Cosmology with a Master Coupling in Flipped SU(5) $\times$ U(1): The $\lambda_6$ Universe

We propose a complete cosmological scenario based on a flipped SU(5) $\times$ U(1) GUT model that incorporates Starobinsky-like inflation, taking the subsequent cosmological evolution carefully into account. A single master coupling, $\lambda_6$, connects the singlet, GUT Higgs and matter fields, controlling 1) inflaton decays and reheating, 2) the gravitino production rate and therefore the non-thermal abundance of the supersymmetric cold dark matter particle, 3) neutrino masses and 4) the baryon asymmetry of the Universe.

It is common lore that the Universe may have been in a symmetric state soon after the Big Bang, but its subsequent evolution to the present-day universe with its content of matter, dark matter and neutrinos remains problematic. Typical grand unified theory (GUT) models require many seemingly unrelated couplings to explain various physical observables. In this Letter we develop a complete cosmological scenario based on a detailed flipped SU(5)×U(1) GUT model [1,2] incorporating Starobinsky-like inflation [3], and relate a host of cosmological observables through a single master coupling, denoted by λ 6 .
In addition to quark, lepton and Higgs fields, the model contains four gauge singlets that drive inflation, provide a μ-term for the mixing of the electroweak Higgs doublets, and a seesaw mechanism [4,5] for neutrino masses. Among the superpotential couplings of the singlet fields there is one that couples the singlet, GUT Higgs fields and matter, denoted by λ 6 . Remarkably this one coupling controls 1) inflaton decays and therefore the reheat-* Corresponding author. ing temperature, 2) the gravitino production rate and therefore the non-thermal abundance of the lightest supersymmetric particle (LSP) that is a candidate for cold dark matter, 3) neutrino masses, and 4) the baryon asymmetry of the Universe through leptogenesis [6]. This Letter explores the deep correlations between these apparently disparate quantities that are all related by the master coupling λ 6 -the λ 6 Universe.
In the flipped SU(5) × U(1) GUT [7][8][9]   of the right-handed quarks and leptons are "flipped" with respect to those in standard SU (5). The minimal supersymmetric standard model (MSSM) Higgs fields H d and H u are in 5 (−2) and 5 (2) representations, denoted by h and h , respectively. The GUT gauge group is broken into the SM gauge group by 10 (1) and 10 (−1) Higgs representations of SU (5), which are denoted by H and H , respectively. The four singlet chiral multiplets are denoted φ a (a = 0, . . . , 3), and we assume that the inflaton can be identified with one of these, which we denote by φ 0 . The superpotential of this model [1] is given by where we impose a Z 2 symmetry: H ↔ −H, which forbids the mixing between the SM matter fields, Higgs color triplets, and the Higgs decuplets, and suppresses the supersymmetric mass term for H and H . Owing to the absence of these terms, rapid proton decay due to colored Higgs exchange is avoided. In addition, the doublet-triplet splitting problem is solved by the missing-partner mechanism [9,11]. Without loss of generality, we take λ is the weak scale vacuum expectation value (VEV) of h . A more detailed discussion of this model is given in Ref. [1]. In order to describe cosmology, such a supersymmetric model must be embedded in a supergravity theory, which requires the specification of a Kähler potential K . In this model K has the no-scale form [12] that emerges from string theory [13]. Denoting μ 00 = m s /2 and assuming λ 000 is the reduced Planck mass, the asymptoticallyflat Starobinsky-like potential is realized for the inflaton field φ 0 [14]. The value m s 3 × 10 13 GeV reproduces the measured value of the primordial power spectrum amplitude [1].
The inflaton φ 0 couples directly to the fields F i via the couplings λ i0 6 , which play a central role in our analysis. Two other singlet fields, φ 1 and φ 2 , also couple to F i . The remaining singlet field does not couple to F i , and develops a supersymmetry breaking scale VEV which generates a μ mixing term for the MSSM Higgs doublets. We assume that λ a 7 = 0 (a = 0, 1, 2), so as to suppress R-parity violation. This setup was introduced in Ref. [1], where it was called "Scenario B". In this case, R-parity is violated in the singlet sector, which is sufficiently sequestered from the observable sector that the LSP has a lifetime much longer than the age of the Universe [2]. A general challenge in supersymmetric GUTs is the presence of multiple degenerate vacua [15,16]. While inflation might have left the Universe in the correct vacuum state, one should follow the dynamic evolution of the universe, showing that the GUT phase transition occurred. Finite-temperature effects break the vacuum degeneracy through differences in the numbers of degrees of freedom associated with the different phases [15][16][17]. Although the global minimum generally lies in the symmetric state at temperatures of order the GUT scale, a GUT like SU(5) confines at lower temperatures T ∼ 10 10 GeV. This raises the GUT-symmetric vacuum energy, and opens the way towards successful cosmological evolution.
GUT symmetry breaking in our model occurs along one of the We focus on the portion of parameter space where the strong reheating scenario discussed in Ref. [2] is realized. As shown in Ref. [2], in this case the GUT symmetry is unbroken at the end of inflation. We further assume that the system remains in the unbroken phase during reheating, as is confirmed in the following analysis. The phase transition is triggered by the difference in the number of light degrees of freedom, g, between the broken and unbroken phases [1,2,[15][16][17]. Massless superfields provide a thermal correction to the effective potential of −gπ 2 T 4 /90, where T denotes the temperature of the Universe. Since the number of light degrees of freedom in the unbroken phase (g = 103) is larger than that in the Higgs phase (g = 62), is kept at the origin at high temperatures. However, once the temperature drops below the confinement scale of the SU(5) gauge theory, c , the number of light degrees of freedom significantly decreases (g ≤ 25), and thus the Higgs phase becomes energetically favored [1]. We have found that in this strong reheating scenario the incoherent component indicating a direct relation between T reh and λ 6 .
During reheating, gravitinos are produced via the scattering/decay of particles in the thermal bath . For the calculation of the gravitino production rate, we use the formalism outlined in [36], but using the group theoretical factors and couplings appropriate to flipped SU(5)×U(1).
These gravitinos eventually decay into LSPs, and the resultant "non-thermal" contribution to the LSP abundance is given by The total dark matter abundance is obtained by adding this nonthermal component to the thermal relic density of the LSP, which is reduced by a dilution factor . Thus the LSP relic density is also directly related to λ 6 .
The neutrino mass structure in this model was studied in Refs. [1,2]. As we noted above, only three singlet fields, including the inflaton, couple to the neutrino sector. The masses of the heavy states are approximately (m s , μ 1 , μ 2 )/2, and the mass matrix of the right-handed neutrinos is obtained from a first seesaw mechanism: where we take ν cH = 10 16 GeV in this paper. We diagonalize the mass matrix (5) using a unitary matrix U ν c : m D ν c = U T ν c m ν c U ν c . The light neutrino mass matrix is then obtained through a second seesaw mechanism [4,5]: This mass matrix is diagonalized by a unitary matrix U ν as m D ν = U * ν m ν U † ν . We note that, given a matrix λ ia 6 , the mass eigenvalues of m ν are uniquely determined as functions of μ 1 and μ 2 via Eqs. (5) and (6).
On the other hand, as discussed in Ref. [39], the PMNS matrix differs from U ν by an additional factor of a unitary matrix U l : ν . This prevents us from predicting the PMNS matrix in this framework. We note, however, that we can instead use this equation to determine U (given U PMNS ). It was found in [39] that the matrix U affects the ratios between proton decay channels; which is in general different from the ratio predicted in an ordinary SU(5) GUT. A more detailed discussion of proton decay will be given in a forthcoming paper [40].
As can be seen from Eq. (5), right-handed neutrinos become massive after H develops a VEV. In the strong reheating scenario, therefore, right-handed neutrinos are massless and in thermal equilibrium right after the reheating is completed. They become massive and drop out of equilibrium almost instantaneously at the time of the GUT phase transition and eventually decay non-thermally [2,16] to generate a lepton asymmetry [6]. The lepton asymmetry is then converted to a baryon asymmetry via the sphaleron process [41]. The resultant baryon number density is given by where [2,39] with [42] g(x) ≡ − √ It is important to note that the sign in (7) is fixed: in order to obtain n B /s > 0, we must require i i < 0.
As we see in Eqs. (3) and (4), the coupling λ 6 determines the reheating temperature and the non-thermal component of the dark matter abundance. This coupling also controls the neutrino mass and baryon asymmetry through the right-handed neutrino mass matrix in Eq. (5).
We now investigate numerically the effect of the λ 6 coupling on these physical observables. To this end, we perform a parameter scan of λ 6 . We first write it in the form λ 6 = r 6 M 6 , where r 6 is a real constant and M 6 is a complex 3 × 3 matrix. We then scan r 6 logarithmically over the range (  These predicted values are below the current limit imposed by Planck 2018 [44], i m ν i < 0.12 eV, but can be probed in future CMB experiments such as CMB-S4 [45]. Moreover, the IO case can be probed in future neutrino-less double beta decay experiments, whereas testing the NO case in these experiments is quite challenging [46]. We show in Fig. 1 the distribution of the non-thermal dark matter density produced by gravitino decays in these solutions for λ 6 . We find that many parameter solutions predict DM h 2 10 −2 for m LSP = 1 TeV, corresponding to T reh 10 12 GeV (see Eq. (4)), while some solutions yield DM h 2 10 −1 corresponding to a reheating temperature as high as T reh 10 13 GeV. In both cases, the reheating temperature is much higher than the SU(5) confinement scale c , satisfying the strong reheating condition [2].
In Fig. 2 we show the distribution of n B /s for = 10 4 , where we see that both positive and negative baryon asymmetries can be obtained. In particular, the observed value (in both magnitude and sign) of the baryon asymmetry n B /s = 0.87 × 10 −10 [44], which is shown as the vertical solid line, can easily be explained in our scenario.
In Fig. 3, we plot the non-thermal contribution to the LSP abundance from gravitino decay against the baryon asymmetry predicted at the same parameter point, assuming = 10 4 . The vertical black and horizontal green lines show, respectively, the observed values of baryon asymmetry and dark matter abundance  DM h 2 = 0.12 [44] for m LSP = 1 TeV. We find that most of the points predict n B /s O(10 −9 ) and DM h 2 O(10 −2 ), where the typical values of λ 6 are O(10 −4 ) and |μ 1 | |μ 2 | m s . The predicted value of n B /s is found to be larger than that estimated in Refs. [1,2]; this is due to an enhancement in the mass function g(x) in Eq. (9) for a degenerate mass spectrum, which was neglected in the previous estimation. On the other hand, we find many solutions where the non-thermal component of the LSP abundance from gravitino decays accounts for the entire dark matter density DM h 2 0.12. In this case, λ 6 = O(10 −3 ), and the singlet μ parameters are hierarchical, m s |μ 1 | |μ 2 |. For such parameter points, one must ensure that the thermal relic of the LSP is sufficiently depleted, which is obtained easily if ∼ 10 4 , as we have assumed.
There are also many solutions where the abundance is found to be smaller than the observed value (particularly for IO). Therefore we expect the observed dark matter abundance in these cases should be explained mainly by thermal relic LSPs. Notice, however, that the freeze-out density of the LSP can be much larger than in a standard cosmological scenario due to the presence of the dilution factor . This may revive a wide range of parameter space in supersymmetric models where the thermal relic of the LSP would otherwise be overabundant. A detailed study of this possibility will be given elsewhere [40].
In summary: we have examined the correlations between inflationary reheating, the non-thermal dark matter abundance produced by gravitino decays, neutrino masses, and the baryon asymmetry in a simple model based on a single master superpotential coupling λ 6 involving a gauge singlet, a heavy Higgs breaking the GUT gauge symmetry and the (flipped) 10 matter representation. Using the known neutrino mass-squared differences as a constraint, we find that the typical reheating temperature is 10 12 GeV and the typical baryon-to-entropy ratio lies between n B /s ∈ (10 −13 − 10 −7 ), embracing the observed value near 10 −10 .
For the preferred value of the baryon asymmetry, we find that, for NO neutrino masses, the non-thermal LSP abundance may saturate the measured relic density of dark matter, but may be significantly lower, leaving open the possibility of a dominant thermal contribution. With IO masses, the non-thermal component is typically subdominant. In this case, because of late entropy production, regions of parameter space that would yield DM h 2 ∼ 1000 in standard cosmology are preferred, opening new regions of supersymmetric parameter space for experimental searches.