Cosmography of chameleon gravity

We begin with the action of chameleon gravity given by,

where the matter fields $\psi^{(i)}$ are coupled to scalar field $\phi$ by the definition $g_{\mu\nu}^{(i)}\equiv e^{2\beta_{i}\phi/M_{Pl}}g_{\mu\nu}$. The $\beta_{i}$ are dimensionless coupling constants, one for each type of matter. In the following, we assume a single matter energy density component $\rho_{m}$ with coupling $\beta$ Khourym (Khoury & Weltman 2004). Assuming that the universe is filled with cold dark matter, i.e. $\gamma=0$, the variation of action (I) with respect to the metric tensor components in a spatially flat FRW cosmology yields the field equations,

Where, the chameleon effective potential is defined by,

providing the wave equation of chameleon scalar field $\phi$.

One can also easily find the mass associated with field $\phi$:

For $\gamma=0$, if $\beta$ in the second term of $V_{eff}(\phi)$ is positive, the effective potential monotonically decreases to a minimum at a finite field value $\phi=\phi_{min}$, where $\frac{d}{d\phi}V_{eff}(\phi)|_{\phi=\phi_{min}}=0$, and $m_{ch}=m_{ch_{min}}$. From equation (23)

Note that $m_{ch_{min}}$ in the above equations is the inverse of the characteristic range of chameleon force in a given medium. By differentiating both sides of the equation (21) with respect to time

The equation (I) can be rewritten as

From equations (20) and 21 we can also obtain

Using equations (28) and (29) and by considering exponential potential $V=V_{0}e^{\frac{\alpha}{M_{pl}}\phi}$ where dimensionless constant $\alpha$ characterizing the slope of potential, one can obtain

Also, the mass of chameleon field is expressed by

So, using equations (23) and (30) the second time derivative of scalar $\phi$ would be

By combining the equations (I)and (I), the second derivative of Hubble parameter gives

Here, we rewrite the cosmographic parameters as follows

Here for more simplicity, we also define the following dimensionless parameters

Where equation (20) puts a constraint on the variables as

Equation (I) can be rewritten in terms of the cosmographic parameters $(q,Q)$ and the model parameters $(x,\alpha,\beta)$ as

This is a cubic polynomial in terms of $x$, as $ax^{3}+bx^{2}+cx+d=0$, where
$\left\{\begin{array}[]{ll}a=(\alpha-\beta)\\ b=3\\ c=\Big{(}(4\alpha-2\beta)(1+q)+2\alpha\Big{)}\\ d=2\Big{(}1-Q\Big{)}\\ \end{array}\right.$

Solving the equation (I), therefore, gives $x$ in terms of the cosmographic $(q,Q)$ and constant $(\alpha,\beta)$ parameters.

For a simple case where $Q=1$, one can find the following three solutions
$\left\{\begin{array}[]{ll}x=0\\ x=\frac{-3+(9+16\alpha^{2}q-24\alpha\beta q-8\alpha^{2}+8\beta^{2}+8\beta^{2}q)^{\frac{1}{2}}}{2(\alpha-\beta)}\\ x=-\frac{3+(9+16\alpha^{2}q-24\alpha\beta q-8\alpha^{2}+8\beta^{2}+8\beta^{2}q)^{\frac{1}{2}}}{2(\alpha-\beta)}\\ \end{array}\right.$
It is a striking and slightly puzzling fact that almost all current cosmological observations can be summarized by a simple statement: The jerk of the Universe is equal to one” $(Q=1).$ Maciej ,Alam ,Kun . Also VisserVisse , have investigated in some details the jerk condition $(Q=1)$ . From equations $(\ref{vv})$ and $(\ref{vv1})$, the variables $(y,z)$ can also be specified in terms of $x,q$;

If the values of $(\alpha,\beta)$ are known, consequently, all model variables would be reconstruct in terms of cosmographic parameters $(q,Q)$. In addition, the equation (31) gives the chameleon mass $m_{ch}$ in terms of $(H,q,Q,\alpha,\beta)$

Equations (I) and (43) imply the possibility of determining the chameleon mass in terms of coupling parameters $(\alpha,\beta)$ and first three derivatives of scale factor $(H,q,Q)$. Note that l.h.s of equation (44) is positive, hence $x^{2}<2(1+q)$. Exerting this condition on equation (43) gives an upper bound for chameleon mass as $m_{ch}^{2}<\alpha^{2}(2-q)H^{2}$. If the parameters $(\alpha,\beta)$ are known, chameleon mechanism can be interpenetrated by only the first three derivatives of scale factor $(H,q,Q)$. As it was pointed out by the original chameleon article that Khourym (Khoury & Weltman 2004), in harmony with string theory, $\beta$ must be of the order unity; so that we got $\beta=1$. Furthermore, we get $\alpha$ of the order of unity with negative sign (because the potential $V(\phi)$) is assumed to be of the runaway form for monotonically decreasing function of $\phi$), then the solution of equation (I) gives

Where,
$A=6\Big{(}1830+5292q+4941q^{2}+1296q^{3}+486Q+972qQ+324Q^{2}\Big{)}^{\frac{1}{2}}-81-162q-108Q$
Finally, using equations (44), (41) and (42), the chameleon gravity is expressed on the way of time derivatives of scale factor measuring the $(H,q,Q)$ parameters.

However, if nothing can be proposed about $(\alpha,\beta)$, the chameleon parameters will not be determined by considering the first three derivatives of scale factor, as its forth and fifth derivatives must also be measured. By differentiating both sides of the equation (21) with respect to time, the third time derivative of Hubble parameter would be obtained as

Differentiating equation (23) also yields

Their composition is expressed by

Corresponding to what was previously achieved for the equation (47), by differentiating both sides of equations (I) and (46) and using equations (34) to (37), we take

As it goes to describe the behaviour of a dynamical system, we implement this approach to trace the dynamics of the universe in the presence of chameleon field in deferent epochs of its evolution.

We are interested to quantify the chameleon parameters such as chameleon mass $m_{ch}$ at the critical points of the system, points that represent all important epochs in the evolution of the universe. The dynamics of the universe in chameleon gravity can be simplified by introducing the following dimensionless variables,

Then using equations (20)-(23), the evolution equations of these variables become,

Where prime indicates from now on differentiation with respect to $N=lna$. Note that $\zeta_{1}=\frac{x}{\sqrt{6}},\zeta_{2}=\frac{y}{3},\zeta_{3}=\frac{z}{3}$. The Friedmann equation (20) also becomes

Using the constraint (53), the equations (50)-(52) are converted to

In terms of the new dynamical variable, we also have

One can also rewrite the equations (I) and (43) as

It is more convenient to investigate the properties of the dynamical system, namely Eqs.(II) and (II) rather than Eqs.(50)-(52). We obtain the fixed points (critical points) and study the stability of these steady states that are always exact constant solutions in the context of autonomous dynamical systems. Those are often the extreme points of the orbits and therefore describe the asymptotic behavior. In the following we find fixed points by solving $\frac{d\zeta_{1}}{dN}=0$ and $\frac{d\zeta_{2}}{dN}=0$ simultaneously. Two eigenvalues $\lambda_{i}(i=1,2)$ are obtained by substituting linear perturbations $\zeta_{1}^{\prime}\rightarrow\zeta_{1}^{\prime}+\delta\zeta_{1}^{\prime}$, $\zeta_{2}^{\prime}\rightarrow\zeta_{2}^{\prime}+\delta\zeta_{2}^{\prime}$ about the critical points into the two independent equations (II) and (II), to the first order of perturbations. Stability requires that the real part of all eigenvalues be negative. There are also five fixed points which some of them explicitly depend on $\beta$ and $\alpha$, as shown in Table 1.

Critical points, $A_{\pm}$, corresponding to two kinetic-dominated solutions. These are equivalent to the stiff-fluid-dominated evolution with $a=t^{\frac{1}{3}}$, irrespective of the nature of the potential. The kinetic-dominated solution for $A_{+}$ has two eigenvalues, $\lambda_{+}=3+\beta\sqrt{6},\lambda_{-}=6+\sqrt{6}\alpha$,

and is stable for $\beta<\frac{-\sqrt{6}}{2}$,$\alpha<-\sqrt{6}$. The solution $A_{-}$ also has two eigenvalues $\lambda_{+}=3-\beta\sqrt{6}$, and $\lambda_{-}=6-\sqrt{6}\alpha$, stabilized for $\beta>\frac{\sqrt{6}}{2}$ and $\alpha>\sqrt{6}$. The phase space of the system, in which these critical points are stable, has been shown in Figs. (1) and (2). Also, from equations 56 and (I) to (48), the values of the cosmographic parameters at these points are as follows:
$\left\{\begin{array}[]{ll}q_{c}=2\\ Q_{c}=10\\ X_{c}=-80\\ Y_{c}=880\\ \end{array}\right.$
where both $V(\phi)$ and effective $V_{eff}$ potentials are zero. So, the chameleon mass $m_{ch}$ at those points is zero, as confirmed by equation (58).

Critical point, $B$, corresponding to a potential-kinetic-scaling solution. This solution exists for all kinds of potentials, and has two eigenvalues depending on the slope of the potential and coupling constant $\beta$: $\lambda_{+}=-3+\frac{\alpha^{2}}{2}\,,\qquad\lambda_{-}=-\beta\alpha+\alpha^{2}-3.$
As, the solution is stable for
$\left\{\begin{array}[]{ll}\beta<\frac{-3+\alpha^{2}}{\alpha},-\sqrt{6}<\alpha<0\\ \beta>\frac{-3+\alpha^{2}}{\alpha},0<\alpha<\sqrt{6}\hbox{$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $}\end{array}\right.$
which means that the potential-kinetic-dominated solution is stable for a sufficiently flat potential ($\alpha^{2}<6$). The phase space of the system has been shown in Fig.(3), in which the critical point is stable. The set of the values of the cosmographic parameters at this point is as follows $\left\{\begin{array}[]{ll}q_{c}=-1+\frac{1}{2}\alpha^{2}\\ Q_{c}=\frac{1}{2}\alpha^{4}-\frac{3}{2}\alpha^{2}+1\\ X_{c}=-\frac{3}{4}\alpha^{6}+\frac{11}{4}\alpha^{4}-3\alpha^{2}+1\\ Y_{c}=\frac{3}{2}\alpha^{8}-\frac{25}{4}\alpha^{6}+\frac{35}{4}\alpha^{4}-5\alpha^{2}+1\\ \end{array}\right.$
The chameleon mass $m_{ch}$ would also be

indicating that the stability condition $(\alpha^{2}<6)$ leads to an upper bound for chameleon mass as $m_{ch}^{2}<\frac{9H^{2}}{2}$. For $\alpha=\pm\sqrt{2}$ at this point, corresponding to the $deceleration-acceleration$ phase of the universe, $m_{ch}=2H$ and all cosmographic parameters are zero. For $\alpha\rightarrow 0$ at this point, in addition, the potential tends to $V(\phi)\rightarrow M^{4}$ and the cosmographic parameters would be $\{q_{c}=-1,Q_{c}=X_{c}=Y_{c}=1\}$ , representing the cosmographic parameters of $\Lambda CDM$ model.
Critical point $C$, corresponds to the fluid-kinetic-scaling solution. This solution depends on the coupling constant $\beta$ and exists for all potentials. It has two eigenvalues depending on both $\alpha$ and $\beta$: $\lambda_{+}=-3/2+\beta^{2}$ and $\lambda_{-}=3+2\beta^{2}-2\beta\alpha$. The solution is stable for
$\left\{\begin{array}[]{ll}\alpha<\frac{3+2\beta^{2}}{2\beta},-\sqrt{\frac{3}{2}}<\beta<0\\ \alpha>\frac{3+2\beta^{2}}{2\beta},0<\beta<\sqrt{\frac{3}{2}}\hbox{$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $}\end{array}\right.$
in appropriate phase space shown in Fig.(4). The set of the values of cosmographic parameters at this point is:
$\left\{\begin{array}[]{ll}q_{c}=\frac{1}{2}+\beta^{2}\\ Q_{c}=1+3\beta^{2}+2\beta^{4}\\ X_{c}=-\frac{7}{2}-\frac{27}{2}\beta^{2}-16\beta^{4}-6\beta^{6}\\ Y_{c}=\frac{35}{2}+\frac{163}{2}\beta^{2}+134\beta^{4}+94\beta^{6}+24\beta^{8}\\ \end{array}\right.$