Positivity in the sky

Abstract

Positivity bounds - the consequences of requiring a unitary, causal, local UV completion - place strong restrictions on theories of dark energy and/or modified gravity. We derive and investigate such bounds for Horndeski scalar-tensor theories and for the first time pair these bounds with a cosmological parameter estimation analysis, using CMB, redshift space distortion, matter power spectrum and BAO measurements from the Planck, SDSS/BOSS and 6dF surveys. Using positivity bounds as theoretical priors, we show that their inclusion in the parameter estimation significantly improves the constraints on dark energy/modified gravity parameters. Considering as an example a specific class of models, which are particularly well-suited to illustrate the constraining power of positivity bounds, we find that these bounds eliminate over 60% of the previously allowed parameter space. We also discuss how combining positivity requirements with additional theoretical priors has the potential to further tighten these constraints: for instance also requiring a subluminal speed of gravitational waves eliminates all but 1% of the previously allowed parameter space.

Figures (3)

• Figure 1: Cosmological parameter constraints for the quartic Horndeski theory \ref{['quarticA']}, using $\alpha_i = c_i \Omega_{\rm DE}$\ref{['oParam']} and different combinations of positivity \ref{['posprior']} and (sub-)luminality priors \ref{['lumprior']}. The positivity priors are derived from $\phi\phi \to \phi\phi$ scattering. Contours mark $68\%$ and $95\%$ confidence intervals, computed using CMB, RSD, BAO and matter power spectrum measurements. Dotted lines mark $c_i=0$ (the GR value), $c_T \geq -1$ (real GW speed) and $c_B < 2 c_T/(1+c_T)$ (positivity). The positivity prior eliminates over $60\%$ of the $2\sigma$ parameter space. If also combined with a (sub-)luminality prior, only $\lesssim 1\%$ of the $2\sigma$ parameter space survives.
• Figure 2: $\phi \phi \to \phi \phi$ scattering amplitude for Horndeski using the above vertices (below each diagram we include only the shift-symmetric parts of the $G_n$ for brevity), where $+...$ includes all permutations of external legs. The top line is the leading order result, while the bottom line is subleading in $M_P$ (and violates the Galileon symmetry, leading to a non-zero $c_{ss}^{\phi\phi}$).
• Figure 3: $\phi h \to \phi h$ scattering amplitude for Horndeski using the above vertices (below each diagram we include only the shift-symmetric part of $G_n$ for brevity), where $+...$ includes all permutations of external legs. The diagrams in blue vanish when the external gravitons are taken on-shell, and the remaining two terms in black exactly cancel.