# Constraints on Scalar and Tensor spectra from $N_{eff}$

## TL;DR

This paper derives a new indirect bound on the tensor spectrum from scalar fluctuations using CMB data and $N_{eff}$ constraints, impacting inflation models, primordial black holes, and gravitational wave observations.

## Contribution

It introduces a novel bound on the tensor tilt sourced by scalar fluctuations based on $N_{eff}$ and CMB data, refining inflationary parameter constraints.

## Key findings

- Bounds on scalar spectral runnings are tightened by an order of magnitude.
- Positive runnings are disfavored at at least 1 sigma.
- Implications for primordial black hole dark matter and gravitational wave constraints.

## 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 $r\sim 10^{-6}$. 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 \textit{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, $N_{eff}$. 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 $\alpha,\beta \lesssim (n_s-1)^2$, 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 $\alpha$ and/or $\beta$ are disfavored at least at $1\sigma$. 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.