  The generic expectation in string/supergravity models is that there are
multiple moduli fields with masses of the order of the supersymmetry breaking
scale. We study the cosmology that arises as a result of vacuum misalignment of
these moduli fields (in contrast to previous studies which mostly focussed on
the single modulus case). We show that the dark radiation produced from the
heavier moduli undergoes significant dilution. This happens even if there is a
small splitting between the masses of the lightest and the heavier moduli. On
the other hand, in the absence of fast annihilation processes decay of heavier
moduli generically leads to overproduction of dark matter. We discuss a
scenario where the problem can be addressed with a prompt dark matter
annihilation to dark radiation. This can lead to realistic dark matter
abundances, and the additional dark radiation produced as a result of this
mechanism undergoes sufficient dilution as long as the annihilation is prompt.
