(abridged)Prompt GRB emission is often interpreted as synchrotron radiation from high-energy electrons accelerated in internal shocks. Fast synchrotron cooling predicts that the photon index below the spectral peak is alpha=-3/2. This differs significantly from the observed median value alpha approx -1. We quantify the influence of inverse Compton and adiabatic cooling on alpha to understand whether these processes can reconcile the observations with a synchrotron origin. We use a time-dependent code that follows both the shock dynamics and electron energy losses. We investigate the dependence of alpha on the parameters of the model. Slopes between -3/2 and -1 are reached when electrons suffer IC losses in the Klein-Nishina regime. This does not necessarily imply a strong IC component in the Fermi/LAT range because scatterings are only moderately efficient. Steep slopes require that a large fraction (10-30%) of the dissipated energy is given to a small fraction (<~1%) of the electrons and that the magnetic energy density fraction remains low (<~ 0.1%). Values of alpha up to -2/3 can be obtained with relatively high radiative efficiencies (>50%) when adiabatic cooling is comparable with radiative cooling (marginally fast cooling). This requires collisions at small radii and/or with low magnetic fields. Amending the standard fast cooling scenario to account for IC cooling naturally leads to alpha up to -1. Marginally fast cooling may also account for alpha up to -2/3, although the conditions required are more difficult to reach. About 20% of GRBs show spectra with slopes alpha>-2/3. Other effects, not investigated here, such as a thermal component in the electron distribution or pair production by HE photons may further affect alpha. Still, the majority of observed GRB prompt spectra can be reconciled with a synchrotron origin, constraining the microphysics of mildly relativistic internal shocks.
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