Astrophysical constraints on baryonic Q-balls

Volume 246, number 1, 2 PHYSICS LETTERS B 23 August 1990

Astrophysical constraints on baryonic Q-balls

Jes Madsen Institute of Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark

Received 26 April 1990

The cosmic abundances of baryonic Q-balls and similar neutron absorbers are severely constrained by big bang nucleosyn-

thesis and capture in neutron star progenitors. Astrophysical limits are orders of magnitude more restrictive than limits from

cosmic ray detectors, but some of the limits assume that Q-stars cannot sustain neutron star glitches. If Q-balls are stable

and Q-stars can explain glitches, most neutron stars are predicted to be Q-stars.

The properties of so-called non-topological solitons have recently been studied by a number of authors. Non-topological solitons are stable solutions

of classical field theories, stabilized because of a conserved quantum number carried by fields confined to a finite spatial region. Here we shall consider the

subset of non-topological solitons where the conserved quantum number is the baryon number, A.

Examples are strange quark matter [ 1,2] and baryonic

Q-balls [3-71. Non-topological solitons can play an important

role in astrophysics and cosmology. They may have formed in the very early universe and could be candi- dates for the non-luminous dark matter. However, small solitons most likely evaporated in the first hot milliseconds of history [8]. Should baryonic non- topological solitons survive to the era of nucleosyn-

thesis, they would play a role as neutron absorbers, thus reducing helium production. This can be used to constrain their cosmic abundance as described below. Another possible origin for baryonic solitons is the interior of neutron stars. Lynn and coworkers [ 5,7] have even speculated that baryonic Q-balls with stellar masses (called Q-stars when gravity is important) could be an alternative to ordinary neutron stars, just as strange stars have been proposed previously [1,9,10], but with the difference that baryonic Q-stars could have masses or rotation frequencies significantly exceeding those of neutron stars. The pur- pose of the present investigation is to point out that

some of the detector searches [ 11,121 and astrophys-

ical limits [13-151 previously used to constrain the properties of quark nuggets, in particular their cosmic abundance and local galactic flux, can be generalized

to the case of baryonic Q-balls. The astrophysical limits are applicable to any non-topological solitons capable of absorbing neutrons. This excludes major ranges of Q-ball parameters, or (in case a model for Q-star glitches can be invented) predicts that most neutron stars are Q-stars.

In the following we consider baryonic Q-balls large enough to be treated as bulk matter (A b lo’@-‘), but small enough that gravity is unimportant [A 6

1055(p/~)3’2]. The mass and radius of these systems are given by MQ = amA, and R, = PA’/‘/m = 2.1 x lo-l4 cm pA1’3, where m = 939.566 MeV is the neutron mass and (Y and p are constants (unless otherwise noted we work in units where h = c = k, =

1, and we neglect the electron mass and the neutron- proton mass difference). Defining the nucleon binding energy by I, = m + M,(A) - M,(A + 1) one sees that (Y = 1 -In/m. For stable Q-balls 0 < (Y < 1. For strange matter /3 = 5, and (Y = 1. We shall return to the values of LY and p in a particular Q-ball model later, but until then they are treated as free parameters.

Stable baryonic Q-balls consist of neutrons and protons confined by an attractive boson pressure, and electrons bound by electrostatic attraction from the protons to assure charge neutrality. As in the case of

0370-2693/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland) 135


strange quark matter, the different strength of the strong and electrostatic forces will result in a surface structure with a very sharp baryonic surface sur- rounded by a thin electron atmosphere. This makes Q-balls inert in low-energy collisions with ions, since the positively charged nucleus will see a positive electrostatic barrier when it penetrates the electron atmosphere. Neutrons are absorbed efficiently if the Q-ball state is stable.

If baryonic Q-balls form in the early universe and manage to survive high-temperature evaporation [cf. eq. (16)], they will participate in nucleosynthesis at temperatures T ~ 1 - 0 . 1 MeV as neutron absorbers [13]. For homogeneously distributed Q-balls and neutrons the rate of neutron absorption by Q-balls is given per neutron as h. = r/etYnVn, where the Q-ball number density is n Q = p e / M e , with pQ= 1.88 x 10 29 g cm-3 f2Qoh~(T/2.735 K) 3, where I2eo is the present Q-ball density in units of the critical density, ho is the Hubble parameter divided by 100 km s ~ Mpc -1, and T is the photon temperature (presently 2.735 K). The cross-section for neutron absorption is geometrical, cr. = ;rR~, and the relative velocity equals the thermal neutron velocity, v, = (2T/m) 1/2. Thus the absorption rate per neutron is

g]QO h2°/32 T7/2 s 1. (1) A~(T)= l .64x106 aAl/3 --MCV

The competing rate for beta-decay is

10.2min ~d = 1.13 x 1 0 - 3 - - s -1, (2)

T1/2

where ~'1/2 is the neutron half-life. To good approximation the production of 4He at

TN ~0.1 MeV is given by the ratio of the neutron-to- proton abundance, n/p. This is again given by the ratio at weak interaction freeze-out at TF ~ 1 MeV, and by the rates of neutron decay and absorption between TF and TN. It is clear that neutron absorption in Q-balls reduces 4He production in an unacceptable way relative to standard big bang nucleosynthesis

t N if za=-S,F Aadt~>0"l" Using the relation t = 2.42 s g-~/2T~2v, where the total statistical weight is g = 10.75 at weak interaction freeze-out, and g = 3.36 after electron-positron annihilation (below we use a "weighted mean" g-= 9g9), leads to a lower limit on

A for big bang nucleosynthesis to produce sufficient helium, namely

g2Q0h06fl 6 T3N/e). Ami n = 5.5 X 1018 og3g3/2A 3 ( T 3 F / 2 - - (3)

With Tv = 1 MeV, TN = 0.1 MeV, A = g9 ~ 1, and h0 ~> 0.5 nucleosynthesis constraints exclude Q-balls with A<_8xlO16.03Qo/36/a3 from being present during nucleosynthesis. For some choices of c~ and/3 even larger Q-balls are likely to have evaporated at earlier epochs, but this is not always the case. The nucleosynthesis bound is independent of those considerations, only assuming Q-balls to be present at weak interaction freeze-out.

Surviving Q-balls of cosmological origin could be present in our galaxy, perhaps contributing to the dark halo, and if Q-stars exist, one would also expect a background flux of Q-balls in different sizes as a result of energetic emission phenomena or collisions of stars in binaries containing Q-stars. (A system like the binary pulsar PSR 1913+16 is probably not unique; when the stars in such systems collide, they will distribute several tenths of solar masses of material in the galactic plane. Most of this will be in the form of Q-balls, if one of the stars consist thereof.)

For Q-balls in a monoenergetic distribution the number accretion rate on stars with mass M , and radius R , is

F = 27rGM2,n~v2~ 1 R , - (1 + v~R, ~, (4) M, \ 2M, G]

where v~o and n~ = p~o/Me are the characteristic speed and number densities far from the accreting star. With v~-=V2so250kms -1 and p~---p2410-24gcm 3, and neglecting the second (geometrical accretion) term in the parentheses, one gets a flux

M, R , (5) F = 1.39 x 1030 s -I a - l A IP241~210 Mo Ro"

In time t a star is hit by at least one Q-ball if Q-balls with baryon number A ~< A~ contribute with density P24, where [14]

Al = 4 . 4 x 10 37 t O~ Ip24/)2510 M , R , 1 yr M® R®" (6)

Some of these Q-balls penetrate the star while others are stopped and brought to rest in the stellar center due to electrostatic scattering off the electron atmo-

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sphere. For small Q-balls the effective electron cloud radius equals 10-Scm, whereas Q-balls with A ~ 1017/3 -3 have electrons out to an electron Compton wavelength (of order 4 x 10 - " cm) above the baryon surface. In the latter case the effective radius for electrostatic scattering equals the baryonic radius, RQ, quoted previously. The cross-section for electrostatic scattering on the Q-ball electrons is therefore

o" e = 1.39 x 10-27/32A 2/3 cm 2, A ~ 1017/3 -3,

=3.14× 10 - '6 cm 2, A<~ 1017/3 -3. (7)

A Q-ball is s topped if it sweeps up a mass comparable to or larger than its own mass [11,14]. The mass Msweep removed by a Q-ball traversing distance x through a medium with density p(x) is given by

M, weep= i o-ep(x) dx = o-~D(x) (8)

0

where the column density D(x) =- ~o p(x) dx. Msweep exceeds MQ(A) for A < A~op, where

5 Astop = 5 . 7 X 10 -1° D 3, A ~ 1 0 ' 7 / 3 - 3 ,

= l . 9 x 108c~ 1D, A ~ < 1017/3 -3. (9)

Gravi ty will accelerate a Q-ball approaching a stellar surface to a speed ( 2 G M , / R , ) '/2, which in the cases of interest below exceeds the typical halo speed (V25o = 1). Thus it is a good approximat ion to assume that Q-balls penetrate stars on radial orbits. Moving a distance 2 R , through a star the Q-ball will encoun- ter a total column density D = yM,/R2., where y = 5.0 for an n = 3 polytrope (a reasonable model for a star on the upper main sequence).

Figs. 1 and 2 show the Q-ball baryon numbers for which at least one Q-ball is brought to rest in stars at different stages of evolution [the limits on A given by min(A1, mstop ) ] for two sets of parameters, (a , /3) = (1,4.34) and (0.0185, 100). (In the chiral model described below the first set corresponds to a mar- ginally bound Q-ball, whereas the binding corresponding to the second set is slightly larger than permit ted by white dwarf observations.) For upper main sequence stars the radius has been assumed to

0.8 scale as R , = M , , and the main-sequence lifetime was set to 3.7x 109(M,/M®) -L9 yr. Capture in the

post-main-sequence giant phase was supposed to be governed by a 1.4M® core with radius 10-2Ro. Cap- ture in neutron stars may be possible in the molten phase, i.e. the first few months after the formation in a supernova explosion. After solidification of the crust only extremely large Q-balls (A ~> 1032fl 6) have a chance of being gravitat ionally dragged through the resisting lattice, whereas smaller chunks are caught in the crust. Q-balls hit neutron stars with kinetic energies (due to the large surface gravity) of order 140c~ MeV per baryon. The collision is therefore highly inelastic, and it is not obvious that they survive the heating. Therefore neutron star capture has not been included in figs. 1 and 2. The reader is referred to refs. [14,15] where these issues are further discussed in the case of quark nuggets, and where it is shown that capture in neutron stars may be of

40~ / /

30

20

/ /

/ /

/

/ /MS

Y / / / /

/ ./

/ / , 13

/

/ /

/ z

/I

10~ / L_./~'_ _/_ E

/

//BBI~

-20 -10 0

Fig.1. Constraints on Q-ball baryon number versus contribution to galactic halo density for (a,/3) = (1, 4.34) (full lines) and (0.0185, 100) (dashed lines). Capture in upper main sequence stars with masses of 10Mo is denoted by MS, and in giant stars by G. Limits from detector searches [12] (plotted for V25o = 1) are denoted by E, and BBN are the constraints from big bang nucleosynthesis assuming a Q-ball density of g/QO fl24 in the halo.

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\ \ \

\ \ * \ \

E -20 , "~. ~ , \

),( .9. =

o

--~-30

-40

I I I 0 10 20 30 40 Iog~o A 5 0

. . . . \

I \ i \ I \ M S _ j

\

Fig. 2. Limits on the flux of Q-balls reaching the earth [accretion rate given by the geometrical term in eq. (4)]. Astrophysical limits assume v250 = 1. The diagonal curves are the upper flux limits corresponding to the galactic dark matter. The notation is similar to fig. 1.

relevance for a l imited range of parameters near A ~ 1032~ 6.

For compar ison the figures show the flux ranges ruled out by the best ground- and space-based detector searches (derived from the compila t ion of quark nugget limits given by Price [12]). The stellar capture limits can be considered as significant improvements of these ruled-out regions for the reasons descr ibed now. Like strange stars [1,9,10], baryonic Q-stars can sustain a t iny crust of ordinary nuclei because of the electrostatic surface potential . The crust can, however, only reach densities below that of neutron drip (4.3 x 10 it g cm-3), because free neutrons will be absorbed by the stable Q-ball phase. The crust will consequently be much too small to explain neutron star glitches, which are assumed to involve a restruc- turing of the solid lattice in neutron star crusts. Con- sequently (if no al ternative glitch model is invented for Q-stars) glitching pulsars must be ordinary neutron stars, not Q-stars. Since the presence of a tiny lump of Q-ball at neutron star formation will lead to conversion of the whole neutron star into the conjectured lower energy Q-state, one may exclude the presence of such chunks in the progenitors of

pulsars like Crab and Vela. (The Q-balls do not evap- orate at the temperatures and densities in the central regions of a supernova explosion). This excludes the parameter ranges shown in figs. 1 and 2. Q-balls with A<~8x 1025fl6/O~ 3 (~---10 30 and 10 43 in the two cases

shown in the figures) are therefore not significant contr ibutors to the galactic halo density. Indeed, even a single Q-star collision releasing a fraction of the solar mass in Q-balls into the galactic disk would lead to fluxes significantly exceeding the very strin- gent limits for A ~ < 1026f l6 /0 '3. Since it seems hard to avoid spreading of Q-bails in our galaxy if baryonic Q-stars exist, this is an argument against the existence of such systems.

The use of pulsar progenitors as Q-ball detectors depends on the assumed inabil i ty of Q-stars to explain pulsar glitches. If they can explain these phenomena , then the arguments above may be turned a round to predict that if Q-stars and Q-balls are stable, then it is very likely that nearly all neutron stars are in fact Q-stars. The argument being that it is hard to avoid spreading of Q-balls in the galactic disk.

Finally let us look at the Q-ball parameters a and /3 in the simple chiral model discussed in ref. [5].

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This model gives the Q-ball equation of state as

E - 3 P - 4 U o + Crv(e - P - 2 Uo) 3/2 = 0, (10)

where av is a vector repulsion strength given by Walecka 's mean field theory by fitting to nuclear propert ies [16], Uo is a scalar potential , the value of which can be chosen freely in the model, and energy density • and pressure P are given by

4 4 4 2 3 3 2 k F , n + k F , p q - k F , e + l ' v (kF,n+kF,p'~

• = 4,rr 2 2 m 2 \ 371.2 ,]

+Uo, (11)

4 4 k 4 ~_lg2v___ 3 3 2 p _ k F . n + k F . p + F,e 12~r 2 2 m E \ 3~r 2 ]

- U 0 . (12)

Here kF, i are the Fermi momenta of neutrons, protons, and electrons, and gv and my denote vector boson coupling and mass respectively, related to the repulsion strength by av--- 3t/E(gv/mv)2/77". Walecka 's fit

to the propert ies of nuclei [16] gives av = 1.23 x 10 -4 MeV -2. Bahcall et al. [5] investigate the

propert ies of Q-stars descr ibed by eqs. (10)-(12) by solving the Oppenhe imer -Volkov equation. Here we concentrate on Q-balls with negligible gravity, and may thus set P = 0 . This gives a baryon number density

- k3F'n+k3F'p-- - ( 3 ' / 2 ( 4 U ° - • ) ) 1/2 (13) na 3 ~r 2 7ray '

and the Q-ball mass- and radius-scaling parameters

a and /3 are given by

a = (14) m n B

/3 = m \4~rnB/ " (15)

The relat ions between Q-ball parameters are shown in fig. 3. Q-ball stability without gravity requires that Uo<(173 MeV) 4 to have a < 1. With gravity (or in other models with different parameter fits), larger values of Uo are permitted, cf. the Q-star model with Uo = (200 MeV) 4 described in ref. [5]. Note that /3 is a decreasing function of a with minimum value/3 = 4.34 for a = 1 (this set of parameters closely resembles that for strange quark matter; for a v = 0 one can

10 3

10 ~

lO

i . . . . . . . . . , . . . . . . . , . . . . . . . .

, L , [ . . . . . . . . I . . . . . . . 1 lO-3 10-2 10-1 o.

Fig. 3. Relations between Q-ball parameters a,/3 and U 1/4 (in MeV) in the chiral model of ref. [5], and the baryon number, A~ . . . . of Q-balls surviving surface evaporation in the early universe.

interpret Uo as a dynamical ly given bag constant in the MIT bag model) .

Also shown in fig. 3 is the minimum baryon number of Q-balls that can survive surface evaporat ion of nucleons starting at a temperature of 50 MeV in the early universe, calculated according to the eq. (8)

Asurv ~ 1053 f16 exp[ -3 (1 - a ) m / T ] (16) [l + ( l - a ) m / T ] 3

Eq. (16), where T denotes the evaporat ion starting temperature, is likely to overestimate the effect of evaporat ion because it does not take account of absorpt ion and problems related to nucleon t ransport away from the surface. One notes from fig. 3 that /Lurv stays close to its minimum value of 1035 over a significant range of values for a and/3. For /3 = 100 (one of the examples used in figs. 1 and 2) A . . . . = 1037, which is 6 orders of magni tude below the A-l imit for

139


main sequence capture. In genera l the as t rophys ica l

l imits on the Q-bal l a b u n d a n c e are most restr ict ive

for low c~, where the cosmic survivabi l i ty chance is

largest !

In conc lus ion , big bang nuc leosynthes i s and the

con jec tu re that pu lsar gl i tches involve sol id crusts

unob ta inab l e a round Q-stars severely l imit the cosmic

a b u n d a n c e and local ga lac t ic flux o f ba ryon ic Q-bal ls

in a m a n n e r s imilar to the case o f s t range quark

nuggets . A n a l o g o u s results will app ly to o the r non-

topo log ica l sol i tons capab le o f neu t ron absorp t ion .

Gene ra l l imits on the a l lowed ba ryon number s were

expressed in terms of mass- and radius-sca l ing param-

eters, and the re la t ions a m o n g these pa ramete r s were

i l lus t ra ted in the case o f the chiral m o d e l o f ref. [5].

Shou ld Q-bal ls turn out to be stable, and a m o d e l is

f ound for Q-s tar gl i tches, then the limits shou ld be

in te rpre ted as p red ic t ing that a lmos t all neu t ron stars

are in fact Q-stars. Even in this case eq. (3) exc ludes

signif icant p a r a m e t e r ranges no t p robed by de tec to r

searches.

I thank S tephen Sel ipsky for in t roduc ing me to

ba ryon ic Q-bal ls , and for p rov id ing useful c o m m e n t s

toge ther with Bryan Lynn.

References

[1] E. Witten, Phys. Rev. D 30 (1984) 272. [2] E. Farhi and R.L. Jaffe, Phys. Rev. D 30 (1984) 2379. [3] B.W. Lynn, Nucl. Phys. B 321 (1989) 465. [4] S. Bahcall, B.W. Lynn and S.B. Selipsky, Nucl. Phys. B

325 (1989) 606. [5] S. Bahcall, B.W. Lynn and S.B. Selipsky, Nucl. Phys. B

331 (1990) 67. [6] B.W. Lynn, A.E. Nelson and N. Tetradis, Stanford pre-

print Stanford-ITP-860 (1989). [7] S. Bahcall, B.W. Lynn and S.B. Selipsky, Stanford preprint

SU-ITP-866 (1989). [8] J.A. Frieman, A.V. Olinto, M. Gleiser and C. Alcock,

Phys. Rev. D 40 (1989) 3241. [9] P. Haensel, J.L. Zdunik and R. Schaeffer, Astron.

Astrophys. 160 (1986) 121. [10] C. Alcock, E. Farhi and A. Olinto, Astrophys. J. 310 (1986)

261. [11] A. DeRujula and S.L. Glashow, Nature 312 (1984) 734. [12] P.B. Price, Phys. Rev. D 38 (1988) 3813. [13] J. Madsen and K. Riisager, Phys. Lett. B 158 (1985) 208. [14] J. Madsen, Phys. Rev. Lett. 61 (1988) 2909. [ 15] J. Madsen, Quark nuggets, dark matter and pulsar glitches,

in: Proc. IXth Moriond Astrophysics Meeting (Les Arcs, March 1989), eds. J. Audouze and J. Tran Thanh Van (Editions Fronti6res, Gif-sur-Yvette, 1990) p. 119.

[16] B.D. Serot and J.D. Walecka, in: Advances in nuclear physics, Vol. 16, eds. J.W. Negele and E. Vogt (Plenum, New York, 1985) p. 1.

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Astrophysical constraints on baryonic Q-balls