Download Scale-invariant theory of gravitation

Keynote Address Presented in National Conference on
Recent Advances in Mathematical Sciences at Amravati
University, Amravati on
Cosmological Models in Modified Theories of
Gravitation – A bird’s eye view
By
Dr. D.R.K. Reddy, Professor (Retd.)
Dept. of Applied Mathematics
Andhra University (A.U.)
Visakhapatnam
This Lecture Consists of the Following
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Mathematical modeling.
General relativity (Einstein’s theory).
Cosmology and Cosmological models.
Modified theories of gravitation.
Various physical sources used to obtain cosmological models
in modified theories of gravitation.
Steps in constructing cosmological models
Dark energy and dark matter
A quick review of various cosmological models obtained by
several authors in modified theories of gravitation.
Conclusions
Mathematical Modeling
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All the investigations that we are doing in Mathematical
sciences, Biological sciences and Social sciences come
under mathematical modeling.
Given a physical situation, we form the differential equation.
Solve the differential equation.
Determine the constants using initial conditions.
Interpret the solution.
Draw conclusions.
Thus differential equations play a vital role in every
branch of Science.
Einstein’s Theory of Gravitation
 Special theory of relativity formulated by Einstein in 1905
takes care of the relativity of uniform translatory motion in
a region of free space wherein the effects of gravity is
neglected.
 Einstein (1915) formulated general relativity which
describes relativity of all kinds of motion and which
generalizes Newtonian theory of gravitation. This is known
as Einstein’s theory of gravitation.
 Einstein’s theory of gravitation is based on Riemannian
metric tensor gij which describes not only the gravitational
field but also the geometry.
Principle of Covariance
It helps to write the physical laws of the Universe in covariant form
Principle of Equivalence
It incorporates gravitational effects in general relativity
Mach’s Principle
The geometry of the universe is uniquely determined by the
distribution of matter and energy of the momentum.
Einstein’s Field Equations
1
Rij  R gij   gij  8 Tij
2
Rij isthe Ricci Tensor , R is the scalar curvature
Tij is the Energy Momentum Tensor
 is a cosmological constant ,
gij is a metric tensor.
Cosmology
 Cosmology is the scientific study of the large scale structure
of the universe.
 The main aim of these models with the present day Universe
as observed by astronomers.
 The study of cosmology is based on the cosmological
principle; which states that on a sufficiently large scale the
Universe is homogeneous and isotropic.
Cosmological models
 The theory of cosmological models began with Einstein’s
development of the static universe in 1917.
 Friedman (1922) was the first to investigate the most general
non-static, homogeneous and isotropic space-time described by
the Robertson-Walker metric
2
dr

2
2
2
2
2
2
2
2
ds  dt  a ( t ) 
 r d  r sin  d 
2
1  kr

where a(t) is the scale factor, k is a constant which by a suitable
choice of r can be chosen to have values +1, 0 or -1 according
as the universe is closed, flat or open respectively.
Anisotropic Bianchi space times
 Bianchi type-I, III, V and VI0 metrics
ds 2  dt 2  A2dx 2  B2e2mxdy 2  C 2e2nxdz 2
Bianchi type-I if m= n= 0 ; Bianchi type-III if n = 0;
Bianchi type-V if m = n ; Bianchi type- VI0 if m = -n
 Bianchi type-II, VIII and IX metrics
ds2  dt 2  R 2 d 2  f 2 ( )d 2   S 2 d  h( )d  2


The above line element represents
 Bianchi type-II if f ( )  1 and h    
 Bianchi type-VIII if f ( )  cosh   and h    sinh  
 Bianchi type- IX if f ( )  sin and h    cos
Necessity of Modified theories of gravitation
 Mach’s principle is not fully incorporated in Einstein’s field
equations.
 Einstein’s theory has singularity problem.
 The recent scenario of accelerated expansion of the universe
is not explained by Einstein’s theory.
 Hence there has been a need for modifying Einstein theory
of gravitation.
 The following are modifications (list not exhausted) of
Einstein theory of gravitation.
 The complete details of the modification and derivation are
not presented here.
Modified theories of Gravitation
 Brans-Dicke (1961) scalar-tensor theory of gravitation.
 Nordtvedt’s (1971) theory of gravitation.
 Scale covariant theory of gravitation (Canuto et al.1977).
 Scale-invariant theory of gravitation (Wesson 1981).
 Barber’s (1982) self-creation theory of gravitation.
 Saez-Ballester (1986) scalar-tensor theory of gravitation.
 f(R) theory of gravity (Odintsov 2000).
 f(R,T) theory of gravity (Harko et al. 2011).
Brans-Dicke scalar-tensor theory
Variational principle

,i ,i 
4
  R  16GL 
 g d x0
 

Brans-Dicke field equations are given by
1
1


Rij  R gij  8  1 Tij    2  , i , j  gij , k  , k    1 i ; j  gij 
2
2



where
  8  3  2  1 T
Energy conservation equation is T;ijj  0

Nordtvedt’s general scalar-tensor theory
Variationa l principle

  
'i
  16L  R 
 ' i   g d 4 x  0.



Nordtvedt’s general scalar-tensor field equations are given by
1
1

1
2 
Rij  R gij  8  Tij     , i , j  gij , k  , k    1 i ; j  gij 
2
2



where
8 T
1 d


,i ,i
3  2 3  2 d
Energy conservation equation is T;ijj  0.

Scale covariant theory of gravitation
The field equations in the scale covariant theory are
1
Rij  R gij  fij  Tij   gij
2
where
 fij  2ij  4i  j   g ab  a b  2 g ab  ab 
in which β is a scalar or gauge function satisfying 0 < β < ∞.
Scale-invariant theory of gravitation
The field equations of the scale invariant theory are
1
Rij  R gij  fij  0  2 gij   Tij
2
where
;ij
,i , j  ab , a ,b
ab ;ij 
fij  2
4 2 g
 2g
 gij
2

 



Saez-Ballester scalar-tensor theory
Variationa l principle
  R   n ,  ,   GLm   g dx dy dz dt  0

Saez-Ballester field equations are given by
1
1


n
Rij  Rgij   8 Tij    , i , j  gij , k  , k 
2
2


Scalar field satisfies the condition
2 n i; j  n  n  1, k  , k  0
The energy conservation equation
T ij ; j  0 is a consequence of the above field equations.
f(R) theory of gravity
Variational principle
   g  f ( R)  Lm  d 4 x  0
The f(R) field equations resulting from this action are
F  R  R 
1
f  R  R     F  R   g 
2
where
F  R 
d
f  R ;

dR
    .
 i is the covariant derivative
F  R   T
f(R,T) theory of gravity
Variational principle

4
f
R
,
T

L

g
d
x0




m


16
Field equations of f(R,T) gravity


1
f R  R,T  Rij  f  R,T  gij  gij k k  i j f R  R,T   8 Tij  fT  R,T  Tij  fT  R,T  ij
2
where ij  2Tij  gij Lm  2 g
lk
 i is the covariant derivative
 2 Lm
g ij g
; fR 
lm
f  R, T 
f  R, T 
; fT 
R
T
Various Physical Sources to obtain
cosmological models
 Dust distribution for which the energy momentum tensor is
given by
Tij   ui u j
 Perfect fluid distribution with energy momentum tensor as
Tij  ( p   )ui u j  pgij
 Electromagnetic field with energy momentum tensor
1
Tij  Fij Fi  gij Fkj F kj
4
 Cosmic string sources with Tij   ui u j   xi x j
j
 Bulk viscous fluid containing one dimensional string
Tij  ( p   )ui u j  pgij   xi x j
Steps in constructing cosmological models
The following are usual steps in constructing and studying the
cosmological models in any theory of gravitation:
 Choosing the space-time metric.
 Taking the physical source for gravity.
 Deriving gravitational field equations of the theory under
consideration.
 Solving the field equations using suitable mathematical or physical
conditions.
 Presenting the cosmological model.
 Studying various physical and kinematical parameters which are
important in the discussion of cosmology.
 Then comparing the model with other existing models and relating
to observational data.
Dark energy and Dark matter
 Dark matter and dark energy are a cosmological mystery in
modern cosmology.
 The supernova observations of Riess (1998) and Perlmutter
(1998) •first indicated that the universe is undergoing an accelerated
expansion.
 Cosmological observations and cosmic microwave background
data suggest that the universe is spatially žflat and is dominated by
an exotic component with large negative pressure dubbed as dark
energy.
 Wilkinson Microwave Anisotropy Probe (WMAP) measures that
dark energy, dark matter, and baryonic matter occupy 73%, 23%,
and 4%, respectively, of the energy-mass content of the universe.
 In order to explain this accelerated expansion of the universe,
two different approaches have been advocated: to construct
different dark energy candidates and to modify Einstein’s theory
of gravitation.
 Several candidates have been put forward to understand dark
energy.
 Cosmological constant: But it fails serious problems like five
tuning and cosmic coincidence.
 Quintessence: (Martix 2008) with eqn. of state pde  de  de
 Phantom : A scalar field with negative kinetic energy de  1
Universe will end with a big rip-life time finite.
 k-essence: It is based on the idea that dark energy
components is due to a minimally coupled scalar field with
non-local kinematic energy resulting negative pressure.
 Tachyon: based on scalar field 0  de  1
 Chaplygin gas: perfect fluid satisfying eqn. of state
p
A

 Holographic dark energy: based on holographic principle
which state that entropy of a system scales not with its
volume but with its surface area L2

 Pilgrim dark energy:   1H   2 H

2 
 Modified Ricci dark energy:    31HH 1  3 2 H  33 H 2
 Another possible way of exploiting dark energy is to
modify geometric part of the Einstein-Hilbert action.
 In recent years, there have been several further investigations
on dark matter and dark energy cosmological models in
alternative theories of gravitation.
 Some relevant alternatives to Einstein’ s theory of gravitation
are Brans-Dicke (BD) (1961), Saez-Ballester (1986), and other
scalar-tensor theories of gravitation, f(R) gravity, f(R,T) gravity,
and f(T) gravity.
A quick review of various cosmological models obtained
by several authors in modified theories of gravitation
 Several authors have investigated various cosmological models in
the above modified theories of gravitation using the different
physical sources.
 It is a Herculean task to present here all the models and the work
done in the above theories by several authors.
 However, we mention some of the investigations of cosmological
models in the name of various research groups.
 Models with perfect fluid source have been obtained in several
modified theories of gravitation by (The list is exhaustive)
 Reddy and his group
 Adhav and Katore and their group
 Rao and his group
 Sharif and his group
 Shri Ram and his group
 Pradhan and his group
 Mishra and Sahoo group
 Mohanthy and his group
 Models with cosmic strings and domain walls source have
been discussed in modified theories of gravitation by




Reddy and his group
Adhav, Katore and his group
Rao and his group
Shri ram and his group
 Models with bulk viscous string source have been discussed in
modified theories of gravitation by




Reddy and his group
Rao and his group
Shri ram and his group
Pradhan and his group
 Models with electromagnetic source have been discussed in
modified theories of gravitation by




Reddy and his group
Rao and his group
J. K. Singh and his group
Shri ram and his group
 Dark energy models in modified theories of gravitation by
 Sharif and his group








Reddy and his group
Rao and his group
Adhav, Katore and their group
Pradhan and his group
T. Singh and his group
Shri ram and his group
Mishra and Sahoo group
Yadav and his group
Conclusions
 We live in this universe, which consists of stars, star clusters,
galaxies nebulae, pulsars, quasars etc.
 It is but natural that we will be interested to know the type of
matter that fills the universe, how rapidly the universe is
expanding today, how old is the universe- open, flat, closed or
otherwise and ultimate fate of the universe.
 This could be effectively achieved by constructing cosmological
models and studying their physical behavior.
 We have seen that several investigations have been carried out
to study the physical behavior of the universe through various
cosmological models in Einstein and other modified theories of
gravitation.
 The test of any model to represent the present day universe
will largely depend on the availability of experimental data.
 Hence, the origin, the geometry, physical content and the
ultimate fate of the universe will be only decided by comparing
the cosmological models obtained in various theories of
gravitation when advanced astronomical data available.
Thank you