# Big Bang Nucleosynthesis with an Inhomogeneous Primordial Magnetic Field Strength

###### Abstract

We investigate the effect on the Big Bang Nucleosynthesis (BBN) from the presence of a stochastic primordial magnetic field (PMF) whose strength is spatially inhomogeneous. We assume a uniform total energy density and a gaussian distribution of field strength. In this case, domains of different temperatures exist in the BBN epoch due to variations in the local PMF. We show that in such case, the effective distribution function of particle velocities averaged over domains of different temperatures deviates from the Maxwell-Boltzmann distribution. This deviation is related to the scale invariant strength of the PMF energy density and the fluctuation parameter . We perform BBN network calculations taking into account the PMF strength distribution, and deduce the element abundances as functions of the baryon-to-photon ratio , , and . We find that the fluctuations of the PMF reduces the Be production and enhances D production. We analyze the averaged thermonuclear reaction rates compared with those of a single temperature, and find that the averaged charged-particle reaction rates are very different. Finally, we constrain the parameters and for our fluctuating PMF model from observed abundances of He and D. In this model, the Li abundance is significantly reduced. We also discuss the possibility that the baryon-to-photon ratio decreased after the BBN epoch. In this case, we find that if the value during BBN was larger than the present-day value, all produced light elements are consistent with observational constraints.

National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
\move@AU\move@AF\@affiliationDepartment of Astronomy, Graduate School of Science,
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
\move@AU\move@AF\@affiliationSchool of Physics and Nuclear Energy Engineering,
and International Research Center for Big-Bang Cosmology and Element Genesis, Beihang University 37, Xueyuan Rd., Haidian-qu, Beijing 100083 China
^{†}^{†} ydong.luo@nao.ac.jp

National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan \move@AU\move@AF\@affiliationDepartment of Astronomy, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan \move@AU\move@AF\@affiliationSchool of Physics and Nuclear Energy Engineering, and International Research Center for Big-Bang Cosmology and Element Genesis, Beihang University 37, Xueyuan Rd., Haidian-qu, Beijing 100083 China

National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan \move@AU\move@AF\@affiliationSchool of Physics and Nuclear Energy Engineering, and International Research Center for Big-Bang Cosmology and Element Genesis, Beihang University 37, Xueyuan Rd., Haidian-qu, Beijing 100083 China

National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan \move@AU\move@AF\@affiliationCenter for Astrophysics, Department of Physics, University of Notre Dame, Notre Dame, IN 46556, U.S.A.

## 1 Introduction

Light element synthesis in the early universe is well described by the standard model of Big Bang Nucleosynthesis (BBN). A comparison of predicted isotopic abundances with observation is essential to constrain cosmological models and the physical processes during the BBN epoch (2006NuPhA.777..208F; 2007ARNPS..57..463S; Cyburt:2016cr; 2017IJMPE..2641001M). The standard BBN (SBBN) model evolves a network of nuclear reactions among primordial elements (mainly D, He, He and Li) in a space-time characterized by general relativity, while the microphysics is characterized by particle interactions described within the standard model of particle physics (Bertulani:2016ci).

Theoretical calculations of light element abundances in SBBN are now well defined and precise (Cyburt:2016cr; 2017IJMPE..2641001M). The only parameter is the baryon-to-photon ratio (), which is now well determined from the power spectrum of the cosmic microwave background temperature fluctuations. For the value of derived from Planck or the WMAP-9yr analysis (2013ApJS..208...20B; 2016A&A...594A..13P), there is excellent agreement between BBN and the observed primordial abundances of D and He (2003PhLB..567..227C; Cyburt:2016cr). However the observed abundance of Li in metal-poor halo stars, (1982A&A...115..357S; 2010A&A...522A..26S) implies Li/H= which disagrees with the theoretical prediction by about a factor of 3 (Li/H) (Cyburt:2016cr).

A number of suggestions have been proposed to solve this problem. One is that a better understanding of the diffusive transport may be needed to understand the lithium abundances of the metal-poor halo stars on the Spite plateau (Fu:2015uua). Others have argued for the existence of a stellar mass-dependent mechanism to deplete stellar lithium (Richard:2005et). Motivated by recent observations (Piau:2006kb) suggest that the interstellar medium (ISM) is in fact quite dynamic, it has also been suggested that the Li depleted ejecta from massive Population III stars may be mixed inefficiently with the proto-Galactic ISM prior to the formation of the MPH stars of the galactic halo.

In addition to explanations from astrophysical processes, it has been proposed that the current uncertainties in the cross-sections of relevant nuclear reactions are at the level of 0.2% for He, 5% for D and He and 15% for Li (Descouvemont:2004ks). Therefore, a partial solution from the nuclear reaction side might be possible once more accurate measurements are achieved (Broggini:2012kj). Experimentally, a recent measurement of the Be(,)Li reaction suggests that the final state involving the first excited state of Li can contribute up to 20% of the total cross section (private; newdata). Theoretically, detailed nuclear reaction network calculations up to the CNO cycle have been carried out (Coc:2011jh; Acoc2017), as well as a Monte Carlo likelihood analysis to make a rigorous approach of the theoretical BBN nuclear reaction networks (Iliadis:2016ej). Those results, however, do not give a solutions to the lithium problem. On the other hand, the possibility of new resonance reactions to destroy Li such as BeC and Be(He,2)He will be explored in the near future, although it has been found that these resonances must have unrealistically large decay widths (Chakraborty:2011ju; Civitarese:2013jl; Hammache:2013it).

Beside these, a variety of nonstandard BBN models have also been proposed such as an Inhomogeneous BBN (Applegate:1987hp; 1987ApJ...320..439A; 1988PhRvD..37.1380F; Kajino:1991jf; Orito:1997wn; Lara:2006es; 2017IJMPE..2641003N), dark matter decay (2013PhLB..718..704K), sterile neutrinos (Esposito:2000iz; 2014PhRvD..90h3519I) and super symmetric particles (2008PhLB..669...46A; 2011JPhCS.312d2012K; MK2017). Those possibilities are discussed in KurkiSuonio:2017wl; 2017IJMPE..2641001M; 2018AIPC.1947b0014M.

Recently, non-extensive (non-Maxwellian) statistics (Tsallis statistics) have also been proposed as a solution to the lithium problem (Hou:2017ed). In this framework, an extra parameter characterizes the deviation from a Maxwell-Boltzmann (MB) distribution. When , the distribution function is the classical Maxwell-Boltzmann distribution (Bertulani:2013cp). Several physical sources of the parameter have been discussed (Lutz:2003cs; 2007EL.....7819001B; Rossani:2009cy; Livadiotis:2010eh; 2012arXiv1203.4003P). Among them is the possibility of entropy and/or temperature fluctuations (Wilk:2000jx; Wilk:2002kj) in the equilibrium state. This motivates the interesting speculation that during the BBN epoch, the background photon energy density or temperature may not have a universal homogeneous value as assumed in previous SBBN studies. Here, we explore this possibility in a phenomenological model whereby sub-horizon isocurvature temperature fluctuations arise from fluctuations in a primordial magnetic field (PMF). Previous studies (Yamazaki:2012jz) introduced a constant scale invariant (SI) PMF strength within a certain co-moving radius during the BBN epoch. However, as the magnetic field evolves, the strength may not always be homogeneous (e.g. 2017PhRvD..96l3525M). That can affect the temperature on large scales. In section 2, we discuss a primordial stochastic magnetic field and an ansatz for its strength distribution. We take into account this inhomogeneity of the PMF strength in a BBN calculation. In section 3, we discuss its effects on primordial-element abundances and analyze the averaged thermonuclear reactions rates in this inhomogeneous PMF model.

## 2 Stochastic Magnetic Field

### 2.1 Homogeneous magnetic energy density

The origin and evolution of the galactic magnetic field has been a subject of interest for a number of years. From one view point, the galactic magnetic field might be a fossil remnant of PMFs amplified through the galactic dynamo process (1998PhRvL..81.3575S; 2004PhRvD..70l3003B). Several mechanisms have been proposed to explain the origin of a PMF from early cosmological phase transitions (PhysRevLett.95.121301; Ichiki827; Durrer:2013ec; 2016RPPh...79g6901S; 2016PhRvD..93d3004Y). However, these scenario cannot account for a large scale magnetic field. The co-moving correlation length scale for these models is at most given by the horizon during the phase transition, which is much smaller than a typical galaxy size at the present day. One possible solution of this problem is a super-horizon PMF generated during inflation (1988PhRvD..37.2743T; 1993PhRvD..48.2499D; 2009JCAP...08..025D). This kind of magnetic field is “frozen-in” with the dominant fluids. Previous studies have shown that such a PMF can slightly change the weak reaction rate and the electron-positron distribution function while their main effect is the enhancement of the cosmic expansion rate (1996PhLB..379...73G). In this sub-section, we first consider this case of a super-horizon scale magnetic field.

A statistically homogeneous and isotropic magnetic field must have a two-point correlation function for the co-moving wave vector (2011PhR...505....1K)

(1) |

Here, as in previous studies, we assume that the power spectrum of the PMF energy density is a power law (PL) spectrum.

(2) |

where is the power-law index. This PL spectrum is the most common assumption for magnetic fields on cosmological scales.

One can then derive the normalization coefficient from the variance of the magnetic fields in real space. The co-moving PMF strength inside a spherical Gaussian radius should be (Mack:2002ew)

(3) |

where is a typical co-moving length scale for the present-day, usually set as 1 Mpc. Then, applying a Fourier transform to space and integrating this together with a window function, the co-moving strength becomes

(4) |

Here, the window function is required to constrain large values of the wave number. This means that large spatial scales of the PMF are taken into account while the smallest scales are cut off. A lower cutoff of the PMFs results from decay of the magnetic field on small scales. Magneto-hydrodynamical (MHD) turbulence generates such a cutoff (Durrer:2013ec). It has been pointed out (Brandenburg:1996dj) that with random initial conditions for the magnetic field, turbulence can have an inverse cascade that transfers the magnetic energy density from small scales to large scales. From Eqs. (1)(2.1), the final result (Yamazaki:2012jz) for the energy density contributed from a PMF is

(5) |

where and are the maximum and minimum wave numbers, respectively. Their values depend on . For example, for an averaged magnetic field strength with with the radiation energy density after the epoch of annihilation, the magnetic field energy density would be given by

(6) |

where is the effective number of statistical degrees of freedom after the epoch of annihilation. For at the present day, . With this amount of magnetic energy, the magnetic field would have a present RMS amplitude of .

We note that this magnitude of the PMF strength is much greater than the upper limit of a few nG on a 1 Mpc co-moving scale inferred from the Planck analysis (2016A&A...594A..19P). However, in the present analysis, we adopt a lower cutoff of the correlation scale below the large scales that are constrained by the CMB power spectrum. Previous studies (Durrer:2013ec; Brandenburg:1996dj) pointed out that MHD turbulence can lead to such a cutoff. Moreover, the fluid-viscosity due to neutrinos and photons can induce damping of magnetic fields (1998PhRvD..57.3264J; 1998PhRvD..58h3502S). In fact, the MHD modes with wavelengths smaller than the mean free path of neutrinos and(or) photons are in such a diffusion regime. Such damping process suggests that a PMF on the smaller scales associated with BBN would dissipate by the time of photon last scattering. Hence, a PMF would only affect the CMB power spectrum via the expansion rate. The Planck constraint of (95% C.L.) is consistent with the upper limit of adopted here.

Previous studies (2008PhRvD..77d3005Y) used Eq. (6) to constrain the ratio of the SI energy density contributed from the PMF based upon the CMB power spectrum. The primordial element abundances can also be computed (2013PhRvD..88j3011Y; 2016PhRvD..93d3004Y) by introducing this amount of extra energy density contribution to the total energy density. The present day co-moving length scale 1 Mpc corresponds to a length of cm during the BBN epoch, which is well beyond the horizon ( cm) during BBN. Hence, within the horizon volume, the averaged magnetic energy density mainly affects BBN through the expansion rate

(7) |

(8) |

where is the gravitational constant and we use natural units, i.e. . The quantities and are the energy density and the pressure respectively. Since , the epoch of weak decoupling () occurs when

(9) |

The right hand side results from the fact that the weak-reaction cross sections scale as and the background particle number density is proportional to . Then if the magnetic energy density is included, the left hand side of Eq. (9) will be smaller (). This leads to a shorter decoupling time or a higher decoupling temperature . Since a larger corresponds to larger n/p ratio, the He abundance will increase consequently.

Fig. 2.1 shows the primordial element abundances as a function of . The vertical band shows the limits on the baryon to photon ratio derived from Planck analysis (2016A&A...594A..13P). The horizontal shaded bands indicate observational constraints on the element abundances.

There remain some ambiguities in the primordial abundances (e.g. Cyburt:2016cr). Hence, in this work, for each element, we list two observational constraints to compare with our calculations: (1) (mass fraction of He) : (2010JCAP...05..003A), or (Izotov:2014kt); (2)D/H: (Cooke:2014br), or (2012MNRAS.426.1427O); (3)Li/H: (1 from 2010A&A...522A..26S), or (2 from 2010A&A...522A..26S).

In Fig. 2.1, increases as expected when a PMF is introduced. The other primordial elements D, He and Li are also slightly affected. The amount of PMF energy density is constrained to be % of the total energy density in the figure. This is based upon the upper limit to from the observations of Izotov:2014kt. This is equivalent to a co-moving PMF field strength .

### 2.2 Inhomogeneous magnetic energy density

In addition to the effect of a homogeneous PMF energy density, fluctuations of the magnetic field over a wide range of sub-horizon scales will serve as a non-linear driving force that induces the metric fluctuations (Journal:uj). As has already been proposed (1997PhLB..390...87D; 1999PhRvD..59f3008S; Dolgov:2001kq; 2004PhRvD..70l3003B), it is possible to have an inhomogeneous sub-horizon PMF in the early Universe. Once the turbulence is produced, an induced MHD dynamo can amplify the field exponentially until equipartition between the plasma turbulent kinetic energy and the PMF energy is eventually reached. This can consequently lead to an inhomogeneity in the energy density (Brandenburg:1996dj; 2001ApJ...550..824B; 2001PhRvE..64e6405C; Dolgov:2001kq).

For a magnetic field on small scales, the strength can be damped due to photon and neutrino viscosities. This means that the magnetic field on scales with (Durrer:2000jy) dissipates rapidly, where is the age of the universe and is the magnetic diffusivity. An estimate of the damping scale due to the viscosity in the magneto-hydrodynamic evolution process is given by splitting long and short wavelength fluctuations in the B field separately (Brandenburg:1996dj). Moreover, a magnetic field with scale is not easy to generate, while that with will not dissipate, and the magnetic field is frozen-in with the dominant fluids (dendy1990plasma). Thus, the survival length scale for the PMF during the BBN epoch with temperature set as MeV is cm (2012PhR...517..141Y). This is much smaller than the co-moving length scale for which a constraint on the field amplitude is given from the CMB power spectrum, i.e., Mpc cm for the BBN redshift of . Therefore, we cannot exclude the possibility of fluctuations in the PMF length scales of the same order as . We can also consider that the energy density of the PMF could have some distribution rather than the ideal case with . The effect of a PMF on baryons and the plasma has also been studied (1996PhLB..379...73G; Kawasaki:2012kn). However, the effect they discussed is not very important for the present application since the modification to the distribution functions is proportional to . However, if a distribution function exists, then the associated radiation energy density fluctuations can modify the nuclear reaction yields after averaging over all local regions.

Most BBN network calculations have considered the photon energy density to be homogeneous during the entire epoch. Here, however, we consider large-scale energy density fluctuations in the temperature (or equivalently photon energy density). The nuclear reactions occur locally, this means that the local velocity distribution function for baryons is,

(10) |

Here, refers to the inverse temperature and corresponds to the local temperature. This is just the classical MB distribution which refers to the velocity distribution function of particles for a certain temperature in equilibrium. Since the nucleon gas in the early Universe was dilute, two-body nuclear reactions dominate. The local two-body reaction rate per unit volume can be written as

(11) |

where and are the number densities of reacting particles 1 and 2, respectively, is the Kronecker’s delta function for avoiding the double counting of identical particles 1 and 2, and is the averaged thermonuclear reaction rate for a given temperature written as

(12) |

where is the reduced mass of the system 1+2. Because local fluctuations of the energy density occur due to the inhomogeneous PMF, locally nuclei obey a classical MB distribution with inverse temperature equal to . The thermonuclear reaction rates averaged over the set of temperature fluctuations is then given by

In the last equation, we defined a new function which is independent of as an effective distribution function averaged over the set of temperature fluctuations^{2}^{2}2Note: is the average velocity distribution function over a length scale much longer than the typical size of magnetic domains’ but not a real particle velocity distribution. In principle, the evolution of nuclear abundances should be solved inhomogeneiously, i.e. the abundance at a given time depends on locations, i.e., Y. But in the present calculation, the inhomogeneity of nuclear abundances is neglected, i.e., Y. Then, an average distribution function can be defined.

(13) |

Here, is the distribution function of generated from averaging over fluctuations of the energy density. The derivation of this deviation from a classical MB distribution is similar to that deduced in Beck:2001bq in terms of Tsallis statistics. Now, we can invoke the central limit theorem and simply assume that the distribution function of magnetic energy density follows a gaussian distribution with a peak located at the mean value ( in Eq. (2.1))

(14) |

We then introduce the fluctuation parameter as a dimensionless quantity, i.e., to describe the fluctuations of the PMF. In the limit of , this is a delta function which corresponds to the homogeneous case. Now we assume that the total energy density is uniform for all volumes, but with some fraction contributed from the magnetic energy density:

(15) |

an effective temperature can be defined as

(16) |

Since is constant, the magnetic energy density can not exceed in which case would obtain an unphysical negative value. Here, we impose a cut-off to the distribution function ().

Fig. 2.2 shows Gaussian functions for various values of . Since we do not expect a very large inhomogeneity in the magnetic energy density strength during BBN, a narrow distribution is required. For , is consistent with our cut-off range for . The photon temperature determines the radiation energy density as , so Eq. (15) becomes

(17) |

The final expression for the distribution function for is then

(18) |

### 2.3 Effect on reaction rates

Adopting this as the distribution function, we show that the averaged charged particle reactions are affected significantly by the inhomogeneous temperature distribution. For neutron induced reactions, the transmission probability of a neutron through the nuclear potential surface is proportional to the inverse of the velocity within the assumption of a sharp potential surface (Bertulani:2016ci). Hence, the cross section is usually expressed as , where is a smooth function. Therefore, the change of reaction rates is mainly determined by the deviation of the average distribution function from a MB distribution function. This is not a large effect as shown in Fig. 2.3 (solid straight line and dashed straight line).

For charged particle reactions, the astrophysical S-factor is introduced to rewrite the cross section in terms of a much smoother dependence on the center of mass energy :

(19) |

where approximately expresses the probability to penetrate the Coulomb barrier. This is also known as Gamow factor, . Eq (12) is peaked at the so called Gamow energy .The deviation from a MB distribution function in the inhomogeneous PMF model is not large. However, the impact on reaction rates can increase when we take into account the factor of for charged particle reactions as shown in Fig. 2.3. The distribution function (shown by straight lines) in our PMF model looks similar to MB distribution function. However, is also a energy dependent function, and the inhomogeneous PMF model suggests an effective reduction of the Gamow window derived by multiplying this term with the average distribution function .

In conclusion, for the case of an inhomogeneous PMF during BBN epoch, the effect generated from the distribution of PMF energy density can be divided into two parts: 1) changes in the Hubble expansion rate (see Section 2.1) and 2) changes within nuclear reaction rates due to an effective non-MB averaged distribution function when we calculate the sum of averaged thermonuclear reaction rates in all domains.

## 3 Results and Discussion

### 3.1 Standard case

We have encoded the temperature averaged reaction rates as described in Eqs. (2.2) and (18) to calculate the BBN reaction network and compare the results with the observationally inferred abundances for D, He and Li. We use the current Particle Data Group world average value s for the neutron lifetime (2014ChPhC..38i0001O). The baryonic density of the Universe or is now deduced to be (2016A&A...594A..13P) from the observations of the anisotropies of the CMB radiation.

Fig. 3.1 shows the parameter dependence of the final primordial light element abundances as a function of (left panel) and (right panel). In the left panel, element abundances are presented as a function of mean magnetic energy density () for a fixed value of . The effect of on the primordial element abundances is consistent with a PMF model with a homogeneous energy density in the previous study of Yamazaki:2012jz: He is most sensitive to the changes of the cosmic expansion rate, which is equivalent to a change of . The constraint from the observed value of and D/H implies that the PMF mean energy density has an upper limit of . The right panel shows the element abundances as a function of the fluctuation parameter . For this panel we set which is the upper limit from the observations. In the case that the fluctuation parameter approaches (i.e no fluctuation occurs), the result is consistent with the homogeneous energy density PMF model of Yamazaki:2012jz. As increases, the inhomogeneity enhances. This affects the element abundances. It is also generally true that as the D abundance increases, the Li production is reduced. In this case, the other primordial abundances are strongly dependent on , while remains nearly the same as in the case of the homogeneous PMF model. This is a completely new effect on BBN from a PMF model which includes spatial inhomogeneities in the energy density. Finally, from the and D constraints, we obtain and without violating the observational constraints on the He and D abundances.

In Fig. 3.1, we illustrate the light element abundances as a function of with the allowed parameter values of and . In the grey region, the D/H and calculations are consistent with observations, and the Li/H value is reduced to compared with SBBN. However, this is still above the Spite Plateau (1982A&A...115..357S; 2010A&A...522A..26S). The calculated primordial element abundances for are shown in Table.3.1. Finally, by keeping which is the upper limit for the mean magnetic energy density, we find that the predicted Li/H abundance reduces to with a fluctuation parameter (dash-dotted line in Fig.3.1). Since this parameter region is inside the allowed region of observed , the ’Lithium Problem’ may be solved in this model. However, the D abundance is D/H which is inconsistent with the observational upper limits (2012MNRAS.426.1427O; Cooke:2014br).

The thermonuclear reaction rates are key factors in determining the final primordial abundances. As shown in Fig. 3.1, D/H is enhanced and Li/H reduced as a result of an inhomogeneous PMF energy density model. The H reaction is the main production mechanism for deuterium, while the H(,)He and H(,)H reactions are the main destruction channels. For Li (or Be), the main production reaction is HeBe. The main destruction process is BeLi. In Fig. 3.1, we show the reduction fraction for charged particle reaction rates in our PMF model compared with the SBBN results as a function of temperature. For lower temperature, the reduction is larger than that at higher temperature. Since the reaction He()Be has the largest Coulomb barrier, the reduction is large compared to the D destruction reactions. For deuteron destruction reactions, i.e., H(,n)He (dotted line) and H(,)H (dash-dotted line), we see the same trend in the low energy region since they have the same Gamow energy . The solid line shows a larger reduction for the beryllium production rate HeBe at low temperature. Because of the stronger Coulomb repulsion for this reaction, a large contributes to a steeper exponential term for charged particle reaction rates (cf. Fig. 3.1). Hence, we conclude that all 3 reaction rates which determine the D and Li abundance are reduced by an inhomogeneous magnetic energy density.

### 3.2 An evolved value

The above discussion is based on the presumption that no other physical process occurs between BBN and the photon last scattering epoch so that the value from the Planck analysis is the same as that during BBN. In Fig. 3.2 we explore the possibility to find a parameter region with a concordance for all light element abundances with a higher value for the baryon-to-photon ratio. The fraction is chosen as 0.11 which is the mean magnetic field strength constrained from the observed mean He abundance. In the left panel, the calculated element abundances are shown as functions of for the fluctuation parameter . Although there is no solution to the Li problem within the range of Planck (light blue vertical band), at (light orange vertical band), all of the elements fall into a region that is consistent with the observational constraints. In the right panel we expand this result to a parametric study of the fluctuation parameter . This panel shows contours for light nuclear abundances in the plane of and . The upper limit of satisfies our upper limit on the contribution of the PMF for the case of (see Section 2.2). Here, for a larger fluctuation parameter which is taken to be , there is an area (grey-shaded area) in which the abundances of all light elements D, and Li are consist with observations. However, the baryon-to-photon ratio in this region is , which is larger than the Planck observational constraints (light blue vertical band).

We note however, that a dissipation of the PMF between BBN and the last scattering of the background radiation can results in an evolution of the value (see Sec. IV-B in Yamazaki:2012jz). For example, an 11% increase in the total energy density by the PMF leads to a 13.5 % increase in the photon number density from the case without dissipative heating. As a result, the value during BBN would be 13.5% larger than the value after the dissipation. Although this change is not enough to explain the 30% increase required for the high value in Fig. 3.2 (left panel), a change in the value can also be induced by other mechanisms such as the radiative decay of exotic particles (1979MNRAS.188P..15L; 1985NuPhB.259..175E; 1988ApJ...330..545D; 1992NuPhB.373..399E; 1995PThPh..93..879K; 1995ApJ...452..506K; 2000PhRvL..84.3248J; 2001PhRvD..63j3502K; 2003PhRvD..67j3521C; 2005PhRvD..71h3502K; 2005PhLB..619...30E; 2006PhRvD..74j3509J; 2006PhRvD..74b3526K; 2013PhRvD..87h5045K; 2014PhRvD..90h3519I; 2015PhRvL.114i1101P). It has been found that for special values of the particle mass, Be nuclei are preferentially destroyed and the Li problem is solved (2013PhRvD..87h5045K; 2014PhRvD..90h3519I; 2015PhRvL.114i1101P). The radiative decay generates additional background radiation, and the photon number is increased (e.g., 2003PhRvD..68f3504F; 2014PhRvD..90h3519I). Depending upon the lifetime and the abundance of the decaying particle, a preferential destruction of deuterium and Be via photodisintegration is possible: for example, previous studies have shown that Be and D abundances are significantly affected if the lifetime of the decaying particle is s and the parameter for the abundance of is GeV (2003PhRvD..67j3521C; 2006PhRvD..74b3526K). This parameter set corresponds to 10 % change in the eta value by the decay (2003PhRvD..68f3504F).
Thus, the radiative particle decay can further lead to an agreement between observed and calculated D and Li abundances. Also, the value increases if some mechanism for photon cooling occurs, e.g. via the formation of an axion condensate after the BBN (2013PhLB..718..704K).

## 4 Conclusion

In this work, an inhomogeneous PMF model is introduced during the BBN epoch, and has been explored. The PMF is described by a stochastic field constrained by the observed CMB power spectrum under the assumption of a power-law correlation function. However, the strength of the magnetic field varies spatially once the magnetic field is generated before weak decoupling. We adopt a PMF energy density characterized by a Gaussian dispersion in local field strength. This model implies the existence of an inhomogeneous PMF during the BBN epoch [Eq. (14)]. We assume a homogeneous value of total energy density in the universe, and inhomogeneity of temperature along with that of the PMF. Locally, primordial baryons are in equilibrium with the same temperature which determines the photon energy density. Globally, due to the existence of an inhomogeneous PMF energy density, the temperature is inhomogeneous. This causes an effective non-MB distribution function for baryonic velocities during the BBN epoch. We derived an expression for the temperature distribution function [Eq. (18)] and calculated the effective baryonic distribution function in our PMF model [Eqs. (13)–(17)]. We analyzed the reaction rates and concluded that charged particle reactions are affected most due to the Coulomb barrier while neutron induced reactions are not [Fig. 3.1]. The inhomogeneous PMF energy density was also added to the BBN network. We find that He abundance is most sensitive to [Fig. 3.1]. We verified that under the limit of , the abundances obtained from a homogeneous PMF strength are naturally recovered [Fig. 3.1].

In our model, the D and Li abundances are the most sensitive to the fluctuation parameter [Fig.3.1]. By comparing our results with the constrains, we find that is less than of the total energy density, and the range of provides the best fit to the observed abundances of for both and D. This amount of magnetic energy density corresponds to a present PMF of . With the constraints from both He and D/H, we conclude that a PMF with a fluctuation parameter can exist during the BBN epoch without violating the and D constraints. Moreover, the Li abundance is reduced in our model to a value of , which is still above the Spite Plateau [Table 3.1]. For a non-standard case in which the baryon-to-photon ratio decreased after primordial nucleosynthesis, we find that for a baryon to photon ratio of during BBN, and a fluctuation parameter of , there is a possible solution to the Li problem [Table 3.2].

TK was supported by Grants-in-Aid for Scientific Research of JSPS KAKENHI (Grant Numbers JP15H03665 and JP17K05459). MK was partially supported by the visiting scholar program of NAOJ. Work of GJM supported in part by DOE nuclear theory grant DE-FG02-95-ER40934 and in part by the visitor program at NAOJ.

PMF_reference