N-Body Simulations for Coupled Scalar Field Cosmology

TL;DR

This work develops and validates a fully consistent N-body framework for coupled scalar field cosmologies with an inverse power-law potential and exponential dark-matter coupling. It solves the scalar-field equation alongside the Poisson equation, incorporating a fifth force on dark matter while keeping baryons uncoupled, and demonstrates that the fifth force is effectively proportional to gravity with a scale set by $f \approx 2\gamma^2 g$. The study shows significant nonlinear effects: early-time deviations in the growth of structure require model-specific initial conditions, and nonlinear power spectra exhibit notable enhancements with stronger coupling, accompanied by heavier halos and modified inner density profiles relative to $\Lambda$CDM. The results, supported by a modified halo finder, provide a practical route to confront coupled scalar-field scenarios with observations and establish the robustness of a rescaled gravitational constant approach in these models.

Abstract

We describe in detail the general methodology and numerical implementation of consistent N-body simulations for coupled scalar field cosmological models, including the background cosmology and the generation of initial conditions (with the different couplings to different matter species taken into account). We perform fully consistent simulations for a class of coupled scalar field models with an inverse power-law potential and negative coupling constant, for which the chameleon mechanism does not operate. We find that in such cosmological models the scalar-field potential plays a negligible role except in the background expansion, and the fifth force that is produced is proportional to gravity in magnitude, justifying the use of a rescaled gravitational constant G in some earlier N-body simulations of similar models. We study the effects of the scalar coupling on the nonlinear matter power spectra and compare with linear perturbation calculations to investigate where the nonlinear model deviates from the linear approximation. For the first time, the algorithm to identify gravitationally virialized matter halos is adapted to the scalar field cosmology, and then used to measure the mass function and study the properties of virialized halos. We find that the net effect of the scalar coupling helps produce more heavy halos in our simulation boxes and suppresses the inner (but not the outer) density profile of halos compared with those predicted by lambda-CDM, while this suppression weakens as the coupling between the scalar field and dark matter particles increases in strength.

Figures (14)

• Figure 1: (Color Online) Figures to illustrate the background evolution of our coupled scalar field models. Upper Left Panel: The fractional energy densities for radiation (green), coupled dark matter (black), baryons (blue) and the scalar field (red); note that $\Omega_{\mathrm{cc}}=8\pi GC(\varphi)\rho_{\mathrm{CDM}}/3H^2$. Upper Right Panel: The equation of state of the scalar field, $w\equiv p_{\varphi}/\rho_{\varphi}$ in which $p_{\varphi}=\frac{1}{2}\dot{\varphi}^2-V(\varphi)$ and $\rho_{\varphi}=\frac{1}{2}\dot{\varphi}^2+V(\varphi)$. Lower Left Panel: The ratio between the Hubble expansion rate in the coupled scalar field model and that in the $\Lambda$CDM paradigm, other physical parameters such as $\rho_{\mathrm{B}}, \rho_{\mathrm{RAD}}, \rho_{\mathrm{CDM}}$ being held the same, as a function of the scale factor $a$. Lower Right Panel: The "varying mass" of the dark matter particles as a function of $a$ -- here $m_0$ is the constant bare mass of the particles and $m=e^{\gamma\varphi}m_0$. In all figures we have chosen $\alpha=0.1$ and the solid, dotted, dashed and dot-dashed curves represent the models with $\gamma=-0.05, -0.10, -0.15$ and $-0.20$ respectively.
• Figure 2: (Color Online) The linear power spectra of the coupled scalar field model. Upper Left Panel: The Cosmic Microwave Background (CMB) power spectra for the four models with $\alpha=0.1$. Upper Right Panel: The same but for the models with $\alpha=0.5$. Lower Left Panel: The matter power spectra at present day (redshift $z=0$) for the four models with $\alpha=0.1$. Lower Right Panel: The same but for the models with $\alpha=0.5$. In all figures the solid (black), dotted (blue), dashed (green) and dot-dashed (red) curves represent the models with $\gamma=-0.05, -0.10, -0.15$ and $-0.20$ respectively.
• Figure 3: (Color Online) Upper Panels: The matter power spectra for baryons only (Left) and for dark matter only (Right), at current time ($z=0$). Lower Panels: The same but at an earlier time $z=49$ where the initial condition for the $N$-body simulations are computed. In all figures $\alpha=0.1$ and the solid (black), dotted (blue), dashed (green) and dot-dashed (red) curves represent the models with $\gamma=-0.05, -0.10, -0.15$ and $-0.20$ respectively.
• Figure 4: Upper Left Panel: The fractional change of the growth factor $D_{+}$ of the dark matter density perturbation in the coupled scalar field model as compared with the $\Lambda$CDM prediction; for clearness shown are only the results for the two models with $\alpha=0.1,\gamma=-0.05$ and $\alpha=0.1,\gamma=-0.15$, as indicated above the curves, and for each model the solid, dotted, dashed and dot-dashed curves represent respectively the result for $k=0.0001, 0.001, 0.01$ and $0.1~\mathrm{Mpc}^{-1}$. Upper Right Panel: The same as above, but for the fractional change of $\dot{D}_{+}$ as a function of the scalar factor $a$. Lower Left Panel: The time evolution of the fractional energy density of the kinetic energy of the scalar field, $\Omega_{\mathrm{kin}}=4\pi G\dot{\varphi}^2/3H^2$ for the two models with $\alpha=0.1,\gamma=0.05$ (solid curve) and $\alpha=0.1,\gamma=-0.15$ (dotted curve) respectively. Lower Right Panel: The evolution of the density contrast of the scalar field, of which the energy density is $\rho_{\varphi}=\frac{1}{2}\dot{\varphi}^2+V(\varphi)$; the solid, dotted, dashed and dot-dashed curves are for four different length scales $k=0.0001, 0.001, 0.01$ and $0.1~\mathrm{Mpc}^{-1}$ respectively, and for each style of curve the upper (lower) one is for the model with $\alpha=0.1,\gamma=-0.15$ ($\alpha=0.1,\gamma=-0.05$).
• Figure 5: (Color Online) Snapshots of the particle distribution in our four coupled scalar field models as indicated by the subtitles of the panels. $a$ is the scale factor and $a=1$ is the present time. For clearness we only pick out a slice of the simulation box with $30~h^{-1}\mathrm{Mpc}<z<30.3~h^{-1}\mathrm{Mpc}$ and $0~h^{-1}\mathrm{Mpc}<x,y<64~h^{-1}\mathrm{Mpc}$. The blue dots represent dark matter particles and red dots baryons.