Diagnostic of Horndeski theories

Abstract

We study the effects of Horndeski models of dark energy on the observables of the large-scale structure in the late time universe. A novel classification into Late dark energy, Early dark energy and Early modified gravity scenarios is proposed, according to whether such models predict deviations from the standard paradigm persistent at early time in the matter domination epoch. We discuss the physical imprints left by each specific class of models on the effective Newton constant µ, the gravitational slip parameter η, the light deflection parameter Σ and the growth function f σ 8 and demonstrate that a convenient way to dress a complete portrait of the viability of the Horndeski accelerating mechanism is via two, redshift-dependent, diagnostics: the µ(z) − Σ(z) and the f σ 8 (z) − Σ(z) planes. If future, model-independent, measurements point to either Σ − 1 < 0 at redshift zero or µ − 1 < 0 with Σ − 1 > 0 at high redshifts or µ − 1 > 0 with Σ − 1 < 0 at high redshifts, Horndeski theories are effectively ruled out. If f σ 8 is measured to be larger than expected in a ΛCDM model at z > 1.5 then Early dark energy models are definitely ruled out. On the opposite case, Late dark energy models are rejected by data if Σ < 1, while, if Σ > 1, only Early modifications of gravity provide a viable framework to interpret data.

Effective field theory of dark energy: A review

“…The validity of the conjecture at all times is in tension with the previous work where the sign agreement is lost for z < 1.5 [167]. The reasons stems from how viable are the models considered: using only stability conditions (ghost and gradient) leads to the violation of the conjecture at small redshifts [167], while requiring additional priors based on observational constraints allows for the conjecture to be restored [155]. In particular, Solar System bounds on the variation of the Newton's constant translate into a stringent prior on f at present time [168].…”

“…The results showed that the conjecture holds very well within the QS approximation, but the exact behaviors of µ and Σ obtained with EFTCAMB violate the conjecture at k = 0.001 h/Mpc, where the full dynamics of the Horndeski models allows Σ > 1 and µ < 1 [131]. The validity of the conjecture at all times is in tension with the previous work where the sign agreement is lost for z < 1.5 [167]. The reasons stems from how viable are the models considered: using only stability conditions (ghost and gradient) leads to the violation of the conjecture at small redshifts [167], while requiring additional priors based on observational constraints allows for the conjecture to be restored [155].…”

“…The sign agreement in µ ∞ − 1 and Σ ∞ − 1 is more obvious because they have the same pre-factor and the additional contributions from α T and the fifth force are of the same order [155]. The existence of a correlation between µ and Σ was also found using a Monte-Carlo approach to generate a large sample of viable Horndeski model with a fixed wCDM background 11 and the QS approximation on sub-Compton scales to model µ ∞ and Σ ∞ [167]. This diagnostic suggests that Horndeski theories would be strongly disfavored if [167]: i) µ ∞ and Σ ∞ are observed to have opposite sign for z 1.5; ii) µ ∞ < 1 at z = 0.…”

“…The existence of a correlation between µ and Σ was also found using a Monte-Carlo approach to generate a large sample of viable Horndeski model with a fixed wCDM background 11 and the QS approximation on sub-Compton scales to model µ ∞ and Σ ∞ [167]. This diagnostic suggests that Horndeski theories would be strongly disfavored if [167]: i) µ ∞ and Σ ∞ are observed to have opposite sign for z 1.5; ii) µ ∞ < 1 at z = 0. A complementary analysis allowed to investigate the validity of the conjecture not only at different redshift but also at different scales as shown in Figure 2 [131].…”

“…The selection of the viable parameter space induced by the stability conditions has non trivial implications for EFT predictions. We already discussed in Section 3.2 the µ − Σ conjecture [155] and its implications [167,131]. Other general behaviors can be identified using a Monte-Carlo approach to generate a vast sample of healthy pure Horndeski models and explore the phenomenology of µ, η, Σ and f σ 8 across z. f σ 8 is the so-called growth function, i.e.…”