Non-linear Evolution of Matter Power Spectrum in Modified Theory of Gravity

Abstract

We present a formalism to calculate the non-linear matter power spectrum in modified gravity models that explain the late-time acceleration of the Universe without dark energy. Any successful modified gravity models should contain a mechanism to recover General Relativity (GR) on small scales in order to avoid the stringent constrains on deviations from GR at solar system scales. Based on our formalism, the quasi non-linear power spectrum in the Dvali-Gabadadze-Porratti (DGP) braneworld models and $f(R)$ gravity models are derived by taking into account the mechanism to recover GR properly. We also extrapolate our predictions to fully non-linear scales using the Parametrized Post Friedmann (PPF) framework. In $f(R)$ gravity models, the predicted non-linear power spectrum is shown to reproduce N-body results. We find that the mechanism to recover GR suppresses the difference between the modified gravity models and dark energy models with the same expansion history, but the difference remains large at weakly non-linear regime in these models. Our formalism is applicable to a wide variety of modified gravity models and it is ready to use once consistent models for modified gravity are developed.

Figures (8)

• Figure 1: Fractional change in the non-linear power spectrum relative to the linear power spectrum in the self-accelerating branch of DGP. The solid (black) line is the analytic solutions and the circles are numerical solutions obtained by solving the closure equation. The dashed (red) line shows the analytic solutions obtained by neglecting the non-linear interaction terms ${\cal I}$. The triangles represent the numerical solutions in this case. We used the best fit cosmological parameters for the flat universe $\Omega_{m}=0.258, \Omega_{b}= 0.0544, h=0.66, n_s=0.998$.
• Figure 2: The same in the normal branch as in Fig.1. We only show the analytic solutions. The numerical solutions agree with them very well. The cosmological parameters are the same as the self-accelerating universe but in addition there is a cosmological constant $\Omega_{\Lambda}=1.5$.
• Figure 3: Fractional change in the non-linear power spectrum in $f(R)$ gravity models relative to the GR models with the same expansion history. $P_{\rm GR}$ is the non-linear power spectrum in $\Lambda$CDM model with the same cosmological parameters. The solid (black) lines are the solutions in the perturbation theory obtained by solving the closure equation numerically. The circles show the results of N-body simulations. The dashed (red) line is the perturbation theory solutions obtained by neglecting the non-linear interaction terms ${\cal I}$. The triangles represent the corresponding N-body solutions. The arrow indicate the valid regime of the perturbation theory. The parameters are taken as $|f_{R0}|=10^{-4}, n_s=0.958, \Omega_{m}=0.24, \Omega_{b}=0.046, \Omega_{\Lambda}=0.76, h=0.73$.
• Figure 4: A fractional change in the power spectrum in the DGP self-accelerating solution relative to the GR model which has the same expansion history as the DGP. The solid (black) line shows the perturbation theory solution and the dashed (red) line shows the perturbation theory solution without the non-linear interaction terms in the Poisson equation. The dotted (blue) line shows the PPF fitting. By allowing the redshift dependence of $c_{\rm nl}$, we can fit the power spectrum very well within the validity regime of the perturbation theory indicated by arrows. The right panel shows the results at $z=0$ obtained from the fitting formula by Smith et.al. for $P_{\rm non-GR}$ and $P_{\rm GR}$. If $c_{\rm nl}=0.3$ obtained by the perturbation theory is applicable, the solid (black) line is our prediction on non-linear scales. The cosmological parameters are the same as in Fig 1.
• Figure 5: Comparison between the PPF prediction and N-body simulations. In the left panel, Smith et.al. fitting formula is used to predict $P_{\rm non-GR}$ and $P_{\rm GR}$. We used $c_{\rm nl}$ determined by the perturbation theory $c_{\rm nl}=0.3$ at $z=0$. In the right panel, we fitted N-body results with the linear Poisson equation to derive $P_{\rm non-GR}$.