K-mouflage at high $k$: extending the reach of $\texttt{Hi-COLA}$

TL;DR

This work extends the Hi-COLA Horndeski simulation framework to include K-mouflage gravity, enabling rapid, non-linear predictions of the matter power spectrum within these models. By formulating K-mouflage in the Jordan frame and detailing the background, linear growth, fifth-force, and conformal-factor contributions, the authors assess how the power-spectrum boost relative to GR-ΛCDM arises and evolves. They find that the modified background and conformal weakening of gravity dominate large-scale clustering, while the conformal factor reduces small-scale boosts, yielding a non-monotonic scale dependence distinct from Vainshtein-screened theories. Convergence tests and parameter-space explorations demonstrate Hi-COLA’s utility for probing K-mouflage with upcoming surveys and for building emulators, while highlighting frame considerations essential for fair cross-frame comparisons.

Abstract

The $\texttt{Hi-COLA}$ code is an efficient dark matter simulation suite that flexibly handles the Horndeski family of modified gravity models. In this work we extend the scope of $\texttt{Hi-COLA}$ to accommodate Horndeski theories with K-mouflage screening, allowing for the computation of matter power spectra in the non-linear regime in these models. We explore the boost of the dark matter power spectrum relative to GR-$Λ$CDM in K-mouflage gravity, and also discuss how large-scale structure computations change between the Einstein and Jordan frames. A dissection of the relative contributions of the modified background, linear growth, fifth force, and the conformal factor (a new inclusion to $\texttt{Hi-COLA}$) to the boost factor is presented. The ability of $\texttt{Hi-COLA}$ to run with general Horndeski models and multiple screening mechanisms makes it an ideal tool for testing gravity with upcoming galaxy survey data.

Figures (10)

• Figure 1: Left panel: Ratio of the Hubble expansion rates in the Einstein (solid orange line) and Jordan (dashed green line) frames for K-mouflage gravity with the the Hubble rate in $\Lambda$CDM. In this case, the K-mouflage parameters are: $n=2, K_0=1$ and $\beta_K = 0.2$. Right panel: Fractional energy densities for K-mouflage in the Jordan frame, for the same model parameters as in the left panel. Notice that it is the effective energy densities, defined in eqs. \ref{['eq:effective_omegam']} and \ref{['eq:effective_omegaphi']}, that sum to 1 in the Jordan frame. The need to define effective energy densities to recover regular closure is due to the transformation of energy densities between the Jordan and Einstein frame.
• Figure 2: Left: Growing mode solutions, $D_+$, of the growth equation for K-mouflage in the Einstein frame (blue), and GR-$\Lambda$CDM (red). The modifications to the K-mouflage growth factor equation from the coupling term in the Poisson equation and the conformal term are shown in green, while the enhancing effects from the expansion history are shown in yellow. The non-linear counterpart of this plot is Fig. \ref{['fig:bk_kmou_breakdown_evolving_redshift']}. Right: The opposing contributions of the conformal term in the K-mouflage growth factor equation in the Jordan frame [eq. \ref{['eq:growth_jordan']}] and Einstein frame [eq. \ref{['eq:growth_einstein']}], and the transformation of the expansion history [eq. \ref{['eq:E2J_hubble']}], resulting in negligible frame differences.
• Figure 3: Components affecting the growth equation, besides the expansion: the conformal factor $A$ [eq.\ref{['eq:conformal_factor']}], and the modification to the Poisson equation $\mu$ [eq. \ref{['eq:kmou_mu']}]. The dashed vertical line corresponds to a redshift of $z_E = 0.5$, while the dotted vertical line corresponds to a redshift of $z_E = 1.0$. These will be referred to in the discussion of Fig. \ref{['fig:bk_kmou_breakdown_evolving_redshift']}.
• Figure 4: Diagrammatic representation of the radii of significance for $X^m$ terms compared to $X^p(\Box \phi)^q$ terms. $r_{X^m}$ is the radius of significance for the perturbation sourced by all terms like $X^m$ in action \ref{['eq:galileon_action']}. $r_{p,q}$ corresponds to the radius of significance for terms like $X^p(\Box \phi)^q$. The $p=q=1$ term is the canonical Vainshtein radius $r_V$. Notice the $X^p(\Box \phi)^q$ terms have differing radii of significance, in contrast to the $X^m$ terms.
• Figure 5: A comparison of the effect of modifications to the background (yellow curves), and modifications to the gravitational forces (green curves) experienced by dark matter particles, on the deviations from GR-$\Lambda$CDM at the level of the power spectrum. On the left is K-mouflage with $n=2$, $K_0=1$ and $\beta_K=0.2$ (the K-mouflage model with the strongest boost signal), and on the right the Cubic Galileon for reference. As the Cubic Galileon possesses Vainshtein screening, there is an extra curve (red), showing the effects of the unscreened fifth force on the Cubic Galileon boost. For both models, the modified background is a significant contributor to the enhancement of the boost. However, unlike the Cubic Galileon, where the Vainshtein mechanism suppresses the enhancing effects of the linear fifth force, in K-mouflage the strong modified background-enhancement competes with the conformal weakening of the gravitational force. This leads to a suppression of the boost, below $1$ in K-mouflage's case.