Testing and extending the inflationary consistency relation for tensor modes

Latham Boyle, Kendrick M. Smith, Cora Dvorkin, and Neil Turok

Phys. Rev. D 92, 043504 – Published 3 August 2015

Abstract

If observations confirm BICEP2’s claim of a tensor-scalar ratio $r \approx 0.2$ on CMB scales, then the inflationary consistency relation $n_{t} = - r / 8$ predicts a small negative value for the tensor spectral index $n_{t}$ . We show that future CMB polarization experiments should be able to confirm this prediction at several sigma. We also show how to properly extend the consistency relation to solar system scales, where the primordial gravitational wave density $Ω_{gw}$ could be measured by proposed experiments such as the Big Bang Observer. This would provide a far more stringent test of the consistency relation and access much more detailed information about the early Universe.

Received 5 September 2014

DOI:https://doi.org/10.1103/PhysRevD.92.043504

Authors & Affiliations

Latham Boyle¹, Kendrick M. Smith¹, Cora Dvorkin², and Neil Turok¹

¹Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
²Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, New Jersey 08540, USA

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Vol. 92, Iss. 4 — 15 August 2015

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Images

Figure 1
Gravity wave and lensing contributions to $C_{ℓ}^{B B}$ , before and after delensing with $Δ_{P} = 1 μ K$ -arcmin and $θ_{FWHM} = 2$ arcmin. Delensing allows the gravity wave signal to be measured to higher $ℓ$ , improving the lever arm for measuring $n_{t}$ .
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Figure 2
Forecasted statistical error $σ (n_{t})$ , marginalized over $r$ , for varying instrumental noise $Δ_{P}$ , with or without delensing, computed using the Fisher matrix and noise power spectrum estimation machinery from [10].
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Figure 3
To quantify the accuracy of the extrapolations discussed in the text, we compare $Δ Ω_{gw}^{BBO} = 10^{- 17}$ , the 1-sigma error bar of “standard BBO” after one year of operation, with $Δ Ω_{gw}^{extrap} = | Ω_{gw}^{exact} - Ω_{gw}^{extrap} |$ , the difference between the exact value $Ω_{gw}^{exact}$ on BBO scales (predicted by a given potential) and the extrapolated value $Ω_{gw}^{extrap}$ based on the zeroth-order (black), first-order (red), second-order (green) or third-order (blue) traditional extrapolation (2), or the improved extrapolation (3) (pink). We compare with three families of standard inflationary potentials: (i) the monomial potentials $V (φ) = λ φ^{n}$ with $1 < n < 4$ (solid curves); (ii) the axionlike (“natural inflation”) potentials $V (φ) = V_{0} [1 - \cos (φ / f)]$ (dashed curves); and (iii) the Higgs-like potentials $V (φ) = V_{0} [1 - (φ / f)^{2}]^{2}$ (dotted curves). The zeros in the pink solid curve at $r \approx 0.14$ and $r \approx 0.27$ correspond to the potentials $V = (1 / 2) m^{2} φ^{2}$ and $V = λ φ^{4}$ , where our improved extrapolation is essentially exact. The axion curve dips to zero at $r \approx 0.14$ , too: this happens because, in the limit $f \to \infty$ , the final 60 $e$ -folds of inflation occur near the bottom of the potential well, where the potential may be well appoximated by $(1 / 2) m^{2} φ^{2}$ . Finally, the improved extrapolation is also nearly exact for the Higgs-like potential (iii), so the pink dotted curve lies below the bottom of the figure, and is not shown.
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Figure 4
$Ω_{gw}$ (on BBO scales) vs $r$ (on CMB scales). The solid black diagonal line shows the relationship if $P_{t}$ is exactly scale invariant ( $n_{t} = 0$ ). The filled cyan band shows the third-order traditional extrapolation (2), for Planck’s best-fit value $n_{s} = 0.9603$ , and $α_{s} = 0 \pm 0.0003$ (this range is chosen to make the width of the cyan band comparable to the size of the BBO error bar). The orange dotted curves show the improved extrapolation based on (3), the same $n_{s}$ , and a future $1 σ$ error bar ( $n_{s} = 0.9603 \pm 0.0017$ ). The vertical dashed lines show the range of $r$ predicted by $V = (1 / 2) m^{2} φ^{2}$ inflation, for $49 < N < 62$ , cf. [32]. For reference, we also show the $1 σ$ error bar for “standard BBO” after one year of operation.
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