Thermodynamical behaviors of nonflat Brans-Dicke gravity with interacting new agegraphic dark energy Xue Zhang Department of Physics, Liaoning Normal University, Dalian, China

Outline Motivation The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe The First Law and the generalized second law of Thermodynamics Interacting Entropy-Corrected Conclusion

1 Motivation ★ Since the discovery of black hole thermodynamics in 1970s, [1, 2] physicists have been speculating that there should be some relations between black hole thermodynamics and Einstein equations, because black hole solutions are derived from the Einstein equations. -- the surface gravity of the cosmological horizon Hawking temperature entropy -- the cosmological horizon area thermodynamical quantities geometric quantities Thermodynamic behavior have been extensively discussed in various gravity models, such as Gauss–Bonnet gravity, Lovelock gravity and f(R) gravity. 海南三亚。 装机量？规模？ [1] S. W. Hawking, Commun. Math. Phys. 43 (1975) 199. [2] J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333. 3

1 Motivation ★ The Brans–Dicke theory is a simple extension of the Einstein’s GR theory. [3] ① where the gravitational constant is replaced with the inverse of a time-dependent scalar field, namely, , and this scalar field couples to gravity with a coupling constant . ★ The new agegraphic dark energy (NADE) has the energy density [4] 海南三亚。 装机量？规模？ ② . [3] C. Brans and R. H. Dicke, Phys. Rev. 124 (1961) 925. [4] H. Wei and R. G. Cai, Phys. Lett. B 660 (2008) 113. 4

1 Motivation Our motivations and purposes ★ whether the field equation of the Brans–Dicke gravity can be cast to the form of the first law and the generalized second law of thermodynamics; ★ what the thermodynamical interpretation of the interaction between NADE and DM would be. 海南三亚。 装机量？规模？ 5

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe The action of the Brans–Dicke gravity is (1) where is the Brans–Dicke scalar field, is the scalar curvature, and is the dimensionless parameter and is the Lagrangian of matter fields. The nonflat FRW metric is given by (2) where is the scale factor of the universe with being cosmic time, is the metric of two-dimensional sphere with unit radius, the spatial curvature constant correspond to a closed, flat and open universe, respectively.

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe Varying action (1) with respect to metric (2) for a universe filled with matter and dark energy yields the following two independent equations: (3) (4) where is the Hubble parameter, and , and are respectively the energy density and pressure of dark energy, as well as is the energy density of pressureless DM.

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe We assume that Brans–Dicke field can be described as a power law of the scale factor [5,6,7,8] (5) , Taking the derivative with respect to time of the relation, we get (6) (7) The Friedmann equation (3) becomes (8) [5] J. H. He and B. Wang, J. Cosmol. Astropart. Phys. 0806 (2008) 010. [6] X. L. Liu and X. Zhang, Commun. Theor. Phys. 52 (2009) 761. [7] A. Sheykhi, Phys. Rev. D 81 (2010) 023525. [8] K. Karami et al., Gen. Relativ. Gravit. 43 (2011) 27.

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe The critical energy density and the energy density of the curvature can be defined as (9) (10) The fractional energy densities are also defined as usual , , . (11) The Friedmann equation (8) can be rewritten as (12)

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe The NADE density can be written as (13) where (14) The fractional energy densities of NADE can be expressed by (15) where (16)

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe We consider the FRW universe filled with dark energy and pressureless matter and use a phenomenological term to describe the energy exchange between dark energy and DM. The corresponding continuity equations can be written as (17) (18) Where , and index is taken as 0, 1, 2, 3.

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe The equation of state for the dark energy can be expressed as . (19) Moreover, by complicated calculations the equation of motion for can be given as (20) where the prime denotes the derivative with respect to . Also, the deceleration parameter can be expressed as (21)

The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe

2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe The smaller the parameter is, the earlier can cross the phantom divide .

The First Law and The Generalized Second Law of Thermodynamics 3 The First Law and The Generalized Second Law of Thermodynamics (1) The First Law of Thermodynamics By means of the spherical symmetry, the metric (2) can be rewritten as , (22) where

3 The First Law and The Generalized Second Law of Thermodynamics The explicit evaluation of the apparent horizon gives the apparent horizon radius . (23) The surface gravity of the apparent horizon is defined by (24) By means of the relation (23) and taking differential to time, we can obtain (4)

3 The First Law and The Generalized Second Law of Thermodynamics , , , . , By complicated calculations we can give the following relation: , (25) where is the energy production term grown up internally due to the nonequilibrium setup within the Brans–Dicke theory at the apparent horizon of FRW universe. and is the effective entropy. the standard form of the first law of thermodynamics

3 The First Law and The Generalized Second Law of Thermodynamics The entropy of NADE and matter inside the apparent horizon can be obtained by the Gibbs’ equation (26) (27) so we obtain (28) (29)

3 The First Law and The Generalized Second Law of Thermodynamics The entropy of the apparent horizon in Brans–Dicke theory can be written as , then we have (30) By complicated calculations we can get the GSL in Brans–Dicke theory as (31) where .

3 The First Law and The Generalized Second Law of Thermodynamics

Interacting Entropy-Corrected 4 Interacting Entropy-Corrected The entropy has been assigned an extra logarithmic correction [9] (32) where the superscript (0) denotes the noninteracting case and the superscript (1) denotes the logarithmic correction in interacting case. [9] A. Sheykhi and M. R. Setare, Mod. Phys. Lett. A 26 (2011) 1897.

Interacting Entropy-Corrected 4 Interacting Entropy-Corrected In the noninteracting case, we get . (33) Accordingly, we can further obtain (34) and (35)

Interacting Entropy-Corrected 4 Interacting Entropy-Corrected Finally, by using (28) the interaction term can be derived in the view of thermal fluctuation as follows: (36) In the limiting cases of and , as well as , all previous results of the NADE in Einstein gravity are restored.

5 Conclusion (1) By considering the interacting term , we have derived the evolutions of the fractional energy density , the equation of state of dark energy and the deceleration parameter . behaves like quintessence dark energy behaves like quintom dark energy Compared to the standard form of the first law of thermodynamics, the Friedmann equation in Brans–Dicke gravity can be cast to the similar form, , at the apparent horizon of FRW universe.

5 Conclusion (3) the GSL of thermodynamics always holds for NADE in Brans–Dicke universe the GSL is valid only when , which is dependent onthe parameter . (4) By assuming that the universe is in thermal equilibrium and considering the interaction between the NADE and DM, we have given an expression of the interaction term by means of the logarithmic correction to the equilibrium entropy.

Thank you !