Scale dependence of the primordial spectrum from combining the three-year WMAP, Galaxy Clustering, Supernovae, and Lyman-alpha forests

We probe the scale dependence of the primordial spectrum in the light of the three-year WMAP (WMAP3) alone and WMAP3 in combination with the other cosmological 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's preference on the running of the scalar spectral index on the large scales is now fairly independent of the low CMB multipoles $\ell$. A combination with the galaxy power spectrum from the Sloan Digital Sky Survey (SDSS) prefers a negative 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 the Power Law $\Lambda$CDM cosmology with only six parameters (matter density $\Omega_m h^2$, baryon density $\Omega_b h^2$, Hubble Constant $H_0$, optical depth $\tau$, the spectral index, $n_s$, and the amplitude, $A_s$, of the scalar perturbation spectrum) when we drop the low $\ell$ CMB contributions WMAP3 is consistent with the Harrison-Zel'dovich-Peebles scale-invariant spectrum ($n_s=1$ and no tensor contributions) at $\sim 1\sigma$. When assuming a simple power law primordial spectral index or a constant running, in case one drops the low $\ell$ contributions ($2\le \ell \le 23$) WMAP3 is consistent with 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 running spectral index model, LUQAS$+$ CROFT Ly$\alpha$ and SDSS Ly$\alpha$ exhibit somewhat different preference on the spectral shape.