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Quintessence without the fine tuning problem of the potential
|.
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 87 (2000) 496--497
PROCEEDINGS SUPPLEMENTS www.elsevier.nl/Iocate/ripe
Quintessence without the fine tuning problem of the potential A. B. Kaganovich a* aPhysics Department, Ben Gurion University of the Negev Beer Sheva 84105, Israel An imitation of the present cosmological constant by the slowly-rolling scalar field ¢ (quintessence) requires an extreme free tuning of the potential U(¢), that is the problems appear related to the cosmologicM constant problem and flatness conditions. The field theory is presented which gives rise to the quintessence potential without any fine tuning in the very wide class of models. At the same time the models reproduce equations of Einstein's GR and allow for possibility of inflation in the very early universe.
1. F I N E T U N I N G
PROBLEM
a)For quintessence to be the dominant fraction of the present energy density, its potential must be of order U(¢) N (10-3eV) 4 and the summary contribution of the rest of the fields to the vacuum energy density is zero or significantly less than U(¢). This situation is much more problematic than that with the cosmological constant (CC) problem in the old sense, b)To provide a possibility of a slow-rolling, the potential U(¢) has to obey the flatness conditions. The exponential potential[l] U(¢) = m4e ~/MPz obeys the flatness conditions if e << 1. The inverse power low potential[2] U(¢) = m 4 + n / ¢ n satisfies the flatness conditions as ¢ >> n M p t . However it is not clear what happens with other possible terms in the potential, including quantum corrections[3]. In fact, the potential may contain terms that constitute a structure of polynomials in ¢ and Cnin ¢ and they are not negligible as ¢ is big enough, unless an extreme fine tuning is assumed for the mass and self-couplings[3]. For example, the restriction of the flatness conditions on the quartic self-interaction A¢4 is A << 10-12°(~e-t) 2. In this short paper I report that the very wide class of models in the framework the Two Measures Theory (TMT) [41, [5] is able to solve the fine tuning problem and to give a constructive answer to some other questions (like for example why one can expect that the possible couplings * Email address: alexk~bgumaiLbgu.ac.fl
of the quintessence field to the standard matter fields are very weak[6]). 2. W H A T IS T M T T M T was developed with the aim to solve the CC problem. The action in T M T has a general form including two measures and respectively, two Lagrangian densities S = f (+L1 + v/-Z-gL2) d4x
(1)
where the measure ffP is defined using the antisymmetric tensor field A , , ~ (bdax = O [ ~ A ~ ] dx ~ A dx z A dx "r A dx ~
(2)
and the Lagrangian densities L1, L2 do not depend explicitly on A~,,a. In TMT, a wide class of field theory models exists[5] where the CC problem does not appear at all. By this statement we mean that the models possess all the following three features: 1. A conformal Einstein frame exists where equations of motion coincide with appropriate equations of Einstein's GR (Newton constant G, masses and coupling constants are constants in the same frame), that is the models are free of the well known defects of the scalartensor theories. 2. The energy density of the true vacuum state appears to be zero without any sort of fine tuning for infinite number of initial conditions. This happens also in models (including gauge models) where the true vacuum state
0920-5632/00/$ - see front maUer © 2000 Elsevier Science B.V. All rights reserved. PII S0920-5632(00)00729-5
A.B. Kaganovich/Nuclear Physics B (Proc. Suppl.) 87 (2000) 496-497
is realized due SSB. 3. The models allow for the very early universe undergoes an inflationary stage governed by the vacuum energy density. Notice that the measure ~ is a total derivative and therefore a shift L1 ~ L1 + const does not affect equations of motion whereas a similar shift when implementing with L2 is equivalent to a change of the CC term. Let us consider a simple model with the scalar field ¢: 1 ~ ~ L1 = - ~ R ( P , g ) + ~g 9,~,~ - V1(¢), L2 V2(¢) in the first order formalism (metric g~,, and connection F ~ are independent variables as well as the antisymmetric tensor field Au~). Varying the action with respect to A,u~ we obtain eu~ZOzL1 = 0 which means that L1 = M = const. Variation with respect to g~" gives - ~ R ~ ( F ) + ½¢,~,¢,~ - ~-~V2(¢)gu~ = 0 where the scalar field X is defined by X =- ~_g. Consistency condition of obtained equations has the form of the constraint 1/1(¢) + M
2V2(¢) _ 0 X
(3)
In the conformal Einstein frame defined by - ~ , ( x ) = xg~,u(x), all equations of motion takes the canonical GR f o r m with the TMT effective
C-field potential U(¢) =
1 V2(¢) = 4V~(¢) [M +
v1(¢)f
(4)
For the general enough forms of the pre-potentials V1(¢) and V2(¢), the TMT effective potential (4) provides us with the possibility to conclude (see [5]) that TMT solves the CC problem.
3. Q U I N T E S S E N C E W I T H O U T THE TUNING
POTENTIAL N E E D OF FINE
If the pre-potentials are polynomials in ¢, 6~ i.e. V1(¢) = ~-~i=oA(1)iM4~i¢ i and V2(¢) = ~2b i=o A(2)i M4_i.~i Pl v. and b > a :> 0 then for ¢ large enough, only the leading terms of the prepotentials determine the behavior of the TMT effective potential which takes the inverse power ~(1)2 ~ 4 + 2 ( b - o )
low form U(¢) _ ~ ~ . Notice that the term f ~ Z g A d 4 x which would be the CC
497
term in standard GR, is among the nonleading terms which do not contribute to U(¢) in the quintessence regime. Taking the exponential form for the prepotentials V1 = m4e a¢/MPz , V2 -~ m4e 2~¢~/Mpl with ~ > a > 0, we obtain for/3¢ >> MpI the exponential form U(¢) - ~4 mg 2e -2(fl-a)¢/Mp~ Notice that adding the polynomial terms to the exponential form of V1 and V2 does not affect this result. The flatness conditions are satisfied if /3- a << 1. It is interesting that in the case a =/3 the theory possesses the global symmetry[7] that contains global shifts ¢ ~ ¢ + eonst. So, if the exponential potential that provides with a possibility for a slow-rolling, is realized due to a small breaking of the shifts symmetry, then the possibility appears to answer the very difficult question why the direct coupling of ¢ to the usual matter fields is suppressed[6]. One can show that a change in C-field during its slow rolling is of the order of MpI < ¢ and the Xfield is a constant with extremely high accuracy (it is of the order of 106o + 10s°, depending on the model). This allows to define a cosmological background where quantum field theory and particle physics are realized. In particular, quantum corrections to the TMT effective potential U(¢) do not alter the conclusion that there is no necessity of a fine tuning of the potential to achieve a quintessence. "~
REFERENCES 1. B. Ratra and P.J.E. Peebles, Phys. Rev. D37 (1988) 3406. 2. P.J. Steinhardt, L.Wang and I. Zlatev, Phys. Rev. D59 (1999)123504. 3. C. Kolda and D.H. Lyth, Phys. Lett. B458 (1999) 197. 4. E.I. Guendelman and A.B. Kaganovich, Phys. Rev. D53 (1996) 7020; D55 (1997) 5970; D56 (1997) 3548; D57 (1998) 7200. 5. E.I. Guendelman and A.B. Kaganovich, Phys. Rev. D60 (1999) 065004. 6. S.M. Carroll, Phys. Rev. Lett. 81 (1998) 3067. 7. E.I. Guendelman, Mod. Phys. Lett. A14 (1999) 1397; A14 (1999) 1043.
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 87 (2000) 496--497
PROCEEDINGS SUPPLEMENTS www.elsevier.nl/Iocate/ripe
Quintessence without the fine tuning problem of the potential A. B. Kaganovich a* aPhysics Department, Ben Gurion University of the Negev Beer Sheva 84105, Israel An imitation of the present cosmological constant by the slowly-rolling scalar field ¢ (quintessence) requires an extreme free tuning of the potential U(¢), that is the problems appear related to the cosmologicM constant problem and flatness conditions. The field theory is presented which gives rise to the quintessence potential without any fine tuning in the very wide class of models. At the same time the models reproduce equations of Einstein's GR and allow for possibility of inflation in the very early universe.
1. F I N E T U N I N G
PROBLEM
a)For quintessence to be the dominant fraction of the present energy density, its potential must be of order U(¢) N (10-3eV) 4 and the summary contribution of the rest of the fields to the vacuum energy density is zero or significantly less than U(¢). This situation is much more problematic than that with the cosmological constant (CC) problem in the old sense, b)To provide a possibility of a slow-rolling, the potential U(¢) has to obey the flatness conditions. The exponential potential[l] U(¢) = m4e ~/MPz obeys the flatness conditions if e << 1. The inverse power low potential[2] U(¢) = m 4 + n / ¢ n satisfies the flatness conditions as ¢ >> n M p t . However it is not clear what happens with other possible terms in the potential, including quantum corrections[3]. In fact, the potential may contain terms that constitute a structure of polynomials in ¢ and Cnin ¢ and they are not negligible as ¢ is big enough, unless an extreme fine tuning is assumed for the mass and self-couplings[3]. For example, the restriction of the flatness conditions on the quartic self-interaction A¢4 is A << 10-12°(~e-t) 2. In this short paper I report that the very wide class of models in the framework the Two Measures Theory (TMT) [41, [5] is able to solve the fine tuning problem and to give a constructive answer to some other questions (like for example why one can expect that the possible couplings * Email address: alexk~bgumaiLbgu.ac.fl
of the quintessence field to the standard matter fields are very weak[6]). 2. W H A T IS T M T T M T was developed with the aim to solve the CC problem. The action in T M T has a general form including two measures and respectively, two Lagrangian densities S = f (+L1 + v/-Z-gL2) d4x
(1)
where the measure ffP is defined using the antisymmetric tensor field A , , ~ (bdax = O [ ~ A ~ ] dx ~ A dx z A dx "r A dx ~
(2)
and the Lagrangian densities L1, L2 do not depend explicitly on A~,,a. In TMT, a wide class of field theory models exists[5] where the CC problem does not appear at all. By this statement we mean that the models possess all the following three features: 1. A conformal Einstein frame exists where equations of motion coincide with appropriate equations of Einstein's GR (Newton constant G, masses and coupling constants are constants in the same frame), that is the models are free of the well known defects of the scalartensor theories. 2. The energy density of the true vacuum state appears to be zero without any sort of fine tuning for infinite number of initial conditions. This happens also in models (including gauge models) where the true vacuum state
0920-5632/00/$ - see front maUer © 2000 Elsevier Science B.V. All rights reserved. PII S0920-5632(00)00729-5
A.B. Kaganovich/Nuclear Physics B (Proc. Suppl.) 87 (2000) 496-497
is realized due SSB. 3. The models allow for the very early universe undergoes an inflationary stage governed by the vacuum energy density. Notice that the measure ~ is a total derivative and therefore a shift L1 ~ L1 + const does not affect equations of motion whereas a similar shift when implementing with L2 is equivalent to a change of the CC term. Let us consider a simple model with the scalar field ¢: 1 ~ ~ L1 = - ~ R ( P , g ) + ~g 9,~,~ - V1(¢), L2 V2(¢) in the first order formalism (metric g~,, and connection F ~ are independent variables as well as the antisymmetric tensor field Au~). Varying the action with respect to A,u~ we obtain eu~ZOzL1 = 0 which means that L1 = M = const. Variation with respect to g~" gives - ~ R ~ ( F ) + ½¢,~,¢,~ - ~-~V2(¢)gu~ = 0 where the scalar field X is defined by X =- ~_g. Consistency condition of obtained equations has the form of the constraint 1/1(¢) + M
2V2(¢) _ 0 X
(3)
In the conformal Einstein frame defined by - ~ , ( x ) = xg~,u(x), all equations of motion takes the canonical GR f o r m with the TMT effective
C-field potential U(¢) =
1 V2(¢) = 4V~(¢) [M +
v1(¢)f
(4)
For the general enough forms of the pre-potentials V1(¢) and V2(¢), the TMT effective potential (4) provides us with the possibility to conclude (see [5]) that TMT solves the CC problem.
3. Q U I N T E S S E N C E W I T H O U T THE TUNING
POTENTIAL N E E D OF FINE
If the pre-potentials are polynomials in ¢, 6~ i.e. V1(¢) = ~-~i=oA(1)iM4~i¢ i and V2(¢) = ~2b i=o A(2)i M4_i.~i Pl v. and b > a :> 0 then for ¢ large enough, only the leading terms of the prepotentials determine the behavior of the TMT effective potential which takes the inverse power ~(1)2 ~ 4 + 2 ( b - o )
low form U(¢) _ ~ ~ . Notice that the term f ~ Z g A d 4 x which would be the CC
497
term in standard GR, is among the nonleading terms which do not contribute to U(¢) in the quintessence regime. Taking the exponential form for the prepotentials V1 = m4e a¢/MPz , V2 -~ m4e 2~¢~/Mpl with ~ > a > 0, we obtain for/3¢ >> MpI the exponential form U(¢) - ~4 mg 2e -2(fl-a)¢/Mp~ Notice that adding the polynomial terms to the exponential form of V1 and V2 does not affect this result. The flatness conditions are satisfied if /3- a << 1. It is interesting that in the case a =/3 the theory possesses the global symmetry[7] that contains global shifts ¢ ~ ¢ + eonst. So, if the exponential potential that provides with a possibility for a slow-rolling, is realized due to a small breaking of the shifts symmetry, then the possibility appears to answer the very difficult question why the direct coupling of ¢ to the usual matter fields is suppressed[6]. One can show that a change in C-field during its slow rolling is of the order of MpI < ¢ and the Xfield is a constant with extremely high accuracy (it is of the order of 106o + 10s°, depending on the model). This allows to define a cosmological background where quantum field theory and particle physics are realized. In particular, quantum corrections to the TMT effective potential U(¢) do not alter the conclusion that there is no necessity of a fine tuning of the potential to achieve a quintessence. "~
REFERENCES 1. B. Ratra and P.J.E. Peebles, Phys. Rev. D37 (1988) 3406. 2. P.J. Steinhardt, L.Wang and I. Zlatev, Phys. Rev. D59 (1999)123504. 3. C. Kolda and D.H. Lyth, Phys. Lett. B458 (1999) 197. 4. E.I. Guendelman and A.B. Kaganovich, Phys. Rev. D53 (1996) 7020; D55 (1997) 5970; D56 (1997) 3548; D57 (1998) 7200. 5. E.I. Guendelman and A.B. Kaganovich, Phys. Rev. D60 (1999) 065004. 6. S.M. Carroll, Phys. Rev. Lett. 81 (1998) 3067. 7. E.I. Guendelman, Mod. Phys. Lett. A14 (1999) 1397; A14 (1999) 1043.