After recalling why dynamical adjustment mechanisms represent a particularly
attractive possibility for solving the cosmological constant problem, we
briefly discuss their intrinsic difficulties as summarized in Weinberg's no-go
theorem. We then comment on some problems of the recently proposed
`self-tuning' mechanism in 4+1 dimensions. Finally, we describe an alternative
approach which uses the time-evolution of the universe to achieve a dynamical
relaxation of the cosmological constant to zero.Comment: 4 pages LaTeX, talk at PASCOS 2001, Chapel Hill, North Carolina,
  10-15 April 200

Hebecker, A.

English

arXiv

After recalling why dynamical adjustment mechanisms represent a particularly attractive possibility for solving the cosmological constant problem, we briefly discuss their intrinsic difficulties as summarized in Weinberg's no-go theorem. We then comment on some problems of the recently proposed `self-tuning' mechanism in 4+1 dimensions. Finally, we describe an alternative approach which uses the time-evolution of the universe to achieve a dynamical relaxation of the cosmological constant to zero.After recalling why dynamical adjustment mechanisms represent a particularly attractive possibility for solving the cosmological constant problem, we briefly discuss their intrinsic difficulties as summarized in Weinberg's no-go theorem. We then comment on some problems of the recently proposed `self-tuning' mechanism in 4+1 dimensions. Finally, we describe an alternative approach which uses the time-evolution of the universe to achieve a dynamical relaxation of the cosmological constant to zero

Hebecker, Arthur

CERN Document Server

arXiv:hep-ph/0105315v1  30 May 2001ON DYNAMICAL ADJUSTMENT MECHANISMS FOR THECOSMOLOGICAL CONSTANT∗A. HEBECKERTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandE-mail: arthur.hebecker@cern.chAfter recalling why dynamical adjustment mechanisms represent a particularly at-tractive possibility for solving the cosmological constant problem, we briefly discusstheir intrinsic difficulties as summarized in Weinberg’s no-go theorem. We thencomment on some problems of the recently proposed ‘self-tuning’ mechanism in4+1 dimensions. Finally, we describe an alternative approach which uses the time-evolution of the universe to achieve a dynamical relaxation of the cosmologicalconstant to zero.We are used to describing particle physics and cosmology by an appropriate low-energyeffective field theory. For energies presently accessible in the laboratory, this effective fieldtheory contains the quantum fields of the standard model together with classical gravity.In this framework, one expects the vacuum fluctuations of the quantum fields to producea cosmological constant corresponding to an energy density ∼ µ4, where the cutoff µ is atleast O(TeV). This has to be compared with the experimental upper bound ∼ O(meV4).It is hard to imagine that this discrepancy can be resolved by any high-energy symmetryargument, since numerous low-energy standard-model contributions to the vacuum energy,which depend on all the details of the low-energy field theory, exist. An example is providedby the gluon condensate of QCD, the value of which depends on the non-perturbativedynamics. Even small variations of this condensate, associated, e.g., with the light quarkmasses, are sufficient to exceed the experimental upper bound on the cosmological constantby a vast amount. It is difficult to see how a high-energy mechanism, based, e.g., onsupersymmetry or string theory, would ‘know’ enough about all these effective low-energyparameters to compensate their contribution with the high accuracy required.The idea of a dynamical adjustment of the cosmological constant to zero represents anattractive alternative possibility (see 1 for a review). Ideally, such a mechanism would ensurethat, given the sudden appearance of a new contribution to the vacuum energy density (e.g.,through a phase transition in late cosmology), an equal opposite-sign contribution is createdand exponential expansion or collapse are avoided.Unfortunately, as was explained in 1, the most straightforward attempts of constructingsuch a mechanism are bound to fail. More specifically, it was shown that one can not buildan action for gravity, standard model and a finite number of scalar fields which would,without fine-tuning, be extremized by a space-time independent field configuration withzero curvature. A very simple, intuitive argument can be based on the actionS =∫d4x√−g(12M2R+ LSM + L(φ)), (1)which, after integrating out the standard model degrees of freedom and restricting oneself∗ talk at PASCOS 2001, Chapel Hill, North Carolina, 10-15 April 2001adjustment: submitted to Rinton on February 11, 2014 1to constant fields φ, takes the formS =∫d4x√−g(12M2R− ΛSM − V (φ)). (2)It is now clear that one can not design a potential V (φ) such that the minimum value ofΛSM + V (φ) is zero for any value of ΛSM . Furthermore, the situation is not improved bygeneralizing the action according to ΛSM → ΛSMf(φ), as would be the case in Brans-Dicke-like theories∗. As discussed in more detail in 1, the reason for these difficulties lies inthe fact that, for constant fields, the action depends on the metric only through an overallfactor√−g. Thus, the problem is reduced to the requirement that the minimal value of apotential is zero. Given the additive contribution ΛSM , this leads to fine-tuning.To design a working adjustment mechanism, one needs to relax some of the assumptionsof the above no-go theorem. For example, the low energy effective theory of gravity couldbe quite different from Einstein’s general relativity without conflict with experiment. Thishas been demonstrated very impressively in 2, where a non-compact extra dimension givesrise to a continuum of gravitational Kaluza-Klein modes without a mass gap (the Randall-Sundrum II scenario). Indeed, starting with 3, many attempts have been made to constructan adjustment mechanism in this framework.The original model is based on gravity + a scalar field in 4+1 dimensions, with the stan-dard model fields being localized on a 3+1 d boundary (with orbifold boundary conditions).After integrating out the standard model degrees of freedom, the action isS =∫d4xdy√−g5(12M35R−32(∂φ)2)−∫d4x√−g4ΛSM exp(2φM3/25). (3)The essential observation is that, for any value of ΛSM , the equations of motion derivedfrom this action have a static solution with vanishing brane curvature. In the bulk, thissolution has a curvature singularity at finite proper distance from the brane. This givesrise to the hope that low-energy 4d gravity will result since the fifth dimension is effectivelyfinite.The above ‘self-tuning’ scenario has already been criticized on a rather fundamental levelin 4. Nevertheless, we want to describe an objection to 3 which, although close in spiritto 4, might be somewhat simpler and more direct. The point is that the field configurationof 3 does not extremize the action Eq. (3). This is immediately clear since the only non-derivative coupling of φ appears in the factor multiplying ΛSM . Thus, for nonzero ΛSM ,changing φ by an infinitesimal constant leads to a change of S proportional to that constant.All this is not in contradiction to the claim of 3 that their field configuration solves theequations of motion (everywhere outside the singularity). The reason is that, when varyingan action on a finite interval (as is appropriate if a singularity is present), one gets, inaddition to the equations of motion, a boundary term, which has not been considered in 3.Therefore it appears that one either has to give up the action principle or to allow for thepossibility of a non-derivative coupling of φ to the physics at the singularity. In the lattercase, it seems likely that some form of fine-tuning will, after all, be required.In the remainder of this paper, we want to outline a different way of relaxing the as-sumptions of Weinberg’s no-go theorem 5,6. In this approach, one does not insist on a∗ One has to require that [ΛSMf(φ)+V (φ)]′ = 0 implies ΛSMf(φ)+V (φ) = 0. This leads to the conditionf(φ) = const. × V (φ), which does not solve the original problem.adjustment: submitted to Rinton on February 11, 2014 2flat universe as an extremum of the action, but satisfies oneself with a cosmology thatapproaches the flat geometry asymptotically as the universe grows old.A possible mechanism for such a ‘time-dependent’ adjustment of the cosmological con-stant has been suggested in Rubakov’s scenario of a relaxation at inflation 5. Developing hisideas, we present a time-dependent adjustment mechanism for the cosmological constantthat can be at work in a realistic, late Friedmann-Robertson-Walker universe 6.In our model, the energy density is dominated by non-standard-model dark mattertogether with a quintessence field ϕ. As in Rubakov’s scenario, the cosmological constant,characterized by a field χ, rolls down a potential and approaches zero asymptotically. Thisis realized by a kinetic term for χ that diverges as t4 at large t. The effective t4 behaviouris realized with the help of ϕ. The asymptotic stability of this solution is ensured by thecoupling to a Brans-Dicke field σ.The action of our model can be decomposed according toS = SE + SSF + SSM , (4)where SE is the Einstein action, SSF the scalar field action, and SSM the standard modelaction, which is written in the formSSM = SSM [ψ, gµν , χ] =∫d4x√−gLSM (ψ, gµν , χ) . (5)Here ψ stands for all standard-model fields. The scalar χ is assumed to govern the effec-tive UV-cutoffs of the different modes of ψ, thereby influencing the effective cosmologicalconstant. Units are chosen such that 16piGN = 1.Integrating out the fields ψ, one obtains (up to derivative terms)SSM =∫d4x√g V (χ) . (6)Let the potential V (χ) have a zero, V (χ0) = 0 with α = V′(χ0), and rename the fieldaccording to χ → χ0 + χ. Then the action near χ = 0 becomesSSM =∫d4x√g αχ . (7)Due to this potential the field χ will decrease (for α > 0) during its cosmological evolution.It can be prevented from rolling through the zero by a diverging kinetic term 7.First, let the geometry be imposed on the system, i.e., assume a flat FRW universe withH = (2/3) t−1. With a kinetic lagrangianLSF = 12∂µχ∂µχ t4 , (8)one finds a solution where χ = (α/6) t−2, which provides an acceptable late cosmology.Clearly, it requires fine tuning of the initial conditions to achieve the desired behaviorχ → 0 as t → ∞. However, this fine tuning can be avoided by adding a Brans-Dicke fieldσ that ‘feels’ the deviation of χ from zero and provides the appropriate ‘feedback’ to thekinetic term so that χ reaches zero asymptotically independent of its initial value.The field σ has a canonical kinetic term and it is coupled to LSM by the substitutiongµν → gµν√σ in Eq. (5),SSM = SSM [ψ, gµν√σ, χ] =∫d4xσ√g LSM (ψ, gµν√σ, χ) . (9)adjustment: submitted to Rinton on February 11, 2014 3Integrating out the fields ψ, one obtainsSSM =∫d4x√g ασχ (10)near χ = 0. The scalar field lagrangian is now taken to beLSF = 12(∂χ)2σ2t4 +12(∂σ)2 − βσt−2 . (11)This gives rise to the asymptotic solution χ = χ0 t−2 and σ = σ0 = const. This solution isstable, i.e., one finds the same asymptotic behaviour for a range of initial conditions. Thestability does not depend on the precise values of the parameters α and β.What remains to be done is the replacement of the various explicit t-dependent functionsby the dynamics of an appropriate field. This can be achieved using the simplest versionof quintessence 8 with an exponential potential VQ(ϕ) = e−aϕ. One finds the late-timebehaviour ϕ = (2/a) ln t. The explicit time dependence in Eq. (11) can now be replaced bya coupling to ϕ. Technically, this is realized by the substitution t2 → eaϕ in LSF .The complete lagrangian, including the curvature term and the effective standard modelaction, Eq. (10), readsL = R+ 12(∂χ)2σ2 e2aϕ +12(∂σ)2 +12(∂ϕ)2 + ασχ+ (1 − βσ) e−aϕ . (12)We have checked numerically that the resulting late time solution, which gives rise to anacceptable cosmology, is stable with respect to small variations of all parameters and initialconditions. The only problem is the too strong coupling of the Brans-Dicke field to baryons.However, this can probably be avoided in a more carefully constructed model or by the ad-hoc introduction of a kinetic term for σ that grows for large t.References1. S. Weinberg, Rev. Mod. Phys. 61 (1989) 12. L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064].3. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper and R. Sundrum, hep-th/0001197;S. Kachru, M. Schulz and E. Silverstein, hep-th/00012064. S. Forste, Z. Lalak, S. Lavignac and H. P. Nilles, Phys. Lett. B 481 (2000) 360[hep-th/0002164] and JHEP 0009 (2000) 034 [hep-th/0006139].5. V.A. Rubakov, Phys. Rev. D61 (2000) 061501 [hep-ph/9911305]6. A. Hebecker and C. Wetterich, Phys. Rev. Lett. 85 (2000) 3339 [hep-ph/0003287].7. S.M. Barr, Phys. Rev. D36 (1987) 1691;P.J. Steinhardt and F.S. Accetta, Phys. Rev. Lett. 64 (1990) 27408. C. Wetterich, Nucl. Phys. B302 (1988) 668; A&A 301 (1995) 321 [hep-th/9408025];B. Ratra and P.J.E. Peebles, Phys. Rev. D37 (1988) 3406;R.R. Caldwell, R. Dave and P.J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582[astro-ph/9708069]adjustment: submitted to Rinton on February 11, 2014 4

On Dynamical Adjustment Mechanisms for the Cosmological Constant

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