ON COSMOLOGIES WITH NON-MINIMALLYCOUPLED SCALAR FIELD AND

THE "REVERSE ENGINEERING METHOD"

G.S. Djordjevic1 , D..N. Vulcanov2

(1) Department of Physics, Faculty of Science and Mathematics, University of Nis,Visegradska 33, 18001Nis, Serbia

(2) Department of Theoretical and Applied Physics –“ Mircea Zǎgǎnescu”West University of Timişoara, B-dul. V. Pârvan no. 4, 300223,

Timişoara, Romania

The SEENET-MTP Workshop BW2011

Abstract of the presentation

We studied further the use of the so called “reverse engineering”method (REM) in reconstructing the shape of the potential in cosmologies based on a scalar field non-minimally coupled with gravity. We usethe known result that after a conformal transformation to the socalled Einstein frame, where the theory is exactly as we have aMinimally coupled scalar field. Processing the REM in Einstein frame And then transforming back to the original frame, we investigatedgraphically some examples where the behaviour of the scale factoris modelling the cosmic acceleration (ethernal inflation)

Plan of the presentation Introduction – why scalar fields in cosmology Review of the “reverse engineering” method Cosmology with non-minimally coupled scalar field Einstein frame Some examples Conclusions

Plan of the presentation Introduction – why scalar fields in cosmology Review of the “reverse engineering” method Cosmology with non-minimally coupled scalar field Einstein frame Some examples Conclusions

Introduction : Why scalar fields ?

Recent astrophysical observations ( Perlmutter et . al .) shows that the universe is expanding faster than the standard model says. These observations are based on measurements of the redshift for several distant galaxies, using Supernova type Ia as standard candles. As a result the theory for the standard model must be rewritenin order to have a mechanism explaining this ! Several solutions are proposed, the most promising ones are based on reconsideration of the role of the cosmological constant or/and taking a certain scalar field into account to trigger the acceleration of the universe expansion. Next figure (from astro - ph /9812473) contains, sintetically the results of several years of measurements ...

Introduction : Cosmic acceleration

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)

( ) ⎥⎦⎤

⎢⎣⎡ −∇−−= )(

21

161 2 ϕϕπ

VRgL

Review of “REM”

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”

where we denoted with an "0" index all values at the initial actual time. These are the Ellis-Madsen potentials.

Review of “REM”

Review of “REM”

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 :

( ) ⎥⎦⎤

⎢⎣⎡ −−∇−−= 22

21)(

21

161 ϕξϕϕπ

RVRgL

where ξ is the numerical factor that describes thetype of coupling between the scalar field and thegravity.

Cosmology with non-minimally coupled scalar field

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. In [3] we appealed to the numerical and graphical facilites of a Maple platform.

For sake of completeness we can compute the Einsteinequations for the FRW metric.

After some manipulations we have :

Cosmology with non-minimally coupled scalar field

••

+−=+ )])(()(3)()(21[

)(3)(3 22

22 ttHtVt

tRktH φξφ

•••

−+−=+ )])(()(23)()([)(3)(3 222 ttHtVttHtH φξφ

•

••

−−

−−∂∂

=

)()(3)()(12

)()(6)(

6)(

2

2

ttHttH

ttHtR

kVt

φφξ

φξξφ

φ

where 8πG=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

µνµν gg 2^

Ω= 22 81 πϕξ−=Ωwhere

Then we obtain the following equivalent Lagrangian:

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛∇−−= )(

21

161 ^2^

2^^

ϕϕπ

VFRgL

Einstein frame

where variables with a caret denote those in the Einsteinframe, and

22

22

)81(8)61(1

πξϕπξϕξ

−−−

=F

and ^

22 )81()()(

πξϕϕϕ

−=

VV

Einstein frame

Introducing a new scalar field Φ as

∫=Φ ϕϕ dF )(the Lagrangian in the new frame is reduced to the canonical form:

⎥⎥⎦

⎤

⎢⎢⎣

⎡Φ−⎟

⎠⎞

⎜⎝⎛ Φ∇−−= )(

21

161 ^2^^^

VRgLπ

Einstein frame

⎥⎥⎦

⎤

⎢⎢⎣

⎡Φ−⎟

⎠⎞

⎜⎝⎛ Φ∇−−= )(

21

161 ^2^^^

VRgLπ

Main conclusion: we can process a REM in theEinstein frame (using the results from the minimalllycoupling case and then we can convert the results inthe 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 :

RR Ω=^

∫ Ω= dtt^

and

and the new scalar field Φ can be obtained by integrating its above expression, namely

Einstein frame

[ ])61(22(sin)61(4

2

8)61(1)sgn(34tanh)sgn(

23

1

2

1

ςπςϕξξξπ

πξϕξξϕξπξ

π

−−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−=Φ

−

−

where sgn(ξ) represents the sign of ξ – namely +1 or-1

Examples

Φ→ϕ^

VV →

^tt →

Examples : nr. 1 – exponential expansion

ω = 1, ξ = 0 green lineξ=-0.1 (left) and ξ = 0.1 (right) blue line)(ϕV

Examples : nr. 1 – exponential expansion

),( ωϕV ξ=0.1 (left) and ξ = - 0.1 (right)

Examples : nr. 1 – exponential expansion

ξ = 0 green surfaceξ=-0.1 (left) and ξ = 0.1 (right) blue ),( ωξV

Examples : nr. 4 - tn

n = 3, ξ = 0 green lineξ=-0.1 (left) and ξ = 0.1 (right) blue line)(ϕV

Examples : nr. 4 - tn

n = 3, ξ = 0 green surfaceξ=-0.3 (left) and ξ = 0.3 (right) blue surface),( nV ϕ

Examples : ekpyrotic universe

This is example nr. 6 from [3] having :

)sin()(^^

0 tRtR ω=

πωω43cosh2)(

22

−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ Φ

=ΦB

BVand

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

0

2 141

RkB

πwith

Examples : ekpyrotic universe

ω = 1, k=1, ξ = 0 green lineξ=-0.1 (left) and ξ = 0.1 (right) blue line)(ϕV

Examples : ekpyrotic universe

),( ωϕV κ = 1 and ξ = 0.05

Examples : ekpyrotic universe

k=1, ξ = 0 green surfaceξ = 0.1 (left) and ξ = - 0.3 (right) blue ),( ωϕV

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] Padmanabhan T, PRD 66 (2002), 021301(R) [6] Cardenas VH , del Campo S, astro - ph /0401031

The end !!!

Thank you for your attention !