Inflationary expansion of the universe with variable timescale

We explore a cosmological model in which the time scale is variable with the expansion of the universe and the effective spacetime is driven by the inflaton field. An example is considered and their predictions are contrasted between Planck 2018 data. We calculate the spectrum indices and the slow-rolling parameters of the effective potential. The results are in very good agreement with observations.


I. INTRODUCTION AND MOTIVATION
The emergence of quantum spacetime in the universe has been subject of study in the last years [1]. This is an intriguing issue in the history of the universe that remains unsolved [2] [3].
Possible dissipative effects in the context of fundamental theories of gravity have been discussed [4], as an infrared signal of the coupling of matter fields with gauge models propagating in extra-dimensions, while dispersive phenomena in emergent spacetime have been studied in causal set theory [5]. On the other hand, the Friedmann-Robertson-Walker (FRW) metric is used to describe the large-scale expansion of the universe, and it can be considered as an effective (or phenomenological) way to describe a more fundamental line element which has a quantum mechanical origin [6]. In this metric the time scale along all the expansion of the universe is considered as a constant. However, it is sensate to think that this scale of time was not always the same. If the gravitational field was very intense at the beginning of the universe, it is reasonable to think that events spatially very close, could be subjected to a more slow rate of temporal flux. This means that with the expansion of the universe, spatially very separated events should be described by a more intense flux of time, as the gravitational field becomes weaker and weaker with the expansion of the universe.
In this work we shall consider, in a phenomenological approach, that the emergent spacetime grows exclusively due to the energy density transferred by the inflaton field at cosmological scales. In other words, we shall consider that the spacetime can be considered at cosmological scales as a coherent non-radiating condensate [7]. For generality, we shall use a metric in which time scales of events at cosmological scales are not the same during the expansion of the emerging primordial universe. This means that scaling of time will be considered as variable along the expansion.

II. THE MODEL
We consider an expanding universe that is spatially flat, isotropic and homogenous, which is not necessary in a vacuum background. The background metric is described by the line element which describes a non-vacuum Friedman-Lemaitre-Robertson-Walker (FLRW) metric such that H(t) is the Hubble parameter on the background metric and Γ(t) describes the time scale of the background metric. This should be the case in an emergent accelerated universe in which the spacetime is growing and the time scale can be considered variable with the expansion. In this paper we shall consider natural units, so that c = = 1. The case of a background expansion on a vacuum is recovered by setting Γ = 0.
To describe the expansion of the universe, we consider a single scalar field φ which is minimally coupled to gravity and drives the expansion. If its dynamics is governed by a scalar potential V (φ), the action can be written as where the volume of the background manifold isv = √ −ĝ = a 3 0 e − Γ(t) dt e 3 H(t) dt . We shall consider the model of an universe in which all the potential energy of the inflaton field is transferred to the expansion of this spacetime volume. In that case, the dynamics of the scalar field φ is given byφ that describes the dynamics of the background scalar field evolving on the background metric (1). The first dissipative term is due to the expansion of the universe, but de second one is due to the existence of a nontrivial time scale. The background Einstein equations, are where P = −T 1 1 is the pressure and ρ = T 0 0 the energy density due to the inflaton field. The last term in the left side of Eq. (5) describes the contribution of the non-vacuum to the dynamics of the field. The equation of state that describes the dynamics of the system is: Notice that in the case of a vacuum expansion with Γ = 0, the equation of state agrees with that of an ideal fluid. Of course, the most interesting case is Γ = 0. In that case it is possible, for example, to describe inflationary scenarios where dissipative effects are important in the evolution of the universe.

A. Back-reaction effects
A nonperturbative back-reaction formalism was developed in earlier works [9,10]. In those works it was demonstrated that the background metric can be altered by a scalar field σ without the action to be altered: δI = 0, when it is fulfilled where can be demonstrated that [8] and the lagrangian density associated to the field φ is Here, the back-reaction effects are due to the nonzero flux g αβ δR αβ = − Λ 2 δσ, through a gaussian hypersurface, such that the manifold is defined by and the covariant derivative of the metric tensor, on this manifold, is such that the covariant derivative of the metric tensor on the Riemannian manifold: ∇ γ g αβ = 0, is zero. In other words, the metric tensor has null nometricity on the Riemann manifold, but not on the extended manifold defined by (10). From the point of view of the Riemann manifold Λ is a constant, but from the point of view of the Weylian-like manifold: Λ ≡ Λ(σ, σ α ) can be considered a functional, given by Therefore, a geometrical quantum action on the Weylian-like manifold with (31), can be considered such that the dynamics of the geometrical field is given by the Euler-Lagrange equations, after imposing δW = 0. The dynamics of the back-reaction is described by the equation Notice that the term Γσ in (14), takes into account the interaction between the geometric field σ with the background. In order to describe the algebra of σ, we define the scalar invariant Π 2 = Π α Π α . If we require that [Π 2 , σ] = 0, we obtain the algebra [9,10] [ whereÛ α are the components of the Riemannian velocities.

III. AN EXAMPLE: POWER-LAW INFLATION WITH RESCALED TIME
We consider the case where the Hubble parameter and the dissipative coefficient are In this case the dynamics equation for the inflaton field holds where, in the Einstein equations (4,5), we must set T 0 0 e −2 Γ(t) dt =φ 2 2 + V (φ) and . Notice that we are not considering radiation energy density. This is because all the energy of the inflaton field is transferred to expansion of the spacetime.
The geometrical scalar field σ can be expressed as a Fourier expansion where A † k and A k are the creation and annihilation operators: The metric with back-reaction effects included, is where the background scale factor a(t) is given by (31). Notice that its volume is described by (20), is V q = a 3 0 e 3 H(t)dt e − Γ(t)dt e −2σ = √ −ĝ e −2σ . The relativistic quantum algebra is given by the expressions (15), with co-moving relativistic velocities U 0 = e Γ(t)dt , U i = 0, which are calculated on the Riemannian (background) manifold.
Furthermore, as calculated in a previous work [11], the variation of the energy density fluctuations is The equation of motion for the modes ξ k (t) in the expansion (18), is Using the commutation relation (19) and the Fourier expansions (18) in with canonical momentum: Π α = δLq δσ α = − 3 4 √ −ĝ σ α , we obtain the normalization condition for the modes ξ k (τ ) where the asterisk denotes the complex conjugated. The general solution of Eq. (22), is where H (1,2) ν [y(t)] are the first and second kind Hankel functions, with parameter and argument y(t) = k t 0 (p+q) t −(p+q−1) a 0 (p+q−1) . After quantization, the solution for the modes ξ k , hold and the power-spectrum on cosmological scales is with ν 1 = 3p+5q−3 2(p+q−1) , and β = a 0 t −(p+q) 0 . The power of the spectrum is 3 − 2ν, which is related to the spectral index by the expression Furthermore, using the expression (6), we obtain where ν is given by Eq. (26). Notice that for p = 1, we obtain that ν = 3/2, n s = 1 and ω = −1. This means that the model predicts a Harrison-Zel'dovich spectrum [15] for a vacuum expansion of the universe where the amplitude of the spectrum is frozen. However, last experimental data excludes n s = 1 [16], so that we shall contrast our model with the experimental data.
Now we consider the background dynamics given by the Einstein equations (4,5), with the inflaton dynamics (17). From the Einstein equations, we can obtaiṅ From (31), we obtain thatφ Notice that the case p = 1, corresponds to ω = −1 and ν = 3/2, so that n s = 1, and the spectrum is scale invariant. In this case the power of the spectrum for back-reaction effects (28) are independent of time, so that the amplitude of the back-reaction spectrum on cosmological scales is frozen. Furthermore, in this caseφ = 0, so that the inflaton field assumes a constant value. On the other hand, due to the fact that δV δφ =V /φ, using the expressions H(t) = q/t and Γ(t) = p/t in Eq. (17), we obtain the condition which gives us two possible solutions: p = 1 − 3q and p = −1. The case p = 1 − 3q is consistent with a decelerated expansion of the universe for ν = 1.516. The case p = −1, is consistent with a very accelerated expansion of the universe: q = 127 and ω = −0.989, which is in very good agreement with observations [12]. The scalar spectral index is n s = 1−6ǫ+2η and the tensor index is given by n t = −2ǫ, where the slow-roll parameters are given by the expressions [13]. In the second column of table I we calculate the physical parameters for n s = 0.968 for p = −1: meanwhile the tensor-scalar ratio is given by r = −8 n t . The observational cuts of these parameters from WMAP9 [14] are shown in the second column of the  hand, in [16] the authors consider WMAP9 data with the effective number of relativistic species N ef f and the massive neutrinos simultaneously. Due to the strong degeneracy of n s and N ef f , the spectral index increases to give n s = 0.980 ± 0.011, which also excludes values n s ≥ 1. In our model, n s ≥ 1 means that ω ≤ −1 is excluded and therefore p = −1. In the third column of table I, we calculate the resulting physical parameters with this new value of the spectral index and p = −1.

IV. FINAL COMMENTS
We have studied a cosmological model in which the scale of time is variable and the expansion of the universe is driven by a scalar field. The dynamics of the scalar field together with the Einstein equations require that p = −1, so that the physical time is τ = 1 2 t 2 . This means that at the beginning of the expansion the rate of events is much more hysteric, but after certain amount of expansion, the rate of events becomes slower and slower, and we need much more time τ for a physical event to occur. Beyond that, the metric proposed gives us the possibility to describe exactly the back-reaction effects for any equation of state. This is a great advantage with standard cosmological models where the timescale is not variable.
In the Fig. (1) we have plotted the plausible range of q for two different data sources, which corresponds to spectral indices n s = 0.968 and n s = 0.98, respectively. The results shows that there is an intersection: 0.969 ≤ n s ≤ 0.974[see Fig. (1)], which could be the range of spectral index, which corresponds to the range 131.03 < q < 155.85 for the expansion of the universe. For this range of values ω varies in the range: −0.9914 < ω < −0.9898. Notice that with these values for n s , the k-power of the spectrum (28) is close to zero, but negative.
The amplitude for this spectrum decreases with time. In the Fig. (2) we have plotted the curve of ω(n s ), for all possible values that describe an accelerated universe: ω < −1/3.