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Quantum Generation of Dark Energy

We present a type of dark energy models where the particles of dark energy phi are dynamically produced via a quantum transition at very low energies. The scale where the transition takes places depends on the strength g of the interaction between phi and a relativistic field varphi. We show that a g \simeq 10^{-12} gives a generation scale E_gen simeq eV with a cross section sigma simeq 1 pb close to the WIMPs cross section sigma_w simeq pb at decoupling. The number density n_phi of the \phi particles are a source term in the eq. of motion of phi and it generates the scalar potential v(phi) responsible for the late time acceleration of our universe. Since the appearance of phi may be at very low scales, close to present time, the cosmological coincidence problem can be explained simply due to the size of the coupling constant. We unify dark energy with inflation in terms of a single scalar field phi, and we use the same potential v for inflation and dark energy. However, after inflation phi decays completely and reheats the universe at a scale E_RH propto h^2 m_Pl, where h is the coupling between the SM particles and varphi. The field phi disappears from the spectrum during, from reheating until its re-generation, and therefore it does not interfere with the standard decelerating radiation/matter cosmological model allowing for a successful unification scheme. The same interaction term that gives rise to the inflaton decay accounts for the late time re-generation of phi. We present a simple model where the strength of the g and h couplings are set by the inflation scale E_I with g=h^2 propto E_I/m_Pl giving a reheating scale E_RH propto E_I=100TeV and phi-generation scale E_gen propto E_I^2/m_pl=eV<<E_RH.

We present a type of dark energy models where the particles of dark energy φ are dynamically produced via a quantum transition at very low energies. The scale where the transition takes places depends on the strength g of the interaction between φ and a relativistic field ϕ. We show that a g ≃ 10 −12 gives a generation scale Egen ≃ 1 eV with a cross section σ ≃ 1 pb close to the WIMPs cross section σw ≃ pb at decoupling. The number density n φ of the φ particles is a source term in the equation of motion of φ that generates the scalar potential v(φ) responsible for the late time acceleration of our universe. Since the appearance of φ may be at very low scales, close to present time, the cosmological coincidence problem can be explained simply due to the size of the coupling constant. In this context it is natural to unify dark energy with inflation in terms of a single scalar field 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 ERH ∝ h 2 m P l , where h is the coupling between the SM particles and ϕ. The field φ disappears from the spectrum during most of the time, from reheating until its re-generation at late times, and therefore it does not interfere with the standard decelerating radiation/matter cosmological model allowing for a successful unification scheme. We show that the same interaction term that gives rise to the inflaton decay accounts for the late time re-generation of the φ field giving rise to dark energy. We present a simple model where the strength of the g and h couplings are set by the inflation scale EI with g = h 2 ∝ EI/m P l giving a reheating scale ERH ∝ EI and φ-generation scale Egen ∝ E 2 I /m pl ≪ ERH . With this identification we reduce the number of parameters and the appearance of dark energy is then given in terms of the inflation scale EI .

I. INTRODUCTION.
The nature and dynamics of Dark Energy "DE", which gives the accelerating expansion of the universe at present time, is now days one of the most interesting and stimulating fields of physics. It was discovered more than ten years ago [1] and it has been confirmed by further cosmological observations, being now one of the most robust conjecture in modern physics. In fact the acceleration of the present universe has many experimental proofs such as the CMB temperature and fluctuations [4], in the matter power spectrum measured by galaxy surveys [5,6] and in type Ia supoernovae [7,8,9]. The most popular model is the so called ΛCDM model, in which a cosmological constant and some amount of cold dark matter are included "by hand". Despite its extraordinary consistence with observations, ΛCDM is an effective model that leaves many unsolved theoretical question. In fact the existence of the cosmological constant and its order of magnitude have no theoretical justification in ΛCDM. The cosmological coincidence problem, that is why the universe is starting to accelerate right now, is also unsolved. Introducing a cosmological constant at the initial stages of the standard cosmological model is specially troublesome since one has to fine tune its value to one parte in 10 120 . This problem can be ameliorated if we understand when and how dark energy appears in the universe and this is the main motivation of our present work [17]. However, our approach will also help us to unify dark energy with inflation.
An attractive dark energy alternative to the ΛCDM model consists in introducing a "quintessence" scalar field φ that generates the accelerating expansion [13] [14] of the universe due to is dynamics. The dynamics is fixed by its potential v(φ) and it is possible to choose potentials that lead to a late time acceleration of the universe [14]. This scalar field can be a fundamental or composite particle as for example bound states [27]. In the second case, the bound quintessence fields are scalar fields composed of fundamental fermions, such as meson fields, and can be generated at low energies as a consequence of a low phase transition scale due to a strong gauge coupling constant [27]. This allows to understand why DE appears at such late times. On the other hand, in the former case the appearance of fundamental scalar field is right at the beginning of the reheated universe and the acceleration of the universe takes place at a much later time due to the classical evolution of the quintessence field φ. The huge difference in scales between the reheating and dark energy scales requires a fine tuning in the choice of the potential.
Here, we present an interesting alternative, namely that the emergence of the fundamental quintessence particles φ is originated from a quantum transition taking place at low energies, e.g. as low as eV [17]. The scale where this transition takes places depends on the strength of the interaction between φ and a relativistic field ϕ and it is dynamically determined by the ratio Γ/H where Γ is the transition rate and H the Hubble constant. A value of the coupling g ≃ 10 −12 gives a generation scale E gen ≃ 1 eV with a cross section σ = g 2 /32πE gen ≃ 1 pb close to the WIMPs cross section σ w ≃ O(pb) at decoupling [16]. The subsequent acceleration of the universe is due to the classical evolution of φ due to the scalar potential v(φ). Our quantum generation scheme does not aim to derive the potential v(φ) but to understand why dark energy dominates at such a late time. Clearly, by closing the gap between the energy today E o , where the subscript o always refers to present time quantities, and that of φ production E gen , we do not require a fine tuning of the parameters in v(φ). Since the appearance of φ may be at such low scales this offers a new interpretation and solution to the cosmological coincidence problem in terms of the size of the coupling constant g.
Furthermore, this late time production of the φ particles allows to implement in a natural way a dark energyinflaton unification scheme. In this scenario, after inflation the field φ decays completely and reheats the universe with standard model particles. The universe expands then in a decelerating way dominated first by radiation and later by matter. At low energies the same interaction term that gives rise to the inflaton decay accounts for the quantum re-generation of the φ field giving rise to dark energy. In general, it is not complicated to choose a scalar potential such that the universe accelerates in two different regions, at early inflation and dark energy epochs, as in quintessential models [15]. However, the universe requires to be most of the time dominated first by standard model relativistic particles and later by matter. The reheating of the universe and the long period of decelerating phase are usually not taken into account in inflation-dark energy unification models and these features are essential in the standard cosmological Big Bang model. In our case, the inflaton-dark energy field is completely absent during most of the time (from reheating until re-generation) and therefore it does not interfere with the standard cosmological model. We will exemplify our inflation-dark energy unified scheme with a simple model. The scalar potential v(φ) will have only two parameters fixed by the conditions to give the correct density perturbations δρ/ρ and the present time dark energy scale. The two couplings h, g, which give the strength of reheating process with SM particles and the φ re-generation process at low energies, respectively, are free parameters but we may take them as g = h 2 ∝ E I /m pl , where E I is the scale of inflation and it is one of the parameters of v(φ). Therefore, starting with four free parameters we can reduce the number to only two and these two are fixed by observations. This gives a reheating scale E RH ∝ E I and φ-generation scale E gen ∝ E 2 I /m pl ≪ E RH . The paper is organized as follows: in section II we give an overview of the late time quantum generation of the quintessence field φ and its possible unification with the inflaton field. In section III we present the dark energy quantum generation in detail. In section III we show how to unify inflation and dark energy in terms of a sin-gle scalar field in the context of our dark energy quantum generation process and we present a simple model. Finally, in section V we present the main phenomenological consequences of our model while in section VI we resume and conclude.

II. OVERVIEW
To avoid any future confusion we state the terminology used in this work. We define as usual the energy of the universe at any given time as E ≡ ρ 1/4 or for a i-species i . For particles we take their energy E a as E a = (p 2 a + m 2 a ) 1/2 , where p a , m a are the momentum and mass of the a-particle. If the particles are relativistic and in thermal equilibrium then one can define a temperature T with an energy density ρ a = π 2 30 g a T 4 and number density n a = ζ(3)/π 2 g a T 3 , with g a the relativistic degrees of freedom and ζ(3) ≃ 1.2 is the Riemann zeta function of 3. The energy distribution of thermalized particles is strongly piked around the mean valueĒ =r T withr ≡ (ρ/nT ) = π 4 /30ζ(3) ≃ 2.7 and we identify the energy of the particle E a with the average energy E a =Ē a =rT . We will loosely refer then to either the temperature or the energy of the (thermalized) relativistic particles and we will work in natural units with m 2 pl ≡ 1/8πG = 1 but we will reintroduce the correct energy dimensions when convenient.
We will first present the quantum generation process of dark energy field φ and we will later discuss the possibility to unify it with the inflaton field. As mentioned in the introduction, the main goal of this paper is to describe how the dark energy field φ may be generated at a very late time via a quantum transition process and how the scalar potential v(φ), responsible for the late time acceleration, is generated. Once v(φ) is produced, the classical equation of motion gives the dynamics of φ, and choosing an appropriated flat v(φ) at low energies ensures that the universe enters an accelerating epoch close to present time. This work does not aim to derive the potential v(φ) but to understand why dark energy dominates at such a late time. Of course, by closing the gap between the energy today E o and that of φ production E gen there is a no longer a fine tuning of the parameters in v(φ).
We describe now the generation process. We take a universe filled with standard model SM particles and an extra 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 with neutrinos) but we require that at time of φ-generation ρ ϕ (E gen ) > ρ DE (E o ). In the context of inflation-dark energy unification the ϕ must have been in thermal equilibrium with the SM and therefore T ϕ ≃ T γ with Ω ϕ ≃ (g ϕ /g SM rel )Ω SM rel , where g ϕ and g SM rel are the relativistic degrees of freedom of ϕ and the SM respectively and Ω SM rel is the density of SM relativistic particles. Since we 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 3 or L int = g φ 2 ϕ 2 . In order to produce the φ particles at low energies we require that the interaction rate Γ of the quantum process must be initially smaller than the Hubble parameter H and the ratio Γ/H should increases with the expansion of the universe. This will happen if Γ decreases less rapidly than H. For example if we have a 2 ↔ 2 process of relativistic particles the transition rate scale as Γ ∝ g 2 T ϕ and H ∝ T 2 ϕ giving Γ/H ≡ T gen /T ϕ ∝ g 2 /T ϕ , where we have taken T ϕ ≃ T γ . The φ particles production starts then for temperatures T below T gen ∝ g 2 , where Γ/H > 1. The energy scale at which the φ is generated is fixed by the coupling constant g, so the coincidence problem is explained in terms of the strength of the interactions of the φ field. In particular one can generate the φ field at low energies, e.g. E gen ≃ 1 eV with g ≃ 10 −12 giving a fine structure constant α = g 2 /4π ≃ 10 −25 and a cross section σ ≃ g 2 /32πE 2 gen ≃ 1 pb. The value of σ is quite close to cross section of WIMP dark matter with nucleons σ w ≃ pb [16]. Since we can choose the coupling g in such a way that the φ is generated at low energies close to present time, this offers a new interpretation of the cosmological coincidence problem: dark energy domination starts at such small energies because of the size of the coupling constant g. For a φgeneration energy of E gen ≃ 1 eV the ratio in energy densities from the appearance of the fundamental field φ to present time is ρ DE /ρ gen = 10 −12 and should be compared to a case where φ is present at the Planck time ρ pl with ρ DE /ρ pl = 10 −124 , giving a difference in ratios of 112 orders of magnitude. Therefore, the amount of fine tuning in the parameters of the dark energy potential v(φ) is much less sever in our case than in a standard quintessence dark energy model. The production of the φ particles gives rise to the scalar potential v(φ) and the field φ will then evolve classically given by its equation of motion. Of course, we still need to choose v(φ) appropriately 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 the universe is due to the form of the potential v(φ) but the energy E gen at which the φ is generated is fixed by the coupling constant g. An interesting possibility is to unify inflation and dark energy using the same scalar field φ in our dark energy quantum generation picture [17]. We assume that φ has a potential v(φ) which gives an early inflation at E I , as in standard inflationary models [3]. In this case we use the coupling L int = g φ ϕ 3 to allow the φ field to decay into ϕ particles after inflation, via the process φ → ϕ + ϕ + ϕ. This decay process is very efficient [21] and the φ decays completely disappearing entirely from the spectrum of the universe. In order to produce SM particles, the ϕ is coupled with the SM via the standard interactions L int = h ϕ 2 χ 2 or L int = √ h ϕψψ with χ, ψ SM scalar or fermions, respectively. Reheating of the early universe takes place via a 2 ↔ 2 process ϕ + ϕ ↔ χ + χ or ϕ + ϕ ↔ψ + ψ with a transition . We see how at NRH the φ field disappears and the universe is reheated. At Ngen the φ field is re-generated and the ϕ field disappears. We see how the φ field is rapidly diluted and than maintains nearly constant at ρ φ ≃ ρDE. At late times close to present time, i.e. at N ≃ No, the φ field starts to dominate and inflate the universe. rate Γ RH , for Γ RH /H ≡ T RH /T ϕ ∝ h 2 /T ϕ > 1 at an energy scale E RH ∝ h 2 . Because of its couplings, the SM particles and ϕ are in thermal equilibrium at the time of reheating, and as long as ϕ remains relativistic one has Ω ϕ ≃ Ω SM g ϕ /g SM . After reheating ρ ϕ evolves as radiation and eventually it will re-generate the φ field, at an energy E gen ∝ g 2 . The re-generation process of φ can be produced with the same interaction L int = g φ ϕ 3 as the inflaton decay at E I . A main difference in the transition processes at E I and at E gen is the size of the mass of φ. It varies quite significantly and we should have 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) must be chosen to be flat at high energy E I , to give inflation, and at low energy E o to accelerate the universe at present time. The reheating energy E RH ∝ h 2 and the φ generation energy E gen ∝ g 2 are fixed by the coupling constants h and g independently of the potential v(φ). In the model presented in section IV we set g = h 2 ∝ E I /m pl reducing the number of parameters and connecting the inflation scale to the φ-generation and to the reheating scales, E gen ∝ E 2 I /m pl , E RH ∝ E I . As mentioned in the introduction, any unification of inflation and dark energy, such as quintessential models, need to explain how the universe is reheated with SM particles and must account for the long period of decelerating universe. In our model we are able to explain both features in a consistent way.
We give a schematic picture of the inflation-dark energy unified model showing in fig.(1) the evolution of the logarithms of the energy densities of the φ and ϕ fields, radiation and matter. We start with a φ dominated universe and then at N RH , where N ≡ ln(a) with a(t) the scale factor, the φ field decays and disappears, reheating the universe. After reheating the universe is radiation dominated, as in the standard cosmological model, and radiation includes the SM relativistic degrees of freedom plus the extra relativistic field ϕ. At N gen the φ field is re-generated with ρ φ f ≃ ρ ϕi , where ρ φ f is the value of ρ φ at the end of the generation process and ρ ϕi is the value of ρ ϕ at the begin of the φ generation. At the same time the ϕ field decays and ρ rel diminishes. After the generation, ρ φ is rapidly diluted and subsequently maintains nearly constant at ρ φ ≃ ρ DE and it starts to dominate close to present times, i.e. at N ≃ N o , inflating the universe.

III. THE φ GENERATION.
In this section we describe the quantum generation of the quintessence scalar field φ and the subsequent appearance of the dark energy behavior of the late universe due to the classical evolution of φ. In what follows we assume a universe that consists of the particles of the standard model "SM" together with a massless scalar field ϕ and a quintessence scalar field φ coupled together via a four particle interaction, as for example L int = g φ ϕ 3 or L int = g φ 2 ϕ 2 . The ϕ field is also coupled with the SM through the interactions √ h ϕψψ or h ϕ 2 χ 2 , where ψ and χ are some SM fermions or scalars respectively. We assume that at temperatures above 1 T eV all the SM particles and ϕ are relativistic and ϕ is in thermal equilibrium with T ϕ = T rad . As long as ϕ is relativistic where Ω SM rad is the SM radiation density and g ϕ , g SM r are the relativistic degree of freedom of ϕ and SM, respectively. We also suppose that at temperatures T ≫ 1eV the universe does not contain 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 quantum transition ϕ + ϕ → ϕ + φ or ϕ + ϕ → φ + φ with a transition rate Γ ϕ+ϕ→ϕ+φ = Γ ϕ+ϕ→φ+φ = σ gen v n ϕ ≡ Γ gen , where σ gen = g 2 /32πE 2 is the cross section for a 2 ↔ 2 relativistic particle process, v is the relative velocity and n ϕ is the density number of the ϕ particles [25]. This take place at energies below E gen ≃ c gen g 2 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 ϕi , where Ω ϕi is the ϕ density at the begin of the φ particles production, and also Ω ϕ f ≪ Ω ϕi . The production of relativistic particles φ becomes a source term for the generation of the scalar potential v(φ). Once v(φ) has been produced the classical equation of motion gives the evolution of φ. The acceleration of the universe is then due to the form of scalar potential v(φ).
We consider a system composed by two scalar fields φ and ϕ plus SM, in a flat FRW metric, with a density lagrangian were L is the SM lagrangian and , v(φ) and B(ϕ) are the classical potentials 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 evolution of the two scalar fields and it also originates the quantum transitions between φ and ϕ particles that we use to generate the φ field at late times. We divide the φ(t, x) field into a classical background configuration φ c (t) plus a 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 perturbation δφ(t, x) are usually taken as independent variables. However we choose to take as the two independent variables φ(t, x) and δφ(t, x). Moreover we express δφ(t, x) in terms of the number density n φ of φ particles via the relation n φ = E φ δφ 2 , see appendix A. We stress the fact that when n φ = 0 one also has δφ(t, x) = 0 that implies φ(t, x) = φ c (t), so the scalar field φ is in its classical configuration. Following the same argument we write ϕ(t, x) = ϕ c (t) + δϕ(t, x) and describe the ϕ field through the variables ϕ(t, x) and n ϕ = E ϕ δϕ 2 . Let us concentrate on the process of φ particles production in the case that both φ and ϕ particles are relativistic and the number density of φ particles n φ is initially zero. We use a four particles interaction, such as L int = g φ ϕ 3 or L int = g φ 2 ϕ 2 and we consider a 2 ↔ 2 the processes, e.g. ϕ + ϕ ↔ ϕ + φ or ϕ + ϕ ↔ φ + φ. As mentioned previously, the ϕ field is initially thermalized and it has a phase space distribution given by the Bose-Einstein distribution f ϕ (E) = 1/(e E/Tϕ − 1), so we can take all the ϕ particles with the same energy E ϕ =rT ϕ ≃ T rad . Since the 2 ↔ 2 interaction process conserves energy the φ particles become in thermal equilibrium with ϕ and they has the same energy E φ = E ϕ = E. Moreover the two scalar fields φ and ϕ are relativistic so their energy scales as E = E i /a(t). Before the φ generation the density number of ϕ particles evolves as n ϕ = ζ(3)T 3 ϕ /π 2 , so the transition rate for the considered 2 ↔ 2 processes of relativistic particles is expressed in terms of the energy of the quantum particles as Γ gen = σ gen v n ϕ = (ζ(3)/32π 3r3 ) g 2 E and therefore it scales as Γ gen ∼ 1/a(t). As discussed in section II, in order to have a late time dark energy generation, it is fundamental to have a ratio Γ gen /H that grows up with the expansion of the universe. If we are in radiation domination H ∼ a(t) −2 in in matter domination H ∼ a(t) −3/2 . This means that the Γ gen should decrease more slowly than 1/a(t) 2 during radiation domination and than 1/a(t) 3/2 during matter domination, as in our case. Taking a flat FRW metric, the evolution of our system is described by the following equations (see eqs.(A20) in appendix A for details) together with the Friedman equation H 2 = 1 3 ρ T , where ρ T = ρ φ + ρ ϕ + ρ rad + ρ mat , and ρ rad and ρ mat are the energy densities of radiation and matter respectively. Moreover v(φ) and B(ϕ) are the classical potentials of the φ and ϕ fields respectively and v int (φ, ϕ) is the classical interaction potential. Eqs. (2) and (3) are the Boltzmann equations that govern the dynamic of the number densities n φ and n ϕ . They take into account the quantum transition between φ and ϕ particles thanks to the terms proportional toΓ. The quantityΓ in eqs. (2) and (3) is given byΓ ≡ σ gen v n ϕ for the process ϕϕ ↔ ϕφ andΓ ≡ σ gen v (n ϕ + n φ ) for the process ϕϕ ↔ φφ.Γ takes into account the contribution of the two processes ϕϕ → ϕφ and its inverse ϕφ → ϕϕ in one case and ϕϕ → φφ and φφ → ϕϕ in the other case, since one has At the begin of the φ generation one has n φ = 0 and thereforeΓ = Γ gen in both processes. Moreover the two terms 2 E n φφ and 2 E n ϕφ contained in eqs. (2) and (3) respectively, couple the number densities n φ and n ϕ with the scalars φ and ϕ respectively. Eqs. (4) and (5) are the equations of motion of the lagrangian in eq.(1) for the scalar fields φ(t, x) and ϕ(t, x). These equations contains the terms E 3/2 √ n φ and E 3/2 √ n ϕ that couple the two scalar fields φ and ϕ with their number densities n φ and n ϕ and become the source terms for generating the scalar potential v(φ). Note that the terms E 3/2 √ n φ and E 3/2 √ n φ contained in eqs. (4) and (5) represent the spatial derivatives of the two scalar fields, in fact they come out from the relations − ∇ 2 φ a(t) = E 3/2 √ n φ and − ∇ 2 ϕ a(t) = E 3/2 √ n ϕ , see eq.(A4) in appendix A. Of course eqs.(2)-(5) conserve energy-momentum. The energy density and pressure of the φ field are ρ φ = ρ 1φ + ρ 2φ and p φ = p 1φ + p 2φ , where ρ 1φ ≡φ In the same way the energy density and the pressure of the ϕ field are ρ ϕ = ρ 1ϕ + ρ 2ϕ and p ϕ = p 1ϕ + p 2ϕ , where ρ 1ϕ ≡φ Therefore eqs. (6) give the energy-momentum conservation lawρ T + 3H(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 purposes we will not take into account the expansion of the universe, i.e. we will take H = 0, however we show below that the growth of the scale factor is very small (of the order of H/Γ gen ) [40]. Moreover in this presentation we will not consider the contribution from classical interactions and we also take B(ϕ) = 0. We will also work withΓ constant. ObviouslyΓ depends both on the energy of the decaying particles and on the number densities, but we use this approximation to write an analytical solution of eqs.(2)- (5). Of course comparison of the approximated analytical solution with the numerical solution shows a complete agreement and the classical interaction v int (φ, ϕ) and expansion rate do not play a significant role. Under these approximations the system of eqs. (2)(3)(4)(5) reduce to the following equationṡ where E i is the energy of the quantum particles at the begin of the φ generation. The initial conditions on the Now, the first stage starts with (n ϕ − n φ )Γ ≫ −2 E i n φφ in eq.(8), which is clearly verified at the begin since n φi = 0, and the approximate solution of eqs. (8)(9)(10)(11) is This first phase of the generation consists in the growth of n φ due to φ particle production. However the growth of n φ ends at some time t 1 , when (n ϕ − n φ )Γ = −2 E i n φφ > 0 andṅ φ = 0. This point is the end of stage one and n φ reaches a maximum value At the end of this first stage one also has n ϕ /β ≪ 1, n ϕ (t 1 ) ≃ n ϕi ,φ(t 1 ) ≃φ i and n φ (t 1 ) =n φ ≃ n 2 ϕ /2β ≪ n ϕ . The second phase of the generation maintains (n ϕ − n φ )Γ+2 E i n φφ = 0 dynamically, therefore from eq.(8) it follows that n φ remains constant at the equilibrium valuen φ ≪ n ϕi . One can cheek the stability of this solution n φ =n φ analyzing the dynamical equation for small perturbations of n φ aroundn φ that is δṅ φ = −β δn φ with β =Γ − Eī n φφ 2 > 0, that gives an exponential suppression of perturbations, so the solution n φ =n φ is stable. If n φ is maintained constant at its equilibrium valuen φ , the relation 2 E i n φφ = −Γ (n ϕ −n φ ) =ṅ ϕ is satisfied dynamically (see eqs. (8) and (9)), so we can write eq.(11) in the formφ . We takeφ 2 ≃ E n ϕ , which agrees well with the numerical simulation, and Eφ En ϕ ≃ Eφ 2 ≃ 2 E ρ 1ϕ where ρ 1ϕ ≡φ 2 /2. Therefore eqs.(8)- (11) are reduced to the following effective equations From these equations we see that n ϕ decrease until it reaches the equilibrium at n ϕ f ≃ n φ f ≃n φ ≪ n ϕi and that ρ 1ϕ decrease exponentially to zero so at the end of the regeneration one has ρ ϕ f = E inφ ≪ ρ ϕi . Moreover ρ 1φ grows until ρ 1φ f = ρ ϕi − E inφ so one has ρ φ f = ρ ϕi and in conclusion one has Ω φ f ≃ Ω ϕi , Ω ϕ f ≪ Ω ϕi and v(φ) ≃ ρ ϕi , that means that all the energy initially stored into 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)(3)(4)(5) in the case v(φ) = v i /φ. In fig.(2) we show the evolution of n φ /E 3 and in fig.(3) that of n ϕ /E 3 . The evolution of n φ includes the solutions given in eqs. (12) and (14) showing that n φ starts from zero, it reaches its maximum and then it decreases to its equilibrium valuen φ . The evolution of n ϕ is plotted in fig.(3) and we see how n ϕ decrease exponentially to the asymptotic valuen φ as in eqs. (12)and (14). In fig. (6) we plot the evolution of the density parameter Ω ϕ showing that Ω ϕ goes to zero at the end of the φ generation.
We stress some important features of the φ generation process. The first one is that, in spite of starting with a v(φ i ) = 0, at the end of the process one has generated a potential v(φ) ≃ ρ ϕi = 0. In fig.(4) we show how v(φ) grows from zero to its maximum value v(φ) ≃ ρ ϕi . At this point the quantum generation process is completed and the subsequent evolution of φ is given by its classical equations of motion. At the same time in fig.(5) we see the evolution ofφ, where it starts atφ(t geni ) = 0 and takes negative values (implying the growth of v(φ)) and eventually reaches positive values. The time whenφ(t genf ) = 0 corresponds to the end of φ generation and to the maximum of v(φ). Another feature of the φ generation process consists of the existence of an equilibrium valuen φ ≪ n ϕi for the density number of φ particles n φ . Therefore after a first phase of growth, n φ saturates atn φ and then it is maintained constant. This means that all the energy coming from further ϕ particles 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 generation process. First we note that if the φ field is regenerated when the universe is radiation dominated, the scale factor  evolves as a(t) = a i 1 + H i (t − t i ). The φ generation ends atΓ(t − t i ) > ∼ 1 (see eq. (14)) withΓ ≃ Γ gen , giving H(t − t i ) < ∼ H/Γ gen ≤ 1 and a(t) < ∼ √ 2a i (the generation process has Γ gen /H < ∼ 1), thus we see that the expansion of the universe during the φ generation plays no significant role. We also stress the fact that the generation process preserves the homogeneity and isotropy of the universe, as in the reheating process after inflation [3]. In fact it is obtained via a 2 ↔ 2 quantum process in which the energy-momentum is conserved ( p i = p f ) so the amount of homogeneity is preserved. Now we can state something more about the begin and the duration of the generation process. First note that at the begin of the φ generation n φ = 0, that im-pliesΓ = Γ gen ≡ Γ ϕ+ϕ→φ+ϕ = Γ ϕ+ϕ→φ+φ . Therefore, in order to establish the efficiency of φ particles production one should compare the transition rate Γ gen with the expansion rate of the universe H. We assume that before the generation the ϕ field has the same temperature of the radiation, so we take E rad = E ϕ = E. If the generation starts before the matter domination one has H = (ρ rel /3 m 2 pl Ω rel ) 1/2 ≡ c H E 2 where c H ≡ (π 2 g rel /90r 4 m 2 pl Ω rel ) 1/2 and g rel is the total number of relativistic degrees of freedom. There-fore, using n ϕ = ζ(3) π 2 T 3 ϕ before the φ generation one has Γ gen = ζ(3) 32π 3r3 g 2 E and Γ gen /H = c gen g 2 m pl /E, where we have defined c gen ≡ (ζ(3)/32π 4r ) (90 Ω rel /g rel ) 1/2 . We conclude that the φ generation starts at energies E < ∼ E gen with E gen ≡ c gen g 2 m pl = 1.6 g 10 −12 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 SM rel + g ϕ ≃ 4.36 giving c gen ≃ 6 · 10 −4 . Of course, by means of eq.(15) we can conveniently choose the coupling g in such a way that the generation process starts at an energy of about 1eV . This takes place if g ≃ 10 −12 or α = g 2 /4π ≃ 10 −25 , where α is the fine structure constant of the transition process. We stress the fact that Γ gen /H ≡ E gen /E ≥ 1 during the φ generation process but, when the φ field starts to inflate the universe, one has H ∼ a(t) − 3 2 (1+ω φ ) with ω φ ≃ −1, so H will be roughly constant and at the dark energy epoch Egen . We can summarize the φ generation process as follows. At the begin of the generation we have n ϕi = ζ(3) π 2r3 E 3 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 valuen φ ≪ n ϕi after which n φ remains constant. Subsequently all the energy coming from the ϕ field is transferred to the potential v(φ) through the chain reaction n ϕ → n φ → v(φ). In this process n φ remains constant andn φ represent a maximum transfer efficiency value for n φ in such a way that the energy coming from the decaying ϕ particles is immediately stored into the potential v(φ). At the end of the generation on has n φ f ≃ n ϕ f ≃n φ ≪ n ϕi ρ ϕ f ≪ ρ ϕi and ρ φ f ≃ ρ ϕi .
From this point on, the evolution of φ is the standard one, where its dynamics is dominated by the classical potential v(φ). The generation process described before does not depend explicitly on the form of the potential v(φ) and the value of φ at the end of the generation process is given by the condition v(φ gen ) ≃ ρ ϕi ≃ E 4 ϕi = E 4 gen . Since we want the φ field to give dark energy, we must choose a potential v(φ) that slow rolls at late times, i.e. at present time when φ ≃ φ o . Clearly we must impose the condition v(φ gen ) > v(φ o ) = v DE ≃ (10 −3 eV ) 4 . For example one can consider an effective potential v(φ) = vi φ that verifies the slow roll conditions for φ ≥ √ 2 and take v i = (10 −3 eV ) 4 . In this case the value of φ at the end of the generation is φ gen ≃ v i /E 4 gen where we have used E 4 gen = v(φ gen ) = vi φgen . Typically, runaway quintessence potentials have an EoS parameter ω φ that reaches positive values of ω φ ≃ 1 diluting Ω φ and later there is a transition from ω φ = 1 to ω φ = −1 as in fig.(7) where Ω φ starts growing [14]. Therefore the matter domination epoch is unchanged by our φ generation scheme. In fig.(8) we show the evolution of the density parameters Ω φ , Ω SM rad and Ω mat . The φ is generated at N gen at energies E gen ≃ 1eV , for g ≃ 10 −12 , close to radiation matter equality with Ω φi ≃ 0.2 and present time is at Ω φ ≃ 0.74 and Ω m ≃ 0.26 with ω φ ≃ −1.
The cross section for the generation process is given by 90 Ω rel g rel ≃ 6 · 10 −4 and pb = 10 −36 cm 2 . It is interesting to compare the cross section σ g with that of WIMPS. The relic abundance of WIMPS is Ω w h 2 = 3 × 10 −27 cm 3 / < σ w v > [16] giving σ w v = 0.9 c pb, with c the speed of light. If we take that at decoupling the WIMPS have a mass to temperature ratio m/T = 20 [16] we obtain σ w = 2.4pb equivalent to our σ gen for E gen = 1 eV . However, the present time observational upper limit to σ w between WIMPS and nucleons is σ w < ∼ 10 −42 cm 2 consistent with supersymmetric WIMPS [16].

IV. UNIFICATION OF INFLATION AND DARK ENERGY.
In this section we discuss the possibility of unifying inflation and dark energy by means of a unique scalar filed φ that we call uniton, as in [17]. In general it is not difficult to choose the potential v(φ) in such a way that the φ field is responsible for both inflation and dark energy [15]. To achieve inflation and dark energy with the same scalar φ one requires that the potential v(φ) must satisfy the slow roll conditions |v ′ (φ)/v(φ)| < 1 and |v ′′ (φ)/v(φ)| < 1 at high energies for inflation and at low energies for dark energy. Inflationary potentials which have a minimum v min = 0 at a finite value of φ are not useful to unify inflation and dark energy, since this kind of potentials do not inflate at low energies [14]. This kind of potentials may be useful to unify inflation and dark matter. In most inflation-dark energy unified models only the classical evolution of the quintessential scalar field is considered and the reheating and the long period of decelerating universe (between inflation and dark energy) are not taken into account.
In this section we present an inflation-dark energy unified scheme that can be resumed in the following way: as in usual inflationary models, the early universe is dominated by the φ field that inflates for a sufficient number of e-folds. After the end of inflation the φ field decays completely into the extra relativistic field ϕ already introduced in section III. The ϕ couples and produces SM particles at an energy E RH and the universe is reheated. At low energies E gen the φ field is generated via the quantum generation mechanism studied in section III and the universe enters the dark energy epoch at temperatures close to present time. From E RH to E gen we have the standard evolutionary scenario with the extra relativistic degree of freedom ϕ.
The new feature in the unification scheme that we present here, is that the transition between decelerating radiation-matter dominate universe and the dark energy era is due to a quantum process, i.e. the low energy generation of the φ field. As stated in section III, the energy scale E gen at which the uniton φ is generated is fixed by the couplings g of φ. The scales E gen may be many orders of magnitude smaller than E RH without any fine tuning.
We stress the fact that, although we will describe the inflation-dark energy unified scheme choosing a particular form of the potential v(φ), the quantum generation mechanism works well for a large class of potentials. The only requirements on the potential v(φ) are that it should satisfy the slow roll conditions at high and low energies and that the φ mass m φ ≡ v ′′ (φ) should satisfy the condition m φ (t RH ) ≫ m φ (t gen ), where t RH and t gen are the reheating and φ generation times. In addition one has to require that the φ particles are relativistic at t gen , that implies m φ (t gen ) ≪ E gen . In section IV A we will discuss the inflation and reheating scenario and then in section IV B we will consider the inflation and dark energy unified scheme using a simple example.

A. Inflation and reheating
Let us consider an universe that contains the field φ, a second scalar field ϕ and the SM particles. We want the φ field to inflate the early universe, so we assume that the potential v(φ) has at least one flat region corresponding to inflation. As an example one can consider the potential described in Appendix B. During inflation the φ field dominates the universe and slow rolls as long as the slow roll conditions |v ′ (φ)/v(φ)| ≪ 1 and |v ′′ (φ)/v(φ)| ≪ 1 are satisfied and the universe inflates with H 2 ∼ v(φ). After the end of inflation the φ field decays into ϕ particles and reheats the universe. We couple the two scalars φ, ϕ via the interaction (17) where f (ϕ) is a polynomial of ϕ. We know that the interaction in eq.(17) gives a complete reheating since it involves a single φ particle decaying into ϕ particles [21]. At the end of inflation the φ particles are at rest in 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 3 and we consider the process φ → ϕ+ϕ+ϕ. For simplicity we assume that all the ϕ particles are produced with the same energy E ϕ given by E φ = 3E ϕ giving a decay rate [2] Let us remind that a decay process is efficient if Γ d /H > ∼ 1 [3]. The evolution of the energy density ρ φ of the φ field is given by the equatioṅ If we consider ω and Γ d as piecewise constant, the solution is ρ φ ∼ a(t) −3(1+ω) e −Γ d t and the φ energy density vanish exponentially, that means that ϕ particles are produced 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 − Γ d dt and the condition for an efficient decay is Γ d dt >> 1. At the same time we couple ϕ with SM particles. We take the usual interaction terms where χ and ψ are SM scalars and fermions, respectively. As long as SM particles are relativistic, valid at temperatures 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 2 n ϕ /32πE 2 ϕ , and using n ϕ = ζ(3) T 3 ϕ /π 2 one has Γ RH = ζ(3) h 2 E ϕ /32π 3r3 . SM particles are produced at Γ RH /H ≡ E RH /E > 1 where we have defined and c RH ≡ ζ (3) 32π 4r 90 Ω rel gr ≃ 10 −4 . If reheating takes place above 300 GeV we have g SM r = 106.75 and Ω r = 1 so that we can estimate c RH ≃ 10 −4 . Taking h ≤ 1, so that α h = h 2 /4π < 0.1, the maximum energy for reheating would be 10 14 GeV . However the limit for successful reheating scenario is much lower and it may be as low as 10 M eV [11,12] corresponding to h ≃ 10 −8 .
Therefore SM particles are produced at energies E ≤ E RH and ϕ is in thermal equilibrium with SM particles with T ϕ = T γ where T γ is the photon temperature. As long as ϕ is relativistic T ϕ ∝ T γ and if it remains also in thermal equilibrium we have rel Ω SM rel (22) where g SM rel is the number of SM relativistic degrees of freedom and g ϕ = 1.

B. Inflation-Dark Energy Unification
Let us now describe the inflation-dark energy unified picture with an explicit example. Again we start with two scalar fields coupled through eq. (17). To be specific we will choose L int = g φ ϕ 3 and a scalar potential v(φ) that inflates at high and low energies. As mentioned in the introduction in section IV the choice of the scalar potential is not important, there are a wide number of possibilities, and we choose to work with which has two free parameters E I ≡ V 1/4 I and f with mass dimension. The potential in eq.(23) has two regions φ < −(2f /π) 1/3 and φ > √ 2 in which the slow roll Inflation is associated with the high energy region φ < −(2f /π) 1/3 with v(φ) ≃ V I = E 4 I and dark energy with the region φ > √ 2 with v(φ) ≃ vI f πφ . We determine the two free parameters in eq.(23) with the constrains coming from inflation δρ ρ = 5.3 × 10 −4 and from dark energy gives E I ≃ 100 T eV and f = φ o v DE /V I ≃ 10 −39 eV where we have reintroduced the correct mass units in E I and f .
After inflation we want to reheat the universe with the SM particles. To achieve this we couple φ and ϕ via the interaction term L int = g φ ϕ 3 and ϕ with SM particles as in eq. (20). The φ field decays into ϕ via the process φ → ϕ + ϕ + ϕ with a decay rate Γ d = g 2 m φ (2π) 3 72 given in eq. (18). This process starts immediately after inflation with H ≃ E 2 I . The maximum value of Γ d /H is when m φ is also at its maximum at φ ∼ f giving Γ d /H ≃ 10 36 . Notice that Γ d ≃ 10 9 at its maximum and the lifetime τ φ = 1/Γ d of the φ particles is such that τ /τ pl ≪ 1 where τ pl is the Planck time. Therefore all φ particles decay and at the end of the reheating and one has Ω φ = 0 and Ω ϕ + Ω SM = 1. SM particles are produced through the interaction with the ϕ field via the interaction given in eq.(20) and described in section IV A. This process takes place for energies E ≤ E RH ≡ c RH h 2 given in eq.(21) with Γ RH /H ≥ 1 and therefore ϕ and SM particles are in thermal equilibrium giving Ω ϕ = gϕ g SM rel Ω SM rel as long as ϕ remains relativistic. As discussed in section III as long as E RH > E > E gen ≡ c gen g 2 (c.f eq.(15) ) the universe contains 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 = σ gen v n ϕ . The inflation-dark energy unification scheme involves three different quantum processes: The φ decay into ϕ, the SM particles production and the late time φ generation. Only the first one (inflaton decay) depends on the choice of the potential v(φ), through its mass, the other two depend only on the size of the couplings g, h. These processes take place at energies where (26) and c RH ≃ 10 −4 for a reheating temperature above 300 GeV and c gen ≃ 6 · 10 −4 if the φ is generated at energies below 1 M eV . We stress the fact that the φ generation at late times is due to the same interaction term L int = g φ ϕ 3 that gives the φ decay after inflation. The main difference is that at low energies E gen the mass of φ is many orders of magnitude lower than its value at high 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 respectively. We also require that m φ (t gen ) ≪ E gen so that φ is relativistic at t gen as in section III. Notice that in our model the value of the φ mass at generation time is m φ (t gen ) ≃ 10 −14 eV (see appendix B) which is much smaller than E gen ≃ 1 eV . However at present times m φ (t o ) ≃ 10 −33 eV which is a typical mass for a quintessence (dark energy) field. After the production of the φ particles v(φ) is generated and the φ classical evolution will drive the expansion of the universe as described in section III. Concluding we have shown with a simple example how the inflation-dark energy unification takes place. Of course it is possible to choose different scalar potentials v(φ) or interaction terms L int that gives similar results.

C.
ERH and Egen scales The values of E RH and E gen do not depend on the choice of the potential v(φ) but only on the couplings g, h. The values of E RH and E gen are fixed in terms of the couplings g and h. In general g and h are free parameters that should give E I ≥ E RH > 10 M eV and E RH ≫ E gen > E o and are given by (see eq. (25)) and c RH ≃ c gen ≃ 10 −4 . An interesting reduction of parameters is if we take E RH = E gen (in natural units) which gives g = h 2 and Notice that this choice of g, h implies a low reheating temperature and for E gen as small as E o ∼ 10 −3 eV we have E RH ≥ 1.5 T eV . If we set with q a proportionality constant, we have The fine structure constants associated to the two couplings are α g ≡ g 2 4π , α h ≡ h 2 4π and cross sections σ g , σ RH are then The cross section for the generation process are σ gen = g 2 /(32πE 2 gen ) and σ RH = h 2 /(32πE 2 RH ) giving We find the relationship between g and h in eq.(29) interesting but of course it does not need to hold since in principle g, h and therefore E gen , E RH are independent from each other and eqs. (30) and (31) are equivalent to eqs. (15) and (21), respectively.

V. PHENOMENOLOGY
In this section we summarize the main phenomenological consequences of the dark energy quantum generation. Let us first discuss the consequences of having the relativistic field ϕ. If ϕ is not contained in the SM then it represent an extra relativistic degree of freedom. CMB temperature anisotropies as well as SDSS and 2dF Large Scale galaxy clustering, Lyman-α absorption clouds, type Ia Supernovae luminosity distances and BAO data, can be used to determinate the value of the effective relativistic degrees of freedom, usually described in terms of the effective number of neutrinos N ef f ν . The value of N ef f ν affects the matter-radiation equality epoch and thus the ISW effect, so CMB anisotropies are sensitive to deviations from the standard cosmological model value of N ef f ν ≃ 3.04. Analysis of the WMAP data combined with other cosmological data sets allows for values of N ef f ν different from its standard model value. For example in [30] it is found N ef f ν = 4.6 +1.6 −1.5 at 95% c.l., consistently with other analysis [31]. Moreover BBN is also affected by N ef f ν , because the number of relativistic degrees of freedom change the value of the expansion rate of the universe and than influence the expected primordial Helium abundance. The BBN bound is N ef f ν = 3.1 +1.4 −1.2 at 95% c.l. [19,30,33], that seems to be more stringent than bounds coming from CMB data. Anyhow N ef f ν can evolve from the BBN epoch at T ∼ 1M eV to the CMB decoupling era at T ∼ 1eV [33], so the different bounds coming from BBN and CMB are compatible. In our case the extra relativistic degree of freedom is represented by the scalar field ϕ that contributes to ≃ 2.2 that shows some tension with BBN data but is compatible with CMB data. In any case the existence of the extra relativistic degree of freedom coming from the ϕ field is consistent and apparently favored by cosmological data.
Another important phenomenological aspect of the dark energy generation model, concerns the coupling of the ϕ field with SM particles. In principle ϕ may be coupled with electrons, baryons and photons, but strong limits on the strength of such couplings comes from astrophysical considerations and accelerator phyiscs. In fact if coupled with electrons, the ϕ particles are produced in stars and this fact affects the evolution of the stars, as studied in [37]. The coupling strength between ϕ and electrons should satisfy the condition α ϕee < 0.5 × 10 −26 .
In the model that we present, SM particles are produced at E RH = c RH h 2 m P l ≥ 10 M eV that gives h ≥ 10 −8 and a fine structure constant α = h 2 /4π ≥ 10 −17 , 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 possible observable consequences at astrophysical and cosmological level. The upper bound coming from long range force experiments is α ϕB < 10 −47 [39] thus the ϕ coupling with baryons should be excluded. If the ϕ field is coupled with photons via the axion-like interaction term L int = gϕγγ 4 ϕ F µνF µν = −g ϕγγ ϕ E · B with g ϕγγ < 10 −10 GeV −1 [37]. The bound on the transition Taking H = c H T 2 /m pl the field ϕ and photons are coupled for T < ∼ T γ RH ≃ (m ϕ /eV ) 3/2 10 2 GeV when Γ ϕγγ /H ≥ 1. Therefore if ϕ is coupled with photons, SM particles will be produced at low reheating temperatures of about T γ RH ≃ 10 2 GeV for m ϕ ≃ 1 eV . The coupling with photons is not ruled out by experimental data.
In conclusion, the ϕ field could be coupled either to neutrinos, photons, neutral Higgs field or supersymmetric partners of the SM which are currently searched for at LHC. It is particularly interesting the case in witch the ϕ is coupled with neutrinos. In the standard cosmological model the three neutrinos free-stream and interact only gravitationally. Free-streaming lowers the neutrino perturbations and introduce a source of anisotropic stress. On the contrary, if one or more neutrinos are coupled with the scalar field ϕ, the interacting neutrinos behave as a tightly-coupled fluid with density and velocity perturbations but no anisotropy [35]. This fact also affects the adiabatic sound speed c 2 s , that is equal to 1/3 for freestreaming neutrinos and it is 0 < c 2 s ≤ 1/3 for tightlycoupled neutrinos. Therefore the coupling of the ϕ field with neutrinos produces many observable consequences on the Cosmic Neutrino Background (CNB) [36].
In alternative to the dark energy generation scheme presented in section III, one can generate the φ field without any auxiliary field ϕ but coupling φ directly with SM neutrinos via an interaction L int = √ g φνν. Therefore one or more neutrinos will be tightly-coupled to the φ quintessence field and this will produced observable consequences on the CNB [36].
The φ sector of the quantum generation model has also an interesting phenomenology. For example the inflation dark energy unified model presented in section IV has low inflationary and and reheating scale E I ≃ 100 T eV and E RH ≃ 1T eV for h ≃ 10 −6 . This low reheating energy does not affect the reheating efficiency and it also avoids gravitino overabundance problems. From a cosmological point of view, a low inflationary scale may also affect N ef f ν as showed in [12], giving one more possible test of the unified model. Moreover interesting effects may be observed in accelerators, as for example at LHC, with energy scales not so far from the inflationary energy. Then, as discussed above, a rich phenomenology exists, that may be used to constrain or falsify cosmological models that make use of the dark energy generation mechanism.

VI. CONCLUSIONS
We will now present a summary and conclusions of our work. One of the main goals of this paper was to understand why the dark energy is manifested at such a late time. To achieve this we have presented a novel idea, the quantum generation of dark energy, giving a new interpretation of the late time emergence of DE in terms of a late time quantum production of the quintessence φ particles. We take a 2 ↔ 2 quantum process between φ and a relativistic ϕ particles. The scale where the φ field is generated is dynamically determined by the condition Γ/H = E gen /E ≥ 1 giving an energy scale E ≤ E gen with E gen = c gen g 2 m pl and c gen ≃ 6 · 10 −4 . Therefore the smallness of E gen is due to a small coupling g and for g ≃ 10 −12 gives a E gen ≃ 1 eV and a cross section σ gen ≃ 1pb. The acceleration of the universe is then due to the classical evolution of φ and determined by the scalar potential v(φ). We have described in section III a universe that initially contains no φ particles, i.e. n φ = Ω φ =φ = v(φ) = 0, and once the relativistic particles φ are produced they become a source term for the generation of the scalar potential v(φ). Once v(φ) has been produced the classical equation of motion gives the evolution of φ.
We show in section IV that it is possible to unify inflation and dark energy using the same quintessence field φ. To achieve the unification we required that the potential v(φ) has two flat regions, at high energy for inflation and low energy for dark energy. In this scenario, after inflation the field φ decays completely and reheats the universe with standard model particles. The universe expands then in a decelerating way dominated first by radiation and later by matter. At low energies the same interaction term that gives rise to the inflaton decay accounts for the re-generation of the φ field giving rise to dark energy. An important difference in the quantum process between φ and ϕ at high and low energies is the value of transition rate due to the size of the φ mass, . We presented in section IV a simple example on how the inflation-dark energy unification can be implemented. We used a potential v = E 4 I (1−arctan[φ/f ]) which is flat at high and low energies. The two parameters E I , f are determined by the density perturbations δρ/ρ and the value of v o at present time giving E I = 100 T eV, f = 10 −39 eV . The coupling g between φ and ϕ and the coupling h between ϕ and the SM particles are free parameters but can be taken as g = h 2 = q E I /m pl = (q/100)(E I /100 T eV )10 −12 giving a reheating energy E RH = (q/100)(E I /(00 T eV ) T eV and a generation energy E gen = (q/100) 2 (E I /100 T eV ) 2 26 eV . The cross section between φ and ϕ is σ gen = g 2 /32πE 2 gen ≃ pb quite close to cross section of WIMP dark matter with nucleons σ w ≃ pb at decoupling. By fixing g = h 2 = q E I we have determined the coupling, which set the scales of reheating and φ re-generation, in terms of the inflation scale 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 to produce the fundamental quintessence field φ dynamically at low energies. The energy scale is fixed by the strength of the coupling and this offers a new interpretation of the cosmological coincidence problem: dark energy domination starts at such small energies because of the size of the coupling constant g. Finally, our approach allows for an easy implementation of inflation and dark energy unification with the standard long periods of radiation/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 FRW universe containing the φ field coupled with a second scalar field ϕ and let us write down the equations of motion of the φ and ϕ fields. In what follows we assume that both the φ and ϕ particles are relativistic. The φ(t, x) field can be divided into a classical background 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) → 0 one has φ(t, x) = φ c (t). The dynamic of the φ field is then deduced from its lagrangian density , v(φ) and B(ϕ) are the classical potentials of the φ and ϕ fields respectively and v int (φ, ϕ) = −L int (φ, ϕ) where L int (φ, ϕ) is the interaction lagrangian of the scalar fields φ and ϕ. Using ∇φ(t, x) = ∇δφ(t, x) one has The perturbation δφ evolves as a scalar field with mass . The corresponding quantum operator δφ has the following expression A3) where k and E k are respectively the wave number and the energy of the φ particles with E 2 , [29]. The physical momentum is p = k/a(t) and the φ particles are relativistic so E k = | k|/a(t) = | p|. For simplicity we assume that all the φ and ϕ particles have the same energy E φ and E ϕ respectively. For example this assumption is valid if the quantum particles are thermalized. In this case the phase space distributions are the Bose-Einstein distributions f φ (E) = 1/(e E/T φ − 1) and f ϕ (E) = 1/(e E/Tϕ − 1) and one can take the energies of the φ and ϕ particles as the mean values E φ =r T φ and E ϕ =r T ϕ withr ≃ 2.7. Moreover, in the case that the φ and ϕ fields are in thermal equilibrium one has E φ = E ϕ . Therefore we estimate δφ 2 as the average of δφ on the quantum state |N, E φ > containing N φ particles with energy E φ , i.e. < N, E φ | : δφ 2 : |N, E φ >= n φ E φ , where the :: stands for normal ordering of creation and destruction operators. We have then δφ 2 = n φ /E φ , where n φ is the density number of φ particles. Thus we can write where we have used E 2 k = | p| 2 = | k| 2 /a 2 for relativistic particles. Substituting this expression in eq.(A2) we havë Repeating the same considerations for the ϕ field one has The dynamics of the quantum particles is governed by the Boltzmann equations. If f φ (E, t) is the phase space distribution of the φ particles, one has whereL is the Liouville operator and in a FRM metric isL[f φ (E, t)] = E ∂ t f φ (E, t) − H| p| 2 ∂ E f φ (E, t) andĈ is the collision operator (see [28]). If the φ particles are relativistic one haṡ where for a process a 1 + a 2 + ... + a n ↔ b 1 + b 2 + ... + b l , with a n (b l ) initial (final) particles, one has with dΠ ≡ g d 3 p/(2π) 3 2E, g are the internal degrees of freedom and |M ab | 2 the transition scattering matrix of the process. Eq.(A8) is valid in absence of Bose condensation of Fermi degeneracy when 1 ± f i (E, t) ≃ 1 [28]. In the same way one haṡ The terms A and Q introduced in eqs.(A7) and (A9) are necessary for the energy conservation as we will discuss below eqs.(A18) and (A19).
Let us consider the quadratic interactions L int = g φ 2 ϕ 2 or L int = g φ ϕ 3 . In that case the φ field is generated via the 2 ↔ 2 processes ϕ + ϕ ↔ φ + φ or ϕ + ϕ ↔ ϕ + φ respectively. For simplicity we assume 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 thermalized particles. Therefore we take all the ϕ particles with the same energy E ϕ and one has f ϕ (E, t) dΠ ϕ = f ϕ (E, t) . Moreover, from energy conservation it follows that the φ particles are produced with the same energy of the ϕ particles and we take E φ = E ϕ = E, that is also valid when the φ and ϕ fields thermalize. Therefore the energy distribution of the φ particles is piked around the mean energy E φ and one also has f φ (E, t) dΠ φ = n φ 2E φ . The transition rates for the considered processes are Γ ϕϕ→φφ = Γ ϕϕ→φϕ = Γ φϕ→ϕϕ = σ gen v n ϕ ≡ Γ gen and Γ φφ→ϕϕ = σ gen v n φ , where σ gen = g 2 /32πE 2 is the cross section for a 2 ↔ 2 relativistic particle process and v is the relative velocity [25]. Considering the process ϕϕ ↔ ϕφ one has ϕ − Γ φϕ→ϕϕ n ϕ n φ = σ gen v n ϕ (n ϕ − n φ ) (A10) and for the process ϕϕ ↔ φφ one has ϕ − Γ φφ→ϕϕ n 2 φ = σ gen v (n 2 ϕ − n 2 φ ) = = σ gen v (n ϕ + n φ )(n ϕ − n φ ) (A11) We can write eqs.(A10) and (A11) in a compact form as where we have definedΓ ≡ σ gen v n ϕ for the process ϕϕ ↔ ϕφ andΓ ≡ σ gen v (n ϕ + n φ ) for the process ϕϕ ↔ φφ. Note thatΓ is not necessarily a transition rate, but it accounts for the whole contribution of the two 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 reaḋ n φ + 3Hn φ =Γ (n ϕ − n φ ) + A (A13) n ϕ + 3Hn ϕ = −Γ (n ϕ − n φ ) + Q.
We can express the value of φ at the generation time, i.e. φ gen , in terms of φ o as φ gen = v(φDE ) v(φgen) φ o = ( EDE Egen ) 4 φ o and if one choose E gen ≃ 1 eV one has φ gen ≃ 10 −12 φ o . Therefore we can also compare the value of the mass m φ at the reheating, generation and present times. At reheating one has φ ≃ f and m φ (t RH ) ≃ V I /f 2 , at generation time one has φ gen ≫ f and m φ (t gen ) ≃ One has m φ (t RH )/m φ (t gen ) ≃ φ gen /f ≫ 1, therefore there are many orders of magnitudes of differ-ence between the values of m φ at the reheating and generation times, and this is why the reheating is obtained via the decay process φ → ϕ+ϕ+ϕ of massive φ particles into relativistic ϕ particles, and the φ is generated at late times via a 2 ↔ 2 process between relativistic particles.