Rotational splittings for slow to moderate rotators. Latitudinal dependence or higher order effects in Ω?

Context. The unprecedented photometric quality reached by the CoRoT and Kepler space missions opens new prospects for studying stellar rotation. Information about the rotation rate is contained on the one hand in the low frequency part of the power spectra, where signatures of nonuniform surface rotation are expected, and on the other hand in the frequency splittings induced by the internal rotation rate. Aims: We wish to figure out whether the differences between the seismic rotation period determined by a mean rotational splitting, and the rotation period measured from the low frequency peak in the Fourier spectrum - observed for some of CoRoT's targets - can provide constraints on the rotation profile. Methods: For uniform moderate rotators,perturbative corrections to second and third order in terms of the rotation angular velocity Ω, must not be neglected. These effects, in particular, may mimic differential rotation. We apply our perturbation method to evaluate mode frequencies that are accurate up to Ω3 for uniform rotation. The effects of latitudinal dependence are calculated in the linear approximation. Numerical results were obtained for selected models of the upper and lower parts of the main sequence. For the latitudinal dependence, we adopt two types of rotation profile: one with rotation uniform in depth, and one with a solar-like tachocline. Results: Deviations from the first-order splitting for a uniformly rotating star can be due to both cubic-order effects of rotation and latitudinal differential rotation. In models of β Cephei pulsators, which represent upper main sequence stars, third order effects become comparable to that of a horizontal shear similar to the solar one at rotation rates well below the breakup values. These nonlinear effects are strongly mode-dependent. We show how a clean signature of the latitudinal shear may be extracted. Our models of two CoRoT target HD 181906 and HD 181420, which are solar-like pulsators, represent lower main sequence objects. These are slow rotators and nonlinear effects in splittings are accordingly small. We use data for one low frequency peak and one splitting of a dipolar mode to constrain the rotation profile in HD 181420 and HD 181906. Conclusions: The relative influences of the two effects strongly depend on the type of oscillation modes excited in the star and the magnitude of the rotation rate. Given the mean rotational splitting and the frequency of a spot signature, it is possible to distinguish between the two hypotheses. In the case of differential rotation in latitude, we propose a method to determine the type of rotation profile and a range of values for the shear