Improvements in cosmological constraints from breaking growth degeneracy

The key probes of the growth of large-scale structure are its rate $f$ and amplitude ${\sigma }_8$. Redshift space distortions in the galaxy power spectrum allow us to measure only the combination $f{\sigma }_8$, which can be used to constrain the standard cosmological model or alternatives. By using measurements of the galaxy-galaxy lensing cross-correlation spectrum or of the galaxy bispectrum, it is possible to break the $f{\sigma }_8$ degeneracy and obtain separate estimates of $f$ and ${\sigma }_8$ from the same galaxy sample. Currently there are only a handful of such separate measurements, but even this allows for improved constraints on cosmological models. We explore how having a larger and more precise sample of such measurements in the future could constrain further cosmological models. We consider what can be achieved by a future nominal sample that delivers a $\sim 1\mathrm{\%}$ constraint on $f$ and ${\sigma }_8$ separately, compared to the case with a similar precision on the combination $f{\sigma }_8$. For the six cosmological parameters of $\mathrm{\Lambda }$CDM, we find improvements of $\sim \text{\hspace{0.17em}⁣}5$–$50\mathrm{\%}$ on their constraints. For modified gravity models in the Horndeski class, the improvements on these standard parameters are $\sim \text{\hspace{0.17em}⁣}0$–$15\mathrm{\%}$. However, the precision on the sum of neutrino masses improves by 65% and there is a significant increase in the precision on the background and perturbation Horndeski parameters.