Robust end-to-end test of the single-field inflationary tensor consistency relation across CMB and laser-interferometer scales

The single-field slow-roll inflationary consistency relation predicts that the tensor spectral index satisfies $n_t=-r/8$, where $r$ is the tensor-to-scalar ratio. Boyle, Smith, Dvorkin and Turok (2015) showed that combining degree-scale CMB B-mode measurements (with aggressive delensing) and a proposed space-based laser interferometer such as BBO or DECIGO would provide a lever arm of more than 40 e-folds in scale for testing this relation, and proposed an improved extrapolation of the tensor spectrum based on expanding the inflaton potential amplitude $V^{1/2}(\phi)$ in the field rather than expanding $\ln P_t$ in $\ln k$. The paper's forecast acknowledged that transient features in $V(\phi)$ between the wavebands can break any smooth extrapolation scheme, assumed a reheating temperature above $T\sim10^4$ TeV, and showed that smooth potentials degenerate on CMB scales remain nearly indistinguishable even under direct detection; the concern that the assumed delensing performance is undemonstrated was instead raised by later literature (Trendafilova et al. 2024). The central question nevertheless remains open: current data give $r_{0.05}<0.036$ with $n_t$ effectively unconstrained, the delensed-CMB consistency test is out of reach of proposed technology, the primordial background has not been directly detected at any frequency, and no analysis has quantified the full end-to-end error budget for the combined test. The surviving problem is a robust end-to-end determination of whether and how the consistency relation can be verified across the combined CMB and interferometer wavebands.

Background

Cosmic inflation — a period of accelerated expansion in the very early universe — generates, in its minimal form (a single scalar 'inflaton' field rolling slowly down a smooth potential), a stochastic background of gravitational waves (tensor perturbations) alongside density (scalar) perturbations. The ratio of tensor to scalar power is the tensor-to-scalar ratio $r$; the scale dependence of the tensor power spectrum $P_t(k)\propto k^{n_t}$ defines the tensor spectral index $n_t$. Single-field slow-roll inflation predicts the consistency relation $n_t=-r/8$: a small negative tilt fixed by the amplitude. Verifying or falsifying this relation is one of the sharpest tests of the minimal inflationary paradigm, because many alternatives (multi-field dynamics, modified gravity, phase transitions in the early universe) generically violate it.

The tensor background imprints a curl ('B-mode') pattern in the cosmic microwave background (CMB) polarization on degree angular scales. The all-sky sample-variance floor for measuring $n_t$ from this signal is $\sigma(n_t)\approx0.03$, comparable to the predicted magnitude of $n_t$ itself; 'delensing' — statistically removing the B-modes induced by gravitational lensing of E-mode polarization, using non-Gaussian correlations with small-scale data — lowers this floor. Working in the context of the 2014 BICEP2 claim of $r\approx0.2$ (later shown to be dominated by Galactic dust), Boyle, Smith, Dvorkin and Turok (2015) showed that a very ambitious delensing CMB experiment could test the relation at a few sigma, and that the test becomes far more powerful if CMB measurements are combined with a direct detection of the primordial gravitational-wave energy density $\Omega_{\rm gw}$ at solar-system-scale wavelengths (frequencies near 0.1–1 Hz) by a proposed space-based laser interferometer such as the Big Bang Observer (BBO) or DECIGO. Together the two wavebands span a factor of order $10^{18}$ in scale (more than 40 e-folds), and over this lever arm the power-law form $P_t\propto k^{n_t}$ is no longer a good approximation. Boyle et al. therefore proposed an improved extrapolation that expands the potential amplitude $V^{1/2}(\phi)$ to quadratic order in the field $\phi$ (using the measured scalar amplitude, $r$ and $n_s$ at next-to-leading order in slow roll), which automatically enforces the end-of-inflation conditions that slow roll terminates when the first slow-roll parameter reaches $\epsilon_H=1$ and that the field settles at a minimum with $V\simeq0$; companion analyses by Caligiuri and Kosowsky (2014) studied the same program from the CMB side.

The forecast rests on several idealizations, some acknowledged in the paper itself and some clarified by later work. The improved extrapolation assumes smooth single-field dynamics and can be invalidated by transient features in $V(\phi)$ between CMB and interferometer scales; the authors note that any extrapolation scheme can be "broken" in this way, while arguing that their scheme performs at least as well as the traditional power-law extrapolation wherever the latter is reasonable. Propagating the primordial spectrum to a present-day $\Omega_{\rm gw}$ requires the post-inflationary thermal history — the reheating temperature and the effective numbers of relativistic species $g_*(T)$ and $g_{*,s}(T)$ at temperatures of order $10^4$ TeV, where the interferometer-band waves re-entered the horizon; the paper assumed a reheating temperature above this scale and noted that observed deviations of the transfer function could provide qualitatively new clues about the early universe, but none of these quantities is measured. The forecast improvements in $\sigma(n_t)$ rely on idealized delensing assumptions that the paper did not flag as an open issue; later analysis (Trendafilova et al. 2024) found the delensed-CMB consistency test to be out of reach of current and proposed experimental technology. Finally, even with a BBO-class detector, smooth potentials that are indistinguishable on CMB scales (for example axion-like and Higgs-like potentials tuned to the same $n_s$ and $r$) were shown in the paper to remain nearly indistinguishable on interferometer scales, their predicted $\Omega_{\rm gw}$ values differing by less than the resolution of a standard BBO mission. Since 2015 the observational premise has also shifted: the tightest current limit is $r_{0.05}<0.036$ (95% CL) from BICEP/Keck observations through the 2018 season, with $n_t$ essentially unconstrained when the consistency relation is relaxed (Paoletti et al. 2022), and no primordial gravitational-wave background has been directly detected at any frequency.

Problem Statement

Determine, by a quantitatively defensible end-to-end analysis, whether the single-field slow-roll inflationary tensor consistency relation — the CMB-scale relation $n_t=-r/8$ together with its extension across the more-than-40-e-fold lever arm from CMB scales to laser-interferometer frequencies via the improved $V^{1/2}(\phi)$ extrapolation — can be observationally verified or falsified, and with what required experimental and theoretical inputs. Central to this determination are the following aspects of the source forecast and the developments since, each of which the analysis must address; a treatment that omits any one of them is incomplete: (i) transient features in the inflaton potential between CMB and interferometer scales, which break any smooth extrapolation scheme; (ii) an uncertain reheating temperature and possible non-standard thermal history or relativistic content at $T\sim10^4$ TeV, which modify the tensor transfer function and hence the inferred primordial spectrum; (iii) realistic delensing performance, sky coverage, foregrounds and instrumental systematics, as opposed to idealized forecasting assumptions; and (iv) residual near-degeneracies among smooth single-field potentials that agree on CMB scales, which the source showed persist even under direct detection at interferometer frequencies.

References

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