Matter Power Spectra in Viable $f(R)$ Gravity Models with Massive Neutrinos
Chao-Qiang Geng1,2,3, Chung-Chi Lee2*, Jia-Liang Shen1
1Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
2Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan
3Department of Physics, Chongqing University of Posts & Telecommunications, Chongqing, China
* presenting author:Chung-Chi Lee,
We investigate the matter power spectra in the power law and exponential types of viable $f(R)$ theories along with massive neutrinos.
The enhancement of the matter power spectrum is found to be a generic feature in these models.
In particular, we show that in the former type, such as the Starobinsky model, the spectrum is magnified
much larger than the latter one, such as the exponential model. A greater scale of the total neutrino mass, $\Sigma m_{\nu}$,
is allowed in the viable $f(R)$ models than that in the $\Lambda$CDM one.
We obtain the constraints on the neutrino masses by using the CosmoMC package with the modified MGCAMB.
Explicitly, we get $\Sigma m_{\nu} < 0.451 \ (0.214)\ \mathrm{eV}$ at $95\%$ C.L. in the Starobinsky (exponential) model,
while the corresponding one for the $\Lambda$CDM model is $\Sigma m_{\nu} < 0.200\ \mathrm{eV}$.
Furthermore, by treating the effective number of neutrino species $N_{\mathrm{eff}}$ as a free parameter along with $\Sigma m_{\nu}$,
we find that $N_{\mathrm{eff}} = 3.78^{+0.64}_{-0.84}~(3.47^{+0.74}_{-0.60})$ and $\Sigma m_{\nu} = 0.533^{+0.254}_{-0.411}$
($< 0.386) \ \mathrm{eV}$ at $95\%$ C.L. in the Starobinsky (exponential) model.

Keywords: Modified Gravity Theory, Dark Energy, Massive Neutrino