The inflationary gravitational-wave background and measurements of the scalar spectral index

TL;DR

This paper examines how a measured scalar spectral index $n_s$ different from unity informs the amplitude of the inflationary gravitational-wave background (IGWB) across direct-detection scales and CMB polarization. By extending prior work to WM A P3 data and six single-field models, it shows that the IGWB prediction depends crucially on the sign of the slow-roll parameter $\eta$ (curvature of the potential): models with $\eta<0$ can yield a detectable IGWB even with small $r$, while models with $\eta>0$ tie $r$ more tightly to $|1-n_s|$. The study maps CMB constraints to the IGWB parameter space, highlighting that Planck-level precision in $n_s$ could set a lower bound on $r$ (and hence on $\Omega_{\mathrm{gw}} h^2$) for certain potentials, with some models (e.g., Coleman-Weinberg) predicting $r \gtrsim 0.0046$ and $\Omega_{\mathrm{gw}} h^2 \gtrsim 1.61 \times 10^{-17}$. Overall, the work clarifies how current and future CMB data, together with direct GW measurements, can constrain the curvature of the inflaton potential and the prospects for observing the IGWB.

Abstract

Inflation predicts a stochastic background of gravitational waves over a broad range of frequencies, from those accessible with cosmic microwave background (CMB) measurements, to those accessible directly with gravitational-wave detectors, like NASA's Big-Bang Observer (BBO), currently under study. In a previous paper [Phys. Rev. D73, 023504 (2006)] we connected CMB constraints to the amplitude and tensor spectral index of the inflationary gravitational-wave background (IGWB) at BBO frequencies for four classes of models of inflation by directly solving the inflationary equations of motion. Here we extend that analysis by including results obtained in the WMAP third-year data release as well as by considering two additional classes of inflationary models. As often noted in the literature, the recent indication that the primordial density power-spectrum has a red spectral index implies (with some caveats) that the amplitude of the IGWB may be large enough to be observable in the CMB polarization. Here we also explore the implications for the direct detection of the IGWB.

Figures (7)

• Figure 1: Results for the Higgs potential. The upper left panel shows the CMB constraints (68% and 95% contours) imposed by just considering the WMAP third year data Spergel:2006hy; the lower left panel shows the CMB constraints (68% and 95% contours) imposed by considering a suite of data including measurements of galaxy clustering and Lyman-alpha forest constraints as described in Ref. Seljak:2006bg. The dashed lines on the left-hand panels indicate $r=0.01$ roughly the limit for CMBPol Bock:2006yf. The panels on the right show the corresponding predictions for the IGWB given the CMB constraints. The dashed lines on the right-hand panels indicate the sensitivity of the second generation BBO interferometer known as 'BBO correlated' phinney_commSeto:2005qy. The solid black lines indicate directions of constant number of $e$-folds of inflation and the dotted black lines indicate directions of constant minimum field value, $\mu$.
• Figure 2: Same as Fig. 1 but for the Coleman-Weinberg potential.
• Figure 3: Same as Fig. 1 but for the PNGB potential.
• Figure 4: Same as Fig. 1 but for the chaotic potential. As commented in the text, only those models for which the index $p$ is a positive even integer allow for a proper end to inflation. For other choices of $p$ the form of the inflaton potential must change before inflation ends. As a result, we allow for the field value $\phi_{\mathrm{CMB}}$ to be a free parameter, as discussed in the text. In order to indicate the predictions for those models in this class that reach a proper end of inflation (i.e., where $p$ is a positive even integer), the solid black lines correspond to between 62 and 47 $e$-folds of inflation and the dotted lines indicate constant values for the index of the potential. As is commented in Ref. Spergel:2006hy a massive scalar field ($p =2$) is a good fit to the data whereas a quartic potential lies outside of the 2$\sigma$ confidence region using just WMAP3 data. This disagreement is worsened when using the WMAP3+ constraints.
• Figure 5: Similar to Fig. 1 but for the hybrid potential. Unlike Fig. 1 the solid black lines follow curves of constant $y_{\mathrm{CMB}} \equiv \phi_{\mathrm{CMB}}/\mu$. See the text for further discussion.