Accelerating Cold Dark Matter Cosmology ()
A new kind of accelerating flat model with no dark energy that is fully dominated by cold dark matter (CDM) is investigated. The number of CDM particles is not conserved and the present accelerating stage is a consequence of the negative pressure describing the irreversible process of gravitational particle creation. A related work involving accelerating CDM cosmology has been discussed before the SNe observations [Lima, Abramo & Germano, Phys. Rev. D53, 4287 (1996)]. However, in order to have a transition from a decelerating to an accelerating regime at low redshifts, the matter creation rate proposed here includes a constant term of the order of the Hubble parameter. In this case, does not need to be small in order to solve the age problem and the transition happens even if the matter creation is negligible during the radiation and part of the matter dominated phase. Therefore, instead of the vacuum dominance at redshifts of the order of a few, the present accelerating stage in this sort of Einstein-de Sitter CDM cosmology is a consequence of the gravitational particle creation process. As an extra bonus, in the present scenario does not exist the coincidence problem that plagues models with dominance of dark energy. The model is able to harmonize a CDM picture with the present age of the universe, the latest measurements of the Hubble parameter and the Supernovae observations.
A large amount of data relevant to cosmology (involving Supernovae type Ia and cosmic background radiation probes) have provided strong evidence that the observed universe is well described by an accelerating, flat Friedmann-Robertson-Walker (FRW) model Riess (); Riess07 (); CMB (); Ages (). However, the substance or mechanism behind the current cosmic acceleration remains unknown and constitutes a challenging problem of modern cosmology.
In relativistic cosmology, an accelerating regime is obtained by assuming the existence of a dark energy component (in addition to cold dark matter), an exotic fluid endowed with negative pressure in order to violate the strong energy condition review (). The simplest theoretical representation of dark energy is by means of a cosmological constant , which acts on the Einstein field equations (EFE) as an isotropic and homogeneous source with constant equation of state (EoS) .
All observational data available so far seems to be in good agreement with the cosmic concordance model, i.e., a vacuum energy plus cold dark matter (CDM) scenario. Nevertheless, CDM models are plagued with several problems. For instance, it is very difficult to reconcile the small value required by observations () with estimates from quantum field theories ranging from 50-120 orders of magnitude larger weinberg (). Such problem has inspired many authors to propose alternative candidates in the literature Sfield (); decaying (); decaying1 (); list1 (); XM1 (), among them: (i) a relic scalar field slowly rolling down its potential, (ii) a -term or a decaying vacuum energy density, (iii) the “X-matter”, an extra component characterized by equation of state , where may be constant or a redshift dependent function, (iv) a Chaplygin-type gas whose equation of state is , where A and are positive parameters. More recently, some attention has also been paid to a possible interaction between the dark sector components DSI ().
The space parameter of such models are usually highly degenerated and some of them contain the CDM scenario as a particular case. In point of fact, the plethora of possible candidates does not help to identify the nature of this mysterious component since there is no compelling direct evidence yet for dark energy (or its dynamical effects). In other words, the evidence supporting its existence is not strong enough to be considered established beyond doubt (see BLB () for a critical discussion).
Roughly speaking, a realistic cosmological scenario should be in agreement with at least four well established observational results, namely: (i) the existence of a dark non-baryonic component as required by the dynamics of galaxies and clusters, the matter power spectrum and other independent probes like the temperature anisotropies of the cosmic microwave background (CMB) from last scattering surface (ii) the late time cosmic acceleration, (iii) the (nearly) flatness of the Universe, and, finally, (iv) a Hubble parameter with the Universe being older than 12 Gyrs in order to accommodate the oldest observed structures (globular clusters). When confronted with this simple requirements, we see that the CDM or Einstein-de Sitter cosmology is in clear contradiction with results (ii) and (iv). Therefore, if one assumes that the dark energy does not exist, the first task is to explain how a flat CDM dominated Universe can accelerate at late times because, potentially, accelerating cosmologies solve the age problem.
In this concern, we recall that the presence of a negative pressure is the key ingredient required to accelerate the expansion. This kind of stress occurs naturally in many different contexts when the physical systems depart from a thermodynamic equilibrium states LL (). In general, such states are connected with phase transitions (for example, in an overheated van der Waals liquid), and for some systems the existence of negative pressure seems to be inevitable sakharov (). In this connection, as first pointed out by Zeldovich zeld (), the process of cosmological particle creation at the expenses of the gravitational field can phenomenologically be described by a negative pressure and the associated entropy production. In principle, such an approach is completely different from the one developed by Hoyle and Narlikar narlikar () adding extra terms to the Einstein-Hilbert action describing the so-called C-field. In the latter case, the creation phenomenon is explained trough a process of interchange of energy and momentum between matter itself and the C-field as happens, for instance, in vacuum decaying cosmologies decaying ().
The gravitational matter creation processes was investigated from a microscopic viewpoint by Parker and collaborators Parker () by considering the Bogoliubov mode-mixing technique in the context of quantum field theory in curved space-time BirrellD (). Despite being rigorous and well-motivated, those models were never fully realized, probably due to the lack of a well-defined prescription of how matter creation is to be incorporated in the classical EFE.
The consequences of gravitational matter creation have also been macroscopically investigated mainly as a byproduct of bulk viscosity processes near the Planck era as well as during the reheating of the inflationary scenarios Murphy (); Hu (). However, the first self-consistent macroscopic formulation of the matter creation process was put forward by Prigogine and coworkers Prigogine () and somewhat clarified by Calvão, Lima and Waga LCW () through a manifestly covariant formulation. It was also shown that matter creation, at the expenses of the gravitational field, can effectively be discussed in the realm of the relativistic nonequilibrium thermodynamics. Later on, it was also demonstrated that the matter creation is an irreversible process completely different from bulk viscosity mechanism LG92 () (see also SLC02 () for a more complete discussion). Several interesting features of cosmologies with creation of matter and radiation have been investigated by many authors ZP2 (); LGA96 (); LA99 (); Susmann94 (); ZSBP01 (); freaza02 () (see also Makler07 () for recent studies on this subject).
In comparison to the standard equilibrium equations, the irreversible creation process is described by two new ingredients: a balance equation for the particle number density and a negative pressure term in the stress tensor. Such quantities are related to each other in a very definite way by the second law of thermodynamics Prigogine (); LCW (). The leitmotiv of this approach is that the matter creation process, at the expense of the gravitational field, can happen only as an irreversible process constrained by the usual requirements of non-equilibrium thermodynamics.
In this context, we are proposing here a new flat cosmological scenario where the cosmic acceleration is powered uniquely by the creation of cold dark matter particles. As we shall see, the model is consistent with the supernovae type Ia data, and a transition redshift of the order of a few is also obtained. In this extended CDM model, the Hubble parameter does not need to be small in order to solve the age problem and the transition happens even if the matter creation is negligible during the radiation and considerable part of the matter dominated phase. Moreover, the so-called coincidence problem of dark energy models is replaced here by a gravitational particle creation process at low redshifts.
Ii Cosmology and Matter Creation
For the sake of generality, let us start with the homogeneous and isotropic FRW line element
where is the scale factor and is the curvature parameter. Throughout we use units such that .
where an overdot means time derivative and , , , and are the energy density, thermostatic pressure, creation pressure, particle number density and matter creation rate, respectively. The quantity with dimension of is the creation rate of the process. The creation pressure is defined in terms of the creation rate and other physical quantities. In the case of adiabatic matter creation, it is given by Prigogine (); LCW (); LG92 (); ZP2 (); SLC02 () (see also Appendix A for a simplified deduction)
where is the Hubble parameter.
As one may check, by combining the EFE with usual equation of state, , the equation governing the evolution of the scale function is readily obtained:
The above expression shows how the matter creation rate, , modifies the evolution of the scale factor as compared to the case with no creation. Conversely, the cosmological dynamics with irreversible matter creation will be defined once the matter creation rate is given. As should be expected, by taking it reduces to the FRW differential equation governing the evolution of a perfect simple fluid LA ().
Iii Flat CDM model with matter creation and the age of the Universe
In what follows we focus our attention on the flat cold dark matter model () with the previous equation reducing to:
On the other hand, Eq. (4) can be rewritten as
which means that the creation process can effectively be quantified by the dimensionless ratio (see also Eq. (8))
which in general is a function of time. If , that is, , the creation process is negligible leading to and , as should be expected for an Einstein-de Sitter model. The opposite regime () defines an extreme theoretical situation, where the creation process is a phenomenon so powerful that the dilution due to expansion is more than compensated. Probably, this kind of behavior may happen only in the very early universe as happens, for instance, during the reheating stage of inflation. An intermediary (and physically more reasonable situation) occurs if this ratio is smaller or of the order of unity (). In particular, if the dilution due to expansion is exactly compensated and the number density remains constant. From now on we consider that .
In a series of papers LGA96 (); LA99 (), we have investigated some properties of adiabatic matter creation models with , where is a constant parameter contained on the interval [0,1] (). However, that kind of models are always accelerating for or decelerating for , that is, there is not a transition redshift from a decelerating to an accelerating regime as required by the supernovae type Ia observations (see Figure 3a). In order to cure such a difficulty we add a constant term in this expression, that is, we consider the following matter creation rate (see Appendix for a more rigorous argument)
where the parameter (like ) lies on the interval [0,1]. As we shall see, this scenario is compatible with the basic observations listed in the introduction even for . Inserting Eq. (11) into (8) one finds
whose solution reads
and by integrating the above expression we obtain a big-bang solution for the scale factor
where and are the present day values of and , respectively. In the limit , the above expression reduces to
In Figure 1 we display the behavior of the scale factor as a function of time. All the models start their evolution from the initial singularity (). It is worth noticing that the parameter does not contribute at early times. Actually, for only the parameter appears in the equation of motion (12).
In Figure 2 we show the age parameter as a function of and some particular values of . The solid black line yields the age of the Universe as a function of when is zero. In this case,
Note that ages great enough are obtained even for . In particular, for the age parameter is , exactly the same value predicted by the ‘cosmic concordance’ (CDM) model from WMAP3 and complementary observations CMB (). In the limit one obtains as should be expected. The influence of the parameter is apparent from Figure 2, namely, it increases the age of the Universe for a given value of .
At this point, it is interesting to discuss in what sense this simple CDM scenario with creation behaves like an irreversible process. Adiabatic matter creation means that the total entropy S increases, but, the specific entropy (per particle), , where is the corresponding number of particles, remains constant Prigogine (); LCW (). Quantitatively, implies that
Hence, due to the creation processes (), the universe does not expand adiabatically as happens in the standard CDM model. Besides, since up to a constant factor one has , by inserting Eq. (11) into (4) a straightforward integration yields
Further, from Eq. (18), , and using the above expression one may write the entropy of the CDM particles like
where is the present entropy of the CDM fluid. Note that if the standard conserved quantities are recovered.
Iv Decelerating Parameter, Transition Redshift and Supernova Bounds
and inserting this result into (21) it follows that
For , this expression yields , while for we find
In Figure 3 we display the decelerating parameter as a function of the redshift as given by the above expressions. As remarked earlier, the existence of a transition redshift at late times depends exclusively on the parameter (compare Figs. 3a and 3b).
A simple relation uniting , and can be determined by taking . As one may check, Eq. (23) implies that
For the above expression reduces to
and the age of the Universe can be rewritten in terms of the transition redshift. One finds,
iv.1 Constraints from SNe Ia Observations
Let us now discuss the constraints from distant type Ia SNe data on the class of CDM accelerating cosmologies proposed here. Since can be determined from the Hubble Law and , the model has only two independent parameters, namely, and (see Eq. (22) for ).
The predicted distance modulus for a supernova at redshift , given a set of parameters , is
where and are, respectively, the apparent and absolute magnitudes, the complete set of parameters is , and stands for the luminosity distance (in units of megaparsecs),
with being a convenient integration variable, and the expression given by Eq. (22).
In order to constrain the free parameters of the model consider now the latest sample containing 182 Supernovas as published by Riess and coworkers Riess07 (). The best fit to the set of parameters can be estimated by using a statistics with
where is given by Eq. (29), is the extinction corrected distance modulus for a given SNe Ia at , and is the uncertainty in the individual distance moduli. By marginalizing on the nuissance parameter () we find and at of confidence level. The best fit adjustment occurs for values of and with and degrees of freedom. The reduced where (), thereby showing that the model provides a very good fit to these data.
In this paper we have proposed a flat cold dark matter cosmology whose late time acceleration is powered by an irreversible creation of CDM particles. In our scenario there is no dark energy, and, as such, the so-called coincidence problem is also absent. It should be stressed that does not need to be small in order to solve the age problem. Further, the transition from a decelerating to an accelerating regime at late times happens even if the matter creation is negligible during the radiation and considerable part of the matter dominated phase (this is equivalent to take in all the expressions). Therefore, like in flat CDM scenarios, there is just one free parameter, and the resulting model provides an excellent fit to the observed dimming of distant type Ia supernovae (see Figs. 4a and 4b). Note also that the flat model () with creation of CDM particles proposed here can easily be extended to include negative () and positive () spatial curvatures. The same happens with the inclusion of a small (conserved) baryonic component whose density parameter today is severely constrained by the primordial nucleosynthesis and WMAP results. In this case, the value of the transition redshift as derived in section IV will be slightly modified.
On other hand, the existence of such a model also means that the accelerating expansion does not represent a direct evidence for a non-zero cosmological constant or, more generally, to the existence of dark energy as usually assumed by many authors. Naturally, new constraints on the relevant parameters ( and ) from complementary observations need to be investigated in order to see whether the matter creation model proposed here provides a realistic description of the observed Universe. New bounds on these parameters coming from the background and perturbed equations in the presence of a conserved baryonic component will be discussed in a forthcoming communication.
Appendix A Particle Creation and Irreversibility
In this appendix we describe how the creation pressure given by Eq. (5) can be deduced by using the relativistic non-equilibrium thermodynamics. The idea is to show in a simplified way how an irreversible mechanism of quantum origin can be incorporated in the classical Einstein field equations.
A relativistic self-gravitating simple fluid endowed only with gravitational matter creation is characterized by an energy momentum tensor , a particle current , and an entropy current . In the homogeneous and isotropic case, these quantities satisfy the following relations:
where () means covariant derivative, is the creation pressure, is the particle number density, is the particle creation rate (from quantum gravitational origin) is the specific entropy (per particle), and is the entropy source. In what follows it is assumed that the particles spring up into space-time in such a way that they turn out to be in thermal equilibrium with the already existing ones. The entropy production is then due only to the scalar process of matter creation (bulk viscosity has been neglected). Naturally, for we shall expect that the creation pressure vanishes and so also the entropy production.
The basic aim here is to show how the second law of thermodynamics constrains the dependence of on and other quantities specifying the fluid. Following standard lines, the quantities , , and are related to the temperature by the Gibbs law
while the chemical potential is defined by the Euler’s relation
Finally, the case of adiabatic gravitational matter creation means that the entropy increases but the specific entropy remains constant (). Therefore, the above equation implies that with the creation pressure assuming the form adopted in the present work (cf. Eq. (5))
As should be expected, for , the creation pressure and entropy source vanish thereby recovering the perfect fluid description.
Appendix B Matter Creation Rate and the Transition Redshift
In this appendix we show a curious result, namely: the existence of a transition redshift, , at late time determines the simplest form of the matter creation rate. In order to show that we consider the evolution equation (see section III)
which means that the decelerating parameter () can be written as:
The above expression was first obtained by Zimdahl et al. ZSBP01 () using a different notation (see their Eq. (53)).
Now, by taking in the above expression one finds that , the value of the Hubble parameter at the instant of transition. At low redshifts it is natural to take it proportional to , say, , where the factor 3 is introduced for mathematical convenience and the constant parameter, in general, depends on the transition redshift (see Eq. (27)). Note also that the contribution can be thought as the first order correction of this quantity in powers of
and, therefore, we may write
which is the same expression appearing in section IV (see Eq. (21)). For , the resulting scenario was proposed by Lima, Germano and Abramo LGA96 () (see also Refs. LA99 ()) while for , it was first discussed by Zimdahl et al. ZSBP01 (). Clearly, the scenario proposed here is a combination of both approaches. Note also that only in the enlarged form, it may represent a possible solution to the old (and modified versions) of the coincidence problem (see the available space parameter in the () plane as shown in Fig. 4b).
Acknowledgements.The authors would like to thank V. Busti, J. V. Cunha, A. C. C. Guimarães, R. Holanda, J. F. Jesus and L. Sodré for helpful discussions. JASL is partially supported by CNPq and FAPESP under Grants 304792/2003-9 and 04/13668-0, respectively. RCS is supported by CNPq No. 15.0293/2007-0 and EFS by CAPES (Brazilian Research Agencies).
- (1) A. G. Riess et al., Astron. J. 116, 1009 (1998); S. Perlmutter et al., Astrophys. J. 517, 565 (1999); P. Astier et al., Astron. Astrophys. 447, 31 (2006);
- (2) A. G. Riess et al., Astrop. J. 659, 98 (2007).
- (3) P. de Bernardis, et al., Nature 404, 955 (2000); D. N. Spergel, et al., Astrophys. J. Suppl. Ser. 148, 175 (2003); D. N. Spergel et al. Astrophys. J. Suppl. Ser. 170, 377 (2007).
- (4) J. S. Dunlop et al., Nature 381, 581 (1996); L. Krauss, Astrophys. J. 480, 466 (1997); J. S. Alcaniz and J. A. S. Lima, Astrophys. J. 521, L87 (1999), [astro-ph/9902298, ibdem, Astrophys. J. 550, L133 (2001), [astro-ph/0101544].
- (5) P. J. E. Peebles and B. Ratra Rev. Mod. Phys. 75, 559 (2003); T. Padmanabhan, Phys. Rept. 380, 235 (2003); J. A. S. Lima, Braz. Journ. Phys., 34, 194 (2004), [astro-ph/0402109]; E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D15, 1753 (2006).
- (6) S. Weinberg, Rev. Mod. Phys., 61, 1 (1989).
- (7) P. J. E. Peebles, Astrophys. J. Lett. 325, L17 (1988); B. Ratra and P. J. E. Peebles, Phys. Rev D37, 3406 (1988); R. R. Caldwell, R. Dave, P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998); C. Wetterich, Nucl. Phys. B 302 668 (1998); R. Caldwell, R. Dave, P. Steinhardt, Phys. Rev. Lett. D 59, 123504 (1999); R. Caldwell, Braz. J. Phys. Braz. J. Phys. 30, 215 (2000); P. G. Ferreira and M. Joyce, Phys. Rev. D58, 023503(1998); A. Albrecht and C. Skordis, Phys. Rev. Lett. 84, 2076 (2000); C. Rubano et al., Phys. Rev. D69, 103510 (2004); V. Faraoni and M. N. Jensen, Class. Quant. Grav. 23, 3005 (2006); F. C. Carvalho et al., Phys. Rev. Lett. 97, 081301 (2006), [astro-ph/0608439].
- (8) M. Özer and M. O. Taha, Phys. Lett. B 171, 363 (1986); K. Freese et al., Nucl. Phys. B287, 797 (1987); W. Chen and Y-S. Wu, Phys. Rev. D 41, 695 (1990); D. Pavón, Phys. Rev. D 43, 375 (1991); J. C. Carvalho, J. A. S. Lima and I. Waga, Phys. Rev. D 46, 2404 (1992); J. A. S. Lima and J. M. F. Maia, Phys. Rev D 49, 5597 (1994); J. C. Carvalho and J. A. S. Lima, Gen. Rel. Grav. 26, 909 (1994); J. A. S. Lima and M. Trodden, Phys. Rev. D 53, 4280 (1996),[astro-ph/9508049]; J. A. S. Lima, Phys. Rev. D 54, 2571 (1996), [gr-qc/9605055]; A. I. Arbab and A. M. M. Abdel-Rahman, Phys. Rev. D 50, 7725 (1994); J. M. Overduin and F. I. Cooperstock, Phys. Rev. D 58, 043506 (1998); R. G. Vishwakarma, Class. Quant. Grav. 17, 3833 (2000); V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373, 2000, [astro-ph/9904398].
- (9) M. V. John and K. B. Joseph, Phys. Rev. D 61, 087304 (2000); J. A. S. Lima and J. S. Alcaniz Mon. Not. Roy. Astron. Soc. 317, 893 (2000), [astro-ph/0005441]; J. M. F. Maia, J. A. S. Lima, Phys. Rev. D 60, 101301 (1999), [astro-ph/9910568]; M. K. Mak, J.A. Belinchon, and T. Harko, IJMPD 11, 1265 (2002); M. R. Mbonye, IJMPA 18, 811 (2003); J. V. Cunha and R. C. Santos, IJMP D 13, 1321 (2004),[astro-ph/0402169]; S. Carneiro and J. A. S. Lima, IJMPA 20 2465 (2005),[gr-qc/0405141]; I. L. Shapiro, J. Sola and H. Stefancic, JCAP 0501, 012 (2005); E. Elizalde, S. Nojiri, S.D. Odintsov and P. Wang, Phys. Rev. D 71, 103504 (2005); J. S. Alcaniz and J. A. S. Lima, Phys. Rev. D 72, 063516 (2005), [astro-ph/0507372]; A. Friaca, J. S. Alcaniz and J. A. S. Lima, Mon. Not. R. Astron. Soc. 362, 1295 (2005), [astro-ph/0504031]; J. F. Jesus, Gen. Rel. Grav. 40, 791 (2008), [astro-ph/0603142].
- (10) J. A. S. Lima and A. Maia, Phys. Rev. D 52, 5628 (1995); J. A. S. Lima and J. S. Alcaniz, Astron. Astrophys. 357 393 (2000); T. Padmanabhan and T. R. Choudhury, Mon. Not. Roy. Astron. Soc. 344, 823 (2003); P. T. Silva and O. Bertolami, Astrophys. J. 599, 829 (2003); J. A. S. Lima, J. V. Cunha and J. S. Alcaniz, Phys. Rev. D 68, 023510 (2003), [astro-ph/0303388]; J. S. Alcaniz, J. A. S. Lima, and J. V. Cunha, Mon. Not. Roy. Astron. Soc. 340, L39 (2003), [astro-ph/0301226]; Z.-H. Zhu and M.-K. Fujimoto, Astrophys. J. 585, 52 (2003); L. P. Chimento, Phys. Rev. D 69, 123517 (2004); S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 70, 043531 (2004); Y. Wang and M. Tegmark, Phys. Rev. Lett. 92, 241302 (2004); T. R. Choudhury and T. Padmanabhan, Astron. Astrophys. 429 807 (2005); D. Rapetti, S. W. Allen and J. Weller, Mon. Not. Roy. Astron. Soc. 360, 546 (2005).
- (11) R. Colistete Jr. and J. C. Fabris, Class. Quant. Grav. 22, 2813 (2005); B. Feng, X. Wang and X. Zhang, Phys. Lett. B 607, 35 (2005); R. Lazkoz, S. Nesseris and L. Perivolaropoulos, [astro-ph/0503230]; A. Shafieloo, U. Alam, V. Sahni, A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. 366, 1081 (2006); J. V. Cunha, J. S. Alcaniz and J. A. S. Lima, Phys. Rev. D 69, 083501 (2004), [astro-ph/0306319]; B. Wang, Y. Gong, and E. Abdalla, Phys. Lett. B624, 141 (2005); L. R. Abramo and N. Pinto-Neto, Phys. Rev. D 73, 063522 (2006); J. D. Barrow and T. Clifton, Phys. Rev. D 73, 103520 (2006); B. Wang, C. Y. Lin, and E. Abdalla, Phys. Lett. B637, 357 (2006) T. Clifton and J. D. Barrow Phys. Rev. D 73 104022 (2006); J. V. Cunha, L. Marassi and R. C. Santos, IJMP D 16, 403 (2007), [astro-ph/0306319]; R. A. Sussman, arXiv:0801.3324[gr-qc]; R. C. Santos and J. F. Jesus (2008), arXiv:0806.2822[astro-ph].
- (12) L. Amendola, Phys. Rev. D 62, 043511 (2000); J. M. F. Maia and J. A. S. Lima, Phys. Rev. D 65, 083513 (2002); T. Koivisto, Phys. Rev. D 72, 043516 (2005); S. Lee, G. C. Liu and K. W. Ng, Phys. Rev. D 73, 083516 (2006); M. C. Bento, O. Bertolami and N. M. C. Santos, arXiv:gr-qc/0111047; O. Bertolami, F. Gil Pedro and M. Le Delliou, Phys. Lett. B 654, 165 (2007); F. E. M. Costa, J. S. Alcaniz and J. M. F. Maia, arXiv:0708.3800 [astro-ph]; E. Abdalla, L. R. W. Abramo, L. Sodre Jr. and B. Wang, arXiv:0710.1198 [astro-ph]
- (13) A. Blanchard, M. Douspis, M. Rowan-Robinson, and S. Sarkhar, Astron. and Astrophys. 412, 35 (2003); S. Sarkhar (2007), [astro-ph0710.5307v1].
- (14) L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1985).
- (15) A. D. Sakharov, Collected Scientific Works (Dekker, New York, 1982); P. Danielewicz, Nucl. Phys. A 314, 465 (1979).
- (16) Ya. B. Zeldovich, JETP Lett. 12, 307 (1970).
- (17) F. Hoyle and J. V. Narlikar, Proc. Roy. Soc. A 282, 191 (1964); ibdem, 290, 143 (1966); see also Action at a Distance in Physics and Cosmology, 1974 (New York, Freeman).
- (18) L. Parker, Phys. Rev. Lett. 21, 562 (1968); Phys. Rev. 183, 1057 (1969); S. A. Fulling, L. Parker and B. L. Hu, Phys. Rev. 10, 3905, (1974); B. L. Hu and L. Parker, Phys. Rev. 17, 933, (1978); N. J. Papastamation and L. Parker, Phys. Rev. 19, 2283, (1979).
- (19) N. D. Birrell and P. C. Davies, Quantum Fields in Curved Space, Cambridge Univ. Press, Cambridge, (1982).
- (20) G. L. Murphy, Phys. Rev. D 48, 4231 (1973).
- (21) B. L. Hu, Phys. Lett. A90, 375 (1982); Adv. Astrop. 1, 23 (1983); J. D. Barrow, Phys. Lett B180, 335 (1986); M. Morikawa and M. Sasaki, Phys. Lett. B 165, 59 (1985); T. Padmanabhan and S. M. Chitre, Phys. Lett. A 120, 433 (1987); J. A. S. Lima, R. Portugal and I. Waga, Phys. Rev. D 37, 2755 (1988).
- (22) I. Prigogine et al., Gen. Rel. Grav., 21, 767 (1989).
- (23) J. A. S. Lima, M. O. Calvão, I. Waga, “Cosmology, Thermodynamics and Matter Creation”, Frontier Physics, Essays in Honor of Jayme Tiomno, World Scientific, Singapore (1990), [arXiv:0708.3397]; M. O. Calvão, J. A. S. Lima, I. Waga, Phys. Lett. A162, 223 (1992).
- (24) J. A. S. Lima, A. S. M. Germano, Phys. Lett. A 170, 373 (1992).
- (25) W. Zimdahl and D. Pavón, Phys. Lett. A 176, 57 (1993); W. Zimdahl and D. Pavón, Mon. Not. R. Astr. Soc. 266, 872 (1994); W. Zimdahl and D. Pavón, GRG 26, 1259 (1994); J. Gariel and G. Le Denmat, Phys. Lett. A 200 11 (1995).
- (26) J. A. S. Lima, A. S. M. Germano and L. R. W. Abramo, Phys. Rev. D 53, 2953 (1996), [gr-qc/9511006].
- (27) J. A. S. Lima and J. S. Alcaniz, Astron. Astrophys. 348, 1 (1999), [astro-ph/9902337]; J. S. Alcaniz and J. A. S. Lima, Astron. Astrophys. 349, 729 (1999), [astro-ph/9906410].
- (28) R. Silva, J. A. S. Lima and M. O. Calvão, Gen. Rel. Grav. 34, 865 (2002), [gr-qc/0201048].
- (29) R. A. Susmann, Class. Q. Grav. 11, 1445 (1994); J. Gariel, M. Cissoko, Class. Quant. Grav. 13, 2977 (1996); L. R. W. Abramo and J. A. S. Lima, Class. Quantum Grav. 13, 2953 (1996), [gr-qc/9606064]; V. B. Johri and S. K. Pandey, Int. J. Theor. Phys. 38, 1981 (1999); T. Harko and M. K. Mak, Astrophys. Space Sci. 253, 161 (1997); J. A. S. Lima, Gen. Rel. Grav. 29, 805 (1997), [gr-qc/9605056]; K. Desinkhan, Gen. Rel. Grav. 29, 435 (1997); Class. Quant. Grav. 16, 2741 (1999); J. A. S. Lima and L. R. W. Abramo, Phys. Lett A, 257, 123 (1999); [gr-qc/9606067]; M. K. Mak and T. Harko, Class. Quant. Grav. 16, 4085 (1999); ibdem, IJMP D 8 607 (1999); J. A. S. Lima and U. F. Wichoski, Phys. Lett. A 262, 103 (1999), [astro-ph/9708215]; W. Zimdahl, Phys. Rev. D 61 083511 (2000); J. A. S. Lima, A. I. Silva and S. M. Viegas, Mon. Not. R. Astron. Soc. 312, 747 (2000); G. Montanni, Class. Quant. Grav. 18, 193 (2001); G. P. Singh, R. V. Deshpande, T. Singh, Astrop. Space Sci. 282, 489 (2002); J. A. S. Lima and J. A. E. Carrilo, [astro-ph/0201168] (2002); M. de Campos, Gen. Rel. Grav. 35, 899 (2003); M. de Campos and J. A. Souza, Astron. Astrophys. 422, 401 (2004).
- (30) W. Zimdhal, D. J. Schwarz, A. B. Balakin and D. Pavón, Phys. Rev. D 64, 063501 (2001).
- (31) M. P. Freaza, R. S. de Souza and I. Waga, Phys. Rev. D 66, 103502 (2002).
- (32) C. S. Camara, J. C. Carvalho and M. R. G. Maia, IJMP D 16 427 (2007); Y. Quiang, T-J. Zhang and Z-L. Yi, Astrop. Space Sci. 311, 407 (2007); S. S. Costa and M. Makler (2007), astro-ph/0702418.
- (33) M. J. D. Assad and J. A. S. Lima, Gen. Rel. Grav. 20, 527 (1988); J. A. S. Lima, Am. J. Phys. 69, 1245 (2001), [astro-ph/0109215].