Constraining f (T ) teleparallel gravity by Big Bang Nucleosynthesis
S. Capozziello1,2,3, G. Lambiase4,5, and E.N. Saridakis6,7
arXiv:1702.07952v1 [astro-ph.CO] 25 Feb 2017
1
Dipartimento di Fisica ”E. Pancini”, Università di Napoli “Federico II”,
Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy
2
Istituto 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
4
Dipartimento di Fisica E.R. Cainaiello, University of Salerno,
Via Giovanni Paolo II, I 84084-Fisciano (SA), Italy
5
INFN, Gruppo Collegato di Salerno, Sezione di Napoli,
Via Giovanni Paolo II, I 84084-Fisciano (SA), Italy
6
Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece
7
CASPER, Physics Department, Baylor University, Waco, TX 76798-7310, USA
We use BBN observational data on primordial abundance of 4 He 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.
PACS numbers: 04.50.Kd, 98.80.-k, 98.80.Es, 26.35.+c
I.
INTRODUCTION
Cosmological observations coming from Type Ia Supernovae [1], cosmic microwave background radiation [2] and
the large scale structure [3, 4], 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 [5]). 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 [6], 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 [7].
Over the past decade several DE models have been proposed, such as quintessence [8], phantom [9], k-essence [10],
tachyon [11], quintom [7, 12, 13], Chaplygin gas [14], generalized Chaplygin gas (GCG) [15], holographic DE [16, 17],
new agegraphic DE [18], Ricci DE [19] 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 [20–23] (this class is
very efficient in verifying observational and theoretical constraints and explain the Universe acceleration and phantom
crossing [24–27]), or gravity with higher curvature invariants [28], by coupling the Ricci scalar to a scalar field [29],
by introducing a vector field contribution [30], or by using properties of gravity in higher dimensional spacetimes [31]
(for a review see [32, 33]).
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 [34, 35], is
closely related to General Relativity, except for “boundary terms” [36, 37] 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 [38–41]. Hence, one can start
from TEGR and construct various gravitational modifications based on torsion, with f (T ) gravity being the most
studied one [42–44]. 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
[42–69] (for a detailed review, see [70]). The main advantage of f (T ) gravity is that the field equations are 2nd-order
2
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 [71, 72], 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 ei (xµ ), i = 0, 1, 2, 3, which forms an orthonormal basis in the tangent space at each point
xµ of the manifold, i.e. ei · ej = ηij , with ηij the Minkowsky metric with signature −2: ηij = diag(1, −1, −1, −1).
Denoting with eµi , µ = 0, 1, 2, 3 the components of the vectors ei in a coordinate basis ∂µ , one can write ei = eµi ∂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 gµν (x) = ηij eiµ (x)ejν (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:
λ
Tµν
= Γ̂λνµ − Γ̂λµν = eλi (∂µ eiν − ∂ν eiµ ).
(1)
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
Z
1
I=
d4 xeT ,
(2)
16πG
√
with e = det(eiµ ) = −g, and where the torsion scalar T reads as
T = Sρ µν T ρ µν .
(3)
Here, it is
1 µν
(K ρ + δρµ T θν θ − δρν T θµ θ )
(4)
2
1
(5)
K µν ρ = − (T µν ρ − T νµ ρ − Tρ µν ) ,
2
with K µν ρ 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
Z
1
I=
d4 xe [T + f (T )],
(6)
16πG
Sρ µν =
that reduces to TEGR as soon as f (T ) = 0. Considering additionally a matter Lagrangian Lm , variation with respect
to the vierbein gives the field equations [70]
e−1 ∂µ (eeρi Sρ µν )[1 + f ′ ] − eλi T ρ µλ Sρ νµ [1 + f ′ ]
1
+eρi Sρ µν (∂µ T )f ′′ + eνi [T + f ] = 4πG ei ρ Θρ ν ,
4
(7)
3
where f ′ ≡ df /dT , Si µν = ei ρ Sρ µν and Θµν 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
eA
µ = diag(1, a, a, a),
(8)
which corresponds to a flat Friedmann-Robertson-Walker (FRW) background metric of the form
ds2 = dt2 − a2 (t) δij dxi dxj ,
(9)
where a(t) is the scale factor. Equations (1), (3), (4) and (5) allow to derive a relation between the torsion T and the
ȧ
Hubble parameter H = , namely
a
T = −6H 2 .
(10)
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 (7) yields
12H 2 [1 + f ′ ] + [T + f ] = 16πGρ,
(11)
48H 2 f ′′ Ḣ − (1 + f ′ )[12H 2 + 4Ḣ] − (T − f ) = 16πGp.
(12)
while the i = 1 = ν component gives
The equations close by considering the equation of continuity for the matter sector, namely ρ̇ + 3H(ρ + p) = 0. One
can rewrite (11) and (12) in the usual form
H2 =
8πG
(ρ + ρT ),
3
2Ḣ + 3H 2 = −
8πG
(p + pT ) ,
3
(13)
(14)
where
ρT
pT
′
3
Tf
f
=
,
−
8πG 3
6
1 f − T f ′ + 2T 2 f ′′
=
,
16πG 1 + f ′ + 2T f ′′
(15)
(16)
are the effective energy density and pressure arising from torsional contributions. One can therefore define the effective
torsional equation-of-state parameter as
ωT ≡
f − T f ′ + 2T 2 f ′′
pT
=−
.
ρT
(1 + f ′ + 2T f ′′ )(f − 2T f ′ )
(17)
In these classes of theories, the above effective torsional terms are responsible for the accelerated phases of the early
or/and late Universe [70].
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 [73].
1. The power-law model by Bengochea and Ferraro (hereafter f1 CDM) [43] is characterized by the form
f (T ) = β|T |n ,
(18)
where β and n are the two model parameters. Inserting this f (T ) form into Friedmann equation (11) at present,
we acquire
β = (6H02 )1−n
Ωm0
,
2n − 1
(19)
4
where Ωm0 =
8πGρm
is the matter density parameter at present, and
3H02
H0 = 73.02 ± 1.79km/(sec Mpc)
∼ 2.1 × 10−42 GeV
is the current Hubble parameter value. The best fit on the parameter n is obtained taking the CC + H0 +
SN eIa + BAO observational data, and it reads [74]
n = 0.05536 .
(20)
Clearly, for n = 0 the present scenario reduces to ΛCDM cosmology, namely T + f (T ) = T − 2Λ, with Λ = −β/2.
2. The Linder model (hereafter f2 CDM) [44] arises from
√
f (T ) = αT0 (1 − e−p T /T0 ),
p=
1
,
b
(21)
with α and p (b) the two model parameters. In this case (11) gives that
α=
Ωm0
.
1 − (1 + p)e−p
(22)
The CC + H0 + SN eIa + BAO observational data imply that the best fit of b is [74]
b = 0.04095 .
(23)
As we can see, for p → +∞ the present scenario reduces to ΛCDM cosmology.
3. Motivated by exponential f (R) gravity [75], Bamba et. al. introduced the following f (T ) model (hereafter
f3 CDM) [49]:
f (T ) = αT0 (1 − e−pT /T0 ),
p=
1
,
b
(24)
with α and p (b) the two model parameters. In this case we acquire
α=
Ωm0
.
1 − (1 + 2p)e−p
(25)
For this model, and using CC + H0 + SN eIa + BAO observational data, the best fit is found to be [74]
b = 0.03207 .
(26)
Similarly to the previous case we can immediately see that f3 CDM model tends to ΛCDM cosmology for
p → +∞.
The above f (T ) models are considered viable in literature because pass the basic observational tests [70]. They
are characterized by two free parameters. Actually there are two more models with two free parameters, namely the
logarithmic model [49],
r
cT0
T
,
(27)
ln
f (T ) = αT0
cT0
T
and the hyperbolic-tangent model [50],
n
f (T ) = α(−T ) tanh
T0
T
.
(28)
Nevertheless since these two models do not possess ΛCDM cosmology as a limiting case and since they are in tension
with observational data [73], 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.
5
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
π2
g∗ T 4 ,
the radiation dominated era. The energy density of relativistic particles filling up the Universe is given by ρ =
30
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
λpn (T ) = λn+νe →p+e− + λn+e+ →p+ν̄e + λn→p+e− +ν̄e ,
and its inverse λnp (T ). The relevant quantity is the total rate given by
Λ(T ) = λnp (T ) + λpn (T ) .
(29)
Explicit calculations of Eq. (29) lead to (see (A.23) in the Appendix)
Λ(T ) = 4A T 3 (4!T 2 + 2 × 3!QT + 2!Q2 ) ,
(30)
where Q = mn − mp 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 [71]
Yp ≡ λ
2x(tf )
.
1 + x(tf )
(31)
Here λ = e−(tn −tf )/τ , with tf the time of the freeze-out of the weak interactions, tn the time of the freeze-out of
the nucleosynthesis, τ the neutron mean lifetime given in (A.20), and x(tf ) = e−Q/T (tf ) is the neutron-to-proton
equilibrium ratio. The function λ(tf ) is interpreted as the fraction of neutrons that decay into protons during the
interval t ∈ [tf , tn ]. Deviations from the fractional mass Yp due to the variation of the freezing temperature Tf are
given by
Yp
2λ
2tf δTf
δYp = Yp 1 −
ln
−1 −
,
(32)
2λ
Yp
τ
Tf
where we have set δT (tn ) = 0 since Tn is fixed by the deuterium binding energy [76–79]. The experimental estimations
of the mass fraction Yp of baryon converted to 4 He during the Big Bang Nucleosynthesis are [80–86]
|δYp | < 10−4 .
Yp = 0.2476 ,
(33)
δT
Inserting these into (32) one infers the upper bound on Tff , namely
δTf −4
Tf < 4.7 × 10 .
(34)
During the BBN, at the radiation dominated era, the scale factor evolves as a ∼ t1/2 , where t is cosmic time. The
torsional energy density ρT is treated as a perturbation to the radiation energy density ρ. The relation between the
1
32π 3 g∗ 1/2 T 2
cosmic time and the temperature is given by ≃ (
(or T (t) ≃ (t/sec)1/2 MeV). Furthermore, we use
)
t
90
MP
the entropy conservation S ∼ a3 T 3 = constant. The expansion rate of the Universe is derived from (13), and can be
rewritten in the form
r
ρT
(R)
H = HGR 1 +
= HGR + δH ,
(35)
ρ
r
ρT
− 1 HGR ,
(36)
1+
δH =
ρ
r
8πG
where HGR =
ρ (HGR is the expansion rate of the Universe in General Relativity). Thus, from the relation
2
δT
Λ = H, one derives the freeze-out temperature T = Tf 1 + Tff , with Tf ∼ 0.6 MeV (which follows from HGR ≃ qT 5 )
and
r
ρT
− 1 HGR = 5qTf4 δTf ,
(37)
1+
ρ
6
from which, in the regime ρT ≪ ρ, one obtains:
ρT HGR
δTf
≃
,
Tf
ρ 10qTf5
(38)
with q = 4!A ≃ 9.6 × 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
T0 = 2.6 × 10−13 GeV ,
Ωm0 = 0.25 ,
where T0 is the present value of CMB temperature.
1. f1 CDM model.
For the f1 CDM model of (18) relation (15) gives
1 β(2n − 1)(|6H 2 |)n
16πG
4n
3H02
T
=
Ωm0
,
8πG
T0
ρT =
(39)
and then (38) yields
π
δTf
=
Tf
15
r
πg∗
Ωm0
5
Tf
T0
4(n−1)
1
qMP l Tf3
.
(40)
In Fig. 1 we depict δTf /Tf from (40) vs n, as well as the upper bound from (34). As we can see, constraints
from BBN require n . 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 + H0 + SN eIa + BAO observational data in [74].
FIG. 1: δTf /Tf from (40) vs n (thick line) for the f1 CDM model of (18), and the upper bound for δTf /Tf from (34) (dashed
line). As we can see, constraints from BBN require n . 0.94.
2. f2,3 CDM model.
In the case of f2 CDM model of (21) and f3 CDM model of (24), and for the purpose of this analysis, we can
unified their investigation parameterizing them as
i
h
m
,
(41)
f (T ) = αT0 1 − e−p(T /T0 )
with
α=
Ωm0
,
1 − (1 + 2mp)e−p
7
where m =
1
2
for model f2 CDM and m = 1 for model f3 CDM. Inserting (41) into (38) we acquire
r
4
δTf
2πα πg∗ T0
1
=
Tf
15
5
Tf
qMP Tf3
#
)
("
4m
1 −p(Tf /T0 )4m 1
T0
e
−
.
+
·
mp
Tf
2
2
(42)
Hence, using this relation we can calculate the value of |δTf /Tf | 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 + H0 + SN eIa + BAO
observational data in [74], and we present our results in Table I. As we can see, in all cases the value of |δTf /Tf |
is well below the BBN bound (34). Hence, BBN cannot impose constraints on the parameter values of f2 CDM
and f3 CDM models.
m
1/2
1
p = 1/b
1
10
102
1
10
102
|δTf /Tf |
5.723 × 10−38
1.512 × 10−38
1.511 × 10−38
1.4586 × 10−37
1.5131 × 10−38
1.5116 × 10−38
TABLE I: |δTf /Tf | from (42) for different values of p = 1/b, for m = 1/2 (f2 CDM model) and m = 1 (f3 CDM model).
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 . 0.94. Remarkably,
this bound is in agreement with the constraints obtained using CC + H0 + SN eIa + BAO observational data [74].
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 [70], this kind of constraints could contribute
in the debate of fixing the most realistic picture that can be based on curvature or torsion.
Acknowledgments
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: Big Bang Nucleosynthesis
In this Appendix we briefly review the main features of Big Bang Nucleosynthesis following [71, 72]. In the early
Universe, the primordial 4 He was formed at temperature T ∼ 100 MeV. The energy and number density were formed
8
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
νe + n ←→ p + e−
e+ + n ←→ p + ν̄e
n ←→ p + e− + ν̄e .
(A.1)
(A.2)
(A.3)
The neutron abundance is estimated by computing the conversion rate of protons into neutrons, i.e. λpn (T ), and
its inverse λnp (T ). Thus, the weak interaction rates (at suitably high temperature) are given by
Λ(T ) = λnp (T ) + λpn (T ) .
(A.4)
The rate λnp is the sum of the rates associated to the processes (A.1)-(A.3), namely
λnp = λn+νe →p+e− + λn+e+ →p+ν̄e + λn→p+e− +ν̄e .
(A.5)
Finally, the rate λnp is related to the rate λpn as λnp (T ) = e−Q/T λpn (T ), with Q = mn − mp the mass difference of
neutron and proton.
During the freeze-out stage, one can use the following approximations [72]: (i) The temperatures of particles are
the same, i.e. Tν = Te = Tγ = 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 me can be neglected with respect to the electron and
neutrino energies (me ≪ Ee , Eν ).
Having these in mind, the interaction rate corresponding to the process (A.1) is given by
dλn+νe →p+e− = dµ (2π)4 |hM|2 iW ,
(A.6)
where
dµ ≡
d3 pe
d3 pp
d3 pνe
,
3
3
(2π) 2Ee (2π) 2Eνe (2π)3 2Ep
W ≡ δ (4) (P)n(Eνe )[1 − n(Ee )] ,
P ≡ pn + pνe − pp − pe ,
2
gw
M =
[ūp Ωµ un ][ūe Σµ vνe ] ,
8MW
Ωµ ≡ γ µ (cV − cA γ 5 ) ,
Σµ ≡ γ µ (1 − γ 5 ) .
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
2
In (A.10) we have used the condition q 2 ≪ MW
, where MW is the mass of the vector gauge boson W , with q µ = pµn −pµp
the transferred momentum. From Eq. (A.6) it follows that
λn+νe →p+e− = A T 5 Iy ,
(A.13)
where
A≡
gV + 3gA
,
2π 3
(A.14)
and where
Iy =
Z
y
∞
ǫ(ǫ − Q′ )2
with
y≡
A similar calculation for the process (A.2) gives
p
ǫ2 − y 2 n(ǫ − Q)[1 − n(ǫ)]dǫ,
me
,
T
Q′ =
Q
.
T
λe+ +n→p+ν̄e = A T 5 Jy ,
(A.15)
(A.16)
(A.17)
9
with
Jy =
Z
y
∞
ǫ(ǫ + Q′ )2
p
ǫ2 − y 2 n(ǫ)[1 − n(ǫ + Q′ )]dǫ ,
(A.18)
which finally results to
λe+ +n→p+ν̄e = A T 3 (4!T 2 + 2 × 3!QT + 2!Q2 ) .
(A.19)
Lastly, for the neutron decay (A.3) one obtains
τ = λ−1
n→p+e− +ν̄e ≃ 887sec .
(A.20)
Hence, in the calculation of (A.5) 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 [72]
λe+ +n→p+ν̄e = λn+νe →p+e− .
(A.21)
Thus, inserting (A.21) into (A.5), and then into (A.4), allows to derive the expression for Λ(T ), namely
Λ(T ) ≃ 2λnp = 4λe+ +n→p+ν̄e ,
(A.22)
Λ(T ) = 4A T 3 (4!T 2 + 2 × 3!QT + 2!Q2 ) .
(A.23)
which using (A.19) leads to
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