Non-Minimal B-L Inflation with Observable Gravity Waves

Non-Minimal Inflation with Observable Gravity Waves

Nobuchika Okada E-mail:    Mansoor Ur Rehman E-Mail:    Qaisar Shafi E-mail: Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA

We consider non-minimal inflation in a gauged non-supersymmetric model containing the gravitational coupling , where denotes the Ricci scalar and the standard model singlet inflaton field spontaneously breaks the symmetry. Including radiative corrections, the predictions and for the scalar spectral index and tensor to scalar ratio lie within the current WMAP 1- bounds. If the symmetry breaking scale is of order a TeV or so, one of the three right handed neutrinos is a plausible cold dark matter candidate. Bounds on the dimensionless parameters , and the gauge coupling are obtained.


A non-minimal gravitational coupling of inflaton has been known to play an important role in models of chaotic inflation Salopek:1988qh (). Recently, this idea has received a fair amount of attention Bezrukov:2008dt ()-Linde:2011nh () arising from the possibility of taking the Standard Model (SM) Higgs boson as an inflaton. In the simplest scenarios of this kind, the SM Higgs doublet has a relatively strong non-minimal gravitational interaction , where is the Ricci scalar and a dimensionless coupling whose magnitude is estimated to be of order based on WMAP data Komatsu:2010fb (). This SM Higgs-based inflationary scenario is currently mired in some controversy stemming from the arguments put forward in cutoff () that for , the energy scale during non-minimal SM inflation exceeds the effective ultraviolet cut-off scale . Here of order unity denotes the SM Higgs quartic coupling and  GeV represents the reduced Planck mass. This point has been further elaborated in Refs. Burgess:2010zq (); Hertzberg:2010dc (). However, it has recently been argued Bezrukov:2010jz () that if the Higgs field is perturbed around its non-zero classical background, the effective cut-off can become larger than the energy scale of inflation. (See Germani:2010gm () for other possible solutions to the unitarity problem.) As we will see below, the above problem has a negligible impact on our conclusions in this paper. Even though the inflation in our case carries a charge, a satisfactory scenario imposes relatively mild but nonetheless important constraints on the gauge coupling.

In this paper we implement non-minimal inflation by supplementing the standard model with a gauged symmetry Pati:1974yy (). (For inflation with local supersymmetric , see Lazarides:1996dv (); Jeannerot:1997av (); Senoguz:2005bc () and references therein. For global inflation see Shafi:1984tt ()) The well-known advantages of a spontaneously broken gauge symmetry include seesaw physics Seesaw () to explain neutrino oscillations, and baryogenesis via leptogenesis Fukugita:1986hr (); Lazarides:1991wu () arising from the right handed neutrinos that are present to cancel the gauge anomalies. In the inflation model that we consider the symmetry breaking scale of is arbitrary as long as the lower bound from LEP experiment, TeV LEPbound (), is satisfied. One interesting possibility is to break it at the TeV scale Khalil:2006yi (), and it has been shown Iso:2009ss (); Iso:2009nw () that the minimal model with additional classical conformal invariance naturally predicts the symmetry breaking scale to be at TeV. This means that the new particles, the gauge boson, the Higgs boson and the RH neutrinos have TeV scale masses, and they can be observed at Large Hadron Collider (LHC) Emam:2007dy (); Huitu:2008gf (); Basso:2008iv (); Perez:2009mu (). Furthermore, with TeV scale RH neutrinos we can explain the origin of the baryon asymmetry through resonant leptogenesis resonantLG (); resonantLG2 ().

One important feature missing in the above TeV scale model is non-baryonic dark matter (DM). To circumvent this, following Ref. Okada:2010wd (), we introduce an unbroken parity under which one of the three RH neutrinos is taken to be odd, while all other fields are even. In this case the -odd RH neutrino is absolutely stable and a viable DM candidate. Note that the two remaining RH neutrinos are sufficient to reconcile theory with the observed neutrino oscillation data. The model also predicts that one of the three observed neutrinos is essentially massless. Thus, without introducing any additional dynamical degrees of freedom, the DM particle can be incorporated in the minimal gauged model.

In this paper we consider non-minimal inflation by taking (), to be the inflaton field which is charged under . We take into account quantum corrections to the inflationary potential arising from the inflaton interactions with the gauge field. We find that the tensor to scalar ratio and the scalar spectral index . More generally, in this non-minimal inflation model, the predictions and lie within the WMAP 1- bounds for and . Recall that the corresponding tree level predictions for minimal () chaotic inflation, namely and , lie outside the WMAP 2- bounds.

Our inflation model is based on the gauge group and the particle content is listed in Table 1 Iso:2009nw (). The SM singlet scalar () breaks the gauge symmetry down to by its vacuum expectation value (vev), and at the same time generates the right-handed neutrino masses. The Lagrangian terms relevant for the seesaw mechanism are given by


where the first term yields the Dirac neutrino mass after electroweak symmetry breaking, while the right-handed neutrino Majorana mass term is generated by the second term associated with the gauge symmetry breaking. Without loss of generality, we use the basis which diagonalizes the second term and makes real and positive.

SU(3) SU(2) U(1) U(1)
3 2
3 1
3 1
1 2
1 1
1 1
1 2
1 1
Table 1: Particle content. In addition to the SM particle contents, the right-handed neutrino ( denotes the generation index) and a complex scalar are introduced.

Consider the following tree level action in the Jordan frame:


where and are the vevs of the Higgs fields and respectively. To simplify the discussion, we assume that is sufficiently small so it can be ignored, and also .

The relevant one-loop renormalization group improved effective action can be written as RGEIP ()


where and , with


being the anomalous dimension of the inflaton field. denotes the gauge coupling and the renormalization scale. In the presence of the nonminimal gravitational coupling, the one loop renormalization group equations (RGEs) of , , and are given by Iso:2009ss (); Iso:2009nw ()


where the factor is defined as


In the Einstein frame with a canonical gravity sector, the kinetic energy of can be made canonical with respect to a new field Clark:2009dc (),




The action in the Einstein frame is then given by




In our numerical work, we employ above potential with the RGEs given in Eqs. (5-8). However, for a qualitative discussion it is reasonable to use the following leading-log approximation of the above potential:


where we have assumed , , , , , and with . We have checked that for a broad range of parameters the above expression can be regarded as a valid approximation for the potential given in Eq. (13). In our numerical calculations we fix the renormalization scale TeV.

To discuss the predictions of this model it is useful to first recall the basic results of the slow roll assumption. The inflationary slow-roll parameters are given by


where a prime denotes a derivative with respect to . The slow-roll approximation is valid as long as the conditions , and hold. In this case the scalar spectral index , the tensor-to-scalar ratio , and the running of the spectral index are approximately given by


The number of e-folds after the comoving scale has crossed the horizon is given by


where is the field value at the comoving scale , and denotes the value of at the end of inflation, defined by max.

The amplitude of the curvature perturbation is given by


where is the WMAP7 normalization at Komatsu:2010fb (). Note that for added precision, we include in our calculations the first order corrections Stewart:1993bc () in the slow-roll expansion for the quantities , , , and .

Using Eqs. (14)-(23) above we can obtain various predictions of the radiatively corrected non-minimal model of inflation. Once we fix the parameters and , and the number of e-foldings , we can predict , , and . The tree level () predictions for minimal inflation are readily obtained as:


For (), we find (), () and ().

Figure 1: vs. for the radiatively corrected non-minimal potential defined in Eq. (14) with the number of e-foldings (red solid curve) and (blue dashed curve) and = 0. The WMAP 1- (68% confidence level) bounds are shown in yellow. Along each curve we vary either or .

As expected, the predictions of tree level minimal inflation lie outside the 2- WMAP bounds Komatsu:2010fb ().

However, the situation is improved once the radiative corrections are included NeferSenoguz:2008nn (). The impact of these radiative corrections on the tree level predictions of various inflationary models have been studied in Refs. Rehman:2009wv (); Rehman:2010es (). Furthermore, the nonminimal gravitational coupling also plays an important role in making the tree level predictions consistent with the WMAP data. Indeed, the radiative corrections smear out the tree level predictions of nonminimal inflationary models Okada:2010jf (). A similar behavior is observed in our situation. The approximate potential in Eq. (14) effectively behaves as a nonminimal potential with a running coupling constant . In the limit , assuming to be approximately constant, the scalar spectral index, the tensor to scalar ratio and the running of the spectral index for the radiatively corrected non-minimal inflation are given by Okada:2010jf ()

Figure 2: vs. and for radiatively corrected non-minimal inflation with the number of e-foldings (red solid curve) and (blue dashed curve) and = 0.

These results exhibit a reduction in the value of and an increase in the value of compared to their minimally coupled tree level predictions (Eqs. (24-26)), as can be seen in Figs. 1-3. In our analysis, we set limit for simplicity. From the WMAP 1- bounds ( and ), we obtain a lower bound of with e-foldings Bezrukov:2008dt (). The value of receives a tiny correction to its tree level prediction. Note the sharp transitions in the predictions of and in the vicinity of . This can be understood from the expression for the inflationary potential given in Eq. (14) and Eqs. (27) and (28).

Figure 3: vs. and for radiatively corrected non-minimal inflation with the number of e-foldings (red solid curve) and (blue dashed curve) and = 0.

In the large limit, again assuming to be constant, we obtain the following results for , and




We obtain and consistent with the WMAP 1- bounds. The running of the spectral index varies from to . Note the second sharp transitions in the predictions of and around . Actually, in this limit the approximation of the potential given in Eq. (14) does not hold as the value of the gauge coupling becomes large and we can no longer ignore its running.

Figure 4: vs. (TeV) for radiatively-corrected non-minimal inflation with the number of e-foldings (red solid curve) and (blue dashed curve) and = 0.
Figure 5: vs. for radiatively-corrected non-minimal inflation with the number of e-foldings (red solid curve) and (blue dashed curve) and = 0.

Finally in Figs. 4-6 we display the relation among the parameters , (TeV), and . Within 1- bounds of WMAP data, these parameters take values in the range , , and . However, if we require that , then more stringent upper bounds, , , and are obtained. Although, there is some uncertainty in the calculations of cut-off (i.e., is argued by some (Ref. Bezrukov:2010jz ()) to be larger than during inflation), interesting related LHC physics still looks viable even with the more stringent bound on Basso:2009hf ().

To summarize, we have considered non-minimal chaotic inflation in a minimal gauged extension of SM. Among the very well-known attractive features of this model are the natural presence of three RH neutrinos, seesaw mechanism to understand non-zero neutrino masses, and explanation of the baryon asymmetry via leptogenisis. With an extra symmetry one of the three RH neutrino can be a viable dark matter candidate. To realize inflation we utilize the SM gauge singlet inflaton which is charged under . In addition to the non-minimal gravitational coupling, we have also included the effect of inflaton-gauge coupling . For , and we obtain the inflationary predictions and that are consistent with the WMAP 1- bounds and will be tested by the Planck satellite.

Figure 6: vs. for radiatively-corrected non-minimal inflation with the number of e-foldings (red solid curve) and (blue dashed curve) and = 0.
Figure 7: vs. for radiatively-corrected non-minimal inflation with the number of e-foldings (red solid curve) and (blue dashed curve) and = 0.


We thank Joshua R. Wickman for valuable discussions. We also thank Rose Lerner for pointing out the importance of including running. This work is supported in part by the DOE under grant No. DE-FG02-91ER40626 (Q.S. and M.R.) and No. DE-FG02-10ER41714 (N.O.), and by the University of Delaware (M.R.). N.O. would like to thank the Particle Theory Group of the University of Delaware for hospitality during his visit.


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