Simulations of Baryon Acoustic Oscillations I: Growth of Large-Scale Density Fluctuations
Ryuichi Takahashi, Naoki Yoshida, Takahiko Matsubara, Naoshi Sugiyama, Issha Kayo, Takahiro Nishimichi, Akihito Shirata, Atsushi Taruya, Shun Saito, Kazuhiro Yahata, Yasushi Suto
aa r X i v : . [ a s t r o - ph ] J u l Mon. Not. R. Astron. Soc. , 1–8 (2008) Printed 29 October 2018 (MN L A TEX style file v2.2)
Simulations of Baryon Acoustic Oscillations I: Growth ofLarge-Scale Density Fluctuations
Ryuichi Takahashi , Naoki Yoshida , , Takahiko Matsubara , Naoshi Sugiyama , ,Issha Kayo , Takahiro Nishimichi , Akihito Shirata , , Atsushi Taruya , ,Shun Saito , Kazuhiro Yahata , and Yasushi Suto Department of Physics and Astrophysics, Nagoya University, Chikusa, Nagoya 464-8602, Japan Institute for Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa City, Chiba 277-8582, Japan Department of Physics, School of Science, The University of Tokyo, Tokyo 113-0033, Japan Department of Physics, Tokyo Institute of Technology, Tokyo 152-8511, Japan Research Center for the Early Universe, The University of Tokyo, Tokyo 133-0033, Japan
ABSTRACT
We critically examine how well the evolution of large-scale density perturbationsis followed in cosmological N -body simulations. We first run a large volume simulationand perform a mode-by-mode analysis in three-dimensional Fourier space. We showthat the growth of large-scale fluctuations significantly deviates from linear theorypredictions. The deviations are caused by nonlinear coupling with a small number ofmodes at largest scales owing to finiteness of the simulation volume. We then developan analytic model based on second-order perturbation theory to quantify the effect.Our model accurately reproduces the simulation results. For a single realization, thesecond-order effect appears typically as “zig-zag” patterns around the linear-theoryprediction, which imprints artificial “oscillations” that lie on the real baryon-acousticoscillations. Although an ensemble average of a number of realizations approaches thelinear theory prediction, the dispersions of the realizations remain large even for alarge simulation volume of several hundred megaparsecs on a side. For the standardΛCDM model, the deviations from linear growth rate are as large as 10 percent for asimulation volume with L = 500 h − Mpc and for a bin width in wavenumber of ∆ k =0 . h Mpc − , which are comparable to the intrinsic variance of Gaussian randomrealizations. We find that the dispersions scales as ∝ L − / ∆ k − / and that the meandispersion amplitude can be made smaller than a percent only if we use a very largevolume of L > h − Gpc. The finite box size effect needs to be appropriately takeninto account when interpreting results from large-scale structure simulations for futuredark energy surveys using baryon acoustic oscillations.
Key words: cosmology:theory – large-scale structure of Universe – methods:N-bodysimulations
Understanding the nature of dark energy that dominates theenergy content of the universe is one of the main challengesin cosmology. The time evolution of the mysterious darkcomponent is accessible only by astronomical observations.Baryon acoustic oscillations (BAO) can be used as a stan-dard ruler by which precise measurement of the cosmologicaldistance scale is achievable (e.g., Eisenstein, Hu & Tegmark1998; Seo & Eisenstein 2003; Matsubara 2004).Recent large galaxy redshift surveys, the Sloan Dig-ital Sky Survey and the 2-degree Field survey, detected the signature of the baryon acoustic peaks and thus pro-vide constraints on the dark energy (Eisenstein et al. 2005;Cole et al. 2005; Percival et al. 2007; Okumura et al. 2007).Future observational programs will utilize the distribution ofmillions of high-redshift galaxies to detect BAO with higheraccuracy. In order to properly interpret these observations, itis necessary to make accurate theoretical predictions for thelength scale and other characteristic features of BAO (e.g.Nishimichi et al. 2007; Smith, Scoccimarro & Sheth 2008).Theoretically, a crucial issue is the non-linear evolutionof matter and galaxy distributions (e.g., Seo & Eisenstein2005; Angulo et al. 2007; Guzik, Bernstein & Smith 2007; c (cid:13) R. Takahashi et al.
Smith, Scoccimarro & Sheth 2007). One usually resorts tousing cosmological N -body simulations for this, but variouseffects –both physical and numerical– need to be understoodin order to extract useful information. First of all, the powerspectrum for a realization of a Gaussian random field hasintrinsic deviations from expected values at any wavenum-ber, i.e., the mode amplitudes are Rayleigh-distributed (seee.g., Matsubara 2007a). A realization may thus show an ad-ditional oscillatory feature on large scales which compro-mises the true BAO signature (Huff et al. 2007). There arealso a number of numerical issues. Accurate time integrationis necessary in order to follow the evolution of large-scaledensity perturbations which have small amplitudes. Finite-box size limits the sampling of wavenumbers at the largestscales, where the power amplitude is dominated by only afew modes (Bagla & Prasad 2006 studied the finite box sizeeffect on the mass function of dark matter halos.)In this paper, we examine how accurately the evolutionof large-scale density perturbations is followed in standardcosmological N -body simulations. In particular, we studythe characteristic “wiggle” features which are often foundin the matter power spectra calculated from N -body sim-ulations in previous studies. We use an approach based onperturbation theory to study nonlinear effects in detail. Afurther extensive study is presented in a separate paper byNishimichi et al. (in preparation).Throughout the present paper, we adopt the standardΛCDM model with matter density Ω m = 0 . b = 0 . Λ = 0 . n s = 0 . σ = 0 .
76, andexpansion rate at the present time H = 73 . − Mpc − ,consistent with the 3-year WMAP results (Spergel et al.2007). We use the cosmological simulation code Gadget-2(Springel, Yoshida & White 2001; Springel 2005). For ourfiducial runs, we employ 256 particles in a volume of L = 500 h − Mpc on a side. We dump snapshots at a numberof time steps (redshifts) to study the evolution of the den-sity power spectrum. The simulation parameters are chosensuch that sufficient convergence is achieved in the measuredpower spectrum at the present epoch (Takahashi et al., inpreparation).We generate initial conditions for our runs based on thestandard Zel’dovich approximation using the matter transferfunction calculated by CAMB (Code for Anisotropies in theMicrowave Background; Lewis, Challinor & Lasenby 2000).The initial redshift is set to be z in = 30. When we generatea realization for a Gaussian random field, the amplitudeof each k -mode is assigned such that the ensemble followsthe Rayleigh distribution. While the mean of the power isexpected to approach the input value at k for an ensemble oflarge modes, the actual assigned power in a finite k -bin candeviate significantly from the expected value. Note also thata Rayleigh distribution has a positive skew, which causes themedian to be smaller than the mean. k (h/Mpc) P ( k , z ) / P n w ( k , z ) z in =30 (initial)z=3z=1z=0 L=500h −1 Mpc 1stpeak
Figure 1.
We plot the evolution of the power spectrum fromthe initial epoch (black line) to z = 3 (green), z = 1 (blue),and z = 0 (purple). The measured power spectrum is dividedby the no-wiggle model of Eisenstein & Hu (1998).We subtractthe intrinsic deviations from the input power spectrum at theinitial epoch. The numbers indicate integer sums of n + n + n of wavenumber vectors. The dashed lines are the one-loop powerspectra at each redshift (see text). We first compute the density field for each output of the N -body simulation. We use the CIC (cloud-in-cell) interpo-lation when assigning particles on grids. We check that theinterpolation method does not affect the scales of interest( k < ≃ .
1) by comparing various schemes. We then apply aFast Fourier Transform to obtain the density field δ ( k ) inthree-dimensional Fourier space. We will examine both theamplitudes and the phases in detail in subsequent sections.In order to study closely the Fourier mode-coupling, wecalculate the mean amplitude of modes for a given realiza-tion with wavenumber vector k = ( k , k , k ) asˆ P ( k ) = 1 N k X | k | = k | δ ( k ) | , (1)where the summation is for all the wavenumbers of | k | = k = ( k + k + k ) / , N k is the number of modes in k , andthe wavenumber is discretized as k i = (2 π/L ) n i with aninteger n i . An ensemble average of a number of realizationsprovides its expectation value of P ( k ) = h ˆ P ( k ) i .In order to study the evolution of power spectrum, wedivide the measured power spectrum in equation (1) at red-shift z by the initial one at z in = 30, and then multiply it bythe input power spectrum. In this way, the initial randomscatter included in the power spectrum is removed. c (cid:13) , 1–8 imulations of BAO I Fig.1 shows the evolution of power spectrum ˆ P ( k ) for asingle realization. We show the mean amplitude for modeswhich have exactly the same wavevector norm, | k | = k + k + k , rather than binning in k . The vertical axis is thepower spectrum divided by the no-wiggle model of Eisen-stein & Hu (1999). The black line with symbols is the lineartheory prediction with CAMB. The green, blue, and purplelines with dots are the measured mean values at each wavenumber at z = 3 , , and 0, respectively. The numbers in thefigure indicate integer sums of n + n + n of wavenumbervectors.As clearly seen in the figure, the power amplitudes de-viate from the linear theory prediction at low redshifts.The deviations appear to grow in time monotonically.Some modes (e.g. n = 4 , , , ,
27) grow more rapidlythan the linear growth, while other modes (e.g. n =8 , , , ,
32) grow less. These features can be seen evenin higher resolution simulation of Springel at al. (2005) (seetheir Fig.6). Since the initial randomness of the amplitudeof each mode has been already subtracted in the figure asdescribed in section 2.2, the remaining differences plottedin Fig. 1 are due either to numerical integration errors orto some unknown physical effects. The deviations are in-deed large, with the amplitudes being more than 10% at thescale of the first-peak of the BAO. It is thus important tounderstand and correct the apparent oscillatory features ifthese are artificial effects.In the next section, we show that the deviations are not owing to numerical integration errors but due to the finitenumber of modes at the largest scales. We use second-orderperturbation theory to explain the systematic deviations.
Second-order perturbation theory describes the evolution ofa density perturbation as (e.g. Bernardeau et al. 2002) δ ( k , z ) = D ( z ) D in δ ( k ) + (cid:18) D ( z ) D in (cid:19) δ ( k ) , (2)where δ ( k ) and D in are the linear density and the lineargrowth factor evaluated at the initial redshift. The second-order term is given by δ ( k ) = X p F ( p , k − p ) δ ( p ) δ ( k − p ) , (3)with F ( p , q ) = 57 + p · q (cid:18) p + 1 q (cid:19) + 27 ( p · q ) p q . (4)We sum up all the modes up to the Nyquist fre-quency (256 modes in total) in equation (3). Here,equation (4) includes the fastest growing mode.Bernardeau, Crocce & Scoccimarro (2008) recently presentthe correct formula of F including the sub-leading growingmode.Let us explicitly write the amplitude and the phase ofa mode as δ ( k , z ) = | δ ( k , z ) | exp (i φ ( k , z )) , (5)Then the evolution of amplitude in each mode isˆ P ( k, z ) / ˆ P ( k, z in ) D ( z ) /D = 1 + 1 N k X | k | = k δ ( k ) δ ∗ ( k )] × P ( k, z in ) D ( z ) D in , (6)whereas the phase evolution is φ ( k , z ) − φ in ( k ) = sin φ in ( k ) cos φ in ( k ) × (cid:18) Im δ ( k )Im δ ( k ) − Re δ ( k )Re δ ( k ) (cid:19) D ( z ) D in , (7)up to second order. The expressions in equations (6) and(7) are independent of the initial redshift for the late time( D ≫ D in ), since δ ∝ D in and δ ∝ P ( k, z in ) ∝ D . Wedo not distinguish between δ and δ at the initial redshift( z in = 30), since δ is much smaller than δ at that time. and provide more detail analysis.It is clear from equation (4) that nonlinear mode-coupling occurs with particular sets of wavenumber vectorssuch that p + q = k . From equation (4), we obtain F ( p , k − p ) → (cid:16) −
57 cos θ (cid:17) k p , (8)for k ≪ p , and F ( p , k − p ) → kp cos θ, (9)for k ≫ p . Here θ is an angle between k and p . Hencethe coupling to the mode of much smaller scale p ( ≫ k ) isnegligibly weak, while the coupling to much larger scale p ( ≪ k ) is strong. In summary, most of the contribution to thesecond-order evolution of a mode comes from the modes ofcomparable scales or larger. For a Gaussian random field, the mode amplitudes areRayleigh-distributed, and thus there is a finite probabilitythat a mode has a very large or a very small amplitude withrespect to the expected mean value. Some peculiar modes,which have very large or very small amplitudes compared tothe mean, strongly affect the growth of other modes throughthe mode-coupling as described in the above.In an ideal situation where there are infinite number ofmodes, the second term in equation (6) vanishes. In thatcase, the leading correction arises from the forth order of δ . Then the resultant power spectrum with the one-loopcorrection is, P ( k, z ) = (cid:18) D ( z ) D in (cid:19) P ( k )+ (cid:18) D ( z ) D in (cid:19) [ P ( k ) + P ( k )] , (10)where P = (cid:10) | δ | (cid:11) , P = (cid:10) | δ | (cid:11) , P = 2 h Re[ δ δ ∗ ] i (Makino, Sasaki & Suto 1992; Jain & Bertschinger 1994;Jeong & Komatsu 2006). We integrate from k = 2 π/L tothe Nyquist frequency in the calculation of P and P . Nishimichi et al. (in preparation) distinguish δ from δ at the initial epoch with the 2LPT initial condition(Crocce, Pueblas & Scoccimarro 2006) Muecket et al. 1988 examined the growth of the small-scaleperturbation on the background of the large-scale perturbation.c (cid:13) , 1–8
R. Takahashi et al. linear growth factor D [ D ( z ) / D i n ] redshift
30 5 2 1 0.5 0 P ( k , z ) / P ( k , z i n ) n =1−8 0 0.5 10.911.1 linear growth factor D [ D ( z ) / D i n ] n =9−16 redshift
30 5 2 1 0.5 0 P ( k , z ) / P ( k , z i n ) linear growth factor D [ D ( z ) / D i n ] n =17−24 redshift
30 5 2 1 0.5 0 P ( k , z ) / P ( k , z i n )
22 22 linear growth factor D [ D ( z ) / D i n ] n =25−32 redshift
30 5 2 1 0.5 0 P ( k , z ) / P ( k , z i n )
32 29
Figure 2.
Evolution of the deviation of the power amplitude with respect to the linear theory prediction. The dots are the measurementsfrom our simulation, and red solid lines are the model prediction using the second-order perturbation theory. The integers denote n = n + n + n of wavenumbers, and the figures show different range of n , n = 1 − n = 9 −
16 (upper rightpanel), n = 17 −
24 (lower left panel), and n = 25 −
32 (lower right panel).
The dashed lines in Fig.1 are the one-loop power spec-trum at each redshift. It suggests that the linear theory isapplicable for k < . h /Mpc at z = 0. However, the finitemode coupling in the second term of equation (6) signifi-cantly changes the evolution of the power spectrum even inthe linear regime. Fig.2 shows the evolution of the mean amplitude ofmodes with identical wavenumber n in the range of 1 − n ≃
30 corresponds to the position of the first peak(see Fig.1). The four panels are for n = 1 − n = 9 −
16 (upper right panel), n = 17 −
24 (lowerleft panel), and n = 25 −
32 (lower right panel). The dotsare the measurement from simulation outputs, and red solidlines are the theoretical prediction from the initial densityfields at z in = 30 in equation (6). The second-order pertur-bation theory reproduces the simulation results rather well. Seto (1999) also investigated the finite mode effect on the one-loop correction terms, P + P , in equation (10). The theory fits the data within 0 .
5% at z = 2 and 2% at z = 0 for larger scale ( n = 1 − z = 2 and 10% at z = 0 for smaller scale ( n = 25 − n > n > n , becausethe mean of the phase at k , P φ ( k ), is zero (since φ ( k ) + φ ( − k ) = 0). The phase shifts are typically ≈ . z = 0. Perturbation theory well reproduces the results. Evenif there are infinite modes, the right hand side of equation(7) still remains. The phase shift is not due to the finite boxsize effect.Previously Ryden & Gramann (1991) and Gramann(1992) studied the evolution of amplitude and phase in eachmode using two dimensional simulations. They also calcu-lated second-order perturbation theory and found the de-viation from the linear theory grows in proportional to thescale factor in the EdS model. Suginohara & Suto (1991), c (cid:13) , 1–8 imulations of BAO I linear growth factor D n =1−8 φ − φ ( z ) i n redshift
30 5 2 1 0.5 0 0 0.5 1−0.200.20.4 linear growth factor D n =9−16 φ − φ ( z ) i n
12 1099 11131416111012131416 redshift
30 5 2 1 0.5 00 0.5 1−0.200.2 linear growth factor D n =17−24 φ − φ ( z ) i n redshift
30 5 2 1 0.5 0 linear growth factor D n =25−32 φ − φ ( z ) i n redshift
30 5 2 1 0.5 0
Figure 3.
Same as Fig.2, but for phase evolution in units of radians. We plot the results only for modes with n > n > n . Soda & Suto (1992) and Jain & Bertschinger (1998) also ex-amined the nonlinear evolution in each mode. However theydid not compare the theoretical prediction with the simula-tion results in detail. Their motivations were to understandthe evolution of the density fluctuations in the nonlinearregime, whereas our interest here is in the growth of pertur-bations at the linear scale.
The previous section considers second-order effects for a sin-gle realization. In this section we run 100 simulations to cal-culate dispersions of amplitude and phase deviations fromlinear theory. We prepare the 100 realizations for each ofthree box sizes of L = 500 h − Mpc, 1 h − Gpc, and 2 h − Gpc,and z in = 30 ,
20 and 10, respectively.Fig.4 shows the remaining amplitude dispersions fromthe linear theory prediction after correcting for the initialrandomness at z = 0 for L = 500 h − Mpc (top), 1 h − Gpc(middle), and 2 h − Gpc (bottom). Since we already sub-tract the initial deviations due to the Gaussian distribution, the residuals arise from the mode-coupling during the evo-lution. The grey dots with error bars are the means with1 σ scatters. By using a sufficiently large number of realiza-tions, the means converge to the true values (solid line), andthe magnitude of the dispersions is insensitive to the num-ber of realizations. For L = 500 h − Mpc, the dispersions are ∼
10% near the first peak, and ∼
5% even for a very largevolume of 2 h − Gpc on a side. The dashed lines show thetheoretical prediction of the 1 σ scatter, which is the rms(root-mean-square) of the second term in equation (6) : σ ≡ *(cid:18) ˆ P ( k, z ) / ˆ P ( k, z in ) D ( z ) /D − (cid:19) + = 4 P ( k, z in ) P ( k, z in ) 1∆ N k (cid:18) D ( z ) D in (cid:19) . (11)Here, ∆ N k is the number of modes in the bin, ∆ N k =4 πn ∆ n with n = ( L/ π ) k . In this unbinning case, the num-ber of modes is ∆ N k = kL ∆ n (with ∆ n = 1). The dashedlines well reproduce the results.Fig.4 also shows the results for the binned data of ∆ k = c (cid:13) , 1–8 R. Takahashi et al. k (h/Mpc) P ( k , z ) / P n w ( k , z ) L=2h −1 GpcL=1h −1 GpcL=500h −1 Mpcz=0z=0z=0
Figure 4.
The amplitude dispersions of the 100 realizations at z = 0 for L = 500 h − Mpc (top), 1 h − Gpc (middle), and 2 h − Gpc (bottom). The grey dots with error bars are for the un-binned data, while the black big symbols are for the binned dataof ∆ k = 0 . h /Mpc. The value of k for the binned data is theweighted mean of k with the number of wavenumbers in the bin.The dashed lines are the theoretical prediction. . h /Mpc by the black big symbols. In this case, we usethe power spectrum defined as ˆ P ( k ) = (1 / ∆ N k ) P | δ ( k ) | ,summing up all the modes between ( k − ∆ k/ , k + ∆ k/ N k = ( L k ) / (2 π )∆ k. (12)We calculate the means and error bars for the binned ˆ P ( k ).Fig.5 shows the amplitude dispersions calculated fromour simulation outputs for ∆ k = 0 . h /Mpc (filled circle)and the theoretical prediction (solid line). From this figurewith equations (11) and (12), we find that the dispersion isapproximated as σ amp ( z = 0) ≃ (cid:18) L (cid:19) − / (cid:18) ∆ k . (cid:19) − / , (13)at k = 0 . − . h /Mpc. The dispersion is proportional to∆ N − / k ∝ L − / ∆ k − / from equation (12). Note that even −3 −2 −1 L=500h −1 MpcL=1h −1 GpcL=2h −1 Gpc k (h/Mpc) d i s pe r s i on σσσσ z=0 a m p Figure 5.
The amplitude dispersions calculated from our simula-tion outputs (filled circle • ) and the theoretical predictions (solidlines). We also show the dispersions due to the initial Gaussiandistribution (dashed lines). The vertical dotted line is the positionof the BAO first peak. k (h/Mpc) P ( k , z ) / P n w ( k , z ) L=500h −1 Mpcz=0 k=0.005h/Mpc ∆ Figure 6.
We compare two dispersions. Blue points with errorbars show intrinsic scatter around the expected mean power spec-trum for initial Gaussian random density fields. Black points showthe dispersions owing to the finite nonlinear mode-coupling effect. with a large simulation volume of L ∼ k -binning,the dispersions still remain at the level of a few percent.So far we have discussed the amplitude of deviationsfrom linear theory. Here we also consider the intrinsic scat-ter of the initial Gaussian random realizations. In Fig.5the dashed line is the dispersion for the initial distribu-tion, which is given by (∆ N k / − / . Fig.5 shows that thedashed lines decrease as ∝ (∆ N k ) − / ∝ k − , while the solidlines increase because P /P increases (see equation [11]).These two dispersions are comparable at k ≃ . h /Mpcwhere 2 P /P ≃ z = 0. About a half of the dispersionsnear the position of the BAO first peak ( k ∼ . h /Mpc)are attributed to the second-order effects. The result sug-gests that, at large scales, k < . h /Mpc, the dispersionsarise mainly from the initial Gaussian random distribution, The number of modes ∆ N k is divided by 2 because the Fouriermodes of δ ( k ) and δ ( − k ) are not independent.c (cid:13) , 1–8 imulations of BAO I −3 −2 −1 k (h/Mpc) L=500h −1 Mpc < | δ | ( φ − φ ) > / i n L=2h −1 GpcL=1h −1 Gpc z=0 < | δ | > / Figure 7.
The phase dispersion of the 100 realizations. The solidline is the theoretical prediction. while at smaller scale k > . h /Mpc they are from the mode-coupling (based on the second or higher order perturbation)during the evolution. In Fig.6 the blue symbols are the re-sults for our 100 realizations. The black symbols are same asin the top panel of Fig.4 for ∆ k = 0 . h /Mpc. As expected,the initial random realizations have larger scatters aroundthe mean expected power spectrum, especially at the largestscales.We have also performed a similar analysis for the evolu-tion of the mode phases (equation [7]). Fig.7 shows the phasedispersion calculated from our simulations (the dots). Herewe set − π ( φ − φ in ) π and calculate h| δ | ( φ − φ in ) i instead of h ( φ − φ in ) i . This is because ( φ − φ ) ∝ /δ inEq.(7) and its dispersion diverges at δ = 0. We obtain thephase dispersion from equation (7) as, (cid:10) | δ ( k ) | [ φ ( k , z ) − φ in ( k )] (cid:11)(cid:10) | δ ( k ) | (cid:11) = P ( k, z in )6 P ( k, z in ) (cid:18) D ( z ) D in (cid:19) . (14)The solid lines are the theoretical prediction, which fit thesimulation results well. The phase dispersion in equation(14), as well as the amplitude dispersion in equation (11), areindependent of the initial redshift. In the non-linear limit of k → ∞ , the phases are distributed randomly, and the phasedispersion approaches to π/ √ In this paper, we critically examined how accurately cosmo-logical N -body simulations describe the evolution of large-scale density distributions, particularly focusing on the lin-ear and/or quasi-linear scales. For the power spectrum cal-culated from a single realization, we found that the growth oflarge-scale fluctuations significantly deviates from the lineartheory prediction, and the enhanced or suppressed growth Jain & Bertschinger (1996) previously derived equation (14)with an approximation for the long-wave mode coupling. k (h/Mpc) P ( k , z ) / P n w ( k , z ) initial P k (z in =30) before correctionafter correction k z=2 Figure 8.
The power spectrum at z = 2. The green line is thesimulation output. In the red line, we subtract the second-orderperturbation contribution from the simulation output. The blueline is the one-loop power spectrum. of perturbations produces an ugly noisy pattern in the mat-ter power spectrum. This deviation is not due to the nu-merical errors in the N -body code, but due to the non-linear coupling between finite numbers of modes originat-ing from the finite box size. To study the effect of the fi-nite mode-coupling in detail, we developed perturbation the-ory and quantitatively estimated the finite-mode coupling tothe power spectrum amplitude. Mode-by-mode analysis inthree-dimensional Fourier space reveals that the finite mode-coupling from the second-order perturbation is sufficient toexplain the deviation from linear theory prediction on largescales. The dispersion of the mode-coupling effects estimatedfrom second-order perturbation scales as ∝ L − / ∆ k − / ,and this may surpass the intrinsic scatter of the initial Gaus-sian distribution. Since the finite mode-coupling does notvanish even for a large-volume simulation, it is of criticalimportance to correct it properly for high-precision studiesof baryon acoustic oscillations.We show that the perturbative approach is very helpfulto quantify the significance of finite-mode coupling and thiscan be utilized as an efficient and powerful tool to correct thefinite-mode coupling. As an example, in Fig. 8, we evaluatethe power spectrum directly obtained from a single realiza-tion at z = 2, and subtract the finite-mode coupling usingthe second-order perturbation. Compared the result beforesubtraction with that after subtraction, the deviation fromlinear theory is dramatically reduced and the noisy struc-tures are effectively wiped out. As a result, even the singlerealization data of N -body simulation faithfully reproducesthe linear theory prediction on large scales.Although the present paper mainly concerns with thesecond-order perturbation theory, higher-order perturba-tions are also important for the relevant scales of the mea-surement of baryon acoustic oscillations, where the acousticsignature tends to be erased by the effect of non-linear clus-tering (e.g. McDonald 2007; Crocce & Scoccimarro 2007; c (cid:13) , 1–8 R. Takahashi et al.
Matsubara 2007b; Taruya & Hiramatsu 2008). The heightof the first peak is found to be reduced about 2% (J. Wang,A. Szalay et al. in preparation). Thus, the inclusion of thehigher-order terms may be important for the estimation ofthe finite-mode coupling, which would be helpful to furtherreduce the noisy structures on small scales.We note that the variance of the growth of matterpower spectrum with respect to the linear theory pre-diction, h [( ˆ P / ˆ P in ) / ( D/D in ) − i , which we have stud-ied, is different from the variance of the power spectrumitself, h ( ˆ P − P ) i . It remains unclear if the numericaleffects studied here are important in evaluating covari-ance matrices (e.g., Scoccimarro, Zaldarriaga & Hui 1999;Meiksin & White 1999; Neyrinck & Szapudi 2007) In futurework, we will study nonlinear and numerical effects in thepower spectrum covariance using a large set of simulationsand analytic models. ACKNOWLEDGMENTS
We thank Jie Wang, Erik Reese, and Simon White for use-ful comments and discussions. We also thank the anony-mous referee for careful reading and useful suggestions.This work is supported in part by Grant-in-Aid for Sci-entific Research on Priority Areas No. 467 “Probing theDark Energy through an Extremely Wide and Deep Sur-vey with Subaru Telescope”, by the Mitsubishi Founda-tion, and by Japan Society for Promotion of Science (JSPS)Core-to-Core Program “International Research Network forDark Energy”, and by Grant-in-Aids for Scientific Research(Nos. 18740132, 18540277, 18654047). T. N., A. S. and K.Y. are supported by Grants-in-Aid for Japan Society for thePromotion of Science Fellows.
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