CMB and matter power spectra of early $f(R)$ cosmology in the Palatini formulation

We calculate in this article the cosmic microwave background (CMB) and matter power spectra of a class of early $f(R)$ cosmologies, which takes the form of $f(R)=R+{\lambda }_1{H}_0^2\exp [R/({\lambda }_2{H}_0^2)]$. Unlike the late-time $f(R)$ cosmologies such as $f(R)=R+\alpha (-R{)}^{\beta }$ ($\beta <0$), the deviation from $\Lambda \mathrm{CDM}$ of this model occurs at a higher redshift (thus the name “early $f(R)$ cosmology”), and this important feature leads to rather different integrated Sachs-Wolfe (ISW) effect and CMB spectrum. The matter power spectrum of this model is, however, again very sensitive to the chosen parameters, and large scale structure (LSS) observations such as Sloan digital sky survey (SDSS) should constrain the parameter space stringently. We expect that our results are applicable at least qualitatively to other models that produce $f(R)$ modification to GR at earlier times (e.g., redshifts $\mathcal{O}(10)\lesssim z\lesssim \mathcal{O}(1)$) than when dark energy begins to dominate—such models are strongly disfavored by data on CMB and matter power spectra.

Figure 1

The background evolutions. Upper panel: the ratio between the expansion rates in this exponential $f(R)$ cosmology and in $\Lambda \mathrm{CDM}$ as a function of the scale factor $a$. Lower panel: $F(R)-1$ as a function of $a$. The black, red, green, blue, cyan and magenta curves represent cases of ${\lambda }_2=\infty$, 1000, 500, 300, 100, 50, respectively.

Figure 2

Upper panel: The theoretical CMB spectrum of the present model. From bottom to top: ${\lambda }_2=\infty$, 1000, 500. Lower panel: Contributions from the ISW effect. Solid curves are the results of the $f(R)=R+{\lambda }_1{H}_0^2\exp (R/{\lambda }_2{H}_0^2)$ model, from bottom to top: ${\lambda }_2=\infty$, 1000, 500; Dashed curves are those of the $f(R)=R+\alpha (-R{)}^{\beta }$ model for comparison, from bottom to top: $\beta =0$, $-0.05$, $-0.1$. In both cases we use ${\Omega }_m=0.3$.

Figure 3

Upper panel: The potential ${\varphi }_k$ as a function of the scale factor for the different $k$-modes as indicated besides the curves for the $f(R)=R+{\lambda }_1{H}_0^2\exp (R/{\lambda }_2{H}_0^2)$ model. The black, red and green curves represent cases of ${\lambda }_2=\infty$, 1000 and 500, respectively. Lower panel: same as above, but for the $f(R)=R+\alpha (-R{)}^{\beta }$ model, and the black, red and green curves here represent cases of $\beta =0$, $-0.01$ and $-0.05$. In both cases we use ${\Omega }_m=0.3$.

Figure 4

The matter power spectrum in the present model. The black, red, green, blue, and cyan curves represent the cases of ${\lambda }_2=\infty$, ${10}^5$, $5\times {10}^4$, ${10}^4$, and 1000, respectively.

Figure 5

(Color online) The matter power spectra at different redshifts. From bottom to top: $z=0$, $z=5$, $z=10$ and $z=15$. The black, red, and green curves represent the cases of $f(R)=R+{\lambda }_1{H}_0^2\exp (R/{\lambda }_2{H}_0^2)$ (${\lambda }_2=1000$), $f(R)=R+\alpha (-R{)}^{\beta }$ ($\beta =-0.01$) and $\Lambda \mathrm{CDM}$ respectively.