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Balkan Workshop - 2013 Vrnjacka Banja - Serbia
REM -- the Shape of Potentials for f(R) Theories
in Cosmology and Tachyons
G.S. Djordjevic1 , D.N. Vulcanov2 and C. Sporea2
(1) Department of Physics, Faculty of Science and Mathematics, University of Nis,
Visegradska 33, 18001Nis, Serbia(2) Department of Physics
West University of Timişoara, B-dul. V. Pârvan no. 4, 300223,
Timişoara, Romania
Plan of the presentation
Review of the “reverse engineering” method
Computer programs for dealing with REM and cosmology
Processed examples :
“Regular” potentials and tachyonic ones
Cosmology with non-minimally coupled scalar field
Cosmology with f( R ) gravity and scalar field
Conclusions
Review of the “reverse engineering method”
We are dealing with cosmologies based on Friedman- Robertson-Walker ( FRW ) metric
Where R(t) is the scale factor and k=-1,0,1 for open, flat or closed cosmologies. The dynamics of the system with a scalar field minimally coupled with gravity is described by a lagrangian as
Where R is the Ricci scalar and V is the potential of the scalar field and G=c=1 (geometrical units)
)(
2
1
16
1 2
VRgL
Thus Einstein equations are
where the Hubble function and the Gaussian curvature are
Review of “REM”
Thus Einstein equations are
It is easy to see that these eqs . are not independent. For example, a solution of the first two ones (called Friedman equations) satisfy the third one - which is the Klein-Gordon equation for the scalar field.
Review of “REM”
Thus Einstein equations are
The current method is to solve these eqs . by considering a certain potential (from some background physical suggestions) and then find the time behaviour of the scale factor R(t) and Hubble function H(t).
Review of “REM”
Thus Einstein equations are
Ellis and Madsen proposed another method, today considered (Ellis et . al , Padmanabhan ...) more appropriate for modelling the cosmic acceleration : consider "a priori " a certain type of scale factor R(t), as possible as close to the astrophysical observations, then solve the above eqs . for V and the scalar field.
Review of “REM”
Following this way, the above equations can be rewritten as
Solving these equations, for some initially prescribed scale factor functions, Ellis and Madsen proposed the next potentials - we shall call from now one Ellis- Madsen potentials :
Review of “REM”
Review of “REM”
Computer programs for dealing with REM and cosmology
We used Maple platform with GrTensor II
GrTensorII – a free package (see at http://grtensor.org)
embedded in Maple.
- the geometry in |GrTensorII is a spacetime with
Riemannian structure – adapted for Einstein GR-- It can be easily adapted/exended to alternative theories-- It provides facilities for building dedicated libraries-- simple acces to all Maple facilities – symbolic and - algebraic computation, numerical and graphical facilities
We used Maple platform with GrTensor II
Three steps we done for processing REM, namely :
- a library for algebraic computing of Einstein eqs till
Friedmann eqs and calculating the potential and scalar
field time derivative |(as two slides before)
- composing algbraic computations routines for analytic
processing of REM (if possible).If not
- composing of numerical and graphical routines for
processing REM graphically
Computer programs for dealing with REM and cosmology
where we denoted with an "0" index all values at the initial actual time. These are the Ellis-Madsen potentials.
Examples : “regular” potentials
Tachyonic potentials
Recently it has been suggested that the evolution of a tachyonic condensate in a class of string theories can have a cosmological significance. This theory can be described by an effective scalar field with a lagrangian of of the form
where the tachyonic potential has a positive maximum at the origin
and has a vanishing minimum where the potential vanishes
Since the lagrangian has a potential, it seems to be reasonable to expectto apply successfully the method of ``reverse engineering'' for this typeof potentials. As it was shown when we deal with spatially homogeneous geometry cosmology described with the FRW metric above we can use again a density and a negative pressure for the scalar field as
Tachyonic potentials
and
Now following the same steps as explained before we have the new Friedmann equations as :
With matter included also. Here as usual we have
Tachyonic potentials
We also have a new Klein-Gordon equation, namely :
All these results are then saved in a new library, cosmotachi.m which willreplace the cosmo.m library we described in the prevos lecture.
Now following the REM method we have finally :
which we used to process different types of scale factor, same as inThe Ellis-Madsen potentials above
Tachyonic potentials
Tachionic potentials. Here we denoted with R0 the scale factorat the actual time t0 and with a the quantity f(t) – f0
Cosmology with non-minimally coupled scalar field
We shall now introduce the most general scalar field as a source for the cosmological gravitational field, using a lagrangian as :
where x is the numerical factor that describes thetype of coupling between the scalar field and thegravity.
22
2
1)(
2
1
16
1
RVRgL
Cosmology with non-minimally coupled scalar field
For sake of completeness we can compute the Einstein
equations for the FRW metric.
After some manipulations we have :
Although we can proceed with the reverse method
directly with the Friedmann eqs. obtained from this
Lagrangian (as we did in [3]) it is rather complicated
due to the existence of nonminimal coupling. We
appealed to the numerical and graphical facilites of a
Maple platform.
Cosmology with non-minimally coupled scalar field
)])(()(3)()(2
1[
)(3)(3 22
22 ttHtVt
tR
ktH
)])(()(2
3)()([)(3)(3 222 ttHtVttHtH
)()(3)()(12
)()(6)(
6)(
2
2
ttHttH
ttHtR
kVt
where 8pG=1, c=1
These are the new Friedman equations !!!
Einstein frame
It is more convenient to transform to the Einstein
frame by performing a conformal transformation
gg 2^
where 22 81
Then we obtain the following equivalent Lagrangian:
)(
2
1
16
1 ^2^2
^^
VFRgL
where variables with a caret denote those in the Einstein
frame, and
22
22
)81(
8)61(1
F
and
22
^
)81(
)()(
VV
Einstein frame
Introducing a new scalar field Φ as
dF )(
the Lagrangian in the new frame is reduced to the
canonical form:
)(
2
1
16
1 ^2^^^
VRgL
Einstein frame
)(
2
1
16
1 ^2^^^
VRgL
Main conclusion: we can process a REM in the
Einstein frame (using the results from the minimallly
coupling case) and then we can convert the results in
the original frame.
Einstein frame
Before going forward with some concrete results,
let’s investigate some important equations for
processing the transfer from Einstein frame to the
original one. First the main coordinates are :
dtt^
and RR ^
and the new scalar field F can be obtained by
integrating its above expression, namely
Einstein frame
)61(22(sin)61(4
2
8)61(1
)sgn(34tanh)sgn(
2
3
1
2
1
where sgn(x) represents the sign of x – namely +1 or -1
Einstein frame
Examples
^
VV
^
tt
Examples : ekpyrotic universe
This is example nr. 6 from [3] - see also (6) - having :
)sin()(^^
0 tRtR
and4
3cosh2)(
22
B
BV
with
20
2 14
1
R
kB
Examples : ekpyrotic universe
w = 1, k=1, x = 0 green line
x=-0.1 (left) and x = 0.1 (right) blue line)(V
Examples : ekpyrotic universe
),( V k = 1 and x = 0.05
Examples : ekpyrotic universe
),( V k=1, x = 0 green surface
x = 0.1 (left) and x = - 0.3 (right) blue
Cosmology with f( R ) gravity and minimally coupled scalar field
We shall now move to gravity theories with higher order lagrangian, so alled “f( R ) theories” where
Where we have again a scalar field minimally coupled
with gravity and we have also regular matter fields
described in LM
24 41 1
( ) ( ) ( , )2 8 2
PM M
MS d x g f R V d xL g
Cosmology with f( R ) gravity …
Now we restrict ourselves to the case when
2)( RRRf where a is a real constant. Varying the above action
we get the new field equations as (G=c=1) :
2; ; ; ;
, , , ,
1(1 2 ) ( ) 2 ( )
21 1
( )2 2
R R g R R g R g g g R
g g V
Cosmology with f( R ) gravity …
Working again in FRW metric
we obtained the new Friedmann equations
much more complicated, with extra second and higher order
terms …
),...)(),(,...()(2
1)(
4
1)(
4
3)( 2
22 tHtHk
tR
ktHtHV
),...)(),(,...()(4
1)(
4
1 22
2 tHtHktR
ktH
Cosmology with f( R ) gravity …
Here we need to process all three steps …
Here are some examples, we plotted for two
types of unverses :
1) The exponential expansion unverse with
teRtR 0)(
2) The linear expansion unverse with
nttRtR 00 RR(t)or)(
Cosmology with f( R ) gravity …Expponential case :
V(j) in terms of different w at k=0
Cosmology with f( R ) gravity …Expponential case :
Time behaviour of V(j) in terms of different a at k=0,1 and -1
Cosmology with f( R ) gravity …Expponential case :
V(j) in terms of different a at k=1 and w=0.1
Cosmology with f( R ) gravity …Linear case :
Time behaviour of V(j) in terms of different a at k=0,1 and -1
Cosmology with f( R ) gravity …Linear case :
V(j) in terms of different a at k=1
Conclusions….
Conclusions….
References
[1] M.S. Madsen, Class. Quantum Grav., 5, (1988),
627-639
[2] G.F.R. Ellis, M.S. Madsen, Class. Quantum Grav.
8, (1991), 667-676
[3] D.N. Vulcanov, Central European Journal of
Physics, 6, 1, (2008), 84-96
[4] V. Bordea, G. Cheva, D.N. Vulcanov, Rom. Journ.of Physics,
55,1-2 (2010), 227-237
[5] G. S. Djordjevic, C.A. Sporea, D.N. Vulcanov, Proc.
of the TIM10 Conference, Timisoara, Romania, nov.
2010, in AIP proceedings series.
[6] Cardenas VH , del Campo S, astro - ph /0401031
[7] Tsujikawa S., Phys.Rev.D, 62, 043512, 2000 and references
there
The end !!!
Thank you for your attention !