Quantum Generation of Dark Energy
aa r X i v : . [ a s t r o - ph . C O ] O c t Quantum Generation of Dark Energy
A. de la Macorra and F. Briscese
Instituto de F´ısica, Universidad Nacional Autonoma de Mexico,Apdo. Postal 20-364, 01000 M´exico D.F., M´exicoPart of the Collaboration Instituto Avanzado de Cosmologia
We present a type of dark energy models where the particles of dark energy φ are dynamicallyproduced via a quantum transition at very low energies. The scale where the transition takes placesdepends on the strength g of the interaction between φ and a relativistic field ϕ . We show that a g ≃ − gives a generation scale E gen ≃ eV with a cross section σ ≃ pb close to the WIMPscross section σ w ≃ pb at decoupling. The number density n φ of the φ particles is a source term inthe equation of motion of φ that generates the scalar potential v ( φ ) responsible for the late timeacceleration of our universe. Since the appearance of φ may be at very low scales, close to presenttime, the cosmological coincidence problem can be explained simply due to the size of the couplingconstant. In this context it is natural to unify dark energy with inflation in terms of a single scalarfield φ . We use the same potential v ( φ ) for inflation and dark energy. However, after inflation φ decays completely and reheats the universe at a scale E RH ∝ h m Pl , where h is the couplingbetween the SM particles and ϕ . The field φ disappears from the spectrum during most of thetime, from reheating until its re-generation at late times, and therefore it does not interfere withthe standard decelerating radiation/matter cosmological model allowing for a successful unificationscheme. We show that the same interaction term that gives rise to the inflaton decay accounts forthe late time re-generation of the φ field giving rise to dark energy. We present a simple modelwhere the strength of the g and h couplings are set by the inflation scale E I with g = h ∝ E I /m Pl giving a reheating scale E RH ∝ E I and φ -generation scale E gen ∝ E I /m pl ≪ E RH . With thisidentification we reduce the number of parameters and the appearance of dark energy is then givenin terms of the inflation scale E I . I. INTRODUCTION.
The nature and dynamics of Dark Energy ”DE”, whichgives the accelerating expansion of the universe at presenttime, is now days one of the most interesting and stimu-lating fields of physics. It was discovered more than tenyears ago [1] and it has been confirmed by further cos-mological observations, being now one of the most robustconjecture in modern physics. In fact the acceleration ofthe present universe has many experimental proofs suchas the CMB temperature and fluctuations [4], in the mat-ter power spectrum measured by galaxy surveys [5, 6] andin type Ia supoernovae [7, 8, 9].The most popular model is the so called ΛCDM model,in which a cosmological constant and some amount ofcold dark matter are included ”by hand”. Despite itsextraordinary consistence with observations, ΛCDM isan effective model that leaves many unsolved theoreti-cal question. In fact the existence of the cosmologicalconstant and its order of magnitude have no theoreti-cal justification in ΛCDM. The cosmological coincidenceproblem, that is why the universe is starting to accelerateright now, is also unsolved. Introducing a cosmologicalconstant at the initial stages of the standard cosmolog-ical model is specially troublesome since one has to finetune its value to one parte in 10 . This problem can beameliorated if we understand when and how dark energyappears in the universe and this is the main motivationof our present work [17]. However, our approach will alsohelp us to unify dark energy with inflation. An attractive dark energy alternative to the ΛCDMmodel consists in introducing a ”quintessence” scalarfield φ that generates the accelerating expansion [13][14]of the universe due to is dynamics. The dynamics is fixedby its potential v ( φ ) and it is possible to choose poten-tials that lead to a late time acceleration of the universe[14]. This scalar field can be a fundamental or compositeparticle as for example bound states [27]. In the secondcase, the bound quintessence fields are scalar fields com-posed of fundamental fermions, such as meson fields, andcan be generated at low energies as a consequence of alow phase transition scale due to a strong gauge couplingconstant [27]. This allows to understand why DE appearsat such late times. On the other hand, in the former casethe appearance of fundamental scalar field is right at thebeginning of the reheated universe and the accelerationof the universe takes place at a much later time due tothe classical evolution of the quintessence field φ . Thehuge difference in scales between the reheating and darkenergy scales requires a fine tuning in the choice of thepotential.Here, we present an interesting alternative, namelythat the emergence of the fundamental quintessence par-ticles φ is originated from a quantum transition tak-ing place at low energies, e.g. as low as eV [17]. Thescale where this transition takes places depends on thestrength of the interaction between φ and a relativisticfield ϕ and it is dynamically determined by the ratioΓ /H where Γ is the transition rate and H the Hub-ble constant. A value of the coupling g ≃ − givesa generation scale E gen ≃ eV with a cross section σ = g / πE gen ≃ pb close to the WIMPs cross section σ w ≃ O ( pb ) at decoupling[16]. The subsequent acceler-ation of the universe is due to the classical evolution of φ due to the scalar potential v ( φ ). Our quantum gener-ation scheme does not aim to derive the potential v ( φ )but to understand why dark energy dominates at such alate time. Clearly, by closing the gap between the energytoday E o , where the subscript o always refers to presenttime quantities, and that of φ production E gen , we do notrequire a fine tuning of the parameters in v ( φ ). Since theappearance of φ may be at such low scales this offers anew interpretation and solution to the cosmological co-incidence problem in terms of the size of the couplingconstant g .Furthermore, this late time production of the φ parti-cles allows to implement in a natural way a dark energy-inflaton unification scheme. In this scenario, after infla-tion the field φ decays completely and reheats the uni-verse with standard model particles. The universe ex-pands then in a decelerating way dominated first by ra-diation and later by matter. At low energies the sameinteraction term that gives rise to the inflaton decay ac-counts for the quantum re-generation of the φ field givingrise to dark energy. In general, it is not complicated tochoose a scalar potential such that the universe acceler-ates in two different regions, at early inflation and darkenergy epochs, as in quintessential models [15]. How-ever, the universe requires to be most of the time dom-inated first by standard model relativistic particles andlater by matter. The reheating of the universe and thelong period of decelerating phase are usually not takeninto account in inflation-dark energy unification modelsand these features are essential in the standard cosmo-logical Big Bang model. In our case, the inflaton-darkenergy field is completely absent during most of the time(from reheating until re-generation) and therefore it doesnot interfere with the standard cosmological model. Wewill exemplify our inflation-dark energy unified schemewith a simple model. The scalar potential v ( φ ) willhave only two parameters fixed by the conditions to givethe correct density perturbations δρ/ρ and the presenttime dark energy scale. The two couplings h, g , whichgive the strength of reheating process with SM parti-cles and the φ re-generation process at low energies, re-spectively, are free parameters but we may take them as g = h ∝ E I /m pl , where E I is the scale of inflation andit is one of the parameters of v ( φ ). Therefore, startingwith four free parameters we can reduce the number toonly two and these two are fixed by observations. Thisgives a reheating scale E RH ∝ E I and φ -generation scale E gen ∝ E I /m pl ≪ E RH .The paper is organized as follows: in section II we givean overview of the late time quantum generation of thequintessence field φ and its possible unification with theinflaton field. In section III we present the dark energyquantum generation in detail. In section III we showhow to unify inflation and dark energy in terms of a sin- gle scalar field in the context of our dark energy quantumgeneration process and we present a simple model. Fi-nally, in section V we present the main phenomenologicalconsequences of our model while in section VI we resumeand conclude. II. OVERVIEW
To avoid any future confusion we state the terminologyused in this work. We define as usual the energy of theuniverse at any given time as E ≡ ρ / or for a i -speciesas E i ≡ ρ / i . For particles we take their energy E a as E a = ( p a + m a ) / , where p a , m a are the momentum andmass of the a -particle. If the particles are relativistic andin thermal equilibrium then one can define a temperature T with an energy density ρ a = π g a T and number den-sity n a = ζ (3) /π g a T , with g a the relativistic degreesof freedom and ζ (3) ≃ . E = ¯ r T with ¯ r ≡ ( ρ/nT ) = π / ζ (3) ≃ . E a with the average energy E a = ¯ E a = ¯ rT . We will loosely refer then to eitherthe temperature or the energy of the (thermalized) rela-tivistic particles and we will work in natural units with m pl ≡ / πG = 1 but we will reintroduce the correctenergy dimensions when convenient.We will first present the quantum generation processof dark energy field φ and we will later discuss the pos-sibility to unify it with the inflaton field. As mentionedin the introduction, the main goal of this paper is to de-scribe how the dark energy field φ may be generated at avery late time via a quantum transition process and howthe scalar potential v ( φ ), responsible for the late timeacceleration, is generated. Once v ( φ ) is produced, theclassical equation of motion gives the dynamics of φ , andchoosing an appropriated flat v ( φ ) at low energies en-sures that the universe enters an accelerating epoch closeto present time. This work does not aim to derive thepotential v ( φ ) but to understand why dark energy dom-inates at such a late time. Of course, by closing the gapbetween the energy today E o and that of φ production E gen there is a no longer a fine tuning of the parametersin v ( φ ).We describe now the generation process. We take auniverse filled with standard model SM particles and anextra relativistic field ϕ and no φ particles, i.e. Ω φ = 0.The ϕ particle is not necessarily contained in the SM(however, an interesting possibility is to associate ϕ withneutrinos) but we require that at time of φ -generation ρ ϕ ( E gen ) > ρ DE ( E o ). In the context of inflation-darkenergy unification the ϕ must have been in thermalequilibrium with the SM and therefore T ϕ ≃ T γ withΩ ϕ ≃ ( g ϕ /g SMrel )Ω SMrel , where g ϕ and g SMrel are the rela-tivistic degrees of freedom of ϕ and the SM respectivelyand Ω SMrel is the density of SM relativistic particles. Sincewe want to produce the φ particles via a quantum tran-sition process we couple it to ϕ via an interaction term L int , e.g. L int = g φ ϕ or L int = g φ ϕ . In order toproduce the φ particles at low energies we require thatthe interaction rate Γ of the quantum process must beinitially smaller than the Hubble parameter H and theratio Γ /H should increases with the expansion of the uni-verse. This will happen if Γ decreases less rapidly than H . For example if we have a 2 ↔ ∝ g T ϕ and H ∝ T ϕ giving Γ /H ≡ T gen /T ϕ ∝ g /T ϕ , where we havetaken T ϕ ≃ T γ . The φ particles production starts thenfor temperatures T below T gen ∝ g , where Γ /H > φ is generated is fixed bythe coupling constant g , so the coincidence problem isexplained in terms of the strength of the interactions ofthe φ field. In particular one can generate the φ fieldat low energies, e.g. E gen ≃ eV with g ≃ − giv-ing a fine structure constant α = g / π ≃ − anda cross section σ ≃ g / πE gen ≃ pb . The value of σ is quite close to cross section of WIMP dark mat-ter with nucleons σ w ≃ pb [16]. Since we can choosethe coupling g in such a way that the φ is generatedat low energies close to present time, this offers a newinterpretation of the cosmological coincidence problem:dark energy domination starts at such small energies be-cause of the size of the coupling constant g . For a φ -generation energy of E gen ≃ eV the ratio in energydensities from the appearance of the fundamental field φ to present time is ρ DE /ρ gen = 10 − and should be com-pared to a case where φ is present at the Planck time ρ pl with ρ DE /ρ pl = 10 − , giving a difference in ratiosof 112 orders of magnitude. Therefore, the amount offine tuning in the parameters of the dark energy poten-tial v ( φ ) is much less sever in our case than in a standardquintessence dark energy model. The production of the φ particles gives rise to the scalar potential v ( φ ) and thefield φ will then evolve classically given by its equationof motion. Of course, we still need to choose v ( φ ) appro-priately to give dark energy, for example v ( φ ) = v DEo /φ with v DEo the present time dark energy potential [14].We stress the fact that the dark energy behavior of theuniverse is due to the form of the potential v ( φ ) but theenergy E gen at which the φ is generated is fixed by thecoupling constant g . An interesting possibility is to unifyinflation and dark energy using the same scalar field φ inour dark energy quantum generation picture [17]. Weassume that φ has a potential v ( φ ) which gives an earlyinflation at E I , as in standard inflationary models [3]. Inthis case we use the coupling L int = g φ ϕ to allow the φ field to decay into ϕ particles after inflation, via theprocess φ → ϕ + ϕ + ϕ . This decay process is very effi-cient [21] and the φ decays completely disappearing en-tirely from the spectrum of the universe. In order to pro-duce SM particles, the ϕ is coupled with the SM via thestandard interactions L int = h ϕ χ or L int = √ h ϕ ¯ ψψ with χ, ψ SM scalar or fermions, respectively. Reheat-ing of the early universe takes place via a 2 ↔ ϕ + ϕ ↔ χ + χ or ϕ + ϕ ↔ ¯ ψ + ψ with a transition N I N RH N gen N O N = Log @ a D Log @ Ρ RH D Log @ Ρ No D Log @ Ρ gen D Log H Ρ i L FIG. 1: We plot the logarithms of the energy densities ρ i ,with the density of φ field (black line), ϕ field (red dotted-line), radiation (yellow line) and of matter (blue dashed line)against the number of e-folds N = Log [ a ]. We see how at N RH the φ field disappears and the universe is reheated. At N gen the φ field is re-generated and the ϕ field disappears.We see how the φ field is rapidly diluted and than maintainsnearly constant at ρ φ ≃ ρ DE . At late times close to presenttime, i.e. at N ≃ N o , the φ field starts to dominate and inflatethe universe. rate Γ RH , for Γ RH /H ≡ T RH /T ϕ ∝ h /T ϕ > E RH ∝ h . Because of its couplings, the SMparticles and ϕ are in thermal equilibrium at the timeof reheating, and as long as ϕ remains relativistic onehas Ω ϕ ≃ Ω SM g ϕ /g SM . After reheating ρ ϕ evolves asradiation and eventually it will re-generate the φ field,at an energy E gen ∝ g . The re-generation process of φ can be produced with the same interaction L int = g φ ϕ as the inflaton decay at E I . A main difference in thetransition processes at E I and at E gen is the size of themass of φ . It varies quite significantly and we shouldhave m φ ( E I ) ≫ m φ ( E gen ) and also E gen ≫ m φ ( E gen ).In this case the inflaton-dark energy potential v ( φ ) mustbe chosen to be flat at high energy E I , to give inflation,and at low energy E o to accelerate the universe at presenttime. The reheating energy E RH ∝ h and the φ gen-eration energy E gen ∝ g are fixed by the coupling con-stants h and g independently of the potential v ( φ ). In themodel presented in section IV we set g = h ∝ E I /m pl reducing the number of parameters and connecting theinflation scale to the φ -generation and to the reheatingscales, E gen ∝ E I /m pl , E RH ∝ E I . As mentioned in theintroduction, any unification of inflation and dark en-ergy, such as quintessential models, need to explain howthe universe is reheated with SM particles and must ac-count for the long period of decelerating universe. In ourmodel we are able to explain both features in a consistentway.We give a schematic picture of the inflation-dark en-ergy unified model showing in fig.(1) the evolution of thelogarithms of the energy densities of the φ and ϕ fields,radiation and matter. We start with a φ dominated uni-verse and then at N RH , where N ≡ ln( a ) with a ( t ) thescale factor, the φ field decays and disappears, reheatingthe universe. After reheating the universe is radiationdominated, as in the standard cosmological model, andradiation includes the SM relativistic degrees of freedomplus the extra relativistic field ϕ . At N gen the φ field isre-generated with ρ φ f ≃ ρ ϕ i , where ρ φ f is the value of ρ φ at the end of the generation process and ρ ϕ i is the valueof ρ ϕ at the begin of the φ generation. At the same timethe ϕ field decays and ρ rel diminishes. After the gener-ation, ρ φ is rapidly diluted and subsequently maintainsnearly constant at ρ φ ≃ ρ DE and it starts to dominateclose to present times, i.e. at N ≃ N o , inflating the uni-verse. III. THE φ GENERATION.
In this section we describe the quantum generation ofthe quintessence scalar field φ and the subsequent ap-pearance of the dark energy behavior of the late universedue to the classical evolution of φ . In what follows we as-sume a universe that consists of the particles of the stan-dard model ”SM” together with a massless scalar field ϕ and a quintessence scalar field φ coupled together viaa four particle interaction, as for example L int = g φ ϕ or L int = g φ ϕ . The ϕ field is also coupled with theSM through the interactions √ h ϕ ¯ ψψ or h ϕ χ , where ψ and χ are some SM fermions or scalars respectively.We assume that at temperatures above 1 T eV all theSM particles and ϕ are relativistic and ϕ is in thermalequilibrium with T ϕ = T rad . As long as ϕ is relativistic T ϕ ≃ T rad and Ω ϕ ≃ Ω SMrad g ϕ g SMr , where Ω
SMrad is the SMradiation density and g ϕ , g SMr are the relativistic degreeof freedom of ϕ and SM, respectively. We also supposethat at temperatures T ≫ eV the universe does notcontain any φ particles, therefore the number density of φ particles is zero n φ = Ω φ = 0.We will show that the field φ is generated via the quan-tum transition ϕ + ϕ → ϕ + φ or ϕ + ϕ → φ + φ with atransition rate Γ ϕ + ϕ → ϕ + φ = Γ ϕ + ϕ → φ + φ = h σ gen v i n ϕ ≡ Γ gen , where σ gen = g / πE is the cross section for a2 ↔ v is the relative veloc-ity and n ϕ is the density number of the ϕ particles [25].This take place at energies below E gen ≃ c gen g m pl with c gen a constant, when Γ gen /H >
1. At the end of the φ generation one has Ω φ f ≃ Ω ϕ i , where Ω ϕ i is the ϕ den-sity at the begin of the φ particles production, and alsoΩ ϕ f ≪ Ω ϕ i . The production of relativistic particles φ becomes a source term for the generation of the scalarpotential v ( φ ). Once v ( φ ) has been produced the clas-sical equation of motion gives the evolution of φ . Theacceleration of the universe is then due to the form ofscalar potential v ( φ ).We consider a system composed by two scalar fields φ and ϕ plus SM, in a flat FRW metric, with a densitylagrangian L = ∂ µ φ∂ µ φ ∂ µ ϕ∂ µ ϕ − V T ( φ, ϕ ) + L SM (1)were L is the SM lagrangian and V T ( φ, ϕ ) = v ( φ ) + B ( ϕ ) + v int ( φ, ϕ ), v ( φ ) and B ( ϕ ) are the classical poten-tials of the two scalar fields φ and ϕ and v int ( φ, ϕ ) = − L int ( φ, ϕ ), where L int is the interaction lagrangian. L int plays a double role: it affects the classical evolutionof the two scalar fields and it also originates the quan-tum transitions between φ and ϕ particles that we use togenerate the φ field at late times. We divide the φ ( t, x )field into a classical background configuration φ c ( t ) plusa perturbation δφ ( t, x ) as φ ( t, x ) = φ c ( t )+ δφ ( t, x ), where δφ ( t, x ) corresponds to the quantum configuration of the φ field ( φ particles). The background φ c ( t ) and the per-turbation δφ ( t, x ) are usually taken as independent vari-ables. However we choose to take as the two indepen-dent variables φ ( t, x ) and δφ ( t, x ). Moreover we express δφ ( t, x ) in terms of the number density n φ of φ parti-cles via the relation n φ = E φ δφ , see appendix A. Westress the fact that when n φ = 0 one also has δφ ( t, x ) = 0that implies φ ( t, x ) = φ c ( t ), so the scalar field φ is in itsclassical configuration. Following the same argument wewrite ϕ ( t, x ) = ϕ c ( t ) + δϕ ( t, x ) and describe the ϕ fieldthrough the variables ϕ ( t, x ) and n ϕ = E ϕ δϕ . Let usconcentrate on the process of φ particles production inthe case that both φ and ϕ particles are relativistic andthe number density of φ particles n φ is initially zero. Weuse a four particles interaction, such as L int = g φ ϕ or L int = g φ ϕ and we consider a 2 ↔ ϕ + ϕ ↔ ϕ + φ or ϕ + ϕ ↔ φ + φ . As mentionedpreviously, the ϕ field is initially thermalized and it has aphase space distribution given by the Bose-Einstein dis-tribution f ϕ ( E ) = 1 / ( e E/T ϕ − ϕ particles with the same energy E ϕ = ¯ rT ϕ ≃ T rad . Sincethe 2 ↔ φ par-ticles become in thermal equilibrium with ϕ and they hasthe same energy E φ = E ϕ = E . Moreover the two scalarfields φ and ϕ are relativistic so their energy scales as E = E i /a ( t ). Before the φ generation the density numberof ϕ particles evolves as n ϕ = ζ (3) T ϕ /π , so the transi-tion rate for the considered 2 ↔ gen = h σ gen v i n ϕ = ( ζ (3) / π ¯ r ) g E and therefore it scales as Γ gen ∼ /a ( t ). As discussed insection II, in order to have a late time dark energy gen-eration, it is fundamental to have a ratio Γ gen /H thatgrows up with the expansion of the universe. If we arein radiation domination H ∼ a ( t ) − in in matter domi-nation H ∼ a ( t ) − / . This means that the Γ gen shoulddecrease more slowly than 1 /a ( t ) during radiation dom-ination and than 1 /a ( t ) / during matter domination, asin our case. Taking a flat FRW metric, the evolution ofour system is described by the following equations (seeeqs.(A20) in appendix A for details)˙ n φ + 3 Hn φ = ˜Γ ( n ϕ − n φ ) + 2 p E n φ ˙ φ (2)˙ n ϕ + 3 Hn ϕ = − ˜Γ ( n ϕ − n φ ) + 2 p E n ϕ ˙ ϕ (3)¨ φ + 3 H ˙ φ + E / √ n φ + ∂ φ v ( φ ) + ∂ φ v int ( φ, ϕ ) = 0 (4)¨ ϕ + 3 H ˙ ϕ + E / √ n ϕ + ∂ ϕ B ( ϕ ) + ∂ ϕ v int ( φ, ϕ ) = 0 (5)together with the Friedman equation H = ρ T , where ρ T = ρ φ + ρ ϕ + ρ rad + ρ mat , and ρ rad and ρ mat are the en-ergy densities of radiation and matter respectively. More-over v ( φ ) and B ( ϕ ) are the classical potentials of the φ and ϕ fields respectively and v int ( φ, ϕ ) is the classicalinteraction potential.Eqs.(2) and (3) are the Boltzmann equations that gov-ern the dynamic of the number densities n φ and n ϕ . Theytake into account the quantum transition between φ and ϕ particles thanks to the terms proportional to ˜Γ. Thequantity ˜Γ in eqs. (2) and (3) is given by ˜Γ ≡ h σ gen v i n ϕ for the process ϕϕ ↔ ϕφ and ˜Γ ≡ h σ gen v i ( n ϕ + n φ ) forthe process ϕϕ ↔ φφ . ˜Γ takes into account the con-tribution of the two processes ϕϕ → ϕφ and its inverse ϕφ → ϕϕ in one case and ϕϕ → φφ and φφ → ϕϕ in theother case, since one has Γ ϕ + ϕ → ϕ + φ n ϕ − Γ ϕ + φ → ϕ + ϕ n φ = h σ gen v i n ϕ ( n ϕ − n φ ) = ˜Γ ( n ϕ − n φ ) for ϕ + ϕ ↔ ϕ + φ and Γ ϕ + ϕ → φ + φ n ϕ − Γ φ + φ → ϕ + ϕ n φ = h σ gen v i ( n ϕ − n φ ) =˜Γ ( n ϕ − n φ ) for ϕ + ϕ ↔ φ + φ . At the begin of the φ generation one has n φ = 0 and therefore ˜Γ = Γ gen inboth processes. Moreover the two terms 2 p E n φ ˙ φ and2 p E n ϕ ˙ ϕ contained in eqs.(2) and (3) respectively, cou-ple the number densities n φ and n ϕ with the scalars φ and ϕ respectively. Eqs.(4) and (5) are the equations ofmotion of the lagrangian in eq.(1) for the scalar fields φ ( t, x ) and ϕ ( t, x ). These equations contains the terms E / √ n φ and E / √ n ϕ that couple the two scalar fields φ and ϕ with their number densities n φ and n ϕ and be-come the source terms for generating the scalar potential v ( φ ). Note that the terms E / √ n φ and E / √ n φ con-tained in eqs.(4) and (5) represent the spatial derivativesof the two scalar fields, in fact they come out from therelations − ∇ φa ( t ) = E / √ n φ and − ∇ ϕa ( t ) = E / √ n ϕ , seeeq.(A4) in appendix A. Of course eqs.(2)-(5) conserveenergy-momentum. The energy density and pressure ofthe φ field are ρ φ = ρ φ + ρ φ and p φ = p φ + p φ , where ρ φ ≡ ˙ φ + v ( φ ), ρ φ ≡ E n φ , p φ ≡ ˙ φ − v ( φ ) and p φ = ρ φ / ϕ field are ρ ϕ = ρ ϕ + ρ ϕ and p ϕ = p ϕ + p ϕ , where ρ ϕ ≡ ˙ ϕ + v int ( φ, ϕ ) + B ( ϕ ), ρ ϕ ≡ E n ϕ , p ϕ ≡ ˙ ϕ − v int ( φ, ϕ ) − B ( ϕ ) and p ϕ = ρ ϕ /
3. It is easy to show (see appendix A) that˙ ρ φ + 3 H ( ρ φ + p φ ) = 12 E ˜Γ ( n ϕ − n φ ) − ˙ φ ∂ φ v int (6)˙ ρ ϕ + 3 H ( ρ ϕ + p ϕ ) = − E ˜Γ ( n ϕ − n φ ) + ˙ φ ∂ φ v int (7)Therefore eqs.(6) give the energy-momentum conserva-tion law ˙ ρ T + 3 H ( ρ T + p T ) = 0, where ρ T = ρ φ + ρ ϕ and p T = p φ + p ϕ . Now we are ready to describe the φ particles production qualitatively. For presentation pur-poses we will not take into account the expansion of theuniverse, i.e. we will take H = 0, however we show be-low that the growth of the scale factor is very small (ofthe order of H/ Γ gen ) [40]. Moreover in this presenta-tion we will not consider the contribution from classicalinteractions and we also take B ( ϕ ) = 0. We will also work with ˜Γ constant. Obviously ˜Γ depends both onthe energy of the decaying particles and on the numberdensities, but we use this approximation to write an an-alytical solution of eqs.(2)-(5). Of course comparison ofthe approximated analytical solution with the numericalsolution shows a complete agreement and the classicalinteraction v int ( φ, ϕ ) and expansion rate do not play asignificant role. Under these approximations the systemof eqs.(2-5) reduce to the following equations˙ n φ = ˜Γ ( n ϕ − n φ ) + 2 p E i n φ ˙ φ (8)˙ n ϕ = − ˜Γ ( n ϕ − n φ ) + 2 p E i n ϕ ˙ ϕ (9)¨ φ + E / i √ n φ + ∂ φ v ( φ ) = 0 (10)¨ ϕ + E / i √ n ϕ = 0 (11)where E i is the energy of the quantum particles at thebegin of the φ generation. The initial conditions on the ϕ field are ρ ϕ i = E i n ϕ i , n ϕ i = ς (3) π ¯ r E i with h ˙ ϕ i i = E i n ϕ i and Ω ϕ i = 0 .
1. For the φ field we have the initialconditions ρ φ i = 0 that gives Ω φ i = n φ i = ˙ φ i = v ( φ i ) =0. Now, the first stage starts with ( n ϕ − n φ ) ˜Γ ≫− p E i n φ ˙ φ in eq.(8), which is clearly verified at thebegin since n φ i = 0, and the approximate solution ofeqs.(8-11) is n φ ≃ n ϕ i ˜Γ ( t − t i ) (cid:20) − E i ( t − t i ) (cid:21) ,n ϕ ≃ n ϕ i h E i − ˜Γ) ( t − t i ) i , (12)˙ φ ≃ − E / i q n ϕ i ˜Γ ( t − t i ) / , ˙ ϕ ≃ ˙ ϕ i [1 − E i ( t − t i )] . valid for ˜Γ ( t − t i ) ≪ E i ( t − t i ) ≪
1. This firstphase of the generation consists in the growth of n φ dueto φ particle production. However the growth of n φ endsat some time t , when ( n ϕ − n φ ) ˜Γ = − p E i n φ ˙ φ > n φ = 0. This point is the end of stage one and n φ reaches a maximum value n φ = ¯ n φ ≡ n ϕ + β (1 − (1 +2 n ϕ /β ) / ) with β = 2 E i ˙ φ / ˜Γ . At the end of this firststage one also has n ϕ /β ≪ n ϕ ( t ) ≃ n ϕ i , ˙ ϕ ( t ) ≃ ˙ ϕ i and n φ ( t ) = ¯ n φ ≃ n ϕ / β ≪ n ϕ .The second phase of the generation maintains( n ϕ − n φ ) ˜Γ+2 p E i n φ ˙ φ = 0 dynamically, therefore fromeq.(8) it follows that n φ remains constant at the equilib-rium value ¯ n φ ≪ n ϕ i . One can cheek the stability ofthis solution n φ = ¯ n φ analyzing the dynamical equationfor small perturbations of n φ around ¯ n φ that is δ ˙ n φ = − β δn φ with β = ˜Γ − q E i ¯ n φ ˙ φ >
0, that gives an exponen-tial suppression of perturbations, so the solution n φ = ¯ n φ is stable. If n φ is maintained constant at its equilibriumvalue ¯ n φ , the relation 2 p E i n φ ˙ φ = − ˜Γ ( n ϕ − ¯ n φ ) = ˙ n ϕ is satisfied dynamically (see eqs.(8) and (9)), so we canwrite eq.(11) in the form ¨ φ + ∂ φ v ( φ ) ≃ E i ˜Γ n ϕ − ¯ n φ ˙ φ ,or ˙ ρ φ = E i ˜Γ ( n ϕ − ¯ n φ ) where ˙ ρ φ ≡ ˙ φ + v ( φ ). Wetake ˙ ϕ ≃ E n ϕ , which agrees well with the numericalsimulation, and E ˙ ϕ p En ϕ ≃ E ˙ ϕ ≃ E ρ ϕ where ρ ϕ ≡ ˙ ϕ /
2. Therefore eqs.(8)-(11) are reduced to thefollowing effective equations n φ = ¯ n φ , ˙ n ϕ = − ˜Γ ( n ϕ − ¯ n φ ) + 4 ρ ϕ , ˙ ρ φ = 12 ˜Γ E i ( n ϕ − ¯ n φ ) , (13)˙ ρ ϕ = − E ρ ϕ whose solutions are n φ = ¯ n φ ,n ϕ = n ϕ i h e − ˜Γ( t − t ) + 2 E i e − ˜Γ( t − t − e − Ei ( t − t E i − ˜Γ i ++ ¯ n φ h − e − ˜Γ( t − t ) i ,ρ φ = E i ( n ϕ i − ¯ n φ ) h − e − ˜Γ( t − t ) i ++ E i n ϕ i E i (cid:0) − e − ˜Γ( t − t (cid:1) − ˜Γ ( − e − Ei ( t − t ) E i − ˜Γ) ,ρ ϕ = E i n ϕi e − E i ( t − t ) . (14)From these equations we see that n ϕ decrease until itreaches the equilibrium at n ϕ f ≃ n φ f ≃ ¯ n φ ≪ n ϕ i andthat ρ ϕ decrease exponentially to zero so at the end ofthe regeneration one has ρ ϕ f = E i ¯ n φ ≪ ρ ϕ i . Moreover ρ φ grows until ρ φ f = ρ ϕ i − E i ¯ n φ so one has ρ φ f = ρ ϕ i and in conclusion one has Ω φ f ≃ Ω ϕ i , Ω ϕ f ≪ Ω ϕ i and v ( φ ) ≃ ρ ϕ i , that means that all the energy initially storedinto the quantum field ϕ has been transferred to the φ field in accordance with the total energy conservation.We solve numerically the system of eqs.(2-5) in the case v ( φ ) = v i /φ . In fig.(2) we show the evolution of n φ /E and in fig.(3) that of n ϕ /E . The evolution of n φ includesthe solutions given in eqs.(12) and (14) showing that n φ starts from zero, it reaches its maximum and then it de-creases to its equilibrium value ¯ n φ . The evolution of n ϕ is plotted in fig.(3) and we see how n ϕ decrease exponen-tially to the asymptotic value ¯ n φ as in eqs.(12)and (14).In fig. (6) we plot the evolution of the density parame-ter Ω ϕ showing that Ω ϕ goes to zero at the end of the φ generation. We stress some important features ofthe φ generation process. The first one is that, in spiteof starting with a v ( φ i ) = 0, at the end of the processone has generated a potential v ( φ ) ≃ ρ ϕ i = 0. In fig.(4)we show how v ( φ ) grows from zero to its maximum value v ( φ ) ≃ ρ ϕ i . At this point the quantum generation pro-cess is completed and the subsequent evolution of φ isgiven by its classical equations of motion. At the sametime in fig.(5) we see the evolution of ˙ φ , where it startsat ˙ φ ( t geni ) = 0 and takes negative values (implying thegrowth of v ( φ )) and eventually reaches positive values.The time when ˙ φ ( t genf ) = 0 corresponds to the end of φ generation and to the maximum of v ( φ ). Another featureof the φ generation process consists of the existence of anequilibrium value ¯ n φ ≪ n ϕ i for the density number of φ particles n φ . Therefore after a first phase of growth, n φ saturates at ¯ n φ and then it is maintained constant. This N i N gen N = Log @ a D n (cid:143) n Φ (cid:144) E FIG. 2: We show the evolution of n φ /E against the numberof e-folds N during the φ generation in the case v ( φ ) = v i /φ .We see how n φ grows from zero and than reaches the equilib-rium value ¯ n φ . N i N gen N = Log @ a D (cid:143) n j (cid:144) E FIG. 3: We show the evolution of n ϕ /E against the numberof e-folds N during the φ generation in the case v ( φ ) = v i /φ . n ϕ decreases exponentially and reaches the asymptotic value¯ n φ . means that all the energy coming from further ϕ parti-cles decay is transferred directly into the potential v ( φ )without generate any change in n φ .Let us point out more properties of the φ generationprocess. First we note that if the φ field is regeneratedwhen the universe is radiation dominated, the scale factor N gen N o N = Log @ a D v gen v o v H Φ L FIG. 4: We show the evolution of the potential v ( φ ) againstthe number of e-folds N in the case v ( φ ) = v i /φ . N gen N o N = Log @ a D Φ FIG. 5: We show the evolution of the derivative ˙ φ ( N ) againstthe number of e-folds N in the case v ( φ ) = v i /φ . N gen N o N = Log @ a D W j FIG. 6: We show the evolution of Ω ϕ against the number ofe-folds N . We see that Ω ϕ rapidly goes to zero at the end ofthe φ genearion. evolves as a ( t ) = a i p H i ( t − t i ). The φ generationends at ˜Γ( t − t i ) > ∼ ≃ Γ gen , giving H ( t − t i ) < ∼ H/ Γ gen ≤ a ( t ) < ∼ √ a i (the gener-ation process has Γ gen /H < ∼ φ generation playsno significant role. We also stress the fact that the gen-eration process preserves the homogeneity and isotropyof the universe, as in the reheating process after inflation[3]. In fact it is obtained via a 2 ↔ P p i = P p f )so the amount of homogeneity is preserved.Now we can state something more about the beginand the duration of the generation process. First notethat at the begin of the φ generation n φ = 0, that im-plies ˜Γ = Γ gen ≡ Γ ϕ + ϕ → φ + ϕ = Γ ϕ + ϕ → φ + φ . There-fore, in order to establish the efficiency of φ particlesproduction one should compare the transition rate Γ gen with the expansion rate of the universe H . We as-sume that before the generation the ϕ field has thesame temperature of the radiation, so we take E rad = E ϕ = E . If the generation starts before the matterdomination one has H = ( ρ rel / m pl Ω rel ) / ≡ c H E where c H ≡ ( π g rel /
90 ¯ r m pl Ω rel ) / and g rel is thetotal number of relativistic degrees of freedom. There- fore, using n ϕ = ζ (3) π T ϕ before the φ generation one hasΓ gen = ζ (3)32 π ¯ r g E and Γ gen /H = c gen g m pl /E , wherewe have defined c gen ≡ ( ζ (3) / π ¯ r ) (90 Ω rel /g rel ) / .We conclude that the φ generation starts at energies E < ∼ E gen with E gen ≡ c gen g m pl = 1 . (cid:16) g − (cid:17) eV (15)when Γ gen /H ≡ E gen /E > ∼
1. We aim to have a φ particles production at energy scales well below M eV ,where the number of relativistic degrees of freedom is g rel = g SMrel + g ϕ ≃ .
36 giving c gen ≃ · − . Ofcourse, by means of eq.(15) we can conveniently choosethe coupling g in such a way that the generation processstarts at an energy of about 1 eV . This takes place if g ≃ − or α = g / π ≃ − , where α is the finestructure constant of the transition process. We stressthe fact that Γ gen /H ≡ E gen /E ≥ φ gener-ation process but, when the φ field starts to inflate theuniverse, one has H ∼ a ( t ) − (1+ ω φ ) with ω φ ≃ −
1, so H will be roughly constant and at the dark energy epochΓ gen /H ∼ a ( t ) (1+ ω φ ) − →
0. Therefore, φ and ϕ fieldsdecouple at E dec = √ π √ ρ DE g m Pl = √ c gen π E DE E gen .We can summarize the φ generation process as follows.At the begin of the generation we have n ϕi = ζ (3) π ¯ r E ϕi , n φ = Ω φ = v ( φ ) = 0 and at E ϕi = E gen we have Γ gen = H . The first effect is the decay of ϕ and the growth of n φ to its equilibrium value ¯ n φ ≪ n ϕ i after which n φ remainsconstant. Subsequently all the energy coming from the ϕ field is transferred to the potential v ( φ ) through the chainreaction n ϕ → n φ → v ( φ ). In this process n φ remainsconstant and ¯ n φ represent a maximum transfer efficiencyvalue for n φ in such a way that the energy coming fromthe decaying ϕ particles is immediately stored into thepotential v ( φ ). At the end of the generation on has n φ f ≃ n ϕ f ≃ ¯ n φ ≪ n ϕ i ρ ϕ f ≪ ρ ϕ i and ρ φ f ≃ ρ ϕ i .From this point on, the evolution of φ is the standardone, where its dynamics is dominated by the classicalpotential v ( φ ). The generation process described beforedoes not depend explicitly on the form of the potential v ( φ ) and the value of φ at the end of the generation pro-cess is given by the condition v ( φ gen ) ≃ ρ ϕ i ≃ E ϕi = E gen . Since we want the φ field to give dark energy,we must choose a potential v ( φ ) that slow rolls at latetimes, i.e. at present time when φ ≃ φ o . Clearly wemust impose the condition v ( φ gen ) > v ( φ o ) = v DE ≃ (10 − eV ) . For example one can consider an effectivepotential v ( φ ) = v i φ that verifies the slow roll conditionsfor φ ≥ √ v i = (10 − eV ) . In this case thevalue of φ at the end of the generation is φ gen ≃ v i /E gen where we have used E gen = v ( φ gen ) = v i φ gen . Typically,runaway quintessence potentials have an EoS parameter ω φ that reaches positive values of ω φ ≃ φ andlater there is a transition from ω φ = 1 to ω φ = − φ starts growing [14]. Therefore the mat-ter domination epoch is unchanged by our φ generation N gen N o N = Log @ a D - Ω Φ FIG. 7: We show the evolution of the EoS parameter ω φ against the number of e-folds N. scheme. In fig.(8) we show the evolution of the densityparameters Ω φ , Ω SMrad and Ω mat . The φ is generated at N gen at energies E gen ≃ eV , for g ≃ − , close to ra-diation matter equality with Ω φ i ≃ . φ ≃ .
74 and Ω m ≃ .
26 with ω φ ≃ − σ gen = g πE gen = 1 . (cid:18) − g (cid:19) pb = 2 . (cid:18) eVE gen (cid:19) pb (16)where we have used E gen ≡ c gen g m pl and c gen ≡ ζ (3)32 π ¯ r q
90 Ω rel g rel ≃ · − and pb = 10 − cm . It isinteresting to compare the cross section σ g with thatof WIMPS. The relic abundance of WIMPS is Ω w h =3 × − cm / < σ w v > [16] giving h σ w v i = 0 . c pb ,with c the speed of light. If we take that at decou-pling the WIMPS have a mass to temperature ratio m/T = 20 [16] we obtain h σ w i = 2 . pb equivalent toour σ gen for E gen = 1 eV . However, the present time ob-servational upper limit to σ w between WIMPS and nu-cleons is σ w < ∼ − cm consistent with supersymmetricWIMPS [16]. IV. UNIFICATION OF INFLATION AND DARKENERGY.
In this section we discuss the possibility of unifyinginflation and dark energy by means of a unique scalarfiled φ that we call uniton, as in [17]. In general it isnot difficult to choose the potential v ( φ ) in such a waythat the φ field is responsible for both inflation and darkenergy [15]. To achieve inflation and dark energy withthe same scalar φ one requires that the potential v ( φ )must satisfy the slow roll conditions | v ′ ( φ ) /v ( φ ) | < | v ′′ ( φ ) /v ( φ ) | < v min = 0 at a finite value of φ are notuseful to unify inflation and dark energy, since this kindof potentials do not inflate at low energies [14]. Thiskind of potentials may be useful to unify inflation and N gen N O = Log @ a D W FIG. 8: We plot the density parameters Ω φ (black line), Ω SMrad (yellow line), Ω mat (blue dashed line) and Ω ϕ (red dottedline) against the number of e-folds N = Log [ a ]. We see howat N gen the ϕ disappears and the φ field is generated. Presenttimes are at N o when Ω φ ≃ .
74 and Ω m ≃ . dark matter. In most inflation-dark energy unified mod-els only the classical evolution of the quintessential scalarfield is considered and the reheating and the long periodof decelerating universe (between inflation and dark en-ergy) are not taken into account.In this section we present an inflation-dark energy uni-fied scheme that can be resumed in the following way: asin usual inflationary models, the early universe is domi-nated by the φ field that inflates for a sufficient numberof e-folds. After the end of inflation the φ field decayscompletely into the extra relativistic field ϕ already in-troduced in section III. The ϕ couples and produces SMparticles at an energy E RH and the universe is reheated.At low energies E gen the φ field is generated via the quan-tum generation mechanism studied in section III and theuniverse enters the dark energy epoch at temperaturesclose to present time. From E RH to E gen we have thestandard evolutionary scenario with the extra relativisticdegree of freedom ϕ .The new feature in the unification scheme that wepresent here, is that the transition between deceleratingradiation-matter dominate universe and the dark energyera is due to a quantum process, i.e. the low energy gen-eration of the φ field. As stated in section III, the energyscale E gen at which the uniton φ is generated is fixedby the couplings g of φ . The scales E gen may be manyorders of magnitude smaller than E RH without any finetuning.We stress the fact that, although we will describe theinflation-dark energy unified scheme choosing a particu-lar form of the potential v ( φ ), the quantum generationmechanism works well for a large class of potentials. Theonly requirements on the potential v ( φ ) are that it shouldsatisfy the slow roll conditions at high and low energiesand that the φ mass m φ ≡ p v ′′ ( φ ) should satisfy thecondition m φ ( t RH ) ≫ m φ ( t gen ), where t RH and t gen arethe reheating and φ generation times. In addition onehas to require that the φ particles are relativistic at t gen ,that implies m φ ( t gen ) ≪ E gen . In section IV A we willdiscuss the inflation and reheating scenario and then insection IV B we will consider the inflation and dark en-ergy unified scheme using a simple example. A. Inflation and reheating
Let us consider an universe that contains the field φ , asecond scalar field ϕ and the SM particles. We want the φ field to inflate the early universe, so we assume that thepotential v ( φ ) has at least one flat region correspondingto inflation. As an example one can consider the potentialdescribed in Appendix B. During inflation the φ fielddominates the universe and slow rolls as long as the slowroll conditions | v ′ ( φ ) /v ( φ ) | ≪ | v ′′ ( φ ) /v ( φ ) | ≪ H ∼ v ( φ ). Afterthe end of inflation the φ field decays into ϕ particles andreheats the universe. We couple the two scalars φ, ϕ viathe interaction L int = g φ f ( ϕ ) (17)where f ( ϕ ) is a polynomial of ϕ . We know that theinteraction in eq.(17) gives a complete reheating sinceit involves a single φ particle decaying into ϕ particles[21]. At the end of inflation the φ particles are at restin the comoving frame (the velocity is redshifted as v i = e − ∆ N v f ) so E φ = m φ and we take m φ ≫ m ϕ . We take f ( ϕ ) = ϕ and we consider the process φ → ϕ + ϕ + ϕ . Forsimplicity we assume that all the ϕ particles are producedwith the same energy E ϕ given by E φ = 3 E ϕ giving adecay rate [2] Γ d = g m φ (2 π )
72 (18)Let us remind that a decay process is efficient if Γ d /H > ∼ ρ φ of the φ fieldis given by the equation˙ ρ φ + 3 H (1 + ω ) ρ φ = − Γ d ρ φ (19)If we consider ω and Γ d as piecewise constant, the solu-tion is ρ φ ∼ a ( t ) − ω ) e − Γ d t and the φ energy densityvanish exponentially, that means that ϕ particles are pro-duced and the energy of the φ field is transferred to the ϕ field. If one relaxes the hypothesis of constant Γ d one has ρ φ ∼ e − R Γ d dt and the condition for an efficient decay is R Γ d dt >>
1. At the same time we couple ϕ with SMparticles. We take the usual interaction terms L int = h ϕ χ , L int = √ h ϕ ¯ ψψ (20)where χ and ψ are SM scalars and fermions, respectively.As long as SM particles are relativistic, valid at tem-peratures above 1 T eV , the processes ϕ + ϕ ↔ χ + χ or ϕ + ϕ ↔ ¯ ψ + ψ given by the interaction in eq.(20) have a transition rate Γ RH = h n ϕ / πE ϕ , and using n ϕ = ζ (3) T ϕ /π one has Γ RH = ζ (3) h E ϕ / π ¯ r . SMparticles are produced at Γ RH /H ≡ E RH /E > E RH ≡ c RH h m P l = 3 . (cid:18) h − (cid:19) GeV (21)and c RH ≡ ζ (3)32 π ¯ r q
90 Ω rel g r ≃ − . If reheating takesplace above 300 GeV we have g SMr = 106 .
75 and Ω r = 1so that we can estimate c RH ≃ − . Taking h ≤
1, sothat α h = h / π < .
1, the maximum energy for reheat-ing would be 10 GeV . However the limit for successfulreheating scenario is much lower and it may be as low as10
M eV [11, 12] corresponding to h ≃ − .Therefore SM particles are produced at energies E ≤ E RH and ϕ is in thermal equilibrium with SM particleswith T ϕ = T γ where T γ is the photon temperature. Aslong as ϕ is relativistic T ϕ ∝ T γ and if it remains also inthermal equilibrium we haveΩ ϕ = g ϕ g SMrel Ω SMrel (22)where g SMrel is the number of SM relativistic degrees offreedom and g ϕ = 1. B. Inflation-Dark Energy Unification
Let us now describe the inflation-dark energy unifiedpicture with an explicit example. Again we start withtwo scalar fields coupled through eq.(17). To be specificwe will choose L int = g φ ϕ and a scalar potential v ( φ )that inflates at high and low energies. As mentioned inthe introduction in section IV the choice of the scalarpotential is not important, there are a wide number ofpossibilities, and we choose to work with v ( φ ) = V I (cid:18) − π arctan φf (cid:19) (23)which has two free parameters E I ≡ V / I and f withmass dimension. The potential in eq.(23) has two re-gions φ < − (2 f /π ) / and φ > √ | v ′ ( φ ) /v ( φ ) | ≪ | v ′′ ( φ ) /v ( φ ) | ≪ v ( φ ) in thesetwo regions are (see appendix B) v ( φ ) = ( V I (cid:16) fπφ (cid:17) for φ < − f V I fπφ for φ > f (24)Inflation is associated with the high energy region φ < − (2 f /π ) / with v ( φ ) ≃ V I = E I and dark energy withthe region φ > √ v ( φ ) ≃ v I fπφ . We determine thetwo free parameters in eq.(23) with the constrains comingfrom inflation δρρ = 5 . × − and from dark energy0density ρ DE = 3 H o Ω DE . Taking φ o ≃ √ E I ≃ T eV and f = φ o v DE /V I ≃ − eV where we havereintroduced the correct mass units in E I and f .After inflation we want to reheat the universe with theSM particles. To achieve this we couple φ and ϕ via theinteraction term L int = g φ ϕ and ϕ with SM particlesas in eq.(20). The φ field decays into ϕ via the process φ → ϕ + ϕ + ϕ with a decay rate Γ d = g m φ (2 π ) given ineq.(18). This process starts immediately after inflationwith H ≃ E I . The maximum value of Γ d /H is when m φ is also at its maximum at φ ∼ f giving Γ d /H ≃ .Notice that Γ d ≃ at its maximum and the lifetime τ φ = 1 / Γ d of the φ particles is such that τ /τ pl ≪ τ pl is the Planck time. Therefore all φ particles decayand at the end of the reheating and one has Ω φ = 0 andΩ ϕ + Ω SM = 1. SM particles are produced through theinteraction with the ϕ field via the interaction given ineq.(20) and described in section IV A. This process takesplace for energies E ≤ E RH ≡ c RH h given in eq.(21)with Γ RH /H ≥ ϕ and SM particles arein thermal equilibrium giving Ω ϕ = g ϕ g SMrel Ω SMrel as long as ϕ remains relativistic. As discussed in section III as longas E RH > E > E gen ≡ c gen g (c.f eq.(15) ) the universecontains the SM particles plus ϕ and at energies E ≤ E gen the φ particles starts to be produced via the process ϕ + ϕ ↔ ϕ + φ with a decay rate Γ gen = h σ gen v i n ϕ . Theinflation-dark energy unification scheme involves threedifferent quantum processes: The φ decay into ϕ , the SMparticles production and the late time φ generation. Onlythe first one (inflaton decay) depends on the choice of thepotential v ( φ ), through its mass, the other two dependonly on the size of the couplings g, h . These processestake place at energies E I , E RH ≡ c RH h , E gen ≡ c gen g (25)whereΓ d H ≥ , Γ RH H ≡ E RH E ≥ , Γ gen H ≡ E gen E ≥ c RH ≃ − for a reheating temperature above300 GeV and c gen ≃ · − if the φ is generated atenergies below 1 M eV . We stress the fact that the φ generation at late times is due to the same interactionterm L int = g φ ϕ that gives the φ decay after inflation.The main difference is that at low energies E gen the massof φ is many orders of magnitude lower than its value athigh energies E I , i.e. m φ ( t RH ) ≫ m φ ( t gen ) where t RH and t gen are the reheating and φ generation times re-spectively. We also require that m φ ( t gen ) ≪ E gen sothat φ is relativistic at t gen as in section III. Noticethat in our model the value of the φ mass at generationtime is m φ ( t gen ) ≃ − eV (see appendix B) which ismuch smaller than E gen ≃ eV . However at presenttimes m φ ( t o ) ≃ − eV which is a typical mass for aquintessence (dark energy) field.After the production of the φ particles v ( φ ) is gener-ated and the φ classical evolution will drive the expansion of the universe as described in section III. Concluding wehave shown with a simple example how the inflation-darkenergy unification takes place. Of course it is possibleto choose different scalar potentials v ( φ ) or interactionterms L int that gives similar results. C. E RH and E gen scales The values of E RH and E gen do not depend on thechoice of the potential v ( φ ) but only on the couplings g, h . The values of E RH and E gen are fixed in termsof the couplings g and h . In general g and h are freeparameters that should give E I ≥ E RH > M eV and E RH ≫ E gen > E o and are given by (see eq.(25)) E RH ≡ c RH h , E gen ≡ c gen g (27)and c RH ≃ c gen ≃ − . An interesting reduction ofparameters is if we take E RH = p E gen (in natural units)which gives g = h and E RH = (cid:18) E gen eV (cid:19) / × GeV (28)Notice that this choice of g, h implies a low reheatingtemperature and for E gen as small as E o ∼ − eV wehave E RH ≥ . T eV . If we set g = h = q E I m pl = 4 (cid:16) q (cid:17) (cid:18) E I T eV (cid:19) − , (29)with q a proportionality constant, we have E gen = (cid:16) q (cid:17) (cid:18) E I T eV (cid:19) eV (30) E RH = (cid:16) q (cid:17) (cid:18) E I T eV (cid:19)
T eV. (31)The fine structure constants associated to the two cou-plings are α g ≡ g π , α h ≡ h π and cross sections σ g , σ RH are then α g = (cid:16) q (cid:17) (cid:18) E I T eV (cid:19) − (32) α h = 3 (cid:16) q (cid:17) (cid:18) E I T eV (cid:19) − (33)The cross section for the generation process are σ gen = g / (32 πE gen ) and σ RH = h / (32 πE RH ) giving σ gen = 132 πc gen g = (cid:18) q (cid:19) (cid:18) T eVE I (cid:19) . pb (34) σ RH = 132 πc RH h = (cid:18) q (cid:19) (cid:18) T eVE I (cid:19) − pb (35)We find the relationship between g and h in eq.(29) in-teresting but of course it does not need to hold since inprinciple g, h and therefore E gen , E RH are independentfrom each other and eqs.(30) and (31) are equivalent toeqs.(15) and (21), respectively.1 V. PHENOMENOLOGY
In this section we summarize the main phenomenologi-cal consequences of the dark energy quantum generation.Let us first discuss the consequences of having the rela-tivistic field ϕ . If ϕ is not contained in the SM then itrepresent an extra relativistic degree of freedom. CMBtemperature anisotropies as well as SDSS and 2dF LargeScale galaxy clustering, Lyman- α absorption clouds, typeIa Supernovae luminosity distances and BAO data, canbe used to determinate the value of the effective relativis-tic degrees of freedom, usually described in terms of theeffective number of neutrinos N effν . The value of N effν affects the matter-radiation equality epoch and thus theISW effect, so CMB anisotropies are sensitive to devi-ations from the standard cosmological model value of N effν ≃ .
04. Analysis of the WMAP data combinedwith other cosmological data sets allows for values of N effν different from its standard model value. For ex-ample in [30] it is found N effν = 4 . +1 . − . at 95% c.l., con-sistently with other analysis [31]. Moreover BBN is alsoaffected by N effν , because the number of relativistic de-grees of freedom change the value of the expansion rate ofthe universe and than influence the expected primordialHelium abundance. The BBN bound is N effν = 3 . +1 . − . at 95% c.l. [19, 30, 33], that seems to be more stringentthan bounds coming from CMB data. Anyhow N effν can evolve from the BBN epoch at T ∼ M eV to theCMB decoupling era at T ∼ eV [33], so the differ-ent bounds coming from BBN and CMB are compati-ble. In our case the extra relativistic degree of freedomis represented by the scalar field ϕ that contributes to N effν an amount δN effν = (cid:16) T ϕ T ν (cid:17) . If the ϕ decouplesfrom radiation before neutrinos, one has T ϕ ≤ T ν and δN effν ≤ / ≃ .
57, that is in full agrement with bothBBN and CMB data. If ϕ is coupled with photons onehas δN effν = (cid:16) T γ T ν (cid:17) ≃ . ϕ field is consistent and appar-ently favored by cosmological data.Another important phenomenological aspect of thedark energy generation model, concerns the coupling ofthe ϕ field with SM particles. In principle ϕ may becoupled with electrons, baryons and photons, but stronglimits on the strength of such couplings comes from astro-physical considerations and accelerator phyiscs. In factif coupled with electrons, the ϕ particles are producedin stars and this fact affects the evolution of the stars,as studied in [37]. The coupling strength between ϕ andelectrons should satisfy the condition α ϕee < . × − .In the model that we present, SM particles are producedat E RH = c RH h m P l ≥ M eV that gives h ≥ − and a fine structure constant α = h / π ≥ − ,therefore the ϕ field cannot be coupled with electrons.If coupled with baryons, the massless scalar field ϕ could also generate long range forces [38] with possi-ble observable consequences at astrophysical and cos-mological level. The upper bound coming from longrange force experiments is α ϕB < − [39] thus the ϕ coupling with baryons should be excluded. If the ϕ field is coupled with photons via the axion-like interac-tion term L int = g ϕγγ ϕ F µν ˜ F µν = − g ϕγγ ϕ E · B with g ϕγγ < − GeV − [37]. The bound on the transitionrate is Γ ϕγγ = g ϕγγ m ϕ π < ∼ − ( m ϕ /eV ) eV . Taking H = c H T /m pl the field ϕ and photons are coupled for T < ∼ T γRH ≃ ( m ϕ /eV ) / GeV when Γ ϕγγ /H ≥ ϕ is coupled with photons, SM particleswill be produced at low reheating temperatures of about T γRH ≃ GeV for m ϕ ≃ eV . The coupling with pho-tons is not ruled out by experimental data.In conclusion, the ϕ field could be coupled either toneutrinos, photons, neutral Higgs field or supersymmet-ric partners of the SM which are currently searched for atLHC. It is particularly interesting the case in witch the ϕ is coupled with neutrinos. In the standard cosmologicalmodel the three neutrinos free-stream and interact onlygravitationally. Free-streaming lowers the neutrino per-turbations and introduce a source of anisotropic stress.On the contrary, if one or more neutrinos are coupledwith the scalar field ϕ , the interacting neutrinos behaveas a tightly-coupled fluid with density and velocity per-turbations but no anisotropy [35]. This fact also affectsthe adiabatic sound speed c s , that is equal to 1 / < c s ≤ / ϕ fieldwith neutrinos produces many observable consequenceson the Cosmic Neutrino Background (CNB) [36].In alternative to the dark energy generation schemepresented in section III, one can generate the φ field with-out any auxiliary field ϕ but coupling φ directly with SMneutrinos via an interaction L int = √ g φ ¯ νν . Thereforeone or more neutrinos will be tightly-coupled to the φ quintessence field and this will produced observable con-sequences on the CNB [36].The φ sector of the quantum generation model has alsoan interesting phenomenology. For example the inflationdark energy unified model presented in section IV haslow inflationary and and reheating scale E I ≃ T eV and E RH ≃ T eV for h ≃ − . This low reheating en-ergy does not affect the reheating efficiency and it alsoavoids gravitino overabundance problems. From a cos-mological point of view, a low inflationary scale may alsoaffect N effν as showed in [12], giving one more possibletest of the unified model. Moreover interesting effectsmay be observed in accelerators, as for example at LHC,with energy scales not so far from the inflationary en-ergy. Then, as discussed above, a rich phenomenologyexists, that may be used to constrain or falsify cosmolog-ical models that make use of the dark energy generationmechanism.2 VI. CONCLUSIONS
We will now present a summary and conclusions of ourwork. One of the main goals of this paper was to under-stand why the dark energy is manifested at such a latetime. To achieve this we have presented a novel idea,the quantum generation of dark energy, giving a new in-terpretation of the late time emergence of DE in termsof a late time quantum production of the quintessence φ particles. We take a 2 ↔ φ and a relativistic ϕ particles. The scale where the φ fieldis generated is dynamically determined by the conditionΓ /H = E gen /E ≥ E ≤ E gen with E gen = c gen g m pl and c gen ≃ · − . Thereforethe smallness of E gen is due to a small coupling g andfor g ≃ − gives a E gen ≃ eV and a cross section σ gen ≃ pb . The acceleration of the universe is thendue to the classical evolution of φ and determined bythe scalar potential v ( φ ). We have described in sectionIII a universe that initially contains no φ particles, i.e. n φ = Ω φ = ˙ φ = v ( φ ) = 0, and once the relativistic par-ticles φ are produced they become a source term for thegeneration of the scalar potential v ( φ ). Once v ( φ ) hasbeen produced the classical equation of motion gives theevolution of φ .We show in section IV that it is possible to unify in-flation and dark energy using the same quintessence field φ . To achieve the unification we required that the poten-tial v ( φ ) has two flat regions, at high energy for inflationand low energy for dark energy. In this scenario, afterinflation the field φ decays completely and reheats theuniverse with standard model particles. The universeexpands then in a decelerating way dominated first byradiation and later by matter. At low energies the sameinteraction term that gives rise to the inflaton decay ac-counts for the re-generation of the φ field giving rise todark energy. An important difference in the quantumprocess between φ and ϕ at high and low energies is thevalue of transition rate due to the size of the φ mass, m φ ( E I ) ≫ m φ ( E o ).We presented in section IV a simple example on howthe inflation-dark energy unification can be implemented.We used a potential v = E I (1 − arctan[ φ/f ]) which is flatat high and low energies. The two parameters E I , f aredetermined by the density perturbations δρ/ρ and thevalue of v o at present time giving E I = 100 T eV, f =10 − eV . The coupling g between φ and ϕ and thecoupling h between ϕ and the SM particles are free pa-rameters but can be taken as g = h = q E I /m pl =( q/ E I / T eV )10 − giving a reheating energy E RH = ( q/ E I / (00 T eV ) T eV and a generation en-ergy E gen = ( q/ ( E I / T eV ) eV . The crosssection between φ and ϕ is σ gen = g / πE gen ≃ pb quite close to cross section of WIMP dark matter withnucleons σ w ≃ pb at decoupling. By fixing g = h = q E I we have determined the coupling, which set the scales ofreheating and φ re-generation, in terms of the inflationscale E I and we can reduced the number of parameters. Of course this is not the only possible choice of g and h .To conclude, we have presented a general framework toproduce the fundamental quintessence field φ dynami-cally at low energies. The energy scale is fixed by thestrength of the coupling and this offers a new interpre-tation of the cosmological coincidence problem: dark en-ergy domination starts at such small energies because ofthe size of the coupling constant g . Finally, our approachallows for an easy implementation of inflation and darkenergy unification with the standard long periods of ra-diation/matter domination. APPENDIX A: EQUATIONS OF MOTION
Here we derive the system of differential eqs.(2)-(5)that rules the φ generation. Let us consider a FRWuniverse containing the φ field coupled with a secondscalar field ϕ and let us write down the equations ofmotion of the φ and ϕ fields. In what follows we as-sume that both the φ and ϕ particles are relativistic.The φ ( t, x ) field can be divided into a classical back-ground configuration φ c ( t ) plus a perturbation δφ ( t, x )corresponding to the quantum configuration of the φ field ( φ particles), in such a way that we can write φ ( t, x ) = φ c ( t ) + δφ ( t, x ). We choose φ ( t, x ) and δφ ( t, x )as independent variables, stressing the fact that when δφ ( t, x ) → φ ( t, x ) = φ c ( t ). The dynamic of the φ field is then deduced from its lagrangian density L = 12 ∂ µ φ∂ µ φ + 12 ∂ µ ϕ∂ µ ϕ − V T ( φ, ϕ ) (A1)where V T = v ( φ ) + B ( ϕ ) + v int ( φ, ϕ ), v ( φ ) and B ( ϕ ) arethe classical potentials of the φ and ϕ fields respectivelyand v int ( φ, ϕ ) = − L int ( φ, ϕ ) where L int ( φ, ϕ ) is the in-teraction lagrangian of the scalar fields φ and ϕ . Using ∇ φ ( t, x ) = ∇ δφ ( t, x ) one has¨ φ + 3 H ˙ φ − ∇ δφa + ∂ φ V T ( φ, ϕ ) = 0 (A2)The perturbation δφ evolves as a scalar field with mass m φ = ∂ φ v ( φ ). The corresponding quantum operator δ ˆ φ has the following expression δ ˆ φ = Z d k (2 π ) √ E k h a k f ( t ) e − i~k~x + a † k f ∗ ( t ) e i~k~x i (A3)where k and E k are respectively the wave number andthe energy of the φ particles with E k = | ~k | /a ( t ) + m φ [25],[29]. The physical momentum is ~p = ~k/a ( t ) and the φ particles are relativistic so E k = | ~k | /a ( t ) = | ~p | . For sim-plicity we assume that all the φ and ϕ particles have thesame energy E φ and E ϕ respectively. For example thisassumption is valid if the quantum particles are thermal-ized. In this case the phase space distributions are theBose-Einstein distributions f φ ( E ) = 1 / ( e E/T φ −
1) and f ϕ ( E ) = 1 / ( e E/T ϕ −
1) and one can take the energies of3the φ and ϕ particles as the mean values E φ = ¯ r T φ and E ϕ = ¯ r T ϕ with ¯ r ≃ .
7. Moreover, in the case that the φ and ϕ fields are in thermal equilibrium one has E φ = E ϕ .Therefore we estimate δφ as the average of δ ˆ φ on thequantum state | N, E φ > containing N φ particles withenergy E φ , i.e. < N, E φ | : δ ˆ φ : | N, E φ > = n φ E φ , wherethe :: stands for normal ordering of creation and destruc-tion operators. We have then δφ = n φ /E φ , where n φ isthe density number of φ particles. Thus we can write − ∇ δφa = | ~p | δφ = E / φ √ n φ (A4)where we have used E k = | ~p | = | ~k | /a for relativisticparticles. Substituting this expression in eq.(A2) we have¨ φ + 3 H ˙ φ + E / φ √ n φ + ∂ φ v ( φ ) + ∂ φ v int ( φ, ϕ ) = 0 . (A5)Repeating the same considerations for the ϕ field one has¨ ϕ + 3 H ˙ ϕ + E / ϕ √ n ϕ + ∂ ϕ B ( ϕ ) + ∂ ϕ v int ( φ, ϕ ) = 0 . (A6)The dynamics of the quantum particles is governed by theBoltzmann equations. If f φ ( E, t ) is the phase space dis-tribution of the φ particles, one has n φ = R f φ ( E, t ) d p (2 π ) , ρ φ = R E f φ ( E, t ) d p (2 π ) and p φ = R | ~p | E f φ ( E, t ) d p (2 π ) .In the same way one has n ϕ = R f ϕ ( E, t ) d p (2 π ) , ρ ϕ = R E f ϕ ( E, t ) d p (2 π ) and p ϕ = R | ~p | E f ϕ ( E, t ) d p (2 π ) where f ϕ ( E, t ) is the phase space distribution of the φ parti-cles. The evolution of the phase space density f φ ( E, t )is governed by the Boltzmann equation ˆ L [ f φ ] = ˆ C [ f φ ],where ˆ L is the Liouville operator and in a FRM metricis ˆ L [ f φ ( E, t )] =
E ∂ t f φ ( E, t ) − H | ~p | ∂ E f φ ( E, t ) and ˆ C is the collision operator (see [28]). If the φ particles arerelativistic one has˙ n φ + 3 Hn φ = Z ˆ C [ f φ ( E, t )] d p (2 π ) E + A (A7)where for a process a + a + ... + a n ↔ b + b + ... + b l ,with a n ( b l ) initial (final) particles, one has R C [ f φ ( E, t )] d p (2 π ) E = − Z d Π a ...d Π a n d Π b ...d Π b l × (2 π ) | M ab | δ (Σ ni P a i − Σ lj P b j ) (A8) × [ f a ( E, t ) ...f a n ( E, t ) − f b ( E, t ) ...f b l ( E, t )]with d Π ≡ g d p/ (2 π ) E , g are the internal degrees offreedom and | M ab | the transition scattering matrix ofthe process. Eq.(A8) is valid in absence of Bose conden-sation of Fermi degeneracy when 1 ± f i ( E, t ) ≃ n ϕ + 3 Hn ϕ = Z ˆ C [ f ϕ ( E, t )] d p (2 π ) E + Q (A9)The terms A and Q introduced in eqs.(A7) and (A9) arenecessary for the energy conservation as we will discussbelow eqs.(A18) and (A19). Let us consider the quadratic interactions L int = g φ ϕ or L int = g φ ϕ . In that case the φ field isgenerated via the 2 ↔ ϕ + ϕ ↔ φ + φ or ϕ + ϕ ↔ ϕ + φ respectively. For simplicity we as-sume that the phase space distribution f ϕ ( E, t ) of the ϕ particles is piked around the mean energy E ϕ of the ϕ particles. Of course this is true in the case of ther-malized particles. Therefore we take all the ϕ particleswith the same energy E ϕ and one has R f ϕ ( E, t ) d Π ϕ = R f ϕ ( E, t ) d p ϕ E ϕ (2 π ) ≃ E ϕ R f ϕ ( E, t ) d p ϕ (2 π ) = n ϕ E ϕ . More-over, from energy conservation it follows that the φ par-ticles are produced with the same energy of the ϕ par-ticles and we take E φ = E ϕ = E , that is also validwhen the φ and ϕ fields thermalize. Therefore the en-ergy distribution of the φ particles is piked around themean energy E φ and one also has R f φ ( E, t ) d Π φ = n φ E φ .The transition rates for the considered processes areΓ ϕϕ → φφ = Γ ϕϕ → φϕ = Γ φϕ → ϕϕ = h σ gen v i n ϕ ≡ Γ gen andΓ φφ → ϕϕ = h σ gen v i n φ , where σ gen = g / πE is thecross section for a 2 ↔ v is the relative velocity [25]. Considering the process ϕϕ ↔ ϕφ one has R C [ f φ ( E, t )] d p (2 π ) E = − R C [ f ϕ ( E, t )] d p (2 π ) E == Γ ϕϕ → φϕ n ϕ − Γ φϕ → ϕϕ n ϕ n φ = h σ gen v i n ϕ ( n ϕ − n φ )(A10)and for the process ϕϕ ↔ φφ one has R C [ f φ ( E, t )] d p (2 π ) E = − R C [ f ϕ ( E, t )] d p (2 π ) E == Γ ϕϕ → φφ n ϕ − Γ φφ → ϕϕ n φ = h σ gen v i ( n ϕ − n φ ) == h σ gen v i ( n ϕ + n φ )( n ϕ − n φ ) (A11)We can write eqs.(A10) and (A11) in a compact form as Z C [ f φ ( E, t )] d p (2 π ) E = − Z C [ f ϕ ( E, t )] d p (2 π ) E == ˜Γ ( n ϕ − n φ ) (A12)where we have defined ˜Γ ≡ h σ gen v i n ϕ for the process ϕϕ ↔ ϕφ and ˜Γ ≡ h σ gen v i ( n ϕ + n φ ) for the process ϕϕ ↔ φφ . Note that ˜Γ is not necessarily a transitionrate, but it accounts for the whole contribution of thetwo processes ϕϕ → ϕφ and ϕφ → ϕϕ in one case and ϕϕ → φφ and φφ → ϕϕ in the other case.Therefore eqs.(A7) and (A9) now read˙ n φ + 3 Hn φ = ˜Γ ( n ϕ − n φ ) + A (A13)˙ n ϕ + 3 Hn ϕ = − ˜Γ ( n ϕ − n φ ) + Q. (A14)The energy density and pressure of the system are ρ T = ˙ φ ϕ V T ( φ, ϕ ) + E φ n φ E ϕ n ϕ p T = ˙ φ ϕ − V T ( φ, ϕ ) + E φ n φ E ϕ n ϕ V T ( φ, ϕ ) = v ( φ ) + B ( ϕ ) + v int ( φ,ϕ ) . Note that theterms proportional to the number densities in eqs.(A15)4and (A16) comes from the terms |∇ δφ | /a ( t ) = E φ δφ = E φ n φ and |∇ δϕ | /a ( t ) = E ϕ δϕ = E ϕ n ϕ Therefore eqs.(A15) and (A16), together witheqs.(A5),(A6),(A13) and (A14) give the energy conser-vation in the form˙ ρ T + 3 H ( ρ T + p T ) = − E / φ √ n φ ˙ φ + A E φ −− E / ϕ √ n ϕ ˙ ϕ + Q E ϕ φ and ϕ exists. For example,if the only φ field exists one has v int ( φ, ϕ ) = 0, n ϕ = 0and Q = 0 and form eq.(A17) one has A = 2 p E φ n φ ˙ φ (A18)and in the case in which the only ϕ field exists one obtains Q = 2 p E ϕ n ϕ ˙ ϕ. (A19)Then eqs.(A5),(A6),(A13) and (A14) now read¨ φ + 3 H ˙ φ + E / φ √ n φ + ∂ φ v ( φ ) + ∂ φ v int ( φ, ϕ ) = 0¨ ϕ + 3 H ˙ ϕ + E / ϕ √ n ϕ + ∂ ϕ B ( ϕ ) + ∂ ϕ v int ( φ, ϕ ) = 0˙ n φ + 3 Hn φ = ˜Γ ( n ϕ − n φ ) + 2 p E φ n φ ˙ φ ˙ n ϕ + 3 Hn ϕ = − ˜Γ ( n ϕ − n φ ) + 2 p E ϕ n ϕ ˙ ϕ (A20)The system in eqs.(A20) describe the dynamics of twocoupled relativistic scalar fields with the same energy E φ = E ϕ = E . Note that this system includes the quan-tum interaction between the quantum φ and ϕ particlestrough the term ˜Γ ( n ϕ − n φ ) in the last two equations inthe system (A20). Moreover the density numbers n φ and n ϕ generates a source term for the corresponding fields φ and ϕ in first two equations of the system (A20) and thissource term is responsible of the generation of the classi-cal potential v ( φ ) during the φ generation. It is useful todivide the energy density ρ T in two terms ρ T = ρ φ + ρ ϕ with ρ φ = ρ φ + ρ φ , ρ φ = ˙ φ v ( φ ) ρ φ = E φ n φ ρ ϕ = ρ ϕ + ρ ϕ , ρ ϕ = ˙ ϕ v int ( φ, ϕ ) + B ( ϕ ) , ρ ϕ = E ϕ n ϕ . (A22)In the same way we can write the pressure of the systemas p T = p φ + p ϕ with p φ = p φ + p φ , p φ = ˙ φ − v ( φ ) , p φ = E φ n φ p ϕ = p ϕ + p ϕ , p ϕ = ˙ ϕ − v int ( φ, ϕ ) − B ( ϕ ) , p ϕ = E ϕ n ϕ . (A24) - - Φ (cid:144) f1v H Φ L(cid:144) V I - - FIG. 9: Plot of the potential v ( φ ). It is easy to cheek that these quantities verifies the fol-lowing evolutionary equations˙ ρ φ + 3 H ( ρ φ + p φ ) = − ˙ φ ∂ φ v int − ˙ φE / φ √ n φ ˙ ρ φ + 3 H ( ρ φ + p φ ) = E φ ˜Γ( n ϕ − n φ ) + ˙ φE / φ √ n φ ˙ ρ φ + 3 H ( ρ φ + p φ ) = E φ ˜Γ( n ϕ − n φ ) − ˙ φ ∂ φ v int (A25)and˙ ρ ϕ + 3 H ( ρ ϕ + p ϕ ) = ˙ φ ∂ φ v int − ˙ ϕE / ϕ √ n ϕ ˙ ρ ϕ + 3 H ( ρ ϕ + p ϕ ) = − E ϕ ˜Γ( n ϕ − n φ ) + ˙ ϕE / ϕ √ n ϕ ˙ ρ ϕ + 3 H ( ρ ϕ + p ϕ ) = − E ϕ ˜Γ( n ϕ − n φ ) + ˙ φ ∂ φ v int (A26)Notice that the sum of the first two equations of thesystems in eqs.(A25) or (A26) gives the last equation, re-spectively, while the sum of the last equation in eqs.(A25)and (A26) gives the total energy evolution in eq.(A17) asit should due to energy-momentum conservation. APPENDIX B: THE POTENTIAL v ( φ ) Let us study in more detail the properties of the po-tential v ( φ ) = V I (cid:18) − π arctan φf (cid:19) (B1) v ′ ( φ ) = − V I π f
11 + ( φ/f ) (B2) m φ ≡ v ′′ ( φ ) = V I π f φ (1 + ( φ/f ) ) (B3)where E I = V / I and f are parameters with mass di-mensions. The φ mass is maximum at | φ | ≃ f with m φ ≃ V I /f while v ′ is always negative. The asymp-totic expansion of the potential in eq.(B1) for | φ/f | ≫ v ( φ ) ≃ ( V I (cid:16) fπφ (cid:17) for φ < − f V I fπφ for φ > f (B4)One can easily cheek that the slow roll conditions | v ′ ( φ ) /v ( φ ) | ≪ | v ′′ ( φ ) /v ( φ ) | ≪ φ < − (2 f /π ) / and φ > √
2. There-fore the region φ < − (2 f /π ) / is associated with in-flation at energies E = √ ρ φ ≃ E I . Dark energy is as-sociated to the region φ > √ E = √ ρ φ ≃ E I ( f /φ o ) / ≃ E DE ≃ × − eV where we have cho-sen φ o ≃ √ φ . We canfix the parameters E I , f by imposing δρ/ρ = 5 . × − and from dark energy density ρ DE = 3 H o Ω DE , this gives E I ≃ T eV and f ≃ − eV , which gives a presenttime mass m φ ( t o ) ≃ − eV which is the standard valuefor quintessence field.We can express the value of φ at the generation time,i.e. φ gen , in terms of φ o as φ gen = v ( φ DE ) v ( φ gen ) φ o = ( E DE E gen ) φ o and if one choose E gen ≃ eV one has φ gen ≃ − φ o .Therefore we can also compare the value of the mass m φ at the reheating, generation and present times. At re-heating one has φ ≃ f and m φ ( t RH ) ≃ p V I /f , at gen-eration time one has φ gen ≫ f and m φ ( t gen ) ≃ q V I fπφ gen and at present time time one has φ o ≫ f and m φ ( t o ) ≃ q V I fπφ o . One has m φ ( t RH ) /m φ ( t gen ) ≃ p φ gen /f ≫ m φ at the reheating and gen-eration times, and this is why the reheating is obtainedvia the decay process φ → ϕ + ϕ + ϕ of massive φ particlesinto relativistic ϕ particles, and the φ is generated at latetimes via a 2 ↔ N = ln a f a i = − R φ f φ i v ( φ ) v ′ ( φ ) dφ . During inflationone has φ ≤ − (2 f /π ) / < f , therefore we can use thesecond asymptotic expansion in eq.(B4) to write N = − Z φ f φ i πφ f = π f (cid:0) φ f − φ i (cid:1) (B5)Therefore if one require a minimum number N m ofe-folds during inflation, inflation must start at φ i ≤ (cid:16) φ f − f N m /π (cid:17) / ≃ − ( f /π ) / ( N m + 2) / for φ f ≃− (2 f /π ) / . Note that a reasonable number of e-folds N m ≃ −
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