$f(R)$ gravity: From the Pioneer anomaly to cosmic acceleration

We use metric formalism in $f(R)$ modified gravity to study the dynamics of various systems from the solar system to the cosmological scale. We assume an ansatz for the derivative of action as a function of distance and describe the Pioneer anomaly and the flat rotation curve of the spiral galaxies. Having the asymptotic behavior of action, we propose the action of $f(R)=(R+\Lambda )(1+\ln (R/R_c)/(R/R_0+2/\alpha ))$ where in galactic and solar system scales it can recover our desired form. The vacuum solution of this action also results in a positive late time acceleration for the Universe. We fix the parameters of this model, comparing with the Pioneer anomaly, rotation curve of spiral galaxies, and supernova type Ia gold sample data.

Figure 1

Comparing the rotation curve of a galaxy in Newtonian gravity, $v\propto 1/r$ (dashed line) with the rotation curve from the modified gravity in the weak field regime (solid line), see Eq. (32). The parameters of the galaxy are taken as $M={10}^{11}{M}_{\odot }$ (mass of galaxy), $\alpha \simeq {10}^{-6}$, and $d\simeq 10\mathrm{kpc}$.

Figure 2

Distance modulus of the supernova type Ia new gold sample in terms of redshift. The solid line shows the best fit values with the corresponding parameters of $h=0.64$, ${\Omega }_m=0.31$, ${\Omega }_{\Lambda }=0.69$, and $\alpha \ll {10}^{-3}$ with the corresponding ${\chi }_{\min }^2/N_{\mathrm{dof}}=1.14$.