Structure formation by a fifth force: $N$-body versus linear simulations

We lay out the frameworks to numerically study the structure formation in both linear and nonlinear regimes in general dark-matter-coupled scalar field models, and give an explicit example where the scalar field serves as a dynamical dark energy. Adopting parameters of the scalar field which yield a realistic cosmic microwave background (CMB) spectrum, we generate the initial conditions for our $N$-body simulations, which follow the spatial distributions of the dark matter and the scalar field by solving their equations of motion using the multilevel adaptive grid technique. We show that the spatial configuration of the scalar field tracks well the voids and clusters of dark matter. Indeed, the propagation of scalar degree of freedom effectively acts as a fifth force on dark matter particles, whose range and magnitude are determined by the two model parameters $(\mu ,\gamma )$, local dark matter density as well as the background value for the scalar field. The model behaves like the $\Lambda \mathrm{CDM}$ paradigm on scales relevant to the CMB spectrum, which are well beyond the probe of the local fifth force and thus not significantly affected by the matter-scalar coupling. On scales comparable or shorter than the range of the local fifth force, the fifth force is perfectly parallel to gravity and their strengths have a fixed ratio $2{\gamma }^2$ determined by the matter-scalar coupling, provided that the chameleon effect is weak; if on the other hand there is a strong chameleon effect (i.e., the scalar field almost resides at its effective potential minimum everywhere in the space), the fifth force indeed has suppressed effects in high density regions and shows no obvious correlation with gravity, which means that the dark-matter–scalar-field coupling is not simply equivalent to a rescaling of the gravitational constant or the mass of the dark matter particles. We show these spatial distributions and (lack of) correlations at typical redshifts $(z=0,1,5.5)$ in our multigrid million-particle simulations. The viable parameters for the scalar field can be inferred on intermediate or small scales at late times from, e.g., weak lensing and phase space properties, while the predicted Hubble expansion and linearly simulated CMB spectrum are virtually indistinguishable from the standard $\Lambda \mathrm{CDM}$ predictions.

Figure 1

Upper panels: the potential (dashed curves), coupling function (dotted curves), and effective potential (solid curves) of the strong coupling scalar field model at various cosmic epochs. Lower panels: the same as the upper panel but for a coupling not strong enough so that the scalar field will not reside at the minimum of $V_{\mathrm{eff}}$. Solid circles denote the states of the scalar field and in case that this does not coincide with the minimum of effective potential, the later is indicated by an open circle. See text for explanations.

Figure 2

The evolution of the fractional energy densities in radiation (${\Omega }_{\mathrm{R}}$, red lines), baryons $+$ cold dark matter (${\Omega }_{\mathrm{M}}$, blue lines), and scalar field dark energy component (${\Omega }_{\mathrm{DE}}$, green lines) with respect to $\lg (1+z)$ where $z$ is the redshift. The model parameters for this case are chosen by $\gamma =1$ and $\mu =0.1$, 0.2, 0.3, 0.5, respectively. The physical parameters are ${\Omega }_{\mathrm{R}}=8.475\times {10}^{-5}$ for radiation, ${\Omega }_{\mathrm{B}}=0.05$ for baryons, ${\Omega }_{\mathrm{CDM}}=0.25$ for CDM and $H_0=70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ for the present Hubble expansion rate. To ensure these parameters the value of $V_0$ must be fine-tuned, and here we have used a trial-and-error method to adjust the value of $\lambda =\kappa V_0/3H_0^2$ as 0.53285, 0.44868, 0.38968, 0.30848, respectively, for $\mu =0.1$, 0.2, 0.3, 0.5. Note that given $H_0$ the propagations of the fractional energy densities fix the background evolution.

Figure 3

The total equation of state of the model with different values of $\mu$ as a function of $\lg (1+z)$. The parameters are the same as in Fig. 2 and the values of $\mu$ are respectively 0.1, 0.2, 0.3, and 0.5 from bottom to top on the left end of the figure (indicated beneath the curves).

Figure 4

The CMB power spectra for the couple scalar field model with $\gamma =0.5$. The red, blue, green (from second top to bottom), and black (top) curves correspond to $\mu =0.1$, 0.2, 0.3 with chameleon perturbations and $\mu =0.1$ without chameleon perturbations, respectively. Curves for $\mu \lesssim {10}^{-5}$ are indistinguishable from the $\Lambda \mathrm{CDM}$ result.

Figure 5

The matter power spectra for the couple scalar field model. The red, blue, green (from second bottom to top), and black (bottom) curves correspond to $\mu =0.1,0.2,0.3$ with chameleon perturbations and $\mu =0.1$ without chameleon perturbations. The curves with $\mu \lesssim {10}^{-5}$ are very close to the $\Lambda \mathrm{CDM}$ result.

Figure 6

The simulation results for the $\gamma =0.5$ and $\mu ={10}^{-5}$, ${10}^{-7}$ runs at redshifts $z=0$, $z=1$, and $z=5.5$. Shown are the particle spatial $y\mathrm{\text{−}}z$ distribution in a slab of $x=31.5–32.5\mathrm{Mpc}/\mathrm{fh}$ (upper panel) and the lg-lg diagram of the fifth force vs proper gravitational acceleration (in physical units of $\mathrm{m}/{\mathrm{s}}^2$) in this slab (lower panel). See the text for a detailed description. Each particle is a symbol with its color denoting the value of the effective mass $C(\phi )$, whose minimum/maximum are given on the top of each panel, and correspond to the two ends of the color scale shown on the top. The size of the square symbols in the lower panels is proportional to the misalignment angle between the fifth force and the gravity on a particle; the biggest squares correspond to antialignment, and particles with well-aligned forces are shown as dots. A line showing a fifth-to-gravity ratio of $2{\gamma }^2$ is also drawn to show the (lack of) correlations of the two forces. Note that $C(\phi )$ is generally the biggest in voids where the forces are the weakest, poorly aligned, and less correlated.

Figure 7

The same as Fig. 6 but for the $\gamma =1.0$ and $\mu ={10}^{-5}$, ${10}^{-7}$ runs.