Inflationary gravitational-wave background and measurements of the scalar spectral index

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. D 73, 023504 (2006)] we connected CMB constraints to the amplitude and tensor spectral tilt 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 Wilkinson Microwave Anisotropy Probe 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.

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 [33]; 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. 41. The dashed lines on the left-hand panels indicate $r=0.01$, roughly the limit for CMBPol [60]. 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” [31, 32]. 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 ${\varphi }_{\mathrm{CMB}}$ 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. 33 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 $\mathrm{WMAP}3+$ constraints.

Figure 5

Similar to Fig. 1 but for the hybrid potential. Unlike Fig. 1 the solid black lines follow curves of constant $y_{\mathrm{CMB}}\equiv {\varphi }_{\mathrm{CMB}}/\mu$. See the text for further discussion.

Figure 6

Similar to Fig. 1 but for power-law inflation. The stars indicate values for the power-law index, $p$. For a fixed scalar amplitude, $P_s$, this model occupies only a line in the $n_s-r$ plane because both $ϵ$ and $\eta$ are functions only of the index $p$ and not of the field value.

Figure 7

An upper limit to $r$ along with a measurement of $n_s$ will tell us information on the curvature of the inflaton potential. If we find that $r\lesssim 0.1$ and $0.94\lesssim n_s\lesssim 0.96$ then we may conclude, within the context of single-field slow-roll inflation, that $\eta <0$ which implies $V^{\prime \prime }<0$. Although qualitative, this conclusion would have far reaching implications for inflationary model building.