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Progress in Higgs inflation

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Abstract

We review the recent progress in Higgs inflation focusing on Higgs-\(R^2\) inflation, primordial black hole production and the \(R^3\) term.

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Notes

  1. Neglecting the kinetic term during the inflation, both theories are equivalent since \(\mathcal{L}/\sqrt{-g} \ni (M^2+\xi \phi ^2) R/2 -\lambda \phi ^4/4\) is mapped to \(M^2 R/2+(\xi ^2/4\lambda ) R^2\) by solving the field equation for \(\delta \phi \).

  2. In fact, the form of the action is different when the Palatini formalism is used, which regard the metrics and affine connection independently. In this review, we take the standard metric formalism.

  3. Even in non-Higgs inflation cases, large quantum fluctuation in de Sitter background \( {\mathcal {O}}(H/2\pi ) \) during inflation may cause a problem. See the Ref. [35].

  4. In fact, due to the non-renormalizability of the theory, there exists a dependence on the way to choose the renormalization scale, which is also called ‘prescription’. In this review, we choose \( \mu = \phi \), where \( \phi \) is the Jordan frame Higgs field value. For the meaning of the prescription dependence in detail, see Ref. [39].

  5. In fact, the Higgs boson decay to the longitudinal mode of the gauge boson may depend sensitively on higher order operators. See the Ref. [40].

  6. At linear order, there is a simple relation between the energy density fluctuation and the curvature perturbation:

    $$\begin{aligned} \delta = \frac{4}{9} \left( \frac{k}{aH} \right) ^{2} {\mathcal {R}} \end{aligned}$$
    (38)

  7. In fact, to generate a large enough power spectrum, \(\lambda _\text {min} = \lambda _\text {min}^\text {inf} - \delta c\), with \(\delta c\sim {\mathcal {O}}\left( 10^{-7}\right) \) at the corresponding scale, the \( \lambda _{\text {min}} \) must be smaller than the pure inflection value \( \lambda _{\text {min}} \) by as much as \( {\mathcal {O}}(10^{-7}) \) so that the potential should deviate from a true inflection point.

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Acknowledgements

We thank Kazunori Kohri, Misao Sasaki, Hyun Min Lee, Shi Pi, and Fedor Bezrukov for helpful discussions and Alexei Starobinski, Minxi He, Jun’ichi Yokoyama, Ryusuke Jinno, Kohei Kamada, Kin-ya Oda, and Mio Kubota for valuable collaborations. This work was supported by National Research Foundation grants funded by the Korean government (MSIT) (NRF-2018R1A4A1025334),(NRF-2019R1A2C1089334) (SCP) and (MOE) (NRF-2020R1A6A3A13076216) (SML). The work of SML is supported by the Hyundai Motor Chung Mong-Koo Foundation.

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  1. Department of Physics and IPAP and Lab for Dark Universe, Yonsei University, Seoul, 03722, Korea

    Dhong Yeon Cheong, Sung Mook Lee & Seong Chan Park

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Cheong, D.Y., Lee, S.M. & Park, S.C. Progress in Higgs inflation. J. Korean Phys. Soc. 78, 897–906 (2021). https://doi.org/10.1007/s40042-021-00086-2

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