###### Abstract

In cosmic inflation driven by a scalar gauge singlet field with a tree level Higgs potential, the scalar to tensor ratio is estimated to be larger than , provided the scalar spectral index . We discuss quantum smearing of these predictions arising from the inflaton couplings to other particles such as GUT scalars, and show that these corrections can significantly decrease . However, for , we obtain which can be tested by the Planck satellite.

Higgs Inflation, Quantum Smearing and the Tensor to Scalar Ratio

Mansoor Ur Rehman and Qaisar Shafi

Bartol Research Institute, Department of Physics and Astronomy,

University of Delaware, Newark, DE 19716, USA

In any realistic inflationary cosmology [1] the scalar inflaton field must couple to additional fields in order to implement the transition to a radiation dominated universe. These couplings, through quantum corrections (smearing), modify the tree level inflationary potential and therefore the corresponding predictions for the scalar spectral index , tensor to scalar ratio (measure of gravity waves), and the running of the spectral index . In Ref. [2], it was shown that quantum corrections of the chaotic inflationary potential, computed ala Coleman-Weinberg [3], and induced via couplings of the inflaton to other fields can significantly modify the tree level inflationary predictions. For instance, in the case of chaotic inflation [4] supplemented by a Yukawa interaction, the tree level prediction of and is replaced by and [2]. A similar analysis has also been carried out for a non-supersymmetric hybrid inflationary model in Ref. [5], where inclusion of these quantum corrections is shown to allow even sub-Planckian values of the inflaton field, consistent with the red tilted spectral index exhibited in the WMAP 7 year (WMAP7) results [6]. The presence of a suitable Yukawa coupling, especially one involving right handed neutrinos, has one significant feature as far as realistic inflation model building is concerned. While enabling the transition from an inflaton to a radiation dominated universe, the decay products in this case contain right handed neutrinos whose subsequent out of equilibrium decay can give rise to the observed baryon asymmetry via leptogenesis (thermal [7] or non-thermal [8]). It was shown in [2, 5] that even though the radiative corrections during inflation may be sub-dominant, they can make sizeable corrections to the tree level predictions for and . Therefore, it is interesting to analyze the effects of these radiative corrections for various realistic inflationary models.

Motivated by the above observations, we investigate in this letter the impact of quantum corrections on the predictions of a Higgs inflation model in which a gauge singlet scalar plays the role of the inflaton field. For a tree level treatment of this model, see Refs. [9, 10, 11], where it is shown that field has a trans-Planckian vacuum expectation value. Here we mainly consider the two important interaction terms of in the renormalizable Lagrangian: a quartic interaction between and a GUT symmetry breaking scalar boson (), and a Yukawa interaction between and a right handed Majorana neutrino (). It turns out that in order to obtain significant radiative corrections, both the and fields should be heavier than inflaton, and hence do not significantly contribute to reheating. Therefore, we keep the right handed neutrinos light enough to participate in reheating, while letting the GUT scalar boson be heavy enough to produce considerable radiative corrections. These radiative corrections are then shown to have a significant impact on the tree level predictions of , and d. A very precise measurement of would be an extremely useful first step in the search for the correct inflation model. If can be determined very precisely, say by the Planck satellite experiment, the different predictions for which we report here would be an effective way to look for the quantum smearing effects that we have considered. In practice, we find that for , the scalar to tensor ratio , which can be tested by the Planck satellite.

We consider the Lagrangian density [2]

(1) | |||||

where the subscript ‘B’ denotes bare quantities, and represents the GUT symmetry breaking scalar boson. The GUT symmetry is broken when acquires a non-zero vacuum expectation value (VEV) . In order to keep the discussion simple, we have introduced a single right handed neutrino with Yukawa coupling , and we ignore the bare mass terms for and . In a more realistic scenario, successful leptogenesis requires at least two right-handed neutrinos.

The inflationary potential including one loop corrections [3], in terms of renormalized quantities, can be written as

(2) |

where

(3) |

We have assumed and so that the radiative correction from inflaton self couplings is suppressed [2]. Also, with ( Hubble constant), the ‘flat space’ quantum correction is a good approximation during inflation. Here is the dimensionality of the representation of the field , is the vacuum energy density at the origin and is a constant which can be determined from the minimization condition , where denotes the VEV of at the minimum. Furthermore, if we require the potential to be zero at , i.e. , the value of is determined and the potential then takes the following form

(4) |

where

For the sake of completeness we first review some of the salient features and predictions of tree level Higgs inflation. The tree level Higgs potential can be written as [9, 10, 11]

(5) |

with . As discussed in Refs. [9, 11], inflation may occur above or below the VEV . For shorthand, we henceforth denote these regimes as the BV (below VEV) and AV (above VEV) solutions.

In the leading order approximation the slow-roll parameters are given as [1]

(6) |

where GeV is the reduced Planck mass. 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 given by

(7) | |||||

(8) | |||||

(9) |

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

(10) |

where is the value of the field when the scale corresponding to exits the horizon, and is the value of the field at the end of inflation. The value of is given by the condition , which can be calculated from the Hamilton-Jacobi equation [12]

(11) |

Another expression of which explicitly depends on the thermal history of the universe is give by [13, 2]

(12) |

where is the energy density at the end of reheating, and for [14]. In particular, for and inflation and respectively. In the latter case is independent of . In our numerical calculations we take because the quadratic component dominates near the minimum. Moreover, we assume quantum corrections to to be negligible [2]. We can write is the reheat temperature given by [15] , where

(13) |

Here is the equation of state parameter for the dominant component during the reheating phase, and is the inflaton decay width.

The amplitude of the curvature perturbation is given by

(14) |

where at according to WMAP7 data [6]. Note that, for added precision, we include in our calculations the first order corrections in the slow-roll expansion for the quantities , , , and [16].

(GeV) | (GeV) | (GeV) | (GeV) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

1.3 | 1.3 | 12.7 | 2.03 | 11.7 | 0.944 | 0.020 | 1.5 | 5.1 | 54.9 | -3.6 |

1.4 | 1.4 | 14.0 | 2.86 | 13.0 | 0.950 | 0.028 | 1.6 | 5.3 | 55.0 | -4.3 |

1.5 | 1.4 | 14.8 | 3.42 | 13.8 | 0.953 | 0.033 | 1.7 | 5.3 | 55.0 | -4.6 |

1.6 | 1.5 | 16.8 | 4.93 | 15.9 | 0.958 | 0.045 | 1.8 | 5.0 | 55.1 | -5.2 |

1.7 | 1.6 | 18.1 | 5.94 | 17.1 | 0.960 | 0.051 | 1.8 | 4.8 | 55.2 | -5.4 |

1.8 | 1.6 | 19.5 | 7.15 | 18.6 | 0.961 | 0.058 | 1.9 | 4.6 | 55.2 | -5.7 |

1.9 | 1.7 | 21.2 | 8.61 | 20.3 | 0.962 | 0.065 | 1.9 | 4.2 | 55.2 | -5.9 |

1.9 | 1.7 | 23.2 | 10.3 | 22.2 | 0.963 | 0.072 | 1.9 | 3.9 | 55.2 | -6.0 |

2.0 | 1.8 | 25.4 | 12.3 | 24.4 | 0.964 | 0.078 | 1.9 | 3.6 | 55.2 | -6.1 |

2.4 | 1.9 | 37.2 | 23.6 | 36.2 | 0.965 | 0.101 | 1.9 | 2.4 | 55.2 | -6.4 |

2.8 | 1.9 | 50.0 | 36.0 | 49.0 | 0.965 | 0.112 | 1.8 | 1.7 | 55.2 | -6.5 |

3.2 | 2.0 | 67.0 | 52.8 | 66.0 | 0.965 | 0.121 | 1.8 | 1.3 | 55.1 | -6.6 |

3.7 | 2.0 | 89.5 | 75.2 | 88.5 | 0.964 | 0.127 | 1.8 | 9.2 | 55.1 | -6.6 |

The predicted values of various parameters of the tree level Higgs inflation can be obtained by using Eqs. (5-14) above, and these are displayed in Table 1. As we see, the quantities , and carry trans-Planckian values but still the vacuum energy scale during observable inflation is well below . This implies that the quantum gravity effects are relatively unimportant here. Moreover, very low values of the reheat temperature GeV GeV are realized with the Yukawa coupling . Since we only consider the right handed Majorana neutrinos for reheating, we obtain negligible radiative corrections with their masses (or the Yukawa coupling ). Taking the number of e-foldings between 50 to 60, we obtain and within the WMAP7 1 bounds as shown in Fig. 1. Furthermore, we only show the BV branch in Fig. 1 which lies largely within the WMAP7 1 bounds, whereas, the AV branch remains outside of the WMAP7 1 bounds [11] at tree level. In this letter we mainly restrict our discussion to the WMAP7 1 bounds.

The various limits of tree level Higgs inflation has been discussed in Ref. [11]. For example, for , the BV branch approaches the effective potential of new inflation in a region disfavored by WMAP7. On the other hand, for , the AV branch approaches the effective potential of quartic inflation . As mentioned above this AV branch lies outside of the WMAP7 1- bounds. Finally, near the VEV we obtain an effective potential of quadratic inflation , where denotes the deviation of the field from the minimum and is the inflaton mass. This chaotic inflationary model [4] predicts GeV, , , and , corresponding to GeV. In fact this is the region in which the two branches meet, i.e. both the BV and AV branches converge to quadratic inflation in the high- limit.

Let us now investigate the effect of including quantum smearing to the tree level Higgs inflation. In general we may consider two types of radiative corrections: fermionic radiative corrections (which arise from the Yukawa interaction that couples to fermions) and bosonic radiative corrections (which arise from the quartic interaction between and other bosons). As we see in Eq. (3), these radiative corrections appear in the potential with opposite signs and also exhibit different behavior in various inflationary predictions. These corrections become important around and if we consider the bosonic radiative corrections only, the mass of the gauge scalar boson turns out to be of order the GUT scale GeV, with and . Here GeV is the Planck mass. Similarly, in order to obtain sizable effects with the fermionic radiative corrections we need a Yukawa coupling of order . This, in turn, makes the fermion masses superheavy GeV, larger than the inflaton mass GeV. Thus, these fermions are unable to participate in the reheating process and are also too heavy for seesaw physics. Consequently, we utilize the right handed Majorana neutrinos for reheating, which generates negligible radiative corrections with their masses and the Yukawa coupling . In this regard, the Higgs inflationary model is different from the chaotic inflationary model where the same Yukawa interaction, responsible for the reheating, has also been shown to make sizable contributions, through radiative corrections, in modifications of the tree level predictions [2]. This is related to the fact that the Yukawa couplings in the chaotic inflation are not restricted by a non-zero inflaton VEV. In the following discussion, therefore, we only consider the effect of bosonic radiative corrections from the GUT field .

The predictions of the bosonic radiatively corrected Higgs potential are shown in Fig. 2. The parameter here quantifies the amount of radiative corrections. In order to explore the entire space of smearing, we show in Fig. 2 a plot between and log for the values of which lie within the WMAP7 1 bounds. As we increase the value of , the tensor to scalar ratio is observed to decrease. This is contrary to what we usually expect in the chaotic inflationary models where bosonic (fermionic) radiative corrections increase (reduce) the value of . Actually, the BV branch of the tree level Higgs potential and the added bosonic radiative corrections have slopes with opposite signs, which reduces the magnitude of the total slope of the potential and in turn decreases the value of . In contrast, both the tree level chaotic inflationary potentials and their bosonic radiative corrections share slopes with same signs and this raises the value of . A similar reduction in has also been observed in Ref. [17] by adding higher order even powers of to the tree level Higgs potential. This reduction is prominent (insignificant) for small (large) values of the spectral index and the tensor to scalar ratio as a result of an increase (decrease) in the value of . For example, with , , the reduction in is ( ). Furthermore, the CWP curve provides an upper bound on for a given value of , as is clear from Fig. 2. For instance, the upper bound on is for the WMAP7 central value of spectral index . In the small limit, all curves reproduce the standard tree level Higgs inflation predictions.

In Fig. 3, we fix the value of and vary between zero and its tree level value. We consider, as an example, three different values of the parameter , , and . Each curve of radiatively corrected Higgs potential with constant value of approaches the CWP prediction in the small limit. As is clear from Fig. 3, the CWP curve is more contiguous to the central region of WMAP7 data in comparison to the tree level Higgs potential curve. Therefore, in Fig. 3 we also include a curve of CWP with which further extends the allowed space towards the central region of 1 WMAP7 data. Moreover, both the tensor to scalar ratio and the spectral index are observed to decrease with an increase in as explained above. The predictions of radiatively corrected Higgs potential, in fact, interpolate between the tree level Higgs potential and the CWP curves.

To see the extent of quantum spread, we provide in Table 2 the predicted values of various parameters of this model corresponding to the the WMAP7 central value of the spectral index, . Most of the quantities exhibit a small variation from their respective tree level predictions. However, the the tensor to scalar ratio experiences a significant shift. The tree level prediction of acquires a total quantum spread of with .

The CWP prediction represents the maximal smearing of the radiatively corrected Higgs inflation results. This model has been studied extensively in the past [18, 19, 20] and recently [10, 11, 21]. The predicted values of various parameters of this model are displayed in Table 3 with the number of e-foldings . Similar to tree level Higgs inflation, , and carry trans-Planckian values with vacuum energy scale well below during observable inflation. We obtain and for the WMAP central value of the spectral index .

(GeV) | (GeV) | (GeV) | (GeV) | ||||||
---|---|---|---|---|---|---|---|---|---|

1.91 | 1.72 | 22.6 | 9.79 | 21.6 | 0.070 | 1.9 | 4.0 | -6.0 | |

1.91 | 1.72 | 22.5 | 9.78 | 21.6 | 0.0670 | 1.9 | 4.0 | -6.0 | |

1.90 | 1.72 | 22.4 | 9.72 | 21.5 | 0.069 | 1.9 | 4.1 | -6.0 | |

1.90 | 1.71 | 22.4 | 9.67 | 21.4 | 0.068 | 1.9 | 4.1 | -6.0 | |

1.89 | 1.71 | 22.3 | 9.62 | 21.3 | 0.067 | 1.9 | 4.1 | -6.0 | |

1.87 | 1.70 | 22.1 | 9.53 | 21.2 | 0.066 | 1.9 | 4.2 | -6.0 | |

1.85 | 1.69 | 22.0 | 9.44 | 21.0 | 0.064 | 1.9 | 4.2 | -6.0 | |

1.82 | 1.67 | 21.7 | 9.36 | 20.8 | 0.061 | 1.9 | 4.3 | -6.0 | |

1.82 | 1.66 | 21.7 | 9.35 | 20.7 | 0.060 | 1.9 | 4.3 | -6.0 | |

1.80 | 1.65 | 21.6 | 9.40 | 20.7 | 0.058 | 1.9 | 4.4 | -6.0 | |

1.78 | 1.64 | 21.7 | 9.48 | 20.7 | 0.057 | 1.9 | 4.4 | -6.0 | |

1.76 | 1.62 | 22.0 | 9.91 | 21.0 | 0.055 | 2.0 | 4.4 | -6.1 | |

1.75 | 1.61 | 22.4 | 10.4 | 21.5 | 0.054 | 2.0 | 4.3 | -6.1 | |

1.75 | 1.61 | 23.3 | 11.3 | 22.3 | 0.053 | 2.0 | 4.1 | -6.1 |

(GeV) | (GeV) | (GeV) | |||||||
---|---|---|---|---|---|---|---|---|---|

1.3 | 1.3 | 3.9 | 17.0 | 6.24 | 16.1 | 0.960 | 0.020 | 1.6 | -5.3 |

1.4 | 1.3 | 3.6 | 18.1 | 7.10 | 17.2 | 0.961 | 0.024 | 1.7 | -5.2 |

1.4 | 1.4 | 3.2 | 19.4 | 8.10 | 18.5 | 0.962 | 0.028 | 1.7 | -5.1 |

1.5 | 1.4 | 3.1 | 20.2 | 8.67 | 19.2 | 0.963 | 0.030 | 1.7 | -5.1 |

1.5 | 1.4 | 2.7 | 21.7 | 9.95 | 20.8 | 0.964 | 0.035 | 1.8 | -5.1 |

1.6 | 1.5 | 2.4 | 23.5 | 11.4 | 22.6 | 0.965 | 0.040 | 1.8 | -5.1 |

1.6 | 1.5 | 2.2 | 24.5 | 12.3 | 23.6 | 0.965 | 0.043 | 1.8 | -5.1 |

1.7 | 1.5 | 2.1 | 25.3 | 13.0 | 24.4 | 0.965 | 0.045 | 1.8 | -5.1 |

1.7 | 1.6 | 2.0 | 26.1 | 13.7 | 25.2 | 0.966 | 0.047 | 1.8 | -5.1 |

1.7 | 1.6 | 1.9 | 26.7 | 14.2 | 25.7 | 0.966 | 0.049 | 1.8 | -5.1 |

1.7 | 1.6 | 1.8 | 27.3 | 14.7 | 26.3 | 0.966 | 0.050 | 1.8 | -5.1 |

1.8 | 1.6 | 1.7 | 28.2 | 15.5 | 27.3 | 0.966 | 0.052 | 1.8 | -5.1 |

1.8 | 1.6 | 1.7 | 28.5 | 15.8 | 27.6 | 0.966 | 0.053 | 1.8 | -5.1 |

1.8 | 1.6 | 1.6 | 28.9 | 16.1 | 27.9 | 0.966 | 0.054 | 1.8 | -5.2 |

1.8 | 1.6 | 1.5 | 29.9 | 17.0 | 28.9 | 0.967 | 0.056 | 1.8 | -5.2 |

1.9 | 1.7 | 1.3 | 32.0 | 19.0 | 31.0 | 0.967 | 0.060 | 1.8 | -5.2 |

2.0 | 1.7 | 9.5 | 37.8 | 24.4 | 36.9 | 0.968 | 0.070 | 1.8 | -5.3 |

2.2 | 1.8 | 6.3 | 45.5 | 31.7 | 44.5 | 0.968 | 0.080 | 1.8 | -5.4 |

2.6 | 1.9 | 3.2 | 62.2 | 48.0 | 61.2 | 0.968 | 0.094 | 1.7 | -5.5 |

2.8 | 1.9 | 2.2 | 74.1 | 59.7 | 73.1 | 0.968 | 0.100 | 1.7 | -5.5 |

3.0 | 1.9 | 1.5 | 90.4 | 75.8 | 89.4 | 0.968 | 0.106 | 1.7 | -5.5 |

3.5 | 1.9 | 7.4 | 124 | 110 | 123 | 0.967 | 0.113 | 1.6 | -5.6 |

4.0 | 2.0 | 4.4 | 159 | 144 | 158 | 0.967 | 0.117 | 1.6 | -5.6 |

4.5 | 2.0 | 2.4 | 212 | 197 | 211 | 0.967 | 0.121 | 1.6 | -5.6 |

5.0 | 2.0 | 1.6 | 256 | 241 | 255 | 0.967 | 0.123 | 1.6 | -5.6 |

6.0 | 2.0 | 7.5 | 374 | 358 | 373 | 0.967 | 0.126 | 1.6 | -5.6 |

7.0 | 2.0 | 3.9 | 519 | 504 | 518 | 0.967 | 0.128 | 1.6 | -5.6 |

7.9 | 2.0 | 2.4 | 655 | 640 | 654 | 0.967 | 0.129 | 1.6 | -5.6 |

To summarize, we discuss the effect of including quantum corrections (smearing) in the tree level Higgs inflation. In contrast to chaotic inflation, a Yukawa interaction between inflaton and right handed neutrinos generates negligible radiative corrections owing to light () neutrino masses. On the other hand, the bosonic radiative corrections, which are obtained from a quartic interaction between inflaton and a GUT symmetry breaking scalar boson, can yield noticeable effects. As a result of including these corrections, a reduction in the tensor to scalar ratio is observed. The Coleman-Weinberg potential provides, in this case, the maximal quantum smearing limit. Using the WMAP7 central value of the spectral index , the tree level prediction of is replaced by . We emphasize that while working with high precision observations such as the current Planck satellite experiment we cannot ignore these radiative corrections in analyzing the predictions of various inflationary models.

## Acknowledgments

Q.S. thanks Steve Weinberg for a helpful correspondence regarding inflationary potentials and radiative corrections. We also thank Nefer Şenouz and Joshua R. Wickman for valuable discussions. This work is supported in part by the DOE under grant No. DE-FG02-91ER40626 (Q.S. and M.R.), and by the University of Delaware competitive fellowship (M.R.).

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