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UMN–TH–2424/05,FTPI–MINN–05/52 Big bang nucleosynthesis constraints on scalar-tensor theories of gravity Alain Coc∗ Centre de Spectrométrie Nucléaire et de Spectrométrie de Masse, IN2P3/CNRS/UPS, Bât. 104, 91405 0rsay Campus (France) Keith A. Olive† William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455 (USA) Jean-Philippe Uzan‡ and Elisabeth Vangioni§ Institut d’Astrophysique de Paris, UMR-7095 du CNRS, Université Pierre et Marie Curie, 98 bis bd Arago, 75014 Paris (France) 6 (Dated: February 5, 2008) 0 WeinvestigateBBN in scalar-tensor theoriesof gravitywith arbitrary matter couplingsand self- 0 interaction potentials. Wefirst consider thecase of amassless dilaton with a quadraticcouplingto 2 matter. We perform a full numerical integration of the evolution of the scalar field and compute n the resulting light element abundances. We demonstrate in detail the importance of particle mass a thresholdsontheevolutionofthescalarfieldinaradiationdominateduniverse. Wealsoconsiderthe J simplest extension of this model including a cosmological constant in eitherthe Jordan or Einstein 3 frame. 1 PACSnumbers: PACS 1 v 9 I. INTRODUCTION the dilaton, and other scalar fields, moduli, appear dur- 9 ingKaluza-Kleindimensionalreductionofhigherdimen- 2 sional theories to our usual four dimensional spacetime. 1 The concordance model of cosmology calls for the In cosmology, two main properties make these theo- 0 introduction of a cosmological constant or a dark en- ries appealing. First, an attraction mechanism toward 6 ergy sector. Various candidates have been proposed [1], 0 amongwhichthepossibilitythatgravityisnotdescribed general relativity [8, 9] was exhibited. This implies that / even if the tests of general relativity in the Solar sys- h by general relativity on large cosmological scales. It is tem set strong constraints on these theories, they may p of interest therefore, to test our theory of gravity in a differ significantly from general relativity at high red- - cosmological context. This can be achieved in two com- o shift. Second, it was shown that the general mechanism plementaryways,eitherbydesigningmodelindependent r ofquintessencewasconserved[10,11]ifthe quintessence t tests(seee.g. Refs.[2,3,4]foradiscussionofthevarious s field was non-minimally coupled and that the attraction a possibletests)orbyconsideringaclassofwellmotivated mechanism toward general relativity still held with run- : theoriesanduseallavailabledatatodeterminehowclose v awaypotentials[12,13,14]. Theseextendedquintessence i to general relativity we must be. X models are the simplest theories in which there is a long Among all extensions of general relativity, scalar- rangemodificationofgravity,sincethequintessencefield r a tensor theories are probably the simplest in the sense islight,andtheyallowforaveryinterestingphenomenol- that they consider only the introduction of one [5] (or ogy [15]. many [6]) scalar field(s) universally coupled to mat- Cosmological data give access to various aspects of ter. These theories involve two free functions describing thesemodels. Thecosmicmicrowavebackground(CMB) the coupling of the scalar field to matter and its self- tests the theory in the linear regime [16, 17, 18, 19, 20] interaction potential. They respect local Lorentz invari- while weak lensing opens a complementary window on ance and the universality of free fall of laboratory-size the non-linear regime [21, 22]. Solar system experiments bodies. They are motivated by high-energy theories try- give information on the theory today and big-bang ing to unify gravity with other interactions which gener- nucleosynthesis (BBN) allows us to constrain the at- ically involve a scalar field in the gravitational sector. traction mechanism toward general relativity [8] at very In particular, in superstring theories [7] the supermulti- high redshift. pletofthe10-dimensionalgravitoncontainsascalarfield, BBN is one of the most sensitive available probes of the very early Universe and of physics beyond the standard model. Its success rests on the concordance be- ∗Electronicaddress: [email protected] tween the observational determinations of the light ele- †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] mentabundancesofD,3He,4He,and7Li, andtheir the- §Electronicaddress: [email protected] oretically predicted abundances [23, 24]. Furthermore, 2 measurements of the CMB anisotropies by WMAP [25] helium-4 is sensitive to the modification of gravity con- haveledtoprecisiondeterminationsofthebaryondensity sideredhere. In IV,wewillconsiderthesimplestexten- § orequivalentlythebaryon-to-photonratio,η. Asη isthe sionofthismodelbyintroducingacosmologicalconstant. soleparameterofthestandardmodelofBBN,itispossi- Such a constant can be introduced as a constant poten- ble to make very accuratepredictions [26, 27, 28, 29, 30] tial either in the Einstein frame, hence keeping the dila- andhencefurtherconstrainphysicsbeyondthestandard ton massless, or in the Jordan frame, hence generalizing model [31]. the constant energy density component. Both cases are In particular, the 4He abundance is often used as a considered and we conclude in Section V. Applications sensitive probe of new physics. This is due to the fact tovariouscasesofcosmologicalinterestwillbepresented that nearly all available neutrons at the time of BBN in a follow-up article. end up in 4He and the neutron-to-proton ratio is very sensitivetothecompetitionbetweentheweakinteraction rate andthe expansionrate. Ofinterestto us here is the II. IMPLEMENTING SCALAR-TENSOR THEORIES OF GRAVITY IN A BBN CODE effectofmodificationstogravitywhichwilldirectlyaffect the expansion rate of the Universe through a modified Friedmann equation. A. Scalar-tensor theories in brief TheWMAPbestfitassumingavaryingspectralindex isΩ h2 =0.0224 0.0009andisequivalenttoη = Inscalar-tensortheoriesofgravity,gravityismediated b 10,CMB 6.14 0.25, wher±e η = 1010η. Using the WMAP data not only by a spin-2 gravitonbut also by a spin-0 scalar 10 ± to fix the baryon density, the light element abundances field that couples universally to matter fields. In the [26, 27, 28, 29, 30] can be quite accurately predicted. Jordan frame, the action of the theory takes the form Some BBN results are displayed in Table 1. d4x The effect of scalar-tensor theories of gravity on the S = √ g[F(ϕ)R gµνZ(ϕ)ϕ ϕ 2U(ϕ)] ,µ ,ν 16πG − − − production of light elements has been investigated ex- Z ∗ tensively (see e.g. Ref. [32] for a review). As a first +Sm[gµν;ψ] (1) step, it is useful to consider only the speed up factor, where G is the bare gravitational constant from which ξ = H/H , that arises from the modification of the ∗ GR we define κ = 8πG . This action involves three arbi- value of the gravitational constant during BBN [31, 33]. ∗ ∗ trary functions (F, Z and U) but only two are physical Other approaches considered the full dynamics of the since there is still the possibility to redefine the scalar problembut restrictedthemselves to the particularclass field. F needs to be positive to ensure that the graviton of Jordan-Fierz-Brans-Dicke theory [34], of a massless carries positive energy. S is the action of the matter dilatonwith a quadraticcoupling [35, 36]or to a general m fields that are coupled minimally to the metric g with massless dilaton [37]. It should be noted that a com- µν signature ( ,+,+,+). binedanalysisofBBNandCMBdatawasinvestigatedin − The action (1) can be rewritten in the Einstein frame Ref.[38]andRef.[39]. TheformerconsideredGconstant by performing the conformal transformation during BBNwhile the latter focused ona non-minimally quadratic coupling and a runaway potential. We stress g∗ =F(ϕ)g (2) µν µν that the dynamics of the field can modify CMB results so that one needs to be careful while inferring Ω from as b WMAP. d4x The goal of this article is to implement scalar-tensor S = 16πG √−g∗[R∗−2g∗µν∂µϕ∗∂νϕ∗−4V(ϕ∗)] ∗ theories in an up-to-date BBN code. This will comple- Z +S [A2(ϕ )g∗ ;ψ]. (3) ment our existing set of tools which allows us to con- m ∗ µν front scalar-tensor theories with observations of type Ia The field ϕ and the two functions A(ϕ ) andV(ϕ ) are ∗ ∗ ∗ supernovae and CMB anisotropies [16] as well as weak defined by lensing [21]. In particular, the predictions to be compared with observations can be computed in the same dϕ 2 3 dlnF(ϕ) 2 Z(ϕ) ∗ = + (4) framework for any self-interaction potential and matter- dϕ 4 dϕ 2F(ϕ) coupling function. (cid:18) (cid:19) (cid:20) (cid:21) A(ϕ ) = F−1/2(ϕ) (5) We first recall, in II, the equations describing the ∗ theory to be implemen§ted in our BBN code and we also 2V(ϕ∗) = U(ϕ)F−2(ϕ). (6) discuss local constraints. As a check of our code, we We will denote any Einstein frame quantities by a star consider, in III, the case of a massless dilaton with quadratic cou§pling [35]. In particular, we perform a full (*), e.g. R∗ is the Ricci scalar of the metric gµ∗ν. The strength of the coupling of the scalar field to the matter numericalintegrationuptothepresentthatcanbecom- fields is characterizedby paredwith the analyticalresults of Ref. [35]. We update the constraints on this model by taking into account the dlnA α(ϕ ) (7) latest BBN data discussed above. We reaffirm that only ∗ ≡ dϕ ∗ 3 TABLE I: BBN results for the light element abundances assuming the WMAP-inferred baryon density. Source Yp D/H 3He/H 7Li/H ×10−5 ×10−5 ×10−10 Coc et al. (2004) 0.2479±0.0004 2.60±0.17 1.04±0.04 4.15±0.46 Cyburt et al. (2003) 0.2484+0.0004 2.74+0.26 0.93+0.1 3.76+1.03 −0.0005 −0.16 −0.67 −0.38 and we also define and conservation equations are given by β(ϕ∗)≡ ddϕα . (8) Z(ϕ¨+3Hϕ˙)=3Fϕ H˙ +2H2+ RK2 ∗ (cid:18) (cid:19) 1 Z ϕ˙2 U (12) ϕ ϕ It is useful to study both the Einstein and Jordan −2 − frames. In the Jordan frame, matter is universally cou- ρ˙+3H(ρ+P)=0. (13) pledto the metric. The Jordanmetric defines the length If we define the density parameters today by andtimeasmeasuredbylaboratoryapparatussothatall observations(time,redshift,...) havetheirstandardinter- 8πG ρ ∗ 0 pretationinthisframe. However,todiscussthetheoryit Ω0 ≡ 3H2F , (14) is often better to use the Einstein frame in which the ki- 0 0 netictermshavebeendiagonalizedsothatthespin-2and the evolution of the energy density of a fluid with con- spin-0degreesoffreedomofthetheoryareperturbations stant equation of state w = P/ρ takes the usual form of g∗ and ϕ respectively. The physical properties of µν ∗ both frames are of course identical. For example, when 3H2F Ω we refer to the time variation of the gravitational con- ρ= 0 0 0(1+z)3(1+w) (15) stant (in the Jordan frame), we have assumed fixed par- 8πG∗ ticlemasses. Incontrast,intheEinsteinframe,wewould where z is the redshift defined by 1+z =R /R. 0 infer a fixed gravitational constant and varying masses. In both frames, the quantity Gm2 (which is physically measureable) varies in the same way. 2. Equations in Einstein frame ThescalefactorandcosmictimeinEinsteinframeare B. Friedmann equations related to the ones in Jordan frame by R=A(ϕ )R , dt=A(ϕ )dt (16) ∗ ∗ ∗ ∗ 1. Equations in Jordan frame so that the redshifts are related by WeconsideraFriedmann-Lemaˆıtreuniversewithmet- A 0 1+z = (1+z ). (17) ric in the Jordan frame ∗ A ds2 = dt2+R2(t)γ dxidxj (9) The Friedmann equations in this frame take the form ij − K 3 H2+ =8πG ρ +ψ2+2V(ϕ ) (18) where γij is the spatial metric and R the scale factor. ∗ R2 ∗ ∗ ∗ ∗ The matter fields are describedby a collectionof perfect (cid:18) ∗(cid:19) 3 d2R fluidsofenergydensity,ρandpressureP. Itfollowsthat ∗ =4πG (ρ +3P )+2ψ2 2V(ϕ )(19) the Friedmann equations in Jordan frame take the form −R∗2 dt2∗ ∗ ∗ ∗ ∗ − ∗ where we have introduced H =dlnR /dt and ∗ ∗ ∗ K 1 3F H2+ = 8πG ρ+ Zϕ˙2 3HF˙ +U(10) R2 ∗ 2 − ψ∗ =dϕ∗/dt∗. (20) (cid:18) (cid:19) K 2F H˙ = 8πG (ρ+P)+Zϕ˙2 These equations take the same form as the standard − − R2 ∗ Friedmann equations for a universe containing a perfect (cid:18) (cid:19) +F¨ HF˙ (11) fluidandascalarfield. TheKlein-Gordonequationtakes − the form where a dot refers to a derivative with respect to the dψ dV ∗ +3H ψ = 4πG α(ϕ )(ρ 3P ) (21) cosmic time t and H dlnR/dt. The Klein-Gordon dt ∗ ∗ −dϕ − ∗ ∗ ∗− ∗ ≡ ∗ ∗ 4 while the matter conservation equation is given by 2. Gravitational constant dρ ∗ +3H (ρ +P )=α(ϕ )(ρ 3P )ψ . (22) The Friedmann equations in the Jordan frame define ∗ ∗ ∗ ∗ ∗ ∗ ∗ dt − ∗ an effective gravitationalconstant These equations differ from their standard form due to G =G /F =G A2. (30) thecouplingthatappearsinther.h.s. Thesolutionofthe eff ∗ ∗ evolutionequation(22)canbeobtainedfromtherelation Thisconstant,however,doesnotcorrespondtothegrav- betweenthe energydensity andthe pressureof afluid in itational constant effectively measured in a Cavendish Einstein frame and their Jordan frame counterparts experiment. The constant measured in this type of ex- ρ =A4ρ, P =A4P (23) periment is ∗ ∗ G =G A2(1+α2) (31) which imply, in particular, that cav ∗ 0 0 3H2Ω A 4−3(1+w) where the firstterm, G∗A20, correspondsto the exchange ρ∗ = 8π0G 0 A (1+z∗)3(1+w) (24) of a graviton while the second term, G∗A20α20, is related ∗ (cid:18) 0(cid:19) to the long range scalar force. Assuming fixed particle masses, the time variation of for a fluid with a constant equation of state. the gravitationalconstant is bounded [47] by 1 dG C. Constraints today cav =σ0H0, σ0 <5.86 10−2h−1. (32) G dt | | × cav 1. Post-newtonian constraints Choosing the number of Einstein frame e-folds as a time variable, Thepost-Newtonianparameters(seeRefs.[40,45])can be expressed in terms of the values of α and β today as p= ln(1+z∗), (33) − 2α2 1 β α2 implies that γPPN 1= 0 , βPPN 1= 0 0 . (25) − −1+α2 − 2(1+α2)2 0 0 β σ dϕ 0 0 ∗ 2α 1+ =σ . (34) Solar System experiments set strong limits on these pa- 0 1+α2 − 2 dp 0 (cid:20) 0 (cid:21) (cid:12)0 rameters. The perihelion shift of Mercury implies [41] (cid:12) (cid:12) Notethatthelimitβ = (1+α2)thatc(cid:12)orrespondstothe − so-called Barker theory [48] in which A = cosϕ leads ∗ 2γPPN βPPN 1 <3 10−3, (26) to σ = 0 whatever the value of α and ϕ′ so that the | − − | × ∗ gravitational constant is strictly constant even though the Lunar Laser Ranging experiment [42] sets gravity is not described by general relativity. 4γPPN βPPN 3= (0.7 1) 10−3. (27) − − − ± × D. Numerical implementation Two experiments give a bound on γPPN alone, the Very Long Baseline Interferometer [43] The nuclear reaction network takes its standard form γPPN 1 <4 10−4, (28) in Jordan frame. To compute the light elements abun- | − | × dances during BBN, one only needs to know the expan- and the measurement of the time delay variation to the sion rate history, H(z), from deep in the radiation era Cassini spacecraft near Solar conjunction [44] up to today. It is thus convenient to express the Hubble parameterinthe Jordanframeintermsoftheoneinthe γPPN 1=(2.1 2.3) 10−5. (29) Einstein frame, using Eq. (16), as − ± × These two last bounds imply α0 to be very small, typi- AH =[H∗+α(ϕ∗)ψ∗] (35) cally α2 < 10−5 while β can still be large [46]. Binary 0 0 pulsar observations impose that β & 4.5. Note that where ψ is defined by Eq. (20). Eq. (35) can also be 0 ∗ − even though β is not bounded above by experiment, we expressed in the simple form 0 will assume that it is not very large, typically we assume β0 . 100, so that the post-Newtonian approxima- AH =H 1+α(ϕ )dϕ∗ . (36) tion scheme makes sense. ∗ ∗ dp (cid:20) (cid:21) 5 Itfollowsthat,intermsofthecosmictimet,theequa- as tions of evolution can be recast as 2 ddϕt∗ = A−1(ϕ∗)ψ∗ (37) 3−ϕ′∗2ϕ′∗′+(1−w)ϕ′∗ =−α(ϕ∗)(1−3w). (44) dψ As emphasized in Ref. [8], this is the equation of mo- ∗ = A−1(ϕ )[3H ψ dt − ∗ ∗ ∗ tion of a point particle with a velocity dependent in- ertial mass, m(ϕ ) = 2/(3 ϕ′2), evolving in a po- +4πG∗α(ϕ∗)A4(ϕ∗)Xi (1−3wi)ρi+ ddϕV∗#(38) t−e(n1ti−alwα)(ϕϕ′∗∗.)(D1u−ri∗n3gwt)haencdossmu−boljoegc∗itcatoleavodluatmiopnintghefofirecled, 8πG 1 2 K is driven toward the minimum of the coupling function. H2 = ∗A4(ϕ ) ρ + ψ2+ V(ϕ ) (39) ∗ 3 ∗ i 3 ∗ 3 ∗ −R2 Ifβ >0,itdrivesϕ∗ toward0,thatisα 0,sothatthe Xi ∗ scalar-tensortheorybecomes closerandc→loserto general ρ = ρ (1+z)3(1+wi) (40) relativity. When β < 0, the theory is driven way from i i0 H = A−1[H +α(ϕ )ψ ]. (41) general relativity and is likely to be incompatible with ∗ ∗ ∗ local tests unless ϕ was initially arbitrarily close to 0. ∗ The numerical integration is performed as follows. Thus, we will restrict our analysis to β >0. First we choose some initial value ϕin∗, ψin∗ = 0 deep We need to considerthree regimes: (i) deepin the ra- in the radiation era (typically, zin = 1012 and we inte- diation era, (ii) the effect of particle annihilation during grate the system (37-41) to z = 0. We perform a shoot- the radiation era (electron-positron annihilation in par- ing method so that the solution reaches the value ΩΛ0 ticular)and(iii)thetransitionbetweentheradiationand and GN today, which fixes G∗ and the energy scale of matter era. the potential. At this stage the value of ϕ and α are ∗0 0 known. We also keep track of ψ to infer the time vari- 0∗ ation of the gravitational constant, Gcav, and check its 1. Deep radiation era compatibility with the constraint(32). Subsequently, we perform a second integration of the same system includ- Deep in the radiation era, w = 1/3 and the coupling ing the nuclear reaction network. to ϕ is not efficient. The equation of evolution reduces ∗ to III. MASSLESS DILATON WITH QUADRATIC 2 2 ϕ′′+ ϕ′ =0. (45) COUPLING 3 ϕ′2 ∗ 3 ∗ − ∗ This can be integrated to give The simplest model to consider consists of a massless dilaton with a quadratic coupling to matter. That is, V(ϕ∗)=0, A=ea(ϕ∗), a(ϕ∗)= 21βϕ2∗. (42) ϕ∗ =ϕ∗i−√3ln"αie−(p−pαi)i++p11++αα2i2ie−2(p−pi)#(46) p It follows that where α is defined by i α0 =βϕ0∗, β0 =β. (43) α = ϕ′∗i (47) i This model has been studied in detail in the literature, 3 ϕ′ 2 bothintermsofitsdynamics[8,9]andofitsBBNpredic- − ∗i q tions [35, 36]. We use it as a test model to check our nu- and where ϕ and ϕ′ are the values of ϕ and its p- ∗i ∗i ∗i merical scheme. In particular, the analytical behaviour derivative at the initial time p . From Eq. (3), ϕ is i ∗ of the field during the radiation and matter eras after expressedin Planckunits andwe will allow values ofϕ ∗i BBN was obtained for a flat universe without cosmo- to be of order unity. Interestingly, we see that in the logical constant in Ref. [35]. The numerical integration radiation dominated era, ϕ rapidly tends to a constant ∗ through BBN was also matched to this solution. Since value. The field derivative is just wewouldliketousethe sameintegrationschemeforany potential and coupling, we can not rely on a particular 3 atonaclhyetcickstohleutaicocnu.raIctyisofusoeudr cinodteh.is particular case only ϕ′∗ =s1+α2iαie−(p−pi) (48) sothatitisdividedbyenin∆p=ne-folds. Inparticular if we send p then its variation between p and A. General study i → −∞ i some time in the radiation era is As long asV =0,the Klein-Gordonequation(21) can ∆ϕ √3ln α + 1+α2 . (49) berewrittenintermsofthevariablepdefinedbyEq.(33) ∗ →− i i (cid:18) q (cid:19) 6 Itfollowsthat,aslongasϕ′ √3, ∆ϕ ϕ′ andthe ∗i ≪ | ∗|∼ ∗i Slices at constant b fieldgetsfrozenataconstantvalueduringradiationera. These properties can be recovered easily from the form aout 1 ofEq. (21)oftheevolutionequationsinceitimpliesthat ϕ˙∗ decreases as R∗−3. This behavior is quite general for 1 dilaton-like fields [49]. In conclusion, deep in the radia- -1 10 tion era (much before nucleosynthesis) the initial condi- 10 tion can be chosen to be ϕ˙ =0 and ϕ =constant. ∗in ∗in 40 30 80 -2 10 2. Mass thresholds 50 90 70 100 20 The previous analysis ignores an interesting effect [9] that appears when the universe cools below the mass of -3 10 some species χ, T m . This species becomes non- 60 χ ∼ relativistic and induces a non-vanishing contribution to the r.h.s. of Eq. (21). For example, during electron- positron annihilation, the r.h.s. of Eq. (21) depends on -4 10 Σe =(ρe 3Pe)/ρrad. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 − In the Jordan frame the total energy density of the a radiation is in π2 ρrad =g∗(T) T4 (50) FIG. 1: aout as a function of ain for different values of β 30 between 1and 100. Wesee that aout <ain which reflectsthe where g is the effective number of relativistic degreesof attraction towardsgeneralrelativityduringelectron-positron ∗ freedom, annihilation. 7 g (T)= g + g , (51) ∗ 8 i i 3. Details of the field dynamics near threshold fermion bosons X X and T is the Jordanframe temperature of the radiation, Let us now investigate the dynamics of the attrac- as long as the particles are in thermal equilibrium with tion toward GR during a mass threshold in more detail. theradiationbath. Thetermρe 3Pe takestheform[50] In general, the temperature is related to the integration − g ∞ q2 dq variable p by ρ 3P = e m2 . (52) e− e 2π2 eZ0 eE/T +1 q2+m2e T[ϕ ,p]=T A0 qγ(T0) 1/3e−p (55) ∗ 0 Intoducing x≡E/T and ze ≡me/T, wepdeduce that A(ϕ∗)(cid:20)qγ(T)(cid:21) Σe(T) = π154gg(eT)ze2 ∞ exx2+−1ze2dx (53) wenhteerreinqgγthisetdheefineiffteiocntivoeftnhuemebnterroopfy,rtealaktiinvgisitnitcopaacrctoiculnest ∗ Zze p only particles in equilibrium with the photons. so that the Klein-Gordon equation (44) takes the form ThedependenceT[ϕ ,p]andthenon-lineartermϕ′2/3 ∗ ∗ make Eq. (54) difficult to integrate. In regimes where β 2 2 3 ϕ′2ϕ′∗′+ 3ϕ′∗+Σe(T)βϕ∗ =0. (54) and ϕ∗in are not too large then it can safely be approxi- − ∗ mated by The force term depends on the temperature which de- 3 pends on A(ϕ ) and p. When this term is no longer ef- ϕ′′+ϕ′ + Σ (ep)βϕ =0. (56) ∗ ∗ ∗ 2 e ∗ fective, the field evolves according to Eq. (45) and tends to another constant, ϕ . The relation between ϕ Thisapproximateequationassumesthatthe fieldisslow ∗out ∗in and ϕ , or equivalently between a = a(ϕ ) and rolling and that A(ϕ ) does not vary much during the ∗out in ∗in ∗ a , has a complicated structure. Eq. (44) is almost transition. Itisalinearequationinϕ sothatitssolution out ∗ a damped oscillator (because of the non-linear term). is proportional to ϕ . ∗in When Σ (T)βϕ is small, the field does not have the Fig.3comparesthesolutionsofthetwoequations(ap- e ∗in time to oscillate while Σ is non-negligible, and the rela- proximateandexact)forasinglemassthreshold. Indeed e tion a -a is linear. For larger values of β and/or a as long as β is small, the field is slow rolling and A re- in out in onegetsdampedoscillationssothata <a . Figure1 mainsalmostconstantduringthetransition. Thisisseen out in depicts the relation a -a for various values of the pa- in the top panel of Fig. 3 for β =1 and the field evolves in out rameter β and Figure 2 illustrates the complexity of the to ϕ 0.84ϕ . Howeverwhenβ is largeandas a con- ∗ ∗in ≈ full solution of this equation. sequenceA(ϕ )is alsolarge,the variationofϕ during ∗in ∗ 7 aout -1 10 -2 10 -3 10 -4 10 100 80 2 60 1.5 1.75 1.25 40 0.75 1 0.5 b 20 0 0.25 ain ain b - aout FIG.2: Thegeneralstructureofaout asafunctionofain and β. Thisillustrates thecomplexityofthesolutionsofEq.(44) . the transitionimplies,becauseoftherelation(55)thata given width, ∆T, corresponds to a larger ∆p, while this latter is fixed for the approximate solution. This implies that the attraction toward ϕ = 0 is more important. ∗ This progression is seen in the middle and lower panels FIG. 3: Evolution of the scalar field in phase space during a of Fig. 3. transitionforβ=1,5,50fromtoptobottomwhenweassume Fig. 4 compares the value aout/ain as a function of β. ϕ∗in =1. The solid line corresponds to the exact solution of Eq. (54) while the dashed line corresponds to the solution of When Eq. (56) is used, we recover the result of Ref. [9]. theapproximate equation (56). In this case, since this equation is linear, a /a does out in not depend on the initial value of ϕ and is a universal ∗in function of β. This is compared to the ratio obtained fromthe integrationofEq.(54). As longasϕ is small the initial value of a . ∗in in (typically of order 0.1), both results agree (because ϕ′ It is difficult to predict the value of a from general in remains small and A does not vary significantly). How- arguments. For instance, if we expect ϕ 1 (in Planck ∗ ever, aout/ain is typically 10 times smaller when ϕ∗in is units) at the end of inflation, this means∼that αin β of order unity. This agrees with the results depicted in and a β/2. In this case, one would indeed lik∼e to in Fig. 1. investiga∼tea withvaluesuptoroughly50. Ontheother in In conclusion, we see that both the depedence of the hand, if we expect a deviation from general relativity of source termfor ϕ∗ and the non-linear termin ϕ′∗ lead to order one at the end of inflation, then we might expect significantmodificationsofthedynamicswhentheinitial α 1, or ϕ β−1 and a β−1/2. In that latter in ∗ in ∼ ∼ ∼ value of the scalar field or β are large. caserestrictingtoa 3wouldbesafe. Clearly,without in ∼ a detailed model of the inflationary period it is difficult to determine the “natural” range of variation of a . in 4. Expected value of ain To get some insight on the expected order of magni- tudeofa justbeforetheperiodofelectron-positronan- in nihilations, we must investigatethe effect of higher mass Inthepreviousanalysiswehaverestrictedourselvesto thresholds. To that end, we consider an extention of a between 0 and 3, mainly for numerical reasons. We in Eq. (54) in which the source term is replaced by a sum can now in a position to justify this choice. Indeed, and as noted earlier, because the scalar field is frozen during the radiation dominated era, we need only specify ϕ∗in Σ(T)= Σi(T), (57) as an initial condition. For a given value of β, this fixes species X 8 isalmostdecoupledfromthepreviousones. Thus,wewill be able to compute the state of the scalar field just before the last transition which is of primary importance for BBN. FIG.5: Thesourcefunction,Σ(T),enteringtheKlein-Gordon equationwhenthemassthresholdscorrespondingtothepar- ticleslistedinthetext. Thedashedcurvesshowtheindividual particle contributions, Σi(T), and the solid curve shows the sum, Σ(T). Fig. 6 describes the dynamics of this multi-threshold phase (including the electron-positron annihilation). As long as β or ϕ remain small, we see that each of the ∗in FIG.4: Evolutionofaout/ain asafunctionofβ. (Top): when four peaks of Σ, corresponds to a well defined departure weusedtheapproximateequation(56),itdoesnotdependon from ϕ′ =0 with movement towards smaller ϕ propor- ∗ ∗ theinitialvalueofthescalarfield,ϕ∗in. (middleandbottom): tionaltotheinitialvalueϕ∗in. Forlargervalues,thefield we use the exact equation (54). This equation being non- is firstslow-rollingandthen oscillatesaroundϕ =0. In ∗ linear theratio dependson theinitial value of ϕ∗in. this case, we see that the effect of the four peaks cannot be considered separately because the field does not have time to settle back to ϕ′ =0 between two transitions. where ∗ The evolution of ϕ , when mass thresholds are non- 15 g ∞ x2 z2 ∗ Σi(T) = π4g∗(iT)zi2Zzi pex+−ǫ i dx (58) tnoegtlhigeibpleer,ioaldloowfseulescttroond-epteorsmitrinone tahnenivhailluaetioonf.aiWn eprdioer- note this value by a . Fig. 7 shows a /a , that is the with z = m /T, ǫ = +1 for fermions and ǫ = 1 for ee ee in bosonsi. i − value of a(ϕ∗) just before electron-positron annihilation comparedtoitsinitialvalueatveryhightemperature,as In principle, all massive standard model particles will a function of β. The attractiontowardgeneralrelativity play a role. In addition to electrons and positrons, is very drastic. In the case where ϕ is of order unity, we must consider the effects of muons, pions, charmed ∗in weconcludethata .10−4 a 10−4β/2.5 10−3. quarks, taus, bottom quarks, W± bosons, Z0 boson and ee × in ∼ × Itfollowsthatrestrictingtoa =0 3atbeforeelectron- the top quarks. The role of lighter quarks is tied to the in − annihilation is a safe limit even if ϕ (1) at the end quark hadron transition whose effect we do not include. ∗ ∼ O of the inflationary phase. Nor do we include the effect of the Higgs boson due to its as yet uncertain mass. Fig. 5 depicts the evolutionof Note also that phase transitions are another source of Σ with (the Jordan frame) temperature. In particular, attraction toward general relativity. We have not in- it showsthat the effects of the variousthresholds cannot cluded either the quark-hardon transition or the elec- be considered separately because the transitions overlap troweak transition in the previous analysis. During a andthescalarfield,ϕ ,doesnothavetimetosettleback phase transition, there is generally a significant modifi- ∗ toϕ′ =0betweentwotransitions. Alsonotethat,fortu- cationoftheequationofstatewhichwillinduce asource ∗ nately,thelastthreshold(electron-positronannihilation) term in the Klein-Gordon equation. 9 0 a to a . We allow the code to integrate the evolution out 0 -0.005 equationuptothe present,sothatweobtaina directly. 0 -0.01 For the particular case of a vanishing potential or as (cid:143)!!!'3j(cid:144)-0-.00.1052 ilmonagteas3t−heϕfi′∗2el∼d i3s ssloowthraotllEinqg.,(ϕ4′4≪) ta1k,eosnethceansimappplirfioexd- -0.025 form -0.03 -0.035 d2ϕ∗ 1 dϕ∗ 3 y(y+1) + (5y+4) + βϕ =0, (59) 0.6 0.7 0.8 0.9 1 dy2 2 dy 2 ∗ j(cid:144)jin where we have introduced the variable y R /R ∗ dec∗ ≡ and used the fact that the gas is a mixture of presure- 0 less matter and radiation and the equation of state is -0.02 w = 1/3(1+y). This equation allows us to relate the (cid:143)!!!'3(cid:144) value of the scalar field deep in the radiation era but af- j-0.04 ter BBN, ϕ , to its value today, ϕ . Its solution is out∗ 0∗ a hypergeometric function, f (y)= F [s,s∗,2; y] with -0.06 β 2 1 − s=3/4 i 3(β 3/8)/2 so that − − 0 0.2 0.4 0.6 0.8 1 j(cid:144)jin p ϕ =ϕ f (y ) (60) 0∗ out∗ β 0 FIG. 6: Dynamical evolution of ϕ∗ when the source term is where the matching to the analytical solution has been described by Fig. 5. (Top) we have set β = 1 and ϕ∗in = performed after the end of nucleosynthesis at a time 0.1,0.5,1 (solid, dashed, dotted). (Bottom) We have fixed where ϕ is constant. y is given by ϕ∗in =1andtakenβ=1,15,20,50(thinsolid, solid,dashed, ∗ 0 dotted). R A y = 0 eq =(1+z )exp[(α2 α2)/2β]. (61) 0 R A eq eq− 0 dec 0 Thismethodavoidsintegratingthesystem(37-41)tothe present but requires a determination of y . Indeed when 0 ϕhasnotvariedsignificantlybetweenBBNandequality, then y (1+z )exp[(α2 α2)/2β]. (62) 0 ≃ eq out− 0 However,thissolutioncannotbegeneralizedtoaΛ-CDM or to extended quintessence models. For this reason we donotusethismethodandintegratethesystemnumeri- callyfromz toz =0. Figure8comparesournumerical in integration, from which we determine the exact value of y andthe analyticsolution(60). We see that the agree- 0 ment is almost perfect. It can be checked that an error smaller than 10% on the evaluation of y left a almost 0 0 unchanged. Letusemphasizethatinmoregeneralcases, i.e. for different potentials and coupling functions, such an analytic solution is in general not known so that the full numerical approach is necessary. The solution (60) implies that ϕ′ = ϕ g (y ) with ∗0 out β 0 g (y ) = 3βy F (1+s,1+s∗,3; y )/4. It is then FIG.7: Evolutionofaee/ainasafunctionofβwhenthesource β 0 − 02 1 − 0 term is described by Fig. 5 just before the electron-positron possible to estimate, from Eq. (32), the value of σ0 as a annihilation. (Top): β=1 (Bottom): β =0.1. function of (αout,β), 1+ β σ =2α g (y ) (1+α2out)fβ2(y0) . (63) 5. Radiation-matter transition 0 out β 0 1+ α2outf (y )g (y ) β β 0 β 0 Inprinciple,BBNwillplaceaconstraintonthevalueof As shown by Fig. 9, as soon as α . 1, the constraint out a . As such, our constraint will in effect be dependent on σ is satisfied. This means that for the quadratic out 0 ona whichisunknown. Tocomparetheseconstraintsto coupling model, nearly all parameter choices satisfy this in the ones obtained in the Solar system, we need to relate constraint. 10 6. Equivalent speed-up factor -6 a0 10 As long as V = 0 and one can neglect the curvature 10-7 numerical term,the Friedmannequation(18)canbe written, using Eq. (35) as analytical -8 10 1 ϕ′2/3 10-9 3(1−+αϕ′)2H2 =8πG∗ρA2. (64) ∗ -10 10 Comparing this to the standard Friedmann equation, 10-11 3HG2R = 8πG∗A20(1 + α20)ρ, one obtains the speed-up factor defined to be the ratio of the Hubble parameters, -12 10 A(ϕ )1+α(ϕ )ϕ′ 1 ξ = ∗ ∗ ∗ . (65) -13 10 A0 1−ϕ′2/3 1+α20 10-14 Figure10showsthevarpiationofthepspeed-upfactordur- ing BBN for various values of β, taking into account the -15 10 effects of electron-positron annihilation. ξ is constant 1 10 102 above z 2 1011 and below z 109. For large val- b ues of β,∼typi×cally β & 5, the attr∼action toward general relativity is so efficient that ξ 1 for z . 109. For ∼ FIG.8: Comparison ofthenumericalintegrationandthean- smaller values, ξ is frozen at some constant value ξ > 1 alytical solution for a flat CDM model with ain=1. at the end of BBN and will be driven towards 1 only when the subsequent evolution due to matter domina- tion will be significant. For very small values of β, as 100 pointedoutinRef. [35]andaswehaveshownearlier,ϕ ∗ -(cid:1)2(cid:13) is almost constantduring the electron-positronannihila- tionperiod. Asaresult,ϕ =ϕ andϕ′ 0sothat ∗out ∗in ∗ ∼ 80 (cid:1)-10(cid:13) (cid:1)-5(cid:13) -(cid:1)3(cid:13) b 2lnξ =β−1(α2in−α2out)−ln(1+α20). (66) 60 Inamorecomplexsituation,onecannotapproximatethis factor by a constant and the full dynamics during BBN must be determined. In particular, we see that ξ drops aroundthetimetheneutron-to-protonratio,n/p,freezes 40 out, but generically reaches a constant value during the nucleosynthesis period. More tuned models in which the variation of ξ is not finishedduring BBNmay leadto somesignaturesonthe 20 primordial abundances (see e.g. [51] for a proposal). In- deed,nomodelindependentstatementscanbemadebut ingeneralweexpectthemtobeveryconstrained,inpar- 0 ticular if the mass thresholds prior to electron-positron -4 -3 -2 -1 0 1 annihilationaretakenintoaccount. Suchmodelscaneas- Log [a (cid:2) ] out ily be discussed in future works with the tool presented here. FIG. 9: σ0 as a function of αout and β. The labels on the contour lines give thevalue of log|σ0|. B. Numerical simulations The time evolution of a(ϕ ) is depicted in Figure 11 ∗ The constraint on σ leading to a < 1 is an a pos- for three values of β. It is obtained by numerically in- 0 out teriori argument for not considering very large values of tegrating the equations of IID by a standard Runge– § a at electron-positron annihilation. Very large values Kutta method. In the top panel of Figure 11. The two in of a will in general lead to large values of a that are plateaus at high z correspond to the constant values of in out constrained by local tests on the constancy of the gravi- a during the radiation era before and after BBN. Oscil- tational constant. latory behaviour due to the damped oscillation of the