# Inflationary expansion of the universe with variable timescale

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## Abstract

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.

## 1 Introduction and motivation

The emergence of quantum spacetime in the universe has been a subject of study in the last years [1, 2]. This is an intriguing issue in the history of the universe that remains unsolved [3, 4]. Possible dissipative effects in the context of fundamental theories of gravity have been discussed [5], and so have extra-dimensional models where a condensate of fermion fields drives the expansion of the universe [6]. Other proposals have been studied in a causal extra-dimensional set theory [7]. 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 [8]. In this metric the time scale along the whole of the expansion of the universe is considered as a constant. However, it makes sense to think that this scale of time has not always been 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 slower 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, the emergent spacetime to grow exclusively due to the energy density transferred by the inflaton field at cosmological scales. 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 the scaling of time will be considered as variable along the expansion.

## 2 The model

*H*(

*t*) is the Hubble parameter on the background metric and \(\Gamma (t)\) 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 \(c=\hbar =1\). The case of a background expansion on a vacuum is recovered by setting \(\Gamma =0\).

^{1}

*P*and \(\rho \) are the solutions for \(\Gamma =0\), for which we recover the expressions for a perfect fluid: \(T^{\mu }_{\nu }=\mathrm{{diag}}(\rho ,-P,-P,-P)\). The results (10) can be obtained by using the relationship between \(\bar{V}\) and

*V*, and by equating Eqs. (4) and (7), so that the following conditions must hold:

### 2.1 Back-reaction effects

## 3 An example: power-law inflation with variable timescale

*p*and

*q*to be determined by the observation parameters. In this case the dynamics equation for the inflaton field holds,

*a*(

*t*) is given by (36). The relativistic quantum algebra is given by the expressions (20), with co-moving relativistic velocities \(U^0=e^{\int \Gamma (t)dt}\), \(U^i=0\), which are calculated on the Riemannian (background) manifold.

Observational cuts for slow-roll parameters for \(n_s=0.965\pm 0.004\) and \(p=-\,1\)

Parameters | For \(n_s=0.968\) |
---|---|

\(\epsilon =\left. {1-p\over q}\right| _{(p=-1)}={1-n_s\over (3-n_s)} \) | \( 0.01526 \,<\epsilon < \,0.01912 \) |

\(\eta =\left. {3\over 3q+p}\right| _{(p=-1)}={3(n_s-1)\over 5n_s-17} \) | \( 0.0076 \le \eta \le 0.0096 \) |

\(n_s=\left. {3p+q-3\over p+q-1}\right| _{(p=-1)}={q-6\over q-2}\) | \( 0.961 \,< n_s < \, 0.969\) |

\(n_t=\left. -\,2\epsilon \right| _{(p=-1)}={-2(1-n_s)\over (3-n_s)}\) | \( -\,0.0382 \,< n_t < \,-\,0.0305\) |

\(\left. r\le 16 \epsilon \right| _{(p=-1)} ={16(1-n_s)\over (3-n_s)}\) | \( 0.2442\,<r <\,0.3060 \) |

\(q={2(3-n_s)\over 1-n_s} \) | \(104.56 \,< q < \, 131.03\) |

\(\omega ={n_s-7\over 3(3-n_s)} \) | \(- \, 0.9872 \,< \omega < \, -\,0.9898\) |

## 4 Final comments

*t*, because the physical time evolves as \(d\tau =U_0\,\mathrm{d}x^0=\sqrt{g_{00}}\mathrm{d}x^0=\left( t/t_0\right) \,\mathrm{d}t\), for \(c=1\). Beyond that, the proposed metric gives the possibility to exactly describe back-reaction effects for any equation of state. This is a great advantage over standard cosmological models where the timescale is not variable. In Fig. 1 we have plotted the plausible range of

*q*for \(p=-\,1\), which corresponds to a spectral index \(n_s=0.965\pm 0.004 \). The results show the range \( 0.961 \le n_s \le 0.969 \) (see Fig. 1), for which the rate of expansion of the universe is in the range \(104.56< q < 131.03 \). In this range of

*q*-values, \(\omega \) takes the values \(-\, 0.9872<\omega <-\,0.9898\). The results obtained for \(n_s\) agree with a

*k*-power of the spectrum (33) that is close to zero, but negative. The amplitude for this spectrum decreases with time. In Fig. 2 we have plotted the curve of \(\omega (n_s)\), for all possible values that describe an accelerated universe: \(\omega < -\,1/3\). Our results exclude constraints from \(WMAP9 + N_\mathrm{eff}\) data [18] [with the effective number of relativistic species \(N_\mathrm{eff}\) and the massive neutrinos simultaneously], but the results agree with constraints from Planck 2018 data [15], which are compatible with

*WMAP*9 data without considering the effective number of relativistic species and the mass of the neutrinos.

## Footnotes

- 1.
The new stress tensor is \({T}_{\mu \nu }=-\left[ \phi _{,\mu }\phi _{,\nu }-g_{\mu \nu }\left( \frac{1}{2} g^{\alpha \beta }\phi _{,\alpha }\phi _{,\beta }\bar{V}(\phi )\right) \right] \) and the new Lagrangian density corresponding to \(\phi \) is \({\mathcal{L}}_{\phi }= -\left[ \frac{1}{2} g^{\alpha \beta } \phi _{,\alpha }\phi _{,\beta } - \bar{V}(\phi )\right] \), from which, due to the spatial isotropy and homogeneity, the action (6) can be obtained explicitly.

## Notes

### Acknowledgements

The authors acknowledge CONICET, Argentina (PIP 11220150100072CO) and UNMdP (15/E810), for financial support.

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