1. 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

2. 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

3. 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
ϕϕ
π
VRgL

4. Thus Einstein equations are
where the Hubble function and the Gaussian
curvature are
Review of “REM”

5. 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”

6. 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”

7. 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”

8. 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”

9. Review of “REM”

10. 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

11. 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

12. where we denoted with an "0" index all values at the initial
actual time. These are the Ellis-Madsen potentials.
Examples : “regular” potentials

13. 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 expect
to apply successfully the method of ``reverse engineering'' for this type
of 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

14. 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

15. Tachyonic potentials
We also have a new Klein-Gordon equation, namely :
All these results are then saved in a new library, cosmotachi.m which will
replace 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 in
The Ellis-Madsen potentials above

16. Tachyonic potentials
Tachionic potentials. Here we denoted with R0 the scale factor
at the actual time t0 and with a the quantity f(t) – f0

17. 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 the
type of coupling between the scalar field and the
gravity.
( ) 



−−∇−−= 22
2
1
)(
2
1
16
1
ϕξϕϕ
π
RVRgL

18. 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.

19. Cosmology with non-minimally coupled
scalar field
••
+−=+ )])(()(3)()(
2
1
[
)(
3)(3 22
2
2
ttHtVt
tR
k
tH φξφ
•••
−+−=+ )])(()(
2
3
)()([)(3)(3 222
ttHtVttHtH φξφ
•
••
−−
−−
∂
∂
=
)()(3)()(12
)()(6
)(
6)(
2
2
ttHttH
ttH
tR
kV
t
φφξ
φξξ
φ
φ
where 8pG=1, c=1
These are the new Friedman equations !!!

20. Einstein frame
It is more convenient to transform to the Einstein
frame by performing a conformal transformation
µνµν gg 2
^
Ω= where
22
81 πϕξ−=Ω
Then we obtain the following equivalent Lagrangian:








−





∇−−= )(
2
1
16
1 ^2^
2
^^
ϕϕ
π
VFRgL

21. where variables with a caret denote those in the Einstein
frame, and
22
2
2
)81(
8)61(1
πξϕ
πξϕξ
−
−−
=F
and
22
^
)81(
)(
)(
πξϕ
ϕ
ϕ
−
=
V
V
Einstein frame

22. Introducing a new scalar field Φ as
∫=Φ ϕϕ dF )(
the Lagrangian in the new frame is reduced to the
canonical form:








Φ−





Φ∇−−= )(
2
1
16
1 ^2^^^
VRgL
π
Einstein frame

23. 







Φ−





Φ∇−−= )(
2
1
16
1 ^2^^^
VRgL
π
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

24. 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 :
∫Ω= dtt
^
and RR Ω=
^
and the new scalar field F can be obtained by
integrating its above expression, namely
Einstein frame

25. [ ])61(22(sin)61(
4
2
8)61(1
)sgn(34
tanh)sgn(
2
3
1
2
1
ςπςϕξξ
ξπ
πξϕξ
ξϕξπ
ξ
π
−−+








−−
=Φ
−
−
where sgn(x) represents the sign of x – namely +1 or -1
Einstein frame

26. Examples
Φ→ϕ
^
VV →
^
tt →

27. Examples : ekpyrotic universe
This is example nr. 6 from [3] - see also (6) - having :
)sin()(
^^
0 tRtR ω=
and
π
ωω
4
3
cosh2)(
22
−










 Φ
=Φ
B
BV
with 





+= 2
0
2
1
4
1
R
k
B
π

28. Examples : ekpyrotic universe
w = 1, k=1, x = 0 green line
x=-0.1 (left) and x = 0.1 (right) blue line)(ϕV

29. Examples : ekpyrotic universe
),( ωϕV k = 1 and x = 0.05

30. Examples : ekpyrotic universe
),( ωϕV
k=1, x = 0 green surface
x = 0.1 (left) and x = - 0.3 (right) blue

31. 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
2
4 41 1
( ) ( ) ( , )
2 8 2
P
M M
M
S d x g f R V d xL gµ
µ µνφ φ φ
π
 
= − + ∇ ∇ − + Ψ 
 
∫ ∫

32. Cosmology with f( R ) gravity …
Now we restrict ourselves to the case when
2
)( RRRf α+=
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 ( )
2
1 1
( )
2 2
R R g R R g R g g g R
g g V
µν µν λσ
αβ αβ µν α β αβ µν λ σ
µν
α β αβ µ ν
α α α
φ φ φ φ φ
+ − + − − =
 
= − − 
 

33. 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
2
2
tHtHk
tR
k
tHtHV  +++=
πππ
ϕ
),...)(),(,...(
)(4
1
)(
4
1 2
2
2
tHtHk
tR
k
tH  ++−=
ππ
ϕ

34. 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
t
eRtR ω
0)( =
2) The linear expansion unverse with
n
ttRtR 00 RR(t)or)( ==

35. Cosmology with f( R ) gravity …
Expponential case :
V(j) in terms of different w at k=0

36. Cosmology with f( R ) gravity …
Expponential case :
Time behaviour of V(j) in terms of different a at k=0,1 and -1

37. Cosmology with f( R ) gravity …
Expponential case :
V(j) in terms of different a at k=1 and w=0.1

38. Cosmology with f( R ) gravity …
Linear case :
Time behaviour of V(j) in terms of different a at k=0,1 and -1

39. Cosmology with f( R ) gravity …
Linear case :
V(j) in terms of different a at k=1

40. Conclusions….

41. Conclusions….

42. 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

43. The end !!!
Thank you for your attention !