---
abstract: 'An interesting cosmological history was proposed by Prigogine [*et al.*]{}  who considered the Universe as a thermodynamically open system.  This scenario is characterized by a process of matter creation, which corresponds to an irreversible energy flow from the gravitational field to the pressureless matter fluid.  Here, we show that the gravitationally induced particle production may arise from a nonminimal curvature-matter coupling.  By considering the equivalent scalar-tensor theory, the cosmological implications of the model are discussed.  As all known natural systems tend to a state of thermodynamic equilibrium, and assuming the universe is not different in this respect, we also discuss the conditions to attain the equilibrium state.' 
address:
- |
    $^1$Instituto de Astrofísica e Ciências do Espaço & Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal\
    E-mail: fslobo@fc.ul.pt; jpmimoso@fc.ul.pt
- '$^2$Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom.  E-mail: t.harko@ucl.ac.uk'
- '$^3$Departamento de Física, Universidad Autónoma de Barcelona, 08193 Bellaterra (Barcelona), Spain.  E-mail: diego.pavon@uab.es'
author:
- 'Francisco S. N. Lobo$^1$, Tiberiu Harko$^2$, José P. Mimoso$^1$, and Diego Pavón$^3$'
bibliography:
- 'ws-pro-sample.bib'
title: 'Irreversible matter creation processes through a nonminimal curvature-matter coupling'
---

Introduction
============

An interesting type of cosmological history, including large-scale entropy production, was proposed by Prigogine [*et al.*]{} [ @Pri0; @Pri], who considered the cosmological thermodynamics of open systems.  The role of irreversible processes corresponding to matter creation in General Relativity was investigated, and the usual adiabatic energy conservation laws were modified.  The corresponding cosmologies were based on a reinterpretation of the matter stress-energy tensor in Einstein’s equations, with the matter creation corresponding to an irreversible energy flow from the gravitational field to the created matter fluid [@Cal].  It was shown that the 2nd law of thermodynamics allows spacetime to transform into matter, forbidding the inverse process.  It appears that the conventional initial big bang singularity is structurally unstable with respect to irreversible matter creation.  Thus, the cosmological history in the framework of the thermodynamics of open systems starts from an instability of the quantum vacuum rather than from a singularity.  A remarkable fact is that the de Sitter stage appears to be an attractor independently of the initial fluctuation [@Pri0; @Pri]. 

The final de Sitter phase is particularly relevant due to the recent discovery of the late-time cosmic acceleration [@1; @2], where the simplest explanation is to invoke a cosmological constant.  The latter is usually associated to the quantum vacuum energy, which may not be constant but decays into radiation and matter particles.  In this work, we consider an alternative mechanism for the gravitational particle production, namely, in modified gravity, through a nonminimal curvature-matter coupling [@Bertolami:2007gv].  A general property of these theories is the non-conservation of the energy-momentum tensor [@Bertolami:2007gv; @Harko:2008qz; @Harko:2010hw; @Harko:2011kv].  Thus, the coupling between the matter and the higher derivative curvature terms may be interpreted as an exchange of energy and momentum between both, and we argue that this mechanism naturally induces a gravitational particle production [@Harko:2015pma]. 

Thermodynamic open systems and curvature-matter couplings
=========================================================

We consider the spatially-flat Friedmann-Robertson-Walker (FRW) metric, $ds^2=dt^2-a^2(t)\left(dx^2+dy^2+dz^2\right)$, where $a(t)$ denotes the scale factor.  Using the thermodynamics of open systems, the number of particles is not conserved, i.e., the usual expression for the evolution of the number density, $\dot{n} \, +
\, 3 H n = 0$, where $H = \dot{a}/a$, generalizes to $\dot{n}+3nH=\Gamma n$. Here, the quantity $\Gamma$ stands for the particle creation rate, which as imposed by the second law of thermodynamics obeys $\Gamma \geq 0$. Therefore, the energy conservation equation describing an irreversible particle creation can be rewritten as an effective energy conservation equation, $$\dot{\rho}+3\left( \rho +p+p_{c}\right) H=0, \qquad {\rm with} \qquad p_{c} = -\frac{\rho +p}{3}\frac{\Gamma }{H}.  
  \label{comp}$$where $p_{c}$ is a new thermodynamic quantity, $p_{c}$, denoted the creation pressure [@Pri].  In the following, we argue that an explicit coupling between curvature and matter generates an irreversible energy flow from the gravitational field to newly created matter constituents, with the second law of thermodynamics requiring that space-time transforms into matter.  The gravitational theory, with a nonminimal coupling between matter and curvature[@Bertolami:2007gv], is given by the Lagrangian $L=\frac{1}{2}f_1(R)+\left[1+\lambda f_2(R)\right]{\ L}_{m}$, where the factors $f_i(R)$ (with $i=1,2$) are arbitrary functions of the Ricci scalar $R$. The coupling constant $\lambda$ determines the strength of the interaction between $f_2(R)$ and the matter Lagrangian density, ${L}_{m}$.

The curvature-matter coupling is equivalent to a two-potential scalar-tensor Brans-Dicke type theory [@Faraoni:2007sn], with the following action $$\label{300}
S=\int  \left[ \frac{\psi R }{2} -V(\psi)\, +U(\psi) L_m \right]\,
\sqrt{-g} \, d^4x  \, ,$$ where the two potentials $V(\psi)$ and $U(\psi)$ are defined as $$\label{400}
V(\psi) = \frac{\phi(\psi) f_1^{\prime }\left[ \phi (\psi ) \right] -f_1%
\left[ \phi( \psi ) \right] }{2}, \qquad U( \psi) = 1+\lambda
f_2\left[ \phi( \psi ) \right]\,.$$

By varying the action (\[300\]) with respect to $g_{\mu \nu}$ and to the field $\psi$ provides the gravitational field equations $$\label{14}
\hspace{-0.1cm}\psi \left(R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R\right)+
\left(g_{\mu \nu }\nabla _{\alpha}\nabla^{\alpha} -\nabla _{\mu }\nabla _{\nu }\right)\psi =
U(\psi)T_{\mu \nu}+V(\psi)g_{\mu \nu},$$ $$\label{15}
\frac{R}{2}-V^{\prime }(\psi)+U^{\prime }(\psi)L_m=0.$$ The divergence of the energy-momentum tensor provides the following energy balance equation $$\label{eneq}
\dot{\rho}+3H(\rho +p)+\rho \frac{d}{ds}\ln U(\psi )+\frac{2V^{\prime
}\left( \psi \right) -U^{\prime }(\psi )L_{m}}{U(\psi )}\dot{\psi}=0\,.$$ Therefore, in modified gravity with a linear curvature-matter coupling, and confronting with Eq. ( \[comp\]), the creation pressure is given by $$\label{pc}
p_{c}=-\frac{1}{3H}\left\{\rho \frac{d}{dt}\ln U(\psi )+\frac{2V^{\prime
}\left( \psi \right) -U^{\prime }(\psi )L_{m}}{U(\psi )}\dot{\psi}\right\},$$ where the particle creation rate $\Gamma $ is a non-negative quantity defined as $$\label{33}
\Gamma =-\frac{1}{\rho +p}\Bigg\{\rho \frac{d}{dt}\ln U(\psi )+\frac{%
2V^{\prime }\left( \psi \right) -U^{\prime }(\psi )L_{m}}{U(\psi )}\dot{\psi}%
\Bigg\}.$$ Note that from Eq. ( \[eneq\]), the change in the particle number is due to the transfer of energy-momentum from gravity to matter. 

In the present work, we define the entropy through the particle production rate, as depending on the positive particle creation rate $\Gamma $ via the relation $\dot{S}/S=\Gamma \geq
0$. Therefore, in an ever expanding Universe with particle creation, the matter entropy will increase indefinitely.  On the other hand, natural systems, left to themselves, tend to attain a state of thermodynamic equilibrium.  This implies that the entropy of isolated systems never decreases, $\dot{S}\geq 0$, and that it is concave, at least in the last stage of approaching thermodynamic equilibrium. 

The validity of the second law of thermodynamics in cosmology was investigated in detail in [@Mim].  There it was shown that if one defines the entropy, $S$, measured by a comoving observer as the entropy of the apparent horizon plus that of matter and radiation inside it, then the Universe approaches thermodynamic equilibrium as it nears the final de Sitter phase.  Then it follows that $S$ increases, and that it is concave, thus leading to the result that the second law of thermodynamics is valid, as one should expect given the strong connection between gravity and thermodynamics, for the case of the expanding Universe. 

In the case of a spatially-flat FRW Universe filled with dust, one has $$S = S_{ah} + S_m = \frac{\pi}{H^2} + \frac{4\pi}{3H^3}\,n(t)\;,
\label{MP}$$ where we have used the fact that the radius of the apparent horizon is $%
r_{ah}= H^{-1}$ [@Bak].  Therefore, the thermodynamic requirements $S^{\prime}\geq 0 $ and $S^{\prime \prime }\leq 0$ impose specific constraints on the particle creation rate $\Gamma $, and its derivative with respect to the scale factor.  A particularly important case is that of the de Sitter evolution of the Universe, with $H=H_{\star}=\mathrm{constant}$. In this case, we have $$S^{\prime }= \frac{4\pi }{ 3H_{\star}^{4}}\frac{\left[ \Gamma (a)
-3H_{\star} \right]}{a}n(a) \geq 0  \,,$$ $$\begin{aligned}
S^{\prime \prime}= \frac{4 \pi n(a)}{3 a^2 H_{\star}^5}
\Big\{\Gamma ^2(a)+H_{\star} \left[a \Gamma ^{\prime }(a)+12
H_{\star}\right] -7 H_{\star} \Gamma (a)\Big\} \leq 0 \,.\end{aligned}$$ Accordingly, the constraints $\Gamma \geq 3H_{\star}$ and $\Gamma ^{\prime
}(a)\leq \left[7\Gamma (a)-\Gamma
^2(a)/H_{\star}-12H_{\star}\right]/a$ are imposed on the particle creation rate $\Gamma$. For $\Gamma =3H_{\star}$, we obtain $S =
\mathrm{constant}$, showing that in this case the cosmological evolution is isentropic.  Here, $H_{\star}$ denotes the expansion rate of the final de Sitter phase.  Realistically, it must be lower than the Hubble constant value, $H_{0} \simeq 73.8$km/s/Mpc. 

Conclusion
==========

In conclusion, the gravitational theory investigated in the present essay suggests that the curvature-matter coupling may well be a leading feature of our Universe.  The current accelerated cosmic expansion may be seen as a hint for matter creation which offers a welcome alternative to the elusive dark energy.  The existence of some forms of the curvature-matter coupling leading to matter creation processes is not ruled out by the available data.  Presumably, the functional forms of the potentials $V(\psi)$ and $U(\psi)$ that fully characterize the nonminimal curvature-matter coupling gravitational theory may be provided by fundamental quantum field theoretical models of the gravitational interaction, thus opening the possibility to an in-depth comparison of the predictions of the gravitational theory with cosmological and astrophysical data. 

Acknowledgments {#acknowledgments .unnumbered}
===============

FSNL was supported by a FCT Research contract, with reference IF/00859/2012, funded by FCT/MCTES (Portugal).  FSNL and JPM acknowledge financial support of the Fundação para a Ciência e Tecnologia through the grant EXPL/FIS-AST/1608/2013 and UID/FIS/04434/2013.  DP was partially supported by the “Ministerio de Economía y Competitividad, Dirección General de Investigación Científica y Técnica", Grant N$_{o}$. FIS2012-32099. 

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