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

2. 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

3. 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.
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[1] S. W. Hawking, Commun. Math. Phys. 43 (1975) 199.
[2] J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333. 3

4. 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]
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[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

5. 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.
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6. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
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.

7. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
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.

8. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
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 (2008) 010.
[6] X. L. Liu and X. Zhang, Commun. Theor. Phys. 52 (2009) 761.
[7] A. Sheykhi, Phys. Rev. D 81 (2010)
[8] K. Karami et al., Gen. Relativ. Gravit. 43 (2011) 27.

9. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
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)

10. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
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)

11. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
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.

12. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
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)

13. The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe2
The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe

14. 2 The Evolutions of Interacting NADE in Nonflat Brans–Dicke Universe
The smaller the parameter is, the earlier can cross the phantom divide

15. The First Law and The Generalized Second Law of Thermodynamics3
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

16. 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)

17. 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

18. 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)

19. 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

20. 3
The First Law and The Generalized Second Law of Thermodynamics

21. Interacting Entropy-Corrected4
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.

22. Interacting Entropy-Corrected4
Interacting Entropy-Corrected
In the noninteracting case, we get .
(33)
Accordingly, we can further obtain
(34)
and
(35)

23. Interacting Entropy-Corrected4
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.

24. 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.

25. 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.

26. Thank you !