Baryogenesis in the Desert Between Electroweak and Grand Unification Scale

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BARYOGENESIS IN THE DESERT BETWEEN ELECTROWEAK AND GRAND UNIFICATION SCALE

B. P. GONG1 (Revised)

ABSTRACT As the Universe is dominated by of baryons, there must have been a process in early Universe creating such a asymmetry via baryon number violation. However, until recently all the confirmed experimental data indicate that lepton and baryon number are conserved in agreement with Standard Model of quarks and leptons. This paper proposes an alternative path of baryogenesis. Once the over abundant magnetic monopoles predicted by grand unification emerged at universe temperature of 1016 GeV, then their annihilation should first give rise to photons driving the expansion of the early universe. As the universe temperature dropped to the desert scale of around 10 EeV, departure from thermal equilibrium occurred. Thus the annihilation produced photons give rise to electron-positron pairs, which collide with remnant magnetic monopoles and antimonopoles. The resultant unstable intermediate X and antiX particle become asymmetric, and thereby leading to baryon asymmetry. The excess of antimonopoles corresponding to ten times of mass comparing with that of baryons provides a natural candidate for dark matter. The physics of such a desert regime baryogenesis can be simply characterized by three parameters. The collision predicts a ratio of particle and anti-particle equivalent to that of ultra- high-energy neutrino to their anti-particle, which can be tested by future observations, e.g., the constructing IceCube-Gen2. Subject headings: cosmology, inflation, baryogenesis, magnetic monopole

1. INTRODUCTION called the super-cooling, corresponds to a difference in en- The grand unification Theories (GUTs) attempting to unify ergy density which drives the exponential expansion of the the strong, weak, and electromagnetic interactions and the Universe. quarks and leptons have two startling predictions: the exis- The process cannot be simultaneous everywhere, small tence of stable, superheavy magnetic monopoles (MMs), and bubbles of true vacuum, would be carried apart by the ex- interactions that violate baryon number (B) and lepton num- panding phase too quickly for them to coalesce and produce ber (L) and thereby lead to the instability of the proton. Both a large bubble. The resultant Universe would be highly inho- predictions have very significant cosmological consequences. mogeneous and anisotropic, opposite to what is observed in With a net B from the universe beginning with baryon sym- the cosmic microwave background. metry, most GUTs predict baryogenesis through the decay of The successor to the old inflation was new inflation(Linde supermassive (GUT-scale) X particles with interactions vio- 1982; Albrecht et al. 1982). This is again a theory based lating B conservation (’t Hooft 1976b; Kibble 1976; Kolb & on a scalar field. The field is originally in the false vac- Turner 1983; Bhattacharjee 1998; Kolb & Turner 1990). uum state, but as the temperature lowers it begins to roll Such a decay process based on GUTs requires that the su- down into one of the two degenerate minima. There is no perheavy bosons were as abundant as photons at very high potential barrier, so the phase transition is the second order. Problem of such inflationary models is that they suffer from temperatures of ∼ MGUT which is questionable if the heavy X particles are the gauge or Higgs bosons of Grand Unification. severe fine-tuning problem and requires very specific scalar Because the temperature of the universe might always have field(Steinhardt 2011). One of the most popular inflationary models is the chaotic inflation(Linde 1983), which is based on been smaller than MGUT and, correspondingly, the thermally produced X bosons might never have been as abundant as pho- a scalar field, but it does not require any phase transition. In tons, making their role in baryogenesis negligible(Riotto & such cases, the standard model of cosmology is in fact sepa- Trodden 1999). rated from the standard model of particle physics. Therefore, the GUTs based baryogenesis is still short of This paper shows that simply replacing driving energy of suitable environment to invoke the assumed decay of super- inflation by annihilation of magnetic monopoles (MMs), the heavy particles, evenif the B nonconservation is valid. The first order phase transition of Guth’s still work. Moreover, the problem of such baryogenesis may originate in physical pro- subsequent evolution of the universe at the desert regime au- cess before the decay of superheavy X particles, the inflation. tomatically gives rise to sufficient intermediate supermassive The first inflationary model, so called old inflation, was thus particles to account for the baryogenesis. proposed (Guth 1981) based on a scalar field theory undergo- As over abundant MMs predicted by GUTs(’t Hooft 1974; ing a first order phase transition. The scalar field was ini- Carrigan & Trower 1983; Preskill 1984) nucleated in bub- tially trapped in a local minimum of some potential, and then bles at the critical temperature of the phase transition, Tc ∼ leaked through the potential barrier and rolled towards a true MGUT , their annihilation would invoke enormous radiation minimum of the potential. The transition from initial “false to drive the inflation. With such a heat source, the universe vacuum" phase to the lower energy “true vacuum" phase, so would undergo a free expansion which expands exponen- tially by smooth temperature drop. Therefore, supercooling and reheating expected by adiabatic expansion models can be 1 Department of Astronomy, Huazhong University of Science and Technology, Wuhan, 430074, People’s republic of China; bp- avoided, as well as the difficulties of owing to vacuum field [email protected] driven inflation mentioned above.

At the end of such a inflation, the spread out energy has Such a collision results in intermediate X particles as shown in straighten curves and warps of the universe, which resulted in Eq(1) which can rapidly decay into baryons and leptons. And departure from thermal equilibrium. Then begins the baryo- during such collision and decay processes the annihilation of genesis. The continuing annihilation of MMs at much re- relics MMs continues at much small rate comparing with the duced rate would give rise to photons producing electron- primary inflation. position pairs rather than expanding the universe. Therefore, the change of number density of MMs in the de- And the collision of electron-positron with the remnant parture from thermal equilibrium can be described by Boltz- MMs can reproduce not only sufficient but also intermediate mann equation as, superheavy X particles. d(n a3) As a baryon changes sign when charge conjugation (C) and a−3 M = hannii + hcollii + hdecayi . (2) charge conjugation combined with parity(CP) are violated, dt the chemical potentials of the superheavy intermediate X par- 16 ticle and antiX particle have opposite sign, which results in As discussed in detail in Sect(5), from T ∼ 10 GeV to 10 asymmetry on number of X and antiX particles. And the de- EeV, the universe is dominated by the radiation originating cay of such X and antiX particles automatically can account in annihilation of MMs, so that the right hand side of Eq(2) for baryon asymmetry. is governed by the first term. And from 10 EeV to 1 EeV, In such a case more X particles decay into baryons than the rate of annihilation is much reduced, thereby the universe those of antiX particles into antibaryons. Correspondingly the departs from thermal equilibrium and coasting. In such case remnant MMs is less than that of antiMMs. Due to the X par- all the three terms at right hand side of Eq(2) work. ticles decay into quarks, leptons, and pions, so that approx- The collision process, corresponds to the second term at imately 10% of X particles convert to baryons(Bhattacharjee right hand side of Eq(2). In general the Boltzmann equation 1998). Consequently, the total mass of baryons should be only are a coupled set of integral partial differential equations for ∼ 10% of the antiMMs, which automatically account for dark the phase space distributions of all the species present. For matter. a few species of interests to us, like positrons and MMs as In such a scenario, the annihilation of over abundant MMs shown in Eq(1), should have equal phase space distribution can not only predicts sufficient energy to drive the infla- function because their rapid interactions reducing the prob- tion itself, avoiding some difficulties of inflationary models lem to a single integral partial differential equation for the like no ending and very limited parameter spaces(Steinhardt one species of interests. 2011); but also provides superheavy particles, leptons and Therefore, the Boltzmann equation corresponding to the suitable temperature for the occurrence of collision and de- collision of electron or positron (of number density ne) with cay which lead to baryogenesis under baryon conservation. MMs or antiMMs (of number density nM), and thereby pro- It turns out that GUTs incorporating R-parity(Dreiner et al. ducing intermediate superheavy particle (with number density 2012; Corianò et al. 2008), which ensures a stable proton and nX ), and neutrinos (of number density nν ), can be written, achieves baryon and lepton conservation is consistent with the d(n a3) n n n n new path of baryogenesis. Such a process occurs in scale, a−3 M = n(0)n(0)hσvi[ X ν − M e ] , (3) dt M e (0) (0) (0) (0) ≈ 10EeV −1EeV, which is so called the great desert, between nX nν nM ne the Standard Model particle physics describing physics at the electroweak scale of ∼ 100−200 GeV and the standard model where the definition of number density in terms of the phase of cosmology with GUTs scale of ∼ 1016GeV. space density for species in kinematic equilibrium is read, The collision induced baryogenesis includes three steps: Z d3 p n g exp µ /T i exp E /T the annihilation of MMs started before the collision and ended i = i ( i ) 3 (− i ) after the decay, Sect(4); the collision of leptons with MMs, (2π) Sect(2); the decay of intermediate X particle, Sect(3). Z d3 p n(0) g i exp E /T Sect(5) estimates parameters of annihilation, collision and i = i 3 (− i ) decay processes, and predicts that the collision and decay (2π) can give rise to extremely high energy neutrinos, the ratio of with gi denotes the degeneracy of the species, while Ei and µi which can test the validity of the new scenario. The asso- represent the energy and chemical potential of species respec- ciation of baryon asymmetry with charge neutrality, and the tively. remnant antiMMs as possible a candidate of dark matter, are The left hand side of Eq(3) is of order of nM/t or expressed also addressed. The three major parameters characterizing the with the Hubble constant, nMH. The right hand side is of universe evolution from GUT to desert scale are summarized. order of nMnehσvi. The cross section of the collision between positron and MM as shown in Eq(1) is given, 2. THE COLLISION Z (0) (0) −1 4 As shown in Sect(4), the annihilation of over abundant hσ|vi = (n n ) dπMdπedπX dπν (2π) MMs emerged at phase transition of GUTs can lead to an in- M e flation driven by MM annihilation. At the end of inflation 4 2 when the universe temperature is around 10 EeV, begins pro- ×δ (pM + pe − pX − pν )| f (q)| exp(−EM/T)exp(−Ee/T) , (4) cesses out of thermal equilibrium. 3 3 where dπi = gid pi/[(2π) 2Ei]. It seems inevitable that electrons and positrons emerged at The rate of a particular reaction mediated by boson ex- the desert scale of 10 EeV collide with remnant MMs with | f q |2 much reduced number through previous annihilation. The change is proportional to ( ) multiplied by a phase-space collision of positron with MMs is read, factor, which determines the rate of an unstable state, or the cross section for a collision process. The large the value of e+ + M ↔ X + ν¯ . (1) | f (q)|2 corresponds to a high the cross section of collision. 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Consider a particle being scattered by a potential provided µ = 0, which results in F( fb, f¯b) = 0 and thus n˙ B = 0 in Eq(9). by an infinitely massive source, the effect of which is observed Obviously, such a decay process satisfies C or CP invariance. through the angular deflection of the particle or, equivalently, On the other hand, suppose a volume of early universe con- the momentum transfer, q. The potential A(r) of the massive taining equal numbers of X and X¯ bosons. The mean net source in coordinate space will have an associated amplitude, baryon number produced by the decay of an X to quark/lepton ¯ f (q), for scattering of the particle; which is simply the Fourier final states p1 = qq(B = 2/3) and p2 = q¯l(B = −1/3) is BX = transform of the potential(Perkins 2000), r(2/3) + (1 − r)(−1/3), where r is the decay rate. In contrast, ¯ Z the net number generated by decay of an X to p¯1 = q¯q¯ (B=- iq¯·¯r ¯ f (q) = e A(r)e dV . 2/3) and p¯2 = ql(B = 1/3) is BX¯ = r¯(−2/3) + (1 − r¯)(1/3). The mean net baryon number resulted in the decay of an X and ¯ And the integration over volume can be obtained by setting, X pair is BX + BX¯ = r − r¯. The conservation of C and CP lead to r = r¯, and hence baryon invariance, n˙ = 0(Kolb & Turner q¯· r¯= qr cosθ, dV = r2dφsinθdθdr, which yield, B 1990). Z ∞ sinqr Therefore, the models of baryogenesis directly through de- f (q) = 4πe A(r) r2dr . qr cay superheavy particle requires all the three ingredients of 0 Sakharov (1967), (1) B-nonconserving interactions, (2) a vio- The magnetic potential of a MM with charge g satisfies, lation of both C and CP, (3) a departure from thermal equilib- H AdR = 4πg, thereby, A = −2ig/R, substitute into the equation rium. above, one has, f (q) = −2ieg/|~q|2, and square of which yields, In comparison, the scenario of collision induced baryogenesis can be achieved in the absence of the first one, the B 4e2g2 nonconservation, while holding the other two ingredients. f 2(q) = = ¯h2/q4 . (5) |~q|4 Therefore, as shown in Eq(1), from an initial B and L sym- metry Universe (B=L=B-L=0), the collision of lepton with For an electron interacting with a MM, the 4 momentum of an MMs can result in B and L asymmetries while preserving electron becomes, q2 = p2 − E2/c2 ≈ p2, because it gets much both B + L = 0 and B − L = 0. In fact, the new scenario can be greater momentum than its own rest energy, p2 E2/c2, dur- consistent both with experiments on C and CP nonconserva- ing the interaction. The momentum relates with the impact tion(Christenson et al. 1965; Kleinknecht 1976), and the limit parameter b by, of proton decay(Abe et al. 2014). It appears that scenario favors GUTs incorporating R- egvb Z b dt eg q ≈ ∆P , parity(Dreiner et al. 2012; Corianò et al. 2008), which can y = 2 2 2 3/2 = (6) 4π −∞ (b + v t ) 4πb ensure a stable proton and achieve baryon and lepton conservation at the renormalizable level. Future results of ratio of Substituting Eq(6) into the cross section of Eq(4) yields, extremely high energy neutrino to antineutrino predicted by 1 qc qc q8 the collision may test the validity of such a baryogenesis oc- hσ|vi | f q |2 curring in the desert regime. = 12 8 2 2 ( ) 2 (2 π ) 2Mmc 2MX c c The collision induced baryogenesis can undergo with suffi- 1 G2l qc qc cient leptons and superheavy particles at the universe temper- Pl q4 . ature around 10 EeV. Because 10 EeV corresponds to a criti- = 14 8 4 2 2 (7) (2 π ) ¯h c5 Mmc MX c cal temperature, when the inflation transferring from thermal Notice that the cross section of collision of Eq(7), hσ|vi ∝ equilibrium to out of equilibrium, the leftover MMs from the q6, is very sensitive to the momentum, q. Now the Boltzmann prior annihilation of of abundant MMs is sufficient to account equation responsible for positron and MM collision of Eq(1) for the subsequent baryongenesis, which avoids the difficul- becomes, ties of Riotto & Trodden (1999) mentioned in the introduction. 3 d(nMa ) (0) The next section will show that the annihilation of over a−3 = n n(0)hσviexp(±µ /T)exp(±µ /T) , (8) 16 dt M e X ν abundant MMs from the initial temperature of 10 GeV to 10 EeV can invoke a radiation that expands the universe ex- where µX and µν are chemical potentials of the intermediate ponentially to a volume sufficient to account for the flatness particle X and antineutrino respectively, which satisfy, µX problem, so that baryogenesis may occur in the out of equi- µν , so that exp(−µν /T) ≈ 1. According to Eq(8) the chemical librium state with temperature between 10 EeV and EeV. potential determines the particle to antiparticle ratio, which further governs the ratio of baryons to antibaryons through 4. ANNIHILATION OF MMS the decay of superheavy X particles. To explain the unacceptable large number of MMs generated at the early Universe, Preskill (1979) suggested that MMs 3. THE DECAY OF INTERMEDIATE X PARTICLES can be suppressed through annihilation if the phase transition The decay of the intermediate particle X and its antiparticle at the GUT is strongly first order. can be expressed, Here a big step is made, which proposes that the driving energy of the inflation is the energy release of the annihilation X 2 n˙ B ≡ n˙ b − n˙¯b = Γ12|M0| F( fb, f¯b) , (9) of the over abundant MMs rather than vacuum energy. Simply X change to the rate of annihilation of MM(Preskill 1979) from where Γ12 is the integral operator, |M0| is the decay am- adiabatic expansion to free expansion, the annihilation driving ¯ plitude for X and X, and fi is the phase-space density of inflation can be achieved. −(E−µ)/T species i(Harvey & Kolb 1981), given by fb = e and The rate of annihilation of MMs (Preskill 1979) was esti- −(E+µ)/T f¯b = e . Apparently, one have fb = f¯b in the case of mated by the Boltzmann equation, originating in the covariant Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 September 2021

conservation of the energy-momentum tensor in the comoving The work done responsible for a free expansion of the uni- frame, in which the number density, n (where n = ρ/m, and m verse initiating at temperature 1016GeV can be estimated, is the mass of the particle), and the annihilation term, Dn2 are related by, 1 ? 103 W ≈ g NmkBTi ≈ 2 × 10 (16) dn/dt + (3a˙/a)n = −Dn2 , (10) γ − 1

where D is the velocity-averaged product of the cross-section where kB is the Boltzmann constant, and γ is the polytropic 28 and the velocity, characterizing the annihilation process, with index. With an initial temperature of inflation of Ti ∼ 10 K; ¯ 88 the number density of monopoles, M, and anti-monopoles, M MMs number NM ∼ 4 × 10 , γ ≈ 0.1 and the effective num- assumed to be equal. ber of degrees of freedom of g? = 100, one gets, W ≈ 2×10103 The relationship between the scale factor and temperature by Eq (16). can be written in a general form of Taβ = const, which reduces This again indicates that the annihilation of MMs of num- 88 to the widely used adiabatic expansion in the case of β = 1, ber NM ∼ 4 × 10 , corresponds to a radiation field which is ˙ 2 repulsive to gravity, so that the negative pressure required in βa˙/a = −T/T = T /(CmP) . (11) driving inflation can be satisfied automatically. Thus the in- The estimation of MM annihilation(Preskill 1979) corre- flation of the Universe can be described as free expansion, sponds to β = 1; whereas, the free expansion of Universe with temperature and volume related by TV 1/3ω = const where corresponds to β =6 1. In the case of thermodynamic equi- 1/(3ω) = γ − 1. By e.g., ω = 3.35 (corresponding to γ = 0.1), librium D relates with the temperature by a power law, and it can have a huge change in volume, with e−foldings of N thus Eq(10) and Eq (11) can be integrated, scale factor (a f /ai ∼ e ) accounting for solving flatness problem(Weinberg 2008), while keeps relative smooth change in 1 4π m n r T 2 , temperature. ( ) = 2 [ 2 ] = 3 (12) κ¯h ¯h (βCmp) T Such a free expansion of the early universe can be well described by a relationship of scale factor, time and temperature where κ = 3 ζ(3)P (¯hq /4π)2 with the sum over all spin 4π2 i i of the Universe(subscript P denotes Planck value), states of relativistic particles, and ζ(3) = 1.202 is the Riemann zeta function. By the number density of MMs given by Eq a t T = = ( P )ω , (17) (12), the energy release of such an annihilation in a comoving a t T volume corresponding to the right hand side of Eq (10) is P P 2 where the Planck temperature, time, and scale factor, are TP = ρ˙m = −MmDn , (13) 32 −43 −32 10 K, tP = 10 s and aP = 10 cm respectively. Due to the number of MMs reduces as their annihilation pro- The annihilation of MMs generated a radiation equivalent ceeds, the number of MMs in a comoving volume decreases to a repulsive gravitational force that drove space to swell with the time of evolution, so that ρ˙m < 0, which is analogy to rapidly momentarily. For that to occur, the field’s energy den- the decay of a massive particle species dominating the energy sity has to vary with strength corresponding to value change density of the universe (Kolb & Turner 1990). of ω in Eq (17), such that it had a high-energy plateau, 1016 Similar to the release due to decay of a massive particle GeV and a low-energy valley, e.g., 1 EeV (as discussed later). species, the annihilation of MMs transfer energy from the Correspondingly such an expansion of universe can be rep- heavy particles to radiation also, so that the first law of ther- resented by a polytropic process of AB0, which is infinitively modynamics, dU + pdV = dQ (where U is the internal energy close to the ideal isothermal process, AB, denoting the irre- of the volume, pdV denotes work done to the surrounding, versible process of free expansion. and Q respresents energy supply to the volume as heat), be- As a result, the energy exchange of universe with surround- comes, ings outside the universe predicted by adiabatic expansion can 3 3 3 2 3 be avoided by free expansion. d(a ρR) + pRd(a ) = −d(a ρm) = −3a a˙ρmdt − a ρ˙mdt . (14) Moreover, the free expansion of the universe with Taβ = As the early universe is radiation dominant, Eq (14) can be const, where β = 1/ω < 1 (due to 3 < ω < 4 as shown later), rewritten as, allows many e-foldings expansion of volume at very smooth ρ˙R + 12H pR = −ρ˙m , (15) change of temperature. In contrast, the adiabatic expansion corresponds to β = 1, which gives rise to enormous volume where H = a˙/a. The decrease of energy density of MMs in a expansion at the expense of supercooling in universe temper- comoving volume, ρ˙m < 0 as shown in Eq (13), plays the role ature, to avoid too low a temperature a reheating process is as- of heat source to the energy density of the universe residing sumed. This requires that after many e-foldings of expansion, in two components: non-relativistic MMs and radiation. Con- the inflaton energy that drove the inflation must be rapidly sequently, we have −ρ˙m > 0 in Eq (15), which corresponds to converted to radiation to reheat the universe. The outcome a negative pressure repulsing the gravity and resulting in an of a huge volume with too high a density and wrong distri- expansion of the universe with H = a˙/a > 0. Notice that the bution of galaxies are resulted in such special inflaton energy annihilation term at right hand side of Eq (15) deviating from curve(Steinhardt 2011). Apparently, such problems are natu- the linear relationship of normal matter with p = wρ. rally avoided in the free expansion of the universe. With Em denoting the overall energy release of annihilation The essential scenario is that at around the critical temper- 103 of MMs during the inflation, it requires Em ∼ 4 × 10 (erg) ature, Tc ∼ MGUT , quantum tunneling occur and nucleation of in order to solve the flatness problem(Weinberg 2008), which bubbles (containing huge number of MMs) of the true vacuum 88 corresponds to a total number of MMs of Nm ∼ 4 × 10 in the sea of false vacua begins. At a particular temperature annihilated during the inflation (with each MM of mass of below Tc, the annihilation of MMs in bubbles begins which in- 16 Mm ∼ 1 × 10 GeV). vokes the first generation of bubble collisions, with resultant Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 September 2021

radiation repulsing gravity and driving exponential expansion In the second stage, most photons produced by annihilation of the universe. of MMs of number 1079 give rise to electron and positron pairs Later on, MMs residing in remnant condensations get an- and colliding with remnant MMs of number 1079 rather than nihilated which leads to the second generation of bubble col- driving the inflation. lisions and expansion, and so on until the universe expansion As the bubble collision undergoes in the second stage of in- satisfying the flatness condition(Weinberg 2008). flation, the bubble walls continue passing points in unbroken As the bubble walls pass each point in space, the order pa- phase, CP violation and the departure from equilibrium occur rameter changes rapidly, as do the other fields, so that depar- while the Higgs field is changing, until the point of the true ture from thermal equilibrium occurs. vacuum at 1 EeV, thus ends the baryogenesis. In such a new scenario, the large structure of the universe The collision of leptons with MMs, and thereby the decay at the end of inflation was formed by the annihilation induced of intermediate particles occur in the second stage of inflation, collision of bubbles for enormous number of generations. from 10 EeV to 1 EeV, reduces the number of MMs from 79 65 In comparison, both the first order phase transition of NM ≈ 10 to NM ≈ 10 , the remnant of which is limited by Guth’s and later second order phase transition predict bubbles the total mass of current epoch. expanding with the universe and we are in one of them, which The electrons and positrons generated at universe of tem- are difficult to account for the uniformity problem. And with perature, T ≈ 10EeV correspond to a thermal energy of the adiabatic assumption these models lead to problems of no −6 T = qc ≈ 10 Mm ≈ 1(J). Colliding with MMs of mass ending and extremely narrow range of parameter space (Stein- 16 Mm ≈ 10 GeV, results in the cross section of collision of hardt 2011). Apparently, these problems can be avoided in the hσvi ≈ 10−82m3s−1, which is very sensitive to the temperature, annihilation driven inflation. Moreover, the three longstand- T (or q), e.g., 50% change of T can vary the cross section for ing problem, over abundant MMs, flatness and uniformity can 11 times by Eq(7). also be explained in a simple and natural way. The energy release of the collision corresponding to Eq(8) 5. PARAMETER ESTIMATION AND DISCUSSION is read,

As shown in Preskill (1979), the limit of annihilation of E N + N hσvi = ( e )( e )( M )( ) · τ · exp(µ/T) , (19) MMs can be defined by the dimensionless ratio between the X 1J 1079 1079 10−82 number density and temperature, as shown in Eq (12), which −3 −3 −2ω 2ω can be estimated by the relation of free expansion of Eq (17), where parameter, τ ≡ a t = tPaP TP T , is composed of the magnitude of scale factor a and time t of the universe −3 −3 −3 −3ω (3ω−3) r = Nma T = NM(aP tP )T . (18) related by Eq (17). The criteria of Preskill (1979) is that once the MMs abun- The collision induced energy release of Eq (19) at tempera- dance is comparable to r ∼ 10−10 or smaller, MM annihilation ture, 10EeV, is equivalent to a conversion of number of MMs 65 cannot further reduce the MM density per comoving volume. of 10 to energy of other forms, which can quickly decay At the beginning of the inflation with temperature T ∼ into a shower of secondary particles, with an efficiency of 1028K and of number of MMs, N ∼ 1088, one can have 10%(Bhattacharjee 1998). If the resultant number of quarks M 79 15 r ∼ 1060 which corresponds to an exponential expansion of and leptons is of ∼ 10 which is ∼ 10 times of the number the universe. of MMs (the mass of a MM is ∼ 1015 times of a baryon), then As temperature drops to T≈ 10 EeV,the scale factor reaches the total baryons at current epoch can be explained, a ≈ 50cm (with ω = 3.35) by Eq (17), which corresponds to 26 15 79 an expansion of scale factor of a/a0 ∼ 10 . Then begins the X → NM → NM · 10 · 10% → Nb ≈ 10 , (20) departure from equilibrium, in which case a reduced number 70 µ/T 79 9 where NM = 10 τe denotes the equivalent number of of MMs of NM ∼ 10 corresponds to a ratio of r ∼ 10 , so that 15 the annihilation of MMs continues along with the collision of MMs corresponding to the energy release of Eq(19); and 10 denotes the ratio of baryon number to that of MM. leptons with remnant MMs, and the decay of intermediate X 79 particles leading to baryogenesis. Furthermore, when temper- In fact, the baryon number of Eq (20), Nb ≈ 10 , requires 65 ature further drops to T ≈ 1 EeV, a remnant number of MMs MM number of NM ≈ 10 , which can be satisfied by different 67 −10 −7 −3 of e.g., NM ∼ 10 corresponds to a ratio of r ∼ 10 , which combinations of parameters, like ω = 3.30, τ ≈ 10 m s and means that the annihilation is coming to an end. eµ/T ≈ 102; or ω = 3.35, τ ≈ 10−8m−3s and eµ/T ≈ 103. Therefore, the inflation can be divided into two stages by The baryogenesis estimated by the collision of Eq (19) and the ratio of r given by Eq (18). The first stage started from decay of Eq (20) depend on three parameters, one is the num- 16 79 the temperature 10 GeV, followed by the annihilation of a ber of remnant MM, NM ∼ 10 , which thereby generate num- 88 79 number of MMs of NM ∼ 4 × 10 . Correspondingly, pho- ber of leptons of, Ne− = Ne+ ∼ 10 . And the other is the out of 88 tons of approximately Nγ ∼ 10 were produced which drove equilibrium temperature, assumed to be T ≈ 10 EeV, which exponential expansion of the universe. determines the magnitude of collision cross section of Eq(7) 79 and parameter τ as shown under Eq(19). And the value of When the total number of MMs reduces to NM ∼ 10 at the universe temperature of 10 EeV, the inflation enters its parameter τ depends on the index ω characterizing the free second stage, departure from thermal equilibrium. Owing to expansion as shown by the relation of Eq (17). On the other hand, T ≈ 10EeV also corresponds to a ratio the change in NM and temperature, the work done from T ≈ 10 EeV to 1EeV via annihilation of MMs is significantly reduced of annihilation, r ∼ 109, as shown in Eq (18), which is much as shown in Eq (16), which also results in a profound change less than that of first stage of inflation, r ∼ 1060, and much of the ratio of Eq (18). Thus the universe expands at much greater than that of the end, r ∼ 10−10. In other words, the 16 79 slower rate (coasting) than that of the first stage from 10 assumption of NM ∼ 10 and T ≈ 10EeV can be consistent GeV to 10 EeV. with both the requirements of baryogenesis and the inflation. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 September 2021

12 From the point of the new baryogenesis, the possibility of Eq(25) requires n ≈ ±10 , in the case of Np/NM ∼ 79 79 65 −8 −3 Ne− and Ne+ much larger than 10 , and reproducing number of 10 /10 , which can be achieved by e.g., τ = 10 m s and baryons and antibaryons much larger than 1079, which finally e−µ/T ≈ 10−3. This establishes the relationship of charge achieve a ratio satisfying current epoch through the annihila- quantization condition of Dirac with the evolution of the uni- tion of baryons and antibaryons are very unlikely. verse and baryogenesis. In contrast to the reaction of positrons and MMs of Eq(1), The ideal case of charge neutrality, QM = −Qe = Qp, may the collision of electrons with anti-MMs, disturbed by other factors, which corresponds to a charge ratio of, e− + M− ↔ X − + ν , (21) 0 70 −µ/T ∆Q QM − eNe QM − e[N − 10 τ · e ] is suppressed by the chemical potential with an opposite sign = = e ≈ 10−9τe−µ/T , from that of Eq(19), Q eNe eNe (26) where the term in the bracket can be understood as follows. Ee Ne− NM¯ hσvi ¯ · τ · exp µ/T . 79 X = ( )( 79 )( 79 )( −82 ) (− ) (22) When electrons of number Ne− ∼ 10 emerged at temperature 1J 10 10 10 10 EeV, the universe did generate equal number of protons to Although the decay of antiparticles is of no difference from achieve charge neutrality with the original electrons through that of particles (both are B conservative), the two different e.g., adjusting the value of τ. However, during such a baryo- parameter combinations under Eq (20) predict different pro- genesis, the original number of electrons before the collision 0 70 −µ/T duction of antiparticles, have reduced by a small value, ∆Ne = 10 τ ·e due to the collision as shown in Eq(21), so that deviation in the charge 15 X¯ → NM¯ → NM¯ · 10 · 10% → N¯b , (23) ratio of Eq (26) is made. −7 −3 70 −µ/T −8 −3 Apparently, in the case of τ = 10 m s, the charge neutral- where NM¯ ≈ 10 τe . In the case of τ ≈ 10 m s and −20 −µ/T −4 −µ/T −3 59 73 ity, ∆Q/Q ≤ 10 (Collins et al. 1989), requires e ≤ 10 , e ≈ 10 , one gets NM¯ ≈ 10 and N¯b ≈ 10 . ¯ which is inconsistent with the requirement of baryogenesis of Consequently, different chemical potential of X and X par- Eq (20). ticles owing to C and CP violation, result in different rate In contrast, when τ = 10−8m−3s, the ratio of e−µ/T ≈ 10−3 of collision and thus different number of intermediate parti- can both consists with the charge neutrality, ∆Q/Q ≤ 10−20; cles and finally expecting a ratio of baryons to antibaryons, −2µ/T and the baryogenesis of Eq (20), which corresponds to a ratio NX¯/NX = N¯/Nb ≈ e as shown in Eq(19) and Eq(22) re- −2µ/T −6 b of antiparticle over particle of N¯/N = e ≈ 10 . spectively. b b By Sect(2) and Sect(3), the ratio of baryon to antibaryon; In fact, more MMs have been converted to baryons, Nb ≈ 79 electron to positron; antiMM to MM; and high energy an- 10 (with an efficiency of 10%) which can account for matter tineutrino to neutrino satisfy, of current epoch. This implies that more anti-MMs leftover 65 N N − N ¯ N after the collision, the number of which can be up to 10 , b = e = M = ν¯e = e2µ/T , (27) which naturally explains the mass of dark matter (of mass of + N¯b Ne NM Nνe ten times of baryons). The presence of such dark matter may play important role in the formation of primordial black holes the resultant ratio is most likely around e2µ/T ∼ 106. The and structure formation of the universe. scenario expects a number of superheavy antineutrinos of 65 65 On the other hand, as more positrons have been consumed Nν¯e ≤ 10 and antiMMs of NM¯ ≤ 10 . in the collision than those of electrons, one would expect more Antiprotons are seen in cosmic rays at about the 10−4 level electron leftover to current universe than those of positrons if compared to protons(Stephens 1989). However, such a mag- there are no other reactions to change them. nitude of the antiproton flux is thought to be consistent with If a MM of strength g exists anywhere in the universe, then the hypothesis that the antiprotons are secondaries produced all electric charge must be quantized, which gives so called by cosmic ray collisions with the ISM, and does not seem to Dirac’s quantization condition, indicate that the presence of antimatter in the galaxy, even at the 10−4 level. g 2πn = , n = 0,±1,±2,... (24) While from the point of the new scenario as shown in ¯h e −4 Eq(26) and Eq(27), the possibility of N¯b/Nb ∼ 10 cannot −6 With the fine structure constant, e2/¯hc = 1/137, the electric be excluded although N¯b/Nb ∼ 10 is more likely. and magnetic charge are related by g/e = 137n/2. Therefore, Moreover, the antineutrino has been recorded on Dec. 8, the electric charge corresponding to the number of post colli- 2016, at the speed close to light. Deep inside the ice sheet that sion MMs is QM = gNM = 137nNMe/2. With an initial number covers the South Pole, it smashed into an electron in the ice of electron and positron pair of 1079 at temperature 10 EeV, and produced a heavy charged particle that quickly decayed electrons correspond to a charge of Q = −eN0. into a shower of secondary particles(IceCube Collaboration & e e Abbasi 2021). The detector, IceCube-Gen2, under construc- The universe can achieve charge neutrality, QM = −Qe through adjusting of parameters of expansion like τ (com- tion may measure the excess of antineutrino over neutrino, posed of scale factor and time of the universe) as shown in Eq which can test the relationship of Eq(27). Together with fu- 70 µ/T 0 ture measurement of ratio of matter and antimatter and ultra- (19)-Eq(23), so that QM = gNM = g(10 τe ) = −Qe = eNe . high energy of cosmic rays, the new scenario can be directly And further assuming the conversion of MMs to baryons sat- tested. isfies, QM = Qp, we have, The collision of leptons with MMs of number 1079 at uni- N N g 137 verse temperature of 10EeV, and with a free expansion of in- e = p = = n (25) dex ω ≈ 3.35 can be consistent with both the requirement of NM NM e 2 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 September 2021

baryogenesis and inflation as shown in Figure 1. data(Abe et al. 2014) will continue to increase. (1) The inflation driven by annihilation of MM and an- (6) The first stage of inflation from 1016GeV to 10 EeV an- 88 88 tiMMs can account for a number of puzzles of inflation both nihilated MMs of NM ∼ 10 , generated photons of Nγ ∼ 10 , 79 old and new. and thereby reproduced baryons of Nb ∼ 10 , which corre- (2) The annihilation of MMs induced bubble collisions tend 88 spond to the ratio of baryon to photon, η ≈ ( baryons )( 10 ) ∼ to produce voids and filaments, the relics of which could ac- 1079 photos count for large structure of universe at current epoch. 10−9. The universe could have held such a ratio from the midst (3) Such a bubble collisions may give rise to primordial of inflation to current epoch. gravitational waves. (7) It can consist with the charge neutrality of the universe, (4) The remnant antiMMs may provide a candidate of dark and correspond to a large number in the Dirac’s quantization matter at current epoch and seeds of primordial black holes. condition. (5) Baryon asymmetry is achieved by different cross section (8) It predicts a lower limit of proton lifetime much larger of collision among MMs and leptons stemming from C and than that of the current one, > 5.9 × 1033 years. CP violation, instead of baryon nonconservation. The new (9) The collision and decay expect the generation of ex- model predicts an infinite long lifetime of protons, which extremely high energy Cosmic rays and neutrinos (of number pects that the lower limit of the proton lifetime of > 5.9 × ≤ 1065), as well as ratio of other particles, which can be tested 1033 years at 90% confidence level by Super-Kamiokande by further observations.

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FIG. 1.— Schematic summary of annihilation of MMs induced inﬂation and baryogenesis.