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We probe the scale dependence of the primordial spectrum in the light of thethree-year WMAP (WMAP3) alone and WMAP3 in combination with the othercosmological observations such as galaxy clustering and Type Ia Supernova(SNIa). We pay particular attention to the combination with the Lyman $\alpha$(Ly$\alpha$) forest. Different from the first-year WMAP (WMAP1), WMAP3'spreference on the running of the scalar spectral index on the large scales isnow fairly independent of the low CMB multipoles $\ell$. A combination with thegalaxy power spectrum from the Sloan Digital Sky Survey (SDSS) prefers anegative running to larger than 2$\sigma$, regardless the presence of low$\ell$ CMB ($2\le \ell \le 23$) or not. On the other hand if we focus on thePower Law $\Lambda$CDM cosmology with only six parameters (matter density$\Omega_m h^2$, baryon density $\Omega_b h^2$, Hubble Constant $H_0$, opticaldepth $\tau$, the spectral index, $n_s$, and the amplitude, $A_s$, of thescalar perturbation spectrum) when we drop the low $\ell$ CMB contributionsWMAP3 is consistent with the Harrison-Zel'dovich-Peebles scale-invariantspectrum ($n_s=1$ and no tensor contributions) at $\sim 1\sigma$. When assuminga simple power law primordial spectral index or a constant running, in case onedrops the low $\ell$ contributions ($2\le \ell \le 23$) WMAP3 is consistentwith the other observations better, such as the inferred value of $\sigma_8$.We also find, using a spectral shape with a minimal extension of the runningspectral index model, LUQAS$+$ CROFT Ly$\alpha$ and SDSS Ly$\alpha$ exhibitsomewhat different preference on the spectral shape.Comment: 16 pages, 13 figures Revtex

Scale dependence of the primordial spectrum from combining the three-year WMAP, galaxy clustering, supernovae, and Lyman-alpha forests