ASTROPHYSICS
APPLYING THE “REVERSE ENGINEERING” METHOD IN CERTAIN
COSMOLOGIES WITH SCALAR FIELD
VALENTIN BORDEA, GABRIEL CHEVA, DUMITRU N. VULCANOV1
The West University of Timişoara,
Theoretical and Applied Physics Department,
B-dul V. Pârvan no. 4, 300223 Timişoara, România
1
E-mail: vulcan@physics.uvt.ro
Received February 23, 2009
The method of “reverse engineering” for designing potentials in cosmologies
with massless scalar field and tachyonic scalar field minimally coupled with gravity is
used for certain types of cosmologies having the time behavior of the scale factor as
R(t) = R0 tan(ωt). The study is done using special Maple libraries designed for
cosmological purposes previously published [13].
Key words: cosmology, scalar fields.
PACS: 98.80.-k; 04.25.Dm.
1. INTRODUCTION
Modern cosmology is in search for a solution to explain the recent discovered
“cosmic-acceleration” effect [1], [2] i.e. the actual universe is in an accelerated
inflation period. One of the solutions to this problem is to trigger this acceleration
through one or more scalar fields minimally coupled to gravity (“quintessence”).
Usually one solves the Friedmann equations (the Einstein equations for cosmological
environment – see [3], [4]) for a certain type of potential prescribed within a
theoretical background and then one obtains the time behavior of the Universe (the
scale factor function R(t)) to be compared with the experimental data.
Recently a different method was proposed ([5]–[7]) called today the “reverse
engineering” method (REM). Here one starts with a certain type of the scale factor
R(t) designed to be as close possible with the experimental data. Then, solving the
Einstein-Friedmann equations one obtains the shape of the potential for the scalar
field. Different versions of the theory were proposed ([6],[7]) with various final
potentials bringing new physical interpretation of the theory.
In a recent article [13] was done a systematic use of this method for as many
possible examples of initial scale factor functions including also a generalization of REM
Rom. Journ. Phys., Vol. 55, Nos. 1–2, P. 227–237, Bucharest, 2010
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Valentin Bordea, Gabriel Cheva, Dumitru N. Vulcanov
2
for universes with matter (other than the scalar field) and for tachyonic scalar
fields. A special Maple library designed for analytical calculations in cosmology
was used, described previously in [8]. In this view, our article is a natural
continuation of [13] applying the REM to two new examples, one for a simple scalar
field, the other for a tachyonic massles field, both examples starting with a scale
factor being a time function as R(t) = R0 tan(ωt). This time the REM was processed
using a slightly different method mainly based on graphical methods using Maple
platform, as no simple analytical solution was possible to obtain. We intensively used
algebraic programming methods and graphical facilities of Maple platform, using the
cosmology dedicated library (described in details in [8] and [13]).
The article is organized as follows: next section no. 2 briefly describe the
method of “reverse engineering” (REM) as it was proposed by Ellis and Madsen in
their article [5]. Section no. 3 is doing the same but for tachyonic scalar field as
described in [13]. Section no. 4 is dedicated to our results obtained for a pure scalar
field and Section no. 5 describe our results for the tachyonic scalar field.
The article ends with a section dedicated to the main conclusions and
possible further developments of the method.
2. THE “REVERSE ENGINEERING” METHOD
The starting point of the theory are the Einstein equations:
1
8π G
Rµν − g µν R = 4 Tµν
(1)
2
c
We will use here and furthermore the geometrical units, so G = c = 1. The Greek
indices will run between 0 and 3, Rµν is the Ricci tensor and gµν the metric tensor.
The stress-energy tensor for the massless scalar field (as the only matter minimally
coupled with gravity) will be:
1
Tµν = φ, µφ,ν − g µν (φ ,α φ,α + V (φ ) )
(2)
2
where V(φ) is the potential.
Next we have to consider the Friedman-Robertson-Walker (FRW) metric as
2

2  dr
+ r 2 ( dθ 2 + sin 2 θ dφ 2 ) 
ds 2 = −c 2 dt 2 + R ( t ) 
(3)
2
1 − kr

where R(t) is called “scale factor” and it is only time depending as being spatially
homogeneous. Same is valid from now one for the scalar field φ = φ(t). Thus, the
Einstein equations above and the Klein-Gordon equation for the scalar field, are
called Friedmann equations:
2
3H ( t ) + 3K ( t ) = 4π V + φ2
(4)
(
)
3
“Reverse engineering” method in certain cosmologies
(
3H ( t ) + 3H ( t ) = 4π V − 2φ2
2
)
229
(5)
1 ∂V
=0
(6)
2 ∂φ
Here the Hubble function H(t) and the purely spatial part of the scalar curvature are
defined as, respectively:
R ( t )
k
H (t ) =
; K (t ) = 2
(7)
R (t )
R
φ + 3φH ( t ) +
and we have denoted the time derivatives with an over-dot.
In their 1991 paper, Ellis and Madsen [5], followed by other authors (see
[6],[7]) proposed a simple method for reconstructing a scalar field potential given a
particular form of evolution of the scale factor R(t). This method (called later
“reverse engineering” by Ellis) starts by solving eqs. no. (4) and (5) and we have:
1 
2
(8)
V (t ) =
H ( t ) + 3H ( t ) + 2 K ( t ) 

4π 
1
 − H ( t ) + K ( t ) 
φ2 =
(9)
4π 
The next step is to integrate the above eq. (9) for a certain case, prescribing first the
time behavior of the scale factor R(t) and expressing then all quantities in terms of
the φ–φ0. The examples initially processed by Ellis and Madsen in [5] and other
examples (see [7] and [13]) are summarized in the Table 1 (actually reproducing
the Table 1 from [13]).
2.1. REM FOR 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 ([9]–[12]).
Table 1
Ellis-Madsen potentials – Here we denoted with R0 the scale factor
at the actual cosmic time t0 and φ (t) – φ0 with α
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Valentin Bordea, Gabriel Cheva, Dumitru N. Vulcanov
4
This theory can be described by an effective scalar field with a lagrangian of the
form L = −V (φ ) 1 + ∂ iφ∂ iφ where the tachyonic potential V(φ) has a positive
maximum at the origin (i.e. V(φ) = V0 at φ = 0) and has a vanishing minimum
where the potential vanishes (i.e. V(φ) = 0 φ → +∞ ) . 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 in [9] and [13]
when we deal with spatially homogeneous geometry cosmology described with the
FRW metric (3) and φ = φ (t) we can follow the same steps as in Section 2 and we
have finally:
2 K ( t ) − H ( t )
(10)
φ2 =
3 K ( t ) + H ( t )2
V (t ) =
3
8π
H (t ) +
2
2 2
H (t ) + K (t ) H (t ) + K (t )
3
(11)
restricting for the moment to the pure scalar field case.
With these results in hand we can proceed now in processing different types
of scale factor behavior, as for the pure scalar field potential examples. The results
we obtained in [13] are summarized in the next Table 2 (reproduced from the
Table 2 in [13]).
3. REM FOR A PURE SCALAR FIELD WITH R(t) = R0 tan(ωt)
In this section we will process the REM for a universe having the time
evolution of the scale factor as
(12)
R(t) = R0 tan(ωt)
Table 2
Tachyonic potentials: here we denoted with R0 the initial scale factor and with α = φ(t) – φ0
5
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“Reverse engineering” method in certain cosmologies
showing off an expansion behavior as it can be seen from the graphs plotted in the
next Figure 1 for the time evolution of the scale factor (in the left panel) and of the
deceleration factor (right panel) defined as usual:
q (t ) = −
( t )
R
R (t ) H 2 (t )
=
1 − tan (ωt )
(
2
tan (ωt ) 1 + tan (ωt )
2
)
It can be observed a fast decreasing of the deceleration factor to zero, a more
detailed analysis of it showing that at certain time the deceleration goes to negative
values, i.e. to accelerated expansion. This justifies why we choose this type of
model for describing the late accelerated expansion of the universe.
Fig. 1 – Time behavior of the scale factor (left panel) and deceleration factor (right panel)
for ω = 1.
Replacing now (12) in (8) and (9) we finally obtain, after some manipulations
certain simple expressions for the potential and first derivative of the scalar field, namely:
2
V (t ) =
(
ω 2 + k + sin (ωt ) ω 2 − 2k + k sin (ωt )
2π cos (ωt ) cos (ωt )
2
φ ( t ) =
2
k + ω 2 − ω 2 tan (ωt )
2
π tan (ωt )
2
)
(13)
2
(14)
In the next series of Figures 2 and 3 we plotted the time evolution of the
potential for two different geometries – namely for closed (k=1) and open (k=–1)
universes for a certain value of the parameter ω (left panel) and for different values
of the ω in 3D representation (right panels).
Now integrating the expression of φ ( t ) from equation (14) above we get a
rather complicated expression for the scalar field, namely:
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Valentin Bordea, Gabriel Cheva, Dumitru N. Vulcanov
6
Fig. 2 – Time evolution of the potential for ω = 1 (left) and for a range of ω between 0.1 and 1 for
closed universes – k=1.
Fig. 3 – Time evolution of the potential for ω = 1 (left) and for a range of ω between 0.1 and 1 for
open universes – k= –1.
φ ( t ) = φ0 +
1
4ω π ( k + ω 2 )
−
 k ln ( 2 ) −
4
− k ln  k + ω 2 + k + ω 2 k + ω 2 − ω 2 ( tan (ωt ) )  + 2k ln ( tan (ωt ) ) −


4
−ω 2 ln ( 2 ) − ω 2 ln  k + ω 2 + k + ω 2 k + ω 2 − ω 2 ( tan (ωt ) )  +


2

ω ( tan (ωt ) )

2
+2ω ln ( tan (ωt ) ) − arctan 
4
 k + ω 2 − ω 2 ( tan (ωt ) )

(

ω k + ω 2 +



+ k k + ω 2 ln ( 2 ) + k k + ω 2 ln k + ω 2 + ω 2 ( tan (ωt ) ) +
2
(
)
4
2 
+ k k + ω 2 − ω 2 ( tan (ωt ) )  − k k + ω 2 ln 1 + ( tan (ωt ) ) 


(15)
7
233
“Reverse engineering” method in certain cosmologies
Graphically the behavior of the scalar field in time is showed off in the next
Figure 4 for a fixed value of the parameter ω = 1 (left panel) and in a 3D image for
a range of values of ω = 0.1 – 1 for a closed universe (k=1) and for Phi0 = 0.
Fig. 4 – Time evolution of the scalar field for ω = 1 (left) and for a range of ω between 0.1 and 1 for
closed universes – k=1.
Now theoretically we need to eliminate the time between the two equations,
one for the potential (13) and the above one for the scalar field (15). This was done
(see [5] and [13]) solving (15) for the time and replacing it in
Fig. 5 – Scalar field shape in terms of the potential for a fixed value of the parameter ω = 1 (left) and
for a range of values for ω = 0.1 – 0.5 (right) for closed (k=1) and open (k= –1) universes – solution
with + for tan(ωt).
the potential to obtain the potential in terms of the scalar field. The shape of the
scalar field in (15) kills any hope to do in this manner. Thus we proceeded in a
different way, actually solving the potential expression in (13) for the time, more
precisely for one of the trigonometric functions appearing there. We done this for
tan ω t and we had, after some straightforward calculation in Maple:
2k + 2V ( t ) π − ω 2 ± ω 4 − 12V ( t ) πω 2 + 4V ( t ) π 2 − 8ω 2 k
2
tan (ω t ) =
ω 2 + 2V ( t ) π + ω 4 − 12V ( t ) πω 2 + 4V ( t ) π 2 − 8ω 2 k
2
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Valentin Bordea, Gabriel Cheva, Dumitru N. Vulcanov
8
The only thing we need to do from now one is to replace this in the scalar
field expression (15). We obtained a very long and complicated analytic expression
for both solutions above. To save some space we do not show here the complete
expression, the only important conclusion we can have is that there it exist an
analytic solution and this can be graphically analyzed, which we actually done, in
the next two Figures 5 and 6.
4. REM FOR A TACHYONIC SCALAR FIELD WITH R(T) = R0 tan(ωt)
In this case, replacing directly (12) in (11) and (10) we get, after some
manipulations
Fig. 6 – Scalar field shape in terms of the potential for a fixed value of the parameter ω = 1 (left) and
for a range of values for ω = 0.1 – 0:5 (right) for closed (k=1) and open (k= –1) universes – solution
with – for tan(ω t).
2
2
3 k ( cos (ωt ) ) + 5ω − 4ω ( cos (ωt ) )
4
V (t ) =
φ ( t ) =
π
( sin ( 2ωt ) )
k + ω 2 − ω 2 ( tan (ωt ) )
2
ω 2 + k ( cos (ωt ) )
4
2
4
(16)
6
3 ω 2 + 2ω 2 ( tan (ωt ) )2 + ω 2 ( tan (ωt ) )4 + k
We can graphically study this potential, restricting ourselves to the at case (k=0)
which will be the case we worked mainly. The results for ω = 1 and for a range of
values of ω = [0.01..2] are plotted in the Figure 7.
Proceeding now further with the REM method, we need to integrate the
above expression for φ ( t ) and thus we have
2

1 − tan (ωt )
6
φ (t ) =
tan (ωt )
+
2
3ω 
(17)
1
tan
ω
t
+
(
)

+ EllipticF ( tan (ωt ) , i ) − EllipticE ( tan (ωt ) , i ) 
9
“Reverse engineering” method in certain cosmologies
235
In spite of the elliptic functions in the expression we get for the scalar field,
this is still possible to represent graphically as we done in the next Figure 8, where
we have the tachyonic scalar field behavior in time for a flat universe (k=0).
This opens the hope for further processing of the REM in this case. Of course
again we cannot use the direct method, thus as we done in the previous
Fig. 7 – The potential for a tachyonic scalar field versus time for a fixed value of the parameter ω = 1
(left) and for a range of values for ω = 0.1 – 2(right) for at (k=0).
Fig. 8 – The tachyonic scalar field versus time for a fixed value of the parameter ω = 1 (left) and for a
range of values for ω = 0.1 – 2(right) for at (k=0).
Fig. 9 – The tachyonic scalar field versus the potential for a fixed value of the parameter ω = 0.1 (left)
and for a range of values for ω = 0.1 – 1(right) for at (k=0).
section we solve the potential equation (16) above for tan(ω t) to have:
236
Valentin Bordea, Gabriel Cheva, Dumitru N. Vulcanov
tan (ω t ) =
4
1
3ω
2 V ( t ) π + 3ω 2
10
(18)
To obtain this simple expression we done some approximations taking the equation
(16) expanded in series around t = 0 and keeping only the first terms. We think that
there is still possible to have an exact solution but for our graphical purposes this is
enough for the moment as we restrict ourselves for short period of times after t = 0
considered as the actual moment in the universe development.
Of course again we have a huge complicated expression for the scalar field in
terms of the potential, but again we can have its graphical shape, which we output
in the next Figure 9. This last one contains the graph of the φ (t) in terms of the
potential again for a fixed value of the parameter ω = 0.1 (left panel) and for a
range of values of ω = 0.1 – 1.
5. CONCLUSIONS. FURTHER DEVELOPMENTS
The method of “reverse engineering” was described and used in order to
design potentials in cosmologies with a real scalar field (“quintessence”) or a
tachyonic scalar field for the case of a universe having the scale factor as R(t) = R0
tan(ω t). We investigated both cases in a different way as was done in ([5]) and ([13])
by obtaining the shape of the scalar field in terms of his potential, eliminating the
time between the time expressions of the potential and the scalar field. This was
done doe to the very complicated expression of the scalar field in terms of the
potential but even so we were able to graphically investigate the shape of the
potential, proving that the REM is feasible even in more complicated examples
than those already processed in ([5]) and ([13]). We obtained smooth and analytical
solutions, at least for certain interval values of the parameter ω.
It is in our view to extend REM to cases with matter (other than the scalar
field), as it is sketched in ([13]).
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