Dynamics of minimally coupled dark energy in spherical halos of dark matter

Abstract

We analyse the evolution of scalar field dark energy in the spherical halos of dark matter at the late stages of formation of gravitationally bound systems in the expanding Universe. The dynamics of quintessential dark energy at the center of dark matter halo strongly depends on the value of effective sound speed $c_s$ (in units of speed of light). If $c_s\sim 1$ (classical scalar field) then the dark energy in the gravitationally bound systems is only slightly perturbed and its density is practically the same as in cosmological background. The dark energy with small value of sound speed ($c_s<0.1$), on the contrary, is important dynamical component of halo at all stages of their evolution: linear, non-linear, turnaround, collapse, virialization and later up to current epoch. These properties of dark energy can be used for constraining the value of effective sound speed $c_s$ by comparison the theoretical predictions with observational data related to the large scale gravitationally bound systems.

Appendices

Appendix 1: Non-adiabatic part of pressure perturbation in scalar field dark energy

The non-adiabatic part of pressure perturbation in scalar field dark energy is caused by intrinsic entropy perturbation $\varGamma$ and is equal:

$$\begin{aligned} \delta p_{de}^{n-ad}\equiv p_{de}\varGamma =\delta p_{de}-w\bar{\varepsilon }_{de}\delta _{de}, \end{aligned}$$

(38)

where $w\equiv p_{de}/\varepsilon _{de}$ and the last term is adiabatic part of pressure perturbation. The entropy perturbation can be expressed via density perturbation $\hat{\delta }_{de}$ and effective speed of sound $c_s^2$ in the rest frame as follows

$$\begin{aligned} p_{de}\varGamma =(c_s^2-w)\bar{\varepsilon }_{de}\hat{\delta }_{de}. \end{aligned}$$

(39)

In the rest frame ($\hat{t},\,\hat{r},\,\hat{\vartheta },\,\hat{\varphi }$) of dark energy its 4-velocity and density are as follows

$$\begin{aligned} \hat{u}_i=\{e^{\nu /2},\,0,\,0,\,0\}, \quad \hat{\varepsilon }_{de}(\hat{t},\,\hat{r})=\bar{\varepsilon }_{de}(\hat{t})\left[ 1+\hat{\delta }_{de}(\hat{t},\,\hat{r})\right] , \end{aligned}$$

(40)

where $\nu$ is the same function as in (5) but in the “hat” coordinates. In the conformal Newtonian (CN) frame ($t,\,r,\,\vartheta ,\,\varphi$) they are

$$\begin{aligned} u_i=\{u_0,\,u_1,\,0,\,0\}, \quad \varepsilon _{de}(t,r)=\bar{\varepsilon }_{de}(t)[1+\delta _{de}(t,r)] \end{aligned}$$

(41)

where $u_0$ and $u_1$ can be expressed as in (6). In the case of small perturbations the coordinates of both frames are connected by simple transformations:

$$\begin{aligned} \hat{t}=t+\xi ^0(t,r), \quad \hat{r}=r+\xi ^1(t,r). \end{aligned}$$

(42)

The density is scalar variable and transforms as $\bar{\varepsilon }_{de}(\hat{t})=\bar{\varepsilon }_{de}(t)+\dot{\bar{\varepsilon }}_{de}\xi ^0(t,r)$. The density perturbations $\delta _{de}$ is scalar variable too, but since $\delta _{de}\ll 1$ the first and next order expansion terms are second and next order of infinitesimality, so, $\hat{\delta }_{de}(\hat{t},\hat{r})\approx \hat{\delta }_{de}(t,r)$. A dot denotes the partial derivative with respect to time. Taking into account the conservation law for background density

$$\begin{aligned} \dot{\bar{\varepsilon }}_{de}=-3\frac{\dot{a}}{a}(1+w)\bar{\varepsilon }_{de} \end{aligned}$$

(43)

we obtain the relation

$$\begin{aligned} \hat{\delta }_{de}(t,r)=\delta _{de}(t,r)+3\frac{\dot{a}}{a}(1+w)\xi ^0(t,r). \end{aligned}$$

(44)

The unknown function $\xi ^0(t,r)$ can be found from velocity transformation

$$\begin{aligned} u_1=\frac{\partial {\hat{t}}}{\partial {r}}\hat{u}_0+\frac{\partial {\hat{r}}}{\partial {r}}\hat{u}_1. \end{aligned}$$

(45)

Taking into account (40), (41) and (6), we obtain

$$\begin{aligned} \xi ^0(t,r)=-a\int {e^{(\mu -\nu )/2}\frac{v_{de}}{\sqrt{1-v^2_{de}}}\mathrm{d}r}. \end{aligned}$$

(46)

Here we suppose also $\nu (\hat{t},\hat{r})=\nu (t,r)$, $\mu (\hat{t},\hat{r})=\mu (t,r)$ since they are small. For inhomogeneities of galaxies and clusters of galaxies scales $\nu \ll 1$, $\mu \ll 1$ and $v_{de}\ll 1$, so, the squared term $v^2_{de}$ in the denominator and exponent can be omitted, so

$$\begin{aligned} \hat{\delta }_{de}(t,r)=\delta _{de}(t,r)-3\dot{a}(1+w)\int {v_{de}\mathrm{d}r}. \end{aligned}$$

(47)

Gathering all together (38), (39) and (47) we find

$$\begin{aligned} \delta p_{de}=\bar{\varepsilon }_{de}\left[ c_s^2\delta _{de}-3\dot{a}(1+w)(c_s^2-w)\int {v_{de}\mathrm{d}r}\right] . \end{aligned}$$

(48)

Other approaches and details of deducing of the non-adiabatic part of pressure perturbation of scalar field can be found in [59–62]. The contribution of non-adiabatic part of pressure perturbation last term in (48), fifth term in (18) and fourth term in (31) is important at superhorizon linear stage and practically disappears when perturbation enters into horizon. That is why we do not generalized this term for the non-linear stage.

At last we would like to note here, that some researchers use the term “effective sound speed of dark energy” as speed of propagation of perturbation in any frame (see, for example, [63]). It is related with our $c^2_s$ by simple equation $c^2_{eff}=c_s^2-3\dot{a}(1+w)(c_s^2-w)\tilde{v}_{de}/\tilde{\delta }_{de}.$ Therefore, $c^2_{eff}$ is variable for constant $c^2_s$.

Appendix 2: Linearized system of differential equations for evolution of perturbations in three-component medium

The system of ordinary differential equations which describes the evolution of Fourier amplitudes of cosmological linear perturbations of metric, densities and velocities of three-component zero-shear medium in the conformal-Newtonian gauge is as follows

$$\begin{aligned}&\dot{\tilde{\delta }}_\mathrm{r}-2\dot{\tilde{\nu }}-\frac{4k^2}{3a^2H}\tilde{v}_\mathrm{r}=0,\quad \dot{\tilde{v}}_\mathrm{r}+\frac{\tilde{\nu }}{2a^2H}+\frac{\tilde{\delta }_\mathrm{r}}{4a^2H}=0, \end{aligned}$$

(49)

$$\begin{aligned}&\dot{\tilde{\delta }}_{m}-\frac{3}{2}\dot{\tilde{\nu }}-\frac{k^2}{a^2H}\tilde{v}_{m}=0,\quad \dot{\tilde{v}}_{m}+\frac{\tilde{v}_{m}}{a}+\frac{\tilde{\nu }}{2a^2H}=0, \end{aligned}$$

(50)

$$\begin{aligned}&\dot{\tilde{\delta }}_{de}+\frac{3}{a}(c_s^2-w)\tilde{\delta }_{de}-(1+w)\left[ \frac{k^2}{a^2H}\tilde{v}_{de} +9H(c_s^2-w)\tilde{v}_{de}+\frac{3}{2}\dot{\tilde{\nu }}\right] =0, \end{aligned}$$

(51)

$$\begin{aligned}&\dot{\tilde{v}}_{de}+(1-3c_s^2)\frac{\tilde{v}_{de}}{a}+\frac{c_s^2\tilde{\delta }_{de}}{a^2H(1+w)}+\frac{\tilde{\nu }}{2a^2H}=0, \end{aligned}$$

(52)

$$\begin{aligned}&\dot{\tilde{\nu }}+\left( 1+\frac{k^2}{3a^2H^2}\right) \frac{\tilde{\nu }}{a}= -\frac{H^2_0}{H^2}\left( \varOmega _ma^{-3}\tilde{\delta }_m+ \varOmega _\mathrm{r}a^{-4}\tilde{\delta }_\mathrm{r}+\varOmega _{de}a^{-3(1+w)}\tilde{\delta }_{de}\right) .\qquad \quad \end{aligned}$$

(53)

The system has well known analytical solutions for two special cases—radiation-dominated epoch ($\varOmega _\mathrm{r}=1$, $\varOmega _m=\varOmega _{de}=0$) and matter-dominated one ($\varOmega _{m}=1$, $\varOmega _\mathrm{r}=\varOmega _{de}=0$). For two or three component case it can be solved numerically for given initial conditions, for which we have designed the FORTRAN code dedmhalo-l.f.

Fig. 6

Evolution of density perturbations of matter and dark energy (transformed to synchronous gauge) obtained by integration of system of Eqs. (49)–(53) by our code dedmhalo.f (solid black lines), and the same obtained by integration of Boltzman-Eistein code CAMB in synchronous gauge (dashed blue lines)

The results of numerical integration of this system of equations with initial conditions (25)–(27) are presented in the Figs. 1 and 4 by dotted lines. Here in Fig. 6 we present the evolution of density perturbations of dark matter and dark energy obtained by integration of system of Eqs. (49)–(53) by our code dedmhalo-l.f and transformed to synchronous gauge according to known relation $D_N=\tilde{\delta }_N-3(1+w_N)v_N$ [42]. The evolution of $D_m$ and $D_{de}$ obtained by integration of system of Boltzman-Eistein equations in synchronous gauge by code CAMB [53] is also presented there. This illustrates well agreement of results obtained in the different approaches and by different codes.