# Quantum thermodynamics in a static de Sitter space-time and initial state of the universe

## Abstract

Using Relativistic Quantum Geometry we study back-reaction effects of space-time inside the causal horizon of a static de Sitter metric, in order to make a quantum thermodynamical description of space-time. We found a finite number of discrete energy levels for a scalar field from a polynomial condition of the confluent hypergeometric functions expanded around \(r=0\). As in the previous work, we obtain that the uncertainty principle is valid for each energy level on sub-horizon scales of space-time. We found that temperature and entropy are dependent on the number of sub-states on each energy’s level and the Bekenstein–Hawking temperature of each energy level is recovered when the number of sub-states of a given level tends to infinity. We propose that the primordial state of the universe could be described by a de Sitter metric with Planck energy \(E_p=m_p\,c^2\), and a B–H temperature: \(T_{BH}=\left( \frac{\hbar \,c}{2\pi \,l_p\,K_B}\right) \).

## 1 Introduction and motivation

A de Sitter space-time is the maximally symmetric vacuum solution of Einstein’s field equations with a positive cosmological constant \(\Lambda \), which corresponds to a positive vacuum energy density and negative pressure. In the cosmological context, it describes the exponential accelerated expansion of the universe governed by the vacuum energy density. There is evidence that the very early universe had a period of rapid expansion, called inflation [1, 2, 3, 4], well approximated by de Sitter space-time. Our tiny present-day cosmological constant currently accounts for about \(68 \, \%\) of the energy density of the universe, and this fraction is growing as the universe continues to expand. This means that we are entering a second de Sitter phase. The early, inflationary de Sitter phase had a large cosmological constant and correspondingly tiny radius of curvature. The future dark energy de Sitter will have an energy set by today’s cosmological constant, and enormous radius of curvature close to today’s Hubble scale [5].

In the standard relativistic description, matter (which is described by the matter Lagrangian: \(\hat{{{\mathcal {L}}}}\), in the Einstein-Hilbert (EH) action), is responsible for the spatial curvature of space-time, which is represented in the Einstein’s equations through \(G_{\alpha \beta }\). However, this description only takes into account the expectation value in the physical system under consideration. A better description must consider the effects that quantum fluctuations produce in the background space-time due to the retro-reaction, due to the fields that we are considering in \(\hat{{{\mathcal {L}}}}\). Such description must be non-perturbative, because these effects could be very important when we deal with strong fields, under extreme physical conditions. Because of this, back-reaction effects are very important in general relativity, and in particular, they are essential to make a correct and accurate description of the initial state of the universe, which is believed to be given by a de Sitter metric. In this work we shall make a semiclassical description of such back-reaction effects, without consider non-commutative aspects of space-time, or the origin of quantum spinor fields that originate these quantum effects [6, 7]. However, our semiclassical treatment should be sufficient to describe correctly the quantum thermodynamics inside the horizon of a de Sitter metric. A 4D de Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric: \(R_{\mu \nu } = {3 \over \alpha ^2} \, g_{\mu \nu }\). It describes a vacuum solution of the Einstein’s equations with a cosmological constant given by \(\Lambda = {3 \over \alpha ^2}\) and a scalar curvature \(R=4 \Lambda =12/\alpha ^2\), such that \(\alpha \) is the cosmological horizon. Therefore, a de Sitter space-time describes an hyperbolic space for \(r<\alpha \).

Many years ago, Bekenstein has argued that isolated stable thermodynamic systems in asymptotically flat space-times satisfies the universal entropy bound [8]: \(S \le {2\pi \alpha E \over \hbar c}\), where \(\alpha \) is the radius of an enclosed system with energy *E*. In this work we shall use a recently introduced thermodynamic description of space-time [9] in the study a Schwarzschild black-hole, but now with the aim to explore the interior of a de Sitter space-time (i.e., in the range \(r<\alpha \)). We shall use the formalism of Relativistic Quantum Geometry (RQG) described in [10] and [11], which was revisited in Sect. 2. In Sect. 3 we study back-reaction effects inside the causal horizon of a de Sitter metric, with the aim to explore a quantum thermodynamical description of energy, lengths, entropy and the temperature. Finally, in Sect. 4, we develop some final comments and conclusions.

## 2 Revisited back-reaction effects from boundary conditions in the variation of the EH action

*D*closed manifold defined on an arbitrary region of the background manifold, which is considered as Riemannian and is characterized by the Levi-Civita connections. As in a previous work [9] we must describe the variation of the connections with respect to the background manifold, which is a Riemannian one. We shall consider no-metricity on the extended manifold. To extend the Riemann manifold we shall consider the connections

## 3 Back-reaction solution in a de Sitter metric

*H*is the Hubble parameter,

*c*is the light velocity in the vacuum, and \(\alpha = c/H\) is the Hubble horizon, which is related to the cosmological constant \(\Lambda = 3 (H/c)^2\). In order for describe the back-reaction effects in the interior of the de Sitter space, we must consider solutions of the equation \(\Box \sigma =0\), for \(r<\alpha \), where the space-time is 4D hyperbolic with signature \((+,-,-,-)\), due to the fact \(f(r)>0\). The massless scalar field \(\sigma \) for the line element (15) and \(r<\alpha \) is described by the equation

*a*or

*b*is a non-positive integer \(-n\), in which case the function is reduced to a polynomial of order

*n*. Our aim is using this condition in order for motivate validity of the uncertainly principle for each energetic level and the discretization of the \(\alpha \) values, in order for relate the solutions (19) to the recently studied Schwarzschild black hole’s mass case [9].

### 3.1 Uncertainly principle, energy levels and the cosmological constant

*c*/

*H*. Furthermore, this also provides a discretization of the cosmological constant \(\Lambda =\frac{3}{\alpha ^2}\): \(\Lambda _{(n,l)}\), in terms of the eigenvalues

*n*and

*l*. Therefore, if we introduce \(\Lambda =\frac{3}{\alpha ^2}\) and

*C*is a constant to be determined by normalization.

*l*-values:

The limit case for the previous expression corresponds to \(N(\alpha )=1\) and \(l=2n\), and it’s consistent with the definition for \(\alpha _{(n,l)}\) in (21).

### 3.2 de Sitter temperature from RQG

*n*,

*l*)-values, is

*n*,

*l*)-energy. After specializing, we obtain that the value of a generic \(T_{(n,l)}\) corresponds to

*l*, corresponding to each

*n*-value, will be

*n*,

*l*must take the values \(l \ge 2n\). Furthermore, all combinations \(l=2n\) guarantee \(T_{(n,l=2n)}=T_{BH}\), where \(T_{BH}=\frac{\hbar \,c}{2\pi \alpha \,K_B}\) is the Bekenstein–Hawking (B–H) temperature [15, 16, 17] and \(K_B\) is the Boltzmann constant. In particular, \(T_{(0,0)}=T_{(1,2)} = \cdots = T_{BH}\).

*l*we have \(T_{m\rightarrow \infty }=2\,T_{BH}\). This is the same behavior that energy levels in the Schwarzschild’s Black Hole interior [9]. The important here is that this behavior is repeated for all the possible values of

*m*(

*l*), on each (\(l=2n\))-level.

*L*is the maximum value of

*m*. This is a very important result that replies whole obtained in [9], but here on each (\(l=2n\))-level. Notice that the extreme case where \(\alpha =l_p\) is the Planck length, is a good candidate to describe the initial state of the universe in a de Sitter metric, with a B–H temperature:

*H*is a cosmological observable, so that one could immediately calculate the B–H temperature in the universe.

## 4 Final comments

In this work we have studied back-reaction effects in the interior of a de Sitter space-time (i.e., for \(r<c/H\)), using the RQG formalism in which we take into account, when we variate the EH action, the flux that cross the 3D-gaussian hypersurface. The extended manifold is obtained by making a displacement from the background Riemann manifold to the new extended manifold (4). This flux is described by a scalar field \(\sigma \) (more precisely by their partial derivatives: \(\sigma ^{\alpha }\)), that describes back-reaction effects of the space-time, so that the metric tensor with back-reaction effects included are given by (14). We have applied this formalism to study the back-reaction effects on sub-horizon scales of a static de Sitter metric, and, for the radial solution, we have found a finite number of discrete energy levels for the \(l=2n\) values, such that, for each energy level, we have a number *L* of possible values \(L \ge l\ge 2n \), for a scalar field solution, obtained from a polynomial condition of the confluent hypergeometric functions, which is expanded around \(u=0\). The interesting is that we recover the same structure for the temperature values, that in the interior of the Schwarzschild black-hole [9], but here for each (\(l=2n\))-level. Furthermore, the uncertainty principle (21), is valid for each energy level on sub-horizon scales of the space-time, and the temperature and entropy are dependent on the number of sub-states with different *m*(*l*), on such scales. When this number tends to infinity: \(L \rightarrow \infty \), we recover the B–H temperature for this (\(l=2n\))-level: \(\lim _{L\rightarrow \infty } \sum \nolimits _{m=0}^{L} \Delta T_m =T_{BH}\). Therefore, we propose that the primordial universe could be described by a Planck energy and a B–H temperature: \(T_{BH}=\left( \frac{\hbar \,c}{2\pi \,l_p\,K_B}\right) \) in a de Sitter space-time [18, 19, 20]. Our approach could be extended to another in which we describe a evolving geometry such that \(\Lambda \) is a time-decaying cosmological parameter on the Riemann manifold. This issue, which is beyond the scope of this work, will be developed in the future.

## Notes

### Acknowledgements

The authors acknowledge CONICET, Argentina (PIP 11220150100072CO) and UNMdP (EXA852/18) for financial support.

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