Effects of a time-varying color-luminosity parameter $\beta$ on the
  cosmological constraints of modified gravity models
  It has been found that, for the Supernova Legacy Survey three-year (SNLS3)
data, there is strong evidence for the redshift-evolution of color-luminosity
parameter $\beta$. In previous studies, only dark energy (DE) models are used
to explore the effects of a time-varying $\beta$ on parameter estimation. In
this paper, we extend the discussions to the case of modified gravity (MG), by
considering Dvali-Gabadadze-Porrati (DGP) model, power-law type $f(T)$ model
and exponential type $f(T)$ model. In addition to the SNLS3 data, we also use
the latest Planck distance priors data, the galaxy clustering (GC) data
extracted from Sloan Digital Sky Survey (SDSS) data release 7 (DR7) and Baryon
Oscillation Spectroscopic Survey (BOSS), as well as the direct measurement of
Hubble constant $H_0$ from the Hubble Space Telescope (HST) observation. We
find that, for both cases of using the supernova (SN) data alone and using the
combination of all data, adding a parameter of $\beta$ can reduce $\chi^2$ by
$\sim$ 36 for all the MG models, showing that a constant $\beta$ is ruled out
at 6$\sigma$ confidence level (CL). Moreover, we find that a time-varying
$\beta$ always yields a larger fractional matter density $\Omega_{m0}$ and a
smaller reduced Hubble constant $h$; in addition, it significantly changes the
shapes of 1$\sigma$ and 2$\sigma$ confidence regions of various MG models, and
thus corrects systematic bias for the parameter estimation. These conclusions
are consistent with the results of DE models, showing that $\beta$'s evolution
is completely independent of the cosmological models in the background.
Therefore, our work highlights the importance of considering the evolution of
$\beta$ in the cosmology-fits.
