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RECONSTRUCTION PROCEDURE IN MODIFIED GRAVITY COSMOLOGICAL MODELS
S.Yu. Vernov

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University
QFTHEP'2013, Repino, 24.06.2013

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MODIFIED GRAVITY MODELS
Modified gravity cosmological models have been proposed in the hope of finding solutions to the important open problems of the standard cosmological model. There are lots of ways to deviate from Einstein's gravity: F (R ) gravity Addition of higher-derivative terms to the Einstein­Hilbert action Nonlocal gravity S. Capozziello and V. Faraoni, Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics, Fund. Theor. Phys. 170, Springer, NY, 2011

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MODIFIED GRAVITY MODELS
Modified gravity cosmological models have been proposed in the hope of finding solutions to the important open problems of the standard cosmological model. There are lots of ways to deviate from Einstein's gravity: F (R ) gravity Addition of higher-derivative terms to the Einstein­Hilbert action Nonlocal gravity S. Capozziello and V. Faraoni, Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics, Fund. Theor. Phys. 170, Springer, NY, 2011

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MODIFIED GRAVITY MODELS
Modified gravity cosmological models have been proposed in the hope of finding solutions to the important open problems of the standard cosmological model. There are lots of ways to deviate from Einstein's gravity: F (R ) gravity Addition of higher-derivative terms to the Einstein­Hilbert action Nonlocal gravity S. Capozziello and V. Faraoni, Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics, Fund. Theor. Phys. 170, Springer, NY, 2011

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MODIFIED GRAVITY MODELS
Modified gravity cosmological models have been proposed in the hope of finding solutions to the important open problems of the standard cosmological model. There are lots of ways to deviate from Einstein's gravity: F (R ) gravity Addition of higher-derivative terms to the Einstein­Hilbert action Nonlocal gravity S. Capozziello and V. Faraoni, Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics, Fund. Theor. Phys. 170, Springer, NY, 2011

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Formulation of Nonlocal Gravity via Scalar Fields
A mo dification that assumes the existence of a new dimensional parameter M can be of the form S= d 4 x -g
2 MP 1 R + RF 2 2

M

2

R -

(1)

where M is the mass scale at which the higher derivative 2 terms in the action become important, 8 GN = 1/MP . 2 n An analytic function F ( /M ) = fn .
n0

Biswas T., Mazumdar A., and Siegel W., 2006, JCAP 0603 009 (hep-th/0508194) Biswas T., Koivisto T., and Mazumdar T., 2010, JCAP 1011 008 (arXiv:1005.0590) Biswas T., Koshelev A.S., Mazumdar T., Vernov S.Yu., JCAP 1208 (2012) 024 (arXiv:1206.6374)
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By virtue of the field redefinition one can transform the non-local gravity action (1) as follows: S= d 4 x -g
2 MP 1 (1 + )R + F 2 2

M

2

-

2 MP - 2 (2)

with two new scalar fields and . Variation w.r.t. gives = R and, therefore, the connection (2) with action (1) is obvious.

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A modification that does not assume the existence of a new dimensional parameter in the action
S2 = d 4 x -g
-1

1 R 1+f( 16 GN

-1

R ) - 2 + Lm

, (3)

The term R is dimensionless and it can appear prefactor for the Newtonian gravitational constant, weakening of gravity at cosmological scales. The action (3) can be rewritten by introducing two fields and in the following form: -g 4 ~ S2 = d x {[R (1 + f ( )) + ( - R ) - 16 GN By the variation over , we obtain = R . Substituting = -1 R into (4), one reobtains (3).
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as a and explain scalar 2] + Lm } . (4)


For the model, describing by the initial nonlocal action, a technique for choosing the distortion function so as to fit an arbitrary expansion history has been derived in C. Deffayet and R.P. Woodard, JCAP 0908 (2009) 023, [arXiv:0904.0961]. For the local model, contained a perfect fluid with a constant state parameter wm , a reconstruction procedure has been made in T.S. Koivisto, Phys. Rev. D 77 (2008) 123513, [arXiv:0803.3399] and E. Elizalde, E.O. Pozdeeva, and S.Yu. Vernov, Class. Quantum Grav. 30 (2013) 035002, [arXiv:1209.5957].

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SUPERPOTENTIAL METHOD
The Hamilton­Jacobi formulation (superpotential method) has been proposed in the cosmological models with minimally coupling scalar field: A.G. Muslimov, Class. Quant. Grav. 7 (1990) 231­237; D.S. Salopek, J.R. Bond, Phys. Rev. D 42 (1990) 3936­3962; and has been develop in: I.Ya. Aref 'eva, A.S. Koshelev, S.Yu. Vernov, Phys. Rev. D 72 (2005) 064017, astro-ph/0507067; D. Bazeia, C.B. Gomes, L. Losano, R. Menezes, Phys. Lett. B 633 (2006) 415­419; astro-ph/0512197; K. Skenderis, P.K. Townsend, Phys. Rev. D 74 (2006) 125008, hep-th/0609056; A.A. Andrianov, F. Cannata, A.Yu. Kamenshchik, and D. Regoli, JCAP 0802 (2008) 015, arXiv:0711.4300
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SUPERPOTENTIAL METHOD
The Hamilton­Jacobi formulation (superpotential method) has been proposed in the cosmological models with minimally coupling scalar field: A.G. Muslimov, Class. Quant. Grav. 7 (1990) 231­237; D.S. Salopek, J.R. Bond, Phys. Rev. D 42 (1990) 3936­3962; and has been develop in: I.Ya. Aref 'eva, A.S. Koshelev, S.Yu. Vernov, Phys. Rev. D 72 (2005) 064017, astro-ph/0507067; D. Bazeia, C.B. Gomes, L. Losano, R. Menezes, Phys. Lett. B 633 (2006) 415­419; astro-ph/0512197; K. Skenderis, P.K. Townsend, Phys. Rev. D 74 (2006) 125008, hep-th/0609056; A.A. Andrianov, F. Cannata, A.Yu. Kamenshchik, and D. Regoli, JCAP 0802 (2008) 015, arXiv:0711.4300
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The key point in this method is that the Hubble parameter is considered as a function of the scalar field. For models with non-minimally coupling scalar field this method has been delevoped: A.Yu. Kamenshchik, A. Tronconi, G. Venturi, and S.Yu. Vernov, Phys. Rev. D 87 (2013) 063503, arXiv:1211.6272 The superpotential method is actively used in models with extra spatial dimensions: A. Brandhuber, K. Sfetsos, J. High Energy Phys. 9910 (1999) 013; hep-th/9908116; O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Phys. Rev. D 62 (2000) 046008; hep-th/9909134; A.S. Mikhailov, Yu.S. Mikhailov, M.N. Smolyakov, I.P. Volobuev, Class. Quant. Grav. 24 (2007) 231­242, hep-th/0602143.
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MODELS WITH NON-MINIMALLY COUPLING SCALAR FIELDS
Let us consider the model with the following action S= 1 d 4 x -g U ( )R - g µ ,µ , - V ( ) , 2
2 2 ds 2 = - dt 2 + a2 (t ) dx1 + dx2 + dx 2 3

In FLRW metric: ,

we get the following equations: 1 6UH 2 + 6U H = 2 + V , 2 2U 2H + 3H 1 ¨ + 4 U H + 2U = - 2 + V . 2 + 3H + V, = 6 H + 2H 2 U, . ¨
2
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(5) (6)


Combining Eqs. (5) and (6) we obtain: ¨ 4U H - 2U H + 2U + 2 = 0. (7)

This equation plays a key role in the reconstruction procedure. Let H = Y ( ), and = F ( ). Substituting H , and = F, F into (7), we obtain: ¨ 4UY, + 2(F, - Y )U, + (2U, + 1)F = 0. Equation (8) contains three functions. If two of them are given, then the third one can be found as the solution of a linear differential equation. The potential V ( ) can then be obtained from (5): 1 V ( ) = 6UY 2 + 6U, FY - F 2 . 2
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(8)

(9)


If U ( ) and F ( ) are given, then 2F, U, + (2U, ~~ Y ( ) = - 4U 3/2

~~

+ 1)F

d + c0 U (10) ~



For given Y ( ) and U ( ), we obtain U, Y - 2UY, ~ ~ F ( ) = e d + c0 e ~~ U, ~ where 1 ( ) 2


-( )

,

(11)

2U, + 1 ~~ d . ~ U, ~

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For the case of induced gravity U ( ) = 2 the reconstruction procedure has been proposed in A.Yu. Kamenshchik, A. Tronconi, G. Venturi, Reconstruction of scalar potentials in induced gravity and cosmology, Phys. Lett. B 702 (2011) 191­196, arXiv:1104.2125. They have not used the superpotential method and got a lot of potential for different types of the Hubble behaviors. There are two main reasons to use the superpotential method: U ( ) can be arbitrary function. H (t ) can be more complicated than H = Y ( ). A.Yu. Kamenshchik, A. Tronconi, G. Venturi, and S.Yu. Vernov, Phys. Rev. D 87 (2013) 063503, arXiv:1211.6272. The two methods supplement each other and together allow one to construct different cosmological models with some required properties.
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For the case of induced gravity U ( ) = 2 the reconstruction procedure has been proposed in A.Yu. Kamenshchik, A. Tronconi, G. Venturi, Reconstruction of scalar potentials in induced gravity and cosmology, Phys. Lett. B 702 (2011) 191­196, arXiv:1104.2125. They have not used the superpotential method and got a lot of potential for different types of the Hubble behaviors. There are two main reasons to use the superpotential method: U ( ) can be arbitrary function. H (t ) can be more complicated than H = Y ( ). A.Yu. Kamenshchik, A. Tronconi, G. Venturi, and S.Yu. Vernov, Phys. Rev. D 87 (2013) 063503, arXiv:1211.6272. The two methods supplement each other and together allow one to construct different cosmological models with some required properties.
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Models with non-minimally coupled scalar fields are interesting because of their connection with particle physics. The are models of inflation, where the role of the inflaton is played by the Higgs field non-minimally coupled to gravity. (F.L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659 (2008) 703­706, arXiv:0710.3755). For such models U ( ) = 2 + J . (12) Y ( ) = -


4 F, + (4 + 1)F ~~ d + c0 ~ 4( 2 + J )3/2 ~ 2 + J ,

F ( ) =

Y - 2 + ~ ~

J

Y,

~

~

1 4

d + c0 ~~



-

1+4 4

.

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Mo dels with de Sitter solutions
Let us consider the general form of the potential V ( ), which leads to the existence of the de Sitter solution H = Y ( ) = H0 = const . Using (11), we obtain (for U = 2 + J ) F ( ) =


(13)

H0 e d + c0 e ~~
-

=

4 H0 + c0 ~ 8 + 1
4 8+1

-

1+4 4

.

(t ) = 0 e

H0 t

c0 (8 + 1) ~ + H0

,

(14)

where 0 is an arbitrary constant.
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V ( ) = 2H -

2 0

3J +
-(4

c0 ~2 2

(3 + 32 )(1 + 12 ) 2 - (8 + 1)2 8(12 + 1) +1)/(2 ) + H0 c0 -1 ~ 8 + 1

/(4 )

.

For c0 = 0, F ( ) is a linear function, ~ V = 2H
2 0

3J +

(3 + 32 )(1 + 12 ) (8 + 1)2

2

.

c0 ~2 . 2 The same result has been obtained by the method proposed in A.Yu. Kamenshchik, A. Tronconi, G. Venturi, Phys. Lett. B 702 (2011) 191. At = -1/4,
2 2 ~ V = -5H0 2 - 4H0 c0 + 6H0 J0 -
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The case = -1/8
In the case = -1/8, we get: F ( ) = H0 ln 0 ,

where 0 is an integration constant, and the corresponding potential has the following form:
V=
2 H0 2 24J - 4 2

ln

2

0

+3+



3

ln

2

0

+3-



3

.

The scalar field evolution is given by (t ) = 0 exp e
H0 (t -t0 )

.

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Solutions with the hyperbolic tangent
Let us construct cosmological models, when the Hubble parameter is a function of the hyperbolic tangent. (t ) = A tanh [ (t - t0 )] , where A, and t0 are constants. Note that t0 can be complex, so the parametrization (15) includes the functions (t ) = A coth [ (t - t0 )] as well. For such functions = A- 1 A
2

(15)

= F ( ).

(16)

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To get the desired Hubble parameter evolution (with constant H attractors in the past and in the future), we assume H = Y ( ) = B - C , where B and C are constants. Equation (8) becomes the following equation for U ( ): 2(A2 - 2 )U, +2 [(C - 2) - B ] U, -4CU +(A2 - 2 ) = 0, where = A. A particular solution of this equation is the second degree polynomial U ( ) = - B 2A2 C + 4A2 2 - B 2 12 + + , 12 6(2 + C ) 12(2 + C )C

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Let us consider, for example, A = B = C = 1. We get U ( ) = - 12 1 (2 + 42 - 1) + + , 12 6(2 + 1) 12(2 + 1)

hence, U ( ) = 0 at 1,
2

1 ± 2 22 + 23 = . 2 + 1

U (1) > 0 for all > 0, so, if we choose as a solution (t ) = tanh(t ), then U (t ) is positive at late times. When = 1, U ( (t )) 0 at any time because -1 (t ) 1.

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U

V





Figure : U ( ) and V ( ) at A = +B = C = = 1.

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By the change of variable = - 1/3, we get ~ U ( ) = - ~ In terms of , we finally obtain ~ Y ( ) = ~ F ( ) = ~ 2 - , ~ 3 12 4 + , ~ 12 27 (17)

(2 - 3 ) (4 + 3 ) ~ ~ . 9 So, we found a model with exact solutions and U ( ) in the ~ form U ( ) = 2 + J . ~ ~

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Non-monotonic behavior of the Hubble Parameter in the Case of Induced Gravity
We put U ( ) = 2 . Let us consider Y ( ) as a quadratic polynomial: Y ( ) = A2 2 + A1 + A0 , where Ak are constants. We obtain that F ( ) does not depend on A1 : 4 ((16 + 1)A0 - (8 + 1)A2 2 ) + c0 ~ (8 + 1)(16 + 1)
1+4 4

(18)

F ( ) =

-

. (19)

We assume that = -1/8 and = -1/16.
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When c0 = 0, F ( ) is a cubic polynomial and the equation ~ = F ( ) has the following general solution: (t ) = ± (16 + 1)A0 (16 + 1)A0 c2 e
- t

+ (8 + 1)A2

,

(20)

where = 8 A0 /(8 + 1), c2 is an arbitrary integration constant. The corresponding potential, V ( ), is the sixth degree polynomial which, for example, when = 1 has the following form: V ( ) = 910 2 6 156 A + A1 A2 5 + 289 2 17 2236 52 910 2 2 + 6A2 + A0 A2 4 + A0 A1 3 + A . 1 153 3 81 0
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The cosmological consequences.
H H H

t

t

t

Figure : The function H (t ) with A1 = -6, A1 = -4, and A1 = 0 (from left to right). At all pictures we use A2 = 1, A0 = 2, and c2 = 100000.

The same functions (t ) is associated with different behaviors of the Hubble parameter. At A1 = -4 we get a non-monotonic behavior of H (t ).
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Conclusion
A gravity model with a non-minimally coupling scalar field and U ( ) = 2 + J has been considered. The superpotential method has been used for the reconstruction procedure. We do not need the expression of the Hubble parameter in terms of the cosmic time or of the scale factor. We have found the potentials and the corresponding evolutions of the associated scalar field leading to de Sitter solutions. We have investigated a few models having a different de Sitter asymptotic behaviour in the past and in the future. Non-monotonic behaviour have been found.
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