Mass-Dependent Baryon Acoustic Oscillation Signal and Halo Bias
aa r X i v : . [ a s t r o - ph . C O ] A p r Draft version October 6, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
MASS-DEPENDENT BARYON ACOUSTIC OSCILLATION SIGNAL AND HALO BIAS
Qiao Wang and Hu Zhan
Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China
Draft version October 6, 2018
ABSTRACTWe characterize the baryon acoustic oscillations (BAO) feature in halo two-point statistics using N -body simulations. We find that nonlinear damping of the BAO signal is less severe for halos in themass range we investigate than for dark matter. The amount of damping depends weakly on the halomass. The correlation functions show a mass-dependent drop of the halo clustering bias below roughly90 h − Mpc, which coincides with the scale of the BAO trough. The drop of bias is 4% for halos withmass
M > h − M ⊙ and reduces to roughly 2% for halos with mass M > h − M ⊙ . In contrast,halo biases in simulations without BAO change more smoothly around 90 h − Mpc. In Fourier space,the bias of
M > h − M ⊙ halos decreases smoothly by 11% from wavenumber k = 0 . h Mpc − to 0 . h Mpc − , whereas that of M > h − M ⊙ halos decreases by less than 4% over the samerange. By comparing the halo biases in pairs of otherwise identical simulations, one with and theother without BAO, we also observe a modulation of the halo bias. These results suggest that precisecalibrations of the mass-dependent BAO signal and scale-dependent bias on large scales would beneeded for interpreting precise measurements of the two-point statistics of clusters or massive galaxiesin the future. Subject headings:
Cosmology: theory — large-scale structure of Universe — dark matter INTRODUCTION
Baryon Acoustic Oscillations (BAO, Peebles & Yu1970; Sunyaev & Zeldovich 1970) produce an im-print on the matter distribution by the perturba-tions in the photon-baryon fluid before recombina-tion. It provides a standard ruler for probing darkenergy (e.g., Eisenstein & Hu 1998; Cooray et al. 2001;Blake & Glazebrook 2003; Hu & Haiman 2003; Linder2003; Seo & Eisenstein 2003), and is considered tobe least affected by systematics (Albrecht et al. 2006).Thus far, the BAO signal has been detected in sam-ples from Sloan Digital Sky Survey (Eisenstein et al.2005; Padmanabhan et al. 2007; Percival et al. 2010),2dF Galaxy Redshift Survey (Cole et al. 2005), 6dFGalaxy Survey (Beutler et al. 2011), WiggleZ Dark En-ergy Survey (Blake et al. 2011), and Baryon OscillationSpectroscopic Survey (Anderson et al. 2012). Futurespectroscopic and imaging surveys, such as BigBOSS,Euclid, and Large Synoptic Survey Telescope, all aimto determine cosmic distances to sub-percent level pre-cision with the BAO technique (e.g., Zhan et al. 2009;Schlegel et al. 2011).Although the BAO signal does not evolve in linear the-ory, nonlinearity causes a slight shift of its scale and sig-nificant damping of its amplitude (e.g., Seo et al. 2008,2010; Sherwin & Zaldarriaga 2012; McCullagh et al.2013). These effects must be modeled or correctedin BAO analyses. Given the unprecedented statisti-cal power of future surveys, it is necessary to cali-brate the nonlinear BAO signal accurately. Effortshave been made for dark matter with both N -bodysimulations (Smith et al. 2008; Padmanabhan & White2009) and perturbation theory (Jeong & Komatsu 2006;Crocce & Scoccimarro 2008; Matsubara 2008).It is reasonable to assume that the halo bias is con- [email protected] stant on large scales according to the “peak-backgroundsplit” or “local bias” model(e.g., Fry & Gaztanaga 1993;Mo & White 1996; Scherrer & Weinberg 1998). For pre-cision cosmology, however, one should quantify the scalesof validity as well as any departure from constancy forhalos of different masses. Investigations of the halo BAOsignal have given hints for scale-dependence in the halobias even at the BAO scale (e.g., Seo & Eisenstein 2005;Noh et al. 2009; Desjacques et al. 2010). In this work,we determine the halo correlation functions and powerspectra with high precision and clearly demonstrate themass-dependent halo BAO signal and behavior of thehalo bias beyond the BAO scale. SIMULATIONS
We run two sets of N -body simulations to study thehalo BAO signal and halo bias at redshift z = 0. The ini-tial matter power spectrum of the first set is calculatedusing camb (Lewis, Challinor, & Lasenby 2000) with theparameters: Ω Λ =0.73, Ω M = 0 . h = 0 . σ = 0 . n s = 1, and Ω b = 0 . h − Gpc on each side. With N p = 640 particles, the mass resolution is 2 . × h − M ⊙ , and the softening length is set to 50 h − kpc.The simulations are run with gadget -2 (Springel 2005)from initial redshift z i = 100 to z = 0.There are 150 realizations in each set of the simula-tions. The initial conditions are generated using 2LPTic(Crocce et al. 2006) with the same random seed for apair of simulations, one in each set. In this way, the ef-fect of sample variance can be reduced in the comparisonbetween the results from the two sets of simulations.Halos are identified using the Friends-of-Friends group -0.001 0 0.001 0.002 0.003 80 100 120 140 160 180 ξ (r) r ( h -1 Mpc)< ξ hh > (M>10 h -1 M ⊙ )< ξ hh > (M>10 h -1 M ⊙ )< ξ mmG > ξ mmL ξ mmRPT Figure 1.
Mass-dependent halo BAO signal at z = 0. The sym-bols represent the correlation functions of M > h − M ⊙ halos(filled circles), M > h − M ⊙ halos (open circles), and darkmatter (crosses) in the BAO simulations. The error bars show 1- σ errors of the mean correlation functions of 150 realizations. Thedashed line is the linear dark matter correlation function, and thesolid line is the nonlinear dark matter correlation function from theRPT fitting formula. These correlation functions are normalizedin the range of 160–180 h − Mpc. finder from University of Washington’s N -Body Shop .The linking length is set to 0.2 times the mean parti-cle separation. Given the mass resolution, we limit ourstudy to halos more massive than 10 h − M ⊙ , whichcorresponds to about 35 simulation particles. The ob-tained comoving halo number density at z = 0 is4 . × − h Mpc − for M > h − M ⊙ and 2 . × − h Mpc − for M > h − M ⊙ . HALO BAO AND HALO BIAS
BAO in Halo Correlation Functions
We calculate the halo two-point correlation functions ξ hh ( r ) with the Landy-Szalay estimator (Landy & Szalay1993) ξ hh ( r ) = N RR ( r ) (cid:20) HH ( r ) N − HR ( r ) N h N r + RR ( r ) N (cid:21) , (1)where N r is the size of a uniform random sample, N h isthe size of the halo sample, HH is the number of halopairs with a separation between r − ∆ r/ r + ∆ r/ HR is the number of pairs between the halo sample andthe uniform random sample, and RR is that within theuniform random sample. Since the halo sample from thesimulations has periodic boundaries and a uniform se-lection function, Equation (1) reduces, for an infinitelydense random sample, to ξ hh ( r ) = HH ( r ) N V box V bin ( r ) − , (2)where V box is the volume of the simulation box, and V bin ( r ) is the volume of the spherical shell between r − ∆ r/ r + ∆ r/ h ξ hh i averaged over the en-semble of the 150 BAO simulations are shown in Fig-ure 1 for halos of mass M > h − M ⊙ (open cir-cles) and M > h − M ⊙ (filled circles). For com- parison, we also plot the nonlinear dark matter correla-tion function from the simulations, i.e., the correlationof all simulation particles ( h ξ mmG i , crosses), the lineardark matter correlation function ( ξ mmL , dashed line), andthe nonlinear dark matter correlation function from aconvolution of ξ mmL with the Gaussian damping factorpredicted by Renormalized Perturbation Theory (RPT,Crocce & Scoccimarro 2006, 2008) ( ξ mmRPT , solid line). Be-cause it is time consuming to estimate the dark mattercorrelation function from 640 particles with the pair-counting method, we assign the particles onto a densitygrid (hence the subscript G) of 200 cells with the cloud-in-cell assignment function (Hockney & Eastwood 1981)and then calculate the correlation function from pairs ofthe cells.Studies have shown that nonlinear evolution damps theBAO feature (e.g., Eisenstein et al. 2007; Smith et al.2008; Seo et al. 2008). Such an effect on dark mattercan be fairly accurately modeled with the RPT formulism(see, e.g., Figure 1). Comparing the correlation functionsfrom the simulations, we find that the damping effect ofthe BAO signal depends on the tracer’s mass, which can-not be scaled away by a constant clustering bias. As ex-plained in Eisenstein et al. (2007), the BAO damping isdue to the motions of matter or tracers relative to theirinitial separation, i.e., the Lagrangian displacements be-tween pairs. The mass-dependent damping of the haloBAO signal suggests that the characteristics of halo mo-tions are also mass-dependent. For a test, we measure thepairwise velocity dispersion σ pv , which is proportional tothe rms Lagrangian displacement in the Zel’dovich ap-proximation, and find that σ pv = 650 . ± . − for M > h − M ⊙ halos and σ pv = 632 . ± . − for M > h − M ⊙ halos over the separations of 90–120 h − Mpc. These results appear to be consistent withFigure 1 in the sense that larger pairwise random motionswould damp the BAO signal more. However, we notethat nonlinear evolution weakens the correspondence be-tweeen the pairwise velocity and the Lagrangian displace-ment. A more thorough investigation is needed to fullyunderstand the mass-dependent BAO damping.
Halo Bias from Correlation Functions
The difference between the halo correlation functionsin Figure 1 can be accommodated with scale-dependentbiases. We estimate the halo bias in two ways: b CC ( r ) ≡ h ξ hh ( r ) /ξ hm ( r ) i , (3) b G ( r ) ≡ h ξ hhG ( r ) i / h ξ mmG ( r ) i , (4)where ξ hm is the cross correlation between the halos anddark matter in each simulation, and ξ hhG is the correlationfunction calculated from the halo density grid (analogousto ξ mmG ). The cross correlation is given by ξ hm ( r ) = HM ( r ) N h N p V box V bin ( r ) − , (5)where HM is the number of halo–simulation particlepairs with a separation between r − ∆ r/ r + ∆ r/ ξ hhG /ξ mmG as an estimator for b . Such a ratio canfluctuate wildly and even be negative in some realizationswhen the correlations are close to zero. Equation (4) is b (r) r ( h -1 Mpc)>10 h -1 M ⊙ >5 × h -1 M ⊙ >2 × h -1 M ⊙ >10 h -1 M ⊙ b CC b G b G (no-BAO) Figure 2.
Halo bias from the correlation functions. Open circlesand filled circles represent, respectively, b CC and b G (see Equa-tions [3] and [4]) in the BAO simulations. Horizontal lines mark theaverage values of b G between 50 h − Mpc and 70 h − Mpc. Boxesrepresent b G in the no-BAO simulations. better behaved, though the covariance between ξ hhG and ξ mmG needs to be accounted for when determining the un-certainties of b G .Figure 2 shows the behavior of the halo bias for fourmass ranges as labeled. Since the halo mass functionis fairly steep at M > h − M ⊙ , each mass range isdominated by halos at its low-mass end. The halo biascannot be determined accurately when the correlationfunctions are close to zero. Therefore, we leave a gapwhere the errors become too large to be informative.For the BAO simulations, the results of the two dif-ferent estimators, b CC (open circles) and b G (filled cir-cles), are consistent with each other over the scales andmasses of interest. They both show a systematic off-set between the average halo bias over 50–70 h − Mpc(dashed lines) and that over the BAO peak (roughly 95–115 h − Mpc) for all the four mass ranges. The decre-ment of the bias below the transition scale depends onthe halo mass: 4% for
M > h − M ⊙ and roughly2% for M > h − M ⊙ . This finding is relevant toBAO measurements from luminous red galaxies, whichare mostly central galaxies in halos of masses from a fewtimes 10 h − M ⊙ to 10 h − M ⊙ (Zheng et al. 2009).To see whether it is a coincidence that the bias dropsnear the trough of the BAO signal, we contrast the re-sults from the BAO simulations (circles) with those fromthe no-BAO simulations (boxes) in Figure 2. Since thelatter changes gradually around 90 h − Mpc, we concludethat the drop of the halo bias in the BAO simulations isassociated with the presence of BAO.The scale dependence of the halo bias around theBAO scale has been studied previously. For exam-ple, one may find a hint for a bump of the halobias around 95 h − Mpc for halos with mass M & h − M ⊙ in Manera, Sheth, & Scoccimarro (2010)and Manera & Gazta˜naga (2011). A peak of the halobias ( M > . × h − M ⊙ ) near 100 h − Mpc is alsodetected in Desjacques et al. (2010).This work improves the statistics of the halo biasaround the BAO scale. The mass-dependent drop of thebias below 90 h − Mpc is well detected for halo massesdown to 10 h − M ⊙ . Our results are consistent with aflat bias over 95–115 h − Mpc. We also extend the scale b ( k ) k ( h Mpc -1 )>10 h -1 M ⊙ >5 × h -1 M ⊙ >2 × h -1 M ⊙ >10 h -1 M ⊙ Figure 3.
Halo bias from the power spectra. The width of eachcurve represents 1- σ uncertainties of the mean bias of the ensemble. of interest up to 180 h − Mpc where halos are anticorre-lated (so is dark matter). Because the correlation func-tions are close to zero in the range of 140–180 h − Mpc,the uncertainties of the halo bias in this range are con-siderably larger than those below 120 h − Mpc. Never-theless, the results are consistent with a flat bias over140–180 h − Mpc.
Halo Bias from Power Spectra
One can also estimate the halo bias from the halopower spectrum P hh ( k ) and the dark matter power spec-trum P mm ( k ) with b PS ( k ) ≡ D(cid:2) P hh ( k ) /P mm ( k ) (cid:3) E . (6)To obtain the power spectra, we assign the ha-los and simulation particles to density grids of 512 cells using the Daubechies D12 wavelet scaling func-tion, which has superior performance for measuringthe power spectrum (Cui et al. 2008). We then take P hh ( k ) = h| ˆ δ h ( k ) | | k | = k i G − V box N − and P mm ( k ) = h| ˆ δ m ( k ) | | k | = k i G − V box N − , where ˆ δ h ( k ) and ˆ δ m ( k ) arethe discrete Fourier modes of the halo grid and the darkmatter grid, respectively, and h . . . i G denotes an averagewithin each grid.Figure 3 presents the halo bias determined from thepower spectra for the same mass ranges as in Figure 2.Although one does not expect the halo bias to be ab-solutely constant on large scales, it is still surprisingthat the bias of M > h − M ⊙ halos falls by 11%from k = 0 . h Mpc − to k = 0 . h Mpc − . For halosof mass M > h − M ⊙ , the bias falls by less than4% over the same scales, consistent with the result of M > . × h − M ⊙ in Pollack, Smith, & Porciani(2012).One may notice that there are slight oscillations in thehalo bias, which are more pronounced for more massivehalos. This behavior is discussed in the next subsection. Damping of BAO and Bias Modulation
We use the no-BAO simulations as references to exam-ine the damping of the BAO signal in Fourier space. Thehalo and dark matter power spectra from each simula-tion with BAO are divided, respectively, by those from P / P N B ( k ) Halos (M>10 h -1 M ⊙ )Dark matter b / b N B ( k ) k ( h Mpc -1 ) Figure 4.
Upper panel : Ratio of the power spectrum from theBAO simulations to that from the no-BAO simulations. Filledcircles represent dark matter, and open circles represent
M > h − M ⊙ halos. Lower panel : Ratio of the halo bias in the BAOsimulations to that in no-BAO simulations for
M > h − M ⊙ halos. the corresponding no-BAO simulation. Because the pairof simulations share the same random seed for their ini-tial conditions, the sample variance is greatly reduced.The power spectrum ratios are shown in the upper panelof Figure 4 for M > h − M ⊙ halos ( P hh /P hhNB , opencircles) and dark matter ( P mm /P mmNB , filled circles). Onecan see that the BAO signal in the massive halo powerspectrum suffers less damping than that in the dark mat-ter power spectrum.There is a subtle difference between Figure 1 and Fig-ure 4. Figure 1 illustrates a mismatch between the shapeof the halo correlation functions and that of dark mat-ter around the BAO scale. It does not associate the verypresence of BAO with the mismatch. The latter could bean intrinsic difference between the halos and dark mat-ter that happens to occur around the BAO scale. Fig-ure 3 establishes such a link, but the uncertainty is alittle high for M > h − M ⊙ halos. The power spec-trum ratios in Figure 4 suppress the intrinsic differencebetween halos and dark matter and reduce uncertaintiesarising from the sample variance. It demonstrates for M > h − M ⊙ halos that the difference between thehalo and dark matter BAO signals is truly due to thedifference in the BAO damping mechanism.By definition, the above effect is attributed to the halobias. The lower panel of Figure 4 shows the ratio ofthe halo bias in the BAO simulations to that in no-BAOsimulations for M > h − M ⊙ halos (same as the halodata divided by the dark matter data in the upper panel).The result oscillates in phase with BAO with an ampli-tude of 0 . DISCUSSION
We have demonstrated with high significance that theBAO signal in the halo distribution is dependent on thehalo mass. In the correlation function, the relative am-plitude of the BAO peak of