Constraining f(T) teleparallel gravity by Big Bang Nucleosynthesis
S. Capozziello G. Lambiase E. N. Saridakis







^1Dipartimento di Fisica "E. Pancini", Università di Napoli
  “Federico II”, Complesso Universitario di Monte Sant'Angelo,
  Edificio G, Via Cinthia, I-80126, Napoli, Italy
  

  ^2Istituto Nazionale di Fisica Nucleare (INFN) Sezione
  di Napoli, Complesso Universitario di Monte Sant'Angelo, Edificio G,
  Via Cinthia, I-80126, Napoli, Italy
  

  ^3 Gran Sasso Science Institute, Viale F. Crispi, 7, I-67100,
L'Aquila, Italy


^4Dipartimento di Fisica E.R. Cainaiello, University of Salerno, Via
Giovanni Paolo
II, I 84084-Fisciano (SA), Italy

^5INFN, Gruppo Collegato di Salerno, Sezione di Napoli, Via Giovanni Paolo II, I
84084-Fisciano (SA), Italy

^6Department of
Physics, National Technical University of Athens, Zografou Campus
GR 157 73, Athens, Greece

^7CASPER, Physics Department, Baylor University, Waco, TX 76798-7310, USA



    α
        We use BBN observational data on primordial abundance of ^4He to constrain
        f(T) gravity. The three most  studied viable f(T) models, namely the
        power law, the exponential and the
        square-root exponential are considered, and  the BBN bounds are adopted in order to
        extract
        constraints on
        their free parameters. For the power-law model, we find that the  constraints
        are in agreement with those acquired using late-time cosmological data. For the
        exponential and the square-root exponential models,  we show that for realiable
        regions of  parameters space they always satisfy the BBN bounds. We conclude that
        viable f(T) models can successfully satisfy the BBN constraints.
        
        04.50.Kd, 98.80.-k,  98.80.Es, 26.35.+c Constraining f(T) teleparallel gravity by   Big Bang Nucleosynthesis
            S. Capozziello^1,2,3, G. Lambiase^4,5, and E.N. Saridakis^6,7
            December 30, 2023
        =====================================================================
        
        
        
        § INTRODUCTION
        
        
        Cosmological observations coming from  Type Ia Supernovae <cit.>,  cosmic microwave
        background
         radiation
        <cit.> and the large scale structure <cit.>, provide
        evidences that the Universe  is currently in an accelerating phase. This result is, in
        general,
        ascribed to the existence of a sort of  dark energy
        (DE) sector in the universe, an exotic energy source characterized by a negative
        pressure. At late times, the dark-energy sector eventually dominates over the cold dark
        matter (CDM), and drives the Universe  to the observed accelerating expansion.
        
        The simplest  candidate for DE is the cosmological constant Λ,
        which has an equation-of-state parameter  w=-1. Although this model is in agreement with
        current observations, it is plagued by some difficulties related to the small
        observational value of DE density with respect to the expected one arising from quantum
        field theories (the well known cosmological constant problem <cit.>). Moreover,
        the ΛCDM paradigm, where cold dark matter (CDM) is considered into the game,  may
        also
        suffer from the age problem, as it was  shown
        in <cit.>, while the present data seem to slightly favor an evolving DE with the
         equation-of-state parameter crossing w=-1 from above to below in the near cosmological
        past <cit.>.
        
        Over the past decade several DE models have been proposed, such as quintessence
        <cit.>, phantom <cit.>, k-essence <cit.>, tachyon
        <cit.>, quintom <cit.>, Chaplygin gas
        <cit.>, generalized Chaplygin gas (GCG) <cit.>, holographic DE
        <cit.>, new agegraphic DE <cit.>,   Ricci DE <cit.> etc. On the
        other hand, there are also numerous models that induce an effective dark energy
        which arises from modifications of the gravitational sector itself, such as
        f(R) gravity <cit.> (this class is very efficient in
        verifying observational and theoretical constraints and explain the Universe acceleration
        and phantom crossing <cit.>), or gravity with higher curvature
        invariants <cit.>, by coupling the Ricci scalar to a scalar field
        <cit.>, by introducing a vector field contribution <cit.>, or by using
        properties of gravity in higher dimensional spacetimes
        <cit.> (for a review see <cit.>).
        
        A possibility that can be explored to explain the accelerated phase of the Universe  is to
        consider a theory of gravity based on the Weitzenböck connection, instead of the
        Levi-Civita one, which deduces that the gravitational field is described by
        the torsion instead of the curvature tensor. In such theories, the torsion tensor is
        achieved from products of first derivatives of tetrad fields, and hence no second
        derivatives appear. This   Teleparallel  approach <cit.>,
        is closely related to General Relativity, except for “boundary terms”<cit.>
        that involve
        total
        derivatives in the action, and thus one can construct the Teleparallel Equivalent of
        General Relativity (TEGR), which is completely equivalent with General Relativity at the
        level of equations but is based on torsion instead of  curvature. Teleparallel gravity
        possesses a
        number of attractive features
        related to  geometrical and physical aspects <cit.>.
        Hence, one can start from TEGR and construct various gravitational modifications based on
        torsion, with f(T) gravity being the most studied one
        <cit.>. In particular, it may
        represent an alternative to inflationary models without the use of the inflaton, as well
        as to effective DE models, in which the Universe acceleration is driven by the
        extra torsion terms
        <cit.>
        (for a detailed review,  see <cit.>). The main advantage of
        f(T) gravity is that the field equations are 2nd-order ones, a property that
        makes these theories simpler   if compared to the dynamical equations of other extended
        theories of gravity, such as f(R) gravity.
        
        The aim of this paper is to explore the implications of f(T) gravity to the formation
        of light elements in the early Universe,  i.e. to the Big Bang Nucleosynthesis (BBN).
        On the other hand, we want to explore the possibility to constrain f(T)  gravity by BBN
        observatio
        nal data.
        
        BBN has occurred between the first fractions of  second after the Big Bang, around ∼
        0.01 sec, and a few hundreds of seconds after it, when the Universe  was hot and
        dense (indeed BBN, together with cosmic microwave background radiation, provides the
        strong evidence about the high temperatures characterizing the primordial Universe ). It
        describes the sequence of nuclear reactions that yielded the synthesis of light elements
        <cit.>, and therefore drove the observed Universe. In general, from BBN
        physics,  one may infer stringent constraints on a given cosmological model. Hence, in
        this
        work,  we shall confront various  f(T) gravity models with BBN calculations based on
        current observational data on primordial abundance of ^4He, and
        we shall  extract constraints on their free parameters.
        
        The layout of the paper is as follows. In Section <ref> we review f(T) gravity
        and the related cosmological models. In Section <ref> we use BBN
        calculations in order to impose constraints on the free parameters of specific f(T)
        gravity models. Conclusions are reported in Section <ref>. Finally,
        in the Appendix we summarize the main notions of
        BBN physics.
        
        
        
        
        
        § F(T) GRAVITY AND COSMOLOGY
        
        
        Let us briefly review f(T) gravity, and  apply it in a cosmological
        framework. In this formulation,  the dynamical variable is the vierbein field
        e_i(x^μ),
        i = 0, 1, 2, 3, which forms an orthonormal basis in the tangent space at each point
        x^μ of the manifold, i.e. e_i · e_j=η_ij, with
        η_ij the Minkowsky metric with signature -2: η_ij=diag(1,-1,-1,-1).
        Denoting with e^μ_i, μ=0,1,2,3 the components of the vectors e_i in a
        coordinate
        basis ∂_μ, one can write e_i=e^μ_i∂_μ. As a convection, here we
        use the Latin indices for the tangent space, and the Greek indices for the coordinates on
        the manifold. The dual vierbein allows to obtain the metric tensor of the manifold, namely
        g_μν(x)=η_ij e^i_μ(x)e^j_ν(x).
        
        In teleparallel gravity, one adopts the curvatureless Weitzenböck connection
        (contrarily to General Relativity which is based on the torsion-less Levi-Civita
        connection), which gives rise to the non-null torsion tensor:
        
            T^λ_μν=Γ̂^λ_νμ-Γ̂^λ_μν
            =e^λ_i(∂_μ e^i_ν - ∂_ν e^i_μ).
        
        Remarkably, the torsion tensor (<ref>) encompasses all the information about the
        gravitational field. The Lagrangian density is built using its contractions,
        
        
        
        and hence the teleparallel action is given by
         
            I = 1/16π G∫ d^4x e T ,
        
        with e=det(e^i_μ)=√(-g),
        and where the torsion scalar T reads as
        
            T=S_ρ^μνT^ρ_μν .
        
        Here, it is
        
            S_ρ^μν   =   1/2(K^μν_ρ+δ^μ_ρT^θν_θ-δ^ν_ρT^θμ_θ) 
            K^μν_ρ   =   
            -1/2(T^μν_ρ-T^νμ_ρ-T_ρ^μν) ,
        
        with K^μν_ρ the contorsion tensor which gives the difference between
        Weitzenböck and Levi-Civita connections. Finally, the variation of action
        (<ref>)
        in terms of the vierbiens gives rise to the field equations, which coincide with those of
        General Relativity. That is why the above theory is called the Teleparallel Equivalent of
        General Relativity (TEGR).
        
        One can now start from TEGR, and generalize action
        (<ref>) in order to construct gravitational modifications based on torsion. The
        simplest scenario is to consider a Lagrangian density that is a function of
        T, namely
        
            I = 1/16π G∫d^4xe[T+f(T)],
        
        that reduces to TEGR as soon as  f(T)=0. Considering additionally a matter Lagrangian
        L_m, variation with respect to the vierbein gives the field equations
        <cit.>
            e^-1∂_μ(e e_i^ρS_ρ^μν)[1+f']-
               e_i^λT^ρ_μλS_ρ^νμ[1+f']
                  
              +e^ρ_i S_ρ^  μν(∂_μ T)f” +
                1/4e^ν_i [T+f]=4π G e_i^ρ Θ_ρ^ν ,
        
        where f'≡ df/dT, S_i^μν=e_i^ρS_ρ^μν and
        Θ_μν is the energy-momentum tensor for the matter sector.
        
        In order to explore the cosmological implications of f(T) gravity, we focus on
        homogeneous and isotropic geometry, considering the usual choice for the vierbiens, namely
        
            e_μ^A= diag(1,a,a,a),
        
        which corresponds to a flat Friedmann-Robertson-Walker (FRW) background
        metric of the form
        
            ds^2= dt^2-a^2(t) δ_ij dx^i dx^j,
        
        where a(t) is the scale factor.
        Equations (<ref>), (<ref>), (<ref>) and (<ref>) allow to
        derive a relation
        between the torsion T and
        the Hubble parameter H=ȧ/a, namely
        
            T=-6H^2.
        
        Hence, in the case of FRW geometry, and assuming that the matter sector corresponds to a
        perfect fluid with energy density ρ and pressure p, the i=0=ν component of
        (<ref>) yields
        
            12H^2[1+f']+[T+f]=16π Gρ,
        
        while the i=1=ν component gives
        
            48H^2f”Ḣ-(1+f')[12H^2+4Ḣ]-(T-f)=16π Gp.
        
        The equations close by considering the equation of continuity for the matter sector,
        namely
        ρ̇+3H(ρ+p)=0.
        One can rewrite (<ref>) and (<ref>) in the usual form
        
            H^2=8π G/3(ρ+ρ_T),
        
            2Ḣ+3H^2=-8π G/3(p+p_T) ,
        
        
        
        
        where
        
            ρ_T    =   3/8π G[Tf'/3-f/6],
            
                p_T   =   1/16π G f-T f'+ 2T^2f”/1+f'+ 2Tf” ,
        
        are the effective energy density and pressure arising from torsional contributions. One
        can therefore define the effective torsional  equation-of-state parameter  as
        
            ω_T≡p_T/ρ_T= -f-T f'+2T^2f”/(1+f'+
            2 T
            f”)(f-2Tf') .
        
        In these classes of theories, the above effective torsional terms are responsible for the
        accelerated phases of the early or/and late Universe <cit.>.
        
        Let us present now  three  specific f(T) forms, which are the
        viable ones amongst the variety of f(T) models  with two parameters out of which
        one is independent, i.e which pass the basic observational tests
        <cit.>.
        
        
        
          *  The power-law model by Bengochea and Ferraro
        (hereafter f_1CDM) <cit.> is characterized by the form
            
            f(T) = β |T|^n,
        
        where β and n are the two model parameters.
        Inserting this f(T) form into  Friedmann equation (<ref>) at present,
        we
        acquire
        
            β=(6H_0^2)^1-nΩ_m0/2n-1,
        
         where   Ω_m0=8π G ρ_m/3H_0^2 is the matter
        density parameter at present, and
         
            H_0    =    73.02± 1.79
               ∼    2.1 × 10^-42
        
        is the current Hubble parameter value. The best fit on the parameter n is obtained
        taking the CC+H_0+SNeIa+BAO observational data, and it reads <cit.>
            n=0.05536 .
        
        Clearly, for n=0  the present scenario reduces to ΛCDM cosmology, namely
        T+f(T)=T-2Λ, with Λ=-β/2.
        
        
        
          *  The Linder model (hereafter f_2CDM) <cit.> arises from
        
            f(T)=α T_0(1-e^-p√(T/T_0)),  
            p=1/b ,
        
        with α and p (b) the two model parameters. In this case
        (<ref>)  gives that
        
            α=Ω_m0/1-(1+p)e^-p .
        
         The CC+H_0+SNeIa+BAO observational data imply that the best fit of b is
        <cit.>
            b=0.04095 .
        
        As we can see,   for p → +∞ the present scenario reduces
        to  ΛCDM cosmology.
        
        
        
         
          *  Motivated by  exponential f(R) gravity <cit.>, Bamba et. al.
        introduced the following   f(T) model (hereafter f_3CDM)
        <cit.>:
         
            f(T)=α T_0(1-e^-pT/T_0),    p=1/b ,
        
         with α and p  (b) the two model parameters.
         In  this case we acquire
         
            α=Ω_m0/1-(1+2p)e^-p.
        
         For this model, and using CC+H_0+SNeIa+BAO observational data, the best fit is found
        to be <cit.>
            b=0.03207 .
        
          Similarly to the previous case  we can immediately see that    f_3CDM model tends
        to   ΛCDM cosmology for p → +∞.
        
        
        
        
        The above  f(T) models are considered viable  in
        literature because pass the basic observational tests <cit.>.  They are
        characterized
        by two free parameters. Actually there are two more models with two
        free parameters, namely the logarithmic model  <cit.>,
        
            f(T)=α T_0√(T/cT_0)ln(cT_0/T ) ,
        
          and the hyperbolic-tangent model
        <cit.>,
        
            f(T)=α(-T)^ntanh(T_0/T) .
        
        Nevertheless since these two models do not possess  ΛCDM
        cosmology as a limiting case and since they are in tension with
        observational data <cit.>, in this work
        we do not consider them.
        
        Finally, let us note that one could also
        construct f(T) models with more than two parameters, for example,  combining the
        above  scenarios.  However, considering  many free parameters would be a
        significant disadvantage concerning the corresponding values of the
        information criteria.
        
        
        
        
        
        § BIG BANG NUCLEOSYNTHESIS  IN F(T) COSMOLOGY
        
        
        
        In the Section,  we examine the BBN in the framework of
        f(T) cosmology. As it is well known, BBN occurs during the radiation
        dominated era. The energy density of relativistic particles filling up the Universe  is
        given by ρ=π^2/30g_*  T^4,
        where g_*∼ 10 is
        the effective number of degrees of freedom and T the temperature (in the
        Appendix we review the main
        features related to the BBN physics). The neutron abundance is computed via the
        conversion rate of protons into neutrons, namely
         
            λ_pn( T)=λ_n+ν_e→ p+e^-+λ_n+e^+→ p+ν̅_e+λ_n→ p+e^- +
            ν̅_e ,
        
        and its inverse λ_np( T). The relevant quantity is the total rate given by
         
            Λ( T)=λ_np( T)+λ_pn( T) .
        
        Explicit calculations of Eq. (<ref>) lead to (see (<ref>) in the
        Appendix)
         
            Λ( T) =4 A  T^3(4!  T^2+2× 3!  Q T+2!
             Q^2) ,
        
        where Q=m_n-m_p is the mass difference of neutron and proton, and A=1.02 ×
        10^-11GeV^-4. The primordial mass
        fraction of ^4 He can be estimated by making use of the relation <cit.>
            Y_p≡λ 2 x(t_f)/1+x(t_f) .
        
        Here λ=e^-(t_n-t_f)/τ, with t_f   the time of the freeze-out of the weak
        interactions, t_n   the time of the freeze-out of the nucleosynthesis,
        τ the neutron mean lifetime given in (<ref>), and
        x(t_f)=e^-
        Q/ T(t_f) is the neutron-to-proton equilibrium ratio.
        The function λ(t_f) is interpreted as the fraction of neutrons that decay into
        protons during the interval t∈ [t_f, t_n]. Deviations from the fractional mass Y_p
        due to
        the variation of the freezing temperature T_f are given by
         
            δ
            Y_p=Y_p[(1-Y_p/2λ)ln(2λ/Y_p
            -1)-2t_f/τ]
                δ T_f/ T_f ,
        
        where we have set δ T(t_n)=0 since T_n is fixed by the deuterium
        binding energy
        <cit.>. The experimental estimations of the mass
        fraction Y_p of baryon
        converted to ^4 He
        during the Big Bang Nucleosynthesis are
        <cit.>
            Y_p=0.2476 ,      |δ Y_p| < 10^-4 .
        
         Inserting these into (<ref>) one infers the upper bound on δ T_f/ T_f, namely
         
            |δ T_f/ T_f| < 4.7 × 10^-4 .
        
        
        During the BBN, at the radiation dominated era, the scale factor evolves as a∼
        t^1/2, where t is cosmic time. The  torsional energy density ρ_T is
        treated as a perturbation to the radiation energy density ρ. The relation between
        the
        cosmic time and the temperature
        is given by 1/t≃ (32π^3 g_*/90)^1/2
        T^2/M_P
        (or
        T(t)≃ (t/sec)^1/2MeV). Furthermore, we use the entropy
        conservation
        S∼ a^3
         T^3=constant. The expansion rate of
        the Universe  is derived from (<ref>), and can be rewritten in the form
          
            H   =   H_GR^(R)√(1+ρ_T/ρ)=H_GR+δ H ,  
            δ H   =   (√(1+ρ_T/ρ)-1)H_GR ,
        
        where H_GR=√(8π G/2ρ) (H_GR is the expansion rate
        of the
        Universe  in General Relativity). Thus, from the relation Λ= H, one derives the
        freeze-out
        temperature
        T= T_f(1+δ T_f/ T_f), with T_f∼ 0.6 MeV (which follows from
        H_GR≃ q  T^5)
        and
        
            (√(1+ρ_T/ρ)-1)H_GR = 5q  T_f^4 δ T_f ,
        
        from which, in the regime ρ_T≪ρ, one obtains:
          
            δ T_f/ T_f≃ρ_T/ρH_GR/10 q
             T_f^5 ,
        
        with q=4! A≃ 9.6× 10^-36GeV^-4.
        
        
        In what follows we shall investigate the bounds that arise from the BBN constraints, on
        the free parameters of the three f(T) models presented in the previous Section.
        These constraint will be determined using Eqs. (<ref>) and (<ref>).
        Moreover, we shall use the   numerical values
         
            Ω_m0=0.25 ,    T_0=2.6× 10^-13 ,
        
        where T_0 is the present value of CMB temperature.
        
        
        
          * f_1CDM model.
        
        For the f_1CDM model of (<ref>) relation (<ref>) gives
         
            ρ_T    =   1/16π G[β (2n-1)(|6H^2|)^n] 
               =   3H_0^2/8π G Ω_m0( T/ T_0)^4n ,
        
        and then (<ref>) yields
        
         
            δ T_f/ T_f=π/15√(π
            g_*/5) Ω_m0( T_f/
            T_0)^4(n-1)1/qM_Pl T_f^3 .
        
        In Fig. <ref> we depict δ T_f/ T_f from
        (<ref>)
        vs n, as well as the upper bound from (<ref>). As we can see,
        constraints from BBN require n≲ 0.94. Remarkably, this bound is in agreement with
          the best fit for n of (<ref>), namely n=0.05536, that was obtained
        using CC+H_0+SNeIa+BAO observational data in <cit.>.
        
          * f_2,3CDM model.
        
        In the case of f_2CDM model of (<ref>) and f_3CDM  model of (<ref>), and
        for the purpose of this analysis, we can unified their investigation  parameterizing them
          as
              
            f(T)= α T_0 [1-e^-p(T/T_0)^m] ,
        
              with
              
            α = Ω_m0/1-(1+2mp)e^-p ,
        
        where m=1/2 for model  f_2CDM  and m=1 for model  f_3CDM.
        Inserting (<ref>) into (<ref>) we acquire
        
            δ T_f/ T_f=2πα/15√(π g_*/5)( T_0/ T_f)^41/qM_P 
            T_f^3
                       ·{[mp(
            T_0/ T_f)^4m+1/2]e^-p( T_f/ T_0)^4m -1/2}.
        
        Hence, using this relation we can calculate the value of |δ T_f/ T_f|
        for various values of p=1/b that span the order of magnitude of the best fit values
        (<ref>) and (<ref>) that were obtained using CC+H_0+SNeIa+BAO
        observational data in <cit.>, and we present our results in Table <ref>.
        As we can see, in all cases the value of   |δ T_f/ T_f| is well below
        the BBN bound  (<ref>). Hence, BBN cannot impose constraints on the
        parameter values of f_2CDM and f_3CDM models.
        
        
        
        
        § CONCLUSIONS
        
        
        In this work we have investigated the implications of f(T) gravity to the formation
        of light elements in the early Universe, i.e. to the BBN.
        In particular, we have examined the three most used and well studied viable f(T)
        models, namely the power law, the exponential and the square-root exponential, and we
        have confronted them with BBN calculations based on current observational data on
        primordial abundance of ^4He. Hence, we were able to extract constraints on
        their free parameters.
        
        Concerning the power-law f(T) model, the obtained constraint on the exponent n,
        is n≲ 0.94. Remarkably, this bound is in agreement with the constraints
        obtained using CC+H_0+SNeIa+BAO observational data  <cit.>. Concerning the
        exponential and the square-root exponential, we showed that, for  realistic regions of
        free parameters,  they always satisfy the BBN bounds. This means that, in these cases,
        BBN cannot
        impose strict
        constraints on the values  of  free parameters.
        
        In summary,  we showed that  viable f(T) models, namely those that pass
        the basic observational tests, can also satisfy the BBN constraints. This feature acts as
        an additional advantage of f(T) gravity, which might be a successful candidate for
         describing the  gravitational interaction. As discussed in <cit.>, this kind
        of
        constraints could contribute in the debate of fixing the most realistic picture that can
        be based
        on curvature or torsion.
        
        
        
        
         
        This article is based upon work from COST Action CA15117 “Cosmology and Astrophysics
        Network
        for Theoretical Advances and Training Actions” (CANTATA), supported by COST (European
        Cooperation
        in Science and Technology).
         *
        
        
        
        § BIG BANG NUCLEOSYNTHESIS
        
        
        In this Appendix we briefly review the main features of Big Bang Nucleosynthesis following
        <cit.>.
        
        
        
        
        
        
        
        
        
        
        
        In the early Universe, the primordial ^4He was formed at temperature T∼
        100 MeV. The energy and number density were formed by relativistic leptons (electron,
        positron and neutrinos) and photons. The rapid collisions maintain all these particles in
        thermal equilibrium. Interactions of protons and neutrons were kept in thermal
        equilibrium by means of their interactions with leptons
         
            ν_e+n    ⟷    p+e^- 
            
                e^++n    ⟷    p + ν̅_e 
            
                n   ⟷    p+e^- + ν̅_e .
        
        
        The neutron abundance is estimated by computing the conversion rate of protons into
        neutrons, i.e. λ_pn( T), and its inverse λ_np( T). Thus,
        the weak interaction rates (at suitably high temperature)
        are given by
         
            Λ( T)=λ_np( T)+λ_pn( T) .
        
        The rate λ_np is the sum of the rates associated to the processes
        (<ref>)-(<ref>), namely
         
            λ_np=λ_n+ν_e→ p+e^-+λ_n+e^+→ p+ν̅_e+λ_n→ p+e^- +
            ν̅_e .
        
         Finally, the rate λ_np is related to the rate λ_pn as
        λ_np( T)=e^- Q/ Tλ_pn( T), with Q=m_n-m_p the mass difference of neutron and proton.
        
        During the freeze-out stage, one can use the following approximations
        <cit.>: (i) The
        temperatures of particles are the same, i.e. T_ν= T_e=
        T_γ= T. (ii) The temperature
        T is lower than the typical energies E that contribute to the
        integrals entering the definition of the rates (one can therefore replace the
        Fermi-Dirac distribution with the Boltzmann
        one, namely n≃ e^-E/ T). (iii) The electron mass m_e can be neglected
        with
        respect to the
        electron and neutrino energies (m_e≪ E_e, E_ν).
        
        Having these in mind, the interaction rate
        corresponding to the process (<ref>) is given by
          
            dλ_n+ν_e→ p+e^-= dμ (2π)^4 |⟨ M|^2⟩ W  ,
        
        where
        
            dμ   ≡   d^3p_e/(2π)^3 2E_ed^3p_ν_e/(2π)^3
            2E_ν_ed^3p_
            p/(2π)^3 2E_p , 
            
              W    ≡   δ^(4)( P)n(E_ν_e)[1-n(E_e)] , 
             P   ≡     p_n+p_ν_e-p_p-p_e ,  
             M   =    (g_w/8M_W)^2 [u̅_pΩ^μ u_n][u̅_eΣ_μ v_ν_e] , 
            Ω^μ   ≡   γ^μ(c_V-c_A γ^5) ,
              
            Σ^μ   ≡   γ^μ(1-γ^5)
             .
        
        In (<ref>) we have used the condition q^2 ≪ M_W^2, where M_W is the mass of the
        vector gauge boson W, with q^μ=p_n^μ-p_p^μ  the transferred
        momentum. From Eq. (<ref>) it follows that
         
            λ_n+ν_e→ p+e^-=A   T^5 I_y ,
        
        where
         
            A≡g_V+3g_A/2π^3 ,
        
        and where
        
            I_y=∫_y^∞ϵ(ϵ- Q')^2√(ϵ^2-y^2)  n(ϵ-
            Q)[1-n(ϵ)]dϵ,
        
         with
         
            y≡m_e/ T ,    Q'= Q/ T .
        
        
        
          A similar calculation for the process (<ref>) gives
          
            λ_e^+ + n → p+ ν̅_e=A   T^5 J_y ,
        
         with
         
            J_y=∫_y^∞ϵ(ϵ+ Q')^2√(ϵ^2-y^2) 
            n(ϵ)[1-n(ϵ+ Q')]dϵ ,
        
         which finally  results to
         
            λ_e^+ + n→ p+ν̅_e=A  T^3(4!  T^2+2× 3! 
            Q T+2! 
            Q^2) .
        
        
        Lastly, for the neutron decay (<ref>) one obtains
         
            τ=λ_n→ p+e^- +ν̅_e^-1≃ 887 sec .
        
         Hence, in the calculation of (<ref>) we can safely neglect the above
        interaction rate of the neutron decay, i.e. during the BBN the neutron can be
        considered as a stable particle.
        
        The above approximations (i)-(iii) lead to
        <cit.>
            λ_e^+ +n→ p+ν̅_e=λ_n+ν_e→ p+e^- .
        
         Thus, inserting   (<ref>) into (<ref>), and then
        into (<ref>), allows to derive the expression for
        Λ( T), namely
         
            Λ( T)≃ 2λ_np=4λ_e^+ +n→ p+ν̅_e ,
        
         which using (<ref>) leads to
          
            Λ( T) =4 A  T^3(4!  T^2+2× 3!  Q T+2!
             Q^2) .
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