Dynamical Screening Effects on Big Bang Nucleosynthesis
Eunseok Hwang, Dukjae Jang, Kiwan Park, Motohiko Kusakabe, Toshitaka Kajino, A. Baha Balantekin, Tomoyuki Maruyama, Chang-Mo Ryu, Myung-Ki Cheoun
aa r X i v : . [ nu c l - t h ] F e b Draft version February 24, 2021
Typeset using L A TEX twocolumn style in AASTeX62
Dynamical Screening Effects on Big Bang Nucleosynthesis
Eunseok Hwang, Dukjae Jang, Kiwan Park, Motohiko Kusakabe,
Toshitaka Kajino,
A. Baha Balantekin,
Tomoyuki Maruyama,
4, 7
Chang-Mo Ryu,
2, 8 and Myung-Ki Cheoun
1, 3,4 — Department of Physics and OMEG Institute, Soongsil University, Seoul 156-743, Republic of Korea Center for Relativistic Laser Science, Institute for Basic Science (IBS), Gwangju 61005, Republic of Korea School of Physics and International Research Center for Big-Bang Cosmology and Element Genesis, Beihang University, Beijing100083, China National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan Physics Department, University of Wisconsin-Madison,1150 University Avenue, Madison, Wisconsin 53706, USA College of Bioresource Sciences, Nihon University, Fujisawa 252-0880, Kanagawa-ken, Japan Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea
ABSTRACTWe study dynamical screening effects of nuclear charge on big bang nucleosynthesis (BBN). A mov-ing ion in plasma creates a distorted electric potential leading to a screening effect which is differentfrom the standard static Salpeter formula. We consider the electric potential for a moving test charge,taking into account dielectric permittivity in the unmagnetized Maxwellian plasma during the BBNepoch. Based on the permittivity in a BBN plasma condition, we present the Coulomb potential for amoving nucleus, and show that enhancement factor for the screening of the potential increases the ther-monuclear reaction rates by a factor order of 10 − . In the Gamow energy region for nuclear collisions,we find that the contribution of the dynamical screening is less than that of the static screening case,consequently which primordial abundances hardly change. Based on the effects of dynamical screen-ing under various possible astrophysical conditions, we discuss related plasma properties required forpossible changes of the thermal nuclear reactions. Keywords: early universe — plasmas — primordial nucleosynthesis INTRODUCTIONThe origin of chemical elements in the universe hasbeen a long research interest under debate in nuclearastrophysics. From t = 1 s to t = 100 s after the bigbang, the universe was in a dense state, and particleswere of energies ∼ O (MeV) appropriate for the nuclearreactions. Gamow (1946) suggested that most light andheavy nuclei should have been generated during this era.However, synthesizing the nuclei heavier than helium isnot an easy process without high enough temperatureto maintain the thermonuclear reactions.To explain the origin of solar element abundances, inmodern physics, various nuclear astrophysical processessuch as r -, s -, and p -processes have been studied with the Corresponding author: Dukjae [email protected] development of precise astrophysical observations andexperimental data (Meyer 1994; K¨appeler et al. 2011;Arnould & Goriely 2003; Kajino et al. 2019). Theoret-ical prediction of the astrophysical processes requiresnuclear network calculations interwoven with relevantthermonuclear reaction rates in evolving astrophysicalenvironments. In the thermonuclear reaction rate, asmany textbooks show (Clayton 1968), the cross sectionis written as σ ( E ) = S ( E ) E exp[ − πη ] , (1)where η = Z Z e /v with the relative velocity v , knownas the Sommerfeld parameter. Note that we adopt thenatural unit, i.e., ~ = c ≡ S ( E ),determined experimentally, can grow rapidly at a res-onance energy level, which leads to the precipitous in-crease of reaction rate. Although the low-lying reso- Hwang et al. nance in the nuclear cross section is one of importanttopics in astrophysical nuclear processes as well as bigbang nucleosynthesis (BBN), we will focus on the re-duction of Coulomb barrier by the dynamical screeningeffects with the consideration of plasma effect during thecosmological era.The electric potential between interacting nuclei inastrophysical plasma behaves differently from the mea-sured one in the laboratory, exhibiting a collective be-havior to maintain the quasi-neutrality. Despite manyattempts to bridge the gap between the nuclear reactionsin astrophysical plasma and laboratory—such as experi-ments using an ultra-intense laser (Wu & P´alffy 2017) aswell as accelerators (Huke et al. 2008; Targosz-´Sl¸eczka et al.2013), deciphering this mechanism has been stubbornlyelusive due to complexity of plasma properties.One of fundamental characteristics in plasma is theDebye shielding. The electron cloud surrounding thecharge of an ion screens other nuclear charges far fromthe own radius, approximately given as Debye radius λ D . From the viewpoint of nuclear reaction, this screen-ing effect reduces the Coulomb barrier in the nuclear re-action, so that the penetration probability is enhanced.Under the weak screening condition, Salpeter formula(Salpeter 1954) well describes the static screening effectsfor the thermonuclear reactions, which result in the en-hancement factor f en in terms of given temperature T as follows: f en = exp (cid:20) Z Z e λ D T (cid:21) , (2)where Z i denotes the charge number of species i and theDebye radius λ D,e is defined as λ D,e = r T πn e e , (3)depending on the temperature and the electron numberdensity n e .Since the screening effects change the thermonu-clear reaction rates, the Salpeter form has been ex-ploited widely in nucleosynthesis such as the stellarnuclear fusion (Potekhin & Chabrier 2013), presuper-nova (Liu et al. 2007), and core-collapsing supernova(Liu et al. 2009). Among various studies, in particu-lar, BBN has been a good testbed for the screeningeffects owing to a relatively small number of main re-actions and precise observational data based on cosmicmicrowave background (CMB) study and astronomi-cal spectroscopy. The first application of the Salpeterformula to the BBN shows the screening effects hardlyaffect the primordial abundances because the densityin BBN environment is too low to reveal the effec-tive charge shielding (Wang et al. 2011). However, it was pointed out that the relativistic corrections areconsiderable in BBN environment due to enough hightemperatures (10 K . T . K) to maintain theelectron-positron plasma. Then, replacing Debye radiuswith Thomas-Fermi one, the relativistic correction ofscreening effects on BBN was performed, but the resultstill shows insignificant fractional changes of primordialabundances within the order of 10 − (Famiano et al.2016). Furthermore, a study of the relativistic screeningeffects with the primordial magnetic field (PMF) showsremarkable changes of primordial abundances, in whichthe PMF was consistently treated in the cosmic expan-sion rate, the temperature evolution, and screening cor-rection for weak reactions, which are constrained withobservational data of primordial abundances (Luo et al.2020).As an extension of the previous studies, this paperfocuses on the screening effects for the moving ions onBBN because the velocity distribution includes large ve-locities for some of the ions in the plasma. When thetest charge moves with an intermediate velocity that isenough to react with background charges in plasma, theCoulomb potential forms a distorted shape (Wang et al.1981; Trofimovich & Krainov 1993). This dynamicalscreening effect was originally proposed to solve the so-lar neutrino problem by applying the Coulomb energyto the thermonuclear reaction rates in solar conditions(Carraro et al. 1988). Although the discovery of theneutrino oscillation receives the credit for the solutionof the solar neutrino problem (Kajita 2016; McDonald2016), it is still worthy to study the dynamical screen-ing effect on the nucleosynthesis in that there are roomsfor the change of thermonuclear reaction rates by themoving ions in the plasma.In the solar condition, it was shown that the dynami-cal screening effects on the thermonuclear reaction rateslead to a slightly reduced enhancement factor for thethermonuclear reaction rates rather than the Salpeterform (Carraro et al. 1988). However, since the massof the ion is much heavier than the solar tempera-ture, it was verified that the thermal velocity of the ionis not too fast to realize significant dynamical effects(Brown & Sawyer 1997).Compared to the solar condition, the BBN occurs ina state of low density, but high temperature. Also,the rapid cooldown of the early universe changesBBN plasma components from electron-positron-ion(EPI) to electron-ion (EI) plasma. This change causesthe cosmic time-dependent dielectric permittivity; es-pecially such a description of the electron-positronplasma plays an important role in active galactic nuclei(Begelman et al. 1984), pulsar and neutron star mag- ynamical screening effects on BBN COULOMB POTENTIAL FOR MOVING IONSIN BBN EPOCHFrom the Poisson equation for a moving test chargewith time t , the Coulomb potential can be written as(Carraro et al. 1988) φ ( r − v t ) = Z e π Z d k k e i k · ( r − v t ) ǫ l ( k , k · v ) , (4)where Z denotes charge number of the test charge, r the spherical coordinate in the rest frame, and v thevelocity of the moving test charge. The longitudinalmode of dielectric permittivity ǫ l ( k , ω ), as a functionof wavevector k and frequency ω , is derived by the firstorder perturbation of the Vlasov-Maxwell equation withunmagnetized equilibrium plasma (Lifshitz & Pitaevskii1981). When we consider the electron ( e − ), positron( e + ), H, and He as components in the early universeplasma, ǫ l ( k , ω ) for test charge moving along x -direction is given as ǫ l ( k , ω ) = 1 − πe g e k Z ∞−∞ df e ( p x ) dp x dp x kv x − ω + 1( kλ D, H ) " F ω √ kv T, H ! + 1( kλ D, He ) " F ω √ kv T, He ! , (5)where g e is the statistical degrees of freedom for elec-trons, i.e., g e = 2, and f e ( p x ) denotes the integration ofthe momentum distribution for the net electron over y and z components, i.e., f e ( p x ) ≡ R f ( p ) dp y dp z . We as-sume that all species have same temperature under thethermal equilibrium condition and adopt Fermi-Diracdistributions for the e − and e + , which defines f e as fol-lows: f e ≡ f e − − f e + (6)= (cid:18) √ p + m e − µT (cid:19) + 1 − (cid:18) √ p + m e + µT (cid:19) + 1 , where p is the momentum, m e the mass of e − (or e + ),and µ the chemical potential determined by the chargeneutrality condition of n e − = n e + + n H + 2 n He . Intime-evolving BBN condition, the chemical potential µ is varied with cosmic time (Pitrou et al. 2018). Herewe neglect contributions of other nuclei such as D, He, Li, and Be due to their small numbers. For eachspecies, we use the Debye radius and thermal veloc-ity as λ D,i = p T / πn i Z i e and v T,i = (
T /m i ) / ,respectively. Nuclear mass fraction X i is related toits number density n i = X i ηn γ /A i , where η , n γ ,and A i stand for the baryon-to-photon ratio, num-ber density of photon, and mass number of species i , respectively. To obtain the mass fraction of pro-tons and He, we use the updated BBN calculationcode in Refs. (Kawano 1992; Smith et al. 1993) with re-actions in Refs. (Descouvemont et al. 2004; Iliadis et al.2016), and adopt parameters as follows: neutron lifetime τ n = 879 . s (Tanabashi et al. 2018); η = 6 . × − (Planck Collaboration et al. 2016); effective neutrinonumber N eff = 3 .
046 (Mangano et al. 2005). Function F ( x ) in Equation (5) is defined as (Lifshitz & Pitaevskii1981) F ( x ) = xπ / Z ∞−∞ e − z z − x dz + iπ / xe − x . (7) Hwang et al.
Note that ǫ l ( k , ω ) goes to unity corresponding to freespace when all components are absent. In other words,the contribution of plasma components makes the per-mittivity deviate from the value in free space, whichaffects the shape of Coulomb potential.Figure 1 shows the number densities of plasma compo-nents during the BBN epoch. In the high temperatureregion corresponding to early cosmic time, the numberdensity of relativistic e − and e + is proportional to T ,and they are dominant components. Over the cosmictime, the universe rapidly cools down by the adiabaticexpansion, and n e − and n e + drastically decrease when e − s and e + s become non-relativistic at T . m e . On theother hand, the He abundance increases by the nucle-osynthesis as follows. In the early phase, BBN conditionmaintains enough high temperature to allow the nuclearstatistical equilibrium (NSE). The NSE continues to in-crease He up to T = 0 . H and He slows down the reaction rate of H( p, γ ) He and He( n, γ ) He. By the slow productionrate, He is decoupled from NSE and follows the NSEcurve of H and He. Similarly, at T = 0 . H( n, γ ) H and H( p, γ ) He, which lead to the decou-pling of H and He from the NSE. At T = 0 .
07 MeV,increase of H, He and He following the NSE curve ofdeuterium stops by decoupling of deuterium from NSE.Consequently, those three kinds of decoupling from NSEat T = 0 . . .
07 MeV make the bro-ken lines for He shown in Figure 1 (See also Smith et al.(1993); Sarkar (1996) for details.). Due to this synthesisof He, a difference between n H and n e − remains so asto satisfy the charge neutrality condition. The changeof dominant plasma components transforms the earlyuniverse from EPI to EI plasma, and as a result, thedielectric permittivity becomes cosmic time dependent.Figure 2 shows the deviation of ǫ l ( k , ω ) from the unityas a function of α ≡ v/v T, H . When the proton has highvelocity ( α ≥ ), as shown in all panels in Figure 2,the dielectric permittivity converges to unity, i.e., onein free space. It means that the ion velocity is so fastthat background plasma cannot react to the moving ion.In this very high velocity region, we should consider therelativistic correction of the moving ion, but the typi-cal temperatures of astrophysical environments hardlyallow the high velocity of ions due to large mass. Onthe other hand, near the thermal velocity region, thedielectric permittivity relys on the background propertydepending on temperature.At T = 1 MeV, relativistic e − s and e + s dominate theBBN plasma as explained above. At this moment, the -30 -25 -20 -15 -10 -5 -3 -2 -1 n i ( T ) [ M e V ] T [MeV] H Hee + e - Figure 1.
The number densities of plasma components as afunction of BBN temperature. Magenta-dashed-single dot-ted, blue-dashed-double dotted, red-solid, and green-dashedlines denote the number densities of H, He, e + , and e − ,respectively. near-equality of number densities e − s and e + s leads thenet current to become null, so the permittivity is rarelychanged. This dominance of e − and e + continues to T = 0 . T = 0 . T ∼ .
01 MeV, by de-creasing of temperature, the number densities of e − sand e + s rapidly reduce to a level comparable to that ofbaryons. As neutrons decay during the BBN, electronsare produced and e + number density reduces in orderto satisfy the charge neutrality. When n e − decreasesto ∼ n H , the difference between n e − and n e + becomessignificant. While positrons continue to annihilate, theelectron annihilation freezes out. Then, a deviation ofthe permittivity from unity develops. However, freeze-out of nuclear reactions does not change the nuclearabundance after T = 0 .
01 MeV, which makes the de-viation at T = 0 .
001 MeV similar to the T = 0 .
01 MeVcase. Therefore, the lower two panels in Figure 2 showthe similar deviations of the permittivity from that ofthe free space.In the imaginary part related to a damping of electricpotential, an oscillation behavior stems from the func-tion F ( x ) defined in Equation (7). At T = 1 MeV, the He synthesis does not start yet, and the fourth term inEquation (5) can be omitted. Then, the only one peakby the proton ion is shown in the upper-left panel of Fig-ure 2. After the He synthesis, the contribution of Heion to the imaginary part appears, by which the secondpeak is seen in other panels.Adopting the obtained dielectric permittivity, we cal-culate the Coulomb potential for a moving proton. Forconvenience, we perform a transformation from the fluid ynamical screening effects on BBN -1 ε l - α Real partImaginary part
T=1 MeV -1 ε l - α Real partImaginary part
T=10 -1 MeV -1 ε l - α Real partImaginary part
T=10 -2 MeV -1 ε l - α Real partImaginary part
T=10 -3 MeV
Figure 2.
Difference of dielectric permittivity between BBN plasma and free space i.e., ǫ l −
1, as a function of α (= v/v T, H )at T = 1 MeV (upper-left), T = 0 . T = 0 .
01 MeV (lower-left), and T = 0 .
001 MeV (lower-right). Red andblue solid lines denote the real and imaginary parts of the dielectric permittivity, respectively. rest frame to the moving proton frame, which leadsEquation (4) to φ ( R ) = Z e π Z d k k e i k · R ǫ l ( k , k · v ) , (8)where R ≡ r − v t . Figure 3 shows the electric potentialnormalized by Coulomb potential, i.e., φ ( R ) / ( Z e/R ),for a proton moving along the x -axis in the plane ofthe two dimensional coordinate as ( ρ x ≡ x/λ D,H , ρ y ≡ y/λ D,H ). According to Equation (8), the Coulomb po-tential for a moving test charge depends on the dielectricpermittivity as well as ion thermal velocity or tempera-ture. Here the velocity of the proton is set to the thermalvelocity. The dependence of the potential on the veloc-ity and the dielectric permittivity implies that the shapeof the potential also evolves with cosmic temperature.In the non-relativistic regime, it is known that thefaster the test charge is, the more distorted the shape ofthe electric field is from a spherical shape (Wang et al.1981; Carraro et al. 1988). At T = 1 MeV, the thermalvelocity of the proton is the highest among the panelsin Figure 3, which effectively polarizes the background charge. As a result, at T = 1 MeV, the valley in thebackward direction of the moving charge has the mini-mum value and the shape of the potential is largely dis-torted, despite the small deviation of dielectric permit-tivity. On the other hand, the potential in the forwarddirection is increased. At T = 0 . T = 0 .
01 MeV (the left-bottom panel of Figure3), the deviation of dielectric permittivity is larger bygrowing up of the heavy ion fractions, and the distor-tion of the electric potential is also larger. After thefreeze-out of the nucleosynthesis at T = O (0 .
01 MeV),the dielectric permittivity is nearly constant while thethermal velocity reduces. A slow nucleus results in analmost spherical shape of electric potential as shown inthe lower-right panel of Figure 3. For ρ y = 0, Figure4 shows the same calculation results with Figure 3, inwhich we can see the apparent difference between for-ward and backward directions of a moving proton de- Hwang et al.
Figure 3.
Normalized electric potential for a moving ion under the BBN condition, i.e., φ ( R ) / ( Z e/R ). As a test charge, weadopt a proton moving along x -direction in dimensionless real space of ρ x ( ≡ x/λ D ) and ρ y ( ≡ y/λ D ) with the thermal velocity,i.e., v = p T /m H . pending on temperatures. In a nutshell, we note thatthe polarizability of the electric field is intensified bythe high velocity of test charge and the deviation of di-electric permittivity. The dynamical screening effect ap-pears by the distorted shape of the electric potential. THERMONUCLEAR REACTION RATES WITHDYNAMICAL SCREENING EFFECTFor a two-body reaction between species 1 and 2, thegeneral form of the thermal averaged nuclear reactionrate is given as N A h σv i = N A Z d v f ( v ) Z d v f ( v ) v r σ ( E ) , (9)where N A is Avogadro’s number, v i the velocity of par-ticle i , f ( v i ) the normalized velocity distribution func-tion of particle i , v r = | v − v | the relative velocity, and σ ( E ) the cross section for the given reaction dependingon the total kinetic energy of E = µv r / µ of particles 1 and 2 in the center of mass(CM) frame. Owing to the low energy condition fortypical temperature of astrophysical environments, the cross section can be expressed in terms of astrophysicalS-factor, S ( E ), as follows: σ ( v , v ) = S ( E ) E P ( v , v ) , (10)where P ( v , v ) is the penetration factor which dependson the Coulomb interaction energy, i.e., P ( v , v ) = exp h − p µ (11) × Z r c r n dr r Z Z e r + W ( r, v , v , θ , θ ) − E , where the dynamical screening contribution to P ( v , v )is involved in W ( r, v , v , θ , θ ). In principle, theCoulomb interaction energy is obtained from the in-tegration of the given potential between the classi-cal turning point r c and nuclear radius r n . How-ever, since the radius of electron cloud—approximatelygiven by Debye radius λ D —is much larger than r c ,one can use the weak screening condition, resulting in ynamical screening effects on BBN -0.2 0 0.2 0.4 0.6 0.8-10 -5 0 5 10 φ ( ρ x ) / ( Z e / ρ x ) | ρ y = ρ x DynamicStatic
T=1MeV -0.2 0 0.2 0.4 0.6 0.8-10 -5 0 5 10 φ ( ρ x ) / ( Z e / ρ x ) | ρ y = ρ x DynamicStatic
T=10 -1 MeV -0.2 0 0.2 0.4 0.6 0.8-10 -5 0 5 10 φ ( ρ x ) / ( Z e / ρ x ) | ρ y = ρ x DynamicStatic
T=10 -2 MeV -0.2 0 0.2 0.4 0.6 0.8-10 -5 0 5 10 φ ( ρ x ) / ( Z e / ρ x ) | ρ y = ρ x DynamicStatic
T=10 -3 MeV
Figure 4.
Normalized electric potential for ρ y = 0 at denoted temperatures. Red and blue solid lines show the potential formoving and static test charges, respectively. the vanish of r dependence in Coulomb energy, i.e., W ( r, v , v , θ , θ ) ≃ W ( v , v , θ , θ ).Precisely, from the following polarized potential: φ pol ( r, v , v , θ , θ ) = 12 [ Z eφ pol , ( r , v , θ ) (12)+ Z eφ pol , ( r , v , θ )] with φ pol ,i ( r , v ) = Z i e π Z d k k (cid:20) ǫ l ( k , k · v i ) − (cid:21) (13) × exp[ i k · ( r i − v i t )] , the W ( v , v , θ , θ ) defined at r = 0 is obtained as fol-lows (Carraro et al. 1988) W ( v , v , θ , θ ) = − Z Z e π Z d k k (cid:20) e − i k · ( v − v ) t (cid:18) ǫ l ( k , k · v ) − (cid:19) + e − i k · ( v − v ) t (cid:18) ǫ l ( k , k · v ) − (cid:19)(cid:21) . (14)We note that this term corresponds to the Dybye-H¨uckelpotential if we take v = v = 0. Namely, the dynamicalscreening termed in this paper already reflects the staticscheme. Finally, with the use of the CM frame, thethermonuclear reaction rate in Equation (9) is written as (see Appendix.) N A h σv i = N A r πµT × Z ∞ σ ( E ) Ef s ( E ) e − E/T dE, (15)
Hwang et al. where the enhancement factor by dynamical screeningis f s ( E ) = Z ∞ Z π E / π − / T − / sin θ × e − E cm /T e W ( E,E cm ,θ ) /T dθdE cm , (16)where E cm is the kinetic energy of the CM and θ is theangle between the relative velocity and the velocity ofthe CM (see Appendix.). Unlike the static screeningeffects, f s ( E ) in Equation (16) depends not only on thecharge number (Equation (14)) but on the reduced massof interacting particles, related to the thermal velocity,as well as the dielectric permittivity of plasma.For the reaction of H( p, γ ) He, the enhancement fac-tor f s ( E ) is shown in Figure 5 for four temperatures.In general, the screening effect decreases with increas-ing energy because we can neglect the screening termwhen the electric potential becomes much lower thanthe kinetic energy of nuclei (See Equation (12).). Note-worthy is that the static screening potential is more de-creased than the dynamic one in the high energy regionbecause the fast ion forms an intense polarization of po-tential. On the other hand, in the low energy region,slow ions cannot effectively cause the polarization of theelectric field, and the enhancement factor by the dy-namical screening effect is reduced more than the staticcase.In the integral over energy in the thermonuclear re-action rates (Equation (15)), at the Gamow energy, thedifferential rate is maximally impacted. Therefore thisis an important energy scale. Although the screeningenergy slightly changes the Gamow peak position, weadopt the standard Gamow energy formula at each giventemperature because the change by a shift of the Gamowpeak is insignificant. Figure 5 shows that the dynami-cal screening effect is smaller than the static one at theGamow window. The fact that the enhancement factoris almost constant with respect to energy allows us thefollowing approximation: h σv i dynamic ≃ f s ( E G ) h σv i bare , (17)where E G and h σv i bare stand for the Gamow energy andthe bare reaction rate, respectively.The deuterium abundance is related to the baryon-to-photon ratio observed in CMB and can be comparedwith astronomical observations of D/H from the anal-ysis of quasar photon spectra (Cooke et al. 2018). Forevaluation of primordial abundances as well as D de-struction in stars, the precise measurements of relatedreaction cross sections are important. A recent ex-periment of H( p, γ ) He reaction performed in Labora- tory for Underground Nuclear Astrophysics (LUNA) re-markably narrows the uncertainty of the cross section(Mossa et al. 2020). According to our calculation, thedynamical screening effect changes the thermonuclearreaction rate of H( p, γ ) He by the order of . − ,which is allowed within the uncertainty of thermonu-clear reaction rates obtained from analysis of LUNA ex-periments.Figure 6 shows the enhancement factors for main reac-tions in BBN by dynamical and static screening effectsas a function of cosmic temperature. As shown in Figure3, the Coulomb potential by dynamic nuclei depends onthe direction; the potential in forward direction of themoving ion increases, while the one in backward direc-tion decreases. In the colliding system where the inter-acting particles approach to each other, the Coulombpotential in the forward direction predominantly affectsthe reaction, which results in the higher Coulomb bar-rier than the one by the Dybye-H¨uckel potential. There-fore, overall temperature region, the dynamical screen-ing enhancement is lower than the static case becausethe thermonuclear reaction rates are hardly changed.Although the early universe has high enough temper-atures to produce feasible velocities of ions, low particledensities cannot effectively change the dielectric permit-tivity. With the eight main charged particle reactions inBBN, we calculate the primordial abundances, and theirtime evolution is shown in Figure 7. Due to the smallchange of thermonuclear reaction rates, the effects of dy-namical screening on main BBN reactions are invisiblein primordial abundances. DISCUSSION AND CONCLUSIONSIn summary, we presented the effects of dynamicalscreening on BBN. In the early phase of the BBN,plasma is dominated by relativistic e − and e + . Theequality between e − and e + number densities leads tocancel out the net current, and the dielectric permittiv-ity is not significantly changed from the free space. Onthe other hand, the high temperature in this phase cor-responds to a high thermal velocity of the proton, andthe motion of nuclei causes a highly polarized potential.However, during the BBN in the adiabatic expansion ofthe universe, fractions of heavy nuclei increase, and a dif-ference in number density between e − and e + becomesrelatively larger at later time when the annihilation pro-ceeds. Hence, the dielectric permittivity deviates fromthe unity, although the low thermal velocity suppressesthe dynamical screening effects.In conclusion, the enhancement of thermonuclear reac-tion rates by dynamical screening effects in the Gamowenergy region is lower than the static screening case, and ynamical screening effects on BBN
0 0.5 1 1.5 2 2.5
T=1 MeV f s ( E )- ( - ) E [MeV]
DynamicStatic
0 0.1 0.2 0.3 0.4 0.5
T=10 -1 MeV f s ( E )- ( - ) E [MeV]
DynamicStatic
0 0.02 0.04 0.06 0.08 0.1 0.12
T=10 -2 MeV f s ( E )- ( - ) E [MeV]
DynamicStatic
0 0.005 0.01 0.015 0.02 0.025
T=10 -3 MeV f s ( E )- ( - ) E [MeV]
DynamicStatic
Figure 5.
Deviation from the unity of the enhancement factor for the reaction of H( p, γ ) He as a function of energy at T =1 MeV (upper-left), 0.1 MeV (upper-right), 0.01 MeV (lower-left), and 0.001 MeV (lower-right), respectively. Red and blue linesdenote the enhancement factors for dynamical and static screening potentials, respectively. Black vertical dashed lines indicatethe Gamow energies at given temperatures. f s ( E G )- ( - ) T [MeV]
T(d,n) HeT( α , γ ) Li Li(p, α ) Hed(p, γ ) He Figure 6.
Deviation of enhancement factor from unity as afunction of cosmic temperature. Solid and dashed lines de-note the dynamical and static screening effects, respectively. its effect is invisible in primordial abundances. Never-theless, in Table 1, we show that the enhancement factorby dynamical screening effects is visible under the solar condition, which indicates a ∼
10% increase of the en-hancement factor in the third column and a reductionby the dynamical screening in the fourth column. Theseresults leave several issues worth discussing for relatedplasma properties in other astrophysical environments(See also Carraro et al. (1988) related to the result inTable 1.).First, a plasma state can be characterized with theplasma coupling parameter, classically defined as Γ = n / e ( Ze ) /T , which means a ratio of mean Coulombenergy to averaged kinetic energy. (For the degener-ate system, quantum plasma parameter depending onthe Fermi energy is required to classify the ideality ofplasma state, but we consider the only classical regimein this paper.) Figure 8 shows the classical plasma cou-pling parameter and trajectories of denoted astrophys-ical environments on the parameter plane of temper-ature and electron number density n e . For the den-sity evolution, we adopt trajectories of BBN calculatedby the updated code from Kawano (1992); Smith et al.(1993), 25 solar mass (M ⊙ ) star from Paxton et al.0 Hwang et al. -12-10-8-6-4-2 0 0.001 0.01 0.1 1 l og ( X , Y , A / H ) T [MeV]
D/HY p3 He/HT/H Be/Hn/H Li/HX p Figure 7.
Primordial abundances as a function of tem-perature. Solid and dashed lines stand for the case withdynamical screening effects and standard BBN calculation,respectively. For the calculation, we adopt the parametersas follows: neutron lifetime τ n = 879 . s (Tanabashi et al.2018); η = 6 . × − (Planck Collaboration et al. 2016);effective neutrino number N eff = 3 .
046 (Mangano et al.2005). This calculation is performed by the updatedcode of (Kawano 1992; Smith et al. 1993) with reactionsin Refs. (Descouvemont et al. 2004; Iliadis et al. 2016) andfinal abundances are D/H= 2 . × − , Y p = 0 . He/H= 1 . × − , and Li/H = 5 . × − , which aresame as the results of the standard BBN calculation.reaction E G /T f s ( E G ) f s ( E G ) /f en ( E G ) p - p He- He 16.5938 1.1078 0.9591 He- He 17.3375 1.1087 0.9599 p - Be 13.8710 1.1187 0.9686 p - N 20.5473 1.2242 0.9514
Table 1.
The comparison of enhancement factors betweendynamical and static screening effects for several importantreactions in the solar environment. The f s (Equation (16))and f en (Equation (2)) stand for the enhancement factorby dynamical and static charges, respectively. For this cal-culation, we adopt the following conditions: mass density ρ = 1 . × g / cm ; temperature T = 1 . × K; massfraction of proton X = 0 .
7; and mass fraction of Helium-4 Y p = 0 . (2011, 2015, 2018, 2019), model of collapsar jet with 5degree ejection angle in Nakamura et al. (2015), TypeIa Supernovae (SNe) for W7 model in Nomoto et al.(1984); Mori et al. (2020), initial preSN for SN1987Amodel from Nomoto et al. (1984); Shigeyama & Nomoto(1990) and the condition of the sun used in Table 1. Ac-cording to our investigation, such astrophysical environ-ments are located in ideal or weakly non-ideal plasmaregion —justifying the weak screening condition. On log T [MeV] l o g n e [ c m ] Non-ideal plasma Ideal plasma = l o g Figure 8.
Classical plasma parameter Γ = n / e ( Ze ) /T andtrajectories of BBN (blue-solid), ejected collapsar jet (white-dotted), the center of Type Ia supernovae (red-dashed-doubledotted), the center of 25 M ⊙ star (purple-dashed), the initialprofile of preSN for SN1987A (green-dashed-single dotted),and the sun (blue star mark) as a function of T and n e . Eacharrow indicates the direction of the evolution. The black-solid line indicates the contour line of Γ = 1 and the openbox corresponds to the region of parameter space discussedin Figure 9. the other hand, for the more extreme conditions satisfy-ing Γ &
1, it may require the physics beyond the weakscreening assumption using quantum electrodynamics inthe finite temperature medium involving many-body in-teractions.Second, even in weakly coupled plasma region, the leftpanel in Figure 9 implies that the enhancement factor bydynamical screening effects can be significant (See alsoopen box in Figure 8.). This alludes that the thermonu-clear reaction rates in other astrophysical environmentscan also be affected by dynamical screening effects underthe weak screening condition. For example, in the solarcondition, contour lines in Figure 9 show the remark-able enhancements for reaction rates of p ( p, e + ν e ) H and N( p, γ ) O (See also Table 1.). Both reactions are im-portant in that the former reaction triggers the solarfusion known as pp chain and the latter is one of theCNO cycle both of which operate for He synthesis inmain-sequence stars. In particular, relevant reactions ofCNO cycle dominating the solar energy generation sig-nificantly impacts on not only the evolution of the solardensity profiles but the CNO neutrinos recently detected(Borexino Collaboration et al. 2020). Although the dy-namical screening effects are lower than the static caseas shown in right panels in Figure 9, the enhancementof reaction rates increases with charge numbers of in-teracting nuclei similar to the static case. This means ynamical screening effects on BBN f s (E G ) p(p,e + ν e ) H -3 -2.5 -2 -1.5 -1 log T [MeV]
24 25 26 27 l og n e [ c m - ] f s (E G )/f en (E G ) p(p,e + ν e ) H -3 -2.5 -2 -1.5 -1 log T [MeV]
24 25 26 27 l og n e [ c m - ] f s (E G ) N(p, γ ) O -3 -2.5 -2 -1.5 -1 log T [MeV]
24 25 26 27 l og n e [ c m - ] f s (E G )/f en (E G ) N(p, γ ) O -3 -2.5 -2 -1.5 -1 log T [MeV]
24 25 26 27 l og n e [ c m - ] Figure 9.
Enhancement factor by dynamical screening (left panels) and ratio of static to dynamical enhancement factor, i.e., f s ( E G ) /f en ( E G ) (right panels) as a function of T and ρ in the region of the open box in Figure 8. The upper and lower panelsshow the results for the reactions of p ( p, e + ν e ) H and N( p, γ ) O, respectively. We adopt the solar condition denoted by thered point and mass fractions of nuclei used in Table 1. that the dynamical screening effects on CNO cycle aremore effective than that on the pp chain, which couldprovide the correction of the CNO cycle for the solarevolution as well as CNO neutrino detection. Further-more, we expect the conceivable changes of electron cap-ture rates (Luo et al. 2020; Mori et al. 2020) and otherthermonuclear reaction rates in various astrophysical en-vironments can result in a change of relevant nuclearprocesses. Such a change would play significant roles inthe theoretical predictions of neutrino signal or rate ofgravitational waves emission from the stellar objects aswell as the elemental yields by the nucleosynthesis in thecosmos.Third, a correlation energy from the electromagneticfluctuation can be produced as the plasma parameterincreases (Opher et al. 2001). The correlation in mi-crostates is one of the motivations to invent the non-extensive statistics, known as Tsallis statistics (Tsallis1988), and we can consider the non-extensive statisticsinvolving the non-Maxwellian distribution in the denseastrophysical environments to properly describe the de- viation from the standard thermodynamics. Althoughthe effects of Tsallis distribution on BBN were investi-gated and those results for thermonuclear reactions arenot consistent with observations (Kusakabe et al. 2019),we expect that the study including the strong electro-magnetic field condition and correlated systems for otherastrophysical environments is valuable as future work.Lastly, as an important topic, we address effects of thetransverse mode of permittivity in astrophysical nuclearprocesses. According to Opher & Opher (1997), by thefluctuation-dissipation theorem, the transverse mode ofdielectric permittivity affects the electromagnetic fluctu-ation or vice versa, by which the electromagnetic spec-tra deviate from the black body form. This change isclearly related to the photo-disintegration rate as wellas the number or energy density of photons. Since thischange can be a solution to the primordial lithium prob-lem (Jang et al. 2018), further research along this direc-tion is indispensable. Such a proper consideration of cor-rections for astrophysical nuclear processes from plasma2 Hwang et al. properties would advance further the understanding ofthe origin of elements.ACKNOWLEDGMENTSWe are grateful to H. Ko for useful discussions andinformation. The work of D.J. and C.M.R. is sup-ported by Institute for Basic Science under IBS-R012-D1. E.H, K.P and M.K.C are supported by the NationalResearch Foundation of Korea (Grant Nos. NRF-2020R1A2C3006177 and NRF-2013M7A1A1075764).The work of M.K. was supported by NSFC ResearchFund for International Young Scientists (11850410441).T.K. is supported in part by Grants-in-Aid for ScientificResearch of JSPS (20K03958, 17K05459) and T.M. issupported by JSPS (19K03833). A.B.B. is supportedin part by the U.S. National Science Foundation grantNo. PHY-1806368 and acknowledges support from theNAOJ visiting professor program. APPENDIX: THERMONUCLEAR REACTIONRATE WITH DYNAMICAL SCREENINGWe here derive the thermonuclear reaction rates withdynamical screening describe in Equation (15) usingtransformation of the collision frame to CM frame. Ourcollision frame is defined as shown in the left panel ofFigure 10, in which the interaction of two nuclei occursat r = 0. Velocities and angles are described using theCM velocity and the angles defined in the CM frame asfollows: v = s(cid:18) m m + m + v cm cos θ (cid:19) + ( v cm sin θ ) ,v = s(cid:18) m m + m − v cm cos θ (cid:19) + ( v cm sin θ ) , (18) θ = π − arcsin (cid:16) v cm sin θv ( v cm ,v r ,θ ) (cid:17) (cid:16) for π ≤ θ ≤ π and − v cm cos( θ ) > m m + m v r (cid:17) , arcsin( v cm sin θv ( v cm ,v r ,θ ) ) (otherwise) ,θ = π − arcsin (cid:16) v cm sin θv ( v cm ,v r ,θ ) (cid:17) (cid:16) ≤ θ < π and v cm cos( θ ) > m m + m v r (cid:17) , arcsin( v cm sin θv ( v cm ,v r ,θ ) ) (otherwise) . (19)With these transformations, arguments of W ( v , v , θ , θ )are changed as W ( v r , v cm , θ ). Recalling the thermonu-clear reaction rate in laboratory frame given as, N A h σv i = N A Z ∞ Z ∞ σ ( E ) v r f ( ~v ) f ( ~v ) × e W ( v ,v ,θ θ T d ~v d ~v , (20) by Jacobian transformation, we can write the thermonu-clear reaction rate in CM frame as follows: N A h σv i = N A Z ∞ Z ∞ σ ( E ) v r (cid:16) m πT (cid:17) / (cid:16) m πT (cid:17) / e − M~v T e − µ~v r T e W ( vr,v cm ,θ ) T d~v r d~v cm = N A Z ∞ σ ( E ) v r (cid:16) µ πT (cid:17) / e − µv r T "Z ∞ (cid:18) M πT (cid:19) / e − M~v T e W ( vr,v cm ,θ ) T d~v cm d~v r = N A Z ∞ σ ( E ) v r f ( ~v r ) (cid:20)Z ∞ f ( ~v cm ) e W ( vr,v cm ,θ ) T d~v cm (cid:21) d~v r = N A r πµT Z ∞ σ ( E ) Ee − ET (cid:20)Z ∞ f ( ~v cm ) e W ( vr,v cm ,θ ) T d~v cm (cid:21) dE, (21) ynamical screening effects on BBN Figure 10.