Cosmological imprints of the stealth scalar field

Abstract

We study some cosmological models endowed with a peculiar scalar field known as stealth field. The models are constructed by considering a scalar field $\phi$ obeying a non-minimal coupling to the gravitational field where the matter content is provided by both dust and a perfect fluid. Moreover, we analyze the stealth field implications in the cosmology by assuming a power-law behavior for the scale factor. Surprisingly, we find that stealth configurations that emerge, which in principle have no back-reaction to the background space-time, do contribute to the cosmological dynamics. We provide analytical expressions of these contributions to the energy density coming from the matter sources considered as well as for the pressure in the power-law scenario. This is a first signal that the stealth fields contributions are quantifiable. In the de Sitter cosmology, we found stealth does not have contributions either to cosmological constant or deceleration parameter. Additionally, we discuss the behaviors of the self-interaction potential for some particular values of the parameters involved.

Appendix: Stealth field equation

We study the consequences of the stealth in cosmology, Einstein’s field equations for a perfect fluid source are furnished by

$$\begin{aligned} G_{\mu \nu }-\kappa T^{m}_{\mu \nu } =0=\kappa T^{s}_{\mu \nu } \end{aligned}$$

(63)

here $G_{\mu \nu }$ is are the Einstein tensor, $T^{m}_{\mu \nu }$ is the energy momentum tensor of a perfect fluid

$$\begin{aligned} T_{\mu \nu }^{m}=(\rho +p)u_{\mu }u_{\nu }+pg_{\mu \nu }, \end{aligned}$$

(64)

where $p=p(t)$ is the pressure, $\rho =\rho (t)$ the energy density, $u_{\mu }$ is the fluid velocity four vector with $u_{\mu }u^{\mu }=1$. $T^{s}_{\mu \nu }$ is the stress energy tensor of the stealth field (3).

The Friedmann-Lemaitre-Robertson-Walker metric for a homogeneous isotropic universe is provided by

$$ ds^{2}=-dt^{2}+\frac{a^{2}}{1-kr^{2}}dr^{2}+a^{2} \bigl(r^{2}d\theta ^{2}+r ^{2} \sin(\theta )^{2} d\phi ^{2}\bigr) $$

(65)

where $k=\pm 1,0$, indicating the spatial curvature constants. In the case $k=0$ the field equations for the background are given by

$$\begin{aligned}& \frac{\dot{a}^{2}}{a^{2}}=\frac{\kappa }{3}\rho (t), \end{aligned}$$

(66)

$$\begin{aligned}& 2\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}=-\kappa p(t), \end{aligned}$$

(67)

and the continuity equation that satisfies the fluid is

$$\begin{aligned} \dot{\rho } +3H(\rho +p)=0. \end{aligned}$$

(68)

Now, we assume that the universe is spatially flat and the dust case, i.e. when the pressure is zero. We go back to Eqs. (66) and (68), to obtain that density and scale factor are given explicitly in the form

$$\begin{aligned}& \rho (t) =\rho _{0}/a^{3}, \end{aligned}$$

(69)

$$\begin{aligned}& a(t) =(a_{1}t+a_{0})^{2/3} \quad \mbox{where } a_{1}^{2}=\kappa \rho _{0}/3, \end{aligned}$$

(70)

in thats case the equations for the stealth are given by Eq. (13).

Expanding Eq. (13), explicitly we get

$$\begin{aligned} \kappa \rho =\kappa \frac{\rho _{0}}{a^{3}}=2 \frac{\ddot{\phi }}{ \phi } + \frac{(2\zeta -1)}{\zeta } \biggl(\frac{\dot{\phi }}{\phi } \biggr) ^{2}-2 \frac{\dot{\phi }}{\phi }\frac{\dot{a}}{a}, \end{aligned}$$

(71)

now, in order to get an analytical solution of the stealth field, we propose $\phi =\phi (a(t))=\phi (a)$ in Eq. (13). Now, the field equation is given by

$$\begin{aligned}& \frac{1}{\phi } \biggl(\frac{d^{2}\phi }{da^{2}} \dot{a}^{2}+ \frac{d \phi }{da}\ddot{a} \biggr) +\frac{(2\zeta -1)}{2\zeta }\frac{1}{\phi ^{2}} \dot{a}^{2} \biggl(\frac{d\phi }{da} \biggr)^{2} \\ & \quad-\frac{1}{\phi a}\frac{d\phi }{da}\dot{a}^{2}= \frac{\kappa \rho _{0}}{2 a^{3}} \end{aligned}$$

(72)

From (70) is easy to shown

$$\begin{aligned} \dot{a}^{2}=\frac{a_{1}^{2}}{a}\quad \mbox{and} \quad \ddot{a}=- \frac{ \dot{a}^{2}}{2 a}, \end{aligned}$$

(73)

then Eq. (72) finally have the form

$$\begin{aligned} \biggl[\frac{1}{\phi }\frac{d^{2}\phi }{da^{2}} -\frac{3}{2 a} \frac{1}{ \phi }\frac{d\phi }{da} +\frac{(2\zeta -1)}{2\zeta } \frac{1}{\phi ^{2}} \biggl(\frac{d\phi }{da} \biggr)^{2} \biggr] \dot{a}^{2} =\frac{\kappa \rho _{0}}{2 a^{3}} , \end{aligned}$$

(74)

factorizing $\frac{1}{\phi }\frac{d\phi }{da}=\frac{d \ln \phi }{da}$ and rearranging terms and by substituting the expressions of $\dot{a}^{2}$ and $a_{1}^{2}$ one arrive at Eq. (14).