Structure formation in modified gravity scenarios

We study the growth of structures in modified gravity models where the Poisson equation and the relationship between the two Newtonian potentials are modified by explicit functions of space and time. This parametrization applies to the $f(R)$ models and more generally to screened modified gravity models. We investigate the linear and weakly nonlinear regimes using the “standard” perturbative approach and a resummation technique, while we use the spherical dynamics to go beyond low-order results. This allows us to estimate the matter density power spectrum and bispectrum from linear to highly nonlinear scales, the full probability distribution of the density contrast on weakly nonlinear scales, and the halo mass function. We analyze the impact of modifications of gravity on these quantities for a few realistic models. In particular, we find that the standard one-loop perturbative approach is not sufficiently accurate to probe these effects on the power spectrum, and it is necessary to use resummation methods even on weakly nonlinear scales, which provide the best observational window for modified gravity as relative deviations from general relativity do not grow significantly on smaller scales where theoretical predictions become increasingly difficult.

Figure 1

Linear growing mode $D_{+}(k,t)$ normalized to the scale factor $a(t)$ for four $(n,m_0)$ models. In each case we show the results for wave numbers $k=1h{\mathrm{Mpc}}^{-1}$ (lower curve) and $k=5h{\mathrm{Mpc}}^{-1}$ (upper curve), as a function of $a(t)$. These two scales are in the nonlinear regime and have only been chosen to exemplify the type of effects obtained in modified gravity.

Figure 2

Linear decaying mode $D_{-}(k,t)$ normalized to $a(t{)}^{-3/2}$ for four $(n,m_0)$ models. In each case we show the results for wave numbers $k=1h{\mathrm{Mpc}}^{-1}$ (upper curve) and $5h{\mathrm{Mpc}}^{-1}$ (lower curve), as a function of $a(t)$. These two scales are in the nonlinear regime and have only been chosen to exemplify the type of effects obtained in modified gravity.

Figure 3

Linear growing mode $D_{+}(k,t)$ normalized to the scale factor $a(t)$ for four $(n,m_0)$ models, at redshift $z=0$ up to nonlinear scales.

Figure 4

Linear decaying mode $D_{-}(k,t)$ normalized to $a(t{)}^{-3/2}$ for four $(n,m_0)$ models, at redshift $z=0$ up to nonlinear scales.

Figure 5

Linear growth rate $f(k,z)=\partial \ln D_{+}/\partial \ln a$ for wave number $k=1h{\mathrm{Mpc}}^{-1}$, for four $(n,m_0)$ models.

Figure 6

Ratio of the power spectrum $P(k)$ to a smooth $\Lambda \mathrm{CDM}$ linear power spectrum $P_{Ls}(k)$ without baryonic oscillations, from Ref. 61. We show our results for three models with $(n,m_0)=(1,0.1)$ (middle red lines), (0, 0.1) (upper black lines), and (0, 1) (lower blue lines). In each case, we plot both the linear power (dashed line) and our nonlinear result (solid line) from Eq. (124), which is based on Eq. (68). For comparison, we also plot the standard one-loop result from Eq. (73) for the case (0, 1) (upper blue dotted line).

Figure 7

Ratio of the equilateral bispectrum, $B_{\mathrm{eq}}(k)=B(k,k,k)$, to the product $3P_{Ls}(k{)}^2$, where $P_{Ls}(k)$ is a smooth $\Lambda \mathrm{CDM}$ linear power spectrum without baryonic oscillations, from Ref. 61. As in Fig. 6, we show our results for three models with $(n,m_0)=(1,0.1)$ (middle red lines), (0, 0.1) (upper black lines), and (0, 1) (lower blue lines). In each case, we plot the tree-level bispectrum (dashed line) from Eq. (79), the one-loop bispectrum (dash-dotted line) from Eq. (80), and our nonlinear result (solid line) from Eq. (128).

Figure 8

Reference linear density contrast ${\delta }_{c(\Lambda )}={\mathcal{F}}_q^{-1}(200)$ associated with a nonlinear density threshold of 200 at redshift $z=0$. We show our results as a function of the halo mass $M$ for four $(n,m_0)$ models, for typical initial profiles of the form (104). In each case, the upper curve is the approximate result from Eq. (102) and the lower curve the exact result from Eq. (96).

Figure 9

Probability distribution of the matter density contrast within spherical cells of radius $5h^{-1}\mathrm{Mpc}$ at $z=0$ (all curves almost fall on each other).

Figure 10

Relative deviation from general relativity of the probability distribution $\mathcal{P}({\delta }_x)$, at redshift $z=0$ for a radius $x=5h^{-1}\mathrm{Mpc}$. For each $(n,m_0)$ model the deviation from GR is positive at low and high densities and negative around $\delta \sim 0$. The solid and dotted lines are the exact results from Eq. (96) for $(n,m_0)=(1,0.1)$ and (0, 0.1). The closest dashed line of the same color is the result from the approximation (102), for the same value of $(n,m_0)$.

Figure 11

Halo mass function at redshift $z=0$.

Figure 12

Relative deviation from $\Lambda \mathrm{CDM}$ of the halo mass function at redshift $z=0$.

Figure 13

Logarithmic power, ${\Delta }^2(k)=4\pi k^{3}P(k)$, at $z=0$ for four $(n,m_0)$ models. In each case we plot the linear power (dashed line) and the nonlinear power (solid line).

Figure 14

Equilateral bispectrum $B_{\mathrm{eq}}(k)=B(k,k,k)$, at $z=0$ for four $(n,m_0)$ models. The bispectrum is multiplied by a factor $k^3$ in this plot to decrease the range spanned by the vertical axis and to make the figure easier to read. In each case we plot the tree-level bispectrum (dashed line) and the full nonlinear bispectrum (solid line).

Figure 15

Relative deviation from $\Lambda \mathrm{CDM}$ of the power spectrum obtained in four models at redshift $z=0$. In each case, we plot the relative deviation both of the linear power (dashed line) and of the nonlinear power (solid line). From left to right we consider the models $(n,m_0)=(0,0.1)$, (1, 0.1), (0, 1), and (1, 1).

Figure 16

Relative deviation from $\Lambda \mathrm{CDM}$ of the bispectrum obtained in four models at redshift $z=0$. In each case, we plot the relative deviation both of the tree-level bispectrum (dashed line) and of the nonlinear bispectrum (solid line). From left to right we consider the models $(n,m_0)=(0,0.1)$, (1, 0.1), (0, 1), and (1, 1).

Figure 17

Relative deviation from $\Lambda \mathrm{CDM}$ of the halo mass function at redshift $z=0$, for $n=1$ and $|f_{R_0}|={10}^{-4}$, ${10}^{-5}$, and ${10}^{-6}$, from top to bottom.

Figure 18

Relative deviation from $\Lambda \mathrm{CDM}$ of the power spectrum at redshift $z=0$, for $n=1$ and $|f_{R_0}|={10}^{-4}$, ${10}^{-5}$, and ${10}^{-6}$. In each case, we plot the relative deviation both of the linear power (dashed line) and of the nonlinear power (solid line). The points are the results of the “no-chameleon simulations” from Ref. 25.

Figure 19

Relative deviation from $\Lambda \mathrm{CDM}$ of the bispectrum at redshift $z=0$, for $n=1$ and $|f_{R_0}|={10}^{-4}$, ${10}^{-5}$, and ${10}^{-6}$. In each case, we plot the relative deviation both of the tree-level bispectrum (dashed line) and of the nonlinear bispectrum (solid line).