Applying the expanding photosphere and standardized candle methods to
  Type II-Plateau supernovae at cosmologically significant redshifts: the
  distance to SN 2013eq
  Based on optical imaging and spectroscopy of the Type II-Plateau SN 2013eq,
we present a comparative study of commonly used distance determination methods
based on Type II supernovae. The occurrence of SN 2013eq in the Hubble flow (z
= 0.041 +/- 0.001) prompted us to investigate the implications of the
difference between "angular" and "luminosity" distances within the framework of
the expanding photosphere method (EPM) that relies upon a relation between flux
and angular size to yield a distance. Following a re-derivation of the basic
equations of the EPM for SNe at non-negligible redshifts, we conclude that the
EPM results in an angular distance. The observed flux should be converted into
the SN rest frame and the angular size, theta, has to be corrected by a factor
of (1+z)^2. Alternatively, the EPM angular distance can be converted to a
luminosity distance by implementing a modification of the angular size. For SN
2013eq, we find EPM luminosity distances of D_L = 151 +/- 18 Mpc and D_L = 164
+/- 20 Mpc by making use of different sets of dilution factors taken from the
literature. Application of the standardized candle method for Type II-P SNe
results in an independent luminosity distance estimate (D_L = 168 +/- 16 Mpc)
that is consistent with the EPM estimate.
