D-brane Disformal Coupling and Thermal Dark Matter
MMI-TH-1763
D-brane Disformal Coupling and Thermal Dark Matter
Bhaskar Dutta a , Esteban Jimenez a , Ivonne Zavala b a Mitchell Institute for Fundamental Physics and Astronomy, Department of Physics and Astron-omy, Texas A&M University, College Station, TX 77843, USA b Department of Physics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK
Abstract:
Conformal and Disformal couplings between a scalar field and matter occurnaturally in general scalar-tensor theories. In D-brane models of cosmology and particlephysics, these couplings originate from the D-brane action describing the dynamics of itstransverse (the scalar) and longitudinal (matter) fluctuations, which are thus coupled. Dur-ing the post-inflationary regime and before the onset of big-bang nucleosynthesis (BBN),these couplings can modify the expansion rate felt by matter, changing the predictionsfor the thermal relic abundance of dark matter particles and thus the annihilation raterequired to satisfy the dark matter content today. We study the D-brane-like conformaland disformal couplings effect on the expansion rate of the universe prior to BBN and itsimpact on the dark matter relic abundance and annihilation rate. For a purely disformalcoupling, the expansion rate is always enhanced with respect to the standard one. Thisgives rise to larger cross-sections when compared to the standard thermal prediction for arange of dark matter masses, which will be probed by future experiments. In a D-brane-likescenario, the scale at which the expansion rate enhancement occurs depends on the stringcoupling and the string scale.
Keywords:
Dark energy theory, dark matter theory, annihilation rate, scalar-tensor the-ories, string theory and cosmology [email protected] [email protected] [email protected] a r X i v : . [ h e p - ph ] N ov ontents ϕ and H C = const. 143.4 Conformal and disformal case C (cid:54) = const 173.5 Effect on the relic abundances and cross section 21 Recent cosmological data support the phenomenological Λ-Cold Dark Matter (Λ
CDM )model for cosmology, that describes the energy density content of the universe in termsof a cosmological constant, Λ and cold dark matter – which together make up 95% ofthe universe’s energy density budget – as well as the ∼
5% made of baryonic standardmodel (SM) particles. This phenomenological model is complemented with the inflationarymechanism, the most successful framework to date to account for the origin of structure inthe universe. After inflation, the universe must reheat providing the initial conditions forthe hot big bang.However, the physics describing the universe’s evolution from the end of inflation tothe onset of big bang nucleosynthesis (BBN) at around t ∼ −
300 s ( T ∼ (cid:104) σv (cid:105) , which is around 3 . × − cm s − (or σ ∼ (cid:104) σv (cid:105) (see Refs. [1, 2]). In the future,HAWC [3] and CTA [4] will probe the annihilation rate for a wide range of dark mattermasses. It is thus worth establishing whether a larger or smaller annihilation rate than thestandard thermal prediction, could still have a thermal origin due to modifications to thestandard cosmological evolution before BBN.On the other hand, string theory approaches to SM particle physics and inflation modelbuilding generically predict the presence of several new ingredients, and in particular newparticles such as scalar fields with clear geometrical interpretations. Type II string theorymodels of particle physics introduce new ingredients such as D-branes, where matter (andDM) is to be localised . In D-brane constructions, longitudinal string fluctuations areidentified with the matter fields such as the SM and/or DM particles, while transversefluctuations correspond to scalar fields, which may play a role during the cosmologicalevolution . These scalars couple conformally and disformally to the matter living on thebrane [6] and thus may change the cosmological expansion rate felt by matter and thestandard predictions for the DM relic abundance [7] as well.The modifications to the relic abundances in conformally coupled scalar-tensor theories(ST) such as generalisations of the Brans-Dicke theory were first discussed by Catena etal. in Ref. [8]. These authors showed a general enhancement on the modified expansionrate, ˜ H with respect to the standard general relativity (GR) rate, H GR , before rapidlydropping back to the GR value well before the onset of BBN. They also found that thisrapid relaxation of the scalar-tensor expansion rate towards H GR led to a reannihilationeffect: after the initial particle decoupling, the dark matter species experienced a subsidiaryperiod of annihilation as the expansion rate of the universe dropped below the interactionrate. In Ref. [7], we established the conditions under which this effect happens. As wediscussed there, a reannihilation phase can occur for a nontrivial set of initial conditions forsuitable conformal couplings. We found that for dark matter particles with large masses( m ∼ GeV) the particles undergo this second annihilation process. Moreover, we alsodetermined that the annihilation rate had to be up to four times larger than that of standardcosmology in order to satisfy the dark matter content of the universe of 27 %. On the otherhand, for smaller masses this reannihilation process does not occur, but we found that formasses of around 130 GeV, the annihilation rate can be smaller than the annihilation ratein the standard cosmological model. Further studies on conformally coupled ST modelshave been performed in recent years in Refs. [9–13] (see also Refs. [14–20]).In scalar-tensor theories the most general physically consistent relation between twometrics in the presence of a scalar field, is given by [21]:˜ g µν = C ( φ ) g µν + D ( φ ) ∂ µ φ∂ ν φ . (1.1) For a review on D-brane models of particle physics see e.g. Ref. [5]. For example, a coupled dark energy - dark matter D-brane scenario was proposed in Ref. [6]. More generally, the functions, C and D can depend on X = ( ∂φ ) as well. We do not consider thiscase in the present paper. – 2 –he first term in Eq. (1.1) is the conformal transformation which characterises the Brans-Dicke class of scalar-tensor theories widely explored in the literature [8–13]. The secondterm is the so-called disformal coupling, which is generic in extensions of general relativity.In particular, it arises naturally in D-brane models, as discussed in Ref. [6] in a naturalmodel of coupled dark matter and dark energy. In Ref. [7], we studied briefly the effectof turning on the disformal term in addition to the conformal one studied in Ref. [8]in a phenomenological setup. In such a case, the functions C and D are in principleindependent functions, so long as they satisfy the causality constraints: C ( φ ) > C ( φ ) + 2 D ( φ ) X > X = ( ∂φ ) ) [21]. However we found that in order to have a realpositive modified expansion rate, ˜ H , the conformal and disformal factors need to satisfya nontrivial relation [7]. Moreover, when turning on a small disformal deformation tothe conformal case, the profile of the modified expansion rate has a similar shape witha comparable enhancement with respect to the standard expansion rate and a possiblereannihilation phase. The net effect is the possibility to have larger and smaller annihilationrates for a large range of masses of the DM candidate for the observed DM content.In the present work, we study in detail the effects on the expansion rate and the DMrelic abundances of the disformal coupling in (1.1), which arises in the case of matter lo-calised on D-branes. In this case, the conformal and disformal terms are closely relatedand dictated by the underlying theory, such as type IIB flux compactifications in stringtheory. The picture we have in mind is the following. After string theory inflation reheat-ing takes place, giving rise to a thermal universe. At this stage standard model particlesand dark matter should be produced. The SM would arise from stacks of D-branes atsingularities or intersecting at suitable angles [5], while DM particles could arise from thesame or a different stack of D-branes, which may be moving towards their final stablepositions in the internal six-dimensional space before the onset of BBN. From the end ofinflation to BBN, a nonstandard cosmological evolution can take place without spoilingthe predictions of BBN, in particular, a change in the expansion rate felt by the matterparticles due to the D-brane conformal and disformal couplings between the scalar field(s)(associated the transverse brane fluctuations) and matter fields (associated to the longi-tudinal brane fluctuations). As we will see, due to the coupling the expansion rate willgenerically be enhanced, allowing for DM annihilation rates larger than the standard pre-diction . Let us stress that a realistic string theory scenario would be more complicatedand may include nonuniversal couplings to baryonic and dark matter. However, it is veryinteresting that scalar couplings present in string theory can give interesting modificationsof the post-inflationary evolution after string inflation.In this paper we show for the first time that this enhancement happens due to a purelydisformal contribution or a combination of conformal and disformal terms. The former case– a disformal enhancement – is particularly interesting as it can be interpreted in termsof an “unwarped” compactification, which is typical of a large-volume compactification ofstring theory, which is needed for perturbative control. When we identify the scale arising It is interesting to notice that a phenomenological model with a faster-than-usual expansion at earlytimes, driven by a new cosmological species, has recently been discussed in Ref. [22]. – 3 –rom the disformal coupling with the tension of a moving D3-brane, where matter is lo-calised, this scale is determined by the string scale and the string coupling. Interestingly,the modification of the expansion rate can take place at different temperature scales, de-pending on the value of the string scale. When we turn on the conformal coupling, whosegeometrical interpretation is a nontrivial warping, the profile of the modified expansion rateresembles the disformal example studied in Ref. [7], allowing for a reanihilation phase forsome DM masses and suitable initial conditions. Compared to the pure conformal case [8],(conformally and) disformally coupled scalar-tensor theories offer a richer phenomenology.The enhancement of expansion due to conformal and disformal terms impact the earlyuniverse cosmology and we study dark matter phenomenology in this paper. We usethe thermal freeze-out picture as an example and study its impact on the correlationof annihilation cross section with the dark matter content. This enhancement can alsoaffect other cosmological phenomena, e.g., leptogenesis, which would require a detailedunderstanding of the model arising from string theory.The rest of the paper is organised as follows. In the next section, we first brieflyintroduce briefly the general D-brane-like setup following the conventions and notation ofRef. [7] (see also Ref. [6]). We then go directly to the cosmological equations and discusshow the expansion rate is modified in general. In section 3 we move on to the D-brane-likecase, where the conformal and disformal functions are related. We start discussing theequations in the Jordan frame as well as the initial conditions and constrains that we useto numerically solve the full equations. We then discuss in detail the solutions for theunwarped case, that is, a purely disformal effect (or C =const.). We numerically computethe modified expansion rate, the enhancement factor and the effects on the relic abundanceand DM annihilation rate. Next we discuss the warped case, using for concreteness thesame conformal function used in Refs. [7, 8]. We also comment on the result of using otherfunctions. Finally in 3.5 we discuss the effect on the DM relic abundances and annihilationrates. We conclude in Section 4 with a discussion and summary of our results. We start this section by outlining our setup, which (as described in the introduction) canarise from a post-string inflationary scenario. At this stage, the universe is already four-dimensional and moduli associated to the compactification have been properly stabilised .However, the relevant parameters in the model will depend on the string theory quantitiessuch as the string scale, string coupling and compactification volume as we will argue.The starting action we consider is given by S = S EH + S brane , (2.1) Though these fields might be displaced from their minima, giving rise to a matter dominated regime,with interesting consequences (see e.g. Ref. [23]). – 4 –here: S EH = 12 κ (cid:90) d x √− g R, (2.2) S brane = − (cid:90) d x √− g (cid:34) M C ( φ ) (cid:115) D ( φ ) C ( φ ) ( ∂φ ) + V ( φ ) (cid:35) − (cid:90) d x (cid:112) − ˜ g L M (˜ g µν ) , (2.3)where in a string setup, κ = M − P = 8 πG is related to the string coupling, scale and overallcompactification volume by M P = V πg s α (cid:48) , where M − s = (cid:96) s = α (cid:48) (2 π ) is the string scale, V is the dimensionless six-dimensional (6D) volume in string units and g s is the stringcoupling. Note also that G is not in general equal to Newton’s constant as measured bye.g. local experiments.In (2.3) we describe the brane dynamics (of transverse and longitudinal fluctuationsassociated to the scalar and matter respectively) given by the Dirac-Born-Infeld (DBI) andChern-Simons actions for a single D3-brane. The DBI part gives rise to the noncanonicallynormalised scalar field φ , associated to the single overall position, r = (cid:80) i y i , of the branein the internal 6D space with coordinates y i . In this case, the scale M is dictated bythe tension of a D3-brane as M = T = ( g s α (cid:48) (2 π ) ) − = M s (2 π ) g − s = g s π V M P andthus by the string scale and coupling. In reality, one would most likely have a stack ofbranes moving in the internal space. However, to study the cosmological evolution afterinflation, it is enough to model all matter living on the moving brane as in Eq. (2.3) (seealso Refs. [6, 24]) via the disformally coupled matter Lagrangian L M .In Ref. (2.3), the disformally coupled metric ˜ g µν is given by the induced metric on thebrane, which for a brane moving along a single internal direction can be written as˜ g µν = C ( φ ) g µν + D ( φ ) ∂ µ φ∂ ν φ . (2.4)where the scalar field is related to the D-brane position by φ = √ T r , and while C ( φ )is dimensionless, D ( φ ) has units of mass − . These functions are specified by the ten-dimensional (10D) compactification and therefore in general will be related to each otheras we see below (see also [6]). Einstein’s equations obtained from Eq. eq2.1 are given by R µν − g µν R = κ (cid:16) T φµν + T µν (cid:17) , (2.5)where the energy-momentum tensors are defined with respect to the Einstein-frame metric g µν and are given by T µν = P g µν + ( ρ + P ) u µ u ν , (2.6) In general, a D3-brane can move in all six of the internal dimensions. For the D3-brane case, one can also consider different dimensionalities, which will add extra factors dueto the internal volumes wrapped by the brane in that case. – 5 –or matter, where ρ , P are the energy density and pressure for matter with equation ofstate P/ρ = ω . For the scalar field, the energy-momentum tensor takes the form: T φµν = − g µν (cid:2) M C γ − + V (cid:3) + M CD γ∂ µ φ ∂ ν φ (2.7)where the energy density and pressure for the scalar field are identified as: ρ φ = M C γ + V , P φ = − M C γ − − V , (2.8)and the “Lorentz factor” γ introduced above is defined by γ ≡ (cid:18) DC ( ∂φ ) (cid:19) − / . (2.9)It will be convenient to rewrite Eq. (2.8) by introducing V ≡ V + C M , as ρ φ = − M CDγ γ + 1 ( ∂φ ) + V , P φ = − M CD γγ + 1 ( ∂φ ) − V . (2.10)The equation of motion for the scalar field is: −∇ µ (cid:2) M DCγ ∂ µ φ (cid:3) + γ − M C (cid:20) D ,φ D + 3 C ,φ C (cid:21) + γ M C (cid:20) C ,φ C − D ,φ D (cid:21) + V φ − T µν (cid:20) C ,φ C g µν + D ,φ C ∂ µ φ∂ ν φ (cid:21) + ∇ µ (cid:20) DC T µν ∂ ν φ (cid:21) = 0 , (2.11)where C ,φ denotes derivative of C with respect to φ , and similarly for D, V . Finally, theenergy-momentum conservation equation, ∇ µ T µνtot = ∇ µ (cid:16) T µνφ + T µν (cid:17) = 0, combined withthe equation of motion for the scalar field allows us to define Q as Q ≡ ∇ µ (cid:20) DC T µλ ∂ λ φ (cid:21) − T µν (cid:20) C ,φ C g µν + D ,φ C ∂ µ φ ∂ ν φ (cid:21) , (2.12)so that, ∇ µ T µνφ = −∇ µ T µν = Q∂ ν φ [7]. Let us now look at the cosmological evolution. We start with a Friedmann-Rrobertson-Walker background metric: ds = − dt + a ( t ) dx i dx i , (2.13)where a ( t ) is the scale factor in the Einstein frame. With this metric, the equations ofmotion become H = κ ρ φ + ρ ] , (2.14)˙ H + H = − κ ρ φ + 3 P φ + ρ + 3 P ] , (2.15)¨ φ + 3 H ˙ φ γ − + C D (cid:18) D ,φ D − C ,φ C + γ − (cid:20) C ,φ C − D ,φ D (cid:21) − γ − C ,φ C (cid:19) + 1 M CDγ ( V ,φ + Q ) = 0 , (2.16)– 6 –here, H = ˙ aa , dots are derivatives with respect to t , γ = (1 − D ˙ φ /C ) − / , and Q = ρ (cid:20) DC ¨ φ + DC ˙ φ (cid:18) H + ˙ ρρ (cid:19) + (cid:18) D ,φ C − DC C ,φ C (cid:19) ˙ φ + C ,φ C (1 − ω ) (cid:21) , (2.17)where we have used the equation of state for matter P = ωρ . The continuity equations forthe scalar field and matter are given by˙ ρ φ + 3 H ( ρ φ + P φ ) = − Q ˙ φ , (2.18)˙ ρ + 3 H ( ρ + P ) = Q ˙ φ . (2.19)Using Eq. (2.19), we can rewrite this as Q = ρ (cid:18) ˙ γ ˙ φ γ + C ,φ C (1 − ω γ ) − Hω ( γ − φ (cid:19) . (2.20)Plugging this into the (non)conservation equation for matter (2.19) gives˙ ρ + 3 H ( ρ + P γ ) = ρ (cid:20) ˙ γγ + C ,φ C ˙ φ (1 − ωγ ) (cid:21) . (2.21) The modified expansion rate felt by matter ˜ H (which will enter into the Boltzmann equa-tion below) is given by the Jordan-frame expansion rate, defined in terms of Jordan (ordisformal) frame quantities, defined with respect to the disformal metric ˜ g µν . In this frame,the Hubble parameter is given by:˜ H ≡ d ln ˜ ad ˜ τ = γC / (cid:20) H + C ,φ C ˙ φ (cid:21) . (2.22)and it is thus a function of the Einstein-frame rate H , the scalar field and its derivatives.The proper time and the scale factors in the Jordan and Einstein frames are related by˜ a = C / a , d ˜ τ = C / γ − dτ . (2.23)Furthermore, the energy densities and pressures in the two frames are related by˜ ρ = C − γ − ρ , ˜ P = C − γP , (2.24)while the equation of state is given by ˜ ω = ωγ . (2.25)One can check that in the Jordan frame, the continuity equation for matter takes thestandard form [7]: d ˜ ρd ˜ τ + 3 ˜ H ( ˜ ρ + ˜ P ) = 0 . (2.26)– 7 –o proceed further, we next swap time derivatives with derivatives with respect to thenumber of efolds, N = ln a/a , so dN = Hdt . We also define a dimensionless scalar field ϕ = κφ . In this case, (2.22) becomes:˜ H = HγC / (cid:2) α ( ϕ ) ϕ (cid:48) (cid:3) , (2.27)where a prime denotes a derivatives with respect to N and we have defined α ( ϕ ) = d ln C / dϕ . (2.28)Note also that in terms of ϕ and N derivatives, the Lorentz factor is now given by γ − = 1 − H κ DC ϕ (cid:48) . (2.29)We want to compare the Jordan-frame expansion rate with that expected in GR, whichis given by H GR = κ GR ρ . (2.30)We can write this in terms of H , ϕ and its derivatives as follows. We first write Eq. (2.14)as (see Refs. [7, 8]): H = κ λ ) B ρ = κ C γ (1 + λ ) B ˜ ρ , (2.31)where λ = V /ρ (= ˜ V / ˜ ρ ), B = 1 − M CDγ γ + 1) ϕ (cid:48) , (2.32)and we have used Eq. (2.24) in the second equality of Eq. (2.31). By inserting Eq. (2.31)into Eq. (2.30), we can write H GR entirely as a function of H, ϕ, ϕ (cid:48) as: H GR = κ GR κ C − B γ − H (1 + λ ) . (2.33)Therefore, once we find a solution for H and ϕ , we can compare the expansion rates ˜ H with H GR using Eqs. (2.27) and (2.33). To measure the departure from the standard expansion,we define the parameter: ξ = ˜ HH GR . (2.34)Notice that ξ can be larger or smaller than one, indicating an enhancement or reduction of˜ H with respect to H GR . This means that ˜ H can grow during the cosmological evolution.However notice that this does not imply a violation of the the null energy condition (NEC).This is because the Einstein-frame expansion rate H is dictated by the energy density ρ and pressure p , which obey the NEC and therefore ˙ H < H and ϕ derived from Ref. (2.15) and Ref. (2.16).– 8 – .4 Coupled equations for ϕ and H The field equations (2.15) and (2.16) can be written as H (cid:48) = − H (cid:20) B λ ) (1 + ω ) + ϕ (cid:48) M CDγ (cid:21) , (2.35) ϕ (cid:48)(cid:48) (cid:20)
1+ 3 H γ − BM CDκ (1 + λ ) DC (cid:21) + 3 ϕ (cid:48) (cid:20) γ − − H γ − BωM CDκ (1 + λ ) DC (cid:21) + H (cid:48) H ϕ (cid:48) (cid:20) H γ − BM CDκ (1 + λ ) DC (cid:21) + 3 Bγ − M CD (1 + λ ) α ( ϕ )(1 − ωγ )+ 3 Bλγ − M CD (1 + λ ) V ,ϕ V + 3 H γ − BM CDκ (1 + λ ) DC (cid:2) ( δ ( ϕ ) − α ( ϕ )) ϕ (cid:48) (cid:3) + κ H CD (cid:2) γ − (5 α ( ϕ ) − δ ( ϕ )) + δ ( ϕ ) − α ( ϕ ) (cid:0) γ − (cid:1)(cid:3) = 0 , (2.36)where δ ( ϕ ) = d ln D / dϕ . (2.37)We notice here that, contrary to the pure conformal case, we cannot eliminate the equationfor H , and we end up with a single master equation for the scalar [8]. Due to the disformalterm, we need to consider the coupled equations for ϕ and H . The cubic equation for H Below we solve the equations numerically, for which we need the initial conditions for H i and ( ϕ i , ϕ (cid:48) i ). Therefore, we need to find an expression for H in terms of all other quantitiesand in particular ˜ ρ . We can obtain this from the Friedmann equation written in terms of ˜ ρ in Eq. (2.31). Recalling that γ depends nontrivially on H (Eq. (2.29)) one obtains a cubicequation for H given by : A H + A H + A H + A = 0 (2.38)where A = Dϕ (cid:48) Cκ , (2.39) A = 2 M CDϕ (cid:48) − , (2.40) A = M C κ (cid:18) M CDϕ (cid:48) − (cid:19) , (2.41) A = (cid:18) M κ C (cid:19) (1 + λ ) ˜ ρM (cid:18) (1 + λ ) ˜ ρM + 2 (cid:19) . (2.42) A similar equation was found in Ref. [7] for the phenomenological disformal case. In that case, A = − A = 0. – 9 –ne of the solutions to Eq. (2.38) can be written as H = 13 A (cid:32) − A + (cid:0) A − A A (cid:1) (cid:18) (cid:19) / + (cid:18) ∆2 (cid:19) / (cid:33) , (2.43)with∆ = − A A + 9 A A A − A + (cid:113)(cid:0) − A A + 9 A A A − A (cid:1) − (cid:0) A − A A (cid:1) ≡ L + (cid:112) L − (cid:96) . (2.44)The other two solutions can be obtained by replacing (cid:18) (cid:19) / → e πi/ (cid:18) (cid:19) / and (cid:18) (cid:19) / → e πi/ (cid:18) (cid:19) / . We are interested in real positive solutions for H . These can be identified by consideringa complex ∆, that is, 4 (cid:96) > L , which implies a condition on ˜ ρ, ϕ (cid:48) , and C . For this choice,the imaginary parts of (∆ / / and (cid:96) (∆ / − / cancel each other . We will use the realpositive solutions in our numerical implementations to find the initial condition for H . As we discussed in Section 2, when considering a probe D3-brane moving in a warped10D space, which is a solution to the 10D equations of motion, C and D are related andgiven in terms of the warp factor of the geometry [6]. In particular, in the normalisationwhere φ becomes canonically normalised once the DBI action is expanded, M CD = 1,(see Appendix C of Ref. [7]). Other normalisations are possible; however the results willbe equivalent. Thus in this section we study solutions for the D-brane conformally anddisformally coupled matter with the choice above, which implies δ ( ϕ ) = − α ( ϕ ). We start bypresenting the equations of motion for this case, followed by a discussion on the constraintsand initial conditions we use in our numerical analysis. We first discuss in detail thenumerical solutions for the C = const or a pure disformal case, followed by the C (cid:54) = constcase. We then analyse the implications for the dark matter relic abundance and associatedannihilation rate. For this we concentrate on the C = const. case, since C (cid:54) = const. givessimilar results to those studied in Ref. [7]. We are interested in the radiation- and matter-dominated eras during which the potentialenergy of the scalar field is subdominant. Therefore in what follows we consider λ ∼
0. Also,to solve the equations Eqs. (2.35) and (2.36), we need to write them in terms of Jordan-frame quantities ˜ ω = ωγ and ˜ ρ = C − γ − ρ . After doing this, the coupled equations abovebecome In this case, we can write Z = ∆2 = L + i √ (cid:96) − L , then Z ¯ Z = (cid:96) and = ¯ Z(cid:96) and thus the imaginaryparts in Eq. (2.43) cancel. – 10 – (cid:48) = − H (cid:20) B ωγ − ) + ϕ (cid:48) γ (cid:21) , (3.1) ϕ (cid:48)(cid:48) (cid:20)
1+ 3 H γ − BM C κ (cid:21) + 3 ϕ (cid:48) γ − (cid:20) − H γ − BM C κ ˜ ω (cid:21) + H (cid:48) H ϕ (cid:48) (cid:20) H γ − BM C κ (cid:21) − H γ − BM C κ α ( ϕ ) ϕ (cid:48) + 3 Bγ − α ( ϕ )(1 − ω ) − M C κ H (cid:2) γ − − γ − + 1 (cid:3) α ( ϕ ) = 0 . (3.2)Furthermore, we also convert derivatives with respect to N to derivatives with respectto ˜ N , the number of e-folds in the Jordan frame [7]. Using Eq. (2.23), we see that N ≡ ln (cid:18) aa (cid:19) = ˜ N + ln (cid:20) C C (cid:21) / . (3.3)where ˜ N ≡ ln (˜ a/ ˜ a ) and the subscript “0” means that the quantity is evaluated at thepresent time. Since we are interested in expressing quantities as functions of temperature,we then use entropy conservation in the Jordan frame. Recalling that the entropy is givenby ˜ S = ˜ a ˜ s , where ˜ s = π g s ( ˜ T ) ˜ T , ˜ N can be expressed as˜ N = ln ˜ T ˜ T (cid:32) g s ( ˜ T ) g s ( ˜ T ) (cid:33) / . (3.4)Therefore, derivatives w.r.t. N transform to derivatives w.r.t. ˜ N (assuming well behavedfunctions) as: ϕ (cid:48) = 1 (cid:16) − α ( ϕ ) dϕd ˜ N (cid:17) dϕd ˜ N , ϕ (cid:48)(cid:48) = 1 (cid:16) − α ( ϕ ) dϕd ˜ N (cid:17) (cid:32) d ϕd ˜ N + dαdϕ (cid:18) dϕd ˜ N (cid:19) (cid:33) . (3.5)To avoid clutter we write down expressions with derivatives with respect to N , but itshould be understood that all our numerical calculations are made using derivatives withrespect to ˜ N .Let us start by discussing Eq. (3.2) to understand the behaviour of the solutions.Similarly to the conformal case [7, 8], the derivative of C ( ϕ ) acts as an effective potential,given by V eff ∼ − ω ) ln C . (3.6)Deep in the radiation-dominated era, the equation of state is given by ˜ ω = 1 / Notice that the last term in (3.2) proportional to α is not part of an effective potential, as it vanisheswhen taking the velocity terms, ϕ (cid:48) to zero (so B = 1 and γ = 1). – 11 –elow the rest mass of each of the particle types, nonzero contributions to 1 − ω arise,activating the effective potential. On the other hand, during the matter-dominated era,˜ ω = 0, and the effective potential is active through it. In Section 3.1.1 of [7] we showedhow to calculate ˜ ω during the radiation-dominated era. We reproduce here the calculationof ˜ ω during the radiation-dominated era in Figure 1. - - - - - T ˜( GeV ) ω ˜ ( T ˜ ) Figure 1 : Equation of state ˜ ω as function of temperature during the radiation-dominatedera. Before we move on to solving the coupled equations (3.1) and (3.2) to find the modifiedexpansion rate, ˜ H and compare it with the standard one, H GR , we stop here to describethe constraints and initial conditions we use in our numerical analysis. Parameter Constraints
In scalar-tensor theories, deviations from GR can be parametrised in terms of the post-Newtonian parameters, γ P N and β P N . In the standard conformal case, these parametersare given in terms of α ( ϕ ) defined in Eq. (2.28) and its derivative, α (cid:48) = dα/dϕ | ϕ as[26, 27] γ P N − − α α , β P N − α (cid:48) α (1 + α ) , (3.7)Solar System tests of gravity – including the perihelion shift of Mercury, lunar laser Rangingexperiments, and the measurements of the Shapiro time delay by the Cassini spacecraft [28–30] –constrain α to very small values of order α (cid:46) − , while binary pulsar observationsimpose that α (cid:48) (cid:38) − .
5. The strongest constraint applies to the the speed-up factor ξ ,which has to be of order 1 before the onset of BBN [31]. Further, the relation betweenthe bare gravitational constant and that measured by local experiments for conformallycoupled theories is given by [32]: κ GR = κ C ( ϕ )[1 + α ( ϕ )] . (3.8)– 12 –or the phenomenological disformal case, Solar System constraints and the ratio (3.8)have been studied for constant D in Ref. [33]. In particular, they found κ GR = κ (1+3Υ / ∝ ϕ (cid:48) . As we will see, all solutions we found have ϕ (cid:48) = 0 at the onset of BBN.Therefore, for the constant conformal case, κ GR = κ . For the C (cid:54) = const case, on theother hand, we will use the constraints on α above requiring that the standard expansionrate is recovered well before the onset of BBN. This is what we need to ensure that thepredictions of the standard cosmological model are not modified. Initial conditions and the scale M
To find the numerical solutions, we need to fix the initial conditions for
H, ϕ, ϕ (cid:48) . Since ϕ is given in Planck units, we take ϕ i , ϕ (cid:48) i (cid:46)
1. To find the initial value for H , we need thereal positive solution to Eq. (2.38) given by Eq. (2.43) for the case CDM = 1, given interms of the initial variables ϕ i , ϕ (cid:48) i , ˜ ρ i . In this case the coefficients A i simplify greatly.Writing as before ∆ = L + i (cid:112) (cid:96) − L , (3.9)we now have L = 2 + 2 ϕ (cid:48) − ϕ (cid:48) + 227 ϕ (cid:48) − ϕ (cid:48) R , (3.10) L = 4 (cid:96) − L = − ϕ (cid:48) R ) (cid:2) ϕ (cid:48) R − (3 + 4 ϕ (cid:48) )( ϕ (cid:48) − (cid:3) , (3.11) (cid:96) = (cid:18) ϕ (cid:48) (cid:19) , (3.12) R = ˜ ρM (cid:18) ˜ ρM + 2 (cid:19) . (3.13)From here it is not hard to see that L can be either positive or negative and we requirethat L > ϕ (cid:48) i , this requirement implies R ≤ (3 + 4 ϕ (cid:48) i )( ϕ (cid:48) i − ϕ (cid:48) i . (3.14)Recalling that during the radiation-dominated era the energy density is given by ˜ ρ ( ˜ T ) = π g eff ( ˜ T ) ˜ T , once we fix ϕ (cid:48) i and the initial temperature T i , the value of M is fixed viaEq. (3.14). Indeed, Eq. (3.14), is satisfied for ˜ ρ i /M in the interval (cid:16) , − ϕ (cid:48) i (cid:113) (3 + ϕ (cid:48) i ) (cid:17) .Or, in terms of T i and ϕ (cid:48) i , the value of M lies in the interval: π g eff ( ˜ T i ) ϕ (cid:48) i − ϕ (cid:48) i + 20 (cid:113) (3 + ϕ (cid:48) i ) / ˜ T i , + ∞ . (3.15)As an example, we show the lower bound for M as a function of ( ϕ (cid:48) i ) in Figure 2for the initial temperature of 1.0 TeV. For simplicity we take C = 1, so that derivatives Recall that in our numerical solutions we take derivatives w.r.t. ˜ N , so ϕ (cid:48) i should be read as ϕ (cid:48) i − α ( ϕ i ) . – 13 –ith respect to N and ˜ N are the same. As can be seen from Eq. (3.15) and Fig. 2, for agiven initial condition T i , the closer ϕ (cid:48) i goes to √
6, the larger the values of M , and viceversa. Also, the larger the value of T i , the larger also the lower bound of M . In terms ofthe D-brane-like scenario as we described in section 2, the scale M is related to the stringcoupling and scale (or the six-dimensional volume) as M = M s (2 πg − s ) / . Therefore, wesee that the scale decreases for small string scales (large compactification volumes) andsmall string couplings, which are needed for the string perturbative description to be valid.We will come back to this point below. Figure 2 : Lower bound for M (Eq. (3.15)) as a function of ( ϕ (cid:48) i ) for C = 1. C = const. We are now ready to discuss in detail the numerical solutions for H and ϕ and use themto compute the modified expansion rate. We start with the case C ( ϕ ) = const which canbe understood as a pure disformal case, which is presented here for the first time. Indeed,notice that in this case γ (cid:54) = 1, which precisely carries the disformal (or derivative) effect,while α = 0 (which carries the conformal effect)Without loss of generality we can take C ( ϕ ) = 1 and therefore D ( ϕ ) = M . Comparingwith the phenomenological case studied in Ref. [7], one could think that an arbitrary choiceof the function D there (with C = 1) would give different results. However, we expect thatthe effects of an arbitrary function in that case can be encoded in the choice of the scale M here, and therefore will give similar results to those presented here.For C = 1, the system of coupled equations reduces to the following form, H (cid:48) = − H (cid:20)
32 (1 + ˜ ωγ − ) B + ϕ (cid:48) γ (cid:21) , (3.16) ϕ (cid:48)(cid:48) (cid:20)
1+ 3 H γ − BM κ (cid:21) + 3 ϕ (cid:48) γ − (cid:20) − H γ − BM κ ˜ ω (cid:21) + H (cid:48) H ϕ (cid:48) (cid:20) H γ − BM κ (cid:21) = 0 . (3.17)As expected, the effective potential is flat, since α = 0 (see discussion above). We solvethese equations numerically to find the dimensionless scalar field ϕ and the Hubble param-– 14 –ter H , as functions of ˜ N . We have explored a wide range of initial conditions for ϕ and ϕ (cid:48) and values of the scale M . To find the initial condition for H ( H i ), we use the appropriatereal positive solution of Eq. (2.38). We find that at most two of the solutions Eq. (2.38)for H i , are real and positive. For these two H i ’s, the corresponding initial value of γ ( γ i )is obtained using (2.29) (setting M CD = 1). We find that one of these γ i ’s is usually oforder one while the other is 1 or 2 orders of magnitude larger (sometimes even larger). Wefind that the solutions to Eqs. (3.16) and (3.17) that obey the necessary constraints arethose with γ i ∼
1. Once we have found the solutions for ϕ and H , we go back to Eq. (2.27)to obtain the expansion rate in the Jordan frame.Before looking into the full numerical solutions, let us take a closer look at the ratiobetween the modified expansion rate and the standard rate, (2.34). For C = const thisbecomes ξ = γ / B / . (3.18)Since γ, B ≥
1, it is clear that ξ ≥
1, that is, ˜ H ≥ H GR . In other words, in this case, theexpansion rate is always enhanced with respect to the standard evolution. Moreover, thisenhancement is driven by γ and B . As soon as γ >
1, there will be a nontrivial disformalenhancement.In Figure 3 we show the resulting modified expansion rate for different values of ϕ (cid:48) andthe mass scale M . In these plots we use ϕ i = 0 .
2, but any value in the interval (0 ,
1) andappropriate choices of ϕ (cid:48) i and the mass scale M , will give similar results. As we can seein Figure 3, ˜ H (colored lines) is always enhanced with respect to the standard expansionrate (black line), H GR , as discussed above. From (3.18) and Figure 4, it is clear that theratio ξ is always greater than or equal to 1. Moreover as the temperature decreases, theratio ξ grows from a value close to 1 (recall that γ i ∼ γ ismaximal and eventually decreases towards one before BBN. The maximum value of theratio increases and moves to lower temperatures as the mass scale M becomes smaller.We can understand this behavior by looking at the evolution of the factor f = H γ − BM κ ,inside the square brackets of Eq. (3.17). We have seen numerically that this factor evolvesas f ( ˜ T ) (cid:119) g eff ( ˜ T )10 (cid:16) ˜ TM (cid:17) as temperature decreases (see Figure 7). For the scale M andtemperatures plotted in Figure 3, f ( ˜ T ) starts much bigger than one (up to f ( ˜ T i ) (cid:119) )and decreases as the temperature decreases. The bigger the scale M , the earlier f ( ˜ T )becomes of order 1. While f ( ˜ T ) is bigger than 1, ξ increases, the velocity of the scalar field ϕ (cid:48) increases slowly, and thus the scalar field increases very slowly too (see Figure 4 and 5).As f ( ˜ T ) approaches 1, ξ reaches a maximum and the scalar field starts increasing faster.Then, as the temperature decreases further, f ( ˜ T ) becomes smaller than 1. Meanwhile, ˜ H starts converging towards H GR (that is, ξ starts decreasing) while the scalar field keepsincreasing. Finally, when ˜ H becomes of order H GR , f ( ˜ T ) is much smaller than 1 and thescalar field starts moving towards a final constant value.We see the behaviour described above in Figures 4 and 5. For instance, for the plotcorresponding to M = 106GeV, f ( ˜ T ) is approximately g eff ( ˜ T )10 (cid:16) ˜ T
106 GeV (cid:17) , which becomes– 15 – igure 3 : Modified expansion rate for the pure disformal case, C = 1. We show differentboundary conditions and values of the scale parameter. The initial value of the scalar fieldfor all the curves is ϕ i = 0 .
2. The black lines in all plots represent the standard expansionrate H GR .1 at around ˜ T = 50 GeV. Between 1000 GeV and 50 GeV, ˜ H differs from H GR andin this range the scalar field increases very slowly, looking almost constant. For lowertemperatures, between 50 GeV and 1 GeV, ˜ H converges towards H GR and the scalar fieldincreases faster while for temperatures smaller than 1 GeV, ˜ H ∼ H GR and the scalar fieldreaches its final value.All the cases shown in Figure 3 satisfy the constraints discussed in Section 3.2. Inparticular, ϕ (cid:48) BBN = 0 (so Υ = 0) and the speed-up factor ξ is equal to 1 prior to BBN asshown in Figure 4. For scales M smaller than 10 GeV the last condition is not satisfied,that is ξ > M smaller than 10 GeV are discarded.As we have mentioned, if we consider larger values of M than the ones presented inFigure 3, the enhancement of the expansion rate will occur earlier at higher temperatures,such that f ( ˜ T ) is much bigger than 1 at around the initial value of the temperature, ˜ T i .To achieve this, one has to consider M smaller than ˜ T i , which happens when the initialvalue of ϕ (cid:48) is much smaller than 1. We illustrate this in Figure 6 were we show a seriesof plots were the mass scale takes values up to order EeV. This figure also shows that the– 16 – igure 4 : Speed-up factor, ξ = ˜ H/H GR , as function of temperature for the expansionrates shown in the bottom left plot in Figure 3. The initial conditions chosen are ϕ i = 0 . ϕ (cid:48) i = 0 . Figure 5 : Scalar field as a function of temperature. The initial conditions chosen are ϕ i = 0 . ϕ (cid:48) i = 0 . T i /M does not change.For instance, for all the green lines ˜ T i /M =58.8. C (cid:54) = const We now move to the case where the conformal coupling is not constant, so both conformaland disformal effects are turned on. For concreteness we consider the same conformalcoupling as that studied in Refs. [7] and [8], which is given by C ( ϕ ) = (1 + b e − β ϕ ) , (3.19)– 17 – igure 6 : Modified expansion rate for the pure disformal case, C = 1, for larger values of M as compared to Fig. 3. For these plots, ϕ i = 0 . ϕ (cid:48) i = 2 × − . Figure 7 : Evolution of the factor f as a function of temperature for C = const case (left)and C (cid:54) = const. ( f C , right). The initial conditions chosen in the left plot are shown inFigure 3, while in the right plot ϕ i = 0 . ϕ (cid:48) i = − . b = 0 . β = 8. We have also analysed other functions such as C = ( b ϕ + c ) with b = 4 , , c = 1. However it is harder to find numerical solutions for this and otherfunctions, which satisfy the phenomenological constraints. In the cases we analysed, theeffect on the expansion rate ˜ H was smaller with respect to the case in Eq. (3.19).– 18 – igure 8 : Scalar field as a function of temperature for different values of M . The conformalcoupling is (1 + 0 . e − ϕ ) and the initial conditions chosen are ϕ i = 0 . ϕ (cid:48) i = − . f C = H γ − BM C κ is much larger than 1, as can be seen from Eq. (3.2). In this regime, theevolution of the scalar field is given by a flat effective potential, and the scalar field staysapproximately constant. When f C is becomes of order 1 or smaller and ˜ ω (cid:54) = 1 /
3, theevolution of the scalar field is driven by the effective potential (3.6) and by the Hubblefriction term.For the conformal coupling (3.19), the effective potential allows for an interestingbehaviour, according to the choice of initial conditions [7]. That is, for negative initialvelocities, ϕ (cid:48) i <
0, the scalar field will start rolling up the effective potential towards smallervalues. After reaching a maximum point, it will turn back down the effective potential,eventually reaching its final value. This behaviour in the scalar field sources a nontrivialbehaviour in C and (importantly) its derivative α , and therefore in the modified expansionrate ˜ H . Indeed, when C (cid:54) = const. we have ξ = κκ GR C / γ / B / (cid:2) α ( ϕ ) ϕ (cid:48) (cid:3) . (3.20)It is not hard to see that for the initial conditions above, due to the factor inside theparentheses, ξ can become less than one during the evolution. Recalling that ξ = ˜ H/H GR , ξ < H < H GR , as shown in the explicit solutions below. This effect gives riseto the possibility of a reannihilation period, as was discussed in Ref. [7] and first pointedout in Ref. [8].Let us now take a closer look at the evolution of f C with temperature. Numerically,we found that when f C (cid:38) f C ( ˜ T ) (cid:119) g eff ( ˜ T )10 (cid:16) ˜ TM (cid:17) . But when f C < f C ( ˜ T ) (cid:119) h ( ˜ T ) g eff ( ˜ T )10 (cid:16) ˜ TM (cid:17) , where h ( ˜ T ) is a function that measures the– 19 – igure 9 : Modified expansion rate for the case C = 1 + 0 . e − ϕ . The initial value of thescalar field for all the curves is ϕ i = 0 .
2. Also, ϕ (cid:48) i = − .
004 for the plot on the left and ϕ (cid:48) i = − . H , which is larger than 1 and depends on the scale M (see right plot inFigure 7). When f C (cid:29)
1, the effective force is negligible and the scalar field stays roughlyconstant. As f C decreases and approaches and/or becomes smaller than 1, the effectiveforce takes over the evolution of the scalar field. The velocity of the scalar field startsdecreasing (we use small negative velocities), and for suitable values, the scalar field goesup the effective potential and comes back down again as described above.In Figure 8 we plot the full numerical solution for the scalar field for ϕ i = 0 . ϕ (cid:48) = − . T i = 1000 GeV) show the scalar field going up the effective potential toward smaller valuesof the field, and then rolling down its terminal value., while for the brown curve ( M =1000GeV), the scalar field stays almost constant because for this value of M its initial velocityis not negative enough to move the field up the effective potential.The effect of the scalar field on the modified expansion rate is shown in Figure 9 (theblack straight line is H GR ). The left plot shows ˜ H corresponding to the scalar field solutionsin Figure 8. For these solutions, the factor f C is initially much bigger than 1 and as thetemperature decreases it passes one (around 200 GeV) and keeps decreasing to very smallvalues. For some values of M , the scalar field goes up and down the effective potential,producing the enhancement and the little notch in ˜ H (blue), where ξ < f C is initially of order 1 and then decreasesto negligible values. The initial velocity used ( ϕ (cid:48) = − .
4) is sufficiently negative to producethe enhancement and notch in ˜ H for some of the M values (green and blue).Let us mention another point about the right plot in Figure 9. For the brown curvecorresponding to M = 5000 GeV, the enhancement is very small, and since the factor f C decreases as the mass scale M increases, choosing larger values of M would give a similarresult, for the same choice of initial conditions. Indeed, as M → ∞ , f C = H γ − BM C κ → M increases, since γ → M increases. So, by dropping allterms proportional to f C and the last term in Eq. (3.2) ones recovers the conformal case– 20 –quations studied in Refs. [7, 8]. For a very large value of M , we will recover the results ofRef. [7, 8], by suitably changing the initial conditions for ϕ i and ϕ (cid:48) i .Let us finally comment on the differences between the present case (conformal plusdisformal) and the pure disformal and pure conformal cases, where there is no derivativeinteraction. We saw in the previous subsection that in the pure disformal case the en-hancement in the expansion rate can be produced at any temperature (see Figure 6), bysuitably changing the value of the scale M . However, in the present C (cid:54) = 1 case, this doesnot happen at any scale since ω (cid:54) = 1 / ω (cid:54) = 1 / ϕ (cid:48) go to zero very fast, effectively making γ ∼ ω can introduce an enhancement at that scale. This will be similar to the case withthe additional M scale associated with the D-brane models The conformal enhancementis effective so long as the effective potential (3.6) is active, that is, whenever ω (cid:54) = 1 / Now that we have computed the modified expansion rate, we move on to discuss its impactand implications on the dark matter relic abundance and cross section. We focus on thecase C = const. since the C (cid:54) = const. case gives similar results to those studied in [7], aswe discuss below.For a dark matter species χ with mass m χ and a thermally averaged annihilation crosssection (cid:104) σv (cid:105) (where v is the relative velocity), the dark matter number density n χ evolvesaccording to the Boltzmann equation dn χ dt = − Hn χ − (cid:104) σv (cid:105) (cid:0) n χ − ( n eqχ ) (cid:1) , (3.21)where ˜ H is the expansion rate in the Jordan frame computed in the previous section (seeFigures 3 and 9), felt by the matter particles and n eqχ is the equilibrium number density.To solve Eq. (3.21), we rewrite it in the standard form in terms of x = m χ / ˜ T , dYdx = − ˜ s (cid:104) σv (cid:105) x ˜ H (cid:0) Y − Y eq (cid:1) , (3.22)where Y = n χ ˜ s is the abundance and ˜ s = π g s ( ˜ T ) ˜ T is the entropy density. As a concreteexample, we solve Eq. (3.22) numerically, for the expansion rate corresponding to M = 12GeV shown in the top left plot of Figure 3 and for dark matter particles with massesranging from 10 GeV to 5000 GeV. Other choices of M would give similar results. InFigure 10 we show the solution for a DM particle of mass m χ = 100 GeV. In this plot, wealso include the abundance Y GR ( x ) calculated in the standard cosmology model and theabundance when dark matter particles are in thermal equilibrium, Y Eq ( x ).In the plot we can see how the modification of the expansion rate gives rise to an earlierthan standard freeze-out (see Figure 10 around x = 20). This is due to the enhancement ofthe expansion rate ˜ H . As the temperature decreases ( x increases), ˜ H becomes comparable– 21 – igure 10 : Abundance ˜ Y for a dark matter particle with a mass of 100 GeV.to the interaction rate ˜Γ and for a small period, between x = 20 and x = 1000, theabundance decreases slowly until it becomes constant. It is interesting to notice that asimilar behaviour was found in Ref. [22] in a phenomenological model where an extrascalar species drives a faster than usual expansion rate, giving rise to a similar behaviourin the relic abundance. The comparison between ˜ H (brown) and ˜Γ (purple) can be seen inFigure 11. Between around 5 GeV ( x = 20) and 0.1 GeV ( x = 1000), ˜ H and ˜Γ are close toeach other as temperature decreases.In Fig. 11, we also show the interaction rate for two other DM particle masses, 600 GeV(green) and 2500 GeV (red). Notice that for the three masses shown, once the interactionrate becomes smaller than the expansion rate ˜ H (brown), it always stays smaller thanit. Therefore, there is no reannihilation effect, as we anticipated in section 3.3. Howeverreannihilation can occur for the C (cid:54) = const. case, where after the first freeze-out ˜Γ canovercome ˜ H due to ξ <
1, and later become smaller again.Let us now turn to the dark matter cross sections we have used when solving theBoltzmann equation (3.22). For this we used the observed dark matter density Ω = 0 . (cid:104) σv (cid:105) required to match thecurrent 27% DM content. The present dark matter content of the universe is determined bythe current value of the relic abundance. This can be obtained from the current value of theenergy density parameter Ω = ρ ρ c, = m Y s ρ c, . Here ρ c, and s are the well-known currentvalues of the critical energy density and the entropy density of the universe, respectively.The resulting annihilation cross sections we determine in this way are shown in Figure12 for dark matter masses between 10 GeV and 5000 GeV, for different values of M andcorresponding expansion rates ˜ H shown in Figure 3. We compare this to the annihilation The interaction rate is defined as ˜Γ ≡ (cid:104) σv (cid:105) ˜ s ˜ Y . – 22 – igure 11 : Expansion rate corresponding to M = 12 GeV shown in the top left plot ofFigure 3, and interaction rates of 100 GeV (purple), 600 GeV (green) and 2500 GeV (red)GeV DM particle masses as a function of temperature. The interaction rate ˜Γ is given by (cid:104) σv (cid:105) ˜ s ˜ Y . Figure 12 : (cid:104) σv (cid:105) as function of dark matter particle mass. (cid:104) σv (cid:105) GR predicted by thestandard cosmology model correspond to the black line, while the colored lines correspondto (cid:104) σv (cid:105) predicted by using the expansion rates (shown in Figure 3), representing massscales of M = 12 (brown), 34 (red), 106 (green) and 333 GeV (blue).cross sections (cid:104) σv (cid:105) GR predicted by the standard cosmology model (black line), which isaround 2 . × − cm /s .The behaviour of the cross section (cid:104) σv (cid:105) in Fig. 12 shows an enhancement with respectto the standard case, with a maximum that moves towards larger dark matter masses as– 23 –he scale M increases. Therefore, the smaller the scale M the larger the annihilation crosssection (cid:104) σv (cid:105) . We can correlate this behaviour with that of ξ in Fig. 4, which shows theenhancement of the expansion rate. For example, for a mass scale of 34 GeV (red) themaximum (cid:104) σv (cid:105) is around 100 × − cm / s for a DM mass of 2000 GeV, for a mass scaleof 12 GeV (brown) the maximum (cid:104) σv (cid:105) is around 200 × − cm /s for a DM mass of 700GeV. For a 600 GeV DM mass the ratio (cid:104) σv (cid:105) / (cid:104) σv (cid:105) GR is almost 1 for M = 333 GeV (blueline), while for M = 12 GeV it is around 100 (brown line). Scalar-tensor theories where the gravitational interaction is mediated by both the metricand scalar fields arise commonly in modifications of standard general relativity theories.The prototype example is the Brans-Dicke theory where the metric and the scalar field arerelated via the conformal coupling as ˜ g µν = C ( φ ) g µν . However, the most general physicallyconsistent relation between two metrics which can be given by a single scalar field includesa disformal (or derivative) coupling [21]: ˜ g µν = C ( φ ) g µν + D ( φ ) ∂ µ φ∂ ν φ .Both couplings C and D can give rise to a different expansion rate from the standardcosmological model in the early universe, and still be in agreement with current constraintsfrom BBN and gravity. In particular, BBN imposes a strong constraint on these couplingsand it is encoded in the speed-up parameter ξ (Eq. (2.34)), which needs to be very close toone before the onset of BBN. It was shown in Ref. [8] that the expansion rate modificationdue to a conformal coupling can change the predictions on the dark matter relic abundances,anticipating freeze-out and producing a reannihilation phase for certain choices of initialconditions, as discussed in Ref. [7].As was shown in Ref. [6], the (conformal and) disformal transformation naturally arisesfrom D-branes and thus in D-brane models of cosmology. In this case, the functions C and D are closely related and are dictated by the UV theory, for example, type IIB string theory.Moreover, the scalar field has a geometrical origin in terms of the transverse fluctuations ofthe D-brane, while matter lives on the brane and it comes from the longitudinal fluctuations.In this paper we have studied the modification to the expansion rate due to the dis-formally coupled scalar arising in D-brane-like models for cosmology, where D = 1 /M C (Figs. 3, 6 and 9). Using the modified expansion rates thus found, we solved the Boltzmannequation to compute the dark matter relic abundances (Fig. 10). To find solutions, we usedthe current cosmological data on the DM content to determine the required thermally av-eraged cross sections (Fig. 12).We numerically solved the coupled equations for H and ϕ (Eqs. (3.1) and (3.2)) andused this to find the modified expansion rate. Note that contrary to the purely conformalcase, in the presence of the disformal term it is not possible to eliminate H from the systemto solve a single master equation, as in Ref. [8]. So we need to carefully take into accountboth equations as well as the initial conditions for H . This introduces a cubic equation for H in terms of the other parameters ( ϕ, ˜ ρ ) and a lower bound for the scale M , given theinitial conditions for ( ϕ i , ˜ ρ i ) (see section 3.2).– 24 –n section 3.3 we presented for the first time the purely disformal case correspondingto C = const. where the modification to the expansion rate is fully driven by the derivativecoupling through γ (see Eq. (3.18)). For this case, the modified expansion rate is alwaysenhanced with respect to the standard one (Fig. 3), which implies an anticipated freeze-out and an enhancement of the cross section (cid:104) σv (cid:105) (Fig. 12). These results are robustagainst different initial conditions and we further studied the dependence on the scale M .We found that the larger the value of M , the earlier in the cosmological evolution theenhancement in the expansion rate (Fig. 6). Therefore, depending on the value of M ,the modified expansion rate can appear at different times in the early universe and theexpansion rate can be ∼
500 times bigger compared to the standard GR case. This willaffect any physical process in the early universe where the Hubble expansion rate is neededto determine the out-of-equilibrium temperature. In this paper we focused on the effect onthe relic abundance and the annihilation rate of dark matter.Our results are also robust compared to the phenomenological disformal models usuallydiscussed in the literature and in Ref. [7]. In that case, the conformal and disformalfunctions are unrelated (up to causality constraints). In that case, a purely disformalcontribution will be obtained by setting C = 1 and letting D be an arbitrary function,fixed only by phenomenological constraints. The disformal enhancement in the expansionrate is similarly encoded in γ in Eq. (3.18) and different choices for the function D wouldbe equivalent to different values of M in the present paper. Therefore, our analysis iscompletely general and also applies to these phenomenological models.For the C (cid:54) = const case (section 3.4) we saw that it is possible to have an enhancementas well as a reduction of the expansion rate with respect to the standard case, that is ξ > ξ < M on the profiles of the expansion rates. We considered in detailfor concreteness only the conformal function used in Refs. [7, 8]. For this function, thenumerical analysis is relatively simple; however, there is in principle no obstruction to findsimilar effects for other functions. We looked at the function C = ( bx + c ) (for b = 4 , , c = 1), which can be a toy model for a smooth warp factor in a string theory setup. Forthis example, we found a relatively small enhancement with ξ ∼
4. We expect that a widersearch of parameters and conformal functions will give rise to a larger enhancement as wellas a decrease in the modified expansion rate.
Post-inflationary string cosmology
The period after the end of inflation, from reheating up to the onset of BBN remains largelyunconstrained. Let us now connect our results with a post-inflationary toy model of stringcosmology and discuss the implications in terms of the parameters of the theory.As described in section 2, we imagine a toy model where matter is coupled conformallyand disformally to a scalar field, associated to an overall position of a (stack of) D-brane(s)in the internal six-dimensional compact space in a warped type IIB string compactification.In this case, we can relate the scale M to the tension of a D3-brane T (for example), and– 25 –hus to the string coupling g s and the string scale M s (or the six-dimensional volume V ) as M = T / = M s (2 πg − s ) / . The pure disformal case C = 1 is very interesting and wouldcorrespond to a large-volume compactification, where the warping due to the presenceof fluxes can be ignored [34, 35]. In this case, the lowest value we used for M that isrelevant for the DM relic abundance was ∼
10 GeV and the largest (with a large effect)was M ∼ g s ∼ − , these give string scales ∼ − V ∼ − , that is, a very low stringscale and very weakly coupled compactification. On the other hand, for larger values of M , or larger values of the string scale, the enhancement on the expansion rate will occurearlier in the universe’s evolution (see Fig. 6). For example, the largest value we used, M ∼ GeV, for g s ∼ − would give M s ∼ GeV, volumes of order V ∼ andthe expansion enhancement occurs at around ˜ T ∼ GeV. Therefore, depending on thestring scale, coupling and compactification, we may expect the early pre-BBN cosmologyto be affected at different epochs with interesting consequences for the post-inflationarystring cosmological evolution. This will also be connected to string inflation, which usuallyrequires large string scales (see Ref. [36] for a review).Of course a more realistic model may involve, for example, other parameters of thetheory in the scale M (due for example to higher-dimensional D-branes wrapping theinternal space), nonuniversal couplings among the scalar (or scalars) to SM matter andDM as briefly discussed in Ref. [13] for the pure conformal case, etc. However, we find itvery interesting that scalar couplings present in string theory can give important predictionsfor the post-inflationary evolution after string inflation.Let us finally stress that the cosmological implications of conformal and disformal cou-plings in scalar-tensor theories are in any case very interesting from a more phenomenolog-ical point of view. Here we have taken a further step in making progress to address theseimplications and have presented the disformal effects for the first time. Acknowledgments
We would like to thank Enrico Nardi and Gianmassimo Tasinato for useful discussionsand Gianmassimo Tasinato for comments on the manuscript. BD and EJ acknowledgesupport from DOE Grant de-sc0010813. IZ is partially supported by the STFC grantsST/N001419/1 and ST/L000369/1. IZ would like to thank the Mitchell Institute for Fun-damental Physics and Astronomy at Texas A&M University for kind hospitality while partof this work was done. Part of this work was performed at the Aspen Center for Physics,which is supported by National Science Foundation grant PHY-1607761.
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