GALAXY BIAS AND ITS EFFECTS ON THE BARYON ACOUSTIC OSCILLATION MEASUREMENTS

We generate high force resolution N-body simulations using the ABACUS code by M. Metchnik & P. Pinto (2011, in preparation). The ABACUS code uses a new method to compute N-body forces for periodic boundary conditions by computing the force on a particle from the rest of the simulation and an infinite sum of periodic images. The force on a particle is calculated by dividing the simulation into near field and far field. The near field force is computed by a direct O(N²) method with a gravitational softening length of 0.0488 h⁻¹ Mpc. The far-field force is computed by summing the forces resulting from the multipole expansion of the mass distribution within a grid of distant cells, aided by a cyclic convolution to speed the calculation.

We assume concordance cosmology consistent with the WMAP5+SN+BAO results (Komatsu et al. 2009): Ω_m = 0.279, $\Omega _\Lambda = 0.721$, h = 0.701, Ω_b = 0.0462, n_s = 0.96, σ₈ = 0.817. We used the second-order Lagrangian perturbation theory (2LPT) based IC code by Sirko (2005) to generate the initial conditions. In particular, we use the linear theory density field at z = ∞ and use 2LPT code to generate the density field at z = 50. The initial (z = ∞) density field is computed by back-evolving the z = 0 output from CMBFAST assuming a matter dominated universe. When we compare our results to the initial density field, we always use the initial linear theory density field. We evolve the density fields from z = 50 to z = 1 using the ABACUS code and calculate the spherically averaged power spectrum in both real space and redshift space. We generate 44 simulations each containing 1024³ particles in a 1 h⁻¹ Gpc box giving us a total volume of 44 h⁻³ Gpc³. The resolution of our simulations gives us a particle mass of 7.2 × 10¹⁰ h⁻¹ M_☉. Note that this is the same simulation set as the G1024 set presented in Seo et al. (2010). Figure 1 shows the effects of structure evolution on the BAOs in the power spectrum at z = 1. In this figure, the power spectrum is divided by the no-wiggle power spectrum P_nwl(k), which is the matter density power spectrum without the BAO harmonic features (Eisenstein & Hu 1998). We see two distinct harmonic acoustic peaks before the acoustic oscillations get erased by nonlinear structure growth on small scales.

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Perturbation theory explains how overdensities in matter grow while dark matter simulations help us better understand nonlinear effects. In real surveys, of course, we do not observe the dark matter particles but the galaxies residing in dark matter halos. We assume that the galaxy population are tracers of the underlying dark matter distribution. Since galaxies form in the highest density regions of dark matter halos, they represent a biased version of the matter density. It is critical that we understand how biased tracers of the matter distribution affects the acoustic scale. To examine the effects of galaxy populations, we generated various HODs and applied them to our simulations (Peacock & Smith 2000; Berlind & Weinberg 2002). The HOD models allow us to use various mass tracers (galaxy populations) to study the acoustic scale. To generate the HODs, we use a simple friends-of-friends (FoFs) program (Davis et al. 1985) to identify collapsed dark matter halos with a linking length of b_FoF = 0.16. We populate each halo with a central and satellite galaxies using the following formula:

$$\begin{equation} N_{g}(M) = \left\lbrace \begin{array}{@{}ll@{}}0 & \mbox{if $M < M_{{\rm cen}}$} \\\\[-8pt] 1 + \mbox{Poisson}(M/M_{{\rm sat}}) & \mbox{if $M > M_{{\rm cen}}$}, \end{array}\right. \end{equation} \tag{ 1 }$$

where M is the halo mass, M_cen is the minimum mass needed to have a central galaxy, Poisson(M/M_sat) is an integer randomly chosen from a Poisson distribution with mean M/M_sat, and M_sat is the minimum mass to have at least one satellite galaxy (Guzik & Seljak 2002; Berlind et al. 2003; Kravtsov et al. 2004; Zheng et al. 2005). Note that P(M/M_sat) = 0 for M < M_sat. To test the effects of various galaxy populations, we vary M_cen and M_sat to generate 12 different HOD models whose properties and definitions are detailed in Table 1. By varying the mass thresholds M_cen and M_sat, we obtain HOD models with different numbers of galaxies. For each central galaxy threshold, M_cen, we vary the satellite mass threshold and then slightly adjust M_cen to give us the total number of galaxies: 2 × 10⁶, 1 × 10⁶, 3 × 10⁵, and 1 × 10⁵. The central galaxy is assigned the halo's center of mass position and velocity. We then randomly pick a corresponding number of halo particles and assign their positions and velocities to the satellites.

Thus, each HOD identifies halos based on the mass threshold and replaces the halo by a central galaxy and satellite galaxies (based on the satellite threshold) and uses these galaxies for any analysis. We use the various HOD models as biased tracers of the density fields and explore their effects on the acoustic scale. Since we seek to understand the effects of various galaxy populations or HOD models, we use the "mass" case, where we do not apply any HOD, to be our base case. A full analysis of the mass case is presented in Seo et al. (2010). The different number densities corresponding to each HOD also allow us to explore the effect of shot noise on the acoustic scale.

We compute the power spectrum for each of the 44 simulations following the method used by Seo et al. (2008, 2010). We use χ² analysis to fit the spherically averaged power spectrum, P_obs, to the template power spectrum P_m(k):

$$\begin{equation} P_{\rm obs}= B(k)P_m(k/\alpha) + A(k), \end{equation} \tag{ 2 }$$

where B(k) represents scale-dependent bias and redshift distortions, A(k) represents anomalous power such as shot noise and additive terms of non-linear growth, and α represents the ratio of the linear acoustic scale to the measured acoustic scale. Therefore, α − 1 measures the shift in the acoustic scale, i.e., the error in the distance inferred if one used Equation (4) as the template. α > 1 implies that the measured acoustic scale is shifted toward smaller scales or larger Fourier wavenumbers. In this paper, we quote our shifts in the acoustic scale as α − 1(%) in percent. Our template power spectrum P_m(k) is constructed by modifying the linear power spectrum. We use a nonlinear parameter Σ_M that degrades the BAO peaks in the linear power spectrum to account for nonlinear effects and redshift distortions. The form of the template power spectrum is given by

$$\begin{equation} P_m(k) = [P_{\rm lin}(k) - P_{\rm nwl}(k)]\exp \big({-}k^2\Sigma _M^2\big/2\big) + P_{\rm nwl}(k), \end{equation} \tag{ 3 }$$

where P_lin(k) is the linear power spectrum and P_nwl(k) is the no-wiggle power spectrum as described by Eisenstein & Hu (1998). Due to the large degree of polynomials used to fit A(k) and B(k) (Equation (4)), our results are not sensitive to our value of Σ_M. We use the fiducial values for Σ_M given by the Zel'dovich approximation: Σ_M5.3 h Mpc⁻¹ for real space and 7.0 h Mpc⁻¹ for redshift space. As shown by Seo et al. (2008), the fitting results are not sensitive over a range of Σ_M values (ΔΣ_M = ±2 h Mpc⁻¹) due to the large degree of polynomials used to fit A(k) and B(k). We use a seventh-order polynomial for A(k) and a second-order polynomial for B(k) for both real space and redshift space and fit over 40.02  h Mpc⁻¹ < k < 0.35 h Mpc⁻¹:

$$\begin{eqnarray} A(k) & = & a_0 + a_1k + a_2k^2 + \cdots \ + a_7k^7\; {\rm and} \nonumber \\ B(k) & = & b_0 + b_1k + b_2k^2. \end{eqnarray} \tag{ 4 }$$

In order to measure the scatter in α and therefore the scatter in the acoustic scale, we use a modified bootstrap method. We generate 1000 subsamples by randomly selecting M boxes without replacement out of a total of N simulations. We then perform a χ² fitting to get the best-fit α for the individual subsamples and find the mean α and the scatter. The scatter in the measured α's is rescaled by a factor of $\sqrt{M/(N-M)}$ to reflect the scatter in the mean α. For our resampling method, we have N = 44 and we choose M = 22 so that the rescaling factor is unity. To analyze the scatter in α, we fit each simulation assuming a covariance matrix of the power spectrum given by a Gaussian random field. This method reflects any non-Gaussianity in the density field in the scatter in the best-fit α's between different simulations.

Large-scale gravitational forces cause large-scale velocity fields between the overdensities. These velocity fields tend to broaden the acoustic peak and degrade the BAO measurement. We employ a simple reconstruction scheme introduced by Eisenstein et al. (2007). This reconstruction scheme is based on the Zel'dovich approximation and models the large-scale velocity fields to attempt to restore the BAO information and restore information about the initial linear density field.

In linear perturbation theory, we can estimate the displacements of mass particles based on the density perturbations using the Zel'dovich approximation. With reconstruction, we estimate the bulk flows based on the large-scale information of the observed nonlinear density field and undo the large-scale velocity fields to recover the portion of BAO that has been degraded. The Lagrangian particle position $\vec{r}$ can be mapped to an Eulerian particle position $\vec{x}$ by the displacement $\vec{q}$:

$$\begin{equation} \vec{x} = \vec{r} + \vec{q}. \end{equation} \tag{ 5 }$$

The density in Eulerian coordinates is transformed to the density in Lagrangian coordinates through the Jacobian:

$$\begin{eqnarray} \rho _0 \; d\vec{r} & = & \rho \; d\vec{x} \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \frac{\rho }{\rho _0} & = & 1 + \delta = \frac{d\vec{r}}{d\vec{x}} = J^{-1} = \frac{1}{{\rm Det}\big(\delta _{ij}^K + \partial q_i/\partial r_j\big)}, \end{eqnarray} \tag{ 7 }$$

where ρ₀ is the Lagrangian density and is the same as the Eulerian mean density, δ is the Eulerian overdensity, and δ^K is the Kronecker delta function. If we expand the expression to linear order, it becomes

$$\begin{equation} \delta = -\nabla \cdot \vec{q}. \end{equation} \tag{ 8 }$$

In the linear regime growing mode in CDM cosmologies, $\vec{q}$ is a curl-free field. Hence, we can define a scalar field ϕ such that

$$\begin{equation} \vec{q} = -\nabla \phi. \end{equation} \tag{ 9 }$$

In our reconstruction scheme, we are looking to undo large-scale velocity flows and thus we smooth the small-scale perturbations by using a Gaussian filter. Thus, we enter only the large-scale information to derive the bulk flows. In our analysis, we use a Gaussian smoothing width of R = 14 h⁻¹ Mpc to obtain the real-space galaxy density field δ^real in configuration space. Using Equations (8) and (9), we can express ϕ in terms of δ^real as follows:

$$\begin{eqnarray} &&\nabla ^2 \phi ^{\rm real} = \frac{\delta ^{\rm real}}{b}\ {\rm and}\qquad\qquad\qquad \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} \phi ^{\rm real} & = & {-}\!\int \frac{d^3k}{(2\pi)^3}\frac{\tilde{\delta }^{\rm real}}{b}\frac{1}{k^2}e^{i\vec{k}\cdot \vec{r}}, \end{eqnarray} \tag{ 11 }$$

where $\tilde{\delta }^{\rm real}$ is the real-space smoothed density field in Fourier space. While we use Equation (11) to solve Equation (9) in Fourier space, it can also be easily solved in configuration space as a central force problem. Hence, we call the displacement field derived from solving Equation (9) the trivial displacement field $\vec{q}_{\rm trivial}^{\rm real}$, which is given by

$$\begin{eqnarray} \nabla \cdot \vec{q}_{\rm trivial}^{\rm real} & = & {-}\frac{\delta ^{\rm real}}{b}\ {\rm and\ therefore} \nonumber \\ \vec{q}_{\rm trivial}^{\rm real} & = & {-}\nabla \phi ^{\rm real}. \end{eqnarray} \tag{ 12 }$$

In real space, this trivial displacement field is the desired displacement field. In our simulations, we displace the data points and the random points by the displacement field $\vec{q}_{\rm trivial}^{\rm real}$. By displacing both the data and random points, we ensure that the amplitude of the overdensities remains the same while removing the effects of the large-scale velocity field.

In galaxy redshift surveys, the density field we observe is subject to redshift distortions. We can relate the redshift-space density field $\tilde{\delta }^{\rm red}$ in Fourier space to that in real space as follows:

$$\begin{equation} \tilde{\delta }^{\rm red} = (1 + \beta \mu ^2)\tilde{\delta }^{\rm real}, \end{equation} \tag{ 13 }$$

where μ = k_z/|k| and z is the line-of-sight direction. We can then compute the redshift-space scalar field ϕ^red from Equations (11) and (13):

$$\begin{equation} \phi ^{\rm red} = -\int \frac{d^3k}{(2\pi)^3}\frac{\tilde{\delta }^{\rm red}}{b}\frac{1}{k^2}\frac{1}{1 + \beta \mu ^2}e^{i\vec{k}\cdot \vec{r}}. \end{equation} \tag{ 14 }$$

Note that ϕ^red is our estimate for the scalar field that generates the real-space displacement. Next we relate ϕ^red to the redshift-space galaxy density field

$$\begin{eqnarray} \nabla \cdot \vec{q}_{\rm trivial}^{\rm red} &=& {-}\frac{\delta ^{\rm red}}{b} = {-}\left(\nabla ^2 + \beta \frac{\partial ^2}{\partial z^2}\right)\phi ^{\rm red}\nonumber\\ & =& {-}\nabla \cdot \left(\nabla \phi ^{\rm red} + \beta \hat{z}\frac{\partial \phi ^{\rm red}}{\partial z}\right). \end{eqnarray} \tag{ 15 }$$

It follows from Equation (15) that $\vec{q}_{\rm trivial}^{\rm red}$ can be expressed as

$$\begin{equation} \vec{q}^{\rm red}_{\rm trivial} = -\nabla \phi ^{\rm red} + \beta \hat{z}\frac{\partial \phi ^{\rm red}}{\partial z}. \end{equation} \tag{ 16 }$$

From Equation (16), we can relate $\vec{q}_{\rm trivial}^{\rm red}$ to $\vec{q}^{\rm real}_{\rm trivial}$ as follows:

$$\begin{eqnarray} \vec{q}_{\rm trivial, z}^{\rm red} & = & (1 + \beta)\vec{q}_{\rm trivial, z}^{\rm real},\\ \nonumber \vec{q}_{\rm trivial, x,y}^{\rm red} & = & \vec{q}_{\rm trivial, x,y}^{\rm real}, \end{eqnarray} \tag{ 17 }$$

where x and y are the directions orthogonal to the line of sight. Finally, we note that the redshift-space displacement field is distorted along the z-direction and hence we must modulate the field by a factor of 1 + f in the z-direction, where f = dln D/dln a is the logarithmic derivative of the growth function. The redshift-space displacement field is then given by

$$\begin{eqnarray} \vec{q}_{ z}^{\rm red} & = & \frac{1 + f}{1 + \beta }\vec{q}^{\rm red}_{\rm trivial; z}, \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} &&\vec{q}_{ x,y}^{\rm red} = \vec{q}^{\rm red}_{\rm trivial; x,y}.\quad\ \ \ \end{eqnarray} \tag{ 19 }$$

Thus, in redshift space, we displace our data points and random points by the displacement field $\vec{q}^{\rm red}$ given by Equation (19). In later sections, we demonstrate that this simple reconstruction technique improves the correlation between density fields, reduces the measured shift in the acoustic scale, and reduces the scatter around the measured shift. We plan to explore the effects of moving the data and random points by different amounts on reconstruction in future work.

In linear perturbation theory, the Fourier modes grow independently of each other, e.g., there is no mode-coupling. In this section, we test how close we are to linear theory on a mode-by-mode basis. We look at the correlation of the initial and final (z = 1) density fields for each mode via the propagator, which compares the amount of the initial density field retained in each mode of the final density field. We derive the propagator in the presence of galaxy bias and compare it to the propagator for the mass case presented by Seo et al. (2010) to estimate the additional BAO signal erased by galaxy bias:

$$\begin{equation} G(k) = \frac{1}{b \times P_{{\rm initial}}(k)} \left\langle \tilde{\delta }_{{\rm initial}}(\vec{k}) \frac{\tilde{\delta }_{{\rm final}}^*(\vec{k})D(z=\infty)}{D(z=1)(1 + \beta \mu ^2)}\right\rangle. \end{equation} \tag{ 20 }$$

Here, δ_initial is the linear theory density field at z = ∞, δ_final is the density field at z = 1, D(z) is the growth function, P_initial(k) is the power spectrum of the initial density field, and b is the clustering bias for a given biased tracer.

Matter growth is linear on large scales and hence we expect that the initial and final density fields should be extremely correlated regardless of the mass tracer used. In the absence of nonlinear effects and galaxy bias, we expect perfect correlation between low redshift and high redshift density fields which is represented by G(k) = 1. Figure 2 shows the propagator, G(k), for real space and redshift space for HOD1a. From Figure 2, we note that at k ≈ 0.1 h Mpc⁻¹, the wavenumber around which most of the acoustic information is contained, both the real and redshift-space density fields are correlated extremely well with the initial density field: G(k) ⩾ 0.9. As expected, the nonlinear effects and redshift distortions become increasingly dominant at smaller scales causing the steep drop-off in the propagator. We also see that the real-space density field is correlated with the initial density field up to much smaller scales (larger k) than redshift space. The correlation in redshift space is lower on large scales than real space due to redshift distortions.

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We next use the HOD models from Section 2.2 to evaluate the impact of galaxy bias on the recovery of the initial density field. By using the fact that the initial and final densities must be correlated with G(k) = 1 at small k, we can derive the bias for every HOD. In this paper, we use the real-space G(k) to derive the bias. The bias is reflected in the propagator as a multiplicative factor. If we force b = 1 in Equation (20), then G(k) will converge to the bias at small wavenumber. We fit a constant function to the real-space propagator, G(k), for k < 0.08  h Mpc⁻¹ to compute the bias for each HOD.

For each HOD, we want to estimate the amount of noise increase due to bias relevant to the signal. We define an effective number density (n_eff) by Equation (21) and take any deviation from linear bias into account rather than merely using the shot noise. By definition, this also parameterizes the degree of the scale dependence of the bias:

$$\begin{eqnarray} \frac{1}{n_{\rm eff}} &=& P_{\rm HOD}(k = 0.2 \,h{\rm \;Mpc^{-1}}) - {\rm bias}^2 \nonumber\\ &&\times P_{\rm lin}(k = 0.2 \,h{\rm \;Mpc^{-1}}), \end{eqnarray} \tag{ 21 }$$

where P_lin is the linear power spectrum and P_HOD is the measured HOD power spectrum. Table 1 gives the definitions, HOD parameters, biases, effective number density, and n_effP(k = 0.2 h Mpc⁻¹) for each HOD. From Table 1, we see that we span a wide range of effective number densities (10⁻³ to 10^−4.1) and a range of n_effP (2.6 − 0.6).

Since we are looking at effects of various HOD models, we calibrate our results against the mass case (no HOD), which along with three of the twelve HOD models are shown in Figure 3 for real space (upper panel) and redshift space (lower panel). The three HOD models shown in Figure 3 represent the spread in the propagators for all HOD models and the other HOD models give very similar results. In real space, we see that the G(k) plots for various HOD models are very similar with each other. The propagators for different HOD models diverge from each other at large k. We attribute this to shot noise which depends on the number density of each HOD. Hence, the effect of bias on the propagator in real space is very small at the acoustic wavenumbers. In redshift space, the separation between the HOD models is more pronounced because of the finger-of-God (FoG) distortions in redshift space. Since different HOD models pick out halos at different mass thresholds, the FoG distortions are different for each HOD model. However, we note that in both real space and redshift space, the separation between the HOD models at the acoustic wavenumbers is small.

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Figure 3 also shows the important result that the propagators for the HOD models are very similar to the mass case. In fact, we see that the biased tracers are slightly better than the mass case. This is consistent with a similar result found by Noh et al. (2009). We can explain this effect as follows. The biased cases replace the dark matter halo with a central galaxy and fewer and satellite galaxies. The central galaxies have the same proper velocity as the center of mass of their halo while the satellite galaxies have the proper velocity of the corresponding dark matter particles. Therefore, the biased cases have experienced less random motion than the mass case. Hence, the halos in the mass case are more extended than in the biased case. In the mass case, we smear out the density field and erase some of the correlation with the initial field compared to the biased cases. We see that this effect is amplified in redshift space due to redshift distortions. While the propagator results are consistent with our explanation, we do not rule out effects of halo clustering as explored by Angulo et al. (2008).

We next investigate how reconstruction affects the density correlations in both real space and redshift space. Figure 4 shows the comparison of the propagator with and without reconstruction for three different HOD models and the mass case. We see that reconstruction restores the correlation between the initial and final (z = 1) density fields to larger wavenumber or smaller scales even for biased tracers with a realistic level of shot noise. In fact for G(k) = 0.8 the wavenumber increases by a factor of 1.5–2 in real space and redshift space. Reconstruction further spreads out the HOD models, which is expected as the HOD models have different shot noise levels. As reconstruction restores the correlation with the initial density fields, we expect that reconstruction will help reduce the shifts in the BAO scale and reduce the scatter on measured shifts. We discuss these effects of reconstruction in the following section.

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In a measurement from a redshift survey, α would be the ratio of the acoustic scale in the fiducial cosmology to the measured scale. α = 1 means that the measured acoustic scale is the same as that predicted by linear perturbation theory. In this section, we investigate the effects of realistic galaxy bias on the mean α and the scatter in α using the method described in Section 2.4. The α calculated for each HOD and the mass case are given in Table 2. Figure 5 plots the scatter in α as a function of bias where b = 1 case is the mass case. Figure 6 plots the scatter in α as a function of n_effP whose values are given in Table 1.

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For the mass case, Seo et al. (2010) used lower mass resolution but higher volume simulations (set G576 in that paper) to compute the shift in the acoustic scale and found a significant shift at z = 1: α_mass − 1(%) = 0.13 ± 0.04 in real space and 0.16 ± 0.06 in redshift space. However, the G576 simulations do not have the mass resolution to sufficiently implement our HODs. Seo et al. (2010) use another set of simulations with higher mass resolution (set G1024). In the G1024 set of simulations, they find a smaller, but statistically consistent, shift with larger errors: α_mass − 1(%) = 0.11  ±  0.16 for real space and 0.002 ± 0.23 for redshift space. In this paper, we will compare our results to the G1024 set of simulations from Seo et al. (2010) to be consistent with the mass case. From Table 2, in real space, the lower biased HOD models (b < 2.6, sets 1, 2, and 3) are consistent with the linear acoustic scale (α = 1) within the scatter set by the sample variance of our simulations. Our highest mass HOD models (set HOD4's) show deviation from α = 1: α_HOD4c − 1(%) = 0.57 ± 0.32. We see a similar effect in redshift space, where HOD sets 1 and 2 are consistent with linear theory (α = 1) and HOD sets 3 and 4 show more deviation from linear theory: α_HOD4c − 1(%) = 0.79 ± 0.40. We also note that for the biased tracers, the scatter in α increases with increasing bias, as we are sampling rarer halos and thus shot noise becomes more dominant in the highly biased cases. The scatter in real space ranges from 0.16% to 0.26% while in redshift space, it ranges from 0.22% to 0.43%. Due to redshift distortions, we see that the scatter in redshift space is greater than in real space.

However, once we apply reconstruction, we see that all the α values are consistent with unity: α_HOD4c − 1(%) = −0.07  ±  0.19 in real space and −0.11 ± 0.20 in redshift space. Another important difference between before and after reconstruction is the scatter in α. We see that in all cases the scatter in α is reduced by a factor of 1.5–2 with reconstruction than without reconstruction in both real space and redshift space which is consistent with the propagators shown in Section 3.3. This confirms that even for biased tracers our reconstruction technique is restoring the information and correlation between initial and final density fields on the acoustic scale. This is because reconstruction corrects for the degradation of the BAO signal due to large-scale bulk flows and the mass and biased tracers bulk flows only differ on small scales. From Figure 5, we see that the scatter around α is significantly reduced in real space and in particular redshift space once we apply our simple reconstruction scheme. We also note that the redshift space with reconstruction errors are very close to the real space with reconstruction errors. We see this effect for all HOD models and hence all biased tracers. This shows us that the reconstruction scheme is bias-independent to within the scatter set by the sample variance of our simulations. We see a similar effect in Figure 6 where the scatter decreases as n_effP increases and reconstruction reduces the scatter in α.

To better understand the effects of galaxy population bias on the acoustic scale, we compare the shift in the acoustic scale (α − 1) to the shift from the mass case. We make use of α_HOD − 1(%) versus α_mass − 1(%) plots. With such plots, not only can we analyze how the shifts vary between each HOD, but we can also see how they correlate with the mass case. In these plots, we plot every α − 1 value calculated from the resampling method described above. We show one such plot of HOD 2a versus mass in Figure 7. From the figure, we see that for both real space and redshift space, there is a correlation between the shifts measured with the HOD models and with the mass case. As expected, we see that redshift space has more scatter than real space. However, we see that reconstruction not only constrains the distribution thereby reducing the scatter around the mean measured shift, but it also increases the correlation between the HOD models and mass case.

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In the previous section, we showed that there is a strong correlation between the shifts measured from the biased tracers and those measured from the mass case. This means that we can measure α_HOD–α_mass more precisely than α_HOD alone. This is useful because we can measure α_mass more precisely using simulations with poorer mass resolution. Seo et al. (2010) measured the acoustic scale at z = 1 in redshift space to be α_mass − 1(%) = 0.158 ± 0.061 before reconstruction and α_mass − 1(%) = 0.002 ± 0.030 after reconstruction using the lower resolution, larger volume simulation set (G576). Combining the sets of simulations and precisely measuring α_mass and α_HOD–α_mass allows us to determine α_HOD.

The α_HOD–α_mass values are given in Table 3 and shown in Figure 8. From the table and the figure, we see some deviation from zero difference for the high biased cases which corresponds to low number density HOD models for both real space and redshift space. The difference in redshift space is slightly larger than in real space: α_HOD4c − α_mass(%) = 0.69 ± 0.24 in real space and 0.79 ± 0.31 in redshift space. However, the difference between the shifts from the biased tracers and the mass case, even for the highest biased cases, is about 2.7σ. We also note that the scatter around the difference increases with increasing bias.

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When we apply our reconstruction scheme, we see that there is no difference between the shifts measured by biased tracers and the mass case for both real space and redshift space: α_HOD4c − α_mass(%) = −0.03 ± 0.16 in real space and −0.05 ± 0.16 in redshift space for HOD4c. The difference between the shifts is consistent with zero with errors ⩽0.08% for HOD models 1 and 2. Equally important, we also note that reconstruction reduces the scatter around the difference in the shifts by a factor of 1.5–2 depending on the HOD model. Also, the scatter around the difference in the shifts after reconstruction are the same for both real space and redshift space. This is consistent with our previous result that reconstruction accounts for nonlinear structure formation and these flows for the biased cases and mass tracers differ only at small scales.

From Figure 7, we see that the α_HOD and α_mass values are correlated. In this section, we investigate the slope of the α_HOD − 1 versus α_mass − 1 distribution. If the HOD models and mass cases are perfectly correlated, then we expect the underlying slope to be unity. However, there are two effects than can lead to non-unity slopes. First, if the shifts are slightly uncorrelated between the mass and HOD models, we can see a non-unity slope. Second, even if the shifts are perfectly correlated (underlying unity slope), a difference in the scatter around the mean between the mass case and the HOD models can lead to a non-unity slope.

In Figure 7, we compute the slope by using a linear least-squares fit. However, we know that this slope and the error for the slope are not correct since the linear least-squares method assumes that all the scatter is around the y-axis variable, α_HOD − 1. We test the statistical significance of a non-unity slope by running multiple Monte Carlo simulations with an underlying distribution with unity slope. We then compare the results of the Monte Carlo simulations to our measurements. In detail, we use Monte Carlo simulations given by Equation (22), which has an underlying slope of unity, to calculate the measured slope. Our model is

$$\begin{eqnarray} \alpha _{\rm mass, MC} - 1 & = {\rm Gaussian(}\mu = 0, \sigma = \sigma _{\rm mass}), \nonumber \\ \alpha _{\rm HOD, MC} - 1 & = \alpha _{\rm mass, MC} + {\rm Gaussian(}\mu = 0, \sigma = \sigma _{\rm HOD - mass}),\nonumber\\ \end{eqnarray} \tag{ 22 }$$

where σ_mass is the scatter around α_mass from Table 2 and σ_HOD-mass is the scatter around α_HOD-mass from Table 3. We use the same resampling method as used for our simulations to generate 1000 subsamples by averaging over 22 randomly selected values from 44 values for each subsample given by Equation (22). In order to compute the mean Monte Carlo slope and the scatter around this slope, we create 500 realizations of 1000 subsamples and use linear least-squares fit. In Table 4, we give the slopes for the α_HOD versus α_mass distribution for α_mass at z = 1 and z = ∞ for four different HOD models with the other HOD models giving very similar results. We see that for both redshift space and real space, the low bias and high number density HOD models (HODs 1 and 2) are all consistent with an underlying slope of unity. At the high bias and low number density HOD models (HODs 3 and 4), we start to see some deviation from the slope of unity at the 1.5σ–2.0σ level for z = 1 mass values. However, with reconstruction, we see that all HOD models are consistent with an underlying slope of unity for both z = 1 and z = ∞ mass values within 1σ.

Seo & Eisenstein (2003, 2007) used Fisher matrix analyses to predict the scatter in the acoustic scale available in surveys of a given number density and bias. These models depend on certain assumptions, such as the Gaussianity of the density field up to a cutoff wavenumber, that we can check with N-body simulations. Here, we compare the scatter in the acoustic scale for our HOD models to those predicted by the Fisher matrix model. The Fisher matrix code takes in the number density (Equation (21)), σ₈ (Table 1), Σ_⊥, Σ_∥, and β. For our cosmology, at z = 1, Σ_⊥ = 5.26 is the rms radial displacement across the line of sight and Σ_∥ = 5.26 (real space) and 9.55 (redshift space) is the rms displacement along the line of sight (Seo et al. 2010). When we apply reconstruction, we choose Σ_⊥, Recon = Σ_⊥/2 and Σ_∥, Recon = Σ_∥/2. We know that the N-body analysis tends to overestimate the scatter in redshift space relative to the Fisher matrix code from Seo & Eisenstein (2007) as shown by Takahashi et al. (2009) and Seo et al. (2010). This is because in our N-body analysis we fit a spherically averaged power spectrum, we do not optimally extract the two-dimensional information. Hence, we expect the scatter around the acoustic scale from the N-body simulations to be a factor of $\sqrt{1.16} = 1.08$ greater than the Fisher matrix estimates at z = 1.

Table 2 provides the α − 1 results from our simulations and the Fisher matrix estimates for the scatter are given in brackets. We see that for the low biased HOD models (HODs 1 and 2), the scatter around the mean measured α is very close to the Fisher matrix estimates for real space and redshift space. For the high biased HOD models (HODs 3 and 4), we see that the measured scatter is larger than the Fisher matrix estimates by 20%–30%, which is more than expected. It is unclear if the HOD sets 3 and 4 show larger scatter due to high bias or low n_effP. However, once we apply reconstruction, we have very good agreement between the measured scatter and the Fisher matrix estimates for all HOD models.

Since we measure galaxies in redshift space, we do not know the real-space positions or velocities of galaxies. In this section, we see how well the shifts measured from redshift space are correlated with real-space values. We use the linear least-squares fit to compute the slopes of the distribution as done in Sections 4.2 and 4.4. However, we know that not only are the shifts measured in real space and redshift space correlated, but the scatter around the shifts are also correlated. In Table 5, we quote the linear least-squares slopes for α_redshift versus α_real distribution with and without reconstruction. We expect a non-unity slope for the distributions since we know that the scatter around the redshift-space values are larger than the real-space values. We see that this is true from the values in Table 5. We also see that reconstruction decreases the slope toward unity. We expect this behavior since reconstruction reduces the noise in redshift space that is uncorrelated with real space and vice versa. Figure 9 shows the α_real versus α_redshift plot for HOD2a and the mass case. From such plots, we can compare the shifts derived from real space and redshift space. We see that reconstruction increases the correlation between real space and redshift space and reduces the scatter for both sets of values. We also see that the HOD and mass cases are very similar in their real and redshift space correlation after reconstruction. The other HOD models give very similar results as shown in Table 5.

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When we implement our reconstruction scheme outlined in Section 2.5, we use a smoothing scale of R = 14 h⁻¹ Mpc. We change the smoothing scale to R = 20 h⁻¹ Mpc and rerun our reconstruction scheme. We see no change in either α or σ_α when we change the reconstruction smoothing scale. Thus, we see that reconstruction gives consistent answers over a range of smoothing scales. We also do not see any differences in α_HOD–α_mass values when using the larger smoothing scale for α_HOD.

To implement reconstruction, we need to estimate the value of the galaxy bias for a survey. We explore the effects of inputting a wrong bias for calculating the acoustic scale and reconstruction techniques. We use biases that are incorrect by ±10%, ±20%, and ±40%. The results are plotted in Figure 10. From the figure, we see that changing the bias obviously does not affect the propagator before reconstruction. After reconstruction, the propagator is very close to the actual bias case at the acoustic scale for ±10%. We see that the ±40% case shows moderate deviation from the correct bias case. Thus, we conclude that our reconstruction scheme is insensitive to a large range (up to ±30%) of incorrect input bias. From future weak-lensing and cluster surveys, we expect to know the amplitude of matter clustering well enough for us to know the bias to less than 10%. Hence, we expect this effect to be negligible.

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