Baryon acoustic oscillations in 2D: Modeling redshift-space power spectrum from perturbation theory

We present an improved prescription for the matter power spectrum in redshift space taking proper account of both nonlinear gravitational clustering and redshift distortion, which are of particular importance for accurately modeling baryon acoustic oscillations (BAOs). Contrary to the models of redshift distortion phenomenologically introduced but frequently used in the literature, the new model includes the corrections arising from the nonlinear coupling between the density and velocity fields associated with two competitive effects of redshift distortion, i.e., Kaiser and Finger-of-God effects. Based on the improved treatment of perturbation theory for gravitational clustering, we compare our model predictions with the monopole and quadrupole power spectra of $N$-body simulations, and an excellent agreement is achieved over the scales of BAOs. Potential impacts on constraining dark energy and modified gravity from the redshift-space power spectrum are also investigated based on the Fisher-matrix formalism, particularly focusing on the measurements of the Hubble parameter, angular diameter distance, and growth rate for structure formation. We find that the existing phenomenological models of redshift distortion produce a systematic error on measurements of the angular diameter distance and Hubble parameter by 1%–2% , and the growth-rate parameter by $\sim 5\%$, which would become non-negligible for future galaxy surveys. Correctly modeling redshift distortion is thus essential, and the new prescription for the redshift-space power spectrum including the nonlinear corrections can be used as an accurate theoretical template for anisotropic BAOs.

Figure 1

Ratio of power spectra to smoothed reference spectra in redshift space, $P_{\ell }^{(\mathrm{S})}(k)/P_{\ell ,\mathrm{no}\mathrm{\text{−}}\mathrm{wiggle}}^{(\mathrm{S})}(k)$. $N$-body results are taken from the wmap5 simulations of Ref. 34. The reference spectrum $P_{\ell ,\mathrm{no}\mathrm{\text{−}}\mathrm{wiggle}}^{(\mathrm{S})}$ is calculated from the no-wiggle approximation of the linear transfer function with the linear theory of the Kaiser effect taken into account. Short dashed and dot-dashed lines, respectively, indicate the results of one-loop PT and Lagrangian PT calculations for the redshift-space power spectrum [Eqs. (5, 6)].

Figure 2

Same as in Fig. 1, but here we plot the results of phenomenological model predictions. The three different predictions depicted as solid, dashed, dot-dashed lines are based on the phenomenological model of redshift distortion (9) with various choices of Kaiser and Finger-of-God terms [Eqs. (10, 11)]. The left panel shows the monopole power spectra ($\ell =0$), and the right panel shows the quadrupole spectra ($\ell =2$). In all cases, the one-dimensional velocity dispersion ${\sigma }_{\mathrm{v}}$ was determined by fitting the predictions to the $N$-body simulations. In each panel, the vertical arrows indicate the maximum wave number $k_{1\%}$ for improved PT prediction including up to the second-order Born approximation [see Eq. (12) for a definition].

Figure 3

Redshift evolution of velocity dispersion ${\sigma }_{\mathrm{v}}$ determined by fitting the predictions of monopole and quadrupole power spectra to the $N$-body results. The solid line represents the linear-theory prediction, while the symbols indicate the results obtained by fitting the models of redshift distortion with various choices of Kaiser and damping terms (see Fig. 2).

Figure 4

Contributions of power spectrum corrections coming from the $A$ and $B$ terms divided by the smooth reference power spectrum, $P_{\ell ,\mathrm{corr}}^{(\mathrm{S})}(k)/P_{\ell ,\mathrm{no}\mathrm{\text{−}}\mathrm{wiggle}}^{(\mathrm{S})}(k)$ [Eq. (22)]. We adopt the Gaussian form of the damping function $D_{\mathrm{FoG}}$ with ${\sigma }_{\mathrm{v}}$ computed from linear theory [see Eq. (7)]. Left and right panels, respectively, show the monopole and quadrupole power spectra at redshifts $z=3$ and 1.

Figure 5

Same as in Fig. 2, but here we adopt a new model of redshift distortion (18). Solid and dashed lines represent the predictions for which the spectra $P_{\delta \delta }$, $P_{\delta \theta }$, and $P_{\theta \theta }$ are obtained from the improved PT including the correction up to the second-order Born correction, and one-loop calculations of the standard PT, respectively. In both cases, the corrections $A$ and $B$ given in Eqs. (19, 20) are calculated from standard PT results (see Appendix ). The vertical arrows indicate the maximum wave number $k_{1\%}$ defined in Eq. (12), for standard PT and improved PT (from left to right).

Figure 6

Contribution of each term in the redshift-space power spectrum. For monopole ($\ell =0$, left) and quadrupole ($\ell =2$, right) spectra of the improved model prediction at $z=1$ shown as solid lines of Fig. 5, we divide the total power spectrum $P_{\mathrm{total}}^{(\mathrm{S})}$ (solid) into the three pieces as $P_{\mathrm{total}}^{(\mathrm{S})}=P_{\mathrm{Kaiser}}^{(\mathrm{S})}+P_{\mathrm{corr},A}^{(\mathrm{S})}+P_{\mathrm{corr},B}^{(\mathrm{S})}$, and each contribution is separately plotted dividing by smoothed reference spectra, $P_{\ell ,\mathrm{no}\mathrm{\text{−}}\mathrm{wiggle}}^{(\mathrm{S})}$. Here, the spectrum $P_{\mathrm{Kaiser}}^{(\mathrm{S})}$ (dotted) is the contribution of the nonlinear Kaiser term (10) convolved with the Finger-of-God damping $D_{\mathrm{FoG}}$, and the corrections $P_{\mathrm{corr},A}^{(\mathrm{S})}$ and $P_{\mathrm{corr},B}^{(\mathrm{S})}$ are those given by Eq. (22).

Figure 7

Same as in Fig. 3, but here we adopt the new model of redshift distortion in estimating ${\sigma }_{\mathrm{v}}$. The filled triangles and circles are the results obtained from predictions based on standard PT and improved PT calculations, respectively (see dashed and solid lines in Fig. 5).

Figure 8

Quadrupole-to-monopole ratios for the redshift-space power spectrum, $P_2^{(\mathrm{S})}(k)/P_0^{(\mathrm{S})}(k)$, given at $z=3$, 2, 1, and 0.5 (from top to bottom). Solid and dashed lines, respectively, represent the predictions based on the new model of redshift distortion combining improved PT and standard PT calculation to estimate the three different power spectra $P_{\delta \delta }$, $P_{\delta \theta }$, and $P_{\theta \theta }$. Dot-dashed lines are the results based on the phenomenological model neglecting the corrections, which correspond to solid lines in Fig. 2 (i.e., nonlinear $P_{\mathrm{Kaiser}}+\mathrm{\text{Gaussian}}$, $D_{\mathrm{FoG}}$). The vertical arrows indicate the maximum wave number $k_{1\%}$ for standard PT (left, green) and improved PT (right, magenta).

Figure 9

Results of MCMC analysis using the model of redshift distortion with and without corrections (depicted as blue and red lines, respectively). Based on the power spectrum template (25), we derive the posterior distribution for the parameters $D_A$, $H$, and $f$ from the monopole and quadrupole spectra of $N$-body simulations at $z=1$, marginalized over the one-dimensional velocity dispersion ${\sigma }_{\mathrm{v}}$ and linear bias parameter $b$. Top left, middle center, and bottom right show the marginalized posterior distribution for $D_A/D_{A,\mathrm{fid}}$, $H/H_{\mathrm{fid}}$, and $f$. Shaded regions indicate the 1% interval around the fiducial values. Middle left, bottom left, and bottom center plot the two-dimensional $1\mathrm{\text{−}}\sigma$ errors on the surfaces $(H/H_{\mathrm{fid}},f)$, $(D_A/D_{A,\mathrm{fid}},H/H_{\mathrm{fid}})$, and $(f,H/H_{\mathrm{fid}})$. Note that in estimating the likelihood function (28), we adopted the linear theory to calculate the covariance matrix ${\mathrm{Cov}}_{\ell ,{\ell }^{\prime }}$, including the shot-noise contribution with ${\overline{n}}_{\mathrm{g}}=5\times {10}^{-4}{h}^3{\mathrm{Mpc}}^{-3}$ and assuming an idealistically large survey volume $V_s=20h^{-3}{\mathrm{Gpc}}^3$ (see Appendix  for an explicit expression).

Figure 10

Expected two-dimensional contours on marginalized errors around the best-fit values of $D_A/D_{A,\mathrm{fid}}$ vs $H/H_{\mathrm{fid}}$ (bottom left), $f$ vs $H/H_{\mathrm{fid}}$ (bottom right) and $D_A/D_{A,\mathrm{fid}}$ vs $f$ (top left) at $z=1$, obtained from the full shape of the redshift-space power spectrum. The maximum wave number for parameter estimation is chosen as $k_{\max }=0.12$ (left panels) and $0.2h{\mathrm{Mpc}}^{-1}$ (right panels), so as to satisfy the condition $k_{\max }<k_{1\%}$ for standard PT and improved PT, respectively. In each panel, open and shaded contours indicate the two-dimensional errors for the surveys with $V_s=4$ and $20h^{-3}{\mathrm{Gpc}}^3$.