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