# WIMP Dark Matter Inflation with Observable Gravity Waves

###### Abstract

We present a successful inflation model based on potential in which a Standard Model (SM) singlet inflaton , with mass of around a TeV or less, also plays the role of a weakly interacting scalar dark matter particle (WIMP). The WIMP relic abundance generated after inflation is in accord with the current observations. The spectral index lies within the WMAP 1- bounds, while the Planck satellite may observe the tensor-to-scalar ratio, a canonical measure of gravity waves, which we estimate lies between 0.003 and 0.007. An unbroken parity ensures that the scalar WIMP is absolutely stable.

The idea that the inflaton, a particle responsible for primordial inflation, also may play the role of scalar WIMP dark matter is most intriguing KofmanEtAl and therefore worth pursuing. Ref. LiddleEtAl attempted to implement this idea in chaotic inflation with potential. However, a satisfactory scenario could not be realized which is, to a large extent, related to the fact that GeV, as demanded by inflation, far exceeds the canonical WIMP mass of a TeV or so. Simply replacing the quadratic potential with a quartic one does not help solve the conundrum for in this case the scalar spectral index and tensor-to-scalar ratio lie outside the WMAP 2- bounds WMAP .

In a recent paper, hereafter called AxionInflation , it was shown that inflation, if supplemented by the non-minimal gravitational coupling between the SM gauge singlet scalar field and the curvature scalar , yields values of (scalar spectral index) and (tensor-to-scalar ratio) that are compatible with the WMAP 1- bounds WMAP . This is to be contrasted with minimal inflation whose predictions for and lie outside the WMAP 2- bounds. A series of earlier papers Bezrukov:2008dt -Einhorn:2009bh have previously raised the possibility that the inflaton field could be identified with the SM Higgs doublet , provided the non-minimal coupling . While intriguing, doubt about the viability of this identification have been raised in Naturalness1 Naturalness2 . It stems from the observation that for , the energy scale of inflation exceeds the effective ultraviolet cutoff scale , with being the reduced Planck scale, assuming the SM Higgs quartic coupling is of order unity. In AxionInflation we easily evade this problem by making a SM gauge singlet field so that the parameter is not all that strongly constrained. Indeed, one finds that consistent with the WMAP 1- bounds on and , and can lie within the relatively wide range, , and .

In a separate development, it has been noted by several authors SDM that a stable SM singlet scalar particle, with mass TeV, is a viable cold dark matter candidate (WIMP), provided it has suitable interactions with the SM Higgs doublet and possibly additional fields. The interaction term plays an especially important role in these considerations. Recent estimates suggest KMNO that with and dark matter mass TeV, the relic WIMP abundance is compatible with the value determined by WMAP WMAP . This parameter region will be further explored in the ongoing and planned direct detection experiments of dark matter particle.

In this letter we propose a successful and relatively simple scenario of WIMP dark matter inflation by merging together ideas from AxionInflation and KMNO . Following AxionInflation , we employ non-minimal quartic inflation in which a gravitational coupling of the inflaton to the curvature scalar is included. The model has a further restriction arising from the relic dark matter abundance. It is shown in KMNO that for TeV mass WIMP dark matter, the coupling strength must be of order 0.1 or so. In our case this means that due to radiative corrections involving , the ’effective’ quartic coupling is of order . An important consequence of this WIMP driven inflation model is that it predicts both and in a fairly narrow range. In particular, values close to 0.007 may be accessible to Planck satellite searches. Another important feature of our model is the appearance of thermal dark matter relic abundance which arises during preheating and subsequent transition to a radiation dominated universe with temperature close to GeV. The energy in the oscillating inflaton field is by then essentially negligible.

Consider the following tree level action in the Jordan frame:

(1) | |||||

where GeV is the reduced Planck mass. Here we have introduced a parity under which is odd, while the SM fields are all even. Hence the scalar is stable and will play the role of both inflaton and dark matter particle.

First we consider the non-minimal inflation in this model AxionInflation . During inflation, with field values close to , dominates the scalar potential. The relevant one-loop renormalization group improved effective potential RGEIP is , where , and , with being the anomalous dimension of the inflaton field. We employ a leading-log approximation for the effective potential

(2) |

where , , and we have taken as the renormalization scale. In the Einstein frame with a canonical gravity sector, the kinetic energy of can be made canonical by defining a new field SHW ,

(3) |

The effective potential in the Einstein frame is then given by

(4) |

The inflationary slow-roll parameters are given by

(5) | |||||

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

(6) |

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

(7) |

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

(8) |

which should satisfy the WMAP normalization, WMAP , at .

Using Eqs. (2)-(8) we can obtain various predictions of the radiatively corrected non-minimal inflation model. Once we fix the parameters and the number of e-foldings , we can predict , , and . Note that with TeV, the coupling in order for the relic density of dark matter to be compatible with the WMAP observations KMNO . In our analysis, we set and TeV as reference values. If , the potential during inflation is dominated by the radiatively corrected part. We impose for an unbroken parity.

The predicted values of and are shown in Figure 1 for the number of e-foldings (solid curve) and (dashed curve). In non-minimal inflation AxionInflation , the (effective) scalar quartic coupling becomes larger according to values, and and approach their asymptotic values, and for and and for . These values correspond to the left edge of each curve in Figure 1. In the present case, for , the radiatively induced term in the effective potential dominates the scalar potential and thus the effective quartic coupling has a minimum value. In the limit , and approach and for ( and for ), which correspond to the right edge of each curve in Figure 1. We find that the running of the spectral index very weakly depends on , and () for (). In Figure 2, we show the ratio of the inflation energy scale () to the effective ultraviolet cutoff scale (). This ratio becomes larger as is raised. The minimum value for this ratio, , is achieved for with . This value marginally exceeds the proposed naturalness bound Naturalness2 . As a more conservative bound, we examine the constraint on the ratio of the Hubble parameter and the effective cutoff scale from the validity of the classical inflationary treatments Naturalness1 , namely, . Since , we find the ratio for , while . Thus, this bound is satisfied.

After inflation, the inflaton field starts to oscillate around its potential minimum. During this period the energy density of the inflaton is transmitted to other relativistic fields. Since a single inflaton cannot decay into other particles because of parity, preheating KofmanEtAl via, in our case, the coupling plays the crucial role in energy transmission. In the first stage of preheating, the inflaton energy density is transmitted to particles by explosive production through parametric resonance effects with potential. The amplitude of the inflaton filed is reducing its amplitude by particle production and the expansion of the universe. When this becomes smaller than , the term in the scalar potential dominates and the Higgs doublets are explosively produced by the (broad) parametric resonance through the coupling . This preheating process ends when the amplitude becomes smaller than KofmanEtAl .

Since the inflaton is stable, its oscillations continue without further energy transfer after the preheating era. The oscillating mode has the equation of state of dust and so, in principle, it can play the role of dark matter. The present ratio of the average energy density of this oscillation mode to the number density of photons is estimated as LiddleEtAl

(9) |

where is the oscillation amplitude at time when , the Hubble parameter. Since for , is basically also the amplitude of the oscillating inflaton today. With , we find for TeV and . Comparing it to the observed value by WMAP WMAP , , we conclude that the oscillating mode has a negligible contribution to the energy density of the present universe. It is worth recalling that successful chaotic inflaton requires GeV, and so is needed to realize the observed value of LiddleEtAl ! In our case, successful inflation is realized by non-minimal inflation and the inflaton mass plays no role during inflation for GeV.

We have seen that the energy density in the remnant inflaton oscillations is tiny compared to the observed dark matter relic density. However, the particle can be a suitable WIMP dark matter candidate if the universe is thermalized with the reheating temperature high enough for the particle to be in thermal equilibrium. In the preheating scenario, thermalization of the universe takes place through decays and multiple scatterings of particles (SM Higgs doublets in our model), during explosively produced preheating. A reasonable estimate for the reheating temperature is KofmanEtAl

(10) |

where is the total decay width of the Higgs boson. For a relatively light Higgs boson with mass GeV for example, the dominant decay mode is , so that

(11) |

where GeV is the appropriate bottom quark mass, and GeV is the VEV of the SM Higgs doublet. We find GeV, which is four orders of magnitude larger than ( TeV). Thus, we expect that the temperature of the universe is high enough for particles to be in thermal equilibrium, and the standard WIMP dark matter scenario consistent with the WMAP observations can be realized, as shown in recent analysis KMNO .

In our analysis above, we set TeV in order to keep the inflaton mass much larger than the SM Higgs boson mass, . This parameter choice makes the preheating process as effective as possible. In general, the preheating process can be reasonably efficient even for KofmanEtAl . If the inflaton mass can be lowered close to half of the Higgs boson mass, the magnitude of the coupling needed to obtain the correct dark matter relic abundance becomes significantly smaller than 0.1 KMNO . In this case, the radiative corrections to the potential is negligible and we can easily obtain for a successful inflation scenario as shown in AxionInflation .

In summary, we have shown that the inflaton and WIMP dark matter can indeed be one and the same particle. In the simplest model this is achieved by supplementing the SM with a stable gauge singlet scalar field. The model overcomes serious challenges faced by chaotic inflation LiddleEtAl and, in addition, turns out be quite predictive. Its dark matter properties will be seriously examined by the ongoing direct detection searches CDMS XENON100 . As far as inflation is concerned the predictions for and lie within the WMAP 1- bounds. With an upper bound of around 0.007 on , the model can be excluded if the Planck satellite observes values that are significantly larger than this.

Finally, one promising extension of our model is to introduce an SU(2) triplet scalar field with unit SM hypercharge. This would nicely incorporate neutrino masses and mixings via the type-II seesaw mechanism typeII . Interestingly, this extension essentially coincides with a model proposed in GOS , and it can also account for the anomalous cosmic-ray positron flux reported by the PAMELA satellite experiment PAMELA . In addition, as analyzed in detail in GOS2 , in the type-II seesaw extension of the SM, the vacuum stability bound on the Higgs boson mass can be reduced to coincide with the current experimental lower bound of 114.4 GeV LEP2 .

Note Added

Although our approach in unifying the inflaton and dark matter particle is inspired by Refs. KofmanEtAl LiddleEtAl , by the recent analysis of SM singlet scalar dark matter KMNO , and by non-minimal inflation AxionInflation , the model presented in this letter turns out to be identical to the ones proposed in Refs. Lerner:2009xg and Park . In Lerner:2009xg , a comprehensive study of certain aspects of this model were presented, and where there is overlap with our work, the results appear to be in broad agreement. However, we emphasize that there are several new and important results in this letter. Thus, (1) we have emphasized the WMAP constraints on the coupling , arising from the thermal relic density of dark matter, and evaluated its direct impact on the effective inflaton quartic coupling. As a result, our inflationary scenario predicts both and in a narrow range. (2) This feature also then plays an important role in discussing the naturalness of the inflationary scenario as shown in Figure 2. (3) We have considered the preheating scenario after inflation, following KofmanEtAl and LiddleEtAl , which allows the transition to a radiation dominated universe. Furthermore, we have shown that the energy density in the remnant inflaton oscillations, which plays the role of dark matter in the original scenario LiddleEtAl , can be ignored. (4) Considering thermalization of the universe via preheating, we have estimated the reheating temperature and shown that it is high enough for the singlet scalar to be in thermal equilibrium. This is crucial for a successful WIMP dark matter scenario.

Acknowledgments

N.O. would like to thank the Particle Theory Group of the University of Delaware for hospitality during his visit. The work of Q.S. is supported in part by the DOE under grant No. DE-FG02-91ER40626.

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