Radiation-dominated era and the power of general relativity

It is well known that Einstein's General Relativity Theory (GRT) can be adopted to describe various astrophysical observations and, at scales of the Solar System, it results consistent with many, very accurately precise, astrophysical measurements, such as the gravitational bending of light, the perihelion precession of Mercury and the Shapiro time delay [1, 2]. On the other hand, at larger scales, several shortcomings are present, like the famous dark-energy [3] and dark-matter [4] problems.

An alternative approach consists in assuming that gravitational interaction could act in different ways at large scales [5]. This different framework does not require to find out candidates for dark energy and dark matter at fundamental level (not detected up to now), but takes into account only the observed ingredients (i.e., curvature, radiation and baryon matter), changing the left-hand side of the field equations [6, 7]. In this way, a room for alternative theories can be introduced and the most popular dark-energy and dark-matter models can be, in principle, achieved by considering Extended Theories of Gravity (ETG), i.e., f(R) theories of gravity, where R is the Ricci curvature scalar, see [5–11] and references within, and scalar tensor teories [12–14], which are generalizations of the Jordan-Fierz-Brans-Dicke theory [15–17].

An ultimate endorsement for the approach of ETG should be the realization of a consistent gravitational wave astronomy [13]. In fact, in the case of ETG, gravitational waves generate different oscillations of test masses with respect to gravitational waves in standard GRT. Thus, an accurate analysis of such a motion can be used in order to discriminate among various theories, see [12] for details.

Another key point is that Solar System tests imply that modifications of GRT in the sense of ETG have to be very weak [5, 12]. In other words, such theories have to be viable. In the framework of viable ETG, the theory arising from the action (in this paper we work with 8πG = 1, c = 1 and $\hslash =1$)

$$\begin{equation} S=\frac{1}{2}\int {\rm d}^{4}x\left(\sqrt{-g}f_{0}R^{1+\varepsilon}+S_{m}\right), \end{equation} \tag{ 1 }$$

where f₀ > 0 has the dimensions of a mass squared and ε is a small real number, has been discussed in various papers in the literature [18–28] and [29]. Equation (1) is a particular choice in f(R) theories of gravity [6–10] with respect to the well-known canonical one of general relativity (the Einstein-Hilbert action [30]) which is

$$\begin{equation} S=\frac{1}{2}\int {\rm d}^{4}x\left(\sqrt{-g}R+S_{m}\right). \end{equation}\noindent \tag{ 2 }$$

Various observational constraints set the limits

$$\begin{equation} 0\leqslant \varepsilon \leqslant 7.2\times 10^{-19} \end{equation} \tag{ 3 }$$

on the parameter ε [18, 19], while the recent work [26] obtained a lower limit,

$$\begin{equation} 0\leqslant \varepsilon \leqslant 5\times 10^{-30}. \end{equation} \tag{ 4 }$$

Gravitational waves in this particular theory have been discussed in [27]. In [28], a spherically symmetric and stationary universe has been analyzed in the tapestry of this theory.

In order to discuss this particular theory in the framework of the radiation-dominated era, the well-known Friedman-Robertson-Walker cosmological line element has to be used [1, 27], and, for the sake of simplicity, we will consider the flat case, because the WMAP data are in agreement with it [31],

$$\begin{equation} {\rm d}s^{2}=-{\rm d}t^{2}+a^{2}({\rm d}z^{2}+{\rm d}x^{2}+{\rm d}y^{2}). \end{equation} \tag{ 5 }$$

We also recall that in the radiation-dominated era the equation of state is [1]

$$\begin{equation} p=\frac{1}{3}\rho \end{equation} \tag{ 6 }$$

and the the energy density is given by [1]

$$\begin{equation} \rho=\frac{f\pi^{2}}{120}k^{4}T^{4}, \end{equation} \tag{ 7 }$$

where k is the Boltzmann constant and f is a parameter depending from the particular radiation, for example f = 8 for electromagnetic radiation, f = 7 for neutrinos, etc., see [1] for details.

By varying the action (1) with respect to g_μν (see [27] for a detailed computation) the field equations are obtained,

$$\begin{eqnarray} G_{\mu\nu}&=&\frac{1}{(1+\varepsilon)f_{0}R^{\varepsilon}}\left\{-\frac{1}{2}g_{\mu\nu}\varepsilon f_{0}R^{1+\varepsilon}+[(1+\varepsilon)f_{0}R^{\varepsilon}]_{;\mu;\nu}\right. \nonumber \\ &&\left. -g_{\mu\nu}\square[(1+\varepsilon)f_{0}R^{\varepsilon}]\right\}+T_{\mu\nu}, \end{eqnarray} \tag{ 8 }$$

where

$$\begin{equation} T_{\mu\nu}=\left|\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \rho & 0 & 0 & 0\\ 0 & p & 0 & 0\\ 0 & 0 & p & 0\\ 0 & 0 & 0 & p\end{array}\right| \end{equation} \tag{ 9 }$$

is the well-known stress-energy tensor of the matter [1, 30]. Taking the trace of the field equations (8) one gets

$$\begin{equation} 3\square(1+\varepsilon)f_{0}R^{\varepsilon}=(1-\varepsilon)f_{0}R^{1+\varepsilon}+T, \end{equation} \tag{ 10 }$$

where T = ρ − 3p is the trace of the stress-energy tensor (9) [30].

Following [1], if one computes the components of eqs. (8) and (10) by using the line element (5) three independent Friedman equations are obtained:

$$\begin{eqnarray} &&3\dot{R}^{2}(1-\varepsilon^{2})\varepsilon f_{0}R^{\varepsilon-2}+3\ddot{R}(1+\varepsilon)\varepsilon f_{0}R^{\varepsilon-1} \nonumber \\ && +9\displaystyle\frac{\dot{a}}{a}\dot{R}(1+\varepsilon)\varepsilon f_{0}R^{\varepsilon-1}-2f_{0}R^{1+\varepsilon}=0, \nonumber \\ &&6\left(\displaystyle\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\right)(1+\varepsilon)f_{0}R^{\varepsilon}+3f_{0}R^{1+\varepsilon}\nonumber \\ && -12\displaystyle\frac{\dot{a}}{a}\dot{R}(1+\varepsilon)\varepsilon f_{0}R^{\varepsilon-1}+6\dot{R}^{2}(1-\varepsilon^{2})\varepsilon f_{0}R^{\varepsilon-2}\nonumber \\ && -6\ddot{R}(1+\varepsilon)\varepsilon f_{0}R^{\varepsilon-1}=\rho, \nonumber \\ &&6\displaystyle\frac{\ddot{a}}{a}(1+\varepsilon)f_{0}R^{\varepsilon}+f_{0}R^{1+\varepsilon}-6\frac{\dot{a}}{a}\dot{R}(1+\varepsilon)\varepsilon f_{0}R^{\varepsilon-1}=\rho.\nonumber \\ \end{eqnarray} \tag{ 11 }$$

We recall that the Ricci scalar is given by [32]

$$\begin{equation} R=-6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}\right]. \end{equation} \tag{ 12 }$$

One can also use the Bianchi identities [1] to get another independent equation,

$$\begin{equation} a\dot{\rho}=-4\rho\dot{a}. \end{equation} \tag{ 13 }$$

In standard general relativity, during the radiation-dominated era, the scale factor is [1]

$$\begin{equation} a\sim t^{\frac{1}{2}}. \end{equation} \tag{ 14 }$$

Hence, in the theory which arises from the action (1) one assumes

$$\begin{equation} a\sim t^{(\frac{1}{2}+\delta)}. \end{equation} \tag{ 15 }$$

By using the second and the third of eqs. (11) and eq. (13) one gets

$$\begin{equation} \varepsilon=2\delta. \end{equation} \tag{ 16 }$$

By deriving eq. (12) and by using eq. (16) we write

$$\begin{equation} \dot{R}=\frac{6\varepsilon(1+\varepsilon)}{t^{3}}. \end{equation} \tag{ 17 }$$

Considering the third of eqs. (11) together with eq. (7) one obtains

$$\begin{eqnarray} &&T=\sqrt[4]{\frac{60}{4\pi^{3}f}}\nonumber \\ &&\times\sqrt[4]{6^{(1{+}\varepsilon)}\left(\frac{1{+}\varepsilon}{2}\right){}^{(1+\varepsilon)}\frac{1}{4}[8(1{+}\varepsilon)-5(1+\varepsilon)^{2}-2](-\varepsilon)^{\varepsilon}}\nonumber \\ &&\times \frac{1}{kt^{\frac{(1+\varepsilon)}{2}}}. \end{eqnarray} \tag{ 18 }$$

Putting

$$\begin{eqnarray} F(\varepsilon)&\equiv&6^{(1+\varepsilon)}\left(\displaystyle\frac{1+\varepsilon}{2}\right){}^{(1+\varepsilon)}\nonumber\\ &&\times \displaystyle\frac{1}{4}\left[8(1+\varepsilon)-5(1+\varepsilon)^{2}-2\right](-\varepsilon)^{\varepsilon}, \end{eqnarray} \tag{ 19 }$$

we need

$$\begin{equation} F(\varepsilon)\geqslant 0 \end{equation} \tag{ 20 }$$

in order for T to be a real value. The constraint (20) is satisfied for

$$\begin{equation} -0.69\leqslant \varepsilon\leqslant 0, \end{equation} \tag{ 21 }$$

see fig. 1.

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Considering the bound (21) togheter with the bounds (3) and (4) one gets immediately ε = 0, i.e., general relativity is recovered and the theory which arises from the action (1) is ultimately ruled out.

In summary, in this work we realized an analysis in the framework of the radiation-dominated era in order to put bounds on the weak modification of general relativity which arises from the action (1). The new bounds together with previous ones in the literature rule out this theory in a definitive way.

I thank a reviewer for useful comments.