Inflation with scalar-tensor theory of gravity

The standard (FLRW) model of cosmology based on the basic assumption of homogeneity and isotropy, known as the ‘cosmological principle’, has successfully been able to explain several very important issues in connection with the evolution of the universe. First of all, it predicts the observed expansion of the universe being supported by the Hubble’s law. It also postulates the existence of cosmic microwave background radiation (CMBR), formed since recombination when the electrons combined to form atoms, allowing photons to free stream, with extreme precession, being verified by Penzias and Wilson for the first time. It also predicts with absolute precession the abundance of the light atomic nuclei ($\mathrm{{{}^{4}He\over H}\sim 0.25,{{}^{2}D\over H}\sim 10^{-3},{{}^{3}He\over H}\sim 10^{-4},{{}^{7}Li\over H}\sim 10^{-9}}$, by mass and not by number) observed in the present universe. Finally, assuming the presence of the seeds of perturbation in the early universe, it can explain the observed present structure of the universe. Despite such tremendous success, the model inevitably suffers from a plethora of pathologies. The problems at a glance are the following.
1. ‘The singularity problem’: Extrapolating the FLRW solutions back in time one encounters an unavoidable singularity, since all the physical parameters viz. the energy density ($\rho$), the thermodynamic pressure ($p$), the Ricci scalar ($R$), the Kretschmann scalar ($R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$) etc. diverge.
2a. ‘The flatness problem’: The model does not provide any explanation to the observed value of the density parameter $\Omega\approx 1$, which depicts that the universe is spatially flat.
2b. ‘The horizon problem’: It also can not provide any reason to the observed tremendous isotropy of the CMBR being split in $1.4\times 10^{4}$ patches of the sky, that were never causally connected before emission of the CMBR.
2c. ‘The structure formation problem’: It does not also provide any clue to the seeds of perturbation responsible for the structure formation.
3. ‘The dark energy problem’: Finally, the standard FLRW model does not fit the redshift versus luminosity-distance curve plotted in view of the observed SN1a (Supernova type a) data.

In connection with the first problem, viz. the so called ‘Big-Bang singularity’ and also to understand the underlying physics of ‘Black-Hole’ being associated with Schwarzschild singularity, it has been realized long ago that ’General Theory of Relativity’ (GTR) must have to be replaced by a quantum theory of gravity when and where gravity is strong enough. However, GTR is not renormalizable and a renormalized theory requires to include higher-order curvature invariant terms in the gravitational action. Despite serious and intense research over several decades and formulation of new high energy physical theories like superstring and supergravity theories, a viable quantum theory of gravity is still far from being realized. In connection with last problem, a host of research is in progress over last two decades. It has been realized that to fit the observed redshift versus luminosity-distance curve, it is either required to take into account some form of exotic matter in addition to the barotropic fluid (ordinary plus the cold dark matter) which violates the strong energy condition ($\rho+p\geq 0,~{}\rho+3p\geq 0$), and is dubbed as ‘dark energy’ (since it interacts none other than with the gravitational field) or to modify the theory of gravity by including additional curvature scalars in the Einstein-Hilbert action, known as ‘the modified theory of gravity’. It has been observed that both the possibilities lead to present accelerated expansion of the universe. The problem is thus rephrased as: why the universe undergoes an accelerated expansion at present? The pathology 2, in connection with the flatness, horizon and structure formation problems has however been solved under the hypothesis called ‘Inflation’, which is our present concern.

$$ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})\right],$$

(1)

the co-moving distance (the present-day proper distance) traversed by light between cosmic time $t_{1}$ and $t_{2}$ in an expanding universe may be expressed as, $d_{0}(t_{i},t_{f})=a_{0}\int_{t_{i}}^{t_{f}}{dt\over a(t)}$, where $a(t)$ is called the scale factor. The co-moving size of the particle horizon at the last-scattering surface of CMBR ($a_{f}=a_{lss}$) corresponds to $d_{0}\sim 100$ Mpc, or approximately $1^{0}$ (one degree) on the CMB sky today. In the decelerated radiation dominated era of the standard model of cosmology (FLRW model), for which $a\propto\sqrt{t}$ the integrand, $(\dot{a})^{-1}\sim 2\sqrt{t}$ decreases towards the past, and there exists a finite co-moving distance traversed by light since the Big Bang ($a_{i}\rightarrow 0$), called the particle horizon. The hypothesis of inflation [2, 3] postulates a period of accelerated expansion, $\ddot{a}>0$, in the very early universe, prior to the radiation-dominated era, administering certain initial conditions [4, 5, 6, 7, 8, 9, 10]. During a period of inflation e.g. a de-Sitter universe ($a\propto e^{\Lambda t}$) driven by a cosmological constant (say), $(\dot{a})^{-1}\sim(\Lambda e^{\Lambda t})^{-1}$ increases towards the past, and hence the integral diverges as ($a_{i}\rightarrow 0$). This allows an arbitrarily large causal horizon dependent only upon the duration of the accelerated expansion. Assuming that the universe inflates with a finite Hubble rate $H_{i}$, (instead of a constant exponent $\Lambda$) ending with $H_{f}<H_{i}$, we may have, $d_{0}(t_{i},t_{f})>({a_{i}\over a_{f}})H_{i}^{-1}(e^{N}-1)$ where $N=\ln\Big{(}{a_{f}\over a_{i}}\Big{)}$ is measured in terms of the logarithmic expansion (or ‘e-folds’), and describes the duration of inflation. It has been found that a $40-60$ e-folds of inflation can encompass our entire observable universe today, and thus solves the horizon and the flatness problem discussed earlier. In some situations, e-fold may range between $25\leq N\leq 70$, depending on the model under consideration.

A false vacuum state can drive an exponential expansion, corresponding to a de-Sitter space-time with a constant Hubble rate on spatially-flat hypersurfaces. However, a graceful exit from such exponential expansion requires a phase transition to the true vacuum state. A second-order phase transition [11, 12], under the slow roll condition of the scalar field (that can also drive the inflation instead of the cosmological constant), potentially leads to a smooth classical exit from the vacuum-dominated phase. Further, the quantum fluctuations of the scalar field, which essentially are the origin of the structures seen in the universe today, provides a source of almost scale-invariant density fluctuations [13, 14, 15, 16, 17], as detected in the CMBR. Accelerated expansion and primordial perturbations can also be produced in some modified theories of gravity (e.g., [2, 18] and also a host of models presently available in the literature), which introduce additional non-minimally coupled degrees of freedom. Such inflationary models are conveniently studied by transforming variables to the so-called ‘Einstein frame’, in which Einstein’s equations apply with minimally coupled scalar fields [19, 20], which we shall deal with, in the present manuscript.

Non-minimal coupling with the scalar field $\phi$ is unavoidable in a quantum theory, since such coupling is generated by quantum corrections, even if it is primarily absent in the classical action. Particularly, it is required by the renormalization properties of the theory in curved space-time background. Recently, in view of a general conserved current, obtained under suitable manipulation of the field equations [21, 22, 23, 24], a non-minimally coupled scalar-tensor theory of gravity has been studied extensively in connection with the cosmological evolution, starting from the very early stage (Inflationary regime) to the late-stage (presently accelerated matter-dominated era) via a radiation dominated era [25]. It has been found that such a theory admits a viable inflationary regime, since the inflationary parameters viz. the scalar-tensor ratio ($r$), and the spectral index $n_{s}$ lie well within the limits of the constraints imposed by Planck’s data, released in 2014 [26] and in 2016 [27]. Further the model passes through a Friedmann-like radiation era ($a\propto\sqrt{t},q=1$,) and also an early stage of long Friedmann-like decelerating matter dominated era ($a\propto t^{2\over 3},~{}q={1\over 2}$) till $z\approx 0.4$, where $a$, $q$ and $z$ denote the scale factor, the deceleration parameter and the red-shift respectively. The universe was also found to enter a recent accelerated expansion at a red-shift, $z\approx 0.75$, which is very much at par with recent observations. Further, the present numerical values of the cosmological parameters obtained in the process are also quite absorbing, since the age of the universe ($13.86<t_{0}<14.26$) Gyr, the present value of the Hubble parameter ($69.24<H_{0}<69.96$) $\mathrm{Km.s^{-1}Mpc^{-1}}$, so that $0.991<H_{0}t_{0}<1.01$ fit with the observation with appreciable precision. Numerical analysis also reveals that the state finder $\{r,s\}=\{1,0\}$, which establishes the correspondence of the present model with the standard $\Lambda$CDM universe. Last but not the least important outcome is: considering the CMBR temperature at decoupling ($z\sim 1080$) to be $3000$, required for recombination, it’s present value is found to be $2.7255$, which again fits the observation with extremely high precision. Thus, non-minimally coupled scalar-tensor theory of gravity appears to serve as a reasonably fair candidate for describing the evolution history of our observable universe, beyond quantum domain.

In the mean time new Planck’s data is released [28, 29], which imposed even tighter constraints on the inflationary parameters. In this manuscript, we therefore pose if the theory [25] admits these new constraints. However, earlier we considered a particular form of coupling parameter along with the potential in the form $V(\phi)=V_{0}\phi^{4}-B\phi^{2}$, where, $V_{0}$ and $B$ are constants [25]. Here instead, we choose different forms of the coupling parameters and also different potentials to study the inflationary regime. In the following section 2, we describe the model, write down the field equations, find the parameters involved in the theory in view of a general conserved current. We also present the scalar-tensor equivalent form of the action in Einstein’s frame to find the inflationary parameters. In section 3, we choose different forms of the coupling parameters and associated potentials to test the viability of the model in view of the latest released data from Planck [28, 29]. We conclude in section 4.

We start with the non-minimally coupled scalar-tensor theory of gravity, for which the action is expressed in the form,

$$A=\int\left[f(\phi)R-{\omega(\phi)\over\phi}\phi_{,\mu}\phi^{{}^{,}\mu}-V(\phi)-\mathcal{L}_{m}\right]\sqrt{-g}d^{4}x,$$

(2)

where, $\mathcal{L}_{m}$ is the matter Lagrangian density, $f(\phi)$ is the coupling parameter, while, $\omega(\phi)$ is the variable Brans-Dicke parameter. The field equations are,

$$\Big{(}R_{\mu\nu}-{1\over 2}g_{\mu\nu}R\Big{)}f(\phi)+g_{\mu\nu}\Box f(\phi)-f_{;\mu;\nu}-{\omega(\phi)\over\phi}\phi_{,\mu}\phi_{,\nu}+{1\over 2}g_{\mu\nu}\Big{(}\phi_{,\alpha}\phi^{,\alpha}+V(\phi)\Big{)}=T_{\mu\nu},$$

(3)

$$Rf^{\prime}+2{\omega(\phi)\over\phi}\Box\phi+\Big{(}{\omega^{\prime}(\phi)\over\phi}-{\omega(\phi)\over\phi^{2}}\Big{)}\phi_{,\mu}\phi^{,\mu}-V^{\prime}(\phi)=0,$$

(4)

where prime denotes derivative with respect to $\phi$, and $\Box$ denotes D’Alembertian, such that, $\Box f(\phi)=f^{\prime\prime}\phi_{,\mu}\phi^{{}^{,}\mu}-f^{\prime}\Box\phi$. The model involves three parameters viz. the coupling parameter $f(\phi)$, the Brans-Dicke parameter $\omega(\phi)$ and the potential $V(\phi)$. It is customary to choose these parameters by hand in order to study the evolution of the universe. However, we have proposed a unique technique to relate the parameters in such a manner, that choosing one of these may fix the rest [21, 22, 23, 24, 25]. This follows in view of a general conserved current which is admissible by the above pair of field equations, briefly enunciated below.

$$Rf-3\Box f-{\omega(\phi)\over\phi}\phi_{,\mu}\phi^{{}^{,}\mu}-2V=T_{\mu}^{\mu}=T.$$

(5)

Now eliminating the scalar curvature between equations (4) and (5), one obtains,

$$\Big{(}3f^{\prime 2}+{2\omega f\over\phi}\Big{)}^{\prime}\phi_{,\mu}\phi^{{}^{,}\mu}+\Big{(}3f^{\prime 2}+{2\omega f\over\phi}\Big{)}\Box\phi+2f^{\prime}V-fV=f^{\prime}T,$$

(6)

which may then be expressed as,

$$\Big{[}\Big{(}3f^{\prime 2}+{2\omega f\over\phi}\Big{)}^{1\over 2}\phi^{;\mu}\Big{]}_{,\mu}-{f^{3}\over 2\Big{(}3f^{\prime 2}+{2\omega f\over\phi}\Big{)}^{1\over 2}}\Big{(}{V\over f^{2}}\Big{)}^{\prime}={f^{\prime}\over 2\Big{(}3f^{\prime 2}+{2\omega f\over\phi}\Big{)}^{1\over 2}}T,$$

(7)

and finally as,

$$\left(3f^{\prime 2}+{2\omega f\over\phi}\right)^{1/2}\left[\left(3f^{\prime 2}+{2\omega f\over\phi}\right)^{1/2}\phi^{~{};\mu}\right]_{;\mu}-f^{3}\left({V\over f^{2}}\right)^{\prime}={f^{\prime}\over 2}T^{\mu}_{\mu}.$$

(8)

Thus there exists a conserved current $J^{\mu}$, where,

$$J^{\mu}_{;\mu}=\left[\left(3f^{\prime 2}+{2\omega f\over\phi}\right)^{1/2}\phi^{~{};\mu}\right]_{;\mu}=0.$$

(9)

for trace-less matter field ($T^{\mu}_{\mu}=T=0$), provided

$$V(\phi)\propto f(\phi)^{2}.$$

(10)

To study cosmological consequence of such a conserved current, let us turn our attention to the minisuperspace model (1), in which the conserved current (9), reads as

$$\sqrt{\left(3f^{\prime 2}+{2\omega f\over\phi}\right)}a^{3}\dot{\phi}=C_{1},$$

(11)

in traceless vacuum dominated and also in radiation dominated eras. In the above, $C_{1}$ is the integration constant. Note that, fixing the form of the coupling parameter $f(\phi)$, the potential $V(\phi)$ is fixed in view of (10), once and forever. Further, we use a relation [25]

$$3f^{\prime 2}+{2\omega f\over\phi}=\omega_{0}^{2},$$

(12)

where, $\omega_{0}$ is a constant, to fix the Brans-Dicke parameter as well. As a result, we obtain the relation

$$a^{3}\dot{\phi}={C_{1}\over\omega_{0}}=C,$$

(13)

$C$ being yet another constant. In the process, all the coupling parameters $f(\phi)$, $\omega(\phi)$ and the potential $V(\phi)$ may be fixed a-priori. Note that the above choice (12) finally leads to the conserved current associated with the canonical momenta conjugate to the scalar field $\phi$, in the absence of a variable Brans-Dicke parameter, i.e. for $\big{(}{\omega(\phi)\over\phi}={1\over 2}\big{)}$ with minimal coupling $\Big{(}f(\phi)=(16\pi G)^{-1}={M_{p}^{2}\over 2}\Big{)}$. We shall work, in the typical unit, ${M_{p}^{2}\over 2}=c=1$, and consider different forms of $f(\phi)$, that fixes the Brans-Dicke parameter $\omega(\phi)$ as well as the potential $V(\phi)$. In view of these known functional forms of the parameters of the theory, we focus our attention to study inflation, which must have occurred in the very early vacuum dominated universe. We relax the symmetry by adding an useful term in the potential $V(\phi)$, so that one of the terms act just as a constant in the effective potential. This ensures a constant value of the potential as the scalar field dies out, and this constant acts as an effective cosmological constant ($\Lambda_{e}$). In the process, the number of parameters increases to three ($\omega_{0},V_{0},V_{1}$), which is essential to administer good fit with observation.

In the non-minimal theory, the flat section of the potential $V(\phi)$ responsible for slow-roll is usually distorted. However, flat potential is still obtainable if the Einstein’s frame potential $V_{E}$ is asymptotically constant [31, 32]. Note that, the symmetry explored in section 2, makes the potential ($V_{E}$) in the Einstein’s frame to be constant, once and forever. Thus, we need to relax the symmetry as required by the condition (10), by taking into account additional term in the potential, viz. a constant term $V_{0}$, or even a functional form, to ensure that the effective potential in the Einstein’s frame ($V_{E}$) is asymptotically constant. At the end of inflation, the universe becomes cool due to sudden large (exponential, in the present case) expansion. Therefore, in order that the structure we live in are formed, the universe must be reheated and take the state of a hot thick soup of plasma (the so called hot Big-Bang). This phenomena is possible if at the end of inflation, the scalar field starts oscillating rapidly on the Hubble time scale, about the minimum of the potential. In the process, particles are created under standard quantum field theoretic (in curved space-time) approach, which results in the re-heating of the universe. The universe then eventually transits to the radiation dominated era. At that epoch, the additional term may be absorbed in the potential if it is a constant term ($V_{0}$), or may even be neglected in case it is a function (since $\phi$ goes below the Planck’s mass), without any loss of generality, to reassure symmetry. The symmetry leads to the first integral of certain combination of the field equations, which helps in solving the field equations leading to a Friedmann-like radiation dominated era, as shown earlier [25]. However, in the present manuscript, we only concentrate upon inflationary regime and of-course study possibility of graceful exit from inflation. In the following subsection, we shall study different power law potentials, while in the next we shall deal with exponential potential. We consider de-Sitter solution in the form $a\propto e^{Ht}$, where the Hubble parameter $H$ is slowly varying during inflation. We repeat that according to our current choice of units (${M_{p}^{2}\over 2}=1$), which although is uncommon, but doesn’t create any problem whatsoever, the value of $f(\phi)$ at the end of inflation must be a little greater than $1$.