In a previous work 1 we have discussed a case study of dissipative Liouville-string cosmology, involving non critical string cosmological backgrounds, with the identification 23 of target time with the world-sheet zero mode of the Liouville field 4567. Such cosmologies were found to asymptote (in cosmic time) with the conformal backgrounds of 8910, which are thus viewed as equilibrium (relaxation) configurations of the non-equilibrium cosmologies.

In terms of microscopic considerations, a model for such departure from equilibrium could be considered the collision of two brane worlds, which results 1112 in departure from conformal invariance of the effective string theory on both the bulk and the brane world, and thus in need for Liouville dressing to restore this symmetry 4567. Dynamical arguments 1314, then, stemming from minimization of effective potentials in the low-energy string-inspired effective field theory on the brane, imply the (eventual) identification of the zero mode of the Liouville mode with (a function) of target time 23. There is an inherent time irreversibility in the process, which is associated with basic properties of the Liouville mode, viewed as a local (dynamical) renormalization group (RG) scale on the world-sheet of the string. This implies relaxation of the associated dark energy of such cosmologies, and a gravitational friction, associated with the conformal theory central charge deficit. For a recent review we refer the reader to 1112, where concepts and methods are outlined in some detail.

For our purposes in this letter we would like to concentrate on one interesting aspect of the non-critical string cosmologies, associated with the off-equilibrium effects on the Boltzmann equation describing relic abundances and the associated particle-physics phenomenology. Indeed, in conventional cosmologies, a study of relic abundances by means of Boltzmann equation that governs their cosmic time evolution yields important phenomenological constraints on the parameters of particle physics (supersymmetric) models using recent (WMAP 15 and other) astrophysical Cosmic Microwave Background (CMB) data. Essentially, the astrophysical data constraint severely in some cases the available phase-space distributions of favorite supersymmetric dark matter candidates such as neutralinos 16171819. When off-equilibrium, non-critical string cosmologies are considered 1, which notably are consistent with the current astrophysical data from supernovae, as demonstrated recently 20, then there are significant modifications to the Boltzmann equation, stemming from extra sources (dilaton) and off-shell (non-critical, non equilibrium) terms, which affect the time evolution of the phase-space density of the species under consideration.

It is the purpose of this paper to derive such modifications in detail, and then use them in order to discuss briefly particle-physics models constraints, especially from the point of view of supersymmetry. With regards to this last issue, in this article we shall present an approximate analytical treatment, which will only provide hints to what may actually happen. A complete analysis requires numerical studies which are postponed for a future publication.

Consider the phase space density of a species X, which is assumed coupled to the off-shell (non-critical string) background terms:

f ( | p → | , t ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzdaqadaqaamaaemaabaGafmiCaaNbaSaaaiaawEa7caGLiWoacqGGSaalcqWG0baDaiaawIcacaGLPaaaaaa@3420@

where quantities refer to the Einstein frame, and the Einstein metric is assumed to be of Robertson-Walker (RW) type. Throughout this work we follow the normalization and conventions of 8910.

For completeness we state the main relationships between the Einstein and σ-model frames 8:

g μ ν = e − 2 Φ g μ ν σ , ∂ t ∂ t σ = e − Φ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaem4zaC2aaSbaaSqaaGGaciab=X7aTjab=17aUbqabaGccqGH9aqpcqWGLbqzdaahaaWcbeqaaiabgkHiTiabikdaYiabfA6agbaakiabdEgaNnaaDaaaleaacqWF8oqBcqWF9oGBaeaacqWFdpWCaaGccqGGSaalaeaajuaGdaWcaaqaaiabgkGi2kabdsha0bqaaiabgkGi2kabdsha0naaCaaabeqaaiab=n8aZbaaaaGccqGH9aqpcqWGLbqzdaahaaWcbeqaaiabgkHiTiabfA6agbaaaaaaaa@49FA@

where Φ is the dilaton field, and the superscript σ denotes quantities evaluated in the σ-model frame.

Since in the RW Einstein frame g₀₀ = -1, and in the cosmic Einstein co-moving frame we have for a generic massive species with mass m (that we shall be interested in this work) pσμ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdchaWnaaDaaaleaaiiGacqWFdpWCaeaacqWF8oqBaaaaaa@3006@ = mdx^μ/dτ = m(dx^μ/dt) (dt/dt_σ) = mp^μe^-Φ, it can be readily seen that

| p → σ | ≡ ( p i , σ , p j , σ g i j σ ) 1 / 2 = ( p i p j g i j ) 1 / 2 ≡ | p → | = a ( t ) ( ∑ i = 1 3 p i p i ) 1 / 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@75F0@

We shall be interested in the action of the relativistic Liouville operator L^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdYeamzaajaaaaa@2C26@, acting on a phase-space density of species X, f(|p→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdchaWzaalaaaaa@2C70@|, t, gⁱ), which in general depends on the off-shell backgrounds gⁱ = (g_ii, Φ), where g_ii are the spatial components of the RW space-time metric in Einstein frame, and Φ is the dilaton. The off-shell dependence comes about due to the interpretation of target time as a local world-sheet renormalization-group scale, the Liouville field.

We commence our analysis by recalling the relativistic form of the Liouville operator in conventional general relativity:

L ^ [ f ] = ( p α ∂ ∂ x α + Γ β σ α p β p σ ∂ ∂ p α ) f MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGmbatgaqcamaadmaabaGaemOzaygacaGLBbGaayzxaaGaeyypa0ZaaeWaaeaacqWGWbaCdaahaaWcbeqaaGGaciab=f7aHbaajuaGdaWcaaqaaiabgkGi2cqaaiabgkGi2kabdIha4naaCaaabeqaaiab=f7aHbaaaaGccqGHRaWkcqqHtoWrdaqhaaWcbaGae8NSdiMae83WdmhabaGae8xSdegaaOGaemiCaa3aaWbaaSqabeaacqWFYoGyaaGccqWGWbaCdaahaaWcbeqaaiab=n8aZbaajuaGdaWcaaqaaiabgkGi2cqaaiabgkGi2kabdchaWnaaCaaabeqaaiab=f7aHbaaaaaakiaawIcacaGLPaaacqWGMbGzaaa@5133@

The second term on the r.h.s. is the relativistic form of the force, following from the geodesic equation. For a RW Universe, only the time-energy part survives from the first term, i.e.

p α = ∂ ∂ x α = E ∂ ∂ t . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaahaaWcbeqaaGGaciab=f7aHbaakiabg2da9KqbaoaalaaabaGaeyOaIylabaGaeyOaIyRaemiEaG3aaWbaaeqabaGae8xSdegaaaaakiabg2da9iabdweafLqbaoaalaaabaGaeyOaIylabaGaeyOaIyRaemiDaqhaaOGaeiOla4caaa@3D2B@

Moreover, the connection part receives non trivial contributions only from the terms ∑iΓii0pipi∂∂E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaaqababaGaeu4KdC0aa0baaSqaaiabdMgaPjabdQgaQbqaaiabicdaWaaakiabdchaWnaaCaaaleqabaGaemyAaKgaaOGaemiCaa3aaWbaaSqabeaacqWGPbqAaaqcfa4aaSaaaeaacqGHciITaeaacqGHciITcqWGfbqraaaaleaacqWGPbqAaeqaniabggHiLdaaaa@3DE1@, i = 1, 2, 3 a spatial index (assuming for concreteness an already compactified string theory, or a theory on a three-brane world), with Γii0=−aa˙ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabfo5ahnaaDaaaleaacqWGPbqAcqWGQbGAaeaacqaIWaamaaGccqGH9aqpcqGHsislcqWGHbqycuWGHbqygaGaaaaa@34CC@, where the overdot denotes derivative w.r.t. cosmic RW time t, identified in our string theory with the Einstein-frame RW time (2.1).

Hence, the conventional part of the Liouville operator in a RW Universe would read 21:

L ^ conv = E ∂ ∂ t − a a ˙ ∑ i = 1 3 p i p i ∂ ∂ E = E ∂ ∂ t − a ˙ a ∑ i = 1 3 | p → | 2 ∂ ∂ E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@68AE@

The presence of off-shell, non-critical (Liouville) string 4567 backgrounds, upon the (dynamical) identification 2311121314 of the Liouville mode with the target time, one will receive extra contributions in the expression for the associated Liouville operator, stemming from the fact that the latter is nothing but a total time derivative.

In the context of our string discussion it is important to specify that the initial frame, from which we commence our discussion, is the so-called Einstein frame. It is in this frame that the usual RW cosmology is obtained in string theory 9, for which the expression (2.5) is valid.

To discuss the non-critical-string (off-shell) corrections to Boltzmann equation, it will be necessary to consider time derivatives in the σ-model frame. This is due to the fact that it is in this frame that the target time X⁰ ≡ t_σ is related simply to the Liouville mode ϕin non-critical string theories 23,

ϕ+ t_σ = 0.

This relation is obtained dynamically from minimization arguments of the low-energy effective potential of some physically interesting cosmological models, for instance those involving colliding brane worlds 1113. To be precise, the initial relation (2.6) derived in the specific model of 13 reads: ϕ/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaeGOmaidaleqaaaaa@2C02@ + t_σ = 0, with the factor of 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaeGOmaidaleqaaaaa@2C02@ arising from (logarithmic) conformal-field-theory considerations, and being crucial for yielding a Minkowski space-time in the scenario of 13, where the coordinate X⁰ assumed an initial Euclidean signature, appropriate for a world-sheet path-integral quantization. Nevertheless, in our approach below we shall absorb the 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaeGOmaidaleqaaaaa@2C02@ into the normalization of the units of the Regge slope α', for convenience.

In such scenarios the target-space dimensionality of the string is extended to D + 1 initially, with two time-like coordinates, t and ϕ. Eventually, these two coordinates are identified (c.f. (2.6)) in the solution of the generalized conformal invariance conditions that express the restoration of the conformal invariance by the Liouville mode, as we discuss below. This implies that, initially, the Liouville operator (2.3) acquire extra Liouville components (ϕ, p^ϕ), with p^ϕ≡ E_ϕ. Hence, there are additional structures on the right hand side of (2.3), of the form Eϕ∂∂ϕ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdweafnaaBaaaleaaiiGacqWFvpGAaeqaaKqbaoaalaaabaGaeyOaIylabaGaeyOaIyRae8x1dOgaaaaa@3338@, together with the corresponding amendments in the Christoffel-symbol dependent ∂/∂p^ϕterms, stemming from the extra Liouville coordinate on the target space time, which implies appropriate extra components of the extended-target-space-time metric. The (dynamical) implementation of (2.6) restricts oneself to a D-dimensional hypersurface in the extended target-space, with the identification of E_ϕ→ E, the ordinary (physical) energy. From now on we restrict ourselves on this hypersurface, bearing in mind however that the identification of the Liouville mode ϕwith the target time should only be implemented at the end, and thus any dependence on ϕshould be kept explicit at intermediate steps of the pertinent calculations.

The connection between time derivatives in the Einstein and stringy-σ-model frames is provided by the chain rule of differentiation ∂∂tσ=∂t∂tσ∂∂t MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaKqbaoaalaaabaGaeyOaIylabaGaeyOaIyRaemiDaq3aaSbaaeaaiiGacqWFdpWCaeqaaaaakiabg2da9KqbaoaalaaabaGaeyOaIyRaemiDaqhabaGaeyOaIyRaemiDaq3aaSbaaeaacqWFdpWCaeqaaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabdsha0baaaaa@3F43@, using (2.1). This will result in multiplicative factors of e^-Φ in front of the appropriate non-critical-string modifications of the Liouville operator (2.5).

Since, as mentioned above, the Liouville operator is essentially a total time derivative operator, and in our case time is related to a world-sheet renormalization group scale, ρ(σ, τ), taken to be local on the two-dimensional surface as a result of world-sheet general covariance 2223, the sought-for non-critical-string modifications of the Boltzmann equation emerge from the implicit dependence of the gⁱ background fields on ρ(σ, τ), that is the corresponding β-functions, which in a critical-string theory would vanish.

We now remark that in the approach of 23, the local RG scale ρ(ξ) was identified with the dynamical Liouville mode ϕ, which implied that in this approach the local conformal invariance of the world-sheet of the string was restored 4567,

ρ ≡ ϕ.

With these in mind we then modify the relativistic form (2.3), (2.4) by replacing

∂ ∂ t → d d t ≡ ∂ ∂ t + ∫ Σ ∂ t σ ∂ t ∂ ρ ( ξ ) ∂ t σ ∂ g “ i ″ ∂ ρ ( ξ ) ∂ ∂ g ℌ i ″ = ∂ ∂ t + η e Φ   β “ i ″ ∂ ∂ g “ i ″ , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaqcfa4aaSaaaeaacqGHciITaeaacqGHciITcqWG0baDaaGccqGHsgIRjuaGdaWcaaqaaiabdsgaKbqaaiabdsgaKjabdsha0baakiabggMi6MqbaoaalaaabaGaeyOaIylabaGaeyOaIyRaemiDaqhaaOGaey4kaSYaa8qeaeaajuaGdaWcaaqaaiabgkGi2kabdsha0naaBaaabaacciGae83WdmhabeaaaeaacqGHciITcqWG0baDaaWaaSaaaeaacqGHciITcqWFbpGCcqGGOaakcqWF+oaEcqGGPaqkaeaacqGHciITcqWG0baDdaWgaaqaaiab=n8aZbqabaaaamaalaaabaGaeyOaIyRaem4zaC2aaWbaaeqabaGaeiOiaiIaemyAaKMaeiOiaicaaaqaaiabgkGi2kab=f8aYjabcIcaOiab=57a4jabcMcaPaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kabdEgaNnaaCaaabeqaaiabckcaIiabdMgaPjabckcaIaaaaaaaleaacqqHJoWuaeqaniabgUIiYdGccqGH9aqpaeaajuaGdaWcaaqaaiabgkGi2cqaaiabgkGi2kabdsha0baakiabgUcaRiab=D7aOjabdwgaLnaaCaaaleqabaGaeuOPdyeaaOGaeeiiaaIae8NSdi2aaWbaaSqabeaacqGGIaGicqWGPbqAcqGGIaGiaaqcfa4aaSaaaeaacqGHciITaeaacqGHciITcqWGNbWzdaahaaqabeaacqGGIaGicqWGPbqAcqGGIaGiaaaaaOGaeiilaWcaaaaa@8219@

where ξ = σ, τ, denotes the world-sheet coordinates, ∫_Σ is a world-sheet integration, and the index "i" runs on both, a (discrete) background field space, {g_ii(y), Φ(y)}, i = 1, 2, 3 and a continuous D-dimensional (target) space-time y. Hence, summation over "i″ includes integration ∫dDy−g(...) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaapeaabaGaemizaq2aaWbaaSqabeaacqWGebaraaaabeqab0Gaey4kIipakiabdMha5naakaaabaGaeyOeI0Iaem4zaCgaleqaaOWaaeWaaeaacqGGUaGlcqGGUaGlcqGGUaGlaiaawIcacaGLPaaaaaa@3797@. Note that since the (physical) target time in the σ-model frame, t_σ, is only the world-sheet zero mode of the corresponding σ-model field X⁰(ξ), eventually the world-sheet integration ∫_Σ in (2.8) disappears, since only the world-sheet zero mode ρ of the local RG scale ρ(ξ) (Liouville mode) will yield a non zero contribution. In the last equality of the right-hand side of (2.8), the quantity η is defined as: η≡∂ρ∂tσ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciab=D7aOjabggMi6MqbaoaalaaabaGaeyOaIyRae8xWdihabaGaeyOaIyRaemiDaq3aaSbaaeaacqWFdpWCaeqaaaaaaaa@36E6@. If we use the identification ρ(ξ) = ϕ(ξ) and Eq. (2.6), then η = -1, but at this stage we keep it general to demonstrate explicitly the renormalization-scheme dependence (i.e. choice of the local scale ρ(ξ)).

The non-critical string contributions ∂gi∂ρ=β˜i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaKqbaoaalaaabaGaeyOaIyRaem4zaC2aaWbaaeqabaGaemyAaKgaaaqaaiabgkGi2IGaciab=f8aYbaakiabg2da9iqb=j7aIzaaiaWaaWbaaSqabeaacqWGPbqAaaaaaa@373D@ are the Weyl anomaly coefficients of the string, but with the renormalized σ-model couplings gⁱ being replaced by the Liouville-dressed 4567 quantities. This is a particular feature of the approach of 23 viewing the Liouville mode as a local world-sheet RG scale, as mentioned previously. In turn the latter is also identified (c.f. (2.6)) with (an appropriate function of) the target time t.

The dynamics of this latter identification is encoded in the solution of the generalized conformal invariance conditions, after Liouville dressing, which read in the σ-model frame 234567:

− β ˜ i = g i ″ + Q g i ′ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHsisliiGacuWFYoGygaacamaaCaaaleqabaGaemyAaKgaaOGaeyypa0Jaem4zaC2aaWbaaSqabeaacuWGPbqAgaGbaaaakiabgUcaRiabdgfarjabdEgaNnaaCaaaleqabaGafmyAaKMbauaaaaaaaa@377E@

where the prime denotes differentiation with respect to the Liouville zero mode ρ, and the overall minus sign on the left-hand side of the above equation pertains to supercritical strings 89, with a time like signature of the Liouville mode, for which the central charge deficit Q² > 0 by convention.

Notice the dissipation, proportional to the (square root) of the central charge deficit Q, on the right-hand side of (2.9), which heralds the adjective Dissipative to the associate non-critical-string-inspired Cosmological model. Moreover, the Weyl anomaly coefficients β˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=j7aIzaaiaaaaa@2CAC@ⁱ, i = {Φ, g_μν}, whose vanishing would guarantee local conformal invariance of the string-cosmology background, are associated with off-shell variations of a low-energy effective string-inspired action, S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jr8tbaa@368D@[g^j]

δ S [ g ] δ g i = G i j β ˜ j G i j = Lim z → 0 z 2 z ¯ 2 〈 V i ( z , z ¯ ) V j ( 0 , 0 ) 〉 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaqcfa4aaSaaaeaaiiGacqWF0oazt0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFse=udaWadaqaaiabdEgaNbGaay5waiaaw2faaaqaaiab=r7aKjabdEgaNnaaCaaabeqaaiabdMgaPbaaaaGccqGH9aqpcqGFge=rdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiqb=j7aIzaaiaWaaWbaaSqabeaacqWGQbGAaaaakeaacqGFge=rdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabg2da9iabbYeamjabbMgaPjabb2gaTnaaBaaaleaacqWG6bGEcqGHsgIRcqaIWaamaeqaaOGaemOEaO3aaWbaaSqabeaacqaIYaGmaaGccuWG6bGEgaqeaiabgMYiHlabdAfawnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemOEaONaeiilaWIafmOEaONbaebacqGGPaqkcqWGwbGvdaWgaaWcbaGaemOAaOgabeaakiabcIcaOiabicdaWiabcYcaSiabicdaWiabcMcaPiabgQYiXdaaaaa@6E79@

with z, z¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdQha6zaaraaaaa@2C8A@ (complex) world-sheet coordinates, G MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=zq8hbaa@3675@_ij the Zamolodchikov metric in theory space of strings, and V_i the σ-model vertex operators associated with the σ-model background field gⁱ. It is this off-shell relation that characterizes the entire non-critical (Q) Cosmology framework, associated physically with a non-equilibrium situation as a result of an initial cosmically catastrophic event, at the beginning of the (irreversible) Liouville/cosmic-time flow 1112.

The detailed dynamics of (2.9) are encoded in the solution for the scale factor a(t) and the dilaton Φ in the simplified model considered in 1, after the identification of the Liouville mode with the target time (2.6). In fact, upon the inclusion of matter backgrounds, including dark matter species, the associated equations, after compactification to four target-space dimensions, read in the Einstein frame 1:

3   H 2 − ϱ ˜ m − ϱ Φ = e 2 Φ 2 G ˜ Φ 2   H ˙ + ϱ ˜ m + ϱ Φ + p ˜ m + p Φ = G ˜ i i a 2 Φ ¨ + 3 H Φ ˙ + 1 4 ∂ V ^ a l l ∂ Φ + 1 2 ( ϱ ˜ m − 3 p ˜ m ) = − 3 2 G ˜ i i a 2 − e 2 Φ 2 G ˜ Φ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeWabaaabaGaeG4mamJaeeiiaaIaemisaG0aaWbaaSqabeaacqaIYaGmaaGccqGHsislcuWI9=VBgaacamaaBaaaleaaieGacqWFTbqBaeqaaOGaeyOeI0IaeSy==72aaSbaaSqaaiabfA6agbqabaGccqGH9aqpjuaGdaWcaaqaaiabdwgaLnaaCaaabeqaaiabikdaYiabfA6agbaaaeaacqaIYaGmaaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGccuGFge=rgaacamaaBaaaleaacqqHMoGraeqaaaGcbaGaeGOmaiJaeeiiaaIafmisaGKbaiaajaaOcqGHRaWkkiqbl2==UzaaiaWaaSbaaSqaaiab=1gaTbqabaGccqGHRaWkcqWI9=VBdaWgaaWcbaGaeuOPdyeabeaakiabgUcaRiqbdchaWzaaiaWaaSbaaSqaaiabd2gaTbqabaGccqGHRaWkcqWGWbaCdaWgaaWcbaGaeuOPdyeabeaakiabg2da9KqbaoaalaaabaGaf4NbXFKbaGaadaWgaaqaaiabdMgaPjabdMgaPbqabaaabaGaemyyae2aaWbaaeqabaGaeGOmaidaaaaaaOqaaiqbfA6agzaadaGaey4kaSIaeG4mamJaemisaGKafuOPdyKbaiaacqGHRaWkjuaGdaWcaaqaaiabigdaXaqaaiabisda0aaadaWcaaqaaiabgkGi2kqbdAfawzaajaWaaSbaaeaacqWGHbqycqWGSbaBcqWGSbaBaeqaaaqaaiabgkGi2kabfA6agbaakiabgUcaRKqbaoaalaaabaGaeGymaedabaGaeGOmaidaaOGaeiikaGIafSy==7MbaGaadaWgaaWcbaGae8xBa0gabeaakiabgkHiTiabiodaZiqbdchaWzaaiaWaaSbaaSqaaiabd2gaTbqabaGccqGGPaqkcqGH9aqpcqGHsisljuaGdaWcaaqaaiabiodaZaqaaiabikdaYaaadaWcaaqaaiqb+zq8hzaaiaWaaSbaaeaacqWGPbqAcqWGPbqAaeqaaaqaaiabdggaHnaaCaaabeqaaiabikdaYaaaaaGccqGHsisljuaGdaWcaaqaaiabdwgaLnaaCaaabeqaaiabikdaYiabfA6agbaaaeaacqaIYaGmaaGccuGFge=rgaacamaaBaaaleaacqqHMoGraeqaaOGaeiOla4caaaaa@A4CF@

where ϱ˜m(p˜m) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbl2==UzaaiaWaaSbaaSqaaGqaciab=1gaTbqabaGccqGGOaakcuWGWbaCgaacamaaBaaaleaacqWGTbqBaeqaaOGaeiykaKcaaa@3462@ denotes the matter energy density (pressure), including dark matter contributions, and ϱ_Φ (p_Φ) the corresponding quantities for the dilaton dark-energy fluid. All derivatives in (2.11) are with respect the Einstein time t which is related to the Robertson-Walker cosmic time t_RW by t = ωt_RW. Without loss of generality we have taken ω = 3 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaeG4mamdaleqaaaaa@2C04@H₀ where H₀ is the Hubble constant. With this choice for ω the densities appearing in (2.11) are given in units of the critical density. Then one can see for instance that if the time t_RW is used the first of equations (2.11) above receives its familiar form in the RW geometry when the contributions of the dilaton and non-critical terms are absent.

The overdots in the above equations denote derivatives with respect to the Einstein time. Their right-hand side contain the non-critical string off-shell terms:

G ˜ Φ = e − 2 Φ ( Φ ¨ − Φ ˙ 2 + Q e Φ Φ ˙ ) G ˜ i i = 2   a 2 ( Φ ¨ + 3 H Φ ˙ + Φ ˙ 2 + ( 1 − q ) H 2 + Q e Φ ( Φ ˙ + H ) ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGaf8NbXFKbaGaadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iabdwgaLnaaCaaaleqabaGaeyOeI0IaeGOmaiJaeuOPdyeaaOGaeiikaGIafuOPdyKbamaacqGHsislcuqHMoGrgaGaamaaCaaaleqabaGaeGOmaidaaOGaey4kaSIaemyuaeLaemyzau2aaWbaaSqabeaacqqHMoGraaGccuqHMoGrgaGaaiabcMcaPaqaaiqb=zq8hzaaiaWaaSbaaSqaaiabdMgaPjabdMgaPbqabaGccqGH9aqpcqaIYaGmcqqGGaaicqWGHbqydaahaaWcbeqaaiabikdaYaaakiabcIcaOiqbfA6agzaadaGaey4kaSIaeG4mamJaemisaGKafuOPdyKbaiaacqGHRaWkcuqHMoGrgaGaamaaCaaaleqabaGaeGOmaidaaOGaey4kaSIaeiikaGIaeGymaeJaeyOeI0IaemyCaeNaeiykaKIaemisaG0aaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcqWGrbqucqWGLbqzdaahaaWcbeqaaiabfA6agbaakiabcIcaOiqbfA6agzaacaGaey4kaSIaemisaGKaeiykaKIaeiykaKIaeiOla4caaaaa@7355@

In the above equations H = (a˙ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdggaHzaacaaaaa@2C49@)/a is the Hubble parameter and q is the deceleration parameter of the Universe q≡−a¨a/a˙2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdghaXjabggMi6kabgkHiTiqbdggaHzaadaGaemyyaeMaei4la8IafmyyaeMbaiaadaahaaWcbeqaaiabikdaYaaaaaa@350F@, and are both functions of (Einstein frame) cosmic time. The potential V^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdAfawzaajaaaaa@2C3A@_all appearing above is defined by V^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdAfawzaajaaaaa@2C3A@_all = 2Q² exp(2Φ) + exp(4Φ)V where, in order to cover more general cases, we have also allowed for a potential term in the four-dimensional action −∫d4y−GV MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgkHiTmaapeaabaGaemizaq2aaWbaaSqabeaacqaI0aanaaGccqWG5bqEdaGcaaqaaiabgkHiTiabdEeahbWcbeaaaeqabeqdcqGHRiI8aOGaemOvayfaaa@3529@ in addition to that dependent on the central charge deficit term Q. Although we have assumed a (spatially) flat Universe, the terms on the r.h.s., which manifest departure from the criticality, act in a sense like curvature terms as being non-zero at certain epochs. The dilaton energy density and pressure are given in this class of models by:

ϱ Φ = 1 2 ( 2 Φ ˙ 2 + V ^ a l l ) p Φ = 1 2 ( 2 Φ ˙ 2 − V ^ a l l ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeGabaaabaGaeSy==72aaSbaaSqaaiabfA6agbqabaGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaakiabcIcaOiabikdaYiqbfA6agzaacaWaaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcuWGwbGvgaqcamaaBaaaleaacqWGHbqycqWGSbaBcqWGSbaBaeqaaOGaeiykaKcabaGaemiCaa3aaSbaaSqaaiabfA6agbqabaGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaakiabcIcaOiabikdaYiqbfA6agzaacaWaaWbaaSqabeaacqaIYaGmaaGccqGHsislcuWGwbGvgaqcamaaBaaaleaacqWGHbqycqWGSbaBcqWGSbaBaeqaaOGaeiykaKIaeiOla4caaaaa@51A8@

Notice that the dilaton field is not canonically normalized in this convention and its dimension has been set to zero.

For completeness, we mention at this point that the dependence of the central charge deficit Q(t) on the cosmic time stems from the running of the latter with the world-sheet RG scale 231112, and is provided by the Curci-Paffuti equation 24 expressing the renormalizability of the world-sheet theory. To leading order in an α' expansion, which we restrict ourselves in 1 and here, this equation in the Einstein frame reads:

d G ˜ Φ d t E = − 6   e − 2 Φ ( H + Φ ˙ ) G ˜ i i a 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKnrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiqb=zq8hzaaiaWaaSbaaeaacqqHMoGraeqaaaqaaiabdsgaKjabdsha0naaBaaabaGaemyraueabeaaaaGccqGH9aqpcqGHsislcqaI2aGncqqGGaaicqWGLbqzdaahaaWcbeqaaiabgkHiTiabikdaYiabfA6agbaakiabcIcaOiabdIeaijabgUcaRiqbfA6agzaacaGaeiykaKscfa4aaSaaaeaacuWFge=rgaacamaaBaaabaGaemyAaKMaemyAaKgabeaaaeaacqWGHbqydaahaaqabeaacqaIYaGmaaaaaOGaeiOla4caaa@53C8@

For future use we also state here the corresponding continuity of the matter stress tensor, which is not an independent equation, but can be obtained from (2.11) by appropriate algebraic manipulations:

d ϱ ˜ m d t E + 3 H ( ϱ ˜ m + p ˜ m ) + Q ˙ 2 ∂ V ^ a l l ∂ Q − Φ ˙ ( ϱ ˜ m − 3 p ˜ m ) = 6   ( H + Φ ˙ ) G ˜ i i a 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKjqbl2==UzaaiaWaaSbaaeaaieGacqWFTbqBaeqaaaqaaiabdsgaKjabdsha0naaBaaabaGaemyraueabeaaaaGccqGHRaWkcqaIZaWmcqWGibascqGGOaakcuWI9=VBgaacamaaBaaaleaacqWFTbqBaeqaaOGaey4kaSIafmiCaaNbaGaadaWgaaWcbaGaemyBa0gabeaakiabcMcaPiabgUcaRKqbaoaalaaabaGafmyuaeLbaiaaaeaacqaIYaGmaaWaaSaaaeaacqGHciITcuWGwbGvgaqcamaaBaaabaGaemyyaeMaemiBaWMaemiBaWgabeaaaeaacqGHciITcqWGrbquaaGccqGHsislcuqHMoGrgaGaaiabcIcaOiqbl2==UzaaiaWaaSbaaSqaaiab=1gaTbqabaGccqGHsislcqaIZaWmcuWGWbaCgaacamaaBaaaleaacqWGTbqBaeqaaOGaeiykaKIaeyypa0JaeGOnayJaeeiiaaIaeiikaGIaemisaGKaey4kaSIafuOPdyKbaiaacqGGPaqkjuaGdaWcaaqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiqb+zq8hzaaiaWaaSbaaeaacqWGPbqAcqWGPbqAaeqaaaqaaiabdggaHnaaCaaabeqaaiabikdaYaaaaaGccqGGUaGlaaa@772F@

This expresses the non-conservation equation of matter as a result of its coupling to both, the dilaton source terms and the off-shell, non-equilibrium (non-critical-string) backgrounds.

A consistent solution of a(t), Φ(t), and the various densities, including back reaction of matter onto the space-time geometry, has been discussed in 1, where we refer the interested reader for further study. Also note that a preliminary comparison of such non-critical strings theories with astrophysical data, demonstrating consistency at present, is given in 20. It should be remarked at this point that the solutions obtained in this study tend asymptotically to those presented in 8910. In such an approach, the dilaton rolls down asymptotically in (Einstein) target time as Φ = const. - ln t, consistent with a vanishing string coupling g_s = e^Φ → 0 as t → +∞. This situation is opposite to other studies in critical (equilibrium) string cosmologies, where the dilaton gets a non-trivial vacuum expectation value (v.e.v.) through an appropriate minimisation of its potential, being thus stabilised at early times. In such on-shell situations, as opposed to our non-equilibrium (off-shell) model studied in this and in previous articles 12526, matter particles may acquire dilaton-v.e.v.-dependent masses, which may be heavy, and as such they decouple from a thermal bath. In contrast, as we shall discuss in the next section, our non-equilibrium rolling dilaton and running string coupling, implies that at late eras of the Universe, one may still evaluate the relic abundances of matter particles as in conventional cosmology, using an appropriately modified Boltzmann equation.

A final, but important comment should be made at this juncture. It is understood that, in order not to disturb the delicate balance between particle interactions and the expansion of the Universe during the nucleosynthesis era, it is necessary to consider models in which the dilaton may be almost constant during that era, in such a way that the associated string coupling and particle decay and/or recombination rates are not affected much by the dilaton and non-critical string terms, as compared with the conventional cosmology situation. This important and by far not complete issue is highly model dependent, and it relies on specific properties of the underlying microscopic string theory, involving detailed scenaria of compactification, moduli field stabilization etc. In the literatute there are toy brane-inspired, non-critical-string cosmological models, where such approximately constant dilatons during early epochs of the Universe could be realised explicitly 131425. We also remark that similar assumptions are made in generic dilaton quintessence scenaria of critical strings 27.

After this necessary digression we now come back to discussing the derivation of the dilaton-source and noncritical-string induced modifications to the Boltzmann equation, governing the cosmic evolution of the various species densities. It is important for the reader to bear in mind already at this stage that the Boltzmann equation does not contain any independent information from the dynamical equations (2.11), but it should be rather viewed as an effective way of describing the cosmic evolution of the density of a given species, consistent with the continuity equation (2.15) for the total matter energy density. We shall come back to this important point later on.

At the moment, let us concentrate first on the dilaton-source and non-critical-string background contributions to the Liouville operator (2.8). The presence of (time-dependent) dilaton source terms implies an explicit dependence of the phase space density of a species f on Φ, while the non-conformal on the world-sheet) nature of the metric and dilaton background, induce a Liouville mode ρ dependence through the corresponding backgrounds:

f ( | p → | , t , Φ ( t , ρ ) , g μ ν ( t , ρ ) ) .