Constraining f(T) teleparallel gravity by Big Bang Nucleosynthesis

TL;DR

This paper uses Big Bang Nucleosynthesis data to constrain three popular f(T) gravity models, finding that they can satisfy early universe bounds and are consistent with late-time cosmological data.

Contribution

It provides the first comprehensive BBN-based constraints on power-law, exponential, and square-root exponential f(T) gravity models, confirming their viability.

Findings

Power-law model constraints align with late-time data.

Exponential models satisfy BBN bounds within certain parameter regions.

Viable f(T) models can meet early universe nucleosynthesis constraints.

Abstract

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.

Tables

Table 1. Table 1: | δ ​ 𝒯 f / 𝒯 f | 𝛿 subscript 𝒯 𝑓 subscript 𝒯 𝑓 |\delta{\cal T}_{f}/{\cal T}_{f}| from ( 2 ) for different values of p = 1 / b 𝑝 1 𝑏 p=1/b , for m = 1 / 2 𝑚 1 2 m=1/2 ( f 2 subscript 𝑓 2 f_{2} CDM model) and m = 1 𝑚 1 m=1 ( f 3 subscript 𝑓 3 f_{3} CDM model).

$m$		$p=1/b$		$\left|\delta {𝒯}_f/{𝒯}_f\right|$
1/2		1		$5.723\times {10}^{-38}$
		$10$		$1.512\times {10}^{-38}$
		${10}^2$		$1.511\times {10}^{-38}$
1		1		$1.4586\times {10}^{-37}$
		10		$1.5131\times {10}^{-38}$
		${10}^2$		$1.5116\times {10}^{-38}$

Equations

$$T_{\mu \nu }^{\lambda }={\hat{\Gamma }}_{\nu \mu }^{\lambda }-{\hat{\Gamma }}_{\mu \nu }^{\lambda }=e_i^{\lambda }({\partial }_{\mu }{e}_{\nu }^i-{\partial }_{\nu }{e}_{\mu }^i).$$

$$I=\frac{1}{16\pi G}\int d^{4}xeT,$$

$$T={S_{\rho }}^{\mu \nu }{T^{\rho }}_{\mu \nu }.$$

$${S_{\rho }}^{\mu \nu }$$

$${K^{\mu \nu }}_{\rho }$$

$$I=\frac{1}{16\pi G}\int d^{4}xe\left[T+f(T)\right],$$

$$e^{-1}{\partial }_{\mu }(e{e}_i^{\rho }{S_{\rho }}^{\mu \nu })[1+f^{\prime }]-e_i^{\lambda }{T^{\rho }}_{\mu \lambda }{S_{\rho }}^{\nu \mu }[1+f^{\prime }]$$

$$+e_i^{\rho }{S_{\rho }}^{\mu \nu }({\partial }_{\mu }T)f^{\prime \prime }+\frac{1}{4}{e}_i^{\nu }[T+f]=4\pi G{e_i}^{\rho }{{\Theta }_{\rho }}^{\nu },$$

$$e_{\mu }^A=\mathrm{diag}(1,a,a,a),$$

$$d{s}^2=d{t}^2-a^2(t){\delta }_{ij}d{x}^{i}d{x}^j,$$

$$T=-6H^2.$$

$$12H^2[1+f^{\prime }]+[T+f]=16\pi G\rho ,$$

$$48H^2{f}^{\prime \prime }\dot{H}-(1+f^{\prime })[12H^2+4\dot{H}]-(T-f)=16\pi Gp.$$

$$H^2=\frac{8\pi G}{3}(\rho +{\rho }_T),$$

$$2\dot{H}+3H^2=-\frac{8\pi G}{3}(p+p_T),$$

$${\rho }_T$$

$$p_T$$

$${\omega }_T\equiv \frac{p_T}{{\rho }_T}=-\frac{f-T{f}^{\prime }+2T^2{f}^{\prime \prime }}{(1+f^{\prime }+2T{f}^{\prime \prime })(f-2T{f}^{\prime })}.$$

$$f(T)=\beta |T{|}^n,$$

$$\beta =(6H_0^2{)}^{1-n}\frac{{\Omega }_{m0}}{2n-1},$$

$$H_0$$

$$n=0.05536.$$

$$f(T)=\alpha T_0(1-e^{-p\sqrt{T/T_0}}),p=\frac{1}{b},$$

$$\alpha =\frac{{\Omega }_{m0}}{1-(1+p)e^{-p}}.$$

$$b=0.04095.$$

$$f(T)=\alpha T_0(1-e^{-pT/T_0}),p=\frac{1}{b},$$

$$\alpha =\frac{{\Omega }_{m0}}{1-(1+2p)e^{-p}}.$$

$$b=0.03207.$$

$$f(T)=\alpha T_0\sqrt{\frac{T}{c{T}_0}}\ln \left(\frac{c{T}_0}{T}\right),$$

$$f(T)=\alpha (-T{)}^n\tanh \left(\frac{T_0}{T}\right).$$

$${\lambda }_{pn}(\mathcal{T})={\lambda }_{n+{\nu }_e\to p+e^{-}}+{\lambda }_{n+e^{+}\to p+{\bar{\nu }}_e}+{\lambda }_{n\to p+e^{-}+{\bar{\nu }}_e},$$

$$\Lambda (\mathcal{T})={\lambda }_{np}(\mathcal{T})+{\lambda }_{pn}(\mathcal{T}).$$

Full text

I Introduction

Cosmological observations coming from Type Ia Supernovae Riess , cosmic microwave background radiation Spergel and the large scale structure Tegmark ; Eisenstein , 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 $\Lambda$, 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 Carroll ). Moreover, the $\Lambda$CDM paradigm, where cold dark matter (CDM) is considered into the game, may also suffer from the age problem, as it was shown in Yang , 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 Feng .

Over the past decade several DE models have been proposed, such as quintessence Caldwell , phantom Caldwell2 , k-essence Armendariz , tachyon Padmanabhan , quintom Feng ; Cai:2009zp ; eli , Chaplygin gas Kamenshchik , generalized Chaplygin gas (GCG) Bento , holographic DE Cohen ; Li2004 , new agegraphic DE Wei , Ricci DE Gao 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 capfra ; Nojiri ; book ; fRrev (this class is very efficient in verifying observational and theoretical constraints and explain the Universe acceleration and phantom crossing noj1 ; noj2 ; noj4 ; noj5 ), or gravity with higher curvature invariants Mota , by coupling the Ricci scalar to a scalar field Farajollahi , by introducing a vector field contribution Zuntz , or by using properties of gravity in higher dimensional spacetimes Camera (for a review see noj3 ; Capozziello:2011et ).

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 Einstein ; Einstein2 , is closely related to General Relativity, except for “boundary terms” seb1 ; seb2 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 Hayashi ; Pereira.book ; Maluf:2013gaa ; krssak . Hence, one can start from TEGR and construct various gravitational modifications based on torsion, with $f(T)$ gravity being the most studied one Ferraro2 ; Ferraro3 ; Linder:2010py . 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 Ferraro2 ; Ferraro3 ; Linder:2010py ; Wu1 ; Chen:2010va ; pertft ; Dent:2011zz ; Bamba:2010wb ; Wu ; Zhang:2011qp ; Bengochea ; Yang2 ; Cai11 ; Li:2011rn ; Bamba:2011pz ; Daouda:2011rt ; Atazadeh:2011aa ; Karami:2012fu ; Cardone:2012xq ; Otalora:2013tba ; Ong:2013qja ; Haro:2014wha ; Harko:2014sja ; Hanafy:2014ica ; Capozziello:2015rda ; Bahamonde:2015zma ; Carloni:2015lsa ; Fazlpour:2016bxo (for a detailed review, see Cai:2015emx ). 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 $\sim 0.01$ 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 kolb ; bernstein , 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 ${}^{4}He$, and we shall extract constraints on their free parameters.

The layout of the paper is as follows. In Section II we review $f(T)$ gravity and the related cosmological models. In Section III we use BBN calculations in order to impose constraints on the free parameters of specific $f(T)$ gravity models. Conclusions are reported in Section IV. Finally, in the Appendix we summarize the main notions of BBN physics.

II $f(T)$ 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 $e_{i}(x^{\mu})$, $i=0,1,2,3$, which forms an orthonormal basis in the tangent space at each point $x^{\mu}$ of the manifold, i.e. $e_{i}\cdot e_{j}=\eta_{ij}$, with $\eta_{ij}$ the Minkowsky metric with signature $-2$: $\eta_{ij}=diag(1,-1,-1,-1)$. Denoting with $e^{\mu}_{i}$, $\mu=0,1,2,3$ the components of the vectors $e_{i}$ in a coordinate basis $\partial_{\mu}$, one can write $e_{i}=e^{\mu}_{i}\partial_{\mu}$. 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 $g_{\mu\nu}(x)=\eta_{ij}e^{i}_{\mu}(x)e^{j}_{\nu}(x)$.

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:

[TABLE]

Remarkably, the torsion tensor (1) encompasses all the information about the gravitational field. The Lagrangian density is built using its contractions, and hence the teleparallel action is given by

[TABLE]

with $e=det(e^{i}_{\mu})=\sqrt{-g}$, and where the torsion scalar $T$ reads as

[TABLE]

Here, it is

[TABLE]

with ${K^{\mu\nu}}_{\rho}$ the contorsion tensor which gives the difference between Weitzenböck and Levi-Civita connections. Finally, the variation of action (2) 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 (2) 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

[TABLE]

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 Cai:2015emx

[TABLE]

where $f^{\prime}\equiv df/dT$, ${S_{i}}^{\mu\nu}={e_{i}}^{\rho}{S_{\rho}}^{\mu\nu}$ and $\Theta_{\mu\nu}$ 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

[TABLE]

which corresponds to a flat Friedmann-Robertson-Walker (FRW) background metric of the form

[TABLE]

where $a(t)$ is the scale factor. Equations (1), (3), (4) and (5) allow to derive a relation between the torsion $T$ and the Hubble parameter ${\displaystyle H=\frac{\dot{a}}{a}}$, namely

[TABLE]

Hence, in the case of FRW geometry, and assuming that the matter sector corresponds to a perfect fluid with energy density $\rho$ and pressure $p$, the $i=0=\nu$ component of (II) yields

[TABLE]

while the $i=1=\nu$ component gives

[TABLE]

The equations close by considering the equation of continuity for the matter sector, namely $\dot{\rho}+3H(\rho+p)=0$. One can rewrite (11) and (12) in the usual form

[TABLE]

[TABLE]

where

[TABLE]

are the effective energy density and pressure arising from torsional contributions. One can therefore define the effective torsional equation-of-state parameter as

[TABLE]

In these classes of theories, the above effective torsional terms are responsible for the accelerated phases of the early or/and late Universe Cai:2015emx .

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 Nesseris:2013jea .

The power-law model by Bengochea and Ferraro (hereafter $f_{1}$CDM) Ferraro3 is characterized by the form

[TABLE]

where $\beta$ and $n$ are the two model parameters. Inserting this $f(T)$ form into Friedmann equation (11) at present, we acquire

[TABLE]

where ${\displaystyle\Omega_{m0}=\frac{8\pi G\rho_{m}}{3H_{0}^{2}}}$ is the matter density parameter at present, and

[TABLE]

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 TJCAP16

[TABLE]

Clearly, for $n=0$ the present scenario reduces to $\Lambda$CDM cosmology, namely ${T}+f(T)=T-2\Lambda$, with $\Lambda=-\beta/2$. 2. 2.

The Linder model (hereafter $f_{2}$CDM) Linder:2010py arises from

[TABLE]

with $\alpha$ and $p$ ($b$) the two model parameters. In this case (11) gives that

[TABLE]

The $CC+H_{0}+SNeIa+BAO$ observational data imply that the best fit of $b$ is TJCAP16

[TABLE]

As we can see, for $p\rightarrow+\infty$ the present scenario reduces to $\Lambda$CDM cosmology. 3. 3.

Motivated by exponential $f(R)$ gravity Linder:2009jz , Bamba et. al. introduced the following $f(T)$ model (hereafter $f_{3}$CDM) Bamba:2010wb :

[TABLE]

with $\alpha$ and $p$ ($b$) the two model parameters. In this case we acquire

[TABLE]

For this model, and using $CC+H_{0}+SNeIa+BAO$ observational data, the best fit is found to be TJCAP16

[TABLE]

Similarly to the previous case we can immediately see that $f_{3}$CDM model tends to $\Lambda$CDM cosmology for $p\rightarrow+\infty$.

The above $f(T)$ models are considered viable in literature because pass the basic observational tests Cai:2015emx . They are characterized by two free parameters. Actually there are two more models with two free parameters, namely the logarithmic model Bamba:2010wb ,

[TABLE]

and the hyperbolic-tangent model Wu ,

[TABLE]

Nevertheless since these two models do not possess $\Lambda$CDM cosmology as a limiting case and since they are in tension with observational data Nesseris:2013jea , 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.

III 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 ${\displaystyle\rho=\frac{\pi^{2}}{30}g_{*}{\cal T}^{4}}$, where $g_{*}\sim 10$ is the effective number of degrees of freedom and ${\cal T}$ 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

[TABLE]

and its inverse $\lambda_{np}({\cal T})$. The relevant quantity is the total rate given by

[TABLE]

Explicit calculations of Eq. (29) lead to (see (65) in the Appendix)

[TABLE]

where ${\cal Q}=m_{n}-m_{p}$ is the mass difference of neutron and proton, and $A=1.02\times 10^{-11}$GeV*-4*. The primordial mass fraction of ${}^{4}He$ can be estimated by making use of the relation kolb

[TABLE]

Here $\lambda=e^{-(t_{n}-t_{f})/\tau}$, with $t_{f}$ the time of the freeze-out of the weak interactions, $t_{n}$ the time of the freeze-out of the nucleosynthesis, $\tau$ the neutron mean lifetime given in (62), and $x(t_{f})=e^{-{\cal Q}/{\cal T}(t_{f})}$ is the neutron-to-proton equilibrium ratio. The function $\lambda(t_{f})$ is interpreted as the fraction of neutrons that decay into protons during the interval $t\in[t_{f},t_{n}]$. Deviations from the fractional mass $Y_{p}$ due to the variation of the freezing temperature ${\cal T}_{f}$ are given by

[TABLE]

where we have set $\delta{\cal T}(t_{n})=0$ since ${\cal T}_{n}$ is fixed by the deuterium binding energy torres ; Lambiase1 ; Lambiase2 ; Lambiase3 . The experimental estimations of the mass fraction $Y_{p}$ of baryon converted to ${}^{4}He$ during the Big Bang Nucleosynthesis are coc ; altriBBN1 ; altriBBN2 ; altriBBN3 ; altriBBN4 ; altriBBN5 ; altriBBN6

[TABLE]

Inserting these into (32) one infers the upper bound on $\frac{\delta{\cal T}_{f}}{{\cal T}_{f}}$, namely

[TABLE]

During the BBN, at the radiation dominated era, the scale factor evolves as $a\sim t^{1/2}$, where $t$ is cosmic time. The torsional energy density $\rho_{T}$ is treated as a perturbation to the radiation energy density $\rho$. The relation between the cosmic time and the temperature is given by ${\displaystyle\frac{1}{t}\simeq(\frac{32\pi^{3}g_{*}}{90})^{1/2}\frac{{\cal T}^{2}}{M_{P}}}$ (or ${\cal T}(t)\simeq(t/\text{sec})^{1/2}$MeV). Furthermore, we use the entropy conservation $S\sim a^{3}{\cal T}^{3}=constant$. The expansion rate of the Universe is derived from (13), and can be rewritten in the form

[TABLE]

where $H_{GR}=\displaystyle{\sqrt{\frac{8\pi G}{2}\rho}}$ ($H_{GR}$ is the expansion rate of the Universe in General Relativity). Thus, from the relation $\Lambda=H$, one derives the freeze-out temperature ${\cal T}={\cal T}_{f}\left(1+\frac{\delta{\cal T}_{f}}{{\cal T}_{f}}\right)$, with ${\cal T}_{f}\sim 0.6$ MeV (which follows from $H_{GR}\simeq q{\cal T}^{5}$) and

[TABLE]

from which, in the regime $\rho_{T}\ll\rho$, one obtains:

[TABLE]

with $q=4!A\simeq 9.6\times 10^{-36}$GeV*-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. (38) and (15). Moreover, we shall use the numerical values

[TABLE]

where ${\cal T}_{0}$ is the present value of CMB temperature.

$f_{1}$CDM model.

For the $f_{1}$CDM model of (18) relation (15) gives

[TABLE]

and then (38) yields

[TABLE]

In Fig. 1 we depict $\delta{\cal T}_{f}/{\cal T}_{f}$ from (40) vs $n$, as well as the upper bound from (34). As we can see, constraints from BBN require $n\lesssim 0.94$. Remarkably, this bound is in agreement with the best fit for $n$ of (20), namely $n=0.05536$, that was obtained using $CC+H_{0}+SNeIa+BAO$ observational data in TJCAP16 .

$f_{2,3}$CDM model.

In the case of $f_{2}$CDM model of (21) and $f_{3}$CDM model of (24), and for the purpose of this analysis, we can unified their investigation parameterizing them as

[TABLE]

with

[TABLE]

where $m=\frac{1}{2}$ for model $f_{2}$CDM and $m=1$ for model $f_{3}$CDM. Inserting (41) into (38) we acquire

[TABLE]

Hence, using this relation we can calculate the value of $|\delta{\cal T}_{f}/{\cal T}_{f}|$ for various values of $p=1/b$ that span the order of magnitude of the best fit values (20) and (23) that were obtained using $CC+H_{0}+SNeIa+BAO$ observational data in TJCAP16 , and we present our results in Table 1. As we can see, in all cases the value of $|\delta{\cal T}_{f}/{\cal T}_{f}|$ is well below the BBN bound (34). Hence, BBN cannot impose constraints on the parameter values of $f_{2}$CDM and $f_{3}$CDM models.

IV 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 ${}^{4}He$. 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 $n\lesssim 0.94$. Remarkably, this bound is in agreement with the constraints obtained using $CC+H_{0}+SNeIa+BAO$ observational data TJCAP16 . 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 Cai:2015emx , this kind of constraints could contribute in the debate of fixing the most realistic picture that can be based on curvature or torsion.

Acknowledgements.

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).

Appendix A Big Bang Nucleosynthesis

In this Appendix we briefly review the main features of Big Bang Nucleosynthesis following kolb ; bernstein . In the early Universe, the primordial ${}^{4}He$ was formed at temperature ${\cal T}\sim 100$ 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

[TABLE]

The neutron abundance is estimated by computing the conversion rate of protons into neutrons, i.e. $\lambda_{pn}({\cal T})$, and its inverse $\lambda_{np}({\cal T})$. Thus, the weak interaction rates (at suitably high temperature) are given by

[TABLE]

The rate $\lambda_{np}$ is the sum of the rates associated to the processes (43)-(45), namely

[TABLE]

Finally, the rate $\lambda_{np}$ is related to the rate $\lambda_{pn}$ as $\lambda_{np}({\cal T})=e^{-{\cal Q}/{\cal T}}\lambda_{pn}({\cal T})$, with ${\cal Q}=m_{n}-m_{p}$ the mass difference of neutron and proton.

During the freeze-out stage, one can use the following approximations bernstein : (i) The temperatures of particles are the same, i.e. ${\cal T}_{\nu}={\cal T}_{e}={\cal T}_{\gamma}={\cal T}$. (ii) The temperature ${\cal T}$ 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 $n\simeq e^{-E/{\cal T}}$). (iii) The electron mass $m_{e}$ can be neglected with respect to the electron and neutrino energies ($m_{e}\ll E_{e},E_{\nu}$).

Having these in mind, the interaction rate corresponding to the process (43) is given by

[TABLE]

where

[TABLE]

In (52) we have used the condition $q^{2}\ll M_{W}^{2}$, where $M_{W}$ is the mass of the vector gauge boson $W$, with $q^{\mu}=p_{n}^{\mu}-p_{p}^{\mu}$ the transferred momentum. From Eq. (48) it follows that

[TABLE]

where

[TABLE]

and where

[TABLE]

with

[TABLE]

A similar calculation for the process (44) gives

[TABLE]

with

[TABLE]

which finally results to

[TABLE]

Lastly, for the neutron decay (45) one obtains

[TABLE]

Hence, in the calculation of (47) 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 bernstein

[TABLE]

Thus, inserting (63) into (47), and then into (46), allows to derive the expression for $\Lambda({\cal T})$, namely

[TABLE]

which using (61) leads to

[TABLE]