Phys. Rev. D 102, 103515 (2020) - Natural inflation, strong dynamics, and the role of generalized anomalies

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Natural inflation, strong dynamics, and the role of generalized anomalies

Mohamed M. Anber and Stephen Baker

Phys. Rev. D 102, 103515 – Published 13 November 2020

Abstract

We revisit models of natural inflation and show that the single-field effective theory described by the potential $V (a) \sim \cos \frac{a}{f}$ breaks down as the inflaton $a$ makes large-field excursions, even for values of $f$ smaller than the Planck scale. To remedy the problem, we modify the potential in order to account for the heavy degrees of freedom (hadrons) that become intertwined with the light inflaton as the latter rolls down its potential. By embedding the low-energy degrees of freedom into an ultraviolet-complete gauge theory, we argue that the intertwining between the two scales can be explained as the result of a generalized mixed ’t Hooft anomaly between the discrete chiral symmetry and background fractional fluxes in the baryon number, color, and flavor directions. Further, we study the multifield inflation and show that it entertains rich dynamics. Inflating near the hilltop excites the hadrons and spoils the slow-roll parameters, in contradistinction with the expectations in the single-field inflation. Nevertheless, we identify a safe zone where inflation can proceed successfully. We determine the conditions under which the Universe inflates by at least 60 $e$ -foldings and inflation leads to a power spectrum and tensor to scalar ratio that are consistent with the cosmic microwave background data.

Received 30 August 2020
Accepted 13 October 2020

DOI:https://doi.org/10.1103/PhysRevD.102.103515

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Anomalies Axions Color confinement Effective field theory Inflation

Particles & FieldsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Mohamed M. Anber^* and Stephen Baker ^†

Department of Physics, Lewis & Clark College, Portland, Oregon 97219, USA

^*manber@lclark.edu
^†sbaker@lclark.edu

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Vol. 102, Iss. 10 — 15 November 2020

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Figure 1
A typical plot of the QCD-axion potential $V (a, σ)$ in the $S U (N_{c} = 2)$ case (but not to scale). Since $f ≫ Λ$ , the potential is very steep in the $σ$ direction. Thus, any small fluctuations near the black dot, which indicates the initial value of $a \approx π f$ , will cause the inflation to proceed very quickly in the $σ$ direction (as indicated by the arrows), and thus inflation ends abruptly. However, the axion can slowly roll down the potential, provided that we start inflating near $a \approx \frac{π}{2} f$ , irrespective of the initial value of $σ$ , which is indicated by the thick green line.
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Figure 2
The phase space of $a$ and $σ$ . We take $Λ = 10^{- 3} \sqrt{8 π} M_{P}$ and $f = 6 \sqrt{8 π} M_{P}$ and start inflation at $a \approx \frac{π}{2} f$ and $σ \approx \frac{π}{2} Λ$ . We find, however, that the dynamics of inflation is insensitive to the initial values of $σ$ . One can easily see that the axion velocity stays small throughout the inflation life span.
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Figure 3
The parametric relation between $σ$ and $a$ . We take $Λ = 10^{- 3} \sqrt{8 π} M_{P}$ and $f = 6 \sqrt{8 π} M_{P}$ . While the axion is rolling down from $\frac{π}{2} f$ to $π f$ , the field $σ$ is almost instantaneously frozen (after six $e$ -folds in this example) at $π Λ$ , which is consistent with the analytical finding Eq. (27).
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Figure 4
The dependence of the number of $e$ -folds on the initial value of $a$ . The scattered black points are the numerical values, we take $Λ = 10^{- 3} \sqrt{8 π} M_{P}$ and $f = 6 \sqrt{8 π} M_{P}$ , while the continuous green line is the analytical expression (29). The maximum number is achieved by starting the inflation at $a (0) \approx \frac{π}{2} f$ . We also checked that the number of $e$ -folds is very insensitive to the value of $Λ$ . It changes by less than 5% as $Λ$ changes between $10^{- 2} M_{P}$ and $10^{- 5} M_{P}$ .
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Figure 5
The numerical values of $ε$ and $η$ (the solid lines) against the analytical expressions (dotted lines) given in Eq. (32) as functions of $N_{e}^{⋆}$ , the number of $e$ -folds remaining before the end of inflation. We take $Λ = 10^{- 3} \sqrt{8 π} M_{P}$ and $f = 6 \sqrt{8 π} M_{P}$ . The discrepancy between the analytical and numerical solutions is less than 2%. We also find that changing the value of $Λ$ has a little effect on both $ε$ and $η$ , in agreement with Eq. (32) that does not depend explicitly on $Λ$ .
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Figure 6
The numerical data for the $S U (N_{c} = 3)$ case. The roots are $α_{1} = (\sqrt{2}, 0)$ , $α_{2} = (- \frac{1}{\sqrt{2}}, \sqrt{\frac{3}{2}})$ , and $α_{3} = (- \frac{1}{\sqrt{2}}, - \sqrt{\frac{3}{2}})$ ; see Ref. [42]. We take $f = 6 \sqrt{8 π} M_{P}$ and $Λ = 10^{- 3} \sqrt{8 π} M_{P}$ . The top template shows the total number of $e$ -folds as a function of the initial position of the axion. An initial position at $\frac{π}{3} f$ yields the largest value of the number of $e$ -folds. The bottom template shows the values of $ε$ and $η$ as functions of the number of $e$ -folds remaining before the end of inflation. We can see that these values are almost identical to the ones in the $S U (N_{c} = 2)$ case and that the color group has a little effect on axion inflation once the axion is in the safe zone.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Natural inflation, strong dynamics, and the role of generalized anomalies

Mohamed M. Anber and Stephen Baker

Phys. Rev. D 102, 103515 – Published 13 November 2020

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Physical Review D

covering particles, fields, gravitation, and cosmology

Natural inflation, strong dynamics, and the role of generalized anomalies

Mohamed M. Anber and Stephen Baker

Phys. Rev. D 102, 103515 – Published 13 November 2020

Abstract

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