On the relative velocity distribution for general statistics and an application to big-bang nucleosynthesis under Tsallis statistics
The distribution function of the relative velocity in a two-body reaction of nonrelativistic uncorrelated particles is derived for general cases of given distribution functions of single particle velocities. The distribution function is then used in calculations of thermonuclear reaction rates. As an example, we take the Tsallis non-Maxwellian distribution, and show that the distribution function of the relative velocity is different from the Tsallis distribution. We identify an inconsistency in previous studies of nuclear reaction rates within Tsallis statistics, and derive revised nuclear reaction rates. Utilizing the revised rates, accurate results of big bang nucleosynthesis are obtained for the Tsallis statistics. For this application it is more difficult to reduce the primordial Li abundance while keeping other nuclear abundances within the observational constraints. A small deviation from a Maxwell-Boltzmann distribution can increase the D abundance and slightly reduce Li abundance. Although it is impossible to realize a Li abundance at the level observed in metal-poor stars, a significant decrease is possible while maintaining a consistency with the observed D abundance.
Deviations of particle distribution functions from the Maxwell-Boltzmann (MB) distribution are often found in geophysical and astrophysical observations of systems out of equilibrium Vasyliunas1968 (); Bame1967 (); Livadiotis2009 (). For example, a power-law distribution has been observed in the electron spectrum of the magnetosphere Vasyliunas1968 (). A power-law distribution called the kappa-distribution is also realized in Tsallis’s statistical model in which a generalization of the entropy is postulated to be , where is a real parameter, is Boltzmann’s constant, and are the probabilities of with the total configuration number Tsallis1988 (). See Ref. Livadiotis2009 () for formulations of the Tsallis statistics and relations between the Tsallis distribution function and the power-law. The Tsallis distribution function for has also been found to be realized by a special pattern of temperature fluctuation Wilk:1999dr (). Moreover, non-relativistic baryons in equilibrium with a thermal bath of relativistic electrons have been proposed Sasankan:2018pfq () to obey modified MB statistics.
Usually, one assumes a MB distribution for nonrelativistic particles in calculations of thermonuclear reaction rates. Especially, during big bang nucleosynthesis (BBN) and stellar nuclear burning, the temperature is high enough that very frequent scatterings quickly realize the MB distribution of nuclei (e.g., Voronchev:2008zz (); Voronchev:2009zz ()). Therefore, the standard BBN (SBBN) theory is based upon the MB nuclear distribution. However, in nonstandard BBN models such as those involving scattering from relativistic electrons Sasankan:2018pfq (), or the injection of non-thermal photons Lindley1979MNRAS.188P..15L () and hadrons Dimopoulos:1987fz (); Reno:1987qw () the particle distribution functions can deviate significantly from a MB distribution.
Bertulani et al. Bertulani:2012sv () analyzed effects of the Tsallis distribution function for nuclear velocities on BBN. They assumed the Tsallis distribution for the whole energy range and found that the thermonuclear reaction rates strongly depend on the Tsallis parameter . They then concluded that the Tsallis parameter should be very close to unity which corresponds to the MB distribution in order to satisfy observational constraints on light element abundances. An extended parameter search was made with modifications to the 2-body reverse reaction rates taken into account Hou:2017uap (). It was found that a slightly softer spectra than the MB distribution results in a decrease of the Li abundance.
In the SBBN model, theoretical primordial abundances are consistent with observational constraints except for the Li abundance Coc:2014oia (); Cyburt:2015mya (); Mathews:2017xht (). The discrepancy between the theoretical prediction of the primordial Li abundance and astronomical observations of metal-poor stars Spite:1982dd (); Ryan:1999vr (); Melendez:2004ni (); Asplund:2005yt (); Bonifacio:2006au (); shi2007 (); Aoki:2009ce (); Hernandez:2009gn (); Sbordone:2010zi (); Monaco:2010mm (); Monaco:2011sd (); Mucciarelli:2011ts (); Aoki:2012wb (); Aoki2012b () is called the Li problem. Therefore, this non-MB distribution is a possible solution to the Li problem of metal-poor stars Coc:2014oia (); Cyburt:2015mya (); Mathews:2017xht ().
In this paper, we derive a new formulation of the relative velocity distribution function for general distributions for reacting nonrelativistic particles. In Sec. II, the relative velocity distribution function is formulated, and an explicit function is shown for the case of Tsallis statistics. In Sec. III, the relative velocity distribution is calculated for Tsallis statistics, and we show that our relative velocity distribution is different from that adopted in previous studies. In addition, thermonuclear reaction rates and primordial light element abundances are calculated under the Tsallis statistics. In Sec. IV, we summarize this study.
We note, however, that the correction derived here applies to a variety of physical mechanisms that can directly modify the particle velocity distribution functions. There are many examples in the literature of physical mechanisms that lead to such altered velocity distribution functions. For example, such Tsallis distribution functions have been shown to arise from the combined effects of local magnetic and electric fields Rossani2009 (), or in space plasmas Pavlos2012 () including the heliosheath Burlaga2006 (), planetary magnetospheres, the solar corona, solar dynamics, and cosmic rays. We note, however, that a modified Tsallis-like distribution may also approximate the effect of relativistic electron scattering during BBN Sasankan:2018pfq () or sub-horizon temperature fluctuations as in Luo2018 (). In this latter case the sub-horizon local velocity distributions are indeed Maxwell-Boltzmann and a Tsallis-like distribution only emerges when averaging over a set of sub-horizon volumes at different temperature. In such models, a direct application of the Tsallis distribution to the relative velocity distribution as in Ref. Bertulani:2012sv () is justified.
Our model for the relative velocity distribution for general statistics assumes three things: 1) Each nucleus is described by a non-Maxwell-Boltzmann velocity distribution; 2) conservation of momentum; and 3) conservation of energy. We show in this section that imposing these conditions leads to a general relative velocity distribution function that does not resemble the original distribution in which velocities are replaced with the relative center-of-mass velocities and masses are replaced by the reduced mass. To begin with the thermal rate for a two-body reaction of nonrelativistic uncorrelated particles is given by
where is the reaction cross section, is the velocity vector of species and 2, is the velocity distribution function of , is the relative velocity, and is the center of mass (CM) energy.
We use the CM parameter transformations as follows:
There is also a conservation of energy relation,
We use the variable transformations from and to and (e.g., Clayton1984 ()). We define the distribution function of the relative velocity that satisfies
The distribution function is then given by
where the quantity in brackets with the subscript is estimated for a fixed .
ii.1 the Maxwell-Boltzmann distribution
where is the temperature. If both of the reacting particles are described by an MB distribution, i.e., , the integration over the velocities is given by
Therefore, the CM distribution function of the relative velocity for the case of MB statistics has the same form as that of the individual particle distribution functions, i.e.
ii.2 Tsallis distribution
The Tsallis distribution is, however, given by
where is a normalization constant determined from the requirement . When both of the reacting particles are described by this distribution with the same value, the product of the two distribution functions is given by
We then have the following transformation:
where and can be expressed as functions of and by
The distribution function of the relative velocity for the Tsallis particles is then given by
The integration ranges of and are derived as follows. For , the velocities for two species are limited Livadiotis2009 () to be
The integration ranges are then given by
For a fixed , the distribution function of is then transformed to
where the function is given by
This distribution function is manifestly different from that employed in previous studies Bertulani:2012sv (); Hou:2017uap (), which adopted exactly the same Tsallis function as the single particle velocity distribution with the mass replaced by the reduced mass. Thus, is replaced by in Eq. (17) of Ref. Bertulani:2012sv (). As a result, they separate the distribution functions of the relative velocity and the CM velocity. Their calculations below Eq. (17) and Eq. (13) which are based upon Eq. (17) are then inconsistent with momentum conservation in Tsallis statistics. 111In addition, Figure 9 in Ref. Bertulani:2012sv () is difficult to interpret physically. For larger values, nuclei with low energies are more abundant and those with high energies are less. Since the value for Be(,)Li is larger for smaller CM energy, the thermal reaction rate is expected to be larger for larger . This expectation is opposite to their Fig. 9. In any case, we do not expect a difference by orders of magnitude in the reaction rate since the value is of the same order of magnitude in the wide energy region MeV and the nuclear number density should be unchanged (see D. Jang, et al., in preparation). We note that in Ref. Bertulani:2012sv (), they obtained a similar large effect on the rate of He(,)H as that of Be(,)Li (their Sec. 4.1). Equations for thermonuclear reaction rates are given in a subsequent paper Hou:2017uap () [their Eqs. (3) and (4)]. These are based upon the same formulation as in Ref. Bertulani:2012sv (). Therefore, for the case of a pure Tsallis distribution the results in Hou:2017uap () are also inconsistent with momentum conservation. As noted above, although the forms of the distribution function are the same for and in the MB case, they are different in the case of general statistics.
The thermonuclear reaction rate for Tsallis statistics is then given by
When the particles are nonrelativistic, i.e., , the factor does not depend on . In the present case of the nuclear distribution during the BBN epoch, nuclei are nonrelativistic. In this case the distribution function of reduces to
where we adopt the natural units of .
ii.3 Reduced equations
ii.3.1 Tsallis statistics
We defined a thermal velocity such that the distribution function for the relative velocity amplitude for MB statistics is maximal at that velocity. The distribution function satisfies under the assumption of isotropy222We note that distribution functions of the velocity and the velocity vector should not be confused throughout the formulation.. The thermal velocity is given by
We then normalize all velocity variables in terms of the thermal velocity as follows:
Using this transformation, we derive the distribution function for Tsallis statistics, i.e.,
ii.3.2 the 1 particle Tsallis distribution for the reduced mass
Equation (12) with the mass replaced with is given by
ii.3.3 MB distribution
For comparison, the MB distribution function can be written as
The quantity corresponding to in the Tsallis statistical case for the MB case is given by
We find that the normalized distribution functions only depend on [and and for the Tsallis case: Eq. (39)] and these shapes do not essentially evolve along with the cosmic expansion.
ii.4 Reverse reaction rates
The detailed balance relations Blatt1991 () between cross sections of forward and reverse reactions for 1(2,)3 and 1(2,3)4 are given by
where the are the statistical weights of the respective nuclear species , while and are respectively the reduced mass and the CM energy of the + system.
ii.4.1 Photodisintegration reactions
Under the assumption that nuclei are nonrelativistic and that photons have a Planckian energy distribution, the photodisintegration rates do not depend on the nuclear distribution function. The photodisintegration rate is then given by
where is the photon energy, is the distribution function of the photon energy, is the photodisintegration cross section, is the light speed, and is the Riemann zeta function of 3. As far as the photon distribution follows the Planck distribution, the photodisintegration rate is the same as that of the SBBN 333Effects of non-Planck distribution of background radiation will be discussed anywhere else (D. Jang, et al. in preparation).. Therefore, we adopt the standard rates.
ii.4.2 Two-nuclear reactions
The thermal reaction rate for the reverse reaction of the type 4(3,2)1 is
where the subscript 34 indicates physical quantities of the 3+4 system, and the subscripts 3 and 4 indicate physical quantities of particles 3 and 4, respectively. When the distribution functions for all nuclei are the Tsallis distribution, this rate is reduced with Eqs. (39) and (40) to
We calculate reverse reaction rates using this equation and the detailed balance relation [Eq. (45)].
iii.1 Relative velocity distribution
Figure 1 shows normalized distribution functions for the CM kinetic energy . The Tsallis parameter is set to Hou:2017uap (). That value has been suggested Hou:2017uap () as the value for which the Li problem is solved. Solid and dash-dotted lines are functions for the Tsallis statistics [Eqs. (39) and (40)] for sets of nuclear mass numbers of , (dash-dotted line), , , , , and from the top to the bottom, respectively. The dashed line corresponds to the previously assumed distribution, i.e., the one-particle Tsallis distribution [Eq. (41)]. The dotted line is the MB distribution [Eq. (42)]. The MB distribution depends only on . The previously assumed distribution also depends only on for a fixed . The distribution function for the Tsallis statistics depends on the nuclear masses as well as for a fixed . This is one of the important differences from MB statistics.
In addition, the Tsallis distribution functions for the relative velocity have extended high energy tails compared with the previously assumed function. The maximum velocity [Eq. (LABEL:eq22)] is always larger than that of the one particle Tsallis distribution for the same reduced mass. Therefore, the extended high energy tails are realized in the exact distribution function.
We note that the assumption of the the same Tsallis distribution for the relative velocity [Eq. (41)] independent of masses of reacting nuclei Bertulani:2012sv (); Hou:2017uap () induces an inconsistency if and therefore unphysical. As shown above, the relative velocity distribution is derived from the velocity distribution functions of nuclei, and its relation to the nuclear distribution functions depends on the nuclear masses. If one supposes that the relative velocity distribution, which is in fact a physical quantity derived from distribution functions of respective nuclear velocities, generally follows the Tsallis form, solutions for distribution functions of respective nuclei cannot be found except for the trivial case of .
iii.2 Thermonuclear reaction rates
Table 1 shows the eleven important reactions of BBN Kawano1992 (); Smith:1992yy () (the first column), and references to the available cross section data that we adopted in this study (the second column).
We calculated the freeze-out temperature  for one chosen nuclide participating in one reaction which satisfies the freezeout condition that its abundance rate of change equals the cosmic expansion rate , i.e.,
where is the number density of nuclide , while and are nuclides in the initial state of the reaction , and is the average reaction rate as a function of . The third and fourth columns in Table 1 show the nuclide whose abundance freezes out and its freeze-out temperature, respectively. The fifth column shows the ratio of the peak energy to , where the peak energy has the largest contribution to the integrand in deriving the average reaction rate. We checked that the change in the peak energy is small if the Tsallis value is not changed significantly, i.e., .
|H(,)H||Ando:2005cz () (Fig. 3)||H||0.53||0.68|
|H(,)He||Coc:2015bhi () (Descouvemont:2004cw () for > 2 MeV)||H||0.74||2.1|
|H(,)H||Coc:2015bhi () (Descouvemont:2004cw () for > 0.6 MeV)||H||0.55||2.0|
|H(,)He||Coc:2015bhi () (Descouvemont:2004cw () for > 0.6 MeV)||H||0.53||2.1|
Figure 2 shows contours of the functions [Eq. (40)] (solid lines) and [Eq. (43)] (dashed lines) in the (,) plane for the He+He system at for the energies of (a) and 3 (b), respectively. From this figure, a difference in the contributions of the parameter regions to the reaction rate between the Tsallis and MB cases is apparent. We considered the Be production reaction and its freeze-out temperature for this figure. In panel (a), it is seen that values are hindered compared to the MB case. In addition, the value significantly depends on the angle between and , which is different from the function . Near parallel or anti-parallel scatterings are hindered as seen from the curved contours of . In the regions of and , or becomes maximally large leading to small values of [Eq. (40)]. For intermediate values, both of and values are intermediate, and the hindrance of is minimal. In panel (b), vertical dashed lines correspond to , , …, , from the left to right, respectively, that is a hindrance factor depending on , exactly the same as those in panel (a). The solid lines show the contours of , , and , respectively. Curvatures are larger than in panel (a), which indicates that the contribution from the intermediate satisfying is exclusively important.
Figure 3 shows the average rates for the reactions H(,)H (upper panel) and He(,)Be (lower panel) as a function of temperature . The former reaction is one of two most important reactions for D destruction. We note that the reaction rate of the other reaction, i.e., H(,)He is very similar to that of H(,)H. The He(,)Be reaction is the most important Be production reaction. Therefore, the two reactions in Fig. 3 determine the final freeze-out abundances of D and Be, respectively. The solid line is for the Tsallis statistics, while the dashed line is based upon the one-particle distribution function as previously assumed. The dotted line corresponds to the MB distribution.
For this specific value and the temperature range of , the H(,)H reaction rate for the Tsallis statistics is smaller than that of the MB statistics. Since the distribution function at high energies is smaller than that of the MB function, the reaction rate is hindered by the Coulomb penetration factor. The difference between cases of the Tsallis statistics and the previously assumed function is small.
The reaction rate of He(,)Be for the Tsallis statistics is also smaller than that of the MB statistics because of the more effective Coulomb suppression factor. We find a significant difference between the reaction rates of the Tsallis statistics and the previously assumed case. For a fixed CM energy, the Coulomb penetration factor is suppressed more than that of the reaction H(,)H because of the larger atomic numbers and reduced mass. Therefore, the thermal reaction rate is contributed from higher energies, where the difference in the distribution function between the Tsallis and the previous case is largest (see Fig. 1). The averaged reaction rates of He(,)Be then differ more than those of H(,)H.
iii.3 BBN calculation
We adopt the SBBN code as described in Refs. Kawano1992 (); Smith:1992yy () and have updated reaction rates of nuclei with mass numbers using the JINA REACLIB Database Cyburt2010 () (updated to December, 2014). The neutron lifetime is the central value of the Particle Data Group, s 2016ChPhC..40j0001P (). The baryon-to-photon ratio is taken to be calculated using a conversion Ishida:2014wqa () of the baryon density in the standard CDM model (TT+lowP+lensing) determined from the Planck observation of the cosmic microwave background, Ade:2015xua (). For eleven important reactions of BBN, the reaction rates are calculated for the different distribution cases. Two-body reverse reaction rates are calculated with the detailed balance relation using Eqs. (44)–(47) and (49). Since the effect of the Planckian distribution is small Mathews:2010aa (), we make the usual approximation of replacing the Planck distribution with an exponential.
Figure 4 shows the evolution of nuclear abundances as a function of . and are mass fractions of H and He in total baryon matter, respectively. For other nuclear abundances, the number ratios to H, i.e., /H are shown. In the upper panel, the solid lines are for the Tsallis statistics, while the dashed lines are for the previously assumed relative velocity distribution function. The dotted lines are results for the MB statistics, i.e., SBBN. The Tsallis parameter is set to .
For , the high energy region of the velocity distribution functions of nuclei is suppressed. Therefore, rates of charged-particle reactions are significantly reduced since the cross sections are larger at high energies. For , the opposite situation is realized. On the other hand, rates of neutron reactions are unaffected because there is no Coulomb penetration factor.
In the upper panel of Fig. 4, since the reaction rate of deuteron destruction via H(,)H and H(,)He is smaller than in SBBN (Fig. 3, upper panel), the freeze-out D abundance is larger Bertulani:2012sv (). Since He and H are produced via the same reactions, the higher D abundance results in higher production rates of He and H. Therefore, the final abundances of He and H are higher than in SBBN. The neutron abundance is slightly larger at late times for which . This is because of the larger D abundance. The neutron abundance is determined from the forward and reverse reactions of H(,)H. The forward rate is slightly larger and the reverse rate is the same as that of the SBBN. The H abundance is almost the same in the Tsallis case and SBBN. Therefore, after the D destruction freezes out at a higher level, the abundance is kept higher via the photodisintegration H(,)H. The final Be abundance is smaller than in SBBN because of the smaller reaction rate for He(,)Be (Fig. 3, lower panel). Since the reaction rate is underestimated in the previous calculations Bertulani:2012sv (); Hou:2017uap (), our Be abundance is larger than the previous estimate. The Li destruction rate via Li(,)He is also smaller than in SBBN. Therefore, the freeze-out Li abundance is larger.
The lower panel shows abundances for Tsallis statistics with (dashed line), 1 (dotted), and 1.1 (solid), respectively. When the -value increases, rates of charged-particle reactions are smaller. For the same reason explained for the upper panel, the abundances of D, H, He, , and Li increase while the Be abundance decreases.
Calculated BBN results are compared to observational constraints on light element abundances. Constraints on the primordial He abundance come from observations of metal-poor extragalactic H II regions. We use two different determinations of Peimbert:2016bdg () and Izotov:2014fga (). The primordial D abundance is constrained with observations of metal-poor Lyman- absorption systems towards quasi-stellar objects. We use the weighted mean value of D/H Cooke:2017cwo (). He abundances in Galactic H II regions are determined using the GHz hyperfine transition of He ion. The primordial He abundance can evolve during Galactic chemical evolution. However, the net effect of Galactic chemical evolution is uncertain since stars can both destroy and synthesize He. Moreover, it is not expected that the He abundance has decreased significantly over Galactic history as this would require that a large fraction of Galactic baryonic material has been absorbed in stars that destroy He, while the present interstellar deuterium abundance limits the amount of astration to not more than about a factor of two. We then only adopt the upper limit from the abundance He/H= Bania:2002yj () in Galactic H II regions. We also use the abundance Li/H) derived by observations of Galactic metal-poor stars using a 3D nonlocal thermal equilibrium model Sbordone2010 ().
Figure 5 shows calculated light element abundances as a function of the parameter . Boxes show the 2 observational limits on D/H and Li/H. The dashed and dotted lines show abundances of Be and Li, respectively, immediately after BBN. Long after BBN, Be nuclei electron capture to produce Li nuclei. Therefore, the sum of Be and Li abundances becomes the primordial Li abundance. The He abundance is predominantly contributed by He, plus a small abundance of H produced during BBN (see Fig. 4) has been added. The vertical line is at and corresponding to the SBBN case. The plotted range is allowed by the 2 limit on the He abundance of Ref. Peimbert:2016bdg () and excluded by that of Ref. Izotov:2014fga (). Also this region is allowed by the 2 upper limits on He/H Bania:2002yj ().
The reasons for the abundance changes of D, He, Be, and Li have been explained above. The one percent level of change for the final He mass fraction is caused by different neutron abundances during the He synthesis. For larger , the D destruction rate is smaller and the D abundance is larger. As a result of the balance of forward and reverse reactions of H(,)H (see above), the abundance is kept higher and more neutrons are lost by -decay before He synthesis is completed. The final He abundance is therefore smaller.
It is seen that the D abundance increases and the Li abundance decreases with increasing value. At –1.02, the theoretical result for the D abundance is consistent with the observation. On the other hand, for , the Li abundance agrees with the observation. However, in this parameter region, the D abundance is enhanced to above D/H, which requires an additional mechanism for later D destruction. Because of their fragility, deuterons can be destroyed easily if there is a source of non-thermal photons in the early universe (e.g., Lindley1979MNRAS.188P..15L (); Kusakabe:2013sna ()). Then, the D destruction by non-thermal photons can reproduce the primordial elemental abundances consistent with observations of all light nuclei. This can happen for example, in a model including photon cooling by an axion condensate Kusakabe:2012ds ().
Unless a later D destruction mechanism is induced, the D enhancement is very problematic. Therefore, the observed D abundance places an upper limit on . We find that in the range of –1.02 where the observed D abundance is reproduced, the Li abundance is smaller than in the SBBN by 30–60 %. This level of Li reduction is significant but not enough. On the other hand, there are other astrophysical processes which can further reduce the Li abundance, i.e., a chemical separation of Li ions during structure formation Kusakabe:2014dta () or a depletion on stellar surfaces Richard:2004pj (); Korn:2007cx (); Lind:2009ta ().
We have reformulated the thermal rate of two-body reactions of a gas of nonrelativistic uncorrelated particles with general velocity distribution functions. Taking the Tsallis distribution as an example of a non-MB distribution, we derived the distribution function of the relative velocity, i.e., . It was found that in general the distribution function contains a complicated integration over the CM velocity . By defining a normalized distribution function , the MB distribution can be expressed in terms of the ratio of the CM energy to the temperature, i.e., . However, we found that the normalized distribution function for the Tsallis statistics has additional dependences on the nuclear masses (Sec. II).
We showed differences in the relative velocity distribution function between the Tsallis and MB statistics (Sec. III.1). Using calculated distribution functions, reaction rates that are important for BBN were derived (Sec. III.2). Finally, we performed the BBN nuclear reaction network calculation, and analyzed effects of changing the Tsallis parameter. An increase of results in softer nuclear spectra, and an upper cutoff of the CM energy appears for . We observed that the increase of reduces rates of reactions between charged particles, and explained reasons that abundances of D, H, He, , and Li increase while the Be abundance decreases. Predicted abundances as a function of the parameter were calculated. We found the following points: (1) A slight deviation from the MB statistics, i.e., –1.02 can lead to D abundances consistent with observations, which are larger than the SBBN prediction; (2) The D observation provides the most stringent constraint on the parameter; (3) In that region, the primordial Li abundance is reduced from the SBBN value by 30–60 % (Sec. III.3).
Acknowledgements.MK acknowledges support by the visiting scholar program of NAOJ during his stay there. This work was supported by Grants-in-Aid for Scientific Research of JSPS (15H03665, 17K05459). Work of GJM supported in part by DOE nuclear theory grant DE-FG02-95-ER40934 and in part by the visitor program at NAOJ.
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