Effects of time-varying $\beta$ in SNLS3 on constraining interacting
  dark energy 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 this paper, adopting the $w$-cold-dark-matter ($w$CDM)
model and considering its interacting extensions (with three kinds of
interaction between dark sectors), we explore the evolution of $\beta$ and its
effects on parameter estimation. In addition to the SNLS3 data, we also take
into account the Planck distance priors data of the cosmic microwave background
(CMB), the galaxy clustering (GC) data extracted from SDSS DR7 and BOSS, as
well as the direct measurement of Hubble constant from the Hubble Space
Telescope (HST) observation. We find that, for all the interacting dark energy
(IDE) models, adding a parameter of $\beta$ can reduce $\chi^2$ by $\sim$ 34,
indicating that $\beta_1 = 0$ is ruled out at 5.8$\sigma$ confidence level
(CL). Furthermore, it is found that varying $\beta$ can significantly change
the fitting results of various cosmological parameters: for all the dark energy
models considered in this paper, varying $\beta$ yields a larger $\Omega_{c0}$
and a larger $w$; on the other side, varying $\beta$ yields a smaller $h$ for
the $w$CDM model, but has no impact on $h$ for the three IDE models. This
implies that there is a degeneracy between $h$ and $\gamma$. Our work shows
that the evolution of $\beta$ is insensitive to the interaction between dark
sectors, and then highlights the importance of considering $\beta$'s evolution
in the cosmology fits.
