html. 7 See http://astro.uchicago.edu/~andrey/Talks/PM/pm.pdf. 890 MA Vol. 665

Fig. 1.—Effect of initial redshift of the simulation on the matter power spectrum at z ¼ 0. The initial redshift zini and the step size da are listed in the legend. 1 Fig. The Nyquist frequency is k Ny ¼ 4 h Mpc . All of our simulations use zini ¼ 50 3.—Comparison of the matter power spectrum, at z ¼ 0, from the unless stated otherwise. [See the electronic edition of the Journal for a color simulation with these from fitting formulae of Smith et al. (2003) and Peacock & version of this figure.] Dodds (1996). The top panel shows the matter power spectra P(k), and the bottom panel shows the fractional difference of P(k) between the higher resolu- 3 1 tion simulation with Np ¼ 512 and the rest. The simulation box is 400 h Mpc, 1 step size, Figure 2 shows that, with zini ¼ 50, the power spectrum and the corresponding Nyquist frequencies are k Ny ¼ 4 and 8 h Mpc for sim- 3 3 at z ¼ 0 differs by less than 0.3% between da ¼ 0:004 and 0.001. ulations with Np ¼ 256 and 512 , respectively. The initial power spectrum Choosing da ¼ 0:004 as our standard step size, a full standard run without BAO peaks is used for the Peacock & Dodds fitting formula. [See the electronic edition of the Journal for a color version of this figure.] takes anywhere between 10 hr and 1 day. The box size of the simulation is more determined by the k range that a particular application requires than anything else. that is resolved by the simulation. The lowest mode is 2/Lbox. 1=3 The Nyquist frequency k Ny Ng /Lbox is the highest mode Because of aliasing, not all of the k-modes in this range are faith- fully calculated. To determine the precision of the power spectrum, 3 we use these from simulations with higher resolution (Np ¼ 512 3 and Ng ¼ 1024 ) with which we compare the results from our 3 3 standard simulation (Np ¼ 256 and Ng ¼ 512 ). With Lbox ¼ 400 h1 Mpc, Figure 3 shows that our standard run agrees with the higher resolution run perfectly for k < 0:6 h Mpc1 and the agreement worsens to about 10% at k ¼ 1:0 h Mpc1. For all of our applications, except for the test of the shifting of peaks in the power spectrum, we use the ratio of power spectra instead of the absolute power. This practice takes care of the sample variance. Figure 4 shows the uncertainty levels of the ratio. Again, we compare our standard run with the one that has 3 3 higher resolution (Np ¼ 512 and Ng ¼ 1024 ). For each simulation setup, we run two cosmological models and take the ratio of their power spectra. One cosmological model is ÃCDM [w0 w(z ¼ 0) ¼1andwa dw/daja¼1 ¼ 0], and the other is the w0 ¼1andwa ¼ 0:3 model. According to Figure 4, the ratios of the power spectra in simulations of different resolutions agree to better than 0.5% all the way to k 3 h Mpc1. 3. THE FATE OF A PEAK IN THE INITIAL POWER SPECTRUM 3.1. Setting Up the Experiment To address the issue of whether nonlinear evolution of the Fig. 2.—Effect of the step size of the simulation on the matter power spec- power spectrum would shift the position of BAO peaks, we con- trum at z ¼ 0. The initial redshift zini ¼ 50 and the step sizes da are listed in the 1 struct the following numerical experiment. We put an artificial legend. The Nyquist frequency is k Ny ¼ 4 h Mpc . All of our simulations use da ¼ 0:004 unless stated otherwise. [See the electronic edition of the Journal sharp peak in the otherwise smooth initial matter power spectrum for a color version of this figure.] and evolve it using a PM simulation (Shandarin & Melott 1990; No. 2, 2007 NONLINEAR MATTER POWER SPECTRUM 891

Fig. 5.—Testing the shift of peaks in the initial power spectrum. The as- Fig. 4.—Ratio of the matter power spectra of the wa ¼ 0:3 model to that of terisks show the linear power spectrum at z ¼ 0. The nonlinear power spectrum the ÃCDM model at z ¼ 0. Different lines are from simulations with differ- at z ¼ 0 is shown by plus signs. While the peak is being smeared by nonlinear ent resolutions, which are labeled in the legend. The Nyquist frequencies are 1 3 3 evolution, the position of the peak is not shifted at all. [See the electronic edition k Ny ¼ 4and8h Mpc for simulations with Np ¼ 256 and 512 , respectively. of the Journal for a color version of this ﬁgure.] [See the electronic edition of the Journal for a color version of this ﬁgure.]

Bagla & Padmanabhan [1997] used similar setups to test the between the second left (k ¼ 0:198 h Mpc1) and second right transfer of power among different scales). The initial peak is (k ¼ 0:211 h Mpc1) points relative to the highest point of the located at k 0:2 h Mpc1 with an FWHM ¼ 0:013 h Mpc1, peak in Figure 5. We also know that the maximum of the input and the maximum at k ¼ 0:2051 h Mpc1. At the maximum, peak is at k ¼ 0:204 h Mpc1, so if the peak has shifted, the P(k) is 10 times what it would be without the peak. The location shift is less than (0:211 0:204)/0:204 ¼ 3:2%. of the peak is at about where the third peak of the BAO sits and One reasonable assumption we can make is that there is only deep in the nonlinear regime today. The reason to use a sharp one peak. This narrows down the maximum to somewhere be- peak is to make it easier to detect peak shifting without using any tween the first left (k ¼ 0:201 h Mpc1) and first right (k ¼ sophisticated analysis. To set the peak amplitude high is to over- 0:207 h Mpc1) points. With the input maximum known, the comethesamplevarianceofthesimulation, which is at least shift of the peak is less than 1.6%. at the level of the BAO peaks (10%). Techniques have been We can certainly do better than these crude estimates. One developed to get around sample variance and make BAO fea- simple method we use is to fit the power spectrum local to the tures easily seen in simulations. For example, the Millennium peak with a power law plus a Gaussian. The fits for the power Simulation group (Springel et al. 2005) corrects the power spec- spectra at z ¼ 30 and 0 are shown in Figures 6 and 7, respec- trum of the actual realization of the initial fluctuations in their tively. The centers of the peak are found to be at 0.205092 and simulation to the expected input power and applies these scaling 0.205133 h Mpc1, respectively. They agree to 0.02%. This factors at all other times. It is not clear that these scaling factors estimation is approaching the theoretical limit, and for any prac- are not affected by nonlinear evolution. To test the physics be- tical purpose this is equivalent to no shift. hind it, our approach is much cleaner. These simple arguments have not taken errors into account. The requirement for the simulation is such that k ¼ 0:2 h Mpc1 Sample variance does not apply here because the initial peak and is well resolved and the peak is sampled with at least a few points the final peak are from the same realization. The Poisson errors in k space. With these considerations, we use a simulation with are on the level of 0.4%. This is too small to change the ordering 1 3 box size of 2 h Gpc, number of particles Np ¼ 512 , and num- of the points in the peak, so the first two estimations are not af- 3 6 berofgridcellsNg ¼ 1024 . The corresponding Nyquist fre- fected. The last estimation would have an uncertainty of 10 , quency is 1.6 h Mpc1 and the peak is in the well-resolved scale. which is too small to be relevant. With this simulation setup, the peak is sampled with five points in k space. 3.3. Other Effects that Could Shift the BAO Peaks There are effects other than nonlinear collapse that could shift 3.2. The Shifting of the Peak the BAO peaks. The experiment above is dark matter simulation, Figure 5 shows the position of the peak at z ¼ 30 and 0. Table 1 but the actual tracers of BAO peaks are galaxies that we know are lists the positions k and the corresponding P(k) for the points biased tracers of dark matter. If galaxy bias is scale dependent, it within the peak. could potentially shift the peaks. As shown in White (2005) and The most conservative estimate of how much the peak shifts is Smith et al. (2006), this is indeed the case and the scale depen- to assume that at z ¼ 0 the maximum of the peak is somewhere dency should be carefully calibrated. 892 MA Vol. 665

TABLE 1 Peak Position and Height

Parameter Second Lefta First Left Peak First Right Second Right

k b...... 0.198 0.201 0.204 0.207 0.211 P(k; z ¼ 30) ...... 5.442 17.75 29.66 26.11 11.53 P(k; z ¼ 0) ...... 2884. 4664. 6265. 5616. 3632. Smearingc ...... 0.96 0.48 0.38 0.39 0.57 iVolumed...... 48870 51554 54090 54314 56910

a Relative to the highest point of the peak. b Units: [k]=h Mpc1;[P(k)] = h3 Mpc3. c 2 2 Smearing is P(k; z ¼ 0)D þ(z ¼ 30)/½P(k; z ¼ 30)Dþ(z ¼ 0),whereDþ(z) is the linear growth function. d The number of cells in k space that contribute to the power spectrum. It determines the Poisson noise.

Another worry is that BAO peaks sit on top of the smooth The result is shown in Figure 8. underlying power spectrum whose slope changes due to nonlinear We see that the peak is washed out as the system evolves and evolution might shift the peaks. As pointed out in Eisenstein et al. the nonlinear power is boosted at all scales smaller than the scale (2006b), this need not be the case. The argument is that one can where the peak is located. The amount of boost in power shows easily make a template of the smooth underlying power spec- no scale dependency. trum against which to achieve an unbiased measurement. The mapping between linear and nonlinear scale proposed by Hamilton et al. (1991) and adopted by Peacock & Dodds (1996) 3.4. Effects of the Peak on Nonlinear Power Spectrum would have predicted that the peak is mapped to a nonlinear scale To see the effects of the peak on the nonlinear power spectrum, k NL that increases as the nonlinear power grows. We do not see we compare the nonlinear power spectra evolved from initial that kind of mapping here. On the other hand, the result is similar power spectra with and without an artificial peak. The smooth to what the halo model approach would predict. In the halo parts of these initial power spectra are set to have the same am- model, the one-halo term (see eq. [6]) is determined by the halo plitude. Note that this is different from requiring the same 8, profile and the halo mass function n(m). The halo profile has which would allocate more power to the no-peak case on scales nothing to do with whether there is a peak in the matter power outside the peak, which would certainly make things more com- spectrum or not. plicated and less clean. The simulation we use for this test is with An extra peak in the linear power spectrum would increase 3 3 1 Np ¼ 256 , Ng ¼ 512 , and Lbox ¼ 400 h Mpc, and the corre- the halo mass function below a certain mass threshold determined 1 sponding Nyquist frequency is k Ny ¼ 4 h Mpc . We start the by the position of the peak. As a result, the nonlinear power is simulation at z ¼ 30, and the step size is da ¼ 0:004. The higher boosted on small scales, which is close to what we observe in this Nyquist frequency is the main reason that we use this run instead test. In detail, it does not agree with the halo model prediction of the one used for the peak position test. perfectly. As shown in Figure 9, the halo model predicts that the fractional boost in power due to the existence of a peak has some

Fig. 6.—Fitting the z ¼ 30 power spectrum near the peak with a power law Fig. 7.—Fitting the z ¼ 0 power spectrum near the peak with a power law plus a Gaussian. The peak position k peak and width k are labeled in the legend. plus a Gaussian. The peak position k peak and width k are labeled in the legend. [See the electronic edition of the Journal for a color version of this ﬁgure.] [See the electronic edition of the Journal for a color version of this ﬁgure.] No. 2, 2007 NONLINEAR MATTER POWER SPECTRUM 893

Fig. 8.—Comparison of power spectra in simulations with or without a peak Fig. 9.—Comparison of power spectra predicted by the halo model for the in the initial power spectrum. The comparison is done at a few redshifts to show cases with or without a peak in the linear power spectrum. The comparison is the evolution of the peak and the nonlinear power spectrum. It is clear that done at z ¼ 0. The effect of the peak on the nonlinear power spectrum is similar Peacock & Dodds (1996)–like mapping between linear and nonlinear k does not to that from simulation ( Fig. 8), but different in details. [See the electronic edi- exist. Instead, the effect of the peak is close to what the halo model would predict. tion of the Journal for a color version of this ﬁgure.] [See the electronic edition of the Journal for a color version of this ﬁgure.]

Operationally the perturbation in wi is achieved by modulating weak scale dependence that is slightly different from what we see the amplitude of the form in the simulation. Since the addition of a spike in the power spectrum only changes the mass function at the low-mass end where u(kjm) is scale independent at k equal to a few tenths h Mpc1,the ðÞz zi Sw / exp 2 ; ð21Þ one-halo term acts like a shot-noise term. The fractional dif- 2z ference of the power spectra is scale dependent due to the addition of this shot-noise–like term to the power spectrum. We do not where ¼ 8 is used to make the shapelet sharp and ¼ 0:05 is yet understand what accounts for this difference between the halo z used (2z ¼ 0:1 is the binning size). The resulting wi binnings model and the simulation. are slightly overlapping, which does not affect the applications 4. LINEAR GROWTH FUNCTION AND NONLINEAR considered in this work. Gi is simply chosen as the growth sup- MATTER POWER SPECTRUM pression at the center of the redshift bin zi. The response matrix Rij is calculated numerically using the formula in Hu & Eisenstein Existing fitting formulae of the matter power spectrum, Peacock (1999). & Dodds (1996), Smith et al. (2003), or a simple halo model, To test the strong assumption, we want to construct a w model have the following assumption: the nonlinear power spectrum that leads to the same growth factor as the w ¼ wconst model at does not depend on the time evolution (history) of the linear z ¼ 0 but different growth in the past. In order to keep the linear power spectrum; it is determined by the linear power spectrum growth factor at z ¼ 0 the same, the w model we are looking for at the same epoch. should have an equation of state greater than wconst during some There is a weaker version of the assumption, which states that epochs and smaller during some other epochs (to offset the effect the linear power spectrum plus its time evolution determine the on linear growth when it is greater). So we choose wconst ¼0:8, nonlinear power spectrum. A halo model incorporating merger instead of 1, to avoid having wi < 1. There are many ways history is a good example of building fitting formulae of the non- to achieve our goal. The response matrix Rij is invertible. Thus, linear matter power spectrum under this weaker assumption. either one could specify the linear growth and derive the corre- In this section, we test both versions of the above assumption sponding wi, or the other way around. Here we use the latter. using carefully chosen dark energy models. First, we know that the w model being sought after satisfies the 4.1. Dark Energy Model Construction constraint G1 ¼ 0 ¼ R1jwj, where we explicitly label the redshift bin at z ¼ 0 as bin number 1. Furthermore, we restrict the We choose the set of binned growth suppression Gi Dþ(ai)/ai variation of w to below redshift 1 where the effect of dark energy and equation of state of dark energy wi as parameters to be con- dominates. Finally, we demand that w(z) > 1. We then assign sidered. Gi and wi share the same binning with Áz ¼ 0:1. The lin- values to wj satisfying the above conditions. One of our choices ear response of Gi to the perturbations of wi can be described by of wi is shown in Figure 10 (model I). Figure 11 shows the comparison of the corresponding linear growth suppression to Gi ¼ Rijwj: ð20Þ that of the w ¼0:8 model. Our construction is successful since 894 MA Vol. 665

Fig. 10.—Time evolution of the equation of state of dark energy. Models I Fig. 12.—Comparison of the growth function of the w ¼0:8modelwith and II have the same linear growth functions at z ¼ 0 as that of the w ¼0:8 that of a wiggling w(z) model III, which is shown in Fig. 10. The linear growth model, but different in the past. Their growth function comparisons are shown in functions of these two dark energy models agree to better than 0.05% at all red- Fig. 11. Model III has the same linear growth function (precise to 0.05% as shifts. [See the electronic edition of the Journal for a color version of this figure.] shown in Fig. 12), at all redshifts, as that of the w ¼0:8model.[See the electronic edition of the Journal for a color version of this figure.] redshifts. We apply singular value decomposition (SVD) to the response matrix R and select the eigenvectors that have small the two dark energy models have the same linear growth at z 0 ij ¼ eigenvalues. These eigenvectors correspond to the combinations and a maximum deviation from each other by 0.8% at z 0:7. of w to which G has the least response. Among the selected To test the weaker version of the assumption, we want to i i modes we pick the ones that involve variations of w at low construct a w model that predicts the same (to certain precision) i redshift where the effect of dark energy dominates. Only one linear growth function as that of the w 0:8 model at all ¼ mode passed the selection criteria and is shown in Figure 10 as model III. The linear growth of the selected w model agrees with that of the w ¼0:8 model better than 0.05% at all redshifts as shown in Figure 12. On the other hand, the expansion histories of these two models differ by 0.6% as shown in Figure 13. 4.2. History Does Matter Figures 14 and 15 summarize the results. As shown in Fig- ure 14, the nonlinear matter power spectra at z ¼ 0 are different between the w ¼0:8 model and the constructed dark energy model I (Fig. 10), which has the same linear growth at z ¼ 0 but different during the past. Hence, the nonlinear power spectrum does care about the time evolution of the linear power spectrum, contrary to what existing fitting formulae assume. Our inter- pretation is the following. The difference in the linear power spectrum during the past causes some differences in the nonlinear power spectrum that would stay there unless some rather delicate adjustments of the history of the linear power spectrum are done to offset the effect. An arbitrary history of the linear power spectrum does not have the ability to undo the difference in the nonlinear power spectrum. One possibility is that different linear growth histories would produce different halo merger histories, on which the halo concentration depends (Weshsler et al. 2005). Hence, the nonlinear power spectrum, which mainly comes from the one-halo term, would be different. It is hard to Fig. 11.—Comparison of the growth function of the w ¼0:8 model with imagine that simply matching the linear growth at z ¼ 0 would those of the constructed dark energy models I and II, which are shown in Fig. 10. erase the difference. Thus, in general the difference would be These dark energy models have the same growth functions at z ¼ 0, but different time evolutions. The fractional differences shown in the bottom panel are rela- there as shown in our test case. tive to the w ¼0:8model.[See the electronic edition of the Journal for a color The effect of linear growth history on the nonlinear power version of this figure.] spectrum is by no means limited to the 0.8% level shown in our No. 2, 2007 NONLINEAR MATTER POWER SPECTRUM 895

Fig. 13.—Comparison of the expansion history H(z)/H0 of the w ¼0:8 Fig. 15.—Comparison of the matter power spectrum at z ¼ 0ofthe model with dark energy models whose equations of state are shown in Fig. 10. w ¼0:8 model with that of dark energy model III, which is shown in Fig. 10. [See the electronic edition of the Journal for a color version of this figure.] The linear growth functions of these two dark energy models agree to better than 0.05% at all redshifts. We conclude that the two P(k) are consistent. [See the electronic edition of the Journal for a color version of this figure.] test case. The nonsmooth nature of w(z) also would not make our argument any weaker. By a little bit of more work, we come up with another example that has a much smoother w(z) curve these examples, we see that the difference of the nonlinear power (Fig. 10, model II) and matching linear growth at z ¼ 0 but spectrum due to different linear growth history is at the level of maximum difference of 1.5% in the past (Fig. 11, dashed line). the maximum deviation of the corresponding linear power spec- As shown in Figure 14 (dashed line), the resulting nonlinear tra in the past. power spectra have maximum difference at the 2% level. From Our test of the weaker assumption validates it as shown in Figure 15. The nonlinear power spectra of the w ¼0:8 model and the constructed dark energy model III (Fig. 10) are consistent with being essentially the same given the small difference of the linear power spectra. This is not a proof that the history of the linear power spectrum fully determines the nonlinear power spectrum, but it hints that it is possibly the case. Note that although there is basically no hope of distinguishing the two ap- parently very different dark energy models using their growth functions, their expansion histories differ by 0.6% (see Fig. 13), which is close to the level that BAO could potentially have some handle on. 5. MATTER POWER SPECTRUM AND THE EQUATION OF STATE OF DARK ENERGY With the ambitious observational efforts focused on measur- ing the equation of state of dark energy to high precision (DES, SNAP, LSST, PanSTARRS), theoretical parameter forecasts will need to be done more carefully than before. In particular, we have to have a better understanding of the nonlinear power spectrum, which is very important for cosmic shear and any other probes that utilize the information of growth of density perturbations. Since numerical simulations are too expensive to run for every application, fitting formulae of the matter power spectrum are widely used. Fig. 14.—Comparison of the matter power spectrum at z ¼ 0ofthew ¼ The ones provided by Peacock & Dodds (1996) and Smith 0:8 model with those of the constructed dark energy models I and II, which are et al. (2003) are calibrated using ÃCDM models. McDonald et al. shown in Fig. 10. These dark energy models have the same growth functions (2005) provided a correction factor to the above-mentioned fit- at z ¼ 0, but different time evolutions. The fractional differences shown in the bottom panel are relative to the w ¼0:8 model. The simulation box size is ting formulae for cosmology with constant w (see also Ma et al. 1 1 Lbox ¼ 100 h Mpc , and the Nyquist frequency is k Ny ¼ 16 h Mpc .[See the 1999). For two-parameter dark energy models parameterized by electronic edition of the Journal for a color version of this figure.] w0 and wa, Linder & White (2005) give a simple procedure to 896 MA Vol. 665

Fig. 16.—Performance of the fitting formulae when applied to non-ÃCDM Fig. 18.—Performance of the fitting formulae when applied to non-ÃCDM cosmology. In this particular case, w0 ¼1:0 and wa ¼ 0:3. The simulation box cosmology. In this particular case, w0 ¼1:0 and wa ¼ 0:1. The simulation box 1 1 1 1 is 400 h Mpc, and the corresponding Nyquist frequency is k Ny ¼ 4 h Mpc . is 400 h Mpc, and the corresponding Nyquist frequency is k Ny ¼ 4 h Mpc . The initial power spectrum without BAO peaks is used for the Peacock & Dodds The initial power spectrum without BAO peaks is used for the Peacock & Dodds fitting formula. [See the electronic edition of the Journal for a color version of this fitting formula. [See the electronic edition of the Journal for a color version of this figure.] figure.]

calculate the nonlinear matter power spectrum with a few percent accuracy. While the method provided by Linder & White (2005) has not yet been widely adopted, a common practice is to apply Peacock & Dodds (1996) and Smith et al. (2003) fitting formulae regardless of the dark energy models. In this work we evaluate how well this common practice works. Two things could go wrong if we used an inaccurate fitting formula. One is that the fitting formula predicts the wrong amplitude of the matter power spectrum. This is the part we study in this work. Another effect of an incorrect fitting formula is that the predicted model parameter degeneracy directions and/or extent of the directions are off. Although not covered in this study, the latter effect is very important when the parameters are highly degenerate. During our study of the effect of photo-z on weak- lensing tomography (Ma et al. 2006), we noticed that with no tomography binning the extent of the degeneracy among dark energy parameters could differ by a factor of 4 depending on whether the fitting forms of Peacock & Dodds (1996) or the halo model are used to calculate the matter power spectrum. For this study we further restrict ourselves to two-parameter dark energy models that are parameterized by w0 and wa. 5.1. The Findings The quantity we use to evaluate the performance of the fitting formulae applied to wa 6¼ 0 dark energy models is PkðÞ; w Q 1; ð22Þ PkðÞ; w ¼1 Fig. 17.—Performance of the fitting formulae when applied to non-ÃCDM cosmology. In this particular case, w0 ¼1:0 and wa ¼ 0:2. The simulation box which is the fractional difference of the matter power spec- 1 1 is 400 h Mpc, and the corresponding Nyquist frequency is k Ny ¼ 4 h Mpc . trum between the w model, P(k; w), and the ÃCDM model, The initial power spectrum without BAO peaks is used for the Peacock & Dodds a fitting formula. [See the electronic edition of the Journal for a color version of this P(k; w ¼1). The fractional difference of the power spectrum figure.] in terms of Q is ÁQ/(Q þ 1), and that of the derivative of the No. 2, 2007 NONLINEAR MATTER POWER SPECTRUM 897

Fig. 19.—Fractional differences of Q (defined in eq. [22]), Q(Btting formula)/ Fig. 20.—Fractional differences of Q (defined in eq. [22]), Q(Btting formula)/ Q(simulation) 1, for different wa. The power spectrum is evaluated at z ¼ 0. Q(simulation) 1, for different wa. The power spectrum is evaluated at z ¼ 0:62. [See the electronic edition of the Journal for a color version of this figure.] [See the electronic edition of the Journal for a color version of this figure.]. power spectrum is ÁQ/Q Q(Btting formula)/Q(simulation) 1. incorrect. Adding more corrections to the Peacock & Dodds Figures 16–18 show the comparison of this quantity Q from the fitting formula would only turn it into another ‘‘theory of epi- simulation with those from the fitting formulae of Peacock & cycles.’’ We do not think that this is the right approach to build a Dodds (1996) and Smith et al. (2003) for wa ¼ 0:3, 0.2, and 0.1, good fitting formula. As a short-term solution, this may be worth respectively. doing. We leave the readers to make the judgment. First of all, the fitting formulae are doing a good job describing 6. DISCUSSION AND CONCLUSIONS wa models. The fractional difference of the power spectrum is ÁQ/(Q þ 1) 1%, and that of the derivative is ÁQ/Q 10% We modify the public PM code developed by A. Klypin and for k < 1:0 h Mpc1. J. Holtzman to simulate cosmologies with arbitrary initial matter In the context of Fisher matrix analysis, the Fisher matrix power spectra and the equation of state of dark energy. Using elements would be 20% off at most. Again, just to remind our- N-body simulation, we test various aspects of the matter power selves, we are only considering the effect of the amplitude of the spectrum, including whether nonlinear evolution would shift power spectrum but not that of the shape of the power spectrum, the BAO peaks, the assumptions that go into constructing fitting which is closely tied to the degeneracy direction of the model formulae of the nonlinear power spectrum, and the precision of parameters. the existing fitting formulae when applied to dark energy models The most surprising result we find is that the fractional dif- parameterized by w0 and wa. ferences in Q from the Peacock & Dodds (1996) fitting formula Any shift of the BAO peaks in the matter power spectrum are independent of wa, as shown in Figure 19. To turn this fact would bias the cosmological distance inferred and hence the cos- into a fitting formula of the matter power spectrum for wa mod- mological parameters derived. To test whether nonlinear evolu- els, the time evolution of this potential correction factor to the tion would shift BAO peaks, we implement an artificial sharp Peacock & Dodds (1996) fitting formula should also be indepen- peak in the initial power spectrum and evolve it using the PM dent of wa. Figure 20 shows this correction factor at z ¼ 0:62. code. We find that the position of the peak is not shifted by non- We see that the peak of the correction shifts to smaller scale linear evolution. An upper limit of the shift at the level of 0.02% (higherk) as one goes to higher redshift, and, amazingly, it shows is achieved by fitting the power spectrum local to the peak using no dependency on wa. Although such regular behavior always a power law plus a Gaussian. This implies that, for any practical seems to have a good reason behind it, at the moment, we do not purpose, the BAO peaks in the matter power spectrum are not have a physical explanation. shifted by nonlinear evolution. There are other effects such as It is very tempting to use such a fact to come up with a correc- galaxy bias that could potentially shift the peaks. These should tion factor to the fitting formula of Peacock & Dodds (1996). The be carefully calibrated as well. hump at the nonlinear scale in Figure 19 and the ones at high We also find that the existence of a peak in the linear power redshift can be fitted very well using a Gaussian in k space. How- spectrum would boost the nonlinear power at all scales evenly. ever, as we have pointed out throughout this work, the physical This is contrary to what the HKLM scaling relation predicts, but arguments supporting the Peacock & Dodds fitting formula do not roughly consistent with that of the halo model. The scale de- seem to hold: the linear and nonlinear scale mapping does not pendence is slightly different from the halo model prediction in hold; the assumption that the history of the linear growth does detail. Further study is required to understand the cause of the not affect the nonlinear power spectrum is also being shown to be difference. 898 MA

All existing fitting formulae of the nonlinear power spectrum includes dark matter. To build fitting formulae with percent level assume that the linear power spectrum uniquely determines the accuracy, one has to include the effect of baryons (White 2004; nonlinear one, regardless of the linear growth history. To test this Zhan & Knox 2004; Jing et al. 2006) and substructure (Hagan assumption, we construct two dark energy models with the same et al. 2005). linear power spectra today but different linear growth histories For simple dark energy models parameterized by w0 and wa, and compare their nonlinear power spectra from the simulation. the existing nonlinear power spectrum fitting formulae, which We find that the resulting nonlinear power spectra differ at the are calibrated for the ÃCDM model, work reasonably well. The level of the maximum deviation of the corresponding linear corrections needed are at the percent level for the power spec- power spectra in the past. Similarly, two constructed dark energy trum and the 10% level for the derivative of the power spectrum. models with the same growth histories result in consistent non- We find surprisingly that, for the Peacock & Dodds (1996) fitting linear power spectra. This is consistent with but not a proof of the formula, the corrections needed for the derivative of the power conventional wisdom that together the linear power spectrum spectrum are independent of wa but changing with redshift. As a and the linear growth history uniquely determine the nonlinear short-term solution, a fitting form could be developed for w0 and power spectrum. wa models based on this fact. Next-generation large-scale structure surveys need better fitting formulae than what are available now. With the high accuracy required, new fitting formulae should be based on solid founda- We thank Wayne Hu, Andrey Kravtsov, Scott Dodelson, Josh tions. Our results suggest that we should abandon the HKLM Frieman, Eduardo Rozo, and Doug Rudd for useful discussions. scaling relation, keep the halo model, and further include the We thank Anatoly Klypin and Jon Holtzman for making their linear growth history to build the next-generation fitting for- PM code public. Z. M. is supported by the David and Lucile mulae of the nonlinear power spectrum. Note that our study only Packard Foundation.

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