Primordial monopoles, proton decay, gravity waves and GUT inflation

We consider non-supersymmetric GUT inflation models in which intermediate mass monopoles may survive inflation because of the restricted number of e-foldings experienced by the accompanying symmetry breaking. Thus, an observable flux of primordial magnetic monopoles, comparable to or a few orders below the Parker limit, may be present in the galaxy. The mass scale associated with the intermediate symmetry breaking is $10^{13}$ GeV for an observable flux level, with the corresponding monopoles an order of magnitude or so heavier. Examples based on $SO(10)$ and $E_6$ yield such intermediate mass monopoles carrying respectively two and three units of Dirac magnetic charge. For GUT inflation driven by a gauge singlet scalar field with a Coleman-Weinberg or Higgs potential, compatibility with the Planck measurement of the scalar spectral index yields a Hubble constant (during horizon exit of cosmological scales) $H \sim 7$--$9\times10^{13}$ GeV, with the tensor to scalar ratio $r$ predicted to be $\gtrsim0.02$. Proton lifetime estimates for decays mediated by the superheavy gauge bosons are also provided.


Introduction
The observed quantization of electric charge is elegantly explained by invoking the presence of magnetic monopoles, as shown by Dirac more than eighty years ago [1]. Contemporary unified theories with electric charge quantization based on groups such as SU (4) × SU (2) L × SU (2) R [2], SU (5) [3], SO (10) or E 6 , predict the existence of topologically stable magnetic monopoles [4], and one expects that these monopoles are produced in the early universe.
Despite the presence of fractionally charged quarks, the lightest SU (5) monopole carries a single unit of Dirac magnetic charge. This comes about because the unbroken gauge symmetry SU (3) c × U (1) em share a Z 3 symmetry [5]. The SU (5) monopole ends up carrying some color magnetic flux that is screened due to color confinement. The SU (5) monopoles are superheavy with a mass about an order of magnitude larger than M GUT ∼ 2 × 10 16 GeV.
In non-supersymmetric GUTs such as SO (10) broken to the Standard Model (SM) via G 422 = SU (4) c × SU (2) L × SU (2) R , there appears a new scenario for monopole charges and masses. The SO(10) breaking to G 422 yields, just as in SU (5), a superheavy monopole with a single unit of Dirac magnetic charge [6]. The subsequent breaking of G 422 at some intermediate mass scale M I yields monopoles that carry two units of Dirac charge and mass that can be a few orders of magnitude smaller than the mass of the SU(5) monopole [6].
It was argued a long time ago by Lazarides and Shafi [7] that within the framework of GUT inflation driven by a gauge singlet scalar inflaton field [8], these somewhat lighter monopoles may not be entirely inflated away. 1 The superheavy monopoles produced during the first stage of symmetry breaking experience at least the 50-60 e-foldings of observable inflation. The somewhat lighter monopoles, produced during the intermediate symmetry breaking with mass determined by M I and comparable to the Hubble constant H during inflation, may undergo a significantly reduced number of e-foldings. Therefore, there arises the exciting possibility that these monopoles, lighter than M GUT , may be present in our galaxy at an observable number density, comparable to or a few orders of magnitude below the Parker bound [10].
In recent years the WMAP [11] and Planck [12,13] satellite experiments have provided a fairly accurate determination of the scalar spectral index n s and an upper bound for the tensor to scalar ratio r 0.1. In the framework of GUT inflation driven by a gauge singlet scalar field, one finds that for n s ≥ 0.96, the energy scale during inflation is of order 10 16 GeV [14,15].
In this brief report we calculate the range of energy scales during non-SUSY GUT inflation such that n s and r are compatible with the Planck 2015 constraints [12]. This determines the magnitude of H which, in turn, provides an estimate for the range for M I that is compatible with an observable flux of primordial magnetic monopoles. These monopoles with mass ∼ 10 14 GeV do not necessarily catalyze nucleon decay with a strong interaction rate, and they should be accessible to current and future large scale detectors. Estimates for the proton lifetime are also provided.

Inflation with Coleman-Weinberg potential
The first new inflation models [16] were proposed in the early eighties immediately after Guth's seminal paper [17]. They were based on SU (5) GUT, with symmetry breaking due to the Coleman-Weinberg mechanism [18] occurring in the adjoint Higgs field. However, it was shown in ref. [8] that obtaining sufficiently small density perturbations was only possible if the scalar field φ is a gauge singlet. In these Shafi-Vilenkin type models the field φ has a quartic potential at tree level, and taking into account radiative corrections the potential becomes (omitting terms that don't play an essential role) [19,20]: where χ represents the field breaking the GUT group, A ∼ (1/16π 2 )β 4 , M and C are normalization parameters and V 0 ≡ V (φ = 0) is the vacuum energy density at the origin. The χ field can be replaced by its vacuum expectation value (VEV) χ = (β/ √ a)φ. The parameter C can be fixed by taking M to be the φ VEV at the minimum. Requiring V (φ = M ) = 0 fixes V 0 = AM 4 /4. Also taking λ β 4 /a, the effective potential takes the standard form [19,20]: The inflationary predictions of this potential were recently analyzed in ref. [14] (see also refs. [21,15]). The magnitude of A and the inflationary parameters can be calculated using the standard slow-roll expressions. The slow-roll parameters may be defined as (see ref. [22] for a review and references): Here and below we use units m P = 2.44 × 10 18 GeV = 1, and primes denote derivatives with respect to the inflaton field φ. The spectral index n s , the tensor to scalar ratio r and the running of the spectral index α ≡ dn s /d ln k are given in the slow-roll approximation by The amplitude of the curvature perturbation ∆ R is given by which should satisfy ∆ 2 R ≈ 2.4 × 10 −9 from the Planck measurement [12] with the pivot scale chosen at k * = 0.002 Mpc −1 . 2 The number of e-folds is given by where the subscript " * " denotes quantities when the scale corresponding to k * exited the horizon, and φ e is the inflaton value at the end of inflation, which we estimate by (φ e ) = 1. For V 1/4 0 2 × 10 16 GeV, observable inflation occurs close to the minimum where the potential is effectively quadratic (V 2AM 2 χ 2 , where χ = φ − M denotes the deviation of the field from the minimum). The inflationary predictions are thus approximately given by (2.7) For V 1/4 0 10 16 GeV, assuming inflation takes place with inflaton values below M , the inflationary parameters are similar to those for new inflation models with Note that in the context of non-supersymmetric GUTs, V 1/4 0 is related to the unification scale M U , and is typically a factor of ∼ √ 4π smaller than the superheavy gauge boson masses due to the loop factor in the Coleman-Weinberg potential. The allowed range of V 1/4 0 (and hence of M U ) can be calculated by comparing the n s and r values with the Planck results [12]. Since the resulting constraints depend sensitively on the number of e-folds N , instead of fixing N to a fiducial value, we calculate it using Here ρ e = (3/2)V (φ e ) is the energy density at the end of inflation, ρ r is the energy density at the end of reheating and ω r is the equation of state parameter during reheating, which we take to be constant. For a derivation of eq. (2.8) see e.g. ref. [23].
To represent a plausible range of N , we consider three cases: In the high-N case ω r is taken to be 1/3, which is equivalent to assuming instant reheating. In the middle-N case we take ω r = 0 and the reheat temperature T r = 10 9 GeV, calculating ρ r using the SM value for the number of relativistic degrees of freedom (g * = 106.75). In the low-N case we take T r = 100 GeV (again with ω r = 0). 3 The n s vs. r curve for each case is shown in Figure 1 along with the contours (at the confidence levels of 68% and 95%) given by the Planck collaboration (Planck TT+lowP+BKP+lensing+ext) [12]. Numerical results for selected values of V 0 and the middle-N case are displayed in Table 1.

Inflation with smeared Higgs potential
A generalization of the model considered in section 2 is to take a tree-level Higgs potential Higgs potential and the Coleman-Weinberg potential considered in section 2 [24]: Following ref. [24], we will call V (φ) in eq. (3.1) the smeared Higgs potential. Depending on the value of m, the inflationary predictions for this potential interpolate between the predictions for the tree-level Higgs potential and for the Coleman-Weinberg potential [24]. The tree-level Higgs potential has been analyzed in several papers, see e.g. refs. [25,21,15]. The inflationary predictions are similar to the predictions for the Coleman-Weinberg potential. For V   At the end of inflation, the GUT symmetry breaking fields have a VEV χ ∼ (β/ √ a)M . Taking a ∼ g 2 , where g is the gauge coupling, the unification scale is given by The n s vs. r curves for tree-level and smeared Higgs potentials (for x = 0.25 and 0.1) are shown in Figure 1. Numerical results for selected values of V 0 and the middle-N case are displayed in Table 1.

Magnetic monopoles and proton decay in non-supersymmetric GUTs
The models discussed in sections 2 and 3 can be realized within the framework of nonsupersymmetric GUTs such as those based on SO(10) as well as SU (5), as discussed in ref. [7]. The breaking of SO(10) to the SM can proceed, for example, via the intermediate group  [6]), whose mass is an order of magnitude larger than M I . These may be present in our galaxy at a flux level that depends on the values of V 0 and M I . Below we will estimate the M I scale that corresponds to an observable flux level following the arguments in Ref. [7]. First let us consider the potential for the χ fields breaking the GUT group to the intermediate group, at scale M U . The potential involves a thermal term where T H ≡ H/2π is the Hawking temperature and the coefficient σ χ ∼ 1. Thus symmetry breaking occurs when β 2 φ 2 (H/2π) 2 , and topological structures are "frozen" in soon afterwards [26]. It can be easily checked that this happens much earlier than the horizon exit of cosmological scales, so as mentioned above any such topological structures are inflated away.
Let X denote the fields whose VEV breaks the intermediate group to the SM at the scale M I . This breaking occurs due to the coupling −(1/2)c 2 φ 2 X 2 , where c ∼ M I /M . Thus, symmetry breaking occurs and subsequently the monopoles are "frozen" in when where If M is small compared to the Planck scale, the inflaton φ is essentially constant until almost the end of inflation, rolling quickly only within the last H −1 [20]. This means for substantial dilution of the monopoles, φ x should be very close to φ * . On the other hand, if M is large compared to the Planck scale both φ * and φ x values will be close to M . Since the Planck constraint on n s is only satisfied for this latter case, we have φ x ≈ M and therefore M I ∼ H x /2π ∼ 10 13 GeV.
To be more specific, let's consider how much dilution of the monopoles is necessary. M I ∼ 10 13 GeV corresponds to monopole masses of order M M ∼ 10 14 GeV. For these intermediate mass monopoles the MACRO experiment has put an upper bound on the flux of 2.8 × 10 −16 cm −2 s −1 sr −1 [27]. For monopole mass ∼ 10 14 GeV, this bound corresponds to a monopole number per comoving volume of Y M ≡ n M /s 10 −27 [28]. There is also a stronger but indirect bound on the flux of (M M /10 17 GeV)10 −16 cm −2 s −1 sr −1 obtained by considering the evolution of the seed Galactic magnetic field [29].
Direct search bounds stronger than the MACRO bound were obtained in refs. [30], but these apply to monopoles that catalyze nucleon decay through the Callan-Rubakov process [31]. There are even more stringent indirect bounds from compact astrophysical objects capturing monopoles [32]. However, the monopoles produced during the intermediate symmetry breaking stage do not necessarily catalyze nucleon decay (at least, not with a strong interaction rate) [33]. This improves the chances of directly observing such monopoles in the future, since the bounds from compact astrophysical objects are avoided. The region between the dashed vertical lines correspond to n s and r values within the 95% confidence level contours given by the Planck collaboration (Planck TT+lowP+BKP+lensing+ext) [12].
At production, the monopole number density n M is of order H 3 x [26,7], which gets diluted to H 3 x e −3Nx , where N x is the number of e-folds after φ = φ x . Using where s = (2π 2 g S /45)T 3 r , we find that sufficient dilution requires N x ln(H x /T r )+20. Thus, for T r ∼ 10 9 GeV, N x 30 yields a monopole flux close to the observable level.
Using eq. (4.3), we calculate φ x , H x and N x values, denoted with subscripts " + " and " − ", corresponding respectively to the flux levels 2.8 × 10 −16 cm −2 s −1 sr −1 which is the MACRO bound and 10 −24 cm −2 s −1 sr −1 which we take as a rough threshold for observability. Then using eq. (4.2), we calculate the corresponding M I values, which are shown in Figure  2 and Table 2. For M I M − , the monopoles are too diluted to be observable, whereas M I M + is excluded from the bound on the flux.
Another key prediction of GUTs besides magnetic monopoles is proton decay. The proton mean life can be estimated as  where M U is estimated using eq. (3.2), m pr is the proton mass, and α G ∼ 1/40 is the GUT coupling constant. Using eq. (4.4), the experimental bound τ p (p → e + π 0 ) > 8.2 × 10 33 years [34] corresponds to M U 4 × 10 15 GeV, whereas a realistically observable τ p (p → e + π 0 ) = 10 35 years [35] corresponds to M U ≈ 8 × 10 15 GeV. Since the Planck constraint on n s is only satisfied for V 1/4 0 10 16 GeV, a glance at eq. (3.2) shows that proton decay is typically too slow to be observed in this class of models, unless the smearing parameter x is close to zero. Proton lifetime estimates are displayed in Figure 2 and Table 2.

Conclusion
In this paper we discussed a class of models where the gauge-singlet inflaton has either a quartic or Higgs potential at tree level. The radiative corrections due to couplings with GUT symmetry breaking fields modify the tree level potential into a Coleman-Weinberg or smeared Higgs potential. If the GUT symmetry breaking to the SM proceeds via an intermediate group, the breaking of the intermediate group to the SM gauge symmetry at intermediate scale M I yields monopoles whose mass is an order of magnitude larger than M I . These may be present in our galaxy at a flux level that depends on the values of V 0 (the vacuum energy density at the origin) and M I .
For both Coleman-Weinberg and smeared Higgs potentials the Planck constraint n s > 0.955 is only satisfied for V 1/4 0 10 16 GeV, which implies r 0.02, a level which can be probed in this decade. Another consequence of the Planck constraint is that an observable level of monopole flux can only occur for M I ∼ 10 13 GeV, with lower values excluded due to excessive monopole flux. Thus, a smoking gun evidence for this class of models would be the observation of monopoles with masses of order 10 14 GeV, together with the observation of a B-mode CMB polarization signal corresponding to r 0.02.
The lower bound on M I poses a severe constraint for SO(10) broken to SM via G 422 , since the typical values obtained from the RG analysis is M I ∼ 10 11 GeV and M U ∼ 10 16 GeV [36]. However, taking threshold effects due to the Higgs sector into account, it is possible to achieve M I as high as 3 × 10 13 GeV with M U ≈ 4 × 10 15 GeV [36,37]. 4 Here the lower bound on M U follows from proton decay. Although for Coleman-Weinberg potential the Planck constraint on V 0 corresponds to M U 4 × 10 16 GeV, lower M U values are possible for the smeared Higgs potential. If the smearing parameter x is close to zero, proton decay could also be observed in this class of models.
Finally we note that E 6 breaking via SU (3) c × SU (3) L × SU (3) R can yield intermediate mass monopoles carrying three units of Dirac charge [39].