Constraining Inflationary Dark Matter in the Luminogenesis Model

Using renormalization-group flow and cosmological constraints on inflation models, we exploit a unique connection between cosmological inflation and the dynamical mass of dark-matter particles in the luminogenesis model, a unification model with the gauge group $SU(3)_C \times SU(6) \times U(1)_Y$, which breaks to the Standard Model with an extra gauge group for dark matter when the inflaton rolls into the true vacuum. In this model, inflaton decay gives rise to dark matter, which in turn decays to luminous matter in the right proportion that agrees with cosmological data. Some attractive features of this model include self-interacting dark matter, which may resolve the problems of dwarf-galaxy structures and dark-matter cusps at the centers of galaxies.


Introduction
Dark matter comprises a large portion of our universe, and yet it still eludes our full comprehension. There are many unsolved mysteries about the nature of dark matter, such as the "missing satellite" problem and the dark-matter cusps at the centers of galaxies found in simulations but not in observations.
What if all matter originally came from the dark sector? One possible way to combine dark matter and the Standard Model is via the luminogenesis model [1,2]. It is a model that undergoes the symmetry breaking SU (3) C × SU (6) × U (1) Y → SU (3) C × SU (4) DM × SU (2) L × U (1) Y × U (1) DM at the DUT (Dark Unified Theory) scale. This breaking occurs when the inflaton slips into the true vacuum of its potential. The inflaton decays to dark matter, while decay to luminous matter is suppressed at the tree level. Below the luminogenesis scale M lum , dark matter decays to radiation and mirror matter [3] (which will be discussed below), and almost all mirror fermions decay to standard fermions. Freeze-out of these conversions occurs at high energy scales, leaving standard Big Bang Nucleosynthesis unaffected. In this model, SU (4) DM is unbroken and is confining at a scale much smaller than the DUT scale. This is the main emphasis of this manuscript.
Dark matter in this model is self-interacting, which may resolve the discrepancies between observations and simulations mentioned above. Another attractive feature of this model is the absence of proton decay, which may explain its lack of observation.
In this paper, we explore the interesting and unique connection between cosmic inflation and dark matter. We first analyze renormalization-group flow from the DUT scale to the confinement scale of SU (4) DM , which provides the dynamical mass of the dark-matter particles. We obtain the SU (4) DM coupling at the DUT scale from the unification with the SU (2) L coupling at the DUT scale, which we get from running the SU (2) L coupling from the electroweak scale to that scale. Then we examine constraints on the DUT scale based on constraints on inflation from Planck for a symmetry-breaking (Coleman-Weinberg) inflation potential and obtain constraints on the dark-matter dynamical mass. Thirdly, although there are serious reservations concerning whether BICEP2's measurement is due mainly to dust foregrounds or an authentic signal of gravitational waves, we discuss briefly our ability to accommodate BICEP2's measurement for the tensor-scalar ratio r and the implications for the dark-matter dynamical mass in our model.

Predictions for the Dark-Matter Dynamical Mass from RG Flow
Combining renormalization-group (RG) flow with constraints on cosmic inflation from Planck [4,5], we can make a prediction for the dark-matter dynamical mass. As discussed in [1,2], SU (6) → SU (4) DM × SU (2) L × U (1) DM . The β-function equation for SU (N ), ignoring the negligible term for the contribution due to scalar fields, is where the second and third terms correspond to the contributions from left-handed and right-handed fermions respectively. The representations and group structure of the luminogenesis model for each of the three families are given below [2]. In passing, the existence of mirror fermions, as proposed by [3], provides a mechanism in which right-handed neutrinos obtain Majorana masses proportional to the electroweak scale, and they could be searched for at the Large Hadron Collider. 3   TABLE I. (1, 2) 2 represents luminous matter while (4, 1) 3 + (4 * , 1) −3 represent dark matter.  The inflaton is represented by (1, 1) 0 of 35. It is assumed that (15, 1) 0 + (1, 3) 0 +(4, 2) −3 + (4 * , 2) 3 of 35 have masses that are on the order of the DUT scale and are therefore not included in our analysis of RG flow. The inflaton decays to dark matter through a coupling via 20 × 20 = 1 s + 35 a + 175 s + 189 a , while decay to luminous matter is suppressed at the tree level. Dark matter can decay to luminous matter through a coupling via 20 ×6 = 15 + 105 and 20 × 6 =15 +1 05 and through the massive gauge boson of U (1) DM , the dark photon. More details are in [2].
The SU (4) DM dark matter fermions are represented by (4, 1) 3 + (4 * , 1) −3 in the 20 representation of SU (6). The (6, 2) 0 part of 20 is assumed to decouple below its mass scale M 2 . Since dark matter should have no U (1) Y charge, the SU (4) DM particles in (4, 1) −1 in the 6 representation of SU (6) cannot be dark matter, and they are assumed to decouple below the mass scale M 1 .
For SU (2) L , the Casimir factor for the representation R for all the right-handed and left-handed contributions is C(2) = 1 2 . The numbers of right-and left-handed fermions are n f R = n f L = (3 + 1)3 = 12 for scales µ < M 2 because, using Tables I and II, we see that (3, 6, 1/6) L,R is (3, 1, 2, 1/6) L,R + (3, 4, 1, 1/6) L,R under SU (3) × SU (4) DM × SU (2) L × U (1) Y and it contains three SU (2) L doublets (due to color) per family, and (1, contains one SU (2) L doublet per family. When µ > M 2 , we have an additional (lefthanded) contribution to the number of fermions of 6 · 3 = 18 from the six SU (2) L doublets per family from (6, 2) 0 . Therefore, following Equation (1) For SU (4) DM , C(4) = 1 2 and C(6) = 1. Below the scales M 1 and M 2 , the numbers of right-and left-handed fermions are n f R = n f L = 3 because, per family, (4, 1) 3 and (4 * , 1) −3 contribute one left-handed and one right-handed SU (4) DM fermion. When the M 1 scale is relevant, (4, 1) −1 of 6 of SU (6) contributes (3 + 1)3 = 12 to n f L and n f R since, per family, (3, 1, 2, 1/6) L,R +(3, 4, 1, 1/6) L,R contributes 3 (due to color) and (1, 1, 2, −1/2) L,R + (1, 4, 1, −1/2) L,R contributes 1. When M 2 is relevant, (6, 2) 0 contributes 2 (due to the SU (2) L doublet) per family to the number of left-handed fermions. The β-function for Solving the β-function equation Using Equation (4) for SU (2) L for different energy ranges and evaluating at µ DU T , we get where µ EW = 246 GeV is the electroweak energy scale, and α 2 (µ EW ) ≈ 0.03. Using Equation (4) for SU (4) DM for different energy ranges and evaluating at µ DU T gives us where µ DM is the dark-matter dynamical mass and M < (M > ) is the lesser (greater) of M 1 and M 2 . We assume that M 1 and M 2 are bigger than µ DM and the electroweak scale or any observable scale. If M 1 or M 2 is on the order of µ DU T (and therefore not affecting RG flow), one may set it equal to µ DU T in Equations (5) and (6) to get the appropriate expression. Using Equations (5) and (6) and the unification of SU (2) L and SU (4) DM at the DUT scale (α 4 (µ DU T ) = α 2 (µ DU T )), we get an expression for the dark-matter dynamical mass: Essentially, we run the SU (2) L gauge coupling from the known electroweak scale up to the DUT scale, which can be observationally constrained, and then we run the SU (4) DM gauge coupling down to its appropriate scale for dark matter. And the dark-matter's dynamical mass should be approximately equal to the energy scale of confinement for SU (4) DM , just as the major contribution to the quarks' masses comes mainly from the SU (3) C confinement scale in the Standard Model. In Figure 2, we show regions of confinement energy from 0.5 ≤ α 4 ≤ 1.5 for the one-loop RG flow for various values of µ DU T , and we see that the variation in energy scale in these regions is not very significant. So we estimate that α 4 (µ DM ) ∼ 1.
In preliminary analysis, we let M 2 vary from 10 5 GeV to µ DU T , and we found that the dark-matter dynamical mass was in general very high, ranging even up to 10 13 GeV. Because we are not interested in such purely academic values of the dark-matter dynamical mass, we assume M 2 ∼ µ DU T in all our analysis that follows.
We emphasize the unique and interesting connection between the DUT scale and the dynamical mass of dark matter we have presented. This connection has been made independently of the model of inflation. In what follows, we specify an inflation model in order to apply inflation constraints from cosmological probes to µ DU T .

Constraints on Inflation from Planck
As discussed in [2], the inflaton is expected to decay to dark matter during entropy generation after inflation. Using the slow-roll approximation, we examine a Coleman-Weinberg potential for inflation used in [6]: As in [6], φ can be thought of as the physical field that is the real part of a scalar field Φ such that Φ † Φ = (φ + v) 2 . The expectation value of φ is < φ >= 0, but < Φ >= v. So the potential in terms of the full field Φ is this Coleman-Weinberg potential with (φ+v) 2 → Φ † Φ, and it has its local maximum centered around Φ = 0, whereas the potential for φ has its local maximum centered around φ = −v (see Figure 3). So we use the potential in terms of φ, which is mathematically equivalent up to a field shift of v, so the potential still displays the dynamics of a Coleman-Weinberg potential.
The phase transition into the true vacuum energy with < Φ >= v will provide the symmetry breaking needed for the breaking of SU (6) → SU (4) DM × SU (2) L × U (1) DM . So we expect µ DU T = v, and we constrain v using inflation constraints from Planck. Two significant constraints from Planck [4,5], assuming no running of the scalar spectral index and no tensor perturbations, come from the scalar spectral index, n s (k ) = 0.9603 ± 0.0073, and the scalar power-spectrum amplitude A s (k ), ln(10 10 A s (k )) = 3.089 +0.024 −0.027 . These values come from temperature power spectrum data from Planck and WMAP polarization at low multipoles and are obtained at the pivot scale k = 0.05 Mpc −1 , and the scalar power spectrum with no running is modeled as P s (k) = A s (k ) k k ns−1 . When tensor perturbations are considered with no running of the scalar or tensor spectral indexes, Planck reports at the 95% confidence level r(k ) ≡ As(k ) At(k ) < 0.12 for the tensor-scalar ratio and n s (k ) = 0.9624 ± 0.0075 using the same data sources mentioned earlier and obtained at the pivot scale k = 0.002 Mpc −1 . Although Planck also gives constraints on slow-roll parameters of higher order when running is allowed, depending on the data set used, they are consistent with no running and are not very helpful in constraining our model, so we do not present analysis concerning these constraints from Planck here.
We use the slow-roll approximation to first order to apply these constraints, and we evaluate at the pivot scale's Hubble radius crossing, which is denoted by as a superscript or subscript. Natural units are used throughout with the reduced Planck mass M P l ≡ (8πG) −1/2 .
The scalar spectral index at the pivot scale in terms of the slow-roll potential parameters is given by and The scalar power spectrum amplitude is given by The tensor-scalar ratio at the pivot scale is and the number of e-folds before the end of inflation when the pivot scale k exited the Hubble radius, N , is given by Plugging in our potential from Equation (8) into Equation (9), we obtain the relation where x ≡ φ v . In our potential in Figure 3, the domain of inflation begins soon after (to the right of) the local maximum at φ = −v (or x = −1), and we determine x end from the condition V = 1, which is when the acceleration of the universe due to inflation ceases, and this happens before entropy generation around the local minimum at the origin.
Using Equations (9) and (12), we get Evaluating the definite integral in Equation (14), we get where li(y) ≡  Using these results and Equation (13), we find that r 10 −8 for all these scenarios, so Planck's constraint on r is satisfied. The precise value of N depends on the energy scale of inflation and the uncertain details of entropy generation at the end of inflation (see, for example, Equation (24) of [5] for more details), but the value of N for each scenario above is greater than the minimum number required to solve the horizon problem for each energy scale.
The constrained values of µ DU T in Equation (18) Table III, using Equation (7) for various scenarios concerning the choice of M 1 , we give the upper and lower bounds for dark-matter dynamical mass based on this range for µ DU T . A wide range of predictions of dark-matter dynamical mass is exemplified in Figure 4 for various scenarios. We even show scenarios in which M 1 is as low as 10 4 GeV since the Boltzmann suppression factor (e −M/T ) in the number density for particles of mass 10 4 GeV in equilibrium would be e −100 and stronger for temperatures of T ∼ 100 GeV (electroweak scale) and lower. Note that the inflaton, with mass m φ = √ 8Av, decays to two dark-matter particles, which are massless until the confinement scale µ DM µ DU T . dynamical mass lower bound on the last row is given without concern for the phenomenological constraints on mirror-fermion masses, which are discussed in [7].

Including BICEP2's Constraint
For simple single-field slow-roll inflation models (with constant r), BICEP2's new constraint on the tensor-scalar ratio [8], r = 0.20 +0.07 −0.05 , implies large-field inflation, and GUT-scale energy of inflation, BICEP2's result, if true, may have significant implications for inflation. As of now, the result still needs confirmation from other cosmological probes. There are also serious concerns about the interpretation of the data leading to BICEP2's large value of r. According to the latest from Planck [14], BICEP2's result can be completely accounted for by dust foregrounds, and they call for further data analysis. The following is our consideration of BICEP2's result if their discovery is confirmed upon further scrutiny and data collection. In fact, perhaps BICEP2's result of r ≈ 0.2 can be at best an upper bound due to the difficulty in disentangling dust foregrounds and a signal of gravitational waves.
The implications of Equation (20) are not necessarily problematic, but those of Equation (19) for constant r may be. According to [15], ∆φ is generically super-Planckian for typical single-field inflation potentials when BICEP2's result is considered, and the symmetry-breaking scale in our Coleman-Weinberg model is bigger than the change in the field, v > ∆φ = |φ end − φ |, since the local maximum is shifted from the origin by an amount of field excursion equal to v. But we would expect v < M P l . And when BICEP2's constraint on r is taken into account along with Planck's constraints in the Coleman-Weinberg model considered in the previous section, the model no longer provides an adequate description of inflation. However, the discrepancy may be resolved via unknown extra terms due to quantum gravity corrections for energy scales near the Planck scale. And as [15] points out, sub-Planckian single-field inflation is not ruled out by their result. For example, one may patch together different single-field potentials for different field ranges to form a potential for an effective quantum field theory with a sub-Planckian field excursion.
If one assumes that Equation (19) is roughly accurate, taking r to be constant at its average value over the span of φ for inflation, then one concludes that if ∆φ has any bearing on the scale of symmetry breaking, µ DU T should be on the order of the Planck scale, and µ DM ∼ 10 10 TeV according to Equation (7). However, the field excursion in a consistent single-field inflation model need not correspond at all to the true vacuum expectation value in such a constructed potential as mentioned in the previous paragraph. Our goal is not to accurately construct such a model so as to make precise predictions in consideration of BI-CEP2's data, especially since the veracity of their results is in question. Most important, the connection between µ DU T and µ DM still remains unchanged and independent of cosmological constraints on the inflation potential and the purported correct model of inflation.

Conclusion
In this paper, we explore the unique connection in the luminogenesis model, a model that consistently combines dark matter and the Standard Model, between cosmic inflation and the creation of dark matter that allows the constraining of the dynamical mass of dark-matter particles. The constraint on µ DU T is obtained by fitting a particular symmetry-breaking potential, although the connection between the unification scale µ DU T and the dark-matter dynamical mass µ DM remains independent of the supposed correct model of inflation.
Since the descent into the true vacuum of inflation triggers the breaking of the DUT symmetry and thus the conversion of the inflaton to dark matter, we can arrive at rough constraints on the DUT scale via constraints on inflation from cosmological probes. Through picking a Coleman-Weinberg potential and reasonable energy scales for inflation, we arrive at constraints on the DUT scale, and we can then derive an upper bound on the dark-matter dynamical mass via RG flow. We run the SU (2) L coupling from the known electroweak scale up to the DUT scale, where it is unified with SU (4) DM . We then run the SU (4) DM coupling down to its confinement scale, taken to be when α 4 ∼ 1, and we arrive at the dynamical mass scale for dark matter. Various dynamical mass values for dark matter are possible depending on the mass scale M 1 and M 2 . The possibility of strongly self-interacting dark matter, as proposed in [1,2], with dynamical masses obtained through the connection µ DU T → µ DM as studied here has wide implications concerning the resolution of dwarf-galaxy structures and dark-matter cusps at the centers of galaxies and its potential detectability. This is under investigation.