The D(p,γ)3H reaction at LUNA: implications in cosmology, particle physics and theoretical nuclear physics

The D(p,γ)3He cross section at low energies affects the primordial deuterium abundance, that in turn depends on fundamental cosmological parameters such as the baryon density and the amount of relativistic particles permeating the early universe. This paper discusses a new measurement of the D(p,γ)3He cross-section in the 30-280 keV energy range, performed at the Gran Sasso Laboratory (LNGS) with the LUNA facility. This measurement provides a new determination of the universal baryon density at the BBN epoch Ωb(BBN) with improved accuracy, in excellent agreement with the baryon density value derived from the Cosmic Microwave Background data Ωb(CMB). Furthermore, the LUNA result allows to better constrain the existence of dark radiation, i.e. the amount of light particles not considered in the standard model of particle physics, such as sterile neutrinos or hot axions. This paper also discusses the results of a new analysis showing that the D(p,γ)3He differential cross section is in excellent agreement with recent ab-initio theoretical calculations.


Introduction
Light elements were produced in the first minutes of the Universe through a sequence of nuclear reactions known as big bang nucleosynthesis (BBN) [1,2].Among the light elements produced during BBN, deuterium is an excellent indicator of cosmological parameters because its abundance is highly sensitive to the primordial baryon density and also depends on the radiation density of the early Universe.Although astronomical observations of primordial deuterium abundance have reached percent accuracy [3], theoretical predictions [4,5,6] based on the BBN model are less precise mainly because of the paucity of data of the deuterium-burning process D(p,γ) 3 He.In this regard, the precision study of the D(p,γ) 3 He at the Laboratory for Underground Nuclear Astrophysics (LUNA) represents a substantial improvement of BBN calculations thanks to the negligible cosmic-ray background of the Laboratori Nazionali del Gran Sasso (LNGS, Italy) and to the accurate commissioning of the experiment [7].

Experiment
The measurement were done using the intense proton beam from the LUNA 400 kV accelerator [8], a windowless deuterium gas target and a high-purity germanium (HPGe) detector to detect the γ-rays from the D(p,γ) 3 He reaction (see figure 1) [7]. Figure 2 (left) shows a typical γ-ray spectrum (black line) obtained with the deuterium gas target at P = 0.3 mbar.The full-energy, single-escape and double-escape peaks from the D(p,γ) 3 He reaction are clearly visible.The blue line is the background spectrum acquired in the control run under the same experimental conditions but with an inert 4 He gas target.Both spectra are normalised to the integrated beam current.Note that at high energy the spectrum is essentially background free owing to the million-fold shielding from cosmic-ray muons attained at the LNGS laboratory.LUNA data represent a substantial improvement compared with previous works [9].In fact this process has been studied in the relevant BBN energy region with a standard accuracy less than 3%, much better than literature data in which the uncertainty is at the 10% level [10,11,12,13].As shown in figure 2 (right), our new S-factor best fit (red solid line) implies a cross section that is higher compared with the best fit of previous experimental data (blue dashed curve) [15] and lower compared with predictions based on ab-initio calculations (black dashed curve) [14].The Astrophysical Factor measured by LUNA, together with other data and theoretical calculation [9].See text.

Peak shape analysis
This section describes the new analysis to derive the D(p,γ) 3 He differential cross section.The D(p,γ) 3 He photon energy is determined by the reaction Q-value (Q=5.5 MeV), by the proton energy and by the Doppler effect: In this equation m p , m D and m He are the masses of nuclides involved in the reaction, p p = E p (E p + 2m p ) is the proton momentum and θ lab is the polar angle in the laboratory system.Equation 1 shows that for a given proton energy E p , the photon energy depends on its angle with respect to the beam axis (Doppler effect).The different direction of detected photons determines the broadening of the full detection peak.While the position and the width of the peak depends on the beam energy and on the setup configuration, its shape is due to the angular distribution of emitted photons.The analysis consists in a novel method named Peak Shape Analysis (PSA), in which a minimisation procedure is applied to the following response function F (E): The peak shape is reproduced with the linear superposition of modified Legendre polynomials P * l (E γ ) obtained with a change of variable (from the polar angle θ cm of photons to their energy E γ , see equation 1) and taking into account the HPGe energy resolution and efficiency along the beam line, as determined with a set of dedicated measurements [7].The minimisation procedure provides the a l coefficients of equation 2 (only the first four Legendre polynomials have been considered in the minimisation procedure).As an example, figure 3 (left) shows the measured full detection peak (black points) at E p = 300 keV and the fit of the data (red curve).Also shown are the modified polynomial weighted with the a l coefficients.The analysis shows that the even polynomials P 0 and P 2 are predominant.Figure 3 (right) shows the ratio between the coefficients of the P 0 and P 2 polynomials for all the beam energies measured [16].Note the good agreement between the experimental and theoretical results [14].

Figure 3.
Left: Full peak detection for E p =300 keV measured by LUNA (black points).Also shown are the fit using the first four Legendre polynomials (red curve) and the a l P * l terms of equation 2. Right: ratio of the coefficients a 0 /a 2 as a function of the energy (green=LUNA data; red=ab-initio calculation) [16].See text.

Cosmological baryon and radiation density
As stated before, the precision measurement of the D(p,γ) 3 He cross section by LUNA substantially improves the calculated deuterium and the baryon density, that is a cosmological parameter of extraordinary importance as it reflects the pristine matter-antimatter asymmetry of the Universe.The baryon density is derived by comparing the measured abundance of deuterium with respect to Hydrogen (D/H) obs with the theoretical abundance (D/H) BBN .In figure 4   Left: Likelihood distribution of the baryon density.Right: Likelihood contours at 68%, 95% and 99% confidence levels combining i) deuterium abundance and CMB data (orange) and ii) deuterium and helium abundance (blue).The red line shows the standard model expectation.

Conclusion
The deuterium abundance (D/H) BBN has been improved by the LUNA measurement of the D(p,γ) 3 He cross section and the cosmological parameter Ω b (BBN) is now derived with a precision of 1.6%.The amount of relativistic particles at BBN epoch (∆N ef f /N ef f ∼7%) is consistent with the Standard Model.To further improve this parameters, it is necessary to increase the data accuracy of the other deuterium-burning reactions at the BBN epoch, namely the d(d,n) 3 He and d(d,p)t processes.A more accurate determination of the 4 He abundance is also highly desirable, either with 4th generation of CMB experiments or with improved 4 He abundance determination from astronomical observations.

Figure 1 .
Figure 1.Sketch of the experimental setup.The proton beam impinges the windowless deuterium gas target.The photons produced by the D(p,γ) 3 He reaction are detected by the HPGe detector faced to the beam line.

Figure 2 .
Figure 2. Left: Black (blue) line shows the energy spectrum of the HPGe detector at E p =50 keV, using D 2 ( 4 He) gas target.Right: The Astrophysical Factor measured by LUNA, together with other data and theoretical calculation [9].See text.
(left) are shown the likelihood of baryon density rescaled to the present day value considering i) Ω b (BBN) with and without the LUNA data and assuming the standard BBN theory, in which only photons and 3 neutrino families contribute to the radiation density (red and grey curves, respectively); ii) Ω b (CMB) obtained with CMB data and assuming the ΛCDM model (blue dotted curve); iii) the result of a global analysis performed by the Planck collaboration, in which are used other experimental input and BBN theoretical assumptions (orange curve).Note the impressive agreement (percent level) between Ω b (BBN) and Ω b (CMB).It is worth to point out that Ω b (BBN) reflects the Universe as it was in the first minutes of Cosmic time, while Ω b (CMB) is relative to the Universe at the re-combination epoch, some 380,000 years after the Big Bang.A simple extension of the CDM model has been also considered, with a likelihood analysis in which the number of neutrino families N ef f is a free parameters (N ef f =3.04 in the Standard Model).

Figure 4 (
Figure 4.Left: Likelihood distribution of the baryon density.Right: Likelihood contours at 68%, 95% and 99% confidence levels combining i) deuterium abundance and CMB data (orange) and ii) deuterium and helium abundance (blue).The red line shows the standard model expectation.