  We present improvements to construction of binary black hole initial data
used in SpEC (the Spectral Einstein Code). We introduce new boundary conditions
for the extended conformal thin sandwich elliptic equations that enforce the
excision surfaces to be slightly inside rather than on the apparent horizons,
thus avoiding extrapolation into the black holes at the last stage of initial
data construction. We find that this improves initial data constraint
violations near and inside the apparent horizons by about 3 orders of
magnitude. We construct several initial data sets that are intended to be
astrophysically equivalent but use different free data, boundary conditions,
and initial gauge conditions. These include free data chosen as a superposition
of two black holes in time-independent horizon-penetrating harmonic and damped
harmonic coordinates. We also implement initial data for which the initial
gauge satisfies the harmonic and damped harmonic gauge conditions; this can be
done independently of the free data, since this amounts to a choice of the time
derivatives of the lapse and shift. We compare these initial data sets by
evolving them. We show that the gravitational waveforms extracted during the
evolution of these different initial data sets agree very well after excluding
initial transients. However, we do find small differences between these
waveforms, which we attribute to small differences in initial orbital
eccentricity, and in initial BH masses and spins, resulting from the different
choices of free data. Among the cases considered, we find that superposed
harmonic initial data leads to significantly smaller transients, smaller
variation in BH spins and masses during these transients, smaller constraint
violations, and more computationally efficient evolutions. Finally, we study
the impact of initial data choices on the construction of zero-eccentricity
initial data.
