Hypermagnetogenesis from axion inflation: model-independent estimates

Article
Report number	arXiv:2111.04712 ; CERN-TH-2021-185
Title	Hypermagnetogenesis from axion inflation: model-independent estimates
Author(s)	Gorbar, E.V. (Taras Shevchenko U. ; BITP, Kiev) ; Schmitz, K. (CERN) ; Sobol, O.O. (Taras Shevchenko U. ; EPFL, Lausanne, LPPC) ; Vilchinskii, S.I. (Taras Shevchenko U. ; Geneva U.)
Publication	2022-02-15
Imprint	2021-11-08
Number of pages	22
In:	Phys. Rev. D 105 (2022) 043530
DOI	10.1103/PhysRevD.105.043530 (publication)
Subject category	hep-th ; Particle Physics - Theory ; astro-ph.CO ; Astrophysics and Astronomy ; hep-ph ; Particle Physics - Phenomenology
Abstract	Axion inflation coupled to the Standard Model (SM) hypercharge gauge sector represents an attractive scenario for the generation of primordial hypermagnetic fields. The description of this scenario is, however, complicated by the Schwinger effect, which gives rise to highly nonlinear dynamics. Hypermagnetogenesis during axion inflation in the absence of nonlinear effects is well studied and known to result in a hypermagnetic energy density that scales like $H^4\,e^{2\pi\xi}/\xi^5$, where $\xi$ is proportional to the time derivative of the axion-vector coupling in units of the Hubble rate $H$. In this paper, we generalize this result to the full SM case by consistently taking into account the Schwinger pair production of all SM fermions. To this end, we employ the novel gradient-expansion formalism that we recently developed in [2109.01651], and which is based on a set of vacuum expectation values for bilinear hyperelectromagnetic functions in position space. We parametrize the numerical output of our formalism in terms of three parameters ($\xi$, $H$, and $\Delta$, where the latter accounts for the damping of subhorizon gauge-field modes because of the finite conductivity of the medium) and work out semianalytical fit functions that describe our numerical results with high accuracy. Finally, we validate our results by comparing them to existing estimates in the literature as well as to the explicit numerical results in a specific inflationary model, which leads to good overall agreement. We conclude that the systematic uncertainties in the description of hypermagnetogenesis during axion inflation, which previously spanned up to several orders of magnitude, are now reduced to typically less than 1 order of magnitude, which paves the way for further phenomenological studies.
Copyright/License	publication: © 2022-2024 authors (License: CC BY 4.0), sponsored by SCOAP³ preprint: (License: arXiv nonexclusive-distrib 1.0)

$Model-independent results for the efficiency of hypermagnetogenesis during axion inflation, specifically, for the hyperelectric energy density $\rho_E$ (column a), hypermagnetic energy density $\rho_B$ (column b), and hypercharge Chern-Pontryagin density $\left|\langle\bm{E}\cdot\bm{B}\rangle\right|$ (column c), in dependence on the gauge-field production parameter $\xi$, Hubble rate $H$, and damping factor $\Delta$. The Hubble rate is varied between $10^8$ and $10^{14}\,\textrm{GeV}$; the dependence on $H$ is shown in the form of bands whose upper (lower) edges correspond to $H = 10^8\,\textrm{GeV}$ ($H = 10^{14}\,\textrm{GeV}$). The gray and purple bands in the first row, respectively, correspond to the maximal and equilibrium estimates derived in Refs.~\cite{Domcke:2018,Domcke:2019bar} (see Secs.~\ref{subsec:maxestimate} and \ref{subsec:eqestimate}). All other colorful bands in the first row correspond to our new results based on the gradient-expansion formalism. For all three types of estimates, we derive fit functions; in rows 2 to 7, we compare these fit functions to the exact numerical results in row 1.$ $Model-independent estimates of the generated gauge fields during axion inflation in a specific inflationary model with the potential (\ref{inflaton-potential}) and axial-coupling function (\ref{axial-coupling}) for the coupling parameter $\beta=10$, compared to the numerical results in the same model obtained by means of a full-fledged numerical analysis in the gradient expansion formalism. The magnitude of the generated hyperelectric energy density $\rho_E$ [column (a)], hypermagnetic energy density $\rho_B$ [column (b)], and hypercharge Chern--Pontryagin density $\left|\langle\bm{E}\cdot\bm{B}\rangle\right|$ [column (c)] are shown in row (1) as functions of the gauge-field production parameter $\xi$ (lower labels on the horizontal axis) and the number of $e$-folds until the end of inflation (upper labels on the horizontal axis). The black solid lines show the numerical results obtained from the full gradient expansion formalism of Ref.~\cite{Gorbar:2021} for the specific model under consideration. The values of the parameters $\xi$, $H$, and $\Delta$ obtained from this numerical analysis are then used to compute the model-independent predictions for the generated fields: the red dashed lines show the model-independent gradient-expansion predictions for given $\xi$, $H$, and $\Delta$; the blue dashed-dotted lines show the model-independent gradient-expansion results for the given values of $\xi$ and $H$, and a fixed value of $\Delta=1$; the green dotted lines give the maximal estimates; and the purple dashed-dotted lines with two dots show the equilibrium estimates for given $\xi$ and $H$. Row (2) shows the accuracy of the model-independent results compared to the exact numerical solution; the types of curves correspond to those shown in row (1).$ $Model-independent (MI) estimates of the generated gauge fields during axion inflation in a specific inflationary model with potential (\ref{inflaton-potential}) and the axial-coupling function (\ref{axial-coupling}) for the coupling parameter $\beta=10$, compared to the numerical results in the same model obtained by means of a full-fledged numerical analysis in the gradient-expansion formalism. The magnitude of the generated hyperelectric energy density $\rho_E$ (column a), hypermagnetic energy density $\rho_B$ (column b), and hypercharge Chern-Pontryagin density $\left|\langle\bm{E}\cdot\bm{B}\rangle\right|$ (column c) are shown in row 1 as functions of the gauge-field production parameter $\xi$ (lower labels on the horizontal axis) and the number of $e$-folds until the end of inflation (upper labels on the horizontal axis). The black solid lines show the numerical results obtained from the full gradient-expansion formalism of Ref.~\cite{Gorbar:2021} for the specific model under consideration. The values of the parameters $\xi$, $H$, and $\Delta$ obtained from this numerical analysis are then used to compute the model-independent predictions for the generated fields: the red dashed lines show the model-independent gradient-expansion predictions for given $\xi$, $H$, and $\Delta$; the blue dashed-dotted lines show the model-independent gradient-expansion results for the given values of $\xi$ and $H$, and a fixed value of $\Delta=1$; the green dotted lines give the maximal estimates; and the purple dashed-dotted lines with double dot show the equilibrium estimates for given $\xi$ and $H$. Row 2 shows the accuracy of the model-independent results compared to the exact numerical solution; the types of curves correspond to those shown in row 1.$ Show more plots

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