KGB-evolution: a relativistic $N$-body code for kinetic gravity braiding models

Abstract

We present KGB-evolution, a relativistic $N$-body simulation code that extends the $k$-evolution code by incorporating an effective field theory parameterization of kinetic gravity braiding, while also including the $k$-essence model as a limiting case. As a first step, we implement the linearized dark energy stress-energy tensor and scalar field equations, providing the groundwork for a future full Horndeski theory extension. We validate KGB-evolution by comparing its power spectra against linear predictions from hi$\_$class, finding excellent agreement on large scales at low redshifts and over all scales at high redshifts. We demonstrate that nonlinear growth of matter and metric perturbations on small scales drives the linearized dark energy field into a nonlinear clustering regime, which in turn feeds back on the growth of cosmic structure. In contrast to the $k$-essence limit, a nonzero braiding considerably amplifies this backreaction, producing a significantly stronger alteration of structure formation in the kinetic gravity braiding model.

Figures (13)

• Figure 1: (a) KGB-evolution: nonlinear metric perturbations directly (via metric potentials and their time derivatives) source a linear dark energy perturbation and vice versa. (b) gevolution: purely linear $\delta_{\rm DE}$ sources metric perturbations. (c) hi_class: all perturbations are treated linearly both ways.
• Figure 2: Matter power spectra at $z=2, 1, 0.5, 0$, with lower sub-plots showing relative differences. (a) $k$-essence model ($\hat{\alpha}_{\rm K}=3000,\ \hat{\alpha}_{\rm B}=0$): linear (dashed, hi_class) vs nonlinear (solid, KGB-evolution). (b) KGB model ($\hat{\alpha}_{\rm K}=3000,\ \hat{\alpha}_{\rm B}=1.5$): linear (dashed, hi_class) vs nonlinear (solid, KGB-evolution). (c) Linear spectra: $k$-essence (dashed) vs KGB (solid). (d) Nonlinear spectra: $k$-essence (dashed) vs KGB (solid). The grey lines indicate 1% and 2% bounds.
• Figure 3: Potential power spectra at $z=2, 1, 0.5, 0$, with lower sub-plots showing relative differences. (a) $k$-essence model ($\hat{\alpha}_{\rm K}=3000,\ \hat{\alpha}_{\rm B}=0$): linear (dashed, hi_class) vs nonlinear (solid, KGB-evolution). (b) KGB model ($\hat{\alpha}_{\rm K}=3000,\ \hat{\alpha}_{\rm B}=1.5$): linear (dashed, hi_class) vs nonlinear (solid, KGB-evolution). (c) Linear spectra: $k$-essence (dashed) vs KGB (solid). (d) Nonlinear spectra: $k$-essence (dashed) vs KGB (solid). The grey lines indicate 1% and 2% bounds.
• Figure 4: Linear (dashed lines) and nonlinear (solid lines) dark energy density contrast power spectra at $z=2,1,0.5, 0$ for (a) $k$-essence model, where modes grow above and decay within the sound horizon (vertical dotted lines), and (b) KGB model, where braiding parameter $\alpha_{\rm B}$, sustains enhanced clustering even inside the horizon.
• Figure 5: Linear (dashed lines) and nonlinear (solid lines) scalar field perturbation power spectra at $z = 2, 1, 0.5, 0$ for (a) $k$-essence and (b) KGB models. In $k$-essence case, the power decays inside the sound horizon and nonlinear contributions exceed linear ones at high $k$ due to $\pi$ being sourced by nonlinear matter clustering. In KGB model, $\pi$ crosses zero at intermediate scales and changes sign, which explains the turnover of the power spectrum at that point. Beyond that point, the spectrum decays gradually toward smaller scales.