On the dark radiation problem in the axiverse
aa r X i v : . [ h e p - ph ] F e b On the dark radiation problem in the axiverse
Dmitry Gorbunov , , Anna Tokareva , Institute for Nuclear Research of Russian Academy of Sciences, 117312 Moscow, Russia Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia Ecole Polytechnique Federale de Lausanne, CH-1015, Lausanne, Switzerland
July 14, 2018
Abstract
String scenarios generically predict that we live in a so called axiverse: the Universewith about a hundred of light axion species which are decoupled from the StandardModel particles. However, the axions can couple to the inflaton which leads to theirproduction after inflation. Then, these axions remain in the expanding Universe con-tributing to the dark radiation component, which is severely bounded from presentcosmological data. We place a general constraint on the axion production rate andapply it to several variants of reasonable inflaton-to-axion couplings. The limit merelyconstrains the number of ultralight axions and the relative strength of inflaton-to-axioncoupling. It is valid in both large and small field inflationary models irrespectively ofthe axion energy scales and masses. Thus, the limit is complementary to those associ-ated with the Universe overclosure and axion isocurvature fluctuations. In particular,a hundred of axions is forbidden if inflaton universally couples to all the fields at re-heating. In the case of gravitational sector being responsible for the reheating of theUniverse (which is a natural option in all inflationary models with modified gravity),the axion production can be efficient. We find that in the Starobinsky R -inflationeven a single axion (e.g. the standard QCD-axion) is in tension with the Planck data,making the model inconsistent with the axiverse. The general conclusion is that aninflation with inefficient reheating mechanism and low reheating temperature may bein tension with the presence of light scalars. Introduction
In order to solve several problems of the hot Big Bang cosmology, the inflationary stage ofthe Universe evolution has been proposed [1, 2]. The simplest way to organize the close-to-exponential expansion of the Universe is to exploit the scalar field (inflaton) slowly rollingtowards the minimum of the potential [3, 4]. Recent data from the Planck satellite putsuch strong constraints on the inflaton potential that the simplest quartic and quadraticcases turn out to be excluded [5]. The best agreement with the data is still exhibitedby the large field inflation provided by the exponentially flat potential. Such a potentialnaturally arises in different models with the modified gravitational sector [1, 6, 7] as well asin the supergravity framework [8–10]. All these models, being non-renormalizable, requirean ultraviolet completion which is often thought to be a string theory.However, the common prediction of many string scenarios is the existence of a plenty oflight scalars (axions) arising as zero modes of antisymmetric gauge fields on the compactifieddimensions [11]. They inherit a perturbative shift symmetry violated by instanton contribu-tions [12, 13]. One of these scalars can play a role of QCD axion explaining the zero valueof the QCD θ -angle. The number of light axions is determined by the topology of extradimensions and is likely to be about hundred [14].Many potentially observable consequences of the string axiverse were discussed in liter-ature [14, 15]. Here we explore whether many light scalars are compatible with inflationarymodels. String axions can contribute to dark matter component, see e.g. [16]. In orderto avoid overproduction and suppress isocurvature fluctuations, one constrains the axionmasses, couplings and initial conditions at the inflationary stage. At the same time, lightscalars produced at reheating can contribute to the dark radiation, which implies an addi-tional bound on the model parameters. Dark radiation amount at Big Bang Nucleosynthesis(BBN) and recombination is now strictly bounded by the Planck data [18]. In generic stringscenarios, the number of effective relativistic degrees of freedom at these epochs is predictedto be much larger than the Planck results [18] allow, the discrepancy is mostly due to thecontribution of moduli decays [15]. One can suppose, however, that the supersymmetrybreaking scale, as well as the masses of all moduli, are larger than the Hubble parameterat inflation. In this case, the moduli are not produced and the axions are created only bythe inflaton decay. We show that in this case, if the inflaton couples to SM particles onlythrough the gravity (i.e. via suppressed by Planck mass operators, which is natural, e.g. for F ( R ) inflationary models), then even one light scalar is in tension with the recent Planck2ata. Thus, in the axiverse the inflaton must couple to matter much stronger. In that case,we put a lower bound on the reheating temperature which makes the existence of hundredsof light scalars consistent with the Planck data. In each model, the inflationary stage must be followed by the reheating process eventuallypopulating the Universe with the hot plasma of SM particles. Therefore, the energy of theinflaton field must be somehow transformed into the usual matter. In general, during thisprocess axion-like particles can well be produced. Let the production rate of each axionbe Γ a while that of the SM particles be Γ SM . Then, the number of additional degreesof freedom at BBN and recombination written traditionally as the number of additionalneutrino components is ∆ N eff = N g reh g ν (cid:18) g BBN g reh (cid:19) / Γ a Γ SM . (1)Here N is the number of axion species, g ν = 2 · / g reh and g BBN are the effective numberof relativistic degrees of freedom at reheating and nucleosynthesis stages, respectively. TheirSM values are g reh = 106 .
75 and g BBN = 10 .
75. According to the latest Planck data, withinthe concordance ΛCDM cosmological model the effective number of relativistic species N eff is bounded as [18] N eff = 3 . ± . . (2)This parameter is expected to be measured in future CMB polarization experiments withmuch higher accuracy [19, 20].One can observe from (1) that if the reheating is due to some universal mechanism (i.e.inflaton decays to all scalars including the SM Higgs with the comparable rates Γ a ∼ Γ SM / N eff ∼ . N which is clearly incompatible with bound (2) in the axiverse . This is themain finding of the present paper.The bound (1), (2) constrains axion coupling to inflaton and the number of axions ul-trarelativistic at BBN and recombination. It is applicable to both large- and small-fieldinflation with any mass pattern in the axion sector, as far as the axions remain ultrarela-tivistic. Thus, the obtained constraint is complementary to another bounds on the axionenergy scales, masses and on the energy scale of inflation. These bounds are inferred from theUniverse overclosure argument and absence of the axion isocurvature fluctuations [16, 17].3ormula (1) is exact if the axion branching ratio Γ a / Γ SM is constant in time at thereheating epoch. Otherwise it is corrected by a numerical factor of order one, which is thecase if the inflaton couplings to the axion and SM-particles are of the different nature, e.g.provided by the operators of different dimensions. Then Γ SM in eq. (1) defines the age of theUniverse at reheating, t U ∼ / Γ SM and Γ a / Γ SM ∼ Γ a t U ≪ The Starobinsky model of inflation historically was the first successful model suggested forthe exponential stage of the Universe expansion [1]. However, it still provides the predictionswhich are in a perfect agreement with the present cosmological data [18]. In a Jordan frame,the model is described by the following action: S = − M Z √− g d x (cid:20) R − R m (cid:21) + S matter . (3)Here the reduced Planck mass is M P = M Pl / √ π = 2 . × GeV and S matter denotesthe action for all matter fields in the theory. The value of m , which is actually mass of anadditional scalar degree of freedom (scalaron) responsible for inflation, is determined by theamplitude of scalar perturbations as m = 1 . × − M P [23].After the Weyl transformation of the metric to the Einstein frame, g µν → e √ / φ/M P g µν , (4)action (3) takes the form [21] S = Z √− g d x (cid:20) − M R + 12 ∂ µ φ∂ µ φ − V ( φ ) (cid:21) + ˜ S matter , (5) V ( φ ) = 3 m M (cid:16) − e − √ / φ/M P (cid:17) . (6)Here ˜ S matter is the Weyl transformed action of the matter fields. Thus, conformal non-invariance in the matter sector naturally implies the interaction between scalaron φ and all4ther particles. Fermions and vector fields are Weyl invariant at the tree level in the highenergy limit. Therefore, a key role in the process of reheating is played by the light scalars:SM Higgs boson and axions in the discussed framework (see Ref. [26] for dilaton).Axions a i , i = 1 , . . . , N , unlike the Higgs, due to the perturbative shift symmetry [13]can not be non-minimally coupled to the gravity via terms Ra i . Thus, their kinetic termsare canonical in the Jourdan frame (3), yielding the universal couplings to the inflaton. Thedecay rates of scalaron to Higgs pair and to axion pair are [22, 27]Γ SM = m (1 + 6 ξ h ) π M , Γ a = m πM , (7)respectively, with the SM Higgs boson possibly non-minimally coupled to gravity via la-grangian term L = ξ h Rh /
2. One observes that the decay rates to the axions and to the SMparticles (Higgs bosons) are comparable.The ultrarelativistic axions produced at reheating remain in the late Universe contribut-ing to the energy density and pressure at BBN and recombination as additional∆ N eff = 0 . N, (8)neutrino flavors (we put (7) with ξ h = 0 into (1)). Hence, even one additional light scalar (for example, standard QCD axion) is already in tension with the Planck bound (2) in theStarobinsky model.To resolve the situation one can add some new N s scalars to the matter content of the SM.Then the r.h.s. of (8) gets suppressed by a factor 4 / (4 + N s ), which makes the Starobinskymodel in the axiverse populated by N ∼
100 axions consistent with the present cosmologicalbounds (2) if N s ∼
200 scalars are added.
Another way to obtain the favored by the Planck results exponentially flat potentials in anatural way is connected with the modification of the kinetic term of the inflaton. Thisidea is widely discussed in the context of supergravity [8, 9, 24, 32] where non-trivial kineticterms come from the Kahler potential. In particular, it was realised as α -attractors inRefs. [28,29], where the inflationary region corresponds to the pole in the kinetic term of theinflaton. In all these models the inflaton may be canonically normalized upon an appropriatefield transformation. In general, one can expect that the kinetic terms of additional scalars5axions) are also non-minimal. Thus, the action consistent with the shift symmetry of axions a i reads S = Z d x √− g − M R + ( ∂ µ φ ) N X i =1 f i ( φ ) ( ∂ µ a i ) − V ( φ ) ! , (9)where f i ( φ ) are some functions of the inflaton field.Similarly one may expect non-renormalizable couplings to the SM fields yielding the mostrelevant for reheating two-body decays of inflaton, S int = Z d x √− g (cid:18) y ( φ ) | D µ H| − g j ( φ ) F µν,j F µνj + z i ( φ ) ¯ ψ i γ µ D µ ψ i (cid:19) . (10)Terms in the last set in (10) are proportional to the fermion masses through the equation ofmotion and hence their role in the reheating is negligible.Near the minimum of V ( φ ) (we take it to be zero) one may anticipate the expansions f i ( φ ) =1 + β i φ Λ + γ i φ Λ + . . . ,g i ( φ ) =1 + δ i φ Λ + . . . , y ( φ ) = 1 + γ φ Λ + . . . . (11)From the point of view of the effective theory considered after inflation such terms suppressedby cutoff scale Λ < M P are naturally expected with β , γ , δ ∼ f i ( φ ) ≈ γ = 0 the reheating process can go through the decay of the inflaton to the Higgs bosons.If the coupling to gauge bosons is suppressed for some reason we are left with the similarcase as in the Starobinsky model where the dark radiation production is highly efficient (8).This model is cosmologically forbidden.If the inflaton couples to the kinetic terms of all the matter fields in the model with γ, δ i ∼
1, then all the bosons will be produced roughly at equal amounts. In this case onecan get an estimate for the axion contribution to the effective relativistic degrees ∆ N eff byputting Γ a / Γ SM ∼ /
30 in (1). In this way one obtains ∆ N eff ∼ N/
10, which is certainlycosmologically forbidden for N ∼
100 given the constraint (2).The problem of axion overproduction may be avoided if the inflaton couples to the SMparticles stronger than to the axions. The former may be parametrized by means of the timeof reheating (as we discuss at the beginning of Sec. 2) or the reheating temperature T reh , i.e.6he temperature of the SM plasma at the moment when a half of the total energy density isalready in the form of radiation.Then, if the inflaton mass is m , one obtains from Eqs. (9), (11),Γ SM ≃ T √ g reh M P , Γ a = β m π Λ . (12)Substituting (12) into the equation (1) we obtain for the amount of dark radiation:∆ N eff = 0 . N β m M P Λ T . (13)For the reheating temperature high enough one can see that N ∼
100 may still be allowedby the Planck constraints. On the contrary, inefficient reheating with low T reh can easilythrow the model out of the viable range (2).Note in passing that the terms of the first order in the expansion (11) may be forbiddendue to some symmetry (the simplest one is Z , φ → − φ ). In this case, the production ofaxions would be inefficient if the Universe is reheated due to the other inflaton couplings tomatter provided by a lower order operators.We discuss this case in more details in the next Section using the inflaton non-minimallycoupled to gravity as a realistic example. Although inflation models with reasonable (e.g. renormalizable without gravity) power-lawpotentials are disfavoured by the Planck data, switching on the non-minimal coupling of theinflaton to gravity can provide with the flat potential suppressing the tensor modes. Modelsof such type are widely discussed in the literature (see e.g., [28,30,31]) and include the Higgsinflation [7]. The action for the inflaton field φ reads (here we neglect the possible mass termfor the inflaton at large field values), S = Z d x √− g (cid:18) − M + ξφ R + ( ∂ µ φ ) − λφ (cid:19) . (14)One can get rid of the non-minimal coupling by making use of the metric redefinitionˆ g µν = Ω g µν , Ω = 1 + ξφ M . (15)7fter that, the action in the Einstein frame takes the form S E = Z d x p − ˆ g ( − M R + ( ∂ µ χ ) − U ( χ ) ) , (16)where canonically normalized field χ is defined by dχdφ = r Ω + 6 ξ φ /M Ω , and U ( χ ) = 1Ω ( χ ) λ φ ( χ ) . (17)The kinetic term of axion gets coupled to the inflaton in the Einstein frame: L a = 12 Ω ( ∂ µ a ) = 12 (cid:18) ξφ M (cid:19) ( ∂ µ a ) . (18)At first sight, this coupling is quadratic in inflaton and seems to be strongly suppressed bysquared Planck scale. However, at and some time after inflationary epoch the inflaton fieldtakes large values, φ ∼ M P . This makes the second term in parenthesis in (18) important tothe extent which depends on the value of nonminimal coupling ξ . Taking into account thisfact we study two different cases for the value of ξ which finally yield different results. Large non-minimal coupling, ξ ≫ . This case includes the model of Higgs inflation[7]. In this limit for large field values of φ > M P / √ ξ one obtains from (17) φ ≃ M P √ ξ exp (cid:18) χ √ M P (cid:19) , U ( χ ) = λM ξ (cid:18) (cid:18) − χ √ M P (cid:19)(cid:19) − . (19)Inflation in models of this type is followed by harmonic oscillations with frequency ω = p λ/ M P /ξ : the scalar potential is effectively quadratic while the amplitude of χ is largeenough, χ ≫ M P /ξ . (20)The Universe is expanding as at the stage of matter domination: a ∝ t / . In the originalHiggs inflation [7] the reheating happens at this stage due to decays of Higgs to the SMparticles [25]. The interaction lagrangian between the inflaton and any additional scalar a coming from the Weyl transformation (15) takes the same form as in the Starobinsky modelof Sec. 2.1: L int = χ √ M P ∂ µ a ∂ µ a . (21)If the reheating happens at this stage (Eqs. (19), (21), (20)) then the decay rates of theinflaton are actually given by eq. (12). Here T reh is the temperature of the SM plasma at the8oment of equality between energy densities of radiation and inflaton excitations. For theHiggs inflation T reh ≃ × GeV [25]. Let us evaluate the amount of dark radiation giventhe reference values of the Higgs inflation:∆ N eff ≃ . × − N (cid:18) ω . × − M P (cid:19) (cid:18) × T reh (cid:19) . (22)One can observe that the axiverse with N ∼ is safe from the overproduction of darkradiation in models with high enough reheating temperature. Such models require non-gravitational couplings between the inflaton and SM particles, like gauge and Yukawa inter-actions between the inflaton (Higgs) and the SM fields in the example of Higgs inflation.A side remark concerns models with the axions coupled to the SM particles in plasmathat provide with more efficient mechanisms of the dark radiation production. The QCDaxion with the decay constant in the range f a ∼ − GeV is a realistic example. Suchaxion couples to the SM particles via dimension-5 operators suppressed by Λ = f a ratherthan Λ = M Pl . Therefore, it thermalizes in the SM plasma of temperature above 10 GeVwhich is the case for the Higgs inflation. Thus, the amount of dark radiation in this case isdefined by the axion number density in thermal equilibrium. This leads to the value [33]∆ N eff = 0 . , (23)which can be measured in future CMB polarization experiments [19, 20]. At the same time,the thermal production of string axions with f a ∼ GeV is still inefficient.It is worth noting that the reference value of the reheating temperature in (22) corre-sponds to the final amplitude of the inflaton oscillations of order χ ∼ M P /ξ [25] . In otherwords, for a quartic inflation with non-minimal coupling to gravity, the reheating tempera-ture cannot be lower than the reference value in (22), if the system is still at the effectivematter domination stage provided by (20). Hence eq. (22) imposes a kind of upper limit onthe impact of axions in the model with efficient reheating.If the reheating is less efficient than in the Higgs inflation, the evolution comes to thesecond stage. There the inflaton amplitude drops down to the value χ ∼ M P /ξ before thereheating started, and the potential and interactions of the canonically normalized inflatonfield χ change the forms: U ( χ ) = λ χ , L int ∼ χ M ∂ µ a ∂ µ a . (24) We do not consider values of ξ much larger than those of the Higgs inflation ( ξ ∼ ) because it wouldlead to strong coupling for the inflaton self-coupling, λ & /a due to the Universe expansion providing with negligibleoverall axion production.Therefore at this stage, the axion production actually terminates providing the overallimpact of axions to be of order (22) calculated for the reference value of the reheatingtemperature T reh ≃ × GeV. One can see that for all reasonable parameter choices weare left with a negligible amount of axion dark radiation. Note that the real reheating, thatis a production of the SM plasma, may happen much later, than the reference temperatureindicates, but it does not change this estimate in the slightest. Inflaton couplings to the SMparticles must be of another form than the kinetic one of Eq. (24), the latter is not sufficientfor successful reheating.
Small non-minimal coupling ξ ≪
1. Here the change of variables (17) may besimplified: dχdφ = 1 p ξφ /M , φ = M P √ ξ sinh (cid:18) √ ξχM P (cid:19) . (25)The potential (14) transforms to U ( χ ) = λM ξ th (cid:18) √ ξχM P (cid:19) . (26)Near the minimum, conformal factor (15) can be approximated as Ω = 1+ ξχ /M providingthe leading interaction term with scalars to be L int = ξ χ M ∂ µ a ∂ µ a . (27)Potential (26) is symmetric with respect to χ → − χ so no linear terms are expected. Theproduction of axions in that case is inefficient due to the M suppression of interaction (27)in accord with the expectations we discussed right below eq. (18). In this paper, we investigate the validity of inflationary models in the string axiverse. Manylight scalars can be produced at reheating and later contribute to the dark radiation com-ponent of the Universe which is strictly bounded by the recent Planck data. We found thegeneral conditions for the efficient production of the light scalars at the Universe reheating.10amely, if the inflaton decays to two axions via the dimension-5 Planck-scale suppressedoperators then the amount of the dark radiation is controlled by the reheating temperature.For example, inflationary models with reheating via Planck suppressed couplings of the infla-ton to the SM particles (which seems to be common in the supergravity framework) predicttoo much dark radiation making them inconsistent with the cosmological observations. Weshould stress that our results are directly applicable not only in the string framework butfor any light scalars (Nambu-Goldstone bosons, dilaton) which may appear in a concretecosmological model.However, there are two ways how to make the inflation consistent with the presence ofextra light scalars. The first way is to provide the additional couplings between the inflatonand SM fields which are not suppressed by the Planck mass. It would raise the reheatingtemperature leaving no time for the axions production after inflation. Another way aroundis to deal with models possessing a symmetry which either forbids or strongly suppressesthe inflaton decay to axions (the symmetry must not prevent the successful reheating, ofcourse).The authors are indebted to S. Dubovsky, S. Sibiryakov and A. Starobinsky for the valu-able correspondence and discussions.
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