The combined analysis of the cosmic microwave background on large scales and Lyman alpha forest on small scales provides a sufficiently long lever arm to obtain strong constraints on the slope and curvature of the power spectrum of primordial density fluctuations. We present results from the combination of the first year WMAP data and the dark matter power spectrum inferred by Viel, Haehnelt & Springel for two different sets of high-resolution and high signal-to-noise ratio quasar absorption spectra: the Croft et al. sample with a median redshift z = 2.72 and the LUQAS sample with a median redshift z = 2.125. The best-fitting value for the rms fluctuation amplitude of matter fluctuations is sigma(8) = 0.94 +/- 0.08 and n(s) = 0.99 +/- 0.03, if we do not include running of the spectral index. The best-fitting model with a running spectral index has parameters n(s) = 0.959 +/- 0.036 and n(run) = -0.033 +/- 0.025. The data are thus consistent with a scale-free primordial power spectrum with no running of the spectral index. We further include tensor modes and constrain the slow-roll parameters of inflation.