# Quintessential Inflation with Dynamical Higgs Generation as an Affine Gravity

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## Abstract

**:**

## 1. Introduction

## 2. The Essence of the Non-Riemannian Volume-Form Formalism

## 3. Quintessential Inflationary Model with Dynamical Higgs Effect

- The scalar curvature $R(g,\Gamma )={g}^{\mu \nu}{R}_{\mu \nu}(\Gamma )$ is given in terms of the Ricci tensor ${R}_{\mu \nu}(\Gamma )$ in the first-order (Palatini) formalism:$${R}_{\mu \nu}(\Gamma )={\partial}_{\alpha}{\Gamma}_{\mu \nu}^{\alpha}-{\partial}_{\nu}{\Gamma}_{\mu \alpha}^{\alpha}+{\Gamma}_{\alpha \beta}^{\alpha}{\Gamma}_{\mu \nu}^{\beta}-{\Gamma}_{\beta \nu}^{\alpha}{\Gamma}_{\mu \alpha}^{\beta}$$
- The non-Riemannian volume-element densities ${\Phi}_{1}\left(A\right),{\Phi}_{2}\left(B\right),{\Phi}_{0}\left(C\right)$ are defined as in (6):$${\Phi}_{1}\left(A\right)=\frac{1}{3!}{\epsilon}^{\mu \nu \kappa \lambda}\phantom{\rule{0.166667em}{0ex}}{\partial}_{\mu}{A}_{\nu \kappa \lambda},\phantom{\rule{1.em}{0ex}}{\Phi}_{2}\left(B\right)=\frac{1}{3!}{\epsilon}^{\mu \nu \kappa \lambda}\phantom{\rule{0.166667em}{0ex}}{\partial}_{\mu}{B}_{\nu \kappa \lambda},\phantom{\rule{1.em}{0ex}}{\Phi}_{0}\left(C\right)=\frac{1}{3!}{\epsilon}^{\mu \nu \kappa \lambda}\phantom{\rule{0.166667em}{0ex}}{\partial}_{\mu}{C}_{\nu \kappa \lambda}\phantom{\rule{0.277778em}{0ex}}.$$
- $\varphi $ is a neutral scalar “inflaton” and $\sigma \equiv \left({\sigma}_{a}\right)$ is a complex $SU\left(2\right)\times U\left(1\right)$ iso-doublet Higgs-like scalar field with the isospinor index $a=+,0$ indicating the corresponding $U\left(1\right)$ charge. The corresponding kinetic energy terms in (9) read:$${X}_{\varphi}\equiv -\frac{1}{2}{g}^{\mu \nu}{\partial}_{\mu}\varphi {\partial}_{\nu}\varphi ,\phantom{\rule{1.em}{0ex}}{X}_{\sigma}\equiv -{g}^{\mu \nu}{\partial}_{\mu}{\sigma}_{a}^{*}{\partial}_{\nu}{\sigma}_{a}\phantom{\rule{0.277778em}{0ex}},$$$${V}_{0}\left(\sigma \right)\equiv {m}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{a}^{*}{\sigma}_{a}\phantom{\rule{0.277778em}{0ex}},$$
- ${f}_{1,2}$ and $\alpha $ are dimensionful coupling constants in the “inflaton” potential. The ${\mathsf{\Lambda}}_{0}$ is a small dimensional constant which will be identified in the sequel with the “late” universe cosmological constant in the dark energy dominated accelerated expansion’s epoch.

- (a) (−) flat “inflaton” region for large negative values of $\varphi $ (and ${\sigma}_{a}$ is finite) corresponding to the “slow-roll” inflationary evolution of the “early” universe driven by $\varphi $. Here the effective potential (26) reduces to (an almost) constant value independent of the finite value of ${\sigma}_{a}$—this is energy scale of the inflationary epoch:$${U}_{\mathrm{eff}}\left(\varphi ,\sigma \right)\simeq {U}_{(-)}=\frac{{M}_{1}^{2}}{4{\chi}_{2}\phantom{\rule{0.166667em}{0ex}}{M}_{2}}+2{\mathsf{\Lambda}}_{0}\phantom{\rule{0.277778em}{0ex}}.$$Thus, in the “early” universe the Higgs-like field ${\sigma}_{a}$ must be (approximately) either massless or constant with no non-zero vacuum expectation value, therefore there is no spontaneous breaking of $SU\left(2\right)\times U\left(1\right)$ symmetry. Moreover, in fact as shown in the Remark below, ${\sigma}_{a}$ does not participate in the “slow-roll” inflationary evolution, so $\sigma $ stays constant there equal to the “false”vacuum value $\sigma =0$.
- (b) (+) flat “inflaton” region for large positive values of $\varphi $ (and ${\sigma}_{a}$—finite) corresponding to the evolution of the post-inflationary (“late”) universe, where:$${U}_{\mathrm{eff}}\left(\varphi ,\sigma \right)\simeq {U}_{(+)}\left(\sigma \right)=\frac{{\left({m}_{0}^{2}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{a}^{*}{\sigma}_{a}-{f}_{1}\right)}^{2}}{4{\chi}_{2}{f}_{2}}+2{\mathsf{\Lambda}}_{0}$$$$|{\sigma}_{\mathrm{vac}}|=\frac{1}{{m}_{0}}\sqrt{{f}_{1}}\phantom{\rule{0.277778em}{0ex}},$$$${f}_{1}\sim {M}_{EW}^{4},\phantom{\rule{1.em}{0ex}}{m}_{0}\sim {M}_{EW}$$
- Thus, the residual cosmological constant ${\mathsf{\Lambda}}_{0}$ in (28) has to be identified with the current epoch observable cosmological constant ($\sim {10}^{-122}{M}_{Pl}^{4}$) and, therefore, according to (27) the integration constants ${M}_{1,2}$ are naturally identified by orders of magnitude as$${M}_{1}\sim {M}_{2}\sim {10}^{-8}{M}_{Pl}^{4}\phantom{\rule{0.277778em}{0ex}},$$$${U}_{(-)}\sim {M}_{1}^{2}/{M}_{2}\sim {10}^{-8}{M}_{Pl}^{4}\phantom{\rule{0.277778em}{0ex}},$$
- Here the order of magnitude for ${f}_{2}$ is determined from the mass term of the Higgs-like field $\sigma $ in the $(+)$ flat region resulting from (28) upon expansion around the Higgs vacuum ($\sigma ={\sigma}_{\mathrm{vac}}+\tilde{\sigma}$):$$\frac{{f}_{1}{m}_{0}^{2}}{{\chi}_{2}{f}_{2}}\phantom{\rule{0.166667em}{0ex}}{\left({\tilde{\sigma}}_{a}\right)}^{*}\left({\tilde{\sigma}}_{a}\right)\phantom{\rule{0.277778em}{0ex}},$$$${f}_{2}\sim {f}_{1}\sim {M}_{EW}^{4}\phantom{\rule{0.277778em}{0ex}}.$$
- Let us specifically note that the viability of the present model (in a slightly simplified form without the Higgs scalar) concerning confrontation with the observational data has already been analyzed and confirmed numerically in Reference [141]. In particular, a graphical plot of the evolution of r (tensor-to-scalar ratio) vs. ${n}_{s}$ (scalar spectral index) has been provided there.

**Remark**

**1.**

## 4. Eddinton-Type No-Metric Gravity and Quintessential Inflation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Qualitative shape of the two-dimensional plot for the effective scalar potential ${U}_{\mathrm{eff}}(\varphi ,\sigma )$ (26).

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**MDPI and ACS Style**

Benisty, D.; Guendelman, E.I.; Nissimov, E.; Pacheva, S. Quintessential Inflation with Dynamical Higgs Generation as an Affine Gravity. *Symmetry* **2020**, *12*, 734.
https://doi.org/10.3390/sym12050734

**AMA Style**

Benisty D, Guendelman EI, Nissimov E, Pacheva S. Quintessential Inflation with Dynamical Higgs Generation as an Affine Gravity. *Symmetry*. 2020; 12(5):734.
https://doi.org/10.3390/sym12050734

**Chicago/Turabian Style**

Benisty, David, Eduardo I. Guendelman, Emil Nissimov, and Svetlana Pacheva. 2020. "Quintessential Inflation with Dynamical Higgs Generation as an Affine Gravity" *Symmetry* 12, no. 5: 734.
https://doi.org/10.3390/sym12050734