Axions and the primordial monopole problem

Volume 124B, number 1,2

PHYSICS LETTERS

21 April 1983

AXIONS AND THE PRIMORDIAL MONOPOLE PROBLEM G. LAZARIDES The Rockefeller University, New York, N Y 10021, USA

and Q. SHAFI Laboratory for High Energy Astrophysics, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA and International Centre for Theoretical Physics, Trieste, Italy

Received 17 January 1983

The inevitable existence of an intermediate mass scalein axion models is used to solve the primordial monopole problem of Grand Unified Theories. It is shown that in the case where axions provide the dark matter of the Universe a measurable magnetic monopole flux may exist in our galaxy.

Recently, it has been argued [ 1] that axion models [2] with a Peccei-Quinn (PQ) symmetry broken spontaneously at the Grand Unification scale M G give rise to an unacceptably large axion density in the present universe. For the axion models to be consistent with standard cosmology, the PQ symmetry must be broken by a vacuum expectation value (VEV) which does not exceed 1012 GeV. This rules out the simplest axion models [3] with a single superheavy mass scale and necessitates the introduction of a new intermediate scale MpQ ~ 1012 GeV. The case where this bound is saturated is particularly interesting since it leads to a universe with axions providing the missing matter. Axion models based on Spin(10) and having a PQ symmetry broken at an intermediate mass scale have been constructed in ref. [4]. The unifying gauge symmetry of these models breaks down to the low energy gauge symmetry SU(3)c X SU(2)L X U(1)y via an intermediate SU(4)c X SU(2)L X U(1)R or SU(3)c × SU(2)L X SU(2)R × U(1)B_ L gauge group. These models also solve the domain wall problem discussed by Sikivie [5]. In this paper, we show that the inevitable existence of the intermediate mass scale in axion models can be exploited to resolve the long standing cosmological 26

monopole problem. In order to achieve this, the Higgs potential for the breaking of the intermediate symmetry should be chosen to be of the Coleman-Weinberg (CW) type [6]. In this case, the transition during which the intermediate symmetry breaks down to SU(3)c × SU(2)L X U(1)y takes place only after some supercooling of the universe in the false vacuum. This transition is rapidly completed at the Hawking temperature of the intermediate phase and is followed by a reheating of the universe. The density of primordial monopoles produced at the GUT transition is then diluted to a cosmologically acceptable [7], but hopefully experimentally detectable level. It is important to note that the baryon asymmetry of the universe can be produced after the monopole dilution. We now show how the above mechanism for suppressing the primordial monopole density can be implemented in the context of the Spin(10) model of ref. [4] with an intermediate gauge symmetry SU(4)c × SU(2)L X U(1)R. The gauge symmetry breaking pattern of this model is as follows: Spin (10)

45'(0), 54(0.) SU(4)c X SU(2)LX U(1)R M x ~ 1015 GeV

126(2), 45(4), 16(1)SU(3) c " fl4R~ I ~ G ~ × SU(2)L× U(1)y 0 031-9163/83/0000

(1)

0000/$ 03.00 © 1983 North-Holland

Volume 124B, number 1,2 10(-2) ,SU(3)c X O(1)e m . M w ~ 102 GeV

PftYSICS LETTERS (1 con'd)

Here the Higgs fields implementing this chain of symmetry breaking are indicated with their PQ charges in parenthesis. The Higgs 45' and 54 are real. The lefthanded fermions belong to three 16's ~b~16(i = 1,2, 3) with PQ charge 1 and two 10's ~k~0 (c~ = 1,2) with PQ charge 2. The PQ symmetry is broken at the intermediate mass scale M R ~ 1012 GeV and the axions comprise all or a significant portion of the dark matter of the universe. This model incorporates the solution of the domain wall problem [5] of axion models devised in ref. [8]. Monopoles are produced at the phase transition where the unifying gauge symmetry Spin(10) breaks down to SU(4)c X SU(2)L × U(1)R. This transition takes place at a critical temperature T c ~ Mx/g 1015 GeV (g is the gauge coupling). We will assume that the initial relative monopole density rin = (nM/T3)i n >~ 10 -9 .

(2)

Here n M is the number density ofmonopoles. Subsequent monopole-antimonopole annihilation reduces [7] the relative monopole density to r ~ 10 -9 ,

(3)

at temperatures of order 1012 GeV. The parameters of the theory can be chosen so that the zero temperature effective potential for the breaking o f S U ( 4 ) c X SU(2)L X U(1)R down to SU(3)c X SU(2)L × U ( 1 ) y is of the C o l e m a n Weinberg type, i.e., has zero curvature at the origin o f the Higgs field space. In this case, as the universe cools below a critical temperature Tcl ~ 1012 GeV, the SU(4)c X SU(2)L × U(1)R symmetric phase becomes metastable. The vacuum energy density of this phase soon dominates over the radiation energy density and the universe enters an exponentially expanding de Sitter state. Gravitational and thermal effects destabilize [9] the SU(4)c × SU(2)L X U(1)R phase at a temperature Tc2 of the order of the Hawking temperature of this phase

T H = H/2n ~ g 2 / M p ~ 10 5 GeV.

(4)

Here H is the Hubble constant of the de Sitter state a n d M e ~ 1.2 X 1019 GeV is the Planck mass. We assume, and this is apparently born out by recent calcu-

21 April 1983

lations [10], that the Higgs fields reach their SU(3)c × SU(2)L X U(1)y symmetric values within a time interval smaller than or of order H -1 , so that there is no "new" inflation [ 11 ] associated with this roll over of the Higgs fields. The transition to the SU(3)¢ × SU(2)L X U(1)y phase is rapidly completed at Tc2 and the latent heat is released. The universe reheates to a temperature T R or order Tcl ~ 1012 GeV and the relative monopole density is further reduced to

r(T R) ~ 10 -9 (Tc2/TR) 3 ~ 1 0 - 9 ( T H/Tel)3 ~ 10 -30 .

(5) This value of r is certainly consistent with the cosmological bounds [7] from nucleosynthesis and from the observed values of the Hubble constant and the deceleration parameter. The primordial monopole problem is clearly solved. Moreover, it is important to note that the predicted monopole density is just enough to sustain an anisotropic galactic monopole flux [12] at the level of the Parker bound [13] for about 1010 yr. Therefore, a monopole flux F~10

7cm 2yr-1 ,

(6)

may still exist in our galaxy. Note that this flux is six to nine orders of magnitude larger than the upper bound [ 14] on F derived from observational limits on the diffuse ultraviolet and X-ray background. This bound is based on the suggestion of Rubakov [15] and Callan [16] that monopoles catalyze nucleon decay with a typical strong interaction cross section. However, it has recently been suggested [ 17] that the Rubakov-Callan effect may be suppressed for actual grand unified monopoles. The very stringent bound on the monopole flux obtained in ref. [14] may, then, become irrelevant. It is interesting that a galactic monopole flux at the level of the Parker bound [eq. (6)] is perhaps measurable. Any baryon asymmetry produced prior to the transition to the SU(3)c X SU(2)L X U(1)y phase is diluted by about twenty one orders of magnitude during the reheating of the universe to a temperature T R 1012 GeV. The observed baryon asymmetry of the universe must then be produced after this reheating. This is possible through decays of the superheavy Higgs bosons with masses of order 1012 GeV and with direct couplings to the light ferinions (see ref. [18]). These bosons decay at temperatures of order 27

Volume 124B, number 1,2

PHYSICS LETTERS

1012 GeV. The model under consideration contains superheavy fermions with masses of order 1012 GeV. The dominant decay mode of these fermions is to a light fermion and a light Higgs boson. They do not therefore contribute to the baryon asymmetry of the universe [19]. Our mechanism for suppressing the primordial monopole density cannot be implemented in the context of the S p i n ( 1 0 ) m o d e l of ref. [4] if the intermediate gauge symmetry is chosen to be SU(3)c × SU(2)L × SU(2)R X U(1)B_ L rather than SU(4)c X SU(2)L X U(1)R. The first stage o f symmetry breaking is achieved by the vacuum expectation value of a Higgs 210 which also leaves [20] unbroken the charge conjugation operator C contained in Spin(10). This leads to the production of Z 2 strings. The operator C breaks at the second stage of symmetry breaking causing the formation of domain walls bounded by the Z 2 strings. The supercooling necessary for diluting the primordial monopoles in this model leads [20] to a gravitational collapse o f these domain walls. Thus, the cosmological monopole problem remains if Spin(10) breaks via SU(3)c X SU(2)L × SU(2)R X U(1)B_L. Note that in the model o f eq. (1) this complication does not appear since C is already broken at the first stage of symmetry breaking by the vacuum expectation value of the Higgs 45'. In particular, one can show that the stability group of C(45') and C(54> = (54) is SU(4)c X U(1)L Z SU(2)R. To summarize, we have shown that the inevitable existence o f an intermediate mass scale in axion models can be used to suppress the primordial monopole density to a cosmologically acceptable level. In the interesting case where the axions provide the missing matter of the universe, it is possible that a monopole flux at the level of the Parker bound is still present in our galaxy. Finally, we note that the mechanism suggested for solving the primordial monopole problem also can be adapted to those theories in which the intermediate mass scale is related either to some family symmetry breaking [21] and/or to local supersymmetry breaking [22]. The research o f G. Lazarides is partially supported by the Department o f Energy, Grant No: DE-AC0282ER40033.B000. Q. Shaft would like to thank NASA/

28

21 April 1983

Goddard Space Flight Center and especially F. Stecker for hospitality~

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