It is shown that any realistic model of quintessence should be based on
Supergravity (SUGRA) since the value of the quintessence field on the attractor
is approximately the Planck mass. Under very general assumptions, the typical
shape of a SUGRA tracking potential is derived. Cosmological implications are
investigated. In particular, it is demonstrated that, generically, the equation
of state parameter is driven to a value close to -1 in agreement with recent
observations.Comment: 4 pages, 2 figures. Contribution to the Proceedings of Moriond 2000
  "Energy Densities in the Universe", Les Arcs, France, January 22-29 200

Brax, Philippe

Martin, Jerome

English

arXiv

4 pages, 2 figures. Contribution to the Proceedings of Moriond 2000 "Energy Densities in the Universe", Les Arcs, France, January 22-29 2000It is shown that any realistic model of quintessence should be based on Supergravity (SUGRA) since the value of the quintessence field on the attractor is approximately the Planck mass. Under very general assumptions, the typical shape of a SUGRA tracking potential is derived. Cosmological implications are investigated. In particular, it is demonstrated that, generically, the equation of state parameter is driven to a value close to -1 in agreement with recent observations

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High Energy Physics and Quintessence

It is shown that any realistic model of quintessence should be based on Supergravity (SUGRA) since the value of the quintessence field on the attractor is approximately the Planck mass. Under very general assumptions, the typical shape of a SUGRA tracking potential is derived. Cosmological implications are investigated. In particular, it is demonstrated that, generically, the equation of state parameter is driven to a value close to -1 in agreement with recent observations.It is shown that any realistic model of quintessence should be based on Supergravity (SUGRA) since the value of the quintessence field on the attractor is approximately the Planck mass. Under very general assumptions, the typical shape of a SUGRA tracking potential is derived. Cosmological implications are investigated. In particular, it is demonstrated that, generically, the equation of state parameter is driven to a value close to -1 in agreement with recent observations

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HIGH ENERGY PHYSICS AND QUINTESSENCEaP. BRAX∗ and J. MARTIN†∗ CERN, Theory division, Gene`ve, Switzerland,and Service de Physique The´orique, CEA-Saclay, F-91191 Gif/Yvette Cedex,†DARC, Observatoire de Paris, UMR 8629 CNRS, 92195 Meudon Cedex, France.It is shown that any realistic model of quintessence should be based on Supergravity (SUGRA)since the value of the quintessence eld on the attractor is approximately the Planck mass.Under very general assumptions, the typical shape of a SUGRA tracking potential is derived.Cosmological implications are investigated. In particular, it is demonstrated that, generically,the equation of state parameter is driven to a value close to −1 in agreement with recentobservations.If conrmed, the discovery that our Universe is presently undergoing a phase of acceleratedexpansion 1 has clearly important implications for Cosmology and High Energy Physics. Inparticular, a crucial question is to identify the physical nature of the dark matter componentresponsible for this acceleration which would represent 70% of the matter content of the Universe.A natural candidate is obviously the cosmological constant. The cosmological constant energydensity includes in fact contributions from two origins. The rst contribution is due to thebared cosmological, 0, which appears in the Einstein equations as a \free parameter" andthe second one is due to the zero-point fluctuations of the quantum elds. This gives rise toenergy densities respectively equal to 0c4/(8piG) and (hc/2)∫d3kk/(2pi)3  hck4max/(16pi2). Inthe last expression, kmax is a cut-o which can naturally be taken as the Planck wavenumber.As a consequence, the second contribution becomes huge. Since the observed value is tiny(0.7 times the critical energy density ρc), one needs to ne-tune very accurately the value ofthe bared constant 0 such that the previous simple theoretical predictions be compatible withobservational data. Let us emphasize that the discovery that the Universe is accelerating rendersthis problem even worse than before. Indeed, it is probably easier to explain why there is anexact cancellation between the two contributions such that ρΛ = 0 (due, for example, to someaCERN-TH/2000-078so far unknown symmetry in quantum gravity) than nding a reason for a non-vanishing tinyvalue ρΛ = 0.7 ρc 6= 0.This problem motivates the search for alternative explanations. Among them is the quintessencehypothesis. It consists in assuming that the dark matter component is an homogeneous scalareld Q(t), named quintessence, such that ΩQ = 0.7 (and therefore ΩΛ = 0). However, this isnot sucient to be an interesting proposal. In addition, the shape of the potential V (Q) mustpossess some typical features. Roughly speaking, the parameter Γ dened by the expressionΓ  (d2V (Q)/dQ2)V (Q)/(dV (Q)/dQ)2 must be greater than one and almost constant 2. Theprototype of such a potential is given by V (Q) = 4+α/Qα 3,4 where  and α > 0 are two freeparameters.Let us now examine briefly what are the main properties of this \tracker" quintessenceeld. In order to have ΩQ = 0.7, we must tune the value of . For example, if α = 6, onehas   106GeV, a scale compatible with the typical scales of High Energy Physics. In thissense, the ne-tuning problem is less important than for the cosmological constant since thefree parameter of the potential can now take a \natural" value. This would not have been thecase for the potential V (Q) = (1/2)m2Q2 where the only free parameter must be chosen suchthat m  10−33eV, certainly an \unnatural" value. For the tracking potential the situation isdierent since the mass, dened as the second derivative of the potential, has also this smallvalue; but, as we have seen, this is not the result of an articial choice. This is rather theconsequence of the shape of the potential and of the reasonable value of the free parameter .Therefore, even if the problem has not been completely overcome, something has been gained.Having xed the value of  and α, we now turn to the study of the dependence of the nalresult on the initial conditions. This is the most interesting property of the tracking potential. Itcan be shown that, due to the presence of an attractor, the nal result is completely independentof the initial conditions. The initial value of the energy density can vary in a range of 100 ordersof magnitude: 1061GeV4 < ρQ < 10−37GeV4. Another nice property of such models is that itturns out that the equation of state parameter, ωQ  pQ/ρQ, is naturally such that −1 < ωQ < 0.However, it is not possible to obtain a value less than ωQ = −0.7 which could be a problemsince observations seem to suggest a value close to −1 5.One of the most interesting challenge is to derive the tracking properties from generic HighEnergy Physics considerations. We now show briefly that this is indeed possible and that thisleads to a dierent shape for the potential for which all the previous nice properties are preserved.In addition some of the problems of the previous approach are overcome. Let us consider severalpossible options for the particle physics origin of the quintessence eld. For that it is relevantto express the mass of the quintessence eld in terms of the Hubble constantd2V (Q)d2Q=921 + αα(1− ω2Q)H2. (1)This has two immediate consequences. First of all, as already mentioned above, the mass ofthe quintessence eld is extremely light compared to the particle physics scales. This impliesthat the quintessence eld could mediate an extremely long range fth force. This violatesexperimental results entailling that the quintessence eld must be decoupled from ordinarymatter. On the other hand, the value of Q now (at vanishing redshift) is of order mPl as can beseen on the previous formula. This strongly suggests that High Energy Physics must be takeninto account in trying to nd the origin of the quintessence eld. We will only consider modelsbeyond the standard model of particle physics which incorporate supersymmetry. Moreoverwe shall explicitly consider the supergravity eects as they cannot be neglected for Q  mPl.Within this scheme several possibilities can be envisaged. The quintessence eld could springfrom the meson eld after the gauge breaking of supersymmetric gauge theory 6, it could alsobe the dilaton or one of the compactication moduli. It is easy to see that the supergravityFigure 1: Evolution of the dierent energy densities. The dashed-dotted line represents the the energy density ofradiation whereas the dashed line represents the energy density of matter. The solid line is the energy densityof quintessence in the SUGRA model with α = 11. The dotted line is the energy density of quintessence for thepotential V (Q) = 4+αQ−α with the same α. The initial conditions are such that equipartition, i.e. ΩQi  10−4is realized just after inflation.potential of the meson eld leads to nonsensical negative energy densities. The potential forthe dilaton S is of the type exp(exp−S) which is far too flat. The potential for the moduliT is of exponential type exp(−λT ) which leads to a small value of ΩQ  0.15 7. We are thuslead to postulate the existence of a quintessence eld whose physical origin is not clear fromthe string point of view. Nevertheless, one can adopt an eective theory point of view andlook for a general class of supegravity models leading to a quintessence behaviour 8,9. For thatit is only necessary to specify two functions, the superpotential W which is a holomorphicfunction of the elds and the Ka¨hler potential governing the kinetic terms. The Lagrangiandepends on the combination G = κK + ln(κ3jW j2) where κ  8pi/m2Pl. The scalar potentialis given by V = κ−2eG(GiGi − 3) + VD where VD is a positive contribution from the gaugesector of the models. From this one can analyse the typical behaviour of the scalar potentialin a supergravity quintessence model. First of all for a general polynomial K the exponentialterm eG will be negligible throughout the cosmic evolution of Q at least until Q  mPl. Thisimplies that one looks for models where the GiGi term is responsible for the tracking potential.It is only at low red shift that the eG supergravity correction starts contributing by slowing theevolution of Q hence pushing the value of the equation of state towards −1. In Refs. 8,9 stringinspired Ka¨hler and superpotentials were proposed along these lines. They lead to the SUGRAtracking potential of the formV (Q) =4+αQαeκ2Q2. (2)The cosmological implications of these potentials follow. First of all, the value of  necessaryto obtain ΩQ  0.7 is still the same, i.e. comparable to the scales of particles physics. Secondly,the mass of this potential is now given by m  pρc/mPle2pi. It is therefore still very smalleven if we have gained a factor e2pi. Thirdly, we see that the coincidence problem is still solvedsince, during almost all the cosmic evolution, the exponential factor plays no role. Hence, theattractor is still present. This behaviour is illustrated in Fig. (1) where the evolution of theenergy density of the quintessence eld is displayed.As already mentioned above, the most interesting cosmological implications concern theequation of state parameter. Its evolution is displayed in Fig. (2). It is clear that the equationof state parameter is driven towards −1 at very small redshifts due to the presence of theFigure 2: The dotted line represents the evolution of ωQ for the potential given by V (Q) = 4+αQ−α with α = 11whereas the dashed line represents the evolution of ωQ for the SUGRA tracking potential with the same value ofα.exponential factor. Numerical integration shows that, for the SUGRA tracking potential, itsprecise value is given byωQ  −0.82. (3)It should be emphasized that this value is independent of the free parameter α.In conclusion, we have shown in this article that a realistic model of quintessence must bebased on SUGRA since the value of the eld is  mPl on the attractor. There are of coursequestions which remain to be addressed, in particular the issue of SUSY breaking 10.References1. N. A. Bahcall, J. P. Ostriker, S. Perlmutter and P. J. Steinhardt, Science 284, 1481 (1999).2. P. J. Steinhardt, L. Wang and I. Zlatev, Phys. Rev. D 59, 123504 (1999). Preprintastro-ph/9812313.3. B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988).4. R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998).5. G. Efstathiou, preprint astro-ph/9904356.6. P. Binetruy, Phys. Rev. D 60, 063502 (1999). Preprint hep-ph/9810553.7. P. G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503 (1998).8. P. Brax and J. Martin, Phys. Lett. B 468, 40 (1999). Preprint astro-ph/9905040.9. P. Brax and J. Martin, to be published in Phys. Rev. D. Preprint astro-ph/9912046.10. P. Brax and J. Martin, in preparation.

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High Energy Physics and Quintessence

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