Regularized cosmological power spectrum and correlation function in modified gravity models

Based on the multipoint propagator expansion, we present resummed perturbative calculations for cosmological power spectra and correlation functions in the context of modified gravity. In a wide class of modified gravity models that have a screening mechanism to recover general relativity (GR) on small scales, we apply the eikonal approximation to derive the governing equation for resummed propagator that partly includes the nonperturbative effect in the high-$k$ limit. The resultant propagator in the high-$k$ limit contains the new corrections arising from the screening mechanism as well as the standard exponential damping. We explicitly derive the expression for new high-$k$ contributions in specific modified gravity models, and find that in the case of $f(R)$ gravity for a currently constrained model parameter, the corrections are basically of the subleading order and can be neglected. Thus, in $f(R)$ gravity, similarly to the GR case, we can analytically construct the regularized propagator that reproduces both the resummed high-$k$ behavior and the low-$k$ results computed with standard perturbation theory, consistently taking account of the nonlinear modification of gravity valid at large scales. With the regularized multipoint propagators, we give predictions for power spectrum and correlation function at one-loop order, and compare those with $N$-body simulations in $f(R)$ gravity model. As an important application, we also discuss the redshift-space distortions and compute the anisotropic power spectra and correlation functions.

Figure 1

Resummed linear propagator in $f(R)$ gravity model. Left panel shows the redshift evolution of the propagator at $k=1h{\mathrm{Mpc}}^{-1}$ (left) and $10h{\mathrm{Mpc}}^{-1}$ (right), while right panel plots the scale dependence of the propagator at $z=0$. Top panels plot the ratio of propagator in $f(R)$ gravity to that in GR for density (red) or velocity-divergence fields (blue). Dashed and solid lines respectively represent the result with and without new correction arising from the screening effect, i.e., $[g_{ab}+\delta g_{ab}^{(3)}{]}^{\mathrm{f}(\mathrm{R})}{u}_b/g_{ab}^{\mathrm{\Lambda }\mathrm{CDM}}{u}_b$ and $g_{ab}^{\mathrm{f}(\mathrm{R})}{u}_b/g_{ab}^{\mathrm{\Lambda }\mathrm{CDM}}{u}_b$. On the other hand, to see the size of the correction, bottom panels show the fractional difference between the propagators with and without the correction, i.e., $[\delta g_{ab}^{(3)}{]}^{\mathrm{f}(\mathrm{R})}{u}_b/g_{ab}^{\mathrm{f}(\mathrm{R})}{u}_b$. For all panels, we assume $f(R)$ gravity of the functional form in Eq. (23), and the model parameter is set to $|f_{R,0}|=10^{-4}$. In computing the propagators, we set the initial redshift to $z_{\text{ini}}=99$, and adopt the cosmological parameters; ${\mathrm{\Omega }}_{\mathrm{m}}=0.24$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.76$, ${\mathrm{\Omega }}_{\mathrm{b}}=0.0481$, $h=0.73$, $n_s=0.961$, ${\sigma }_8=0.801$ [44].

Figure 2

Two-point propagator of density field, ${\mathrm{\Gamma }}_1^{(1)}(k)$, measured in $N$-body simulations at $z=0$ (open circles), 0.5 (filled squares), 1 (filled triangles), and 2 (crosses). To compare with simulations, regularized propagators at tree level and one-loop order [Eqs. (51) and (55)] are also shown in each panel, depicted as solid and dotted lines, respectively. Left and right panels, respectively, show the results in GR and $f(R)$ gravity with $|f_{R,0}|=10^{-4}$. In top panels, the propagators are normally plotted as function of wave number. On the other hand, to clarify the high-$k$ limit behaviors, the normalized propagators ${\mathrm{\Gamma }}^{(1)}/D_{+}$ is plotted in upper part of the bottom panel as function of $k^2$ in semilog scale. For reference, lower part of the bottom panel shows the fractional difference between simulation results and regularized one-loop predictions, ${\mathrm{\Gamma }}_{\text{sim}}^{(1)}(k)/{\mathrm{\Gamma }}_{\mathrm{PT}}^{(1)}(k)-1$.

Figure 3

Ratio of two-point propagator in $f(R)$ gravity to that in GR, ${\mathrm{\Gamma }}_{1,\mathrm{f}(\mathrm{R})}^{(1)}(k)/{\mathrm{\Gamma }}_{1,\mathrm{GR}}^{(1)}(k)$. Symbols and line types are the same as those in Fig. 2.

Figure 4

Power spectrum of the density field in real space multiplied by $k^{3/2}$, $k^{3/2}{P}_{11}(k)$, at $z=0$, 0.5, 1, and 2 (from top to bottom). Left panel shows the results in GR, while right panel presents the cases in $f(R)$ gravity with $|f_{R,0}|=10^{-4}$. Solid and dotted lines are RegPT predictions at one-loop and linear theory predictions, respectively. Note that the error bars indicated in $N$-body results are the dispersion of the power spectrum amplitude over the modes in each Fourier bin.

Figure 5

Difference of the real-space correlation function between $f(R)$ gravity with $|f_{R,0}|=10^{-4}$ and GR, $\mathrm{\Delta }\xi (r)={\xi }_{f(R)}(r)-{\xi }_{\mathrm{GR}}(r)$. From left to right panels, the results at $z=0$, 0.5, and 1 are shown. In each panel, filled circles represent the results from $N$-body simulations, while the solid, dashed, and dotted lines are estimated from the tree-level expression of the correlation function, $\xi (r)\simeq [{\mathrm{\Gamma }}^{(1)}{]}^2\otimes {\xi }_0(r)$. Here, ${\xi }_0$ indicates the initial correlation function for which we computed numerically with the random initial data of $N$-body simulation. For the propagator ${\mathrm{\Gamma }}^{(1)}$, the regularized one-loop propagator computed analytically is used in solid lines, while the dashed lines adopt the one directly measured from $N$-body simulations. Finally, the dotted lines are obtained with the linear theory prediction, just replacing ${\mathrm{\Gamma }}^{(1)}$ with the linear growth factor $D_{+}$. Note that for clarity, the results at $z=0.5$ and 1 are multiplied by the factor 3 and 9, respectively.

Figure 6

Monopole ($\ell =0$, top) and quadrupole ($\ell =2$, bottom) moments of redshift-space power spectrum at $z=0$, 0.5, 1, and 2. Left panel shows the results in GR, while right panel presents the cases in $f(R)$ gravity with $|f_{R,0}|=10^{-4}$. In each panel, dotted lines represent the linear theory calculations based on the Kaiser formula, while the solid lines are the RegPT one-loop results based on the TNS model of RSD.

Figure 7

Prediction of redshift-space correlation functions at $z=0$, 0.5, 1, and 2 in GR (left) and $f(R)$ gravity with $|f_{R,0}|=10^{-4}$ (right). Based on the TNS model, the monopole ($\ell =0$) and quadrupole ($\ell =2$) moments of the correlation function are shown in top and bottom panels, respectively. Note that in computing the correlation function, we used the fitting parameter ${\sigma }_{\mathrm{v}}$ determined from power spectrum (Fig. 6). In each panel, solid and dotted lines are the RegPT and linear theory predictions.

Figure 8

Comparison between RegPT and standard PT results at one-loop order in real and redshift spaces. Left panel plots the auto- and cross-power spectra of density and velocity fields in real space at $z=1$. The results in $f(R)$ gravity with $|f_{R,0}|=10^{-4}$ are shown. Solid and dotted lines are the PT predictions computed with RegPT and standard PT, respectively. In the right panel, redshift-space power spectra at $z=1$ are plotted. Based on the TNS model, the monopole ($\ell =0$) and quadrupole ($\ell =2$) power spectra are computed, and the results are compared with $N$-body simulations. The solid and dashed lines, respectively, represent the PT predictions computed with RegPT and standard PT.