Eur. Phys. J. C (2018) 78:58
https://doi.org/10.1140/epjc/s10052-018-5539-0
Regular Article - Theoretical Physics
Thermodynamics in modified Brans–Dicke gravity with entropy
corrections
Shamaila Rani
1,a
, Abdul Jawad
1,b
, Tanzeela Nawaz
1,c
, Rubab Manzoor
2,d
1
Department of Mathematics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan
2
Department of Mathematics, University of Management and Technology, Johar Town Campus, Lahore 54782, Pakistan
Received: 10 November 2017 / Accepted: 11 January 2018 / Published online: 23 January 2018
© The Author(s) 2018. This article is an open access publication
Abstract In this paper, we investigate the thermodynamics
in the frame-work of recently proposed theory called mod-
ified Brans–Dicke gravity (Kofinas et al. in Class Quantum
Gravity 33:15, 2016). For this purpose, we develop the gen-
eralized second law of thermodynamics by assuming usual
entropy as well as its corrected forms (logarithmic and power
law corrected) on the apparent and event horizons. In order to
analyzed the clear view of thermodynamic law, the power law
forms of scalar field and scale factor is being assumed. We
evaluate the results graphically and found that generalized
second law of thermodynamics holds in most of the cases.
1 Introduction
Dark energy is one of the fascinated issue of modern cos-
mology that has encouraged the modification of Einstein
Hilbert action. The modified gravitational part of Einstein
Hilbert action leads to the notion of modified theories of
gravity which modify the dynamic of the universe at large dis-
tances. In another scenario, modified matter part of Einstein
Hilbert action results dynamical models such as cosmolog-
ical constants, quintessence, k-essence, Chaplygin gas and
holographic dark energy (HDE) models [1–9]. Moreover,
several modified theories of gravity are f (R), f (T ) [10–
15], f (R, T ) [16,17], f (G) [18–23], f (T , T
G
) [24–26],
f (T, T) [27,28] (where R is the curvature scalar, T denotes
the torsion scalar, T is the trace of the energy momentum
tensor and G is the invariant of Gauss–Bonnet defined as
G = R
2
− 4R
μν
R
μν
+ R
μνλσ
R
μνλσ
). For clear review of
DE models and modified theories of gravity, see the reference
a
e-mails: shamailatoor.math@yahoo.com;
drshamailarani@ciitlahore.edu.pk;
b
e-mails: jawadab181@yahoo.com; abduljawad@ciitlahore.edu.pk;
c
e-mail: tanzeela_nawaz@yahoo.com;
d
e-mail: rubab.manzoor@umt.edu.pk
[9]. Some authors [29–46] have also discussed various DE
models in different frameworks and found interesting results.
The scalar tensor models of modified gravity have taken
remarkable attention. The reason is that scalar fields for con-
sistent condition appear in different branches of theoretical
physics such as, the low energy limit of the string theory
leads to a scalar degree of freedom. The Brans–Dicke (BD)
theory [47,48] is a prototype of scalar tensor gravity which is
based upon Dirac hypothesis. It relates scalar field (φ) with
dynamical gravitational constant (G =
G
0
φ
) and involves a
tuneable constant coupling parameter ω. The scalar field is
a fundamental feature of this gravity which is considered as
a dark energy candidate. The parameter ω can adjust results
according to the requirement and in the limit ω →∞,it
reduces BD theory into general relativity (GR).
The standard BD theory remains unable to probe cos-
mic evolution accurately. In this context, many researchers
have generalized BD theory in different scenario like, self-
interacting potential model, model having time- dependent
coupling parameter (ω) [49–51]. Brans–Dicke cosmological
models with constant deceleration parameter in the form of
particle creation [52]. The Friedman models (with zero curva-
ture) under the effects of time dependent bulk viscosity [53].
Similarly, the modification of BD theory also involve interac-
tion with dark matter, such as the concept of dissipative cold
dark matter in BD gravity [54,55], model representing trans-
fer of energy between BD gravity and dark matter [56,57].
Recently, Kofinas et al. [58] introduced the most generalized
or corrected form of BD gravity by relaxing the standard
conservation law of matter contribution (energy momentum
of matter). He used a new dimensionless parameter ν in the
theory. This new version of BD gravity has explored cos-
mic evolution in accords to observational data by involving
matter-scalar field interaction.
In literature, the relation between gravitation and ther-
modynamics has been discussed extensively. Moreover, by
inspiring the black hole theory, there is a deep connection
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