1. ON COSMOLOGIES WITH NON-MINIMALLY
COUPLED 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

2. 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 use
the known result that after a conformal transformation to the so
called Einstein frame, where the theory is exactly as we have a
Minimally coupled scalar field. Processing the REM in Einstein frame
And then transforming back to the original frame, we investigated
graphically some examples where the behaviour of the scale factor
is modelling the cosmic acceleration (ethernal inflation)

3. 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

4. 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

5. 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 rewriten
in 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 ...

6. Introduction :
Cosmic acceleration

7. 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
⎡ 1 ⎤
R − (∇ ϕ ) − V (ϕ ) ⎥
1
L= −g⎢
2
⎣ 16 π 2 ⎦
Where R is the Ricci scalar and V is the potential of the
scalar field and G=c=1 (geometrical units)

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

9. 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.

10. 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).

11. 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.

12. 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 :

13. Review of “REM”

14. Review of “REM”
where we denoted with an "0" index all values at the
initial actual time. These are the Ellis-Madsen potentials.

15. Review of “REM”

16. Review of “REM”

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 :
⎡ 1 ⎤
R − (∇ ϕ ) − V (ϕ ) − ξ R ϕ 2 ⎥
1 1
L= −g⎢
2
⎣ 16 π 2 2 ⎦
where ξ is the numerical factor that describes the
type of coupling between the scalar field and the
gravity.

18. 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 Einstein
equations for the FRW metric.
After some manipulations we have :

19. Cosmology with non-minimally coupled
scalar field
•
k 1 • 2
3 H (t ) 2 + 3 = [ φ ( t ) − V ( t ) + 3ξ H ( t ) (φ ( t ) 2 )]
R (t ) 2 2
• • •
3
3 H ( t ) 2 + 3 H ( t ) = [ − φ ( t ) 2 + V ( t ) − ξ H ( t ) (φ ( t ) 2 )]
2
•• ∂V k
φ (t ) = − 6ξ − 6 ξ H ( t )φ ( t )
∂φ R (t ) 2
•
− 12 ξ H ( t ) φ ( t ) − 3 H ( t ) φ ( t )
2
where 8πG=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
^
g µν = Ω 2 g µν where Ω 2 = 1 − ξ 8πϕ 2
Then we obtain the following equivalent Lagrangian:
⎡ 1 ^ 1 2 ⎛ ^ ⎞2 ^
^ ⎤
L= −g⎢ R − F ⎜ ∇ ϕ ⎟ − V (ϕ ) ⎥
⎢16π
⎣ 2 ⎝ ⎠ ⎥
⎦

21. Einstein frame
where variables with a caret denote those in the Einstein
frame, and
1− (1− 6ξ )8πξϕ 2
F =
2
(1− 8πξϕ )2 2
and ^
V (ϕ )
V (ϕ ) =
(1 − 8 πξϕ )
2 2

22. Einstein frame
Introducing a new scalar field Φ as
Φ = ∫ F (ϕ )dϕ
the Lagrangian in the new frame is reduced to the
canonical form:
⎡ 1 ^ 1 ⎛ ^ ⎞2 ^
^ ⎤
L= −g⎢ R − ⎜ ∇ Φ ⎟ − V (Φ ) ⎥
⎢16π
⎣ 2⎝ ⎠ ⎥
⎦

23. Einstein frame
⎡ 1 ^ 1 ⎛ ^ ⎞2 ^
^ ⎤
L= −g⎢ R − ⎜ ∇ Φ ⎟ − V (Φ ) ⎥
⎢16π
⎣ 2⎝ ⎠ ⎥
⎦
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.

24. 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 :
^ ^
t= ∫ Ω dt and R = ΩR
and the new scalar field Φ can be obtained by
integrating its above expression, namely

25. Einstein frame
3 ⎡ 4 3π ϕξ sgn( ξ ) ⎤
Φ= sgn( ξ ) tanh ⎢ −1
⎥
2 π ⎢ 1 − (1 − 6ξ )8πξϕ 2
⎣ ⎥
⎦
+
4
2
πξ
[ ξ (1 − 6ξ ) sin −1
( 2ϕ 2πς (1 − 6ς ) ]
where sgn(ξ) represents the sign of ξ – namely +1 or-1

26. Examples
ϕ → Φ
^
^ t → t
V → V

27. Examples : nr. 1 – exponential expansion
V (ϕ )
ω = 1, ξ = 0 green line
ξ=-0.1 (left) and ξ = 0.1 (right) blue line

28. Examples : nr. 1 – exponential expansion
V (ϕ , ω ) ξ=0.1 (left) and ξ = - 0.1 (right)

29. Examples : nr. 1 – exponential expansion
V (ξ , ω )
ξ = 0 green surface
ξ=-0.1 (left) and ξ = 0.1 (right) blue

30. Examples : nr. 4 - tn
V (ϕ )
n = 3, ξ = 0 green line
ξ=-0.1 (left) and ξ = 0.1 (right) blue line

31. Examples : nr. 4 - tn
V (ϕ , n)
n = 3, ξ = 0 green surface
ξ=-0.3 (left) and ξ = 0.3 (right) blue surface

32. Examples : ekpyrotic universe
This is example nr. 6 from [3] having :
^ ^
R (t ) = R0 sin(ω t )
2
⎡ ⎛ ωΦ ⎞⎤ 3ω
2
and V (Φ ) = 2 ⎢ B cosh⎜ ⎟⎥ −
⎣ ⎝ B ⎠⎦ 4π
1 ⎛ k ⎞
B =
2
⎜1 + 2 ⎟
⎜
with
4π ⎝ R0 ⎟
⎠

33. Examples : ekpyrotic universe
V (ϕ )
ω = 1, k=1, ξ = 0 green line
ξ=-0.1 (left) and ξ = 0.1 (right) blue line

34. Examples : ekpyrotic universe
V (ϕ , ω ) κ = 1 and ξ = 0.05

35. Examples : ekpyrotic universe
V (ϕ , ω )
k=1, ξ = 0 green surface
ξ = 0.1 (left) and ξ = - 0.3 (right) blue

36. Conclusions….

37. Conclusions….

38. 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

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