Decay of baryon inhomogeneities in an expanding universe
DDecay of baryon inhomogeneities in an expanding universe
Pratik K. Das, Sovan Sau, Abhisek Saha and Soma Sanyal
University of Hyderabad, Prof. C.R. Rao Road, Hyderabad 500046, India
Abstract
Baryon inhomogeneities can be generated very early in the universe. These inhomo-geneities then decay by particle diffusion in an expanding universe. We study the decay ofthese baryon inhomogeneities in the early universe using the diffusion equation in the Fried-mann–Lemaˆıtre–Robertson–Walker (FLRW) metric. We have studied the decay starting fromthe electroweak phase transition. We calculate the interaction cross section of the quarks withthe neutrinos, the electrons and the muons and obtain the diffusion coefficients. The diffusioncoefficients are temperature dependent. We find that the expansion of the universe causes theinhomogeneities to decay at a faster rate. We find that the baryon inhomogeneities generated atthe electroweak epoch have very low amplitudes at the time of the quark hadron phase transition.So unless inhomogeneities are generated with a very high amplitude (greater than 10 times thebackground density), they will have no effect on the quark hadron phase transition. After thequark hadron phase transition, we include the interaction of the muons with the neutrons and theprotons till 100 MeV. We also find that large density inhomogeneities generated during the quarkhadron transition with sizes of the order of 1 km must have amplitudes greater than 10 times thebackground density to survive upto the nucleosynthesis epoch in an expanding universe. Keywords: Inhomogeneities, diffusion, Expanding universe a r X i v : . [ h e p - ph ] J a n . INTRODUCTION Primordial cosmological fluctuations are an important part of modern cosmology as theylink the current Cosmic Microwave Background Radiation (CMBR) data to the early uni-verse. Initially the theory related to fluctuations was developed by Lifshitz [1]. Later onsignificant work has been done by Hawkings [2] and Bardeen [3]. In this work, we are in-terested in the decay of baryon density fluctuations or inhomogeneities generated in theearly universe. Baryon density fluctuations can be generated in the electroweak phase tran-sition [4–6] as well as the QCD phase transition [7, 8]. There are various defect mediatedmechanisms which generate these inhomogeneities at the electroweak scales [9]. Thoughbaryon inhomogeneities have other roles to play in the evolution of the early universe, theprimary method of constraining these inhomogeneities is by studying their effects on the BigBang Nucleosynthesis calculations. The inhomogeneities generated at the QCD scales havea higher chance of surviving to the nucleosynthesis epoch, hence they have been studiedmore extensively. The baryon inhomogeneities generated in the electroweak epoch, have lesschance of surviving till the nucleosynthesis epoch, hence they are often ignored. However ithas been shown that baryon over densities generated in the electroweak epoch can definitelyaffect the quark hadron phase transition [10] and may also survive till the nucleosynthesisepoch [11]. One of the effect of these baryon over-densities is to delay the quark hadronphase transition in regions of the inhomogeneities. This, along with the fact that large scaleinhomogeneities from the electroweak epoch may survive till the nucleosynthesis epoch andaffect the abundances of the light elements makes it important to study the diffusion ofparticles in an inhomogeneity at the electroweak epoch.The only detailed study of the evolution of non-linear sub horizon entropy fluctuationsbetween 100 GeV and 1 MeV was done in ref. [11]. The entropy fluctuations that had beenconsidered in that particular study did not necessarily come from baryon inhomogeneitiesonly, the authors were not interested in the source of the entropy fluctuations. They hadassumed certain amplitudes and length scales and evolved them with time. The evolutionof the inhomogeneity was performed as a succession of pressure equilibrium states, thatdissipate or expand due to neutrino heat transport. The neutrino contribution to the heattransport equation was used and after the quark hadron phase transition the neutron andproton diffusion were taken into account. They concluded that large scale entropy fluctua-2ions at 100 GeV could survive until the nucleosynthesis epoch and affect the nucleosynthesiscalculations. The main difference between their study and our current study is that we lookat baryon inhomogeneities generated at the electroweak scale in the diffusive limit takinginto consideration both the quark and the lepton diffusion. Studies have already shown thatbaryon inhomogeneities can be generated in the electroweak scale. Generally, they have anamplitude of around 10 times the background baryon density. Since, the baryon numberis carried by the quarks at these high temperatures, the baryon inhomogeneities at thesescales will primarily mean an excess of the number density of quarks in a certain region.These quarks diffuse out by colliding with the leptons, which are the electrons, muons andthe neutrinos. In this study we focus on obtaining the diffusion coefficient of these quarks asthey move through the plasma. The diffusion coefficient is then used in the particle diffusionequation to study the decay of baryon inhomogeneities in an expanding universe.The diffusion equation has been studied in the early universe mostly in the context ofcosmic rays [12]. For the case of baryon diffusion it has been studied mostly in the hadronicphase [13–15] when the baryon number is carried by the neutrons and the protons. Here, forthe first time, we use it to study, baryon diffusion in the quark gluon plasma phase. Sincewe are using the diffusion equation pertaining to an expanding universe, we also extend ourstudy to the QCD scales. We do find that the expansion term makes a significant differencein the decay of the inhomogeneities. We find that in a static universe the decay rate is slow,while in an expanding universe, the inhomogeneities decay much faster. In the electroweakcase, the particles that we consider are the quarks, muon, electrons and the neutrinos.For the QCD epoch, we have the hadrons which carry the baryon number. Generally, theinhomogeneities which are generated at the QCD epoch have higher amplitude and sizesthan those generated in the electroweak epoch [16]. Hence we consider these two casesseparately.In this work, we only look at sub horizon fluctuations. We also assume that the size ofthe fluctuations are larger than the mean free path of the relevant particles. The diffusionequation, is re-written in the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric. Weneglect baryon number violating processes at the temperatures around 200 GeV and assumethat the total baryon number density is conserved throughout our calculations. All thebaryon density fluctuations we consider are assumed to be Gaussian fluctuations.The chief motivation of understanding the decay of the baryon inhomogeneities is the3ifferent signatures of Inhomogeneous Big Bang Nucleosynthesis (IBBN) that have beenpredicted [17]. An inhomogeneous BBN will result from patches of region where there is abaryon inhomogeneity [18]. However, all the study so far has concentrated only on inho-mogeneities generated in the quark hadron phase transition. The only study for entropyfluctuations generated in the electroweak epoch has indicated that there will be no signifi-cant decay during the period prior to the quark hadron phase transition. We were motivatedto see whether this result will hold for baryon inhomogeneities also. We however, find thatthis result does not hold for baryon inhomogeneities. The inhomogeneities decay rapidlyduring this period and they decrease by about three orders of magnitude. Since some of theinhomogeneities generated in the electroweak epoch have an amplitude of only 10 , thesewill decay and have no effect on the quark hadron phase transition or the nucleosynthesisepoch. We also study the decay of the inhomogeneities after the quark hadron phase tran-sition. Though these will undergo significant decay, some of them might survive till thenucleosynthesis epoch. The BBN can then be used to constrain the models that generatethese fluctuations [19].In section II, we discuss the amplitude and size of the baryon inhomogeneities that areof interest to us. In section III we discuss the diffusion equation in the FLRW metric, insection IV, we obtain the diffusion coefficient in the quark gluon plasma phase. Since wetake into account the scattering of the quarks with electrons, muons and neutrinos, we havedivided this section into three subsections. In section V we present the numerical results ofthe decay of the baryon overdensities between the temperature 200 GeV and 200 MeV. Thisis the period after the electroweak phase transition and before the quark hadron transition.In section VI, we discuss the decay of the inhomogeneities after the quark hadron transition.Inhomogeneities which survive upto the nucleosynthesis period will affect the light elementabundances. In section VII, we summarize our work and present our conclusions. II. BARYON INHOMOGENEITIES IN THE EARLY UNIVERSE
There are several ways in which baryon over densities can be generated in the earlyuniverse. Topological defects such as electroweak strings are unstable and generate baryonnumber when they decay [20, 21]. These give rise to local baryon density fluctuations. Thescale of these fluctuations will be given approximately by the bubble nucleation distance at4he electroweak scale. Inhomogeneities may thus be generated by these strings over smalllength scales ( 10 − cms). These inhomogeneities have a large amplitude and consequentlydiffuse out. Other than the electroweak strings, superconducting strings are also capableof generating baryon inhomogeneities over small lengthscales [22]. The most detailed studyof baryon inhomogeneity generation, their amplitude and size was done in ref. [5]. Thelengthscales of the inhomogeneities generated were about 10 − cm and they had an amplitudeof 10 over the background baryon density. We will not be considering any specific modelduring this epoch, our focus would be to see if these baryon inhomogeneities can survive atleast upto the QCD epoch. We have taken Gaussian fluctuations to represent the baryoninhomogeneities. These are characterised by the amplitude ( A ) and the full width at halfmaxima ( σ ). We have kept the A ∼ as it is expected that the baryon inhomogeneitiesat the electroweak scale are not too high. The σ is varied over the horizon range.For the inhomogeneities at the QCD scale we have taken a higher amplitude. This isbecause there is a greater probability of generating large over densities at the QCD scale.Inhomogeneities can be generated by moving cosmic strings [16], collapsing Z(3) domainwalls [23] and inhomogeneous nucleation of bubbles in a first order phase transition [24].Inhomogeneities can be spread over an area of one meter and can have amplitudes of theorder of 10 − through these various mechanisms. Larger inhomogeneities may also begenerated but they will have lower amplitudes. Again we do not focus on the mechanismthat will lead to the generation of these baryon inhomogeneities, we only study the inhomo-geneities over different ranges for an approximate amplitude of 10 times the backgroundbaryon density. Diffusion and decay of baryon inhomogeneities in the QCD epoch has beenstudied before. In a previous work [15] we have worked out the decay of baryon inhomo-geneities using the diffusion equation but ignoring the expansion of the universe. In thiswork, we redo the multiparticle diffusion that we had done previously in [15] for the caseof an expanding universe. We find that the expanding universe has a significant contribu-tion to the decay of the large scale inhomogeneities. Whereas in the previous work we hadconsidered inhomogeneities with σ values much smaller than the horizon size, in this workwe consider large scale baryon inhomogeneities and see how they decay in the expandinguniverse. 5 II. THE DIFFUSION EQUATION IN THE FLRW METRIC
The diffusion equation in an expanding universe has been obtained in the context ofcosmic ray propagation [12]. We briefly describe it here in terms of baryon diffusion in theearly universe. The FLRW metric for the flat universe is defined by, ds = c dt − a ( t ) d(cid:126)r (1)Here a ( t ) is the scale factor of the expanding universe and (cid:126)r is the spatial coordinate. Thisis the comoving distance in an expanding universe. Consider a region of the universe withan inhomogeneity given by n ( (cid:126)r, t ). As time evolves, the particles in the over dense regiontend to move towards the lesser dense region to restore equilibrium and a particle flux isgenerated. In this case, we consider the diffusion to be isotropic. The local observer thensees the particle flux as, j k = − D ( t ) ∂∂x k n ( (cid:126)r, t ) (2)The diffusion coefficient D ( t ) depends on the scattering cross section and the velocity ofthe particles. Since there are different kinds of particles in this plasma, we are dealing withmulti-particle diffusion here. The conservation of current gives us, ∂∂x µ ( √ gj µ ) = 0 (3)Here √ g = a ( t ) and using the definition of the Hubble parameter as H ( t ) = ˙ a ( t ) a ( t ) , thediffusion equation can be written as, ∂∂t n ( (cid:126)r, t ) + 3 H ( t ) n ( (cid:126)r, t ) − D ( t ) a ∇ n ( (cid:126)r, t ) = 0 (4)This is the diffusion equation that we will solve numerically for a time dependent diffusioncoefficient. As mentioned before, the diffusion coefficient depends on the scattering crosssection of the particles. Hence it is different at different temperatures. Since the scatteringcross sections are obtained in terms of temperatures, we use the time - temperature relationin the radiation dominated universe to convert our time to temperature. t = (0 . × ) T (5)Here t is in secs and T is in Kelvin. We then solve the diffusion equation in the FLRWmetric numerically over the entire range of temperature from 200 GeV - 200 MeV. In thenext section, we first present the details of calculating the diffusion coefficient in the quarkphase. 6 V. DIFFUSION IN THE ELECTROWEAK SCALE
We start by studying particle diffusion in the electroweak scale. The inhomogeneitiesare formed at 200 GeV during the electroweak phase transition. So the diffusion of particlewill start around the same time. During this epoch the most abundant particles are thequarks, electrons, muons and the neutrinos. Out of all these, it is the quarks which carrythe baryon number. So the baryon over-densities would predominantly have a larger densityof quarks as compared to the background number density. As the quarks diffuse out of theinhomogeneities trying to reach an equilibrium state they will collide with the electrons,muons and the neutrinos. Here we take two cases depending on the mass of the particles.This is because the quarks are lighter than the muons but heavier than the electrons. Soas the quarks move through the muons, we have a lighter gas diffusing into a gas of heavierparticles but as the quarks diffuse through the electrons and neutrinos, we have a heavierparticle diffusing through a lighter gas[25]. Since we are not going into the detailed transportequation of the particles, we choose a distribution function for the particles of the light gas.In the first case, for the quarks moving through the muons, the diffusion coefficient isgiven by, D = 13 N (cid:28) vσ t (cid:29) = (cid:32) Tπm (cid:33) / / σ t (6)In the second case, for a heavier particle moving through a lighter fluid, to obtain thediffusion coefficient, we have to first compute the mobility of the particle in the backgroundfluid. If the velocity of the particle is (cid:126)v , then the mobility b is related to the external force( (cid:126)f ) by, (cid:126)v = b (cid:126)f and the diffusion coefficient is given by, D = bT (7)We assume the distribution of particles to be Maxwellian, then the mobility of the particlesis given by, b − = 16 πT (cid:90) p dp h vp σ t e − E/T = 16 σ t m t π . (8)Here σ t is the scattering cross-section, m is the mass of the particle. Once the scatteringcross section is known, substituting it in the expression for b would enable us to obtain thediffusion coefficient D . Since the scattering cross-sections are temperature dependent, thediffusion coefficient too would be temperature dependent. To find the diffusion coefficient7t these temperatures we therefore, obtain the scattering cross section of the quarks withthe leptons. In the next subsections, we will calculate the different scattering cross sectionfor the different interactions. A. Quark-electron scattering
We start with the motion of quarks through the electron gas. For this we need to findthe scattering cross section for the e − e + −→ q ¯ q interaction. The differential cross section isgiven by, dσd Ω = Q f α s (cid:18) u + t s (cid:19) (9)Here α ∼ − is the fine structure constant and Q f is the momentum transfer in thisinteraction. The variables u , t and s are the Mandelstam variables. This gives, σ t = Q f α s (cid:90) (cid:18) u + t s (cid:19) (1 − cosθ ) d Ω (10)The total scattering cross section can be obtained after integrating over the solid angle. Thenumerical value can be obtained once the energy scale of the colliding particles is known.Since we are working around the electroweak scale, the colliding energy of the particles arealso in the GeV range. The mobility factor is thus given by, b − = 2 σ t m π [8 T (1 − e − E/T ) − E (2 E + 4 T ) e − E/T ] (11)The diffusion coefficient can be calculated numerically after obtaining the mobility at varioustemperatures.
B. Quark-neutrino scattering
Neutrinos do not have any charges, they have weak interactions. Though there are differ-ent flavors of quarks as well as neutrinos, since we only need order of magnitude estimations,we just consider, σ t = G F ˆ sπ (12)Here G F is the Fermi constant given by, G F = 1 . × − GeV − . Numerically, the crosssection turns out to be σ t = 17 . × − cm × E ν GeV [26]. Though we are working at veryhigh temperatures in the GeV scale, the value of the diffusion coefficient is difficult to8andle numerically with this value of σ t . For the numerical calculation we therefore rescalethe variables suitably to obtain a stable numerical solution. C. Quark-muon scattering
In both the previous cases we had a heavier particle moving through a lighter gas ofparticles, however the scenario changes considerably when we consider the quarks movingthrough a gas of muons. For the µ − µ + −→ q ¯ q , though the expression for the interactioncross sections are similar to the electrons, but here the quark is the lighter particle whichis moving through a heavier gas of particles (the muons). This means that, the diffusioncoefficient is given by eqn. 6, D = (cid:18) π (cid:19) (cid:32) s Q f α (cid:33) (cid:18) mT (cid:19) [2 T (1 − e − E/T ) − Ee − E/T ] (13)We have the total cross section given by, σ t = 4 πQ f α s (14)In all these cases, we determine the diffusion coefficient at different temperatures numerically.Similar to the quark neutrino cross section, even at such high temperatures, the diffusioncoefficient are numerically very large quantities hence for all the different cases we need todo some scaling to obtain numerically stable solutions. This we have done by scaling theenergy appropriately so that the value of the diffusion coefficient is of reasonable ordersof magnitude. While plotting we have plotted only the amplitude at the different lengthscales depending on the temperature. The amplitude is dimensionless as it is the ratio ofthe enhanced density to the background density ( ∆ n B n B ). Thus the rescaling does not affectthe change in the overdensity that we are interested in. Only, the length scales are changedappropriately to reflect the decrease in the temperature. V. DECAY OF INHOMOGENEITIES IN THE QUARK GLUON PLASMAPHASE
We now look at baryon inhomogeneities generated during the electroweak phase transi-tion. Since the maximum amplitude of these inhomogeneities is of the order of 10 times9he background density, for our simulations we take a low amplitude Gaussian fluctuation.The amplitude is taken to be of the order of 10 . We consider large inhomogeneities whosedecay will be affected by the expanding universe. The horizon at these temperatures is ofthe order of 10 mm [27]. Though the diffusion coefficients can be obtained numerically, theproblem is that they vary considerably in their numerical values. This indicates that theparticle content in the inhomogeneity would ultimate define how they decay. Though thereare multi particles present in the plasma, we do not go for multi particle diffusion as itbecomes numerically quite challenging. We look at each of these interactions separately andsee how much each of them contributes to the decay of the baryon inhomogeneity. We be-lieve this will give us some idea of how the baryon inhomogeneity decays in this temperaturerange.The other challenge in studying the diffusion of particles over such a large temperaturerange, is the fact that the horizon will also change considerably. From 200 GeV where thehorizon is of the order of 1 cm to 200 MeV where the horizon is of the order of 10 kms. Wedivide it into two parts. We evolve the inhomogeneity from 200 GeV to about 1 GeV andthen again from 1 GeV to 200 MeV. Interestingly, we find that the inhomogeneity decaysconsiderably during this period depending on the particle interactions being considered.However, we find that on the log scale, the amplitude decays quite rapidly irrespective of theinteractions considered. This means that low amplitude inhomogeneities will be completelywiped out before the quark hadron phase transition.Let us first look at the decay of the inhomogeneities between 200 GeV and 1 GeV. Here thediffusion coefficients differ considerably as they are dependent on the temperature and this isquite a large temperature range. So we look at the decay of the inhomogeneities for differentinteractions separately. For all the different figures, we have the baryon inhomogeneity( ∆ n B n B ), on the y axis and the length scale on the x- axis. In figure 1, we see the decay due tothe quarks moving through the electrons. The initial fluctuation is taken at 200 GeV andthe final is taken at 17 GeV. As we see the peak of the inhomogeneities goes down by morethan three orders of magnitude. The inhomogeneity also spreads out. The decay is similar inthe case when the surrounding particles are muons and neutrinos. We have used a differentscaling for the neutrinos but as mentioned before the scaling will not affect the relative decayof the amplitude. Though the final amplitude is lower in the case of the neutrinos, the orderof magnitude is similar. Since all the graphs have similar decay in orders of magnitude, we10 B a r y on I nho m ogene i t y Distance (cm)Initial fluctuation- 200 GeVExpanding Universe- 17 GeV
FIG. 1: The decay of the fluctuation is shown in logscale between 200 GeV - 17 GeV asthe quarks moves through a sea of electrons. B a r y on I nho m ogene i t y Distance (cm)Initial fluctuation- 200 GeVExpanding Universe- 17 GeV
FIG. 2: The decay of the fluctuation is shown in logscale between 200 GeV - 17 GeV asthe quarks are predominantly surrounded by neutrinos.have only shown selected graphs. Fig 1, shows the decay due to the motion of the quarksthrough the electrons while fig 2 shows the decay due to the motion of the quarks throughthe neutrinos. Since the plasma at those high temperatures is predominantly dominated byelectrons, our results clearly show that the baryon inhomogeneities decay by about threeorders of magnitude in the high temperature GeV range. This is true, even if there are alarge number of muons and neutrinos in the plasma.We now look at the decay of the inhomogeneities between the temperatures 1 GeV to 200MeV. We find that the order of magnitude decay is again quite large for the three diffusioncoefficients. The individual numbers vary but we plot only the order of magnitude estimates11 B a r y on I nho m ogene i t y Distance (m)Initial fluctuation- 1 GeVExpanding Universe- 236 MeV
FIG. 3: The decay of the fluctuation is shown in log scale between 1 GeV - 236 MeV for aplasma where a quark is moving through a sea of electrons.as before. In figure 3, we find that the inhomogeneity has decreased by three orders ofmagnitude. In the case of fig. 3, the initial fluctuation is at 1 GeV while the final is plottedat 236 MeV.For the case of the quarks moving through a large number of muons, the amplitudedecay is less than the decay in the case when the particles surrounding the quarks are theelectrons. However, the decay is still quite significant. For the neutrinos, again we havethe inhomogeneity decaying by three orders of magnitude. So independent of the particledistribution in the plasma, the amplitude of the inhomogeneities goes down significantly. Infig 4, we have the quarks moving in a region of muons while in fig 5, the quarks move throughthe neutrinos. We thus find that the inhomogeneity decays by three orders of magnitudebetween 200 GeV to 1 GeV, and again decays by at least two orders of magnitude between1 GeV to 200 MeV. This means that any inhomogeneity generated at the electroweak epochneeds to have an amplitude greater than 10 times the background density to survive tillthe quark hadron transition. Thus if we had an inhomogeneity at the electroweak scalewith an amplitude less than 10 , it would be completely wiped out before the quark hadrontransition. 12 B a r y on I nho m ogene i t y Distance (m)Initial fluctuation- 1 GeVExpanding Universe- 236 MeV
FIG. 4: The decay of the fluctuation is shown in log scale between 1 GeV - 236 MeV for aplasma where a quark is moving through muons. B a r y on I nho m ogene i t y Distance (m)Initial fluctuation- 1 GeVExpanding Universe- 236 MeV
FIG. 5: The decay of the fluctuation is shown in log scale between 1 GeV - 236 MeV for aplasma where a quark is moving through neutrinos.
VI. DECAY OF INHOMOGENEITIES IN THE HADRONIC PHASE
We have studied the decay of inhomogeneities in the hadronic phase previously [15],however in that case we were interested in specific inhomogeneities generated by Z(3) domainwalls whose size was much smaller compared to the horizon size. The expansion of theuniverse was ignored in those cases. We are now interested to see the decay of largerinhomogeneities in the hadronic phase. As mentioned before, inhomogeneities generatedduring or after the quark hadron phase transition are not only larger in amplitude butthey may also be larger in size. Consequently, the decay of these inhomogeneities would be13ffected by the expanding universe. The plasma during this period consists of the muons,neutrons, protons. electrons and neutrinos. In our previous work, we had shown that thepresence of muons enhances the diffusion coefficient of the neutrons/protons, however wehad not factored in the expansion of the universe in the previous work. This time we use thediffusion equation for the expanding universe and find that the expansion of the universecauses the inhomogeneities to decay much faster. In the current section, we briefly describethe diffusion coefficient in the hadronic phase and then proceed to present the results of thedecay of the inhomogeneities in the hadronic plasma.As we have mentioned, the calculation of the diffusion coefficient will depend on whichparticle is moving through the plasma. The baryon number is carried by the neutrons andthe protons, hence here we will be considering the motion of a heavier particle through alighter gas. The heavier particle is the neutron or the proton, while the lighter gas is agas of electrons and neutrinos. The muons only play a role till 100 MeV. We thus have touse eqn.7 and the scattering cross-section of the neutrons with the electron-positron gas toobtain the diffusion coefficient of the neutrons in the electron positron gas. The scatteringcross section is given by, dσd
Ω = α κ q M E sin ( θ/ E (cid:48) E × (cid:20) sin ( θ/ (cid:21) (15)Here θ is the scattering angle, while E is the electron energy before the scattering and E (cid:48) isthe electron energy after the scattering. The values of E and E (cid:48) depend on the temperatureof the surrounding plasma. The transport cross section σ t , is given by σ t = (cid:90) dσd Ω (1 − cosθ ) d Ω (16)We can then substitute the scattering cross-section in the transport cross section to obtain, σ t = 3 π (cid:20) ακM (cid:21) (17)The diffusion coefficient is obtained by substituting the expression for the transport cross-section and we get, D ne = M m ακ e /T T f ( T ) . (18)Here M , is neutron mass, m is electron mass and κ = − .
91 is the anomalous magneticmoment. The temperature in this case is dimensionless as it is scaled by a factor of m e c .Finally, the function f ( T ) is given by, f ( T ) = 1 + 3 T + 3 T .14imilar to the neutron - electron cross section, we can obtain the nucleon-muon scatteringcross-section too. We have assumed that the heavy neutron particle is moving through amuon-antimuon gas. The mobility of the neutron is then given by the force on the neutrondue to the gas. This force is given by the interaction cross section. The differential scatteringcross-section is given by, dσd Ω = α κ q M E sin ( θ/
2) 11 + 2
Esin ( θ/ /M × (cid:20) cos ( θ/ − q / M (cid:18) q M − (cid:19) − sin ( θ/ (cid:21) (19)Here we have assumed that the muon energy and mass are less than the neutron mass.The cross-section calculation is simplified by this assumption and we can write the cross-section as, dσd Ω ≈ K α κ M [1 + cosec ( θ/ K = . We then substitute the crosssection in the diffusion constant. Finally, the diffusion coefficient is given by, D nµ = M m µ ακ e /T (cid:48) T (cid:48) f ( T (cid:48) ) (21)Here T (cid:48) = Tm µ c . After obtaining both D ne and D nµ , we calculate the total diffusion coefficientof the neutron moving through the plasma.Apart from the neutron, we need to find the diffusion coefficient of the proton movingthrough the electron positron gas too. For proton-electron scattering, the Coulomb forcehas to be taken into consideration. The scattering cross section for the proton and electronis then given by, dσd Ω = α m e k sin ( θ/ (cid:20) k m e cos ( θ/ (cid:21) (22)We can obtain the transport cross section from these equations, σ t = 4 πα (cid:20) E e h πk (cid:21) ln ( 2 θ ) (23)where θ is the minimum scattering angle. Substituting all the previous equations, we getthe diffusion coefficient as, D pe = 3 π α ln ( θ ) (cid:20) h πm e (cid:21) T e /T f ( T ) . (24)15 B a r y on I nho m ogene i t y Distance (km)Initial fluctuation- 200 MeVExpanding Universe- 105 MeVNon-expanding Universe- 105 MeV
FIG. 6: The decay of the initial fluctuation is shown in logscale between 200 MeV - 100MeV.Since the muons also constitute a significant part of the plasma till 100 MeV, we calculatethe proton muon cross section too. The differential cross section is given by, dσd
Ω = α E sin ( θ/
2) 11 + 2
Esin ( θ/ /M × (cid:20)(cid:18) − κ q M (cid:19) cos ( θ/ − q M (1 + κ ) sin ( θ/ (cid:21) (25)We obtain the numerical value of this diffusion coefficient by substituting the constants inthe transport cross section. Once we have the diffusion coefficients, we numerically solve thediffusion equation in the FLRW metric.As mentioned before we are considering inhomogeneities whose sizes are in the rangeof 1 km. Since the horizon size is around 10 kms in the hadronic phase, these are largeinhomogeneities. We have considered high amplitudes of the order of 10 as well as smalleramplitudes, we find that the decay rate does not depend significantly on the amplitudes.However, we find that in an expanding universe the overdensity falls far more rapidly thanin an non-expanding universe. We have shown both the cases in figure 6. for comparison.We have checked for the decay separately in the range 200 MeV - 100 MeV as the muon isstill present in the plasma at these temperatures. At lower temperatures the muon densityin the plasma becomes negligible.It seems that large inhomogeneities do decay significantly in an expanding universe but aslong as they have very large amplitude, they may still survive upto the nucleosynthesis epoch.So an inhomogeneity whose amplitude is of the order of 10 will be decreased to an amplitude16 B a r y on I nho m ogene i t y Distance (km)Initial fluctuation- 100 MeVExpanding Universe- 6 MeVNon-expanding Universe- 6 MeV
FIG. 7: The decay of the initial fluctuation is shown in logscale between 100 MeV - 1 MeV.of the order of 10 . Hence inhomogeneities with low amplitudes of the order of 10 will bewiped out. Finally, we look at the temperature range from 100 MeV - 1 MeV. The muons willbe negligible in this epoch but the diffusion coefficients will not change. Figure 7 shows thedecay of the inhomogeneities in this epoch. We find that the amplitude of the inhomogeneitydecreases by an order of 10 in this period. This means that any inhomogeneity with anamplitude less that 10 will be wiped out before the nucleosynthesis epoch. So large baryoninhomogeneities generated during the quark hadron transition must have amplitudes greaterthan 10 times the background density to survive till the nucleosynthesis epoch. VII. SUMMARY
In summary, we have done a detailed study of the decay of the baryon inhomogeneitiesgenerated at the electroweak scale. Baryon inhomogeneities have important consequencesin the early universe. If they survive till the quark hadron phase transition they will affectthe phase transition dynamics. The quark hadron phase transition is very important in thethermal history of the universe. Moreover, baryon inhomogeneities can also be generatedduring the quark hadron phase transition. These will have an effect on the Big BangNucleosynthesis calculations. Thus the decay of baryon inhomogeneities are important in theearly universe. There has been no previous studies of the decay of baryon inhomogeneitiesin the early universe during the electroweak scale. We have studied the decay of theseinhomogeneities in the presence of electrons, muons and neutrinos. The baryon number17s carried by the quarks at these high temperatures, so as the inhomogeneity decays, thequarks diffuse through the electrons, muons and neutrinos. The diffusion coefficients forthe different particle interactions are calculated. We then use these diffusion coefficients tostudy the diffusion of the baryon inhomogeneity using the diffusion equation in the FLRWmetric.We have found that baryon inhomogeneities generated in the electroweak epoch shouldhave an amplitude greater than 10 for them to survive till the quark hadron phase transition.This makes it difficult for the baryon inhomogeneities generated in a first order electroweakphase transition to have any effect on the quark hadron epoch. The baryon inhomogeneitieshave to have a very high amplitude, they decay substantially during the period between200 GeV - 400 MeV. This is because the diffusion coefficient is temperature dependent.The quark hadron transition occurs around 200 MeV. We have found that the amplitude ofthe baryon inhomogeneity decreases to about five orders of magnitude during this period.This means any inhomogeneity with an amplitude of 10 (or less) will be wiped away beforethe quark hadron phase transition. We therefore conclude that any model which generatesinhomogeneities with less than 10 amplitude in the electroweak epoch cannot affect thequark hadron phase transition. They will therefore not contribute to inhomogeneous BBNeither.Finally in a previous work, we had looked at the decay of baryon inhomogeneities in theQCD epoch for a stationary universe. This would work only for small scale inhomogeneitiesfor which the expansion of the horizon does not matter. We have extended that work foran expanding universe where we can work with large baryon inhomogeneities. So we lookat the decay of large baryon inhomogeneities in the QCD epoch. We find that the baryoninhomogeneities decrease by 5-6 orders of magnitude. This means that if large baryoninhomogeneities are generated by collapsing domain walls and other topological defectsduring the quark hadron transition they will survive till the nucleosynthesis epoch. Weconclude that the big bang nucleosynthesis, can thus be used to constrain models whichgenerate large amplitude inhomogeneities in the QCD epoch only.AcknowledgmentsThe authors acknowledge discussions with Soumen Nayak and Salil Joshi. A.S is supportedby the INSPIRE Fellowship of the Department of Science and Technology (DST) Govt. of18ndia, through Grant no: IF170627. [1] E. M. Lifschitz, Phys. (Moscow) 10, 116 (1946)[2] S. W. Hawking, Astrophys. J. 145, 544 (1966)[3] J. M. Bardeen, Phys. Rev. D, 22, 1882 (1980)[4] A. F. Heckler, Phys. Rev. D 51, 405 (1995).[5] A. Megevand and F. Astorga, Phys. Rev. D 71, 023502 (2005).[6] A. Megevand and A. D. Sanchez, Phys.Rev. D 77, 063519 (2008).[7] G.M. Fuller, G.J. Mathews, C.R. Alcock Phys. Rev. D, 37, 1380 (1988); H. Kurki-Suonio,Nuclear Physics B - Proceedings Supplements, Volume 24, Issue 2, 67-73 (1991).[8] B. Layek, S. Sanyal and A.M. Srivastava, Phys. Rev. D 67, 083508, (2003).[9] A. Cohen, D. Kaplan, and A. Nelson, Phys. Lett. B 245, 561 (1990); Nucl. Phys. B349, 727(1991); Phys. Lett. B 263, 86 (1991); L. D. McLerran, M. E. Shaposhnikov, N. Turok, and M.B. Voloshin, ibid. 256, 451 (1991); L. D. McLerran, Phys.Rev. Lett. 62, 1075 (1989); M. E.Shaposhnikov, JETP Lett. 44, 465 (1986); Nucl. Phys. B287, 757 (1987); B299, 797 (1988);N. Turok and P. Zadrozny, Phys. Rev. Lett. 65, 2331 (1990); Nucl. Phys. B358, 471 (1991).[10] S. Sanyal, Phys. Rev. D 67, 074009, (2003).[11] K. Jedamzik and G. M. Fuller, Astrophys. J. 423, 33-49 (1994).[12] V. Berezinsky and A. Z. Gazizov, The Astrophysical Journal, 643:8–13, (2006).[13] N. Sasaki, O. Miyamura, S. Muroya, and C. Nonaka, Phys. Rev. C, 62(1), 011901. (2000).[14] R.J. Scherrer and M.S. Turner, Phys. Rev. D 100.4, 043545 (2019).[15] S. Sau, S. Bhattacharya and S. Sanyal, The European Physical Journal C 79(5) 439 (2019).[16] B. Layek, S. Sanyal and A.M. Srivastava, Phys.Rev.D 63, 083512 (2001).[17] Cherubini, S., Figuera, P., Musumarra, A. et al., Eur. Phys. J. A 20, 355–358 (2004); A.Arbey, J. Auffinger and J. Silk, Phys. Rev. D 102, 023503 (2020).[18] J.H. Applegate, C.J. Hogan, R.J. Scherrer, Phys. Rev. D35 1151-1160, (1987);In-Saeng Suhand G.J. Mathews, Phys.Rev.D 58 123002 (1998).[19] R. Nakamura, M. Hashimoto, R. Ichimasa and K. Arai, International Journal of ModernPhysics E Vol. 26, No. 08, 1741003 (2017); K. Inomata, M. Kawasaki, A. Kusenko and L.Yang, Journal of Cosmology and Astrophysics 12, 003, (2018).
20] J. Dziarmaga, Phys. Rev. D 52, R569 (1995); M. Nagasawa, Astropart. Phys. 5, 231 (1996);M. Sato, Phys. Lett. B 376, 41 (1996); M. Nagasawa and J. Yokoyama, Phys. Rev. Lett. 77,2166 (1996); H. K. Lo, Phys. Rev. D 51, 7152 (1995).[21] M. Barriola, Phys. Rev. D 51, R300 (1995).[22] R. H. Brandenberger and A. Riotto, Phys. Lett. B 445, 323 (1999); T. Matsuda, Phys. Rev.D 64, 083512 (2001).[23] A. Atreya, A. Sarkar and A. M. Srivastava, J. Phys.: Conf. Ser. 484 012053,(2012); A. Atreya,A. Sarkar and A. M. Srivastava Phys.Rev.D 90 4, 045010 (2014).[24] Michael B. Christiansen and Jes Madsen,Phys.Rev.D 53, 5446-5454, (1996).[25] L. D. Landau and E. M. Lifshitz, Physical Kinetics. Pergamon Press Ltd.,(1981).[26] K. S. McFarland, Nuclear Physics. B, Proceedings Supplements 235, 143-148 (2013).[27] D. J. Schwarz, Annalen Phys.12:220-270, (2003).20] J. Dziarmaga, Phys. Rev. D 52, R569 (1995); M. Nagasawa, Astropart. Phys. 5, 231 (1996);M. Sato, Phys. Lett. B 376, 41 (1996); M. Nagasawa and J. Yokoyama, Phys. Rev. Lett. 77,2166 (1996); H. K. Lo, Phys. Rev. D 51, 7152 (1995).[21] M. Barriola, Phys. Rev. D 51, R300 (1995).[22] R. H. Brandenberger and A. Riotto, Phys. Lett. B 445, 323 (1999); T. Matsuda, Phys. Rev.D 64, 083512 (2001).[23] A. Atreya, A. Sarkar and A. M. Srivastava, J. Phys.: Conf. Ser. 484 012053,(2012); A. Atreya,A. Sarkar and A. M. Srivastava Phys.Rev.D 90 4, 045010 (2014).[24] Michael B. Christiansen and Jes Madsen,Phys.Rev.D 53, 5446-5454, (1996).[25] L. D. Landau and E. M. Lifshitz, Physical Kinetics. Pergamon Press Ltd.,(1981).[26] K. S. McFarland, Nuclear Physics. B, Proceedings Supplements 235, 143-148 (2013).[27] D. J. Schwarz, Annalen Phys.12:220-270, (2003).