# The Warm Inflation Story

## Abstract

**:**

## 1. Introduction

## 2. The Basic Dynamics

## 3. First Principles Dynamics

## 4. Background

## 5. Density Perturbations

## 6. Model Building

## 7. Advantages

## 8. Critique

## 9. Model Selection

**D-Brane inflation model [98]:**D-branes are solitonic solutions arising in string theories of type I, IIA and IIB. There is an interaction energy between two parallel brane and anti-branes, and this is the potential energy utilized to drive inflation. The inflaton field is a mode corresponding to a relative motion between two parallel branes. The model relies on the locality of the higher dimensional theory to allow for a sub-Hubble mass as necessary in cold inflation. For D-3 branes, the potential has the form,$${V}_{\mathrm{D}-\mathrm{brane}}\left(\varphi \right)={M}^{4}(1-\alpha /{\varphi}^{4})\phantom{\rule{0.277778em}{0ex}}.$$- -
- Quantum gravity [F]
- -
- Dimensions beyond four (6) [F]
- -
- Sub-Hubble mass inflaton [F]
- -
- Supersymmetry [F]
- -
- Symmetries not of type in SM—choice of compactification and D-p brane (2) [T]
- -
- Symmetries included in the model (2) [T]
- -
- Model building beyond the SM [T]

- This gives a speculation count for Fundamental/Technical properties of 9/5. Here the choice of compactification and D-p brane I include in the technical category and not also as a fundamental attribute for new spacetime symmetries, since this model already has been penalized in the fundamental category for extra dimensions, which is sufficient.
**$\mathbf{\alpha}$-attractor superconformal inflation model [99]**: This is a supergravity model where the parameter $\alpha $ is inversely proportional to the curvature of the inflaton Kähler manifold. A common choice of potential is:$${V}_{\alpha -\mathrm{attractor}}\left(\varphi \right)={tanh}^{2n}\left(\frac{\varphi}{\sqrt{6\alpha}}\right)\phantom{\rule{0.277778em}{0ex}},$$- -
- Quantum gravity [F]
- -
- Sub-Hubble mass inflaton field [F]
- -
- Supersymmetry [F]
- -
- Symmetries not of type in SM—superconformal [T]
- -
- Symmetries included in the model—three chiral multiplets and Kähler potential with superconformal and $SU(1,1)$ symmetries (5) [T]
- -
- Model building beyond the Standard Model [T]

- This gives a speculation count for Fundamental/Technical properties of 3/7.
**${\mathit{R}}^{\mathbf{2}}$ Starobinsky model [39,100]:**This is a type of modified gravity model which has a curvature-squared ${R}^{2}/\left(6{M}^{2}\right)$ term added to the Einstein–Hilbert action, where R is the Ricci scalar and $M<{m}_{p}$. This action is transformed into the Einstein frame leading to an inflaton potential of the form,$${V}_{{R}^{2}}\left(\varphi \right)={\mathsf{\Lambda}}^{4}{\left[1-exp\left(-\sqrt{\frac{2}{3}}\frac{\varphi}{{m}_{p}}\right)\right]}^{2}\phantom{\rule{0.277778em}{0ex}}.$$- -
- Quantum gravity [F]
- -
- Modifications to gravity beyond GR [F]
- -
- Sub-Hubble mass inflaton field [F]
- -
- Symmetry not of type in SM—transform from the Jordan to Einstein frame [T]
- -
- Symmetry included in the model [T]
- -
- Model building beyond the Standard Model [T]

- This gives a speculation count for Fundamental/Technical properties of 3/3. In the original paper by Starobinsky, he had viewed the ${R}^{2}$ term as dynamically generated as a self-consistent solution of the vacuum Einstein equations by one loop corrections due to quantized matter fields. The model can also have quantum gravity interactions treated semiclassically but these become subdominant for sufficient number of matter fields. Thus one could also count the assumptions from such a more first-principles approach, but that would need the details about the matter fields and interactions. Nevertheless in such a case the fundamental assumption added in our above list of modifications to gravity beyond GR would not be included, although assumptions about the underlying matter fields would need to be added.
**Higgs Inflation model [101]:**This model assumes there are no other fields in the universe aside from those in the Standard Model, and the Higgs field has a non-minimal coupling to gravity. In the initial Jordan frame the Higgs field, h, has a standard type of quartic symmetry breaking potential of the form ∼$\lambda {({h}^{2}-{v}^{2})}^{2}$. To get rid of the non-minimal coupling to gravity, a conformal transformation is done to the Einstein frame. The Higgs field is then treated as the inflaton, which at high field value has the potential in the Einstein frame,$${V}_{\mathrm{Higgs}\phantom{\rule{4pt}{0ex}}\mathrm{inflation}}\left(\varphi \right)=\frac{\lambda {m}_{p}^{4}}{4{\xi}^{2}}{\left[1+exp\left(-\frac{2\varphi}{\sqrt{6}{m}_{p}}\right)\right]}^{-2}\phantom{\rule{0.277778em}{0ex}},$$- -
- Quantum gravity [F]
- -
- Modification to gravity beyond GR [F]
- -
- Sub-Hubble mass inflaton field [F]
- -
- Symmetry not of type in SM—transform from the Jordan to Einstein frame [T]
- -
- Symmetry included in the model [T]
- -
- Model building beyond the Standard Model [T]

- This gives a speculation count for Fundamental/Technical properties of 3/3.
**Warm little inflaton model [62,83]:**This model has two complex Higgs fields with identical $U\left(1\right)$ charges. The fields have nonzero vacuum expectation values and the phases of both fields then yield two Nambu–Goldstone bosons. The relative phase of the two fields yields a singlet which is the inflaton. The Higgs fields are coupled to left-handed fermions with $U\left(1\right)$ charge and right-handed counterparts that are gauge singlets. There is an interchange symmetry between the two bosons and two fermions and they have identical couplings. There is an additional chiral fermion and singlet bosonic field to couple with the fermions for the particle creation decay width. The interaction Lagrangian for this model is given in Equation (22). The inflaton potential for this model is simply monomials,$${V}_{\mathrm{warm}\phantom{\rule{4pt}{0ex}}\mathrm{little}}\left(\varphi \right)=\frac{\lambda}{4!}{\varphi}^{4}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{or}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\frac{1}{2}{m}_{\varphi}^{2}{\varphi}^{2}\phantom{\rule{0.277778em}{0ex}}.$$- -
- Effective field theory methods with cutoff below ${m}_{p}$ [T]
- -
- Symmetries included in the model—two Nambu–Goldstone bosons, two $U\left(1\right)$, two interchange (6) [T]
- -
- Extra fields beyond the SM not attributed to any symmetries (2) [T]
- -
- Model building beyond the Standard Model [T]

- This gives a speculation count for Fundamental/Technical properties of 0/10. This is the only model studied here with no speculation counts in the fundamental category. Note that in the weak dissipative regime the model would have at least one and up to two fundamental counts for sub-Hubble mass and quantum gravity, the latter because in cases there can be super-Planckian field excursion of the inflaton field. The speculation count highlights the importance of the strong dissipative regime of warm inflation. Nevertheless even in the weak dissipative regime there are less speculation counts in the fundamental category than all the other models examined here.

## 10. Discussion

## Funding

## Conflicts of Interest

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Fundamental [F] |
---|

Quantum gravity |

Additional spacetime dimensions above four |

Modifications to gravity beyond General Relativity |

Sub-Hubble mass scalar fields |

Supersymmetry/other new spacetime symmetries or adjustments to them |

Technical [T] |

Effective field theory methods with cutoff scale below ${m}_{p}$ |

Symmetries included in the model not of the type in the Standard Model and excluding new spacetime symmetries |

Symmetries included in the model |

Extra fields added beyond the Standard Model and not attributed to any symmetry |

Model building beyond the Standard Model |

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Berera, A.
The Warm Inflation Story. *Universe* **2023**, *9*, 272.
https://doi.org/10.3390/universe9060272

**AMA Style**

Berera A.
The Warm Inflation Story. *Universe*. 2023; 9(6):272.
https://doi.org/10.3390/universe9060272

**Chicago/Turabian Style**

Berera, Arjun.
2023. "The Warm Inflation Story" *Universe* 9, no. 6: 272.
https://doi.org/10.3390/universe9060272