Global fits for the spectral index of the cosmological curvature perturbation

Abstract

Best-fit values of the spectral index of the curvature perturbation are presented, assuming the $Λ$CDM cosmology. Apart from the spectral index, the parameters are the Hubble parameter, the total matter density and the baryon density. The data points are intended to represent all measurements which are likely to significantly affect the result. The cosmic microwave anisotropy is represented by the COBE normalization, and heights of the first and second peaks given by the latest Boomerang and Maxima data. The slope of the galaxy correlation function and the matter density contrast on the $8h^{-1}\Mpc$ scale are each represented by a data point, as are the expected values of the Hubble parameter and matter density. The `low-deuterium' nucleosynthesis value of the baryon density provides a final data point, the fit giving a value about one standard deviation higher. The reionization epoch is calculated from the model by assuming that it corresponds to the collapse of a fraction $f\gsim 10^{-4}$ of matter. We consider the case of a scale-independent spectral index, and also the scale-dependent spectral index predicted by running mass models of inflation. In the former case, the result is compared with the prediction of models of inflation based on effective field theory, in which the field value is small on the Planck scale. Detailed comparison is made with other fits, and other approaches to the comparison with theory.

Figures (4)

• Figure 1: The top panels show nominal 1- and 2-$\sigma$ bounds on $n$. In the left-hand panel the reionization epoch $z_{\rm R}$ is fixed. In the right-hand panel, is fixed instead the fraction $f$ of matter which is assumed to have collapsed when at the epoch of reionization. (The corresponding reionization redshift, at best fit, is in the range $10$ to $26$.) The bottom panels show $\chi^2$, with three degrees of freedom.
• Figure 2: Curves correspond to best fit $\pm$ 1-$\sigma$ for $f\simeq 1$. The top panels show the cmb anisotropy $\widetilde{C}_\ell$, for the case of scale-independent spectral index (left panel) and for the running mass model with coupling $c=0.1$ (right panel). The error bars do not include systematic errors; taking these as the calibration uncertainties they are $20\%$ for Boomerang and $8\%$ for Maxima. The fit used only the two data points nearest each of the peaks (one each from Boomerang and Maxima) and added in quadrature the systematic errors. Other data sets around the first peak (not shown) span a wider range and their rejection in favour of Boomerang/Maxima is somewhat subjective. The bottom panels show the corresponding spectrum of the primordial curvature perturbation, normalized to 1 at the COBE scale, against the scale $\ell(k)$ probed by the cmb anisotropy. The shortest scale shown (biggest $\ell$) corresponds to $k^{-1}\simeq 10 h^{-1}\,\hbox{Mpc}$, at which the galaxy data $\widetilde{\sigma}_8$ and $\widetilde{\Gamma}$ apply.
• Figure 3: The horizontal lines show the 1- and 2-$\sigma$ bounds on $n$, with different panels corresponding to different assumptions about the epoch of reionization. Also shown is the dependence of $n$ on $N_{\rm COBE}$, according to some of the models shown in Table 2. From top to bottom these are the logarithmic potential, new inflation with $p=4$, and new inflation with $p=3$. Significant lower bounds on $N_{\rm COBE}$ are obtained for the new inflation models. Taken seriously, the 1-$\sigma$ bound would practically rule out the $p=3$ model.
• Figure 4: The parameter space for the running mass model. As discussed in cl99 the model comes in four versions, corresponding to the four quadrants of the parameter space. In the left-hand panel, the straight lines corresponding to $n_{\rm COBE}=1.2, 1.0$ and $0.8$, and the shaded region is disfavoured on theoretical grounds. In the right-hand panel, we show the region allowed by observation, in the case that reionization occurs when $f\sim 1$. To show the scale-dependence of the prediction for $n$, we also show in this panel the branches of the hyperbola $8sc=\Delta n\equiv n_8-n_{\rm COBE}$, for the reference value $\Delta n=0.04$.