  Within the context of SUSY GUTs, cosmic strings are generically formed at the
end of hybrid inflation. However, the WMAP CMB measurements strongly constrain
the possible cosmic strings contribution to the angular power spectrum of
anisotropies. We investigate the parameter space of SUSY hybrid (F- and D-
term) inflation, to get the conditions under which theoretical predictions are
in agreement with data. The predictions of F-term inflation are in agreement
with data, only if the superpotential coupling $\kappa$ is small. In
particular, for SUSY SO(10), the upper bound is $\kappa\lsim 7\times 10^{-7}$.
This fine tuning problem can be lifted if we employ the curvaton mechanism, in
which case $\kappa\lsim 8\times 10^{-3}$; higher values are not allowed by the
gravitino constraint. The constraint on $\kappa$ is equivalent to a constraint
on the SSB mass scale $M$, namely $M \lsim 2\times 10^{15}$ GeV. The study of
D-term inflation shows that the inflaton field is of the order of the Planck
scale; one should therefore consider SUGRA. We find that the cosmic strings
contribution to the CMB anisotropies is not constant, but it is strongly
dependent on the gauge coupling $g$ and on the superpotential coupling
$\lambda$. We obtain $g\lsim 2\times 10^{-2}$ and $\lambda \lsim 3\times
10^{-5}$. SUGRA corrections induce also a lower limit for $\lambda$.
Equivalently, the Fayet-Iliopoulos term $\xi$ must satisfy $\sqrt\xi \lsim
2\times 10^{15}$ GeV. This constraint holds for all allowed values of $g$.
