THE ASTROPHYSICAL JOURNAL, 495:539–549, 1998 March 10
© 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

A Matter-Antimatter Universe?

A. G. COHEN, 1 A. DE RÚJULA, 1, 2 AND S. L. GLASHOW 1, 3

Received 1997 July 21; accepted 1997 September 23


ABSTRACT

     We ask whether the universe can be a patchwork consisting of distinct regions of matter and antimatter. We demonstrate that, after recombination, it is impossible to avoid annihilation near regional boundaries. We study the dynamics of this process to estimate two of its signatures: a contribution to the cosmic diffuse γ-ray background and a distortion of the cosmic microwave background. The former signal exceeds observational limits unless the matter domain we inhabit is virtually the entire visible universe. On general grounds, we conclude that a matter-antimatter symmetric universe is empirically excluded.

Subject headings: cosmology: theory—diffuse radiation—elementary particles—large-scale structure of universe

FOOTNOTES

     1 Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215; cohen@bu.edu.

     2 Theory Division, CERN, 1211 Geneva 23, Switzerland; derujula@nxth21.cern.ch.

     

     3 Lyman Laboratory of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138; glashow@physics.harvard.edu.

§1. INTRODUCTION

     The laws of physics treat matter and antimatter almost symmetrically, and yet the stars, dust, and gas in our celestial neighborhood consist exclusively of matter. The absence of annihilation radiation from the Virgo cluster shows that little antimatter is to be found within ∼20 Mpc, the typical size of galactic clusters. Furthermore, its absence from X-ray–emitting clusters implies that these structures do not contain significant admixtures of matter and antimatter.

     Many cosmologists assume that the local dominance of matter over antimatter persists throughout the entire visible universe. A vast literature attempts to compute the baryonic asymmetry from first principles. However, observational evidence for a universal baryon asymmetry is weak. In this regard, searches for antimatter in cosmic radiation have been proposed (Ormes & Streitmatter 1991; Adriani et al. 1995; Ahlen et al. 1994). Early next century, the Antimatter Spectrometer (AMS), deployed aboard the International Space Station Alpha (Ahlen et al. 1994), will search for antimatter in space. Its reach is claimed to exceed 150 Mpc (Ahlen et al. 1982). The detection of cosmic anti–alpha particles would indicate the existence of primordial antimatter; the detection of antinuclei with Z > 2 would imply the existence of antistars.

     The possible existence of distant deposits of cosmic antimatter has been studied before (Steigman 1976; Stecker, Morgan, & Bredekamp 1971; Stecker 1985; Mohanty & Stecker 1984; Gao et al. 1990; Omnés 1970; Dolgov 1993; Dudarewicz & Wolfendale 1994). Steigman (1976) concluded that observations exclude significant matter-antimatter admixtures in objects ranging in size from planets to galactic clusters. Stecker et al. (1971) interpreted an alleged shoulder near 1 MeV 4 in the cosmic diffuse gamma (CDG) spectrum as relic γ-rays from antimatter annihilation. Recently, Dudarewicz & Wolfendale (1994) used similar arguments to reach a contrary conclusion: that the observed CDG spectrum rules out any large antimatter domains. These conflicting results are not based on specific dynamics in a consistent cosmology. Our analysis uses current data and avoids ad hoc assumptions concerning a matter-antimatter universe.

     We explore the possibility of universal (but not local) matter-antimatter symmetry. In what we term the B = 0 universe, space is divided into regions populated exclusively by matter or antimatter. Our conclusions do not depend on how this structure evolved, but it is reassuring to have an explicit model in mind: consider an inflationary cosmology in which baryon (or antibaryon) excesses develop in the manner suggested by Sakharov (1967). In models with spontaneous CP violation, the Lagrangian may be chosen judiciously so that the "sign" of CP violation (determining whether a local baryon or antibaryon excess develops) is randomly and abruptly assigned to regions as they emerge from their horizons during inflation. Soon after baryogenesis, the domain walls separating matter and antimatter evaporate. As regions of matter or antimatter later reenter their horizons, the B = 0 universe becomes a two-phase distribution.

     Let today's domains be characterized by a size d0 such that 1/d0 is their mean surface-to-volume ratio. Because the existence of antigalaxies within a matter-dominated domain is empirically excluded, we must (and can) arrange the distribution of domains to be sharply cut off at sizes smaller than d0. Explicit inflationary models satisfying these constraints exist (Cohen, De Rújula, & Gavela 1996) but are described no further because we find all such models to conflict with observation.

     The current domain size d0 is the only parameter of the B = 0 universe crucial to the confrontation of theory with observation. To agree with constraints from X-ray–emitting clusters, d0 must exceed a minimal value, ∼20 Mpc. For d0 = 20 Mpc, the visible universe would consist of ∼107 domains. We derive a stronger lower limit on d0 comparable to the current size of the visible universe, thereby excluding the B = 0 universe.

     An explicit cosmological model is necessary to estimate the observable signals produced by annihilation. We assume a Robertson-Walker universe and use fiducial values for the relevant cosmological parameters: critical mass density Ω = 1, vanishing cosmological constant ΩΛ = 0, Hubble constant H0 = 75 km s-1 Mpc-1 or h = 0.75, and an average baryon (or antibaryon) number density nB ≡ ηnγ, with η = 2 × 1010.

     In § 6 we show that our conclusions are unaffected by other choices for Ω, ΩΛ, and H0 within their empirically allowed domains. Consequently, we do not express our results explicitly in terms of these cosmological parameters. The annihilation signals we study depend linearly on η. To compute lower limits to the signals, we chose η at the low end of the domain allowed by analyses of primordial element abundances (for a recent discussion, see Hata et al. 1997).

     In § 2 we explain why particle-antiparticle annihilation is unavoidable from the time of recombination to the onset of structure formation. Following a conservative approach, we consider only those annihilations occurring during this period. Our analysis involves known principles of particle, atomic, and plasma physics, but the dynamics of the annihilating fluids (discussed in § 3) is complicated, and the considerations required to reach our results are elaborate.

     The immediate products of nuclear annihilation are primarily pions (π+, π0, and π-) with similar multiplicities and energy spectra. The end products are energetic photons from π0 decay, energetic electrons 5 (e+ and e-) from the decay chain π → μ → e, and neutrinos. Although they are produced at cosmological distances, the annihilation photons and electrons can each produce potentially observable signals:

     1. The energy carried off by annihilation electrons (about 320 MeV per annihilation) affects the CBR spectrum directly (via Compton scattering) and indirectly (by heating the medium). The consequent distortion of the CBR (discussed in § 4) cannot exceed observational limits.

     2. Most of the annihilation photons, although redshifted, are still present in the universe. Their flux (computed in § 5) cannot exceed the observed CDG flux.
Because these signals increase inversely with the domain size, our analysis yields a lower limit for d0. In fact, we obtain no new constraint from comparing the expected distortion of the CBR with its measured limits. However, the CDG flux produced by annihilation far exceeds the observed flux unless d0 is comparable in size to the visible universe. Thus, the B = 0 universe is excluded.

FOOTNOTES

     4 We refer to cosmic diffuse photons by conventional names according to the current photon energy: the night sky, the CDG, and the CBR refer to visible, ∼1 MeV, and microwave photons, respectively.

     5 Relativistic electrons and positrons behave similarly, and we refer to both as electrons.

§2. THE ERA OF UNAVOIDABLE ANNIHILATION

     What if the matter and antimatter domains are and have always been spatially separated? If large empty voids lay between them, there would be no observable annihilation signals. We now show how the observed uniformity of the CBR rules voids out.

     Two events took place at roughly the same time in cosmic history: the transition from charged plasma to neutral atoms (recombination) and the decoupling of radiation and ordinary matter (last scattering). For our fiducial cosmological parameters, these events occurred at a temperature ∼0.25 eV and at a redshift yR ≃ 1100 [we use y ≡ 1 + z = 1/R(t) as a redshift parameter, rather than the conventional z]. The transition to transparency was not instantaneous but evolved during an interval yR ± 100 the halfwidth of which is ∼15 Mpc in comoving (current) distance units. Thus, features at recombination of comoving size smaller than 15 Mpc cannot be discerned in the CBR.

     Large-scale nonuniformities of the matter density, whether dark or baryonic generate variations of the CBR temperature (Sachs & Wolfe 1967). Its observed uniformity (to parts in 10-5) implies a very uniform density of ordinary matter at y = yR ≃ 1100, to within the resolution discussed above. It follows that voids between matter and antimatter domains must be smaller than 15 Mpc.

     The baryon density depletion in voids is damped as photons diffuse toward less dense regions, dragging matter with them. By recombination, inhomogeneities with current size ≲16 Mpc would be destroyed 6 by this mechanism (Silk 1968). This upper bound coincides with the smallest resolvable structure in the CBR. Thus, voids large enough to survive until recombination would have been detected. While matter and antimatter regions may have been separated prior to recombination, they must be in immediate contact afterward. Thus, in determining the minimal signal of a B = 0 universe, we do not consider annihilations occurring at y > yR.

     The mechanism by which the nearly uniform universe at large y evolved today's large-scale structures is not well understood. We cannot confidently assert what effects this will have on annihilation in a B = 0 universe. It could well be that the collapse of baryonic matter into galaxies and stars quenches annihilation unless the collapsing system overlaps a domain boundary, a situation we consider shortly. Our conservative estimate of the annihilation signal includes matter-antimatter annihilation taking place prior to the redshift yS at which the earliest density fluctuations become large (δρ/ρ ∼ 1). We take yS ≃ 20, which is estimated to be the epoch of galactic condensation and earliest star formation (Peebles 1993). We compute the signals due to annihilations taking place during the interval 1100 > y > 20.

     The large-scale density contrast of the visible universe need not coincide with the pattern of matter and antimatter domains. A density fluctuation beginning to collapse could overlap a domain boundary. Successful collapse would yield a structure with a significant mixture of matter and antimatter. In this case annihilation would proceed even more rapidly at the onset of structure formation. Yet we cannot be confident that such mixed structures form.

     In the linear regime (δρ/ρ ≪ 1), the mean annihilation rate is not affected by density fluctuations. But what happens as the fluctuations grow? If an overdensity is to overcome expansion and become a self-gravitating system, it must satisfy the Jeans condition: the sound time across the object l/vs must be greater than the characteristic free-fall time 1/(Gρ)1/2, or Gl$\mathstrut{^{2}}$ρ≥v$\mathstrut{^{2}_{s}}$. Suppose that equality is approached by an overdensity containing both matter and antimatter. Further contraction increases the annihilation rate, thus reducing ρ and driving the system away from collapse. Thus, our conservative estimate of the annihilation signal assumes that density fluctuations straddling domain boundaries either fail to collapse or form separate unmixed structures.

FOOTNOTES

     6 This assumes that such inhomogeneities are not strictly isothermal, a situation considered in § 6.

§3. THE MATTER-ANTIMATTER ENCOUNTER

     The B = 0 universe consists of matter and antimatter domains with almost identical mass densities that, as we have shown, touch one another from recombination to the onset of structure formation. As annihilation proceeds near an interface, a flow develops as new fluid replenishes what is annihilated. This flow must be analyzed to determine the annihilation rate on which our putative signals depend. The analysis involves established and well-understood principles of physics but is complicated by the energy released by nuclear annihilation. (We neglect e+e- annihilation, the energy release of which is much smaller.) The processes by which annihilation electrons lose energy produce crucial effects on the ambient fluid, as well as a potentially observable distortion of the CBR. (High-energy photons from π0 decay, although responsible for the CDG signal, have little effect on the medium through which they pass.)

     The primary energy-loss mechanism of the annihilation electrons is Compton scattering off CBR photons (see the Appendix). This process up-scatters target photons to higher energies. The resultant flux of UV photons heats and ionizes ambient matter throughout much of the universe and for all of the relevant period. Moreover, the annihilation electrons lose a small portion of their initial energies by scattering off ambient electrons in the fluid. This process heats the fluid within the electron range, thereby accelerating the flow and leading to even more annihilation—a feedback mechanism making the matter-antimatter encounter potentially explosive.

     Several length scales characterize the fluid dynamics about a matter-antimatter interface. They are the following:

     1. A, the width of the annihilation zone, wherein both matter and antimatter are present;

     2. D, the width of the depletion zone, wherein fluid flow toward the annihilation zone reduces the density;

     3. L, the width of the reheated zone, wherein electrons produced by annihilations directly deposit energy into the fluid. This is simply the electron range.

     These length scales, computed in the Appendix and later in this section, are shown in Figure 1 along with a comoving domain size of 20 Mpc and the horizon scale. Annihilation takes place in the vicinity of the domain boundary and well within the depletion zone, which itself is much shorter than the electron range. That is, in the relevant redshift domain: A ≪ D ≪ L. This distance hierarchy lets us treat the flow as one-dimensional.

Fig. 1

     Annihilation has a negligible effect on the CBR temperature Tγ(y), which remains as it is in a conventional universe. However, the annihilation debris produce and maintain virtually total ionization, as shown in the Appendix. Therefore, the annihilating fluid consists of photons, protons, antiprotons, electrons, and positrons. 7 The proton and electron number densities coincide, except in the narrow annihilation zone. Consequently our analysis may be put in terms of the total matter mass density ρ ≡ mene + mpnp, the total fluid momentum density ρv, the total fluid pressure p, and the total fluid energy density &epsis; = &epsis;thermal + ρv2/2. The internal energy density and pressure are related as for a nonrelativistic ideal gas: &epsis;thermal = 3p/2.

     The equations describing the flow of matter are conservation laws for particle number (mass in the nonrelativistic limit), momentum, and energy. They must take account of the following phenomena:

     1. The depletion of fluid mass, momentum, and energy by nuclear annihilation.

     2. The effect of the CBR on the fluid momentum and energy.

     3. The effect of the annihilation products on the fluid momentum and energy.

     4. The expansion of the universe.

     The expansion of the universe is taken into account by expressing the conservation laws in a Robertson-Walker universe (Weinberg 1972). The metric is ds2 = dt2 - R2(t)dχ2, with χ a comoving spatial coordinate normal to a domain boundary. The remaining effects are dealt with by including appropriate source terms in the fluid equations:



Here Γ$\mathstrut{_{{\rm ann}}}$≡&angl0;σ$\mathstrut{_{{\rm ann}}}$ v&angr0;$\mathstrut{{\ucpmathaccent{n}{"7016}}}$$\mathstrut{_{p}}$ is the matter annihilation rate, σT is the Thompson cross section, and uγ is the CBR energy density. The terms involving σT describe the transfer of energy and momentum between the fluid and the CBR resulting from Compton scattering; H&epsis;, given by equation (31) in the Appendix, is the rate of change of the energy density of the fluid due to its interactions with the annihilation debris. It receives a direct contribution from the annihilation electrons and an indirect one from UV photons up-scattered by Compton collisions of these electrons with the CBR. We find that the contribution of the electrons dominates within the electron range. Beyond this range, only the UV photons contribute to H&epsis;. In equation (2), we have neglected the small contribution by the annihilation debris to the fluid momentum.

     The signals of a B = 0 universe, the CDG and a distortion of the CBR, are functions of J, the number of annihilations taking place per unit time and area orthogonal to the surface of an annihilation zone,



where the integral extends over a single annihilation zone with χ = 0 at is midpoint and $\mathstrut{{\ucpmathaccent{n}{"7016}}}$$\mathstrut{_{p}}$(χ)=n$\mathstrut{_{p}}$(-χ). The width of the annihilation zone A may be estimated as A ∼ J/&angl0;σ$\mathstrut{_{{\rm ann}}}$ v&angr0;n$\mathstrut{^{2}_{{\infty}}}$, where (Morgan & Hughes 1970) σannv ≃ 6.5 × 10-17 cm3 s-1 c/v and n∞ is the proton density far from the annihilation zone. We must solve equations (1)–(3) to determine J.

§3.1. A Qualitative Solution

     Because our fluid equations do not admit analytic solutions, we begin with a qualitative discussion. The value of Γann is always much greater than the expansion rate, so that the solutions to equations (1)–(3) rapidly reach equilibrium in the annihilation zone. Consequently, taking the limit σann → ∞ yields a good approximation. In this limit, the width of the annihilation zone shrinks to zero and the annihilation terms in the fluid equations may be replaced by a boundary condition at the domain interface. The rate of annihilation per unit surface area is then given by the proton flux at the interface:



     Two effects result from the couplings of the fluid to the CBR. The term in equation (2) proportional to σT tends to damp the fluid motion. The corresponding term in equation (3) tends to keep the fluid temperature near Tγ. For y ≳ 400 these terms dominate, so that the two temperatures are locked together, T ≃ Tγ. The CBR drag on the fluid leads to diffusive motion, and we may define a time-dependent diffusion constant,



The solution to the resulting diffusion-like equations gives an estimate of the annihilation rate J,



with n∞ the proton number density far from the interface. The width of the depletion zone is comparable to the diffusion length D ∼ (Deγt)1/2.

     For redshifts y ≲ 200 the effects of the CBR on the fluid motion are negligible and we may ignore terms proportional to σT. In this case, which we refer to as "hydrodynamic," the motion is controlled by pressure gradients and the fluid flows at a substantial fraction of the speed of sound. The resulting equations are those describing a gas expanding into a semi-infinite vacuum in the presence of an energy source. An analytic solution exists for H&epsis; = 0. In this case the annihilation rate J is



with v∞ the speed of sound and T∞ the fluid temperature T = p/(np + ne) far from the annihilation zone. The coefficient of proportionality is 8 C = ($\mathstrut{\frac {3}{4} }$)4. The width of the depletion zone in this case is comparable to the sound-travel distance D ∼ R(t) &smallint;t dt&arcmin;v∞/R.

     In the intermediate region, 200 ≲ y ≲ 400, neither of the above approximations give a quantitatively accurate picture of the fluid motion.

§3.2. The Numerical Solution

     We have integrated equations (1)–(3) numerically to determine the fluid temperature T and the annihilation rate per unit surface area J near a domain boundary. This diffusive nature of the solution at large y has a welcome consequence: all memory of the initial conditions is lost as the fluid evolves. The post recombination annihilation signal does not depend on the (prerecombination) time at which matter and antimatter domains first come into contact. To solve equations (1)–(3) we choose initial conditions at recombination such that the matter and antimatter domains have constant density, have no peculiar velocity, and touch along the surface χ = 0. Our results are more conveniently presented in terms of y rather than time, according to dy = -yH(y)dt. For our fiducial choice of cosmological parameters H(y) = H0y3/2.

     The dotted curve in Figure 2 is the fluid temperature T(y) in a conventional universe: the solid curve is T(y) within the electron range where heating by relativistic electrons dominates, and the dashed curve is its value outside this region, where UV photons are the only heat source. For y ≳ 400, the CBR is an effective heat bath keeping matter and radiation close to thermal equilibrium. Heating due to the annihilation products plays an important role at lower y: it increases the fluid temperature leading to a larger fluid velocity. According to equation (5) the annihilation rate J is thereby enhanced.

Fig. 2

     The solid curve in Figure 3 is our numerical result for J, the annihilation rate per unit surface area defined by equation (4). The dashed curve is the approximation given by equation (8) using the temperature obtained from the numerical integration. Although its derivation ignored H&epsis;, equation (8) agrees quite well with our numerical result for this choice of T(y). At larger redshifts, the motion is diffusive, and our numerical result should be (and is) substantially less than the qualitative hydrodynamic estimate, 9 as is seen in the figure. Had we used the matter temperature of a conventional universe in equation (8), we would have obtained an annihilation rate nearly 2 orders of magnitude smaller. The heating of the fluid by annihilation debris (described by H&epsis;) has a dramatic effect on the annihilation rate J and a fortiori on the consequent signals of the B = 0 universe.

Fig. 3

     At all redshifts the annihilation rate is determined by the flow of the highly ionized matter and antimatter fluids into the annihilation zone. The momentum-transfer cross section σC in proton-antiproton Coulomb collisions, which controls diffusive mixing of these fluids, is large compared to the annihilation cross section σann. If mixing results only from diffusion, as in quasi-static laminar flow, the annihilation current would be reduced by a factor ∼(σann/σC)1/2 relative to J. However, turbulence produces full mixing in the annihilation zone, while leaving the average flow unaffected, thus justifying our neglect of the Coulomb scattering term in equations (1)–(3).

     An analysis of fluctuations about a laminar flow into the annihilation zone demonstrates an instability toward turbulent mixing. (This is analogous to the instability of a planar flame front in combustive flow; Landau & Lifshitz 1959.) For the width A of the annihilation zone we have obtained, the Reynolds number is RA ∼ AσCn > 105, large enough to ensure a turbulent flow at this and larger scales. Turbulence efficiently mixes the fluids in the annihilation zone but does not significantly retard their mean motion in the depletion zone. Thus turbulence drives the flow toward the solution we have discussed, wherein the annihilation rate is determined solely by the rate at which material can be transported toward the annihilation zone.

     The drag on the fluid exerted by the CBR and the velocity redshift due to expansion suppress turbulence on large scales. The first effect dominates during the redshift range of interest, supressing turbulence for scales λ > vmpc/(σTuγ), which is larger than the width A of the annihilation domain.

FOOTNOTES

     7 We neglect the helium contamination (∼7% by number) and those of larger primordial nuclei.

     8 This is the adiabatic solution. For 100 ≲ y ≲ 200 the process is more nearly isothermal. The corresponding value of C is 1/e.

     9 Our diffusive results for y ≳ 400 are at variance with those of Kinney, Kolb, & Turner (1997), where the annihilation rate prior to recombination is estimated on the basis of proton free streaming.

§4. DISTORTION OF THE CBR

     Measurements of the CBR, being much more precise than those of the CDG, might be expected to provide the most stringent constraint on the B = 0 universe. In this section, we use our conservative calculation of the annihilation rate to estimate the distortion of the CBR spectrum. In performing this calculation, we make several approximations that somewhat overestimate the effect. Nonetheless, the consequent distortion lies well below the observed limit and provides no constraint at all.

     Annihilation produces relativistic electrons and energetic photons. Annihilation electrons have a direct effect on the CBR by scattering photons to higher energies, thereby skewing the CBR spectrum. Moreover, these electrons heat the ambient plasma. The heated plasma produces an additional indirect spectral distortion. (The energetic photons from neutral pion decay have energies too high to have much effect on the cosmic microwave background.)

     To compute the direct effect, we must determine the number of CBR photons scattered from energy ωi to ωf by a single electron. This function, d2N(ωf, ωi)/dωfdωi, is computed in the Appendix. The electron multiplicity per p$\mathstrut{{\ucpmathaccent{p}{"7016}}}$ annihilation is similar to the photon multiplicity, measured (Adiels et al. 1986; Ahmad et al. 1985) to be $\mathstrut{{\ucpmathaccent{g}{"7016}}}$≃3.8. The number of annihilation electrons made per unit volume and time is $\mathstrut{{\ucpmathaccent{g}{"7016}}}$J/d, where 1/d ≡ y/d0 is the average domain surface-to-volume ratio at epoch y. The spectral distortion δuγ(ω) (energy per unit volume and energy) satisfies a transport equation:



We have ignored absorption of UV photons by neutral hydrogen because the B = 0 universe is largely ionized.

     The direct contribution to the CBR distortion is the solution to equation (9) evaluated at the current epoch: δuγ(ω) ≡ δuγ(ω, 1). It is given by



where we have confined the source to 1100 > y > 20, the era of unavoidable annihilation. To evaluate the integral we use the annihilation rate J computed in § 3. Figure 4 displays the result for a current domain size of 20 Mpc. Note that |δuγ(ω)| is always less than 3 × 10-3 cm-3 &sime;1.8×10$\mathstrut{^{-6}}$T$\mathstrut{^{3}_{0}}$. The limit set by COBE-FIRAS (Fixsen et al. 1996) on rms departures from a thermal spectrum is | δu$\mathstrut{_{{\gamma}}}$(ω) |<7.2×10$\mathstrut{^{-6}}$T$\mathstrut{^{3}_{0}}$ throughout the energy range T0 < ω < 10T0. This upper limit is 4 times larger than our computed signal for the minimum domain size. Because larger domains yield proportionally smaller results, we obtain no constraint on the B = 0 universe.

Fig. 4

     The indirect contribution to the CBR distortion results from a temperature difference T - Tγ between the heated ambient fluid and the CBR. It may be described by the Sunyaev-Zeldovich parameter Y (Zeldovich & Sunyaev 1969):



where the integral is along the photon path dl = -cdy/yH(y).

     Within the electron range, collisions between annihiliation electrons and the plasma result in a temperature profile T(y) shown as the solid curve in Figure 2. Outside the electron range, reheating is due to photons up-scattered by these electrons, resulting in the temperature profile shown as the dashed curve. CBR photons may have traversed regions of both types. To compute Y, we use the higher temperature profile (the one within the electron range). We thereby overestimate the signal. Our result is Y &lsim; 9 × 10-7, which is over 1 order of magnitude below the COBE-FIRAS limit (Fixsen et al. 1996) of |Y| < 1.5 × 10-5. We conclude 10 that current observations of the CBR spectrum yield no constraint on the B = 0 universe.

     The energy spectrum of uplifted CBR photons shown in Figure 4 extends into the visible, falling as 1/ω1/2. Most of the energy remaining from nuclear annihilation resides in this tail. Nevertheless, the diffuse intensity of the night sky is well above this level.

FOOTNOTES

     10 An additional contribution to Y arises as CBR photons pass through transitional regions being re-ionized but is 2 orders of magnitude smaller than the effect we discussed.

§5. THE DIFFUSE GAMMA-RAY SPECTRUM

     In this section, we use our conservative calculation of the annihilation rate to determine a lower bound to the CDG signal. We find that annihilation in a B = 0 universe produces far more γ-rays than are observed.

     The relic spectrum of γ-rays consists primarily of photons from π0 decay. Let &phis;(E) denote the inclusive spectrum in p$\mathstrut{{\ucpmathaccent{p}{"7016}}}$ annihilation, normalized to $\mathstrut{{\ucpmathaccent{g}{"7016}}}$, the mean photon multiplicity. 11 The average number of photons made per unit volume, time, and energy is Φ(E)J/d. These photons scatter and undergo redshift, leading to a spectral flux of annihilation photons F(E, y) (number per unit time, area, energy, and steradian) satisfying the transport equation,



The first term on the right-hand side is the annihiliation source, and the second is a scattering sink. We slightly underestimate F(E, y) by treating all scattered photons as effectively absorbed. In this case,



with σγ the photon interaction cross section and ne(y) the electron density. For the relevant photon energies, it matters little whether photons encounter bound or unbound electrons.

     Integration of equations (12)–(13) gives the photon flux today, F(E) ≡ F(E, 1):



     Measurements of the CDG flux are shown in Figure 5. From 2 to 10 MeV, preliminary COMPTEL measurements (Kappadath et al. 1995) lie roughly 1 order of magnitude below 12 the earlier balloon data (Fichtel et al. 1975; Mazets et al. 1975; Trombka et al. 1977; Schonfelder et al. 1980, 1993; White et al. 1977). Figure 5 also shows our computed signal F(E). The upper curve corresponds to the smallest allowed domains, d0 = 20 Mpc, the lower curve to d0 = 1000 Mpc. The signal is linear in 1/d0. The relic photon distribution is redshifted from the production spectrum (which peaks at E ∼ 70 MeV) and is slightly depleted at low energies by attenuation.

Fig. 5

     Our consevative lower limit to the γ-ray signal conflicts with observations by several orders of magnitude and over a wide range of energies, for all values of d0 &lsim; 103 Mpc, comparable to the size of the universe. We could argue that the satellite data exclude even larger domain sizes, but we would soon run into questions of the precise geometry and location of these nearly horizon-sized domains.

FOOTNOTES

     11 The measured photon spectrum can be found in Adiels et al. (1986) and Ahmad et al. (1985) and is further discussed in the Appendix.

     12 This discrepancy is attributed by Kappadath et al. (1995) to a rigidity-dependent background correction that the balloon experiments could not perform.

§6. CLOSING LOOPHOLES

     Can our "no-go theorem" for the B = 0 universe be skirted by changing the input parameters, modifying our hypotheses, or including other effects? Here we examine the sensitivity of our conclusions to the chosen value of cosmological parameters, to the possible existence of primordial magnetic fields, and to the assumed isentropic nature of primordial density fluctuations.

     We used a flat and dark matter–dominated universe with vanishing cosmological constant. For this case, the expansion rate is given by the simple expression H(y) = y3/2H0, with H0 the Hubble constant. Other choices for the cosmological parameters (Ωm ≠ 1 and/or ΩΛ ≠ 0) would alter the y dependence of H(y) as follows:



It is only through the modification of H(y) that H0, Ωm, and ΩΛ affect our results.

     We have recomputed the diffuse gamma background (CDG) for a range of observationally viable values of the cosmological parameters and are unable to suppress the signal by more than a factor of 2. The reason is easily seen. Equation (12) shows that J ∝ 1/H(y), and equation (14) shows that the CDG flux is proportional to J/H(y), and hence to H(y)-2. To suppress the flux, we must increase H(y) beyond its value at Ωm = 1, ΩΛ = 0, and h = 0.75. No sensible value of ΩΛ has much effect at y ∼ 20, when most of the CDG flux arises. For Ωm = 2 or h = 0.5, two borderline possibilities, the CDG flux would be reduced by about a factor of 2, not altering our conclusions.

     We assumed that electrons produced by annihilations travel in straight lines. This would not be true were there primordial (or magnetohydrodynamically generated) magnetic fields in the vicinity of domain boundaries. Fields with sufficiently short correlation lengths and large amplitudes would reduce the electron range. If the magnetically reduced range still exceeds D, the width of the depletion zone, the annihilation rate is increased and our conclusions are strengthened. If the electron range were less than D, electrons would deposit their energy near the annihilation zone rather than throughout the plasma. However, heating by UV photons alone results in the temperature profile plotted in Figure 2. Because J ∼ T1/2, the CDG signal cannot be reduced by more than a factor of 3 relative to our previous results. Thus, the existence of magnetic fields at or after recombination cannot alter our conclusion.

     Finally, we claimed that matter and antimatter domains must touch by recombination, if they are not to produce observable (and unobserved) scars in the CBR. Our argument depended on the absence of strictly isothermal fluctuations at recombination. If this hypothesis is false, matter and antimatter islands could be separated by regions of vanishing baryon density, with a uniform photon distribution throughout. If these isothermal voids are so wide that they persist after recombination, annihilation might be prevented. Annihilation might also be prevented by "wrapping" different regions with domain walls, the properties of which are designed to block the penetration of thermal matter while avoiding cosmological constraints (Zeldovich, Kobzarev, & Okun 1974). We have not pursued these contrived lines of thought further.

§7. CONCLUSIONS

     Neither the notion of a universe containing islands of antimatter nor the exploration of its observable consequences is new. Indeed, the literature includes diametrically opposed views as to the viability of such models. The purpose of this paper is to present a class of models (arguably, the most general) for which the observable universe consists of comparable numbers of domains containing either matter or antimatter. These models are parameterized by the typical domain size today, d0. Direct searches for annihilation radiation show that d0 > 20 Mpc, and future searches for antimatter among cosmic rays may increase this lower bound by 1 order of magnitude.

     We have found constraints on a matter-antimatter universe arising from phenomena taking place at cosmological distances. The potentially observable signals are identified as a distortion of the CBR, and the production of a relic flux of diffuse gamma rays (CDG). We have computed these signals with conservative assumptions and considerations based on empirical evidence but with as little theoretical prejudice as possible. We find that matter-antimatter encounters at domain boundaries are unavoidable from recombination to the onset of structure formation. The detailed dynamics underlying our calculation of the annihilation rate is complicated. The flow of matter into antimatter (and vice versa) is diffusive at large y and hydrodynamic at low y. Furthermore, energy deposition by the annihilation debris plays a crucial role, increasing the annihilation rate by up to 2 orders of magnitude relative to what it would have been if this effect had been neglected.

     Part of the energy released by annihilations at cosmological distances ends up as microwave photons that would appear as a nonthermal correction to the cosmic background spectrum. However, we find that measurements of the CBR spectrum do not lead to a competitive constraint on the B = 0 universe.

     High-energy photons produced by annihilations at cosmological distances (most of which survive to the current epoch) are redshifted to current energies of order 1 MeV, thereby contributing to the diffuse γ-ray spectrum. Our conservative estimate of the relic CDG flux far exceeds its measured value. Thus, we have ruled out a B = 0 universe with domains smaller than a size comparable to that of the visible universe. 13 It follows that the detection of Z > 1 antinuclei among cosmic rays would shatter our current understanding of cosmology or reveal something unforeseen in the realm of astrophysical objects.

     We would like to thank S. Ahlen, M. B. Gavela, J. Ostriker, S. Redner, F. Stecker, and S. C. C. Ting for useful conversations. This work was supported in part by the Department of Energy under grant #DE-FG02-91ER40676 and the National Science Foundation under grant number NSF-PHYS-92-18167.

FOOTNOTES

     13 Of course, it is not possible to exclude the existence of small and distant pockets of antimatter (Dolgov & Silk 1993).

APPENDIX

THE ANNIHILATION DEBRIS

     Each p$\mathstrut{{\ucpmathaccent{p}{"7016}}}$ annihilation produces $\mathstrut{{\ucpmathaccent{g}{"7016}}}$&sime;3.8 electrons and positrons and a similar number of photons. The photon spectrum has been well measured (Adiels et al. 1986; Ahmad et al. 1985) and may be used to infer the electron spectrum. The photon distribution peaks at Eγ &sime; 70 MeV, and the average photon energy is &angl0;Eγ&angr0; &sime; 180 MeV. The mean pion energy is twice that of the photon. About one fourth the energy of a charged pion finds its way to an electron. (The muon retains ∼$\mathstrut{\frac {3}{4} }$ of the charged-pion energy, of which ∼one third passes to the decay electron.) Thus, we expect an electron spectrum peaking at Ee ∼ 35 MeV with &angl0;Ee&angr0; ∼ 90 MeV.

     We must determine various properties of an annihilation electron in the redshift interval 20 < y < 1100: its mean range, its effect on the CBR, the energy it deposits in matter along its trajectory, and the ionizing effect of its passage. Three mechanisms control the motion of the electrons in the fully ionized plasma. With p = βγme and in terms of our fiducial cosmological parameters, they are the following:

     1. Cosmological redshift:



     2. Collisions with CBR photons:



     3. Collisions with ambient plasma electrons:



where ne is the position-dependent electron number density while n∞ is its value far enough from a domain boundary to be unaffected by annihilation and fluid motion.

§A1. THE RANGE OF ANNIHILATION ELECTRONS

     Annihilation electrons lose energy as they redshift, but this mechanism, given by equation (16), is negligible compared with collisional energy loss throughout the interval 20 < y < 1100. Collisions with CBR photons, given by equation (17), for which dp/dt ∝ γ2, dominate over most of the trajectory. As an electron becomes nonrelativistic, collisions with background electrons, given by equation (18), for which dp/dt ∝ 1/β2, come into play. These mechanisms cross over at β3γ2 &sime; 8/y, a point denoted by βeq(γeq). Some typical values are βeq &sime; 0.62, 0.33, and 0.19 at y = 20, 200, and 1100.

     To compute the range L(γ0, y) of an electron with initial energy γ0me, we use equation (17) throughout its trajectory and ignore the small effect of multiple-scattering corrections. For y &gsim; 20 the neglect of other energy-loss mechanisms leads to a negligible overestimate of L. Integrating equation (17), we find



For an initially relativistic electron arcsin β0 &sime; π/2, and the electron range is insensitive to the initial electron energy. The dependence of L on γ0 is hereafter suppressed.

     The previously established limit on domains of uniform composition is d(y) &gsim; 20/y Mpc. For y < 30, the electron range exceeds this minimal size and our one-dimensional approximation breaks down. Because we find a much stronger limit on the minimal domain size, this complication need not be faced. The result for the electron range, including all three sources of energy loss of equations (17)–(19), is plotted in Figure 1. Throughout the relevant redshift interval, L is small compared with the horizon.

§A2. UV PHOTONS

     We compute the spectral distortion caused by the passage of one electron (with initial energy E0 = γ0me) through a thermal bath of CBR photons. Compton scatterings conserve photon number but skew the spectrum toward higher energies. The initial spectral distribution of CBR photons is dn$\mathstrut{_{{\gamma}}}$/dω=(ω/π)$\mathstrut{^{2}}$&Nscr;(ω), with &Nscr;=1/(e$\mathstrut{^{{\omega}/T_{{\gamma}}}}$-1). Let d2N(ωf, ωi)/dωidωf denote the number of photons transferred by one electron from the frequency interval dωi to the interval dωf. Define



The function d2N(ωf, ωi)/dωfdn may be regarded as the spectral distribution of struck photons of frequency ωf produced during the voyage of one energetic electron through an isotropic, monochromatic photon gas of unit density and frequency ωi.

     Let dΩi(&thetas;i, &phis;i) be the differential solid angle about the initial photon direction and vi be the relative speed of the colliding particles. We choose to measure angles relative to the total momentum direction of the colliding particles. The function d2N/dωfdn is obtained by averaging the differential transition rate over target photon directions and integrating in time, along the electron trajectory,



where we have neglected the small effect of stimulated emission.

     The computation is simplified if we note that γTγ &Lt; me, so that the Thomson limit applies and



where r ≡ ωf/ωi and μ ≡ 1 - β cos &thetas;i.

     Carrying out the integrations in equation (21) gives our result for d2N/dωfdn. (The dt integration is most easily performed by trading dt for dp using eq. [17]. This integral extends from p0 &sime; meγ0 to peq &sime; meγeq βeq. The result is insensitive to the y-dependence of peq.)

§A3. IONIZATION

     Here we show that the fluid is almost totally ionized by annihilation electrons at all relevant times. The value of the ionization fraction, x, results from a compromise between the recombination and ionization rates. Annihiliation electrons ionize the material they traverse both directly, via electron-atom collisions as described by equation (18), or indirectly, via the UV showers discussed in Appendix A2. We discuss the latter effect, which is more important.

     Many of the photons up-scattered by annihilation electrons have energies exceeding the hydrogen binding energy (B = 13.6 eV) and can ionize hydrogen atoms via γ + H → e + p. The photoionization cross section for hydrogen atoms in their ground state, σK, falls rapidly from a very large threshold value σK(B) &sime; 8 × 10-18 cm2:



We compute the effective ionization cross section $\mathstrut{{\ucpmathaccent{{\sigma}}{"7016}}}$$\mathstrut{_{K}}$ for the entire UV shower associated with a single electron by integrating the product of σK with the photon number distribution:



This cross section is orders of magnitude larger than σK(B) and reflects the large number of photons scattered by a single electron.

     The total ionization rate is the difference of the photoionization rate and the recombination rate. The former is obtained by multiplying the effective ionization cross section for a single annihilation electron $\mathstrut{{\ucpmathaccent{{\sigma}}{"7016}}}$$\mathstrut{_{K}}$ by the flux of electrons. Because half of the e± produced in an annihilation zone move to either side, the flux is half the multiplicity $\mathstrut{{\ucpmathaccent{g}{"7016}}}$ times the annihilation rate J. The total ionization rate $\mathstrut{{\ucpmathaccent{x}{"705F}}}$ (per second and per baryon) is



where the recombination coefficient to all states, but the ground state is



The coefficient of 1 - x in equation (24) is much greater than the coefficient of x2 at all relevant epochs. Consequently, the ionization is very close to one:



In the previous argument, no allowance was made for photon absorption despite the large photoionization cross section. Because the (quasi-) equilibrium ionization is nearly total, UV photons are unlikely to encounter atoms.

     Near the region of electron production, the UV photon shower has not fully developed, so that the ionization is smaller than equation (26) indicates. Our calculation of dN/dωfdn can be modified to treat this case. We find that the UV flux near the annihilation zone is sufficient to maintain total ionization to within a few percent.

     The UV flux generated by annihilation is sufficient to prevent recombination by producing and sustaining almost total ionization. However, for large values of d0, regions lying far from domain boundaries recombine as in a standard cosmology. A moving front develops between ionized and recombined regions as the UV flux progresses. The velocity of the front is



where ξ is the ratio of the nucleon number density to that of the incident UV flux. We find vf ∼ c/3 at y = 1100 and vf ∼ c at y = 20. The intense energy deposition taking place within the front makes an unobservably small contribution to the Sunyaev-Zeldovich parameter.

§A4. ENERGY DEPOSITION

     We compute the heat function H&epsis;: the energy deposited in the plasma, per unit volume and time, by annihilation electrons and UV photons.

     In regions within the electron range, this function is dominated by the primary electron contribution. For γ > γeq, collisions with CBR photons determine the evolution of the electron velocity according to equation (17). Denoting the energy deposition to matter for this portion of the trajectory by &Escr;$\mathstrut{_{1}}$, we integrate equation (18) to find



Here L&arcmin; is the distance traveled when γ = γeq:



     Most of the remaining energy, &Escr;$\mathstrut{_{2}}$&sime;(γ$\mathstrut{_{{\rm eq}}}$-1)m$\mathstrut{_{e}}$ c$\mathstrut{^{2}}$, is deposited in matter over a relatively small distance interval. About one-third of the energy deposition to matter takes place during this short stopping stage. In the following analysis, we ignore this term, thereby underestimating electron heating by ∼30%, and slightly underestimating the production of CDG photons.

     Electrons arise as an isotropic flux from the thin annihilation zone of width A. The angular average, per electron, of the energy deposition to matter at a distance l &Gt; A from this zone is



where the integration variable is the distance traveled by an electron along its trajectory. Within the depletion zone &angl0;dE/dl&angr0;M is a slowly varying function of l that is roughly proportional to the electron density ne:



where 10 &lsim; a &lsim; 20. For our computations we use the smallest value of a.

     Half of the e± produced in an annihilation zone move to either side. Thus the e± flux is $\mathstrut{{\ucpmathaccent{g}{"7016}}}$J/2, and the electron contribution to the heat function is



The UV photon contribution is small in comparison with that of the electrons.

     Outside the electron range, only UV photons contribute to H&epsis;. In an ionizing collision, γ + H → e + p, the mean kinetic energy δE of the recoiling photoelectron is



where the cross sections and distribution function are those of Appendix § A2. The rate per unit volume of such collisions is J$\mathstrut{{\ucpmathaccent{{\sigma}}{"7016}}}$$\mathstrut{_{K}}$(1-x)n$\mathstrut{_{e}}$ $\mathstrut{{\ucpmathaccent{g}{"7016}}}$/2. Using equation (26) we express this rate as n$\mathstrut{^{2}_{e}}$&angl0;σ$\mathstrut{_{{\rm rec}}}$ v$\mathstrut{_{e}}$&angr0;. Multiplying by the mean recoil energy, we obtain the heat function,



The UV photon flux has disappeared from this expression, reflecting the quasi-equilibrium state of the ionization. As a welcome consequence, H&epsis; is insensitive to additional UV photons arising from annihilation zones other than the nearest.

REFERENCES

FIGURES


Full image (20kb) | Discussion in text
     FIG. 1.—Read from the top at large y: the horizon, the lookback size of a 20 Mpc domain, the widths of the reheated zone, the depletion zone, and the annihilation zone.

Full image (16kb) | Discussion in text
     FIG. 2.—Temperatures (in eV) as functions of redshift y = 1 + z

Full image (10kb) | Discussion in text
     FIG. 3.—Annihilation rate J (in particles cm-2) as a function of redshift y. Solid curve is our numerical solution; dashed curve is an approximate result discussed in the text.

Full image (37kb) | Discussion in text
     FIG. 4.—CBR spectral distortion. Beyond the range shown, δuγ ∝ 1/ω1/2, up to ω/T0 ∼ 104.

Full image (23kb) | Discussion in text
     FIG. 5.—Data (Kappadath et al. 1995) and expectations for the diffuse γ-ray spectrum.