Direct detection of the inflationary gravitational-wave background

Inflation generically 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) or Japan’s Deci-Hertz Interferometer Gravitational-wave Observer (DECIGO), both currently under study. Here we investigate the detectability of the inflationary gravitational-wave background at BBO/DECIGO frequencies. To do so, we survey a range of slow-roll inflationary models consistent with constraints from the CMB and large-scale structure (LSS). We go beyond the usual assumption of power-law power spectra, which may break down given the 16 orders of magnitude in frequency between the CMB and direct detection, and solve instead the inflationary dynamics for four classes of inflaton potentials. Direct detection is possible in a variety of inflationary models, although probably not in any in which the gravitational-wave signal does not appear in the CMB polarization. However, direct detection by BBO/DECIGO can help discriminate between inflationary models that have the same slow-roll parameters at CMB/LSS scales.

Figure 1

Regions in the $n_s\mathrm{\text{−}}r$ parameter space consistent with the CMB-only (orange/medium gray) [50], CMB plus galaxy surveys (red/dark gray), and CMB plus galaxy surveys plus Lyman-$\alpha$-forest constraints (green/light gray) [49]. Here, $r$ is the tensor-to-scalar ratio, and $n_s$ is the scalar spectral index at CMB scales. We plot on top of these regions the parameter space occupied by the four models of inflation we consider: power-law (solid black line), chaotic (dotted red), symmetry-breaking (dash-dotted purple) and hybrid (short-dashed dark red). The right axis shows the energy scale $[V(k_c){]}^{1/4}$ of inflaton.

Figure 2

Current limits and projected sensitivities to a stochastic gravitational-wave background versus the gravitational-wave frequency. The solid curves all indicate current upper limits, while the dashed curves indicate projected sensitivities. The LISA curve is from Ref. 89 and BBO correlated from Ref. 62. The BBO sensitivity is estimated by increasing the BBO-correlated curve by 4 orders of magnitude [see Eq. (29)]. The BBN constraint results from the limit to the number of relativistic degrees of freedom at big-bang nucleosynthesis (e.g., Ref. 90); the “M/R” constraint is from CMB/LSS constraints to matter-radiation equality [91]; the “z. var” curve is from Ref. 92; and the quasar-astrometry limit from Refs. 93, 94. We note that the BBN and M/R constraints assume a scale-invariant gravitational-wave background that extends $\sim 60$ $e$ folds below the current Hubble horizon. LIGO sensitivities, taken from the LIGO Scientific Collaboration White Paper on Detector Research and Development [95] are given in terms of a correlated analysis between the Hanford, WA and Livingston, LA sites [see Eq. (29)]. The run 1 LIGO limit (“S1 LIGO”) is from Ref. 96 and the run 3 LIGO limit (“S3 LIGO”) is from Ref. 97. Also shown are millisecond-pulsar timing constraints (current [98, 99] and sensitivities projected for the Square-Kilometer Array [100]). Curves corresponding to scale-invariant (i.e., $n_t=0$) gravitational-wave backgrounds are shown (dotted curves), labeled by the associated inflationary energy scales at CMB/LSS scales (but keep in mind that slow-roll inflation generically predicts $n_t<0$, less power on small scales). The CMB/LSS currently constrains this value to be below $3.36\times {10}^{16}\mathrm{GeV}$ at CMB/LSS scales. Future CMB measurements may be able to reach energy scales near ${10}^{15}\mathrm{GeV}$ [101, 102, 103, 104].

Figure 3

Regions in the ${\Omega }_{\mathrm{GW}}{h}^2\mathrm{\text{−}}{n}_t$ parameter space for (a) power-law, (b) chaotic, (c) symmetry-breaking, and (d) hybrid inflation. The colored (shaded) regions map out the corresponding regions in Fig. 1. Here, the gravitational-wave density ${\Omega }_{\mathrm{GW}}{h}^2$ and spectral index $n_t$ are both evaluated at DECIGO/BBO scales. Also shown are the sensitivity goals of BBO and DECIGO.

Figure 4

Here we plot the inflationary gravitational-wave background amplitude ${\Omega }_{\mathrm{GW}}{h}^2$ obtained with the exact inflationary dynamics described in Sec. 4 versus the amplitudes obtained with the power-law approximations (with slow-roll parameters fixed by CMB/LSS observations) given in Sec. 2b. The panels show results for (a) power-law, (b) chaotic, (c) symmetry-breaking, and (d) hybrid inflation. The regions are models taken from the $n_s\mathrm{\text{−}}r$ parameter space in Fig. 1 (and shown in Fig. 3) and the blue dashed curves indicate equality.