arXiv:2511.15158v1 [astro-ph.CO] 19 Nov 2025
Prepared for submission to JCAP
Non-thermal processes in standard
big bang nucleosynthesis: III.
Reactions with slow nuclei and the
overall effect
V.T.Voronchev
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Leninskiye Gory 1, Moscow 119991, Russia
E-mail: voronchev@srd.sinp.msu.ru
Abstract. The present paper completes a series of our works on non-thermal nuclear processes
in big bang nucleosynthesis (BBN) started in JCAP05(2008)010 (Part I) and 05(2009)001
(Part II). The processes are triggered by non-Maxwellian particles naturally born in the main
BBN reactions. Half of these reactions generate fast particles k + (= n, p, t, 3 He, α). The other
half, being radiative capture processes, produce slow nuclei k − (= d, t, 3 He, 7 Li, 7 Be) which can
undergo (k − , n) reactions with neutrons having large cross sections. The particle production
rate Rk , thermalization time τk , and effective number density nk are determined. It is shown
that the values of nk at the Universe temperatures T > 65 keV can exceed the number densities of Maxwellian 7 Li and 7 Be ions. To clarify the overall non-Maxwellian effect on BBN, both
types of the non-Maxwellian particles are taken into account in the reaction network. Particular attention is paid to two-step sequential processes like p(n, γ)d− (n, γ)t, d(p, γ)3 He− (n, p)t,
t(α, γ)7 Li− (n, γ)8 Li, 3 He(α, γ)7 Be− (n, p)7 Li, d(t, α)n+ (A, n)a1 a2 , and d(3 He, α)p+ (A, p)a1 a2
with (A, a1 , a2 ) = (7 Li, t, α) and (7 Be, 3 He, α). It is obtained that the non-Maxwellian particles can selectively affect the element abundances, e.g., improve the prediction on 7 Li/H
by ∼ 1.5% and at the same time leave unchanged the 4 He abundance. The main conclusion
however is that these particles are unable to significantly change the standard picture of BBN
in general, and provide a pathway toward a solution of the cosmological lithium problem in
particular.
Contents
1 Introduction
1
2 Non-Maxwellian reaction products in the primordial plasma
2.1 Slow nuclei
2.2 Fast nucleons and nuclei
2
3
6
3 Element abundances
7
4 Conclusion
8
1
Introduction
A key issue in the modeling of big bang nucleosynthesis (BBN) is a proper description of
nuclear reaction kinetics in the primordial plasma. The rate equations for abundances Yi of
elements i involving in forward and reverse processes Ni (i) + Nj (j) ⇄ Nk (k) + Nl (l) are given
by [1, 2]
!
N
YiNi Yj j
YlNl YkNk
Yi X
=
Ni −
[ij]k +
[lk]j ,
(1.1)
dt
Ni !Nj !
Nl !Nk !
j,k,l
where Nm is the number of nuclei m. For a two-body i + j reaction, [ij]k = ρb NA ⟨σv⟩ where
the i+j reactivity ⟨σv⟩ is defined as σ ×|vi −vj | folded over the reactant velocity distribution
functions f (vi ) and f (vj ). This quantity is the major nuclear input in BBN simulations.
Equation 1.1 is universal and can be applied to various scenarios of BBN. However, in
different models the values of [ij]k may differ, depending on the adopted particle distribution functions. In the standard model of BBN, the particle distributions are assumed to be
Maxwellian.
The phenomenon of non-Maxwellian particles in BBN is a widely discussed topic in
the literature. They can appear, e.g., in scenarios beyond the Standard Model due to the
decay of exotic relic particles [3–6]. An acceleration of nucleons in the matter [7] and a
transient phenomenon [8] were also considered to be the sources of nonthermal particles.
Models developed in [9–11] modify standard BBN within a thermal approach: although they
do not involve any nonthermal particles, it is assumed that the primordial plasma species
obey not Maxwellian but non-extensive Tsallis statistics. A combined Tsallis-Maxwellian
approach was proposed in [12]. Here, the photon distribution was parameterised with Tsallis
statistics while particles still remain Maxwellian. A mechanism of non-Maxwellian distortion
for particle distributions in the presence of a stochastic magnetic field was analyzed in [13].
Useful links to other non-standard models of BBN can be found in a recent review [14].
Strictly speaking, even in standard BBN a small fraction of non-Maxwellian particles
should also present in the primordial plasma as the products of nuclear reaction are not
initially Maxwellian. BBN reactions have Q-values in the range of 1–20 MeV, so they generate
energetic particles which can somewhat increase the population of the respective high-energy
distribution tails. This was clearly demonstrated on the example of plasma neutrons [15, 16].
In particular, it was shown that the non-Maxwellian neutron distribution tails created by fast
d + t and d + d neutrons increase the rates of some reactions in the plasma. In addition to
–1–
log
10
-3
-1
[Production rate (cm s )]
24
18
12
6
0
Slow
d: p(n, )
-6
3
-12
-18
250
3
Fast
7
200
7
He: d(d,n)
Li:
7
7
Be(n,p)
150
100
He: d(p, )
Li: t(
Be:
3
, )
He(
50
, )
0
T (keV)
Figure 1. The production rate of slow and fast d, 3 He, 7 Li, and 7 Be as a function of the Universe
temperature.
neutrons, it was found [17, 18] that fast charged products of BBN reactions can also enhance
nuclear processes.
All these studies discuss non-thermal effects triggered by MeV particles. At the same
however half of the main BBN reactions are radiative capture processes i(j, γ)k producing slow
particles k as almost all energy released is carried out by photons γ. These slow products
could potentially enhance the low-energy tail of the k-particle distribution compared to a
Maxwellian function. The enhancement may in principle affect the rates of k √
+ n reactions
with neutrons as their cross sections, usually depending on energy as σ ∝ 1/ E, are large
at low energies. Unlike fast particle-induced reactions, little is still known about a possible
role of these processes in BBN. The only study in the field was published recently [19]. The
situation becomes even more interesting if we remember that deuterons as well as 7 Be are
essentially synthesized in radiative capture reactions, so all these nuclei are initially slow in
the primordial plasma.
The purpose of present paper is to incorporate sequential processes involving slow nuclei
like i(j, γ)k[slow] + n in a BBN reaction network and to draw an overall conclusion on the
contribution of non-Maxwellian reaction products to BBN, completing a serious of our studies
started in [20, 21].
2
Non-Maxwellian reaction products in the primordial plasma
The list of BBN reactions that we will pay special attention to is presented in table 1. The
third and fourth columns identify slow (s) and fast (f) products of the top 11 processes
controlling the production of light elements. All these non-Maxwellian particles will be taken
into account in the BBN reaction network.
–2–
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Reaction
Top 11 reactions
p+n⇄γ+d
d + p ⇄ γ + 3 He
d + d ⇄ n + 3 He
d+d⇄p+t
t+d⇄n+α
3 He + n ⇄ p + t
3 He + d ⇄ p + α
t + α ⇄ γ + 7 Li
3 He + α ⇄ γ + 7 Be
7 Li + p ⇄ α + α
7 Be + n ⇄ p + 7 Li
Reactions with neutrons
d+n⇄γ+t
3 He + n ⇄ γ + α
7 Li + n ⇄ γ + 8 Li
7 Be + n ⇄ α + α
Other processes
t+p⇄γ+α
t+t→n+n+α
3 He + He → p + p + α
7 Be + p ⇄ γ + 8 B
7 Li + α ⇄ γ + 11 B
7 Be + α ⇄ γ + 11 C
d+n→n+n+p
d+p→p+n+p
7 Li + n → n + t + α
7 Li + p → p + t + α
7 Be + n → n + 3 He + α
7 Be + p → p + 3 He + α
Product
Reactant
s
s
f
f
f
f
f
s
s
f
f
f
f
f
f, s
f
f
f
f
f, s
f, s
f, s
f, s
f, s
f
f
f
f
f
f
f
f
f
f
f
f
Table 1. BBN reactions for particles with mass number A ≤ 7. Processes involving rare 6 Li and
low-probability three-body reactions are not listed. The third column identifies slow (s) and fast (f)
products of the key 11 reactions controlling the primordial element production. The fourth column
shows what types of the non-Maxwellian particles are taken into account in the processes of the second
column.
2.1
Slow nuclei
The main source of slow nuclei in the primordial plasma is radiative capture reactions i(j, γ)k.
In these processes, most of the energy released Ei + Ej + Q is carried out by photons. With
a precision of a few keV the photon energy is [22]
Eγ = Q + E ′ + (Q + E ′ )
vk
(Q + E ′ )2
cos θ −
c
2mk c2
(2.1)
while the slow nucleus k is born at the energy
Ek = Ec.m. + E ′ + Q − Eγ .
–3–
(2.2)
Thermalization time (s)
3
10
d
-
Coul (i)
0
10
-
+
Coul (i+e +e e )
-
Comp
Coul (e )
-
+
Coul (e e )
-3
10
-6
10
-9
10
Total
-12
10
Thermalization time (s)
3
3
10
He
-
Coul (i)
0
10
-
+
Coul (i+e +e e )
-
Comp
Coul (e )
-
+
Coul (e e )
-3
10
-6
10
-9
10
Total
-12
10
120
100
80
60
40
20
0
T (keV)
Figure 2. The thermalization time of the deuteron and the 3 He nucleus in the primordial plasma.
The total and partial τk provided by different mechanisms are shown.
In these equations, Ec.m. and E ′ are the energy of the center-off-mass frame (c.m.) and the
(i + j) kinetic energy in the c.m. frame, respectively, vk is the velocity of the particle k in the
laboratory system, and θ is the direction of motion of the c.m. frame relative to the observer.
Almost half of the top BBN reactions produce slow nuclei k = d, 3 He, 7 Li, and 7 Be (see
table 1). Their production rate is
Rk,slow = ni nj ⟨σv⟩ij→γk ,
(2.3)
where ni and nj are the reactant number densities. In the present study, ni and nj were
determined by running the Kawano’s code [23] with the baryon-to-photon ratio η = nB /nγ =
6.09 × 10−10 and the values of ⟨σv⟩NA for the main reactions taken from [24, 25]. The
production rate of slow d, 3 He, 7 Li, and 7 Be as a function of the Universe temperature is
plotted in figure 1. For comparison, the rates Rk,fast of competing nuclear channels generating
the same nuclei at high energies are also shown. It is worth noting here that all deuterons
and all 7 Be are initially slow in the plasma while 3 He and 7 Li are predominantly produced
–4–
Thermalization time (s)
7
Li
3
10
-
Coul (i)
+
Comp
Coul (e )
0
10
-
Coul (i+e +e e )
-
-
+
Coul (e e )
-3
10
-6
10
-9
10
Total
-12
10
Thermalization time (s)
7
Be
3
10
-
Coul (i)
+
Comp
Coul (e )
0
10
-
Coul (i+e +e e )
-
-
+
Coul (e e )
-3
10
-6
10
-9
10
Total
-12
10
120
100
80
60
40
20
0
T (keV)
Figure 3. The same parameters as in figure 2 for the 7 Li and 7 Be nuclei.
through the ‘high-energy’ channels. These nuclei may undergo reactions with neutrons (see
table 1) before their Maxwellization occurs. It is important to note that the cross sections of
these reactions become large at low energies [26].
The abundance of slow d, 3 He, 7 Li, and 7 Be depends on the time of their thermalization
in the plasma. This process is governed by Coulomb interaction with background electrons
(Coul(e− )), electron-positron pairs (Coul(e± )), ions (Coul(i)), and by Compton scattering
off thermal photons (Comp). These mechanisms are described, e.g., in [5, 27, 28]. Besides,
the process of nuclear elastic scattering off ambient nucleons and nuclei may also take place.
However, this mechanism can be neglected as the particles are slow.
The calculated thermalization times τk are presented in figure 2 (for deuterons and 3 He)
and figure 3 (for 7 Li and 7 Be). The total time together with partial times τk corresponding to
the different interaction mechanisms are shown. It is seen that the most effective thermalization occurs due to scattering off electron-positron pairs at high temperatures T ≳ 30 keV and
also due to collisions with ions at lower temperatures, after the e± annihilation is completed.
The effective number densities of the slow particles defined as nk,slow = Rk,slow × τk are
–5–
-3
[Number density (cm )]
18
12
6
0
-6
Maxwellian
log
10
-12
D
d
3
3
7
-18
120
7
100
80
slow
He
7
Li
7
Be
60
40
He
Li
Be
20
0
T (keV)
Figure 4. The number densities of slow and Maxwellian particles in the primordial plasma.
Thermalization time (s)
3
d+ He proton
3
10
Coul (i)
-
10
-
-
+
Coul (i+e +e e )
-
Coul (e )
0
Comp
+
Coul (e e )
-3
10
-6
10
-9
10
Total
-12
10
120
100
80
60
40
20
0
T (keV)
Figure 5. The slowing-down time of the 14.7-MeV proton.
given in figure 4 together with the respective number densities for Maxwellian nuclei. It is
interesting that at an early stage of BBN the slow particle abundances prove to be larger that
the abundances of Maxwellian 7 Li and 7 Be.
2.2
Fast nucleons and nuclei
To get an overall picture of the influence of non-Maxwellian reaction products on BBN,
processes with fast neutrons and charged particles, such as protons, tritons, helions, and
α-particles (see table 1), were also taken into account.
–6–
neutrons
charged particles
12
8
4
0
log
10
-3
[Number density (cm )]
16
-4
120
100
80
60
40
20
0
T (keV)
Figure 6. The number density of fast neutrons and fast charged particles.
The slowing-down process of fast ions in the primordial plasma is mainly governed by
Coulomb scattering off electrons and electron-positron pairs. As an example, the total and
partial thermalization times for 14.7-MeV d + 3 He protons are plotted in figure 5. For fast
neutrons (they are generated in the d + t and d + d reactions), the main slowing-down mechanisms are nuclear elastic scattering off thermal protons and magnetic-moment interaction
with electrons and positrons [21].
The number densities of fast neutrons and fast charged particles are given in figure 6. A
comparison of figure 4 and figure 6 indicates that the most abundant non-Maxwellian particles
in the plasma are neutrons.
3
Element abundances
In the present BBN simulations, all forward and reverse reactions in table 1 induced by both
slow and fast particles were taken into account. Special attention was paid to the following
sequential processes:
a) for the slow particles, these are the production of slow nuclei in i(j, γ)k followed by
their burn-up in n + k reactions with large cross sections
p(n, γ)d[slow](n, γ)t,
(3.1)
d(p, γ)3 He[slow](n, p)t,
(3.2)
7
8
t(α, γ) Li[slow](n, γ) Li,
3
7
7
He(α, γ) Be[slow](n, p) Li
(3.3)
(3.4)
b) for the fast particles, these are the disintegration of loosely bound nuclei A = d, 7 Li,
and 7 Be induced by fast neutrons and protons generated in the d + t and d + 3 He reactions
d(t, α)n[fast](A, n)a1 a2 ,
–7–
(3.5)
Element
D/H (×10−5 )
3 He/H (×10−5 )
Yp (×10−1 )
7 Li/H (×10−10 )
[6 Li + A > 7]/H (×10−14 )
BBN scenario
| Observations
with non-Maxwellian Maxwellian
2.576
2.575
2.547 ± 0.029
1.009
1.011
< 1.1 ± 0.2
2.456
2.456
2.45 ± 0.03
4.365
4.424
1.6 ± 0.3
1.237
1.236
Table 2. The element abundances in BBN with the non-Maxwellian particles in the primordial
plasma. The Maxwellian scenario is also given for comparison. Observational data are taken from
[14].
d(3 He, α)p[fast](A, p)a1 a2 ,
(3.6)
where (A, a1 , a2 ) = (7 Li, t, α) and (7 Be, 3 He, α). The n + A rates were calculated using
the reactivity ⟨σv⟩nA→na1 a2 obtained with the realistic neutron distribution functions having
enhanced non-Maxwellian tails at MeV energies [16].
We note that some inaccuracy in the rates of processes 3.1–3.4 may come from choosing
the initial energy Ek,0 of the particle k as it depends on the (i + j) center-off-mass velocity
Vc.m. = mi vi /(mi + mj ) + mj vj /(mi + mj ). To clarify the point, several values of Vc.m.
corresponding to typical speeds of the reactants m(= i, j) were tested. These are the most
′ following from the condition df (v )/dv = 0, the mean speed ⟨v ⟩, the
probable speed vm
i m
m
m
2
r.m.s. speed ⟨vm ⟩1/2 , and also the average relative velocity ⟨|vi − vj |⟩. It was obtained that
the respective n + k rates differ by no more than 5%.
The element abundances A/H calculated with the baryon-to-photon ratio η = 6.09 ×
10−10 are given in table 2. The scenarios allowing for and neglecting the non-Maxwellian
particles are shown. A rather slight impact on the abundances is observed. In particular, the
non-Maxwellian particles can improve the theoretical prediction on 7 Li/H by ∼ 1.5% and the
same time leave unchanged the 4 He abundance. These changes cannot be seen in figure 7
showing the element production dynamics as the Universe expands and cools. It was also
obtained that the effect is practically independent of the baryon-to-photon ratio in the range
of 6 × 1010 –6.2 × 1010 . This is demonstrated in figure 8 showing the non-Maxwellian Y ′ to
Maxwellian Y abundance ratio δ(7 Li) = Y ′ /Y − 1 at different η.
4
Conclusion
The present paper completes the series of our works on non-thermal nuclear processes in
the primordial plasma triggered by non-Maxwellian particles produced in the BBN reactions.
The purpose was to determine the main characteristics of slow reaction products and to find
whether they, together with fast particles, can affect the primordial element production. The
results can be summarized as follows:
1. It is shown that some amount of non-Maxwellian (both slow and fast) reaction products is continuously present in the plasma. Their number densities at the Universe
temperature T ≳ 50 keV exceed the number densities of Maxwellian 7 Li and 7 Be and
can be comparable to the Maxwellian 3 He number density at T ≃ 70–80 keV. The most
abundant non-Maxwellian particles are neutrons.
–8–
-4
-8
-12
Y
Log
10
(Abundance)
0
p
-16
D/H
3
-20
7
-24
-28
7
3
10
2
1
10
He/H
Li/H
Be/H
0
10
10
T (keV)
Figure 7. The dynamics of light element production as the Universe expands and cools.
-1.34
7
( Li) (%)
-1.36
-1.32
-1.30
6.00
6.05
6.10
6.15
6.20
Figure 8. The non-Maxwellian effect on the 7 Li abundance at different values η10 ≡ η × 1010 (for
δ, see the text).
–9–
2. It is found that these particles can selectively, but only slightly, alter the element
abundances compared to the Maxwellian values. For example, they can improve the
theoretical prediction on 7 Li, reducing its abundance by ∼ 1.5%. The abundance of
4 He remains unchanged.
The overall conclusion however is that neither slow nor fast reaction products are able
to significantly change the standard picture of BBN in general, and provide a pathway toward
a solution of the cosmological lithium problem in particular.
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