  We constrain cosmological models where the primordial perturbations have both
an adiabatic and a (possibly correlated) cold dark matter (CDM) or baryon
isocurvature component. We use both a phenomenological approach, where the
primordial power spectra are parametrized with amplitudes and spectral indices,
and a slow-roll two-field inflation approach where slow-roll parameters are
used as primary parameters. In the phenomenological case, with CMB data, the
upper limit to the CDM isocurvature fraction is \alpha<6.4% at k=0.002Mpc^{-1}
and 15.4% at k=0.01Mpc^{-1}. The median 95% range for the non-adiabatic
contribution to the CMB temperature variance is -0.030<\alpha_T<0.049.
Including the supernova (or large-scale structure, LSS) data, these limits
become: \alpha<7.0%, 13.7%, and -0.048<\alpha_T< 0.042 (or \alpha<10.2%, 16.0%,
and -0.071<\alpha_T<0.024). The CMB constraint on the tensor-to-scalar ratio,
r<0.26 at k=0.01Mpc^{-1}, is not affected by the nonadiabatic modes. In the
slow-roll two-field inflation approach, the spectral indices are constrained
close to 1. This leads to tighter limits on the isocurvature fraction, with the
CMB data \alpha<2.6% at k=0.01Mpc^{-1}, but the constraint on \alpha_T is not
much affected, -0.058<\alpha_T<0.045. Including SN (or LSS) data, these limits
become: \alpha< 3.2% and -0.056<\alpha_T<0.030 (or \alpha<3.4% and
-0.063<\alpha_T<-0.008). When all spectral indices are close to each other the
isocurvature fraction is somewhat degenerate with the tensor-to-scalar ratio.
In addition to the generally correlated models, we study also special cases
where the perturbation modes are uncorrelated or fully (anti)correlated. We
calculate Bayesian evidences (model probabilities) in 21 different cases for
our nonadiabatic models and for the corresponding adiabatic models, and find
that in all cases the data support the pure adiabatic model.
