1. The document describes a research proposal on studying various aspects of interacting dark energy under modified gravity theories. 2. The objectives of the proposed research include studying thermodynamic quantities in generalized gravity theories, validity of generalized second law of thermodynamics in different expansions of the universe, interacting dark energy models in loop quantum cosmology and fractional action cosmology, and dark energy reconstruction in modified gravity theories. 3. The proposal discusses approaches to account for late-time cosmic acceleration through dark energy and modified gravity theories and reviews relevant literature.

1. Study of the Various Aspects of Interacting Dark
Energy under Modified Gravity Theories
SERB Sanction Order No and date:
SR/FTP/PS-167/2011 dated 10-09-2012
DoS: 28th September, 2012
PI (Name & Address): Dr. Surajit Chattopadhyay
Pailan College of Management and Technology,
Sector – 1, Phase I, Bengal Pailan Park,(off D.H. Road), Joka, Kolkata 700 104

2. 1. To study the thermodynamic quantities of the universe in generalized gravity theories like Horava-Lifshitz,
Brans-Dicke, f(R), f(T) and to discern how the various thermodynamic quantities behave in the evolution of the
universe governed by generalized gravity theories.
2. To study the validity of generalized second law of thermodynamics in the emergent, intermediate and
logamediate expansions of the universe bounded by the Hubble, apparent, particle and event horizons in
fractional action cosmology (FAC). In this research I propose to examine the validity of the generalized second
law using as well as without using the first law of thermodynamics.
3. To study interacting dark energy models (with special emphasis on generalized holographic and Ricci dark
energies) in loop quantum cosmology (LQC) and in fractional action cosmology (FAC). In such interactions I
propose to reconstruct the model parameters and subsequently study the thermodynamic properties in emergent,
logamediate, intermediate and power law expansions.
4. To do the cosmological reconstruction of various dark energy candidates in anisotropic universe governed by
modified gravity theories and various choices of scale factors.
5. To study the dark energy based on generalized uncertainty principle. In the case of this dark energy I propose to
study the validity of the generalized second law of thermodynamics in the Einstein as well as in the modified
gravity theories.Also, I propose to carry out the cosmographic analysis this dark energy in emergent,
intermediate and logamediate expansions of the universe.
6. Besides the discussion of the dynamics of DE with our general coupling to DM in the universe described by the
Einstein theory, we will also extend our investigation to the Loop Quantum Cosmology (LQC). I propose to
consider interacting dark energy models in LQC as dynamical systems. In light of this, I propose to investigate
behaviors of various model parameters.
Approved
Objectives

3. Approaches to account for the late time cosmic acceleration fall into two representative categories: One is to
introduce “dark energy” in the right-hand side of the Einstein equation in the framework of general relativity (for
reviews on dark energy, see Copeland et al., 2006;Bamba et al., 2012). The other is to modify the left-hand side of
the Einstein equation, called as a modified gravitational theory. e.g., f(R) gravity (for reviews, see Nojiri and
Odintsov, 2011; Clifton et al., 2012).
Reviews on dark energy:
E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15,
1753 (2006). doi:10.1142/S021827180600942X
K. Bamba, S. Capozziello, S. Nojiri, S.D. Odintsov, Astrophys.
Space Sci. 342, 155 (2012). doi:10.1007/s10509-012-1181-8
Reviews on modified gravity:
T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rep. 513,
1 (2012). doi:10.1016/j.physrep.2012.01.001
S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59 (2011). doi:10.1016/
j.physrep.2011.04.001
Cosmic observations from Supernovae Ia (SNe Ia) (Perlmutter
et al. 1999; Riess et al. 1998) have implied that the expansion
of the universe is accelerating at the present stage.
A.G. Riess et al., Astron. J. 116, 1009 (1998).
doi:10.1086/300499
S. Perlmutter, Astrophys. J. 517, 565 (1999).
doi:10.1086/307221

4. Viewing the modified gravity model as an effective description of the underlying theory of DE,
and
considering the various versions of the HDE (M. Li, PLB, 603, 1-5 (2004)) as pointing in the direction
of the underlying theory of DE,
it is interesting to study how the modified-gravity can describe the various forms of HDE densities as
effective theories of DE models.
This motivated us to establish the different models of modified gravity according to the HDEs.
X. Wu , Z-H. Zhu, Phys. Lett. B, 660, 293 (2008) W. Yang et al., Mod. Phys. Lett. A 26, 191 (2011).
M. R. Setare, Int. J. Mod. Phys. D, 17, 2219 (2008). L. N. Granda, Int. J. Mod. Phys. D, 18, 1749 (2009).
K. Karami, M.S. Khaledian, JHEP 03, 086 (2011). M.R. Setare, Phys. Lett. B, 644 , 99 (2007) .
Interacting DE models have been discussed in :
M.R. Setare, Physics Letters B 644 (2007) 99–103
M.R. Setare · D. Momeni · S.K. Moayedi, Astrophys Space Sci (2012) 338:405–410
M.R. Setare, Int J Theor Phys (2010) 49: 759–765
N. Mahata and S. Chakraborty, Modern Physics Letters A, Vol. 30, No. 2 (2015) 1550009
Interacting DE models are favoured by observed data obtained from the Cosmic Microwave
Background (CMB) (Olivares et al., Phys.Rev. D 71, 063523 (2005)) and matter distribution at large
scales (Olivares et al., Phys.Rev. D 74, 043521 (2006))

5. Reconstruction of modified gravity theories done so far
Holographic
Reconstruction of f (T ) gravity
Modified field equation
where
Holographic dark energy
Future event horizon
In the first and second modified field equations
we replace ρ by ρΛ and we get
Solving which we get
f(T) tends
to 0 as T
tends to 0
Squared speed of sound
Astrophys Space Sci (2013) 344:269–274
(1) (2)

6. QCD-ghost reconstruction of f(T) W<-1/3 that
is consistent with
DE
behavior
EoS behaves like
phantom
(for k=0)
Case I
(N. Ohta, PLB, 695, 41-44)

7. Case II
From conservation equation:
f(T)→0 as T→0 vs
2<0
SEC is violated
EoS tends
To -1 and
Behaves like
quintessence
Eur. Phys. J. Plus (2014) 129: 82

8. Holographic reconstruction of modified f (R) Horava-Lifshitz gravity
Modified
field
equations
Future event horizon is the
enveloping horizon
Holographic DE
EoS parameter
EoS parameter tends to -1.
initially, and roughly up to t ≈ 0.9, v2
s
stays at positive level and after that it
shifts to negative level. This indicates
that the holographic-modified f (R)
Horava-Lifshitz gravity is classically
stable for the present universe and in
later stages it becomes unstable.
Astrophys Space Sci (2013) 346:273–278
(M. Chaichian et al., CQG27:185021,2010 )

9. New holographic reconstruction of modified f (R) Horava-Lifshitz gravity
n = 2 (red), n =
2.2
(green), n = 2.4
(blue), n = 2.6
(black)
EoS crosses
Phantom
boundary
ΛCDM is
attainable
We may remark that r–s plane of the
reconstructed model is the combination
of all possible existing well-known
regions which is an interesting feature.
Astrophys Space Sci (2014) 353:293–299
(Karami and Fehri, PLB, 2010, 684,61-68 )

10. Modified holographic Ricci reconstruction of
modified f (R) Horava-Lifshitz gravity
Statefinder pairs {r, s} calculated for different sets of
parameters have led to trajectories in the {r – s} plane
reaching the fixed point {r = 1, s = 0} corresponding to
ΛCDM phase of the universe.
Quintessence
Can. J. Phys. 92: 200–205 (2014)

11. New agegraphic Reconstruction of f(G) gravity
Eur. Phys. J. Plus (2013) 128: 88
red for n
= 2.4,
green for
n = 2.6,
blue for n
= 2.8,
black for
n = 3
Modified field equations
Considering
New agegraphic DE where
For n=3, the EoS
Parameter is
transiting from
Quintessence to
phantom
{r=1,s=0}
ΛCDM

12. Pilgrim dark energy reconstruction of f (T,TG) cosmology
With a=a0tm form of scale factor
Under reconstruction methodology
Astrophys Space Sci (2014) 353:279–292

13. Transition from
quintessence to
phantom
Phantom
Another case: H=H0+H1/t
H=H0+H1/t
H=H0+H1/t
f→0 as t→0
f→0 as t→0
S=2
Quintessence
Transition from
quintessence to phantom
In case of s = 2,
we have taken n
= 0.91, H1 = 2
(red), 2.2
(green), 2.3
(blue) and H0 =
0.5. For s =−2,
we
have taken n = 4,
H1 = 3 (red), 3.1
(green), 3.2
(blue) and
H0 = 0.89.
Astrophys Space Sci (2014) 353:279–292
a=a0tm
S=-2

14. Extended holographic reconstruction of chameleon Brans–
Dicke cosmology
We again differentiate it with respect to 𝑡 and get a new equation involving and ̇ . Using this in second field equation along with the said
ansatz, we get the following differential equation on 𝑓 with 𝑡 as the independent variable:
V
 f
f(ϕ)→0 as
ϕ→0
ISRN High Energy Physics
Volume 2013, Article ID 14615

15. Consequences of DE in modified gravity theories
Interacting Ricci Dark Energy in f(R, T) Gravity
Modified field Eq.
RDE Interacting DE
and DM
Reconstructed Hubble parameter
Transition from
Quintessence to
phantom
Transition
from deceleration
to acceleration
Transition
from matter to
DE domination
ΛCDM
dust
All other cosmological
Parameters are reconstructed
Based on reconstructed H
Proc. Natl. Acad. Sci., India,
Sect. A Phys. Sci. (2014) 84(1):87–93

16. DE with Higher Time Derivatives of Hubble Parameter in
Fractional Action Cosmology
α = 1.3,
θ= 4, γ = 6 and
ϵ= 0.002
Equation of state parameter is
approaching towards −1 boundary.
However, the boundary is never
crossed.
quintessence
This equation does not have any
analytical solution, for this reason we
decided to solve it numerically.
Int J Theor Phys (2014) 53:435–448
Journal of Theoretical and Applied
Physics 2013, 7:25

17. Holographic Ricci Dark Energy in Scalar
Gauss-Bonnet Gravity
Modified field equations
Interaction
We use ansatz
Emergent
logamediate
intermediate
Scale factor
Emergent
logamediate
intermediate
Deceleration
parameter
We observe that q always remains at negative level. Then, we
can conclude that an accelerated universe can be obtained for all
the three models of scale factor considered.
Int J Theor Phys
(2014) 53:2988–3013

18. With power law form of scale factor
phantom
accelerated phase
Eur. Phys. J. Plus (2014) 129: 31

19. kinetic k-essence dark energy model in fractional action
cosmology
p
Purely kinetic
K-essence
ΛCDM Quintessence
Derivative
with
respect to
x=ln a
The {w–w’} plot
Shows that the
{w–w’}
trajectory tends
towards the
point
{w = –1, w’ = 0}.
Can. J. Phys. 91: 844–849 (2013)

20. modified holographic Ricci dark energy in logarithmic f(T)
gravity
and field equation
Solved numerically
Can. J. Phys. 91: 351–354 (2013)
Indian J Phys (October 2013)
87(10):1053–1057

21. New holographic reconstruction of scalar-field dark-energy
models in the framework of chameleon Brans–Dicke
cosmology

22. Reconstruction of
quintessence DE model
Reconstruction of
tachyon DE model
Reconstruction of DBI-essence DE model
Eur. Phys. J. C (2014)
74:3080

23. Logarithmic Entropy-Corrected Holographic Dark Energy
in Horava-Lifshitz cosmology
Some of the parameters involved in the
equations used and derived in this work
have not a precise value yet. Future works
and considerations can be made fitting the
model studied in this work with available
cosmological data in order to have a better
comprehension of the parameters
involved.
Astrophys Space Sci (2013) 348:541–551
(LECHDE)
cut-off based on purely dimensional
grounds proposed by Granda &
Oliveros
Interaction

24. The scale factor obtained for the HDE withGO cut-off and
for the model with higher derivatives of H are similar to
those obtained in earlier studies. For this reason, we
conclude that, for suitable choices of the parameters
involved, the HDE model with GO cut-off and the model
with higher derivatives of the Hubble parameter H in the
framework of Chern–Simons modified gravity have the
same results obtained from the modified Chaplygin gas.
Modified
field equations
HDE models in dynamical Chern–Simons modified gravity Eur. Phys. J. C (2015) 75:44

25. New Agegraphic Dark Energy Model in the Framework of
Generalized Uncertainty Principle
Int J Theor Phys (2014) 53:495–509

26. Thermodynamics of the Dark
Energy Based on Generalized Uncertainty Principle time derivative of the
total entropy is staying
at positive level
throughout the
evolution of the
universe enveloped by
the apparent horizon.
Thus, the total entropy
of the universe is not
decreasing with cosmic
time and hence the
generalized second law
is valid for the
interaction under
consideration.
The entropy of the universe inside the
horizon can be related to its energy and pressure in
the horizon by the Gibbs equation
Int J Theor Phys (2013)
52:2496–2507

27. GSL of thermodynamics in f(T) gravity
The above results hold for k = −1, +
1 and 0. Therefore, our observations
are the following: i) the EoS
parameter has a phantom-like
behavior, ii) the accelerated
expansion of the universe is
realized and iii) the generalized
second law of thermodynamics
does not work here.
i) the EoS exhibits phantom-like
behavior; ii) the strong energy
condition is violated; iii) we get
accelerating universe; and iv)
generalized second law of
thermodynamics is violated here.
Here also the above results hold
irrespective of the choice of k.
i) the EoS exhibits phantom-like
behavior, ii) the strong
energy condition is violated, iii)
we get an accelerating universe
and iv) the GSL of
thermodynamics holds
irrespective of the curvature of
the universe.
Model I
Model II
Model III
Eur. Phys. J. Plus (2013) 128: 12

28. GSL of thermodynamics in QCD ghost
f (G) gravity
Astrophys Space Sci (2014) 352:937–942

29. Concluding remarks
•We presented reconstruction schemes for different modified gravity theories from
different versions of HDE and studied their cosmological consequences.
•We studied the stability of the background evolutions against small perturbations.
•We studied behavior of different scalar field models of interacting DE in modified
gravity and observed the reconstructed EoS parameters for quintessence, phantom or
quintom behavior.
•We have studied statefinder parameters and in most of the cases they could go
beyond ΛCDM model.
•We have studied GSL of thermodynamics for reconstructed modified gravities and
observed their validity is most of the cases.

30. Ongoing…..
Interacting scalar field models of DE in LQG
The effective modified Friedmann equation in a flat
universe is given by
We are considering interacting scalar field models of DE with
different interaction terms in the LQC.
Also I am working on
•Holographic Polytropic Dark
energy in modified gravity
theories.
•Reconstruction of scalar field
models using a DE model with
higher derivatives of the
Hubble parameter
•Interacting scalar field
models in Kaluza-Klein
cosmology

31. Yet to be done…..
•DE models in anisotropic universe.
•Dynamical system analysis related to interacting scalar
field DE models.
•χ2-test based fitting of model parameters from data.
Studies can be extended to…
Accretion of DE models on blackholes and wormholes

32. Publications in SCI journals from the project:
1. A. Pasqua, R. da Rocha, Surajit Chattopadhyay, “Holographic dark energy models and higher order
generalizations in dynamical Chern–Simons modified gravity”, Eur. Phys. J. C (2015) 75:44 (Latest IF=5.436)
2. Surajit Chattopadhyay, “Generalized second law of thermodynamics in QCD ghost f (G) gravity”, Astrophys
Space Sci (2014) 352:937–942 (Latest IF=2.401)
3. Surajit Chattopadhyay, "A Study on the Interacting Ricci Dark Energy in f(R, T) Gravity“, Proc. Natl. Acad. Sci.,
India, Sect. A Phys. Sci. (January–March 2014) 84(1):87–93 (Latest IF=0.179)
4. Surajit Chattopadhyay, A. Pasqua, M. Khurshudiyan, "New holographic reconstruction of scalar-field dark-energy
models in the framework of chameleon Brans–Dicke cosmology“, Eur. Phys. J. C (2014) 74:3080 (Latest
IF=5.436)
5. Surajit Chattopadhyay, "QCD ghost reconstruction of f(T) gravity in flat FRW universe“, Eur. Phys. J. Plus
(2014) 129: 82 (Latest IF=1.475)
6. A. Jawad, Surajit Chattopadhyay, "New holographic dark energy in modified f (R) Horava-Lifshitz gravity“,
Astrophys Space Sci (2014) 353:293–299 (Latest IF=2.401)
7. Surajit Chattopadhyay and A. Pasqua, “A study on modified holographic Ricci dark energy in modified f(R)
Horˇava–Lifshitz gravity”, Can. J. Phys. 92: 200–205 (2014) (Latest IF=0.928)
8. Surajit Chattopadhyay, A. Jawad, D. Momeni, R. Myrzakulov, "Pilgrim dark energy in f (T,TG) cosmology",
Astrophys Space Sci (2014) 353:279–292 (Latest IF=2.401)
9. A. Pasqua, Surajit Chattopadhyay, "On the Dynamics of Dark Energy with Higher Time Derivatives of Hubble
Parameter in El-Nabulsi Fractional Action Cosmology“, Int J Theor Phys (2014) 53:435–448 (Latest IF=1.188)

33. 10. A. Pasqua, Surajit Chattopadhyay, M. Khurshudyan, A. A. Aly, “Behavior of Holographic Ricci Dark Energy in Scalar
Gauss-Bonnet Gravity for Different Choices of the Scale Factor”, Int J Theor Phys (2014) 53:2988–3013 (Latest IF=1.188)
11. Surajit Chattopadhyay, A. Pasqua, A. A. Aly,“Interacting Ricci dark energy in scalar Gauss-Bonnet gravity”, Eur. Phys.
J. Plus (2014) 129: 31. (Latest IF=1.475)
12. A. Jawad, Surajit Chattopadhyay and A. Pasqua, “Power-law solution of new agegraphic modified f(R) Horava-
Lifshitz gravity”, Eur. Phys. J. Plus (2014) 129: 51. (Latest IF=1.475)
13. A. Pasqua, Surajit Chattopadhyay, “A Study on the New Agegraphic Dark Energy Model in the Framework of
Generalized Uncertainty Principle for Different Scale Factors”, Int J Theor Phys (2014) 53:495–509. (Latest IF=1.188)
14. R. Ghosh, A. Pasqua and Surajit Chattopadhyay “Generalized second law of thermodynamics in the emergent
universe for some viable models of f(T) gravity”, Eur. Phys. J. Plus (2013) 128: 12. (Latest IF=1.475)
15. Surajit Chattopadhyay, A. Pasqua, “Reconstruction of f (T ) gravity from the Holographic Dark Energy”, Astrophys
Space Sci (2013) 344:269–274. (Latest IF=2.401)
16. A. Pasqua, Surajit Chattopadhyay, “Logarithmic Entropy-Corrected Holographic Dark Energy in Hoˇrava-Lifshitz
cosmology with Granda-Oliveros cut-off”, Astrophys Space Sci (2013) 348:541–551. (Latest IF=2.401)
17. A.Jawad, Surajit Chattopadhyay, A. Pasqua,“A holographic reconstruction of the modified f (R) Horava-Lifshitz
gravity with scale factor in power-law form”, Astrophys Space Sci (2013) 346:273–278. (Latest IF=2.401)
18. A. Pasqua, Surajit Chattopadhyay , “Theoretical constraints on kinetic k-essence dark energy model in El-Nabulsi
fractional action cosmology ”, Can. J. Phys. 91: 844–849 (2013). (Latest IF=0.928)
19. A. Pasqua, Surajit Chattopadhyay, “A study on the modified holographic Ricci dark energy in logarithmic f(T)
gravity”, Can. J. Phys. 91: 351–354 (2013). (Latest IF=0.928)
20. A. Pasqua, Surajit Chattopadhyay, and I. Khomenko, “ A reconstruction of modified holographic Ricci dark energy in
f(R, T) gravity”, Can. J. Phys. 91: 632–638 (2013). (Latest IF=0.928)

34. 21. A. Jawad, Surajit Chattopadhyay and A. Pasqua“Reconstruction of f(G) gravity with the new agegraphic dark-energy
model”, Eur. Phys. J. Plus (2013) 128: 88. (Latest Impact Factor=1.475)
22. Surajit Chattopadhyay, A. Pasqua, “Various aspects of interacting modified holographic Ricci dark energy”, Indian J
Phys (October 2013) 87(10):1053–1057. (Latest Impact Factor=1.785)
23. Surajit Chattopadhyay, A. Pasqua, “Holographic DBI-Essence Dark Energy via Power-Law Solution of the Scale
Factor”, Int J Theor Phys (2013) 52:3945–3952. (Latest Impact Factor=1.188)
24. A. Pasqua, Surajit Chattopadhyay, I. Khomenko“On the Stability and Thermodynamics of the Dark Energy Based on
Generalized Uncertainty Principle”, Int J Theor Phys (2013) 52:2496–2507. (Latest Impact Factor=1.188)
Publications in non-SCI journals from the project:
1. Surajit Chattopadhyay, “Extended Holographic Ricci Dark Energy in Chameleon Brans-Dicke Cosmology”, ISRN
High Energy Physics, Volume 2013, Article ID 414615 (2013) .
2. Surajit Chattopadhyay,“A Study on Ricci Dark Energy in Bulk-Brane Interaction”, ISRN High Energy Physics,
Volume 2013, Article ID 895614 (2013) .
3. Surajit Chattopadhyay, A. Pasqua, “Reconstruction of modified holographic Ricci dark energy in El-Nabulsi fractional
action cosmology”, Journal of Theoretical and Applied Physics (2013), 7:22.
4. R. Ghosh, A. Pasqua, Surajit Chattopadhyay, “Behavior of interacting Ricci dark energy in logarithmic f (T) gravity”,
Journal of Theoretical and Applied Physics (2013), 7:48.
5. U. Debnath, Surajit Chattopadhyay, M. Jamil, “Fractional action cosmology: some dark energy models in emergent,
logamediate, and intermediate scenarios of the universe”, Journal of Theoretical and Applied Physics (2013), 7:25.

35. Publications in peer-reviewed conference proceedings:
1. Surajit Chattopadhyay,“Restrictions on Brans-Dicke parameter ω in the case of holographically
reconstructed chameleon Brans-Dicke cosmology”, Published in the Proceedings of ICMS 2014 jointly
organized by Sathyabama University, Chennai, IMSc, Chennai and University of Central Florida, Chennai
during July 17-19, 2014; Published by Elsevier India (Year of publication 2014), pp 592-594, ISBN 978-93-
5107-261-4.
2. Surajit Chattopadhyay, S. Roy, “Dependence of Brans-Dicke parameter on scalar field”, In: Proceedings of
Emerging Trends in Applied Mathematics: International Conference, Kolkata, India, February, 2014, Eds:
Susmita Sarkar, Uma Basu, Soumen De, (Year of publication 2014), pp174-179.
Major conference presentations
1. Surajit Chattopadhyay,“A reconstruction scheme for f(G) gravity based on QCD ghost dark energy and its cosmological
consequences”, 7th International Conference on Physics and Astrophysics of Quark Gluon Plasma, at Variable Energy
Cyclotron Centre, Kolkata during 1-6 February, 2015.
2. Surajit Chattopadhyay,“Holographic reconstruction of scalar field dark energy models in chameleon Brans-Dicke
cosmology”, XXI DAE-BRNS High Energy Physics Symposium, at Indian Institute of Technology Guwahati during 8 – 12
December, 2014.
3. Surajit Chattopadhyay, S. Roy, “Dependence of Brans-Dicke parameter on scalar field” ,Emerging Trends in Applied
Mathematics: International Conference, at the University of Calcutta, Kolkata during February 12-14, 2014.
4. Surajit Chattopadhyay, “A study on the interacting Ricci dark energy in f(R,T) gravity”, International Conference on the
Frontiers of Mathematics and Applications (ICFMA 2014) organized by the Department of Mathematics, University of
Burdwan , January 29-31, 2014