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 !