A new take on the inflationary quintessence
aa r X i v : . [ a s t r o - ph . C O ] F e b A new take on the inflationary quintessence
Zurab Kepuladze , and Michael Maziashvili ∗ Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia Institute of Theoretical Physics, Ilia State University, 3/5 Cholokashvili Ave., Tbilisi 0162, Georgia School of Natural Sciences and Medicine, Ilia State University, 3/5 Cholokashvili Ave., Tbilisi 0162, Georgia
The quintessence field coupled to the cosmic neutrino background (CNB) has been widely dis-cussed as an alternative mechanism to address the coincidence problem. As it is well known, it ispossible to extend such models to obtain quintessential inflation, that is, to incorporate inflationarystage as well. Taking an alternative route, one can start from the well established inflationary modelsand obtain successful quintessence models at the expense of coupling with the CNB. To Follow thisroute, we use a slightly reformulated model addressed in PRD , 123521 (2017). This particularmodel assumes Z symmetry for both scalar field potential and coupling term, which then breaksdown in course of the cosmological evolution. For our discussion, however, the Z symmetry of thepotential is not mandatory the model to work. The conventional mechanism of particle productionby the oscillating inflaton field (and their subsequent thermalization) remains operative. It is plainto see that the proposed construction can be easily applied for many successful models of inflation toincorporate dark energy at the expense of coupling with the CNB. We address the issue of neutrinonuggets from the quantum field theory point of view. Namely, these nuggets are considered as boundstates caused basically by the Yukawa force, which arises in the framework of linear perturbationtheory due to exchange of virtual quanta of quintessence field between the neutrinos. I. PREFACE
One of the principle motivations for quintessence mod-els of dark energy, introduced in the late 1980’s [1–4], isto address the cosmic coincidence problem [5, 6]. Thisproblem has two aspects indeed. One of them is to ex-plain the smallness of the present dark energy density andthe other one is to figure out what caused the dark en-ergy to activate in the present epoch. A particular classof quintessence models, referred to as trackers, avoid theproblem of fine tuning the initial conditions of the scalarfield in order to obtain the desired energy density andequation of state at the present time. It is achieved atthe expense of introduction of a small scale in the poten-tial - the origin of which may indeed be explained [7, 8].Another class of models, which we are going to discussthroughout of this paper, explain the coincidence prob-lem by considering a coupling of the quintessence withthe cosmic neutrino background (CNB) [9–13]. Roughlyspeaking, in such models, the idea is to use neutrinomass scale for explaining the smallness of the presentdark energy density and the present time activation ofquintessence is caused by the back reaction of CNB af-ter neutrinos become non-relativistic. The models of thiskind may be used conveniently to unify quintessence withinflation as we have more freedom in choosing the poten-tial. However, in contrast to the quintessential inflation[14–16], which aims at the construction of successful infla-tionary scenario with the ”existing” quintessence model,we favor the idea of inflationary quintessence, that is, toconstruct a successful quintessence model with the use of”good” inflationary models. To be more precise, under ∗ [email protected] the good inflationary models we understand those havinga plateau, which provides a slow-roll regime, and a mini-mum at φ = 0 around of which the field starts to oscillateafter it exits the slow-roll regime [17, 18]. Such modelsmay be considered as main targets for the near-futureobservational missions. The point is that the upcomingCMB experiments may measure the primordial gravita-tional wave power spectrum and its amplitude in terms ofthe tensor to scalar ratio with the precision 5 × − (5 σ )and also aim to improve constraints on the primordialcurvature perturbation power spectrum and its tilt [19].The basic idea for constructing such inflationaryquintessence models is to use the cosmological symmetrybreaking mechanism triggered by the coupling of scalarfield with the CNB [20]. The schematic picture looks asfollows. Because of this coupling, the equations of motioncontain the Spur of CNB stress-energy tensor which kicksup the scalar field trapped in the minimum φ = 0 afterthe end of preheating and enforces it to roll towards oneof the degenerate minima leading thereby to the sponta-neous breakdown of Z symmetry. As it is shown in sec-tion V, it happens shortly after the preheating - aroundthe time of thermalization. After the symmetry breaking,the scalar field evolves adiabatically - tracking roughlythe minimum of the effective potential. As a result, thescalar field acquires a non-zero energy density but it isset by the neutrino mass scale and is therefore too smallto have any effect at earlier times (see Eq.(19)). How-ever, at later times, when CNB gets non-relativistic, thecompound of scalar field and CNB starts to act as a darkenergy because of adiabatic nature of the scalar field evo-lution. It does not last forever, since in a while the CNBdilutes enough and its back-reaction providing the slowroll regime for the scalar field becomes negligible. Af-ter exiting the slow roll regime, the scalar field, whichwas monotonically approaching the value φ = 0, tends”quickly” towards this point leading to the restoration of Z symmetry.One of the most subtle issues from the conceptual pointof view is the preheating. Indeed, it may look less natu-ral in the sense that the preheating is usually describedby using the quantum theory of a free field with a time-dependent effective mass including the coupling of infla-ton with the matter field [21, 22]. This type of preheat-ing can work for fermions as well [23] but if we are ap-plying this formalism immediately to the neutrinos, wehave to distinguish between the background-field timedependence and the real dependence of mass on time.For instance, the coupling of inflaton with the neutrinosresponsible for mass variation will not cause particle pro-duction at all. On the other hand, the very fact that insome cases the coupling contributes to the real masseswhile in other cases it just provides a time-dependentbackground may sound quite unnatural. Emphasizingagain, the problem is of conceptual nature rather thantechnical. It is obvious that the instant preheating mech-anism [24] also suffers from this conceptual problem.In addition, in the framework of the present model,there are scalar field fluctuations that couple to the CNBresulting in the attractive force between the CNB neutri-nos. It can be viewed as a Yukawa force mediated by thescalar quanta as long as we restrict ourselves to the linearperturbations. Under certain circumstances, this approx-imation may work quite well for describing the formationand subsequent growth of the neutrino nuggets.Throughout of this paper we are using natural units: c = ~ = 1, in which G − / N = M P ≈ . × eV, H = 74 km/sec/Mpc ≈ . × − eV. Also note thatall quantities with subscript or superscript zero refer tothe present values. II. DESCRIPTION OF THE MODEL
We assume a spatially flat FLRW universe with metricd s = d t − a ( t )d x , and consider a minimal model of φ - ν coupling given bythe action functional Z d x √− g (cid:18) g αβ ∂ α φ∂ β φ − U ( φ ) − M P R π + i (cid:2) ¯ ψ ν γ α ( x ) D α ψ ν − (cid:0) D α ¯ ψ ν (cid:1) γ α ( x ) ψ ν (cid:3) − m ( φ ) ¯ ψ ν ψ ν (cid:19) . As a next step, for building up the model, the field ψ ν is quantized and taken at a finite temperature. Thatis, ψ ν describes a Fermi gas at a finite temperature andis understood to stand for the CNB [9–12]. Then theequations of motion for the φ - ν model look as follows ˙ ρ ν + 3 H ( ρ ν + p ν ) = d ln m ν d φ ( ρ ν − p ν ) ˙ φ , (1)¨ φ + 3 H ˙ φ + U ′ ( φ ) = − d ln m ν d φ ( ρ ν − p ν ) , (2) H = 8 π M P ( ρ ν + ρ φ + ρ r + ρ m ) . (3)For the sake of generality, in Eq.(3) we have included thematter and radiation components as well. A remarkablecharacteristic feature of this system is that at early times,that is, at high temperatures (when neutrinos are rela-tivistic) p ν ≈ ρ ν / φ and ψ fields are nearlydecoupled and the dynamics of the scalar field is basicallydriven by the potential U ( φ ). However, in the nonrela-tivistic regime p ν ≪ ρ ν and the equations of motion takethe form ˙ ρ ν + 3 Hρ ν ≈ d ln m ν d φ ρ ν ˙ φ , ¨ φ + 3 H ˙ φ + U ′ ( φ ) ≈ − d ln m ν d φ ρ ν , or, in terms of the CNB number density n ν ≈ ρ ν /m ν ( φ ),˙ n ν + 3 Hn ν ≈ , ¨ φ + 3 H ˙ φ + U ′ ( φ ) ≈ − m ′ ν ( φ ) n ν . (4)The Eq.(4) describes the motion of scalar field in themodified potential U eff = U ( φ ) + m ν ( φ ) n ν ( t ) . (5)Assume that the effective potential has a local minimum, φ + , in which the scalar field is trapped. The presence of n ν ( t ) in the effective potential indicates that the mini-mum itself is time-dependent. The model to work, theeffective potential should provide the slow roll, (cid:12)(cid:12)(cid:12) ¨ φ + (cid:12)(cid:12)(cid:12) ≪ H (cid:12)(cid:12)(cid:12) ˙ φ + (cid:12)(cid:12)(cid:12) , ˙ φ ≪ U eff ( φ + ) . The first model we want to consider is obtained by thereformulation of the one addressed in [20]. It consists ofthe Z symmetric potential and φ - ν coupling of the form U ( φ ) = V (cid:16) − e − αφ /M P (cid:17) , m ν ( φ ) = µ ν e − βφ /M P . (6)This model is clearly motivated by the paper [25]. In [20]it is assumed that V is of the order of the present dark For CNB is in the non-relativistic regime, ρ ν ≈ n ν m ν . energy density. We find this assumption undesirable be-cause if we take the existence of the present cosmologicalconstant for granted - then there is nothing to explain asits existence will not spoil anything in the past [26] andtherefore one may not worry about the hiding of thatsmall cosmological constant in the past [20]. As to thescenario, it looks as follows. It is assumed that φ = 0before neutrinos enter a non-relativistic regime and afterthat φ is driven by the effective potential (5). The effec-tive potential turns the point φ = 0 into the local max-imum, as it is depicted in Fig.1, and field starts movingeither left or right providing the present dark energy V . φVU eff ( φ ) U ( φ )Figure 1. The potentials for the model discussed in [20]. After a while, when n ν ( t ) dilutes enough, the potentialapproaches its initial form and the field, which is rollingback, will reach the point φ = 0 restoring thereby theinitial Z symmetry. In itself, the mechanism used inthis scenario for hiding the dark energy both in the pastand in the future looks quite attractive.In order to unify dark energy and inflation, we refor-mulate the above model by assuming large V in Eq.(6).This way one obtains a typical example of the T-model[27]. Thus, we consider the effective potential U eff ( φ ) = V (cid:16) − e − αφ /M P (cid:17) + n ν µ ν e − βφ /M P , where, the parameter V (which is understood to be large)together with the parameter α is ”determined” from therequirements of inflation, while the parameter β is setfrom the requirements of present dark energy. III. α, V
PARAMETERS
The potential U ( φ ) has an infinitely long plateau forlarge values of | φ | starting roughly at φ ≃ M P /α . Thepotentials with plateau provide perfect conditions for theslow-roll inflation as the field rolling down will arrive atthe attractor trajectory from a very wide range of initialconditions and are favorable by the present cosmologicaldata. In this section we estimate the inflationary ob-servables for the model discussed above. The slow-rollparameters are defined as ǫ = M P π (cid:18) U ′ U (cid:19) , η = M P π U ′′ U , ξ = M P (8 π ) U ′ U ′′′ U . The end of inflation occurs for φ f at which ǫ ( φ f ) ≃ α = 1 ⇒ φ f ≃ . × M P , α = 4 ⇒ φ f ≃ . × M P α = 9 ⇒ φ f ≃ . × M P , α = 16 ⇒ φ f ≃ . × M P . The number of e-foldings is N = − πM P φ f Z φ i d φ UU ′ = − π φ f Z φ i d φ " e αφ /M P αφ − αφ =4 πα " Ei (cid:0) αφ i /M P (cid:1) − Ei (cid:0) αφ f /M P (cid:1) (cid:18) φ i φ f (cid:19) , which after demanding N = 60 determines the initialvalues of the field as α = 1 ⇒ φ i ≃ . × M P , α = 4 ⇒ φ i ≃ . × M P α = 9 ⇒ φ i ≃ . × M P , α = 16 ⇒ φ i ≃ . × M P . The slow-roll parameters can be used to express the spec-tral index, its derivative and tensor-to-scalar ratio as n s = 1 − ǫ + 2 η , d n s d ln k = 16 ǫη − ǫ − ξ , r = 16 ǫ . To fit the present observational data [28] n s = 0 . ± . , r < . , d n s d ln k = − . ± . , the parameter α should satisfy α & . V , it is com-monly expected to lie approximately between the TeVand Planck scales. It is related to the amplitude of ten-sor modes V / ≃ . × r / GeV , indicating that the detectable gravitational waves require V / ≃ GeV. That is the energy scale considered in[27] but, in general, such a big value is not typical for theexisting models of inflation. In what follows we admitthe whole ”possible” range of parameter V but for thediscussion of nuggets it is favorable to take this parameternear the lower bound (see section VI). IV. ONSET OF DARK ENERGY
In order to obtain dark energy, the effective potential(5) should provide the slow roll regime. When α > β ,the extremum φ = 0 gives the only minimum. Putting α < β and at the same time demanding αVβn ν µ ν < , we will have two minimum points φ ± M P = ± s β − α ) ln βn ν µ ν αV , (7)and one maximum at φ = 0 as it is shown in Fig.2. In thiscase the symmetry breaking takes place in accordancewith the scenarios described in [20, 25], however, as it isdiscussed in the following section, it occurs shortly afterthe preheating. Around this time, the effective potentialdevelops the minimums for which U ( φ ± ) . U (0) and fieldmoves to one of them and then follows the dynamics ofthis minimum. For definiteness let us take this minimumto be φ + . Returning to the non-relativistic regime ofCNB, it is plain to see that the neutrino masses increasein such a way m ν ( φ + ) = µ ν e − βφ /M P ≈ αVβn ν , that the neutrino energy density ρ ν = n ν m ν ≈ αVβ = const . . (8)This kind of behavior of CNB in the non-relativisticregime lasts until the symmetry restoration takes place,which in view of Eq.(7) occurs when n ν drops down to n ν = αVβµ ν . (9)After the symmetry restoration, the mass of neutrino µ ν becomes time independent. As it is discussed below, µ ν ≃ × m ν , where m ν stands for the present valueof the mass.Now let us see if the model provides a slow roll regimeat present. For this purpose, we shall verify the condition˙ φ ≪ U ( φ + ) , which is tantamount to φU ( φ ) U eff ( φ ) φ + φ − n ν µ ν Figure 2. The symmetry breaking effective potential. M P H ( β − α )8 ln µ ν m ν ( φ + ) ≪ V (cid:18) − exp (cid:18) − αβ − α ln µ ν m ν ( φ + ) (cid:19)(cid:19) . (10)In view of Eq.(8), we need to demand β ≫ α in orderto ensure ρ ν < ρ c , where ρ c ≡ H M P / π stands for thecritical energy density. Recalling that the parameters V and α are set from having a successful inflationary model, α ∼ , TeV . V / . TeV , one can make the following order of magnitude estimate β ∼ α VH M P ∼ VM P ⇒ β & . In view of this, the Eq.(10) simplifies to M P H V ≪ α ln (cid:0) µ ν /m ν ( φ + ) (cid:1) , (11)and is satisfied with an extremely high accuracy atpresent if we take ln (cid:0) µ ν /m ν ( φ + ) (cid:1) = 9, which fol-lows from the requirement that the equation-of-state-parameter = − .
9. Namely, under assumption of slowroll, the dark energy density is given by [10, 29] ρ dark = U ( φ + ) + m ν ( φ + ) n ν ≈ αVβ (cid:18) ln µ ν m ν + 1 (cid:19) , and correspondingly ω = p dark ρ dark ≈ − U ( φ + ) U ( φ + ) + ρ ν = − ln µ ν /m ν ln µ ν /m ν + 1 = − . , results in: ln (cid:0) µ ν /m ν ( φ + ) (cid:1) = 9.Next, we have to tune the parameters α, β, V in orderto obtain: ρ dark = 0 . ρ c . That is, we must demand10 αVβ = 0 . × M P H π . A crude estimate of the duration of present acceleratedexpansion maybe made by assuming that the expansionhas an exponential character. Then, from Eqs.(8, 9) oneobtains e − H ( t − t ) ≃ e − , ⇒ t − t ∼ H − . It is also important to clarify the question - when doesthe accelerated expansion start? For this purpose, wehave to verify the condition3 p φ < − ρ φ − ρ r − ρ m − ρ ν , which follows from¨ aa = ˙ H + H = − π M P ( ρ φ + 3 p φ + 2 ρ r + ρ m + ρ ν ) . That is, we have to check αVβ (cid:18) − µ ν (1 + z ) m ν (cid:19) < − ρ r (1 + z ) − ρ m (1 + z ) . Since we know that αV /β = 0 . ρ c , ln (cid:0) µ ν /m ν (cid:1) =9 , Ω r = 5 . × − , Ω m = 0 .
31, this relation can beput in the form − .
07 [17 + 6 ln(1 + z )] < − × . × − (1 + z ) − . z ) ⇒ z . . . In the next section we go back to the early universe todescribe the evolution of φ - ν mixture at early times. V. EARLY TIMES
To elucidate the model further, it is expedient to pro-ceed the discussion in terms of the CNB temperature byusing the phase space distribution function for the free-streaming neutrinos [10] ρ ν = g a Z d k (2 π ) ε ν ( k )e k/aT ν + 1 ,p ν = g a Z d k (2 π ) k ε ν ( k ) (cid:16) e k/aT ν + 1 (cid:17) ,ε ν ( k ) = r k a + m ν , where g counts all effectively contributing neutrino de-grees of freedom. If we are restricting ourselves to theone species of neutrino, then g = 4. Evaluating the time-derivative ˙ ρ ν , one obtains ˙ ρ ν = − g ˙ aa Z d k (2 π ) ε ν ( k )e k/aT ν + 1 + g a Z d k (2 π ) ˙ ε ν ( k )e k/aT ν + 1 − g a Z d k (2 π ) ε ν ( k ) e k/aT ν (cid:16) e k/aT ν + 1 (cid:17) dd t kaT ν , where the equationdd t aT ν = 0 ⇒ aT ν = const. , determines the temperature as a function of time and theremaining terms, after substituting˙ ε ν ( k ) = − a − ˙ ak + m ν m ′ ν ˙ φε ν ( k ) , result in Eq.(1)˙ ρ ν + 3 H ( ρ ν + p ν ) = d ln m ν d φ ˙ φ ( ρ ν − p ν ) = g m ν m ′ ν ˙ φa Z d k (2 π ) ε ν ( k ) (cid:16) e k/aT ν + 1 (cid:17) . Thus, one finds that ρ ν − p ν = g T ν π m ν T ν ∞ Z d ξ ξ p ξ + m ν /T ν (e ξ + 1) . (12)In the limit m ν /T ν ≫
1, the expression (12) simplifies to ρ ν − p ν ≃ m ν g T ν π ∞ Z d ξ ξ e ξ + 1 = 3 ζ (3) g m ν T ν π , while in the case m ν /T ν ≪ ρ ν − p ν ≃ g T ν π m ν T ν ∞ Z d ξ ξ e ξ + 1 = g m ν T ν , (13)It is worth paying attention that the phase-space distri-bution function for neutrinos what we have used aboveis valid after the neutrinos are decoupled from the restof the universe. Roughly, the decoupling temperature is1 MeV - much bigger than m ν ≤ µ ν ≃ ρ ν = g a Z d k (2 π ) ε ν ( k )e ε ν ( k ) /T ν + 1 ,p ν = g a Z d k (2 π ) k ε ν ( k ) (cid:16) e ε ν ( k ) /T ν + 1 (cid:17) ,ε ν ( k ) = r k a + m ν , which results in ρ ν − p ν = g T ν π m ν T ν ∞ Z m ν /T ν d ξ p ξ − m ν /T ν e ξ + 1 . (14)One sees from Eq.(14) that in the ultra-relativistic limitwe arrive again at the Eq.(13). We shall first con-sider CNB below the decoupling temperature - that isthe free streaming regime. In this case the temper-ature dependent effective potential can be written as U ( φ ) + ρ ν ( φ, T ν ), U eff ( φ, T ν ) = U ( φ ) + g T ν π ∞ Z d ξ ξ p ξ + m ν /T ν e ξ + 1 = V (cid:16) − e − αφ /M P (cid:17) + g T ν π ∞ Z d ξ ξ e ξ + 1 s ξ + e − βφ /M P ( T ν /µ ν ) . (15)For our discussion we need the point φ = 0 to be a max-imum for the present value of temperature T ν ≪ µ ν . Forthis reason, let us evaluate the second derivative of (15)at φ = 0 U ′′ eff ( φ = 0 , T ν ) = 2 αVM P − β g (cid:0) µ ν T ν (cid:1) π M P ∞ Z d ξ ξ (e ξ + 1) r ξ + (cid:16) µ ν T ν (cid:17) , one finds that φ = 0 corresponds to maximum if2 απ Vβ g (cid:0) µ ν T ν (cid:1) < ∞ Z d ξ ξ (e ξ + 1) r ξ + (cid:16) µ ν T ν (cid:17) . (16)Let us note that if φ = 0 represents a maximum for thepresent value of temperature, then it automatically im-plies that this point is maximum for higher temperaturesas well. Namely, the integral in Eq.(16) increases mono-tonically to the value π / ≈ .
82) as the temperaturegoes to infinity while the left-hand side of this inequalitybecomes decreasing as the temperature increases.Let us look at the ultra-relativistic regime, which, inview of Eqs.(13, 15) enables one to put the effective po-tential in a simple form U eff ≈ V (cid:16) − e − αφ /M P (cid:17) + 7 π g T ν
240 + g ( µ ν T ν ) e − βφ /M P . (17)The minimum points in the ultra-relativistic regime aredefined by φ ± M P = ± s β − α ln β g ( µ ν T ν ) αV , (18)while they look as (see Eq.(7)) φ ± M P = ± s β − α ln 3 ζ (3) β g T ν π αV , in the non-relativistic regime. From Eq.(18) one obtainsthat the neutrino mass in the early universe is muchsmaller than the present value m ν ( φ + ) = s ρ ν g T .
Apart from this, the energy density of the scalar field˙ φ U ( φ + ) ≈ M P H β − α ) ln g ( µ ν T ν ) ρ ν + ρ ν g ( µ ν T ν ) ρ ν , (19)is now negligibly small as compared to the neutrino en-ergy density ρ ν ≈ π g T ν . This conclusion is almost obvious by noting that M P H ≃ g ∗ ( T ) T , where g ∗ ( T ) counts relativistic de-grees of freedom at a given temperature and is slightlybigger than 100 above the temperature 300 GeV in theframework of standard model of particle physics. Onemore point worth paying attention is that the energy den-sity (19) is close to ρ ν even if the temperature is taken ashigh as 1TeV. Recall that the kinetic term is suppressedby the huge parameter β , which is at least of the order of10 . One will easily find that the kinetic term at earliertimes is of the order of the potential one. What hap-pens at later times is that the kinetic term decreases as H gets smaller and the compound of scalar field and theCNB start to act as a dark energy after CNB becomesnon-relativistic.Above the decoupling temperature one has to use theequilibrium distribution, which for the effective potentialgives U ( φ ) − p ν ( φ, T ν ). One can easily derive it by notingthat in this case (see Eq.(14)) m ′ ν m ν ( ρ ν − p ν ) = g a dd φ Z d k (2 π ) (cid:16) ε ν ( k ) − T ν ln h ε ν ( k ) /T ν i(cid:17) , which after using an integration by parts gives4 π ∞ Z d kk (2 π ) (cid:16) ε ν ( k ) − T ν ln h ε ν ( k ) /T ν i(cid:17) = − π ∞ Z d kk (2 π ) dd k (cid:16) ε ν ( k ) − T ν ln h ε ν ( k ) /T ν i(cid:17) = − Z d k (2 π ) k ε ν ( k ) (cid:0) ε ν ( k ) /T ν (cid:1) . Thus, one finds m ′ ν m ν ( ρ ν − p ν ) = − d p ν ( φ, T ν )d φ . Expanding p ν ( φ, T ν ) in a power series in m ν /T ν , p ν ( φ, T ν ) = g T ν (cid:20) π − m ν T ν + O (cid:18) m ν T ν (cid:19)(cid:21) , and comparing it with Eq.(17), one infers that the min-ima of U eff are again given by the Eq.(18). Therefore,the consequent conclusions hold above the decouplingtemperature as well.The thermal equilibrium stage is preceded by the parti-cle production in the post-inflation epoch. In the presentmodel the conventional preheating mechanism [21, 22]can operate successfully for creating the cosmic fluid outof thermal equilibrium which then undergoes the ther-malization. Instant preheating, which is inevitable forthe runaway type potentials of quintessential inflationhaving no oscillation regime [14–16], is not required in the present case. There is, however, a subtle point concern-ing the naturalness. The preheating is usually achievedby introducing a coupling of matter field with the infla-ton that results in a time-dependent mass term. Looselyspeaking, in certain regions of the parameter space, thesolution of the matter field in this time-dependent back-ground grows rapidly corresponding to what is calledparametric resonance. One will find that the particle pro-duction within a broad resonance regime is big enoughdraining rapidly the energy from oscillating inflaton field.However, the coupling of φ with ψ ν in the present modeldoes not result in the neutrino production as it providesa real mass variation of neutrinos. It maybe somewhatunnatural that neutrinos represent exception to the gen-eral rule. This problem of naturalness persists for theinstant preheating as well. VI. NEUTRINO LUMPS
Concerning the perturbations, most subtle and inter-esting issue is the possibility of formation of the neutrinoclumps [30–33]. Instead of deriving instabilities via theeffective sound speed of the compound of scalar field andneutrino fluid, we shall approach this problem from asomewhat different point of view. As the perturbationsare of quantum origin, let us consider quantum fluctu-ations both for scalar and fermion fields. For this pur-pose, the scalar field is split as φ + χ and the fermionnumber operator is shifted as ¯ ψ ν ψ ν → n ν + ¯ ψ ν ψ ν . For-getting about the gravitation, the Lagrangian density forthe perturbations takes the form ∂ α χ∂ α χ − (cid:16) U ′′ ( φ + ) + m ′′ ν ( φ + ) n ν (cid:17) χ i ¯ ψγ α ∂ α ψ − m ′ ν ( φ + ) χ ¯ ψψ + C.C. + H.T. , (20)where C.C. denotes complex conjugate and H.T. standsfor the higher order terms. From Eq.(20) one sees thatthere is an attractive force between the neutrinos me-diated by the exchange of χ quanta. It results in theYukawa potential − (cid:0) m ′ ν ( φ + ) (cid:1) exp (cid:16) − p U ′′ ( φ + ) + m ′′ ν ( φ + ) n ν r (cid:17) πr , (21)where r stands for the physical distance, implying thatthe corresponding attractive force is characterized withthe screening length1 p U ′′ ( φ + ) + m ′′ ν ( φ + ) n ν = 1 q U ′′ eff ( φ + ) ≡ m eff ( φ + ) . It is instructive to compare this force, within its screeningradius, with the gravitational one. The ratio is (cid:0) m ′ ν ( φ + ) (cid:1) M P m ν ( φ + ) = 36 β . One sees that the fifth force exceeds the Newtonian oneby many orders of magnitude. As to the effective mass,it reads m eff ≃ βρ ν M P ≃ βH . (22)Proceeding in the spirit of the above discussion, wetreat the neutrino gas as an ideal pressureless fluid sub-ject to the Newtonian self-gravity and also to the Yukawaforce. In addition, we assume an expanding background, r = a ( t ) x , to avoid the ”Jeans swindle” [34]. That is thewell known formalism one can find in many textbookson cosmology, see for instance [35]. The linear pertur-bations of neutrino velocity field and density contrast, δ ≡ δρ ν /ρ ν , satisfy the equations ∂δ∂t + ∇ x · δ v a = 0 , (23) ∂δ v ∂t + Hδ v + c s ∇ x δa + ∇ x Φ N a + m ′ ν ( φ + ) m ν ( φ + ) ∇ x Φ Y a = 0 , (24)where Φ N,Y denote the Newton and Yukawa potentials,respectively:∆ x Φ N a = 4 πM P ρ ν δ , (25) (cid:18) ∆ x a − m eff ( φ + ) (cid:19) Φ Y = m ′ ν ( φ + ) m ν ( φ + ) ρ ν δ . (26)First we take the divergence of Eq.(24) and substitute init ∇ x · δ v from Eq.(23) ∂ δ∂t + 2 H ∂δ∂t − c s ∆ x δa − ∆ x (cid:0) Φ N + Φ Y m ′ ν ( φ + ) /m ν ( φ + ) (cid:1) a = 0 . (27)Applying now Fourier decomposition for the density con-trast and using Fourier transform of Φ N,Y from Eqs.(25,26), the Eq.(27) takes the form ¨ δ ( k ) + 2 H ˙ δ ( k ) + (cid:18) c s k a − πρ ν M P − (cid:18) m ′ ν ( φ + ) m ν ( φ + ) (cid:19) ρ ν k /a k /a + m eff ( φ + ) ! δ ( k ) = 0 . (28)Much of the essential physics concerning the instabili-ties of neutrino perturbations can be extracted from thisequation. One immediately sees that the Yukawa forcehelps the amplification of density perturbations and, ex-ploiting the idea similar to what was suggested in [31],one can incorporate the scalar-mediated force with thegravitational one by introducing an effective Planck mass1 M eff ≡ πM P + (cid:18) m ′ ν ( φ + ) m ν ( φ + ) (cid:19) k /a k /a + m eff ( φ + ) ≃ M P π + βk /a k /a + m eff ( φ + ) ! . From this expression one sees that for k ≫ m eff theNewton constant is amplified at least by the factor ofthe order of 10 leading to the growth of small neu-trino perturbations. Without the Yukawa amplification,there would be no growing perturbations of reasonablesize (see below). Assuming for simplicity that ρ ν , a, c s are constants, one finds the solution of Eq.(28) δ ( t, k ) = C ( k ) exp t s ρ ν M eff − c s k a ! + C ( k ) exp − t s ρ ν M eff − c s k a ! , manifesting the possibility that even for vanishing grav-ity one may have the sub-horizon modes growing at theexpense of Yukawa force (here we use Eq.(22)) H < k a < ρ ν c s (cid:18) m ′ ν ( φ + ) m ν ( φ + ) (cid:19) − m eff ≃ (cid:18) c s − (cid:19) m eff = (cid:18) c s − (cid:19) βH . Behind this expression, one easily recognizes similar es-timates from the previous works [32, 33]. The only dif-ference is that now it is augmented by the factor c − s − c s ∼ p T ν /m ν makes it easy to see whythe formation of nuggets becomes favorable in the non-relativistic regime. In general, the growing modes, withrespect to Eq.(28), satisfy the condition k a < ( πρ ν M P + (cid:18) m ′ ν ( φ + ) m ν ( φ + ) (cid:19) ρ ν − c s m eff ( φ + ) a + √ D ) c s , where D = c s m eff ( φ + ) a − πρ ν M P − (cid:18) m ′ ν ( φ + ) m ν ( φ + ) (cid:19) ρ ν ! + 16 πρ ν m eff c s a M P . In absence of the fifth force, the Jeans length would be ak & c s H , manifesting the need of the fifth force for creating neu-trino nuggets.The above discussion is intended, on the one hand, toelucidate the qualitative features in a simple and trans-parent way and, on the other hand, our results forma new foundation for future investigations of formationand subsequent implications of nuggets. Let us note thatthe lumpy CNB is one of the direct observational conse-quences of the model and there are many papers devotedto the study of nugget formation process involving thenon-linear dynamics [36–43]. The new feature empha-sized in the above discussion is the appearance of effec-tive mass in Eq.(26) indicating the screened nature ofthe fifth-force. For the sake of comparison, see Eqs.(13-16) in [38] and Eqs.(21-25) in [39]. Further research canbe conducted in the light of recent investigations of theYukawa nuggets [44–47]. VII. DISCUSSION AND CONCLUSIONS
The idea behind the introduction of a non-standardcoupling between the quintessence field and the CNB isto tie the dark energy density to the neutrino mass scale[9], see for instance Eq.(19). There are, however, twocategories of such models. The first category of modelsknown as ”growing neutrino quintessence” [11, 12] as-sume that the scalar field is steadily rolling down the po-tential U ( φ ) before the neutrinos become non-relativisticand stop its motion resulting in the potential-dominateddynamics for the quintessence field. We are interestedin other category of models [20, 25] assuming that thescalar field is initially trapped into the vacuum and ac-quires a non-zero vacuum expectation value, which thenvaries in time adiabatically, as a result of back-reactiondue to neutrinos. The latter scenario allows one to natu-rally incorporate well established inflationary models intoit. Most successful inflationary models are believed to bethose with a plateau [18, 27]. Such models are charac-terized by two independent parameters - the width andthe height of the potential. The example considered byus is a typical representative of such a model. It is char-acterized by the parameters α and V , see section III,where the dimensionless parameter α is of the order of unity. Besides, we have one more dimensionless param-eter, β & , which comes from the coupling term.We see a large discrepancy between these dimensionlessparameters that should be ”explained” somehow. Inter-estingly enough, the broad class of inflationary potentialsderived in [48] as a result of spontaneously broken con-formal symmetry, can be straightforwardly used in theabove discussion with the same φ - ν coupling term (whichis certainly taken by hand). Namely, the T-model poten-tials [27] U tanh φ p αM P ! , are closely analogous to what we have considered and,therefore, it is straightforward to generalize our discus-sion to this kind of models. Also, it is almost obvious thatthe construction similar to what we have discussed maywork without demanding Z symmetry for the inflatonpotential. For instance, one may consider a Starobinsky-like model [49, 50], which would give U eff = V (cid:16) − e − αφ/M P (cid:17) + n ν µ ν e − βφ /M P . From the very outset one can make a simplifying approx-imation αφ + /M P ≪
1, which is well justified as long as β ≫ α . So that, for finding the minimums one can usean approximate expression U eff ≈ V α φ M P + n ν µ ν e − βφ /M P ⇒ φ ± M P ≈ s β ln βn ν µ ν α V .
Hence, one finds m ν = α Vβn ν ⇒ ρ ν = α Vβ . (29)The slow roll condition now takes the form9 M P H ≪ α V ln µ ν m ν ( φ + ) , and is satisfied with a high accuracy at present since inthe Starobinsky model: V ≃ . × − M P , α = p / ≈ . µ ν /m ν ( φ + ) =9. The value of β can be estimated crudely from the0Eq.(29) by noting that ρ ν . ρ c ≃ M P H and corre-spondingly β & VM P H . We shall not continue this discussion as it precisely par-allels what is already done in the text. Along the samelines of reasoning, one can construct E-type inflationaryquintessence models given by the potential [27] U ( φ ) = V (cid:18) − exp (cid:18) − αφM P (cid:19)(cid:19) n . It is worth noting that the Starobinsky model is one ofthe most naturally motivated inflationary model, provid-ing a good fit to the current observational data [28, 51],that maybe considered as one of the target models forfuture observations [18].In the framework of the present approach, we havederived the fifth-force as an Yukawa one arising due toexchange of a single χ boson between the neutrinos. Thisforce gets corrected by the exchange of a pair of χ bosons χχ as long as quadratic perturbations are taken into account.It corresponds to the coupling − m ′′ ν ( φ + ) χ ¯ ψψ , and results in the potential [52] − (cid:0) m ′′ ν ( φ + ) (cid:1) m eff ( φ + ) K (cid:16) m eff ( φ + ) r (cid:17) π r , where K is the modified Bessel function. We have againa potential providing an attractive screened force, whichfor r . m − eff behaves roughly as − (cid:0) m ′′ ν ( φ + ) (cid:1) π r . This force starts to dominate over the Yukawa one (21)at relatively short distances r . √ βM P ≃ − cm . Let us note that the length scale √ β/M P is smaller by afactor of 10 − than the range of Yukawa force: 1 /m eff .Namely, for β ≃ − , one finds that (see Eq.(22))1 m eff ≃ . This is a clear indication that for describing of nuggetsone can safely use the Yukawa force without correctionsunless the inter-neutrino distance is smaller than 10 − cm.However, by increasing the inflation energy scale, the β parameter grows and this distance scale gets larger.One more aspect worth paying attention is that theforce between the neutrinos mediated by the scalar fieldgets corrections if one assumes a finite temperature the-ory for the scalar field fluctuations [52]. That is to assumea thermal equilibrium between the χ field and neutrinosmaintained by the χ - ν coupling. So far there are veryfew papers addressing this issue. At least what we arefamiliar with are the papers [53, 54] where the approachto the dark energy is somewhat different from the con-ceptual point of view but the techniques developed therecan be readily used for the problem posed above. ACKNOWLEDGMENTS
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