Constraints on scalar and tensor spectra from N_eff

Abstract

At the linear level, the gravitational wave (GW) spectrum predicted by inflation, and many of its alternatives, can have arbitrarily small amplitude and consequently an unconstrained tilt. However, at second order, tensor fluctuations are sourced by scalar fluctuations that have been measured in the cosmic microwave background (CMB). These second order fluctuations generically produce a minimum amount of tensor perturbations corresponding to a tensor-to-scalar ratio of r ∼ 10 −6 . Inverting this relationship yields a bound on the tensor tilt sourced by scalar fluctuations. Since this induced GW spectrum depends on the scalar spectrum, we derive a new indirect bound that involves all scales of the scalar spectrum based on CMB observations. This bound comes from the constraint on the number of effective relativistic degrees of freedom, N ef f . We estimate the bound using current data, and the improvements expected by future CMB experiment. The bound forces the running and running of running to conform to standard slow-roll predictions of α, β (n s − 1) 2 where α ≡ dns d ln k and β ≡ d 2 ns s ln k 2 , improving on current CMB measurements by an order of magnitude. This bound has further implications for the possibility of primordial black holes as dark matter candidates. Performing a likelihood analysis, including this new constraint, we find that positive α and/or β are disfavored at least at 1σ. Even using conservative analysis β + 0.074 α > 8.6 × 10 −4 are ruled out at 3σ. Finally, using bounds on the fractional energy density of gravitational waves today obtained by LIGO and the Pulsar Timing Array, we obtain a bound on the primordial scalar spectrum on these scales and give forecast for future measurements.

Sourced scalar fluctuations in bouncing cosmology

“…the projected forecasts for the different experiments are listed in Table 1, [42]. Given that the LISA Ω exp GW > 10 −13 1.5 × 10 12 − 1.5 × 10 13…”

Gravitational Wave Gastronomy

“…This is expected to improve significantly to ∆N eff < 0.03 as a conservative estimate of the sensitivity of next generation experiments [88]. A hypothetical experiment limited only by the cosmic variance limit was found to be sensitive to changes to the number of relativistic degrees of freedom as small as [89] ∆N…”

Gauge independence of induced gravitational waves

“…Gravitational waves (GWs) are a useful probe of small-scale curvature perturbations [3][4][5][6][7][8] since if the latter is enhanced, a sizable amount of GWs is produced at the second order of the cosmological perturbation. We call it the induced GWs 1 (see early works [9][10][11][12] and recent developments [13][14][15][16][17][18][19][20][21][22]).…”