Investigating the Primordial Universe through nuclear physics

Big Bang Nucleosynthesis (BBN) requires several nuclear physics inputs and nuclear reaction rates. An up-to-date compilation of direct cross sections of is given, being these ones among the most uncertain bare-nucleus cross sections. A particular attention is devoted to recently indirectly measured cross-section which give important hints for the nuclear astrophysics community. In reality, a significant experimental effort has been made over the past 10 years to explore reactions important to the BBN and determine their astrophysical S(E)-factor using the Trojan Horse Method (THM). Then, numerical calculations are made in the relevant temperature ranges for BBN (0.01<T9 <10) to determine the reaction rates and the relative error for the four reactions of interest. The effects of these values on the calculated primordial abundances and isotopical composition for H, He, and Li were then assessed by using them as input physics for computations of primordial nucleosynthesis. Additionally, recent findings regarding the 7Be(n,alpha)4He reaction rate were taken into consideration. These were put up against estimates of primordial abundance derived from observation at various astrophysical places. Additionally, perspectives on reactions will be examined.


Introduction
BBN is one of the pillars of the Big Bang theory, along with the Hubble expansion and the Cosmic Microwave Background (CMB) radiation [1].BBN explores the early universe, often known as the radiation dominated era, from a fraction of a second to several minutes.It naturally plays a significant part in establishing the link between cosmology and nuclear physics because it involves events that took place at temperatures below 1 MeV.The Standard Big Bang Nucleosynthesis model (SBBN), which focuses solely on the products of the BBN, predicts that only the production of light nuclei ( 2 H, 3,4 He, 7 Li) in measurable quantities, beginning with protons and neutrons, will occur.The abundances of these isotopes in the proper astrophysical conditions today, with the exception of 3 He and lithium, are largely compatible with SBBN predictions [3].The baryon-to-photon ratio, η, which is the only free parameter in the currently accepted model of the SBBN, is constrained by comparing the primordial abundances inferred from precise WMAP CMB observations with those estimated abundances.The value we use in our computations is η = 6.16 ± 0.15 × 10 −10 , which is based on a recent observation [4].Nuclear reaction rates are one of many nuclear physics inputs needed for BBN nucleosynthesis, and they are crucial.Only 12 reactions significantly contribute to the BBN nuclear reaction network due to the very small number of important nuclear species involved [5].A number of those involve neutrons and radioactive ions and are currently not known with sufficient precision (see table 1).n ↔ p (1) p(n,γ)d (2)  d(p,γ) 3 He (5) 3 He(n, p)t (6) t(d, n) 4 He (7) 3 He(d, p) 4 He ( †) (8)  3 He(α, γ) 7 Be (9) t(α, γ) 7 Li (10) 7 Be(n, p) 7 Li or 7 Be(n,α) 4 He (11)( †) 7 Li(p,α) 4 He ( †) (12)   Table 1.Nuclear reactions of greatest relevance for BBN.The reactions whose cross sections were already measured with the Trojan Horse method are marked with a † symbol.
The reaction rates are determined using the low-energy cross sections that are now accessible for processes that are also a crucial input for a number of other unresolved astronomical issues, such as the so-called "lithium depletion" in the Sun or other galactic stars [6].Cross sections of the order of a few hundred keV should be measured in the Gamow window [7].These reactions have received extensive research over the past few decades, with particular emphasis on direct observations made at the appropriate astronomical energy, sometimes in subterranean labs [8].However, there are few direct data at such low energies for many of the pertinent processes (mainly due to challenges associated with the existence of the Coulomb barrier) and the reaction cross-section within the Gamow window extrapolated from direct measurement at high energies.

The Trojan Horse Method
Alternative methods for determining bare nucleus cross sections of astrophysical interest are needed.In this context a number of indirect methods, e.g. the Coulomb dissociation (CD) [9], the Asymptotic Normalization Coefficient method (ANC) [10] and references therein and the Trojan-horse method (THM) were developed [11].For further information on the development and first principles of the method please refer to [12].The latter has already been applied several times to reactions connected with fundamental astrophysical problems such as primordial nucleosynthesis (as sketched in figure 1 ) [14,15], lithium problem [17,25], light elements depletion [23], AGB [26,27] and Novae nucleosynthesis [28].It was also applied to reactions induced by radioactive ion beams [29,30] and neutrons [31].THM selects the quasi-free (QF) contribution of an appropriate three-body reaction performed at energies well above the Coulomb barrier to extract a charged particle two-body cross section at energies of astrophysical interest.The idea of the THM [11] is to extract the cross section of an astrophysically relevant two-body reaction at low energies from a suitable chosen three-body quasi-free reaction In this approach S acts as a spectator to the A + x → c + C binary interaction.This is done with the help of direct processes theory assuming that the Trojan Horse nucleus a has a strong x ⊕ S cluster structure [34,35].In many applications, this assumption is trivially fulfilled e.g. a = deuteron, x = proton, S= neutron.If the bombarding energy E A is chosen high enough to overcome the Coulomb barrier in the entrance channel of the three-body reaction, both Coulomb barrier and electron screening effects are negligible.The polar approximation, used in the standard THM prescription has been extensively verified [32] and constitutes a powerful validity test for the method which strenghtens the theoretical approach.We refer to [19,20] for further and advanced theoretical approach to the method.We just underline that THM allows to link the three-body cross section which is measured in the laboratory with the half-off-energy shell cross section of the binary process of astrophysical interest.Then after inclusion of the Coulombian effects data are then compared and normalized to direct data, at the higher energies available.After that phase, the reaction rate is calculated according to the standard prescriptions.A summary of results presented to the nuclear astrophysics community by means of THM is reported in table 2.
Several of the reactions that are important for the SBBN, such as 7 Li(p,α) 4 He, 2 H(d,p) 3 H, 2 H(d,n)He and 3 He (d, p) 4 He, 7 Be(n,α) 4 He were investigated using the THM in the relevant energy range during an experimental effort that took place over the past ten years [36,13,18,37,23,38].
Rates have been generated for the four reactions indicated above (from a composition of direct and THM data, as published in [15].This is the standard practice used in earlier works (see, for example, [39,40,41]).We have properly accounted for the experimental measurement errors for the 4 reactions of interest, allowing us to assess the corresponding uncertainties in the reaction rates.Most recently the 7 Be(n,α) 4 He and 7 Be(n,p) 7 Li have been measured by means of THM and data are reported elsewhere [45,46].Even in those cases no real step aiming at the cosmological lithium problem was made, thus suggesting that its solution lies elsewhere.
2.1.The 3 He(n,p) 3 H reaction Most recently the attention was focused on the 3 He(n,p) 3 H reaction, taking advantage of the application of THM to neutron induced reactions.
One of the most important neutron-induced reactions in BBN is the 3 He(n,p) 3 H process, which has a significant effect on the generation of the primordial isotopes of He and Li.The reaction rate is defined by the 3 He(n,p) 3 H cross section in the energy range 0≤E cm ≤0. 4 MeV at the temperatures important for forecasting Big Bang yields.Coon et al. [47] conducted the initial research on this reaction in 1950 in the 0.1≤ E cm ≤30 MeV range using a neutron beam.Errors ended up being about 30%. 7Be(n,α) 4 He 7 Be( 2 H,αα)n 2 H = (p⊕n) [44,45,46] Various measurements, with a particular focus on lower energies, were carried out by different research groups.Batchelor et al. [48] conducted direct measurements within the energy range of 0.1≤ E cm ≤1 MeV, Gibbons et al. [49] performed inverse measurements, and Costello et al. [50] directly measured reactions in the range of 0.3≤ E cm ≤1.1 MeV.Additionally, theoretical predictions, including the most recent work by Drosg et al. [51], covered a wider energy spectrum.For astrophysical applications, reaction rates were subsequently calculated by Brune et al. [52], Adahchour et al. [53], and Smith et al. [54], all of which exhibit a consistent trend at astrophysically relevant temperatures.In contrast, Caughlan & Fowler [55] calculated a significantly higher reaction rate.
In the energy range under consideration, the available data are quite limited and predominantly date back more than half a century, primarily due to formidable experimental challenges.Consequently, these measurements have often yielded errors of up to 30%, depending on the energy level.
The experiment made use of a 3 He beam with a total kinetic energy of 9 MeV, generated by the FN Tandem accelerator at the Nuclear Physics Laboratory of the University of Notre Dame.Detailed information about the experimental setup can be found in [21].
Following the standard THM procedures, the experimental data were compared to previously published data from the literature after appropriate normalization (within the energy range of 350 to 400 keV).The resulting outcomes are presented in Table 3.These results demonstrate a strong concurrence with both direct and indirect measurements already documented in the scientific literature, reaffirming the validity of the THM approach for neutron-induced reactions.

Discussion and Perspectives
The reaction rates for four critical reactions within the Big Bang Nucleosynthesis (BBN) network, spanning the temperature range (0.001 ≤ T 9 ≤ 10), namely, 2 H(d,p) 3 H, d(d,n) 3 He, 3 He(d,p) 4 He, and 7 Li(p,α) 4 He, have been computed numerically, taking into account recent Table 3. Cross section for the process of interest normalized to direct data as discussed in the text, together with the center-of-mass energy as reported in [21].The related error is also reported.
E THM measurements.The uncertainties in experimental data, both from direct measurements and THM data, have been thoroughly incorporated for these reactions.The application of a similar approach to other reactions within the BBN reaction network will be explored in an upcoming publication.
The parameters governing each reaction rate are documented in [15].The derived reaction rates are compared with some of the commonly employed compilations found in the literature, which are used to compute the abundances of 3,4 He, deuterium (D), and 7 Li in the context of BBN.The resulting abundances are found to be consistent, within the experimental uncertainties, with those obtained using the compilation of directly-measured reaction rates.
These findings underscore the efficacy of the THM as a valuable tool for investigating charged particle-induced reactions at energy levels characteristic of BBN.Additional efforts in examining reactions amenable to investigation via THM, including 3 He(n,p) 3 H and 7 Be(n,p) 7 Li, have been initiated, further solidifying the role of THM in the field of primordial nucleosynthesis.A new dataset, which will extend the explored energy range up to 800 keV, is currently undergoing analysis.This extension will enhance the accuracy of the related reaction rates and shed light on potential astrophysical implications.
Nonetheless, these indirect method measurements, in conjunction with numerous others, particularly concerning the 7 Be+n interaction [56] and references therein, suggest the possibility of a non-nuclear solution to the cosmological lithium problem.In this context, additional investigations into the destruction of lithium in stars, with particular attention to population II

Figure 1 .
Figure 1.Standard Big Bang nucleosynthesis network.The reactions recently (2020-2022) measured by the ASFIN group are marked by green arrows by means of THM or ANC method.The red ones represent cross sections measured by the same group in previous years.