The effect of inhomogeneities on dark energy constraints
  Constraints on models of the late time acceleration of the universe assume
the cosmological principle of homogeneity and isotropy on large scales.
However, small scale inhomogeneities can alter observational and dynamical
relations, affecting the inferred cosmological parameters. For precision
constraints on the properties of dark energy, it is important to assess the
potential systematic effects arising from these inhomogeneities. In this study,
we use the Type Ia supernova magnitude-redshift relation to constrain the
inhomogeneities as described by the Dyer-Roeder distance relation and the
effect they have on the dark energy equation of state ($w$), together with
priors derived from the most recent results of the measurements of the power
spectrum of the Cosmic Microwave Background and Baryon Acoustic Oscillations.
We find that the parameter describing the inhomogeneities ($\eta$) is
correlated with $w$. The best fit values $w = -0.933 \pm 0.065$ and $\eta =
0.61 \pm 0.37$ are consistent with homogeneity at $< 2 \sigma$ level. Assuming
homogeneity ($\eta =1$), we find $w = -0.961 \pm 0.055$, indicating only a
small change in $w$. For a time-dependent dark energy equation of state, $w_0 =
-0.951 \pm 0.112$ and $w_a = 0.059 \pm 0.418$, to be compared with $w_0 =
-0.983 \pm 0.127$ and $w_a = 0.07 \pm 0.432$ in the homogeneous case, which is
also a very small change. Current data are not sufficient to constrain the
fraction of dark matter (DM) in compact objects, $f_p$ at the 95$\%$ C.L.,
however at 68$\%$ C.L. $f_p < 0.73$. Future supernova surveys will improve the
constraints on $\eta$, and $f_p$, by a factor of $\sim$ 10.
