How many $e$folds should we expect from highscale inflation?
Abstract
In this study, we address the issue of how many $e$folds we would naturally expect if inflation occurred at an energy scale of order ${10}^{16}\text{}\text{}\mathrm{GeV}$. We use the canonical measure on trajectories in classical phase space, specialized to the case of flat universes with a single scalar field. While there is no exact analytic expression for the measure, we are able to derive conditions that determine its behavior. For a quadratic potential $V\left(\Phi \right)={m}^{2}{\Phi}^{2}/2$ with $m=2\times {10}^{13}\text{}\text{}\mathrm{GeV}$ and cutoff at ${M}_{\text{Pl}}=2.4\times {10}^{18}\text{}\text{}\mathrm{GeV}$, we find an expectation value of $2\times {10}^{10}$ $e$folds on the set of FriedmannRobertsonWalker trajectories. For cosine inflation $V\left(\Phi \right)={\mathrm{\Lambda}}^{4}[1\mathrm{cos}(\Phi /f\left)\right]$ with $f=1.5\times {10}^{19}\text{}\text{}\mathrm{GeV}$, we find that the expected total number of $e$folds is 50, which would just satisfy the observed requirements of our own Universe; if $f$ is larger, more than 50 $e$folds are generically attained. Finally, we conclude that one should expect a large amount of inflation in largefield models and more limited inflation in smallfield (hilltop) scenarios.
 Authors:

 California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics
 Publication Date:
 Research Org.:
 California Institute of Technology (CalTech), Pasadena, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); Betty Moore Foundation
 OSTI Identifier:
 1596472
 Alternate Identifier(s):
 OSTI ID: 1180062
 Grant/Contract Number:
 SC0011632; DGE1144469; 776
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review. D, Particles, Fields, Gravitation and Cosmology
 Additional Journal Information:
 Journal Volume: 90; Journal Issue: 6; Journal ID: ISSN 15507998
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Citation Formats
Remmen, Grant N., and Carroll, Sean M. How many efolds should we expect from highscale inflation?. United States: N. p., 2014.
Web. doi:10.1103/PhysRevD.90.063517.
Remmen, Grant N., & Carroll, Sean M. How many efolds should we expect from highscale inflation?. United States. doi:10.1103/PhysRevD.90.063517.
Remmen, Grant N., and Carroll, Sean M. Mon .
"How many efolds should we expect from highscale inflation?". United States. doi:10.1103/PhysRevD.90.063517. https://www.osti.gov/servlets/purl/1596472.
@article{osti_1596472,
title = {How many efolds should we expect from highscale inflation?},
author = {Remmen, Grant N. and Carroll, Sean M.},
abstractNote = {In this study, we address the issue of how many efolds we would naturally expect if inflation occurred at an energy scale of order 1016 GeV. We use the canonical measure on trajectories in classical phase space, specialized to the case of flat universes with a single scalar field. While there is no exact analytic expression for the measure, we are able to derive conditions that determine its behavior. For a quadratic potential V(Φ)=m2Φ2/2 with m=2×1013 GeV and cutoff at MPl=2.4×1018 GeV, we find an expectation value of 2×1010 efolds on the set of FriedmannRobertsonWalker trajectories. For cosine inflation V(Φ)=Λ4[1cos(Φ/f)] with f=1.5×1019 GeV, we find that the expected total number of efolds is 50, which would just satisfy the observed requirements of our own Universe; if f is larger, more than 50 efolds are generically attained. Finally, we conclude that one should expect a large amount of inflation in largefield models and more limited inflation in smallfield (hilltop) scenarios.},
doi = {10.1103/PhysRevD.90.063517},
journal = {Physical Review. D, Particles, Fields, Gravitation and Cosmology},
number = 6,
volume = 90,
place = {United States},
year = {2014},
month = {9}
}
Web of Science
Works referenced in this record:
Natural Chaotic Inflation in Supergravity
journal, October 2000
 Kawasaki, M.; Yamaguchi, Masahide; Yanagida, T.
 Physical Review Letters, Vol. 85, Issue 17
Attractor solutions in scalarfield cosmology
journal, October 2013
 Remmen, Grant N.; Carroll, Sean M.
 Physical Review D, Vol. 88, Issue 8
Natural inflation in supergravity and beyond
journal, August 2014
 Kallosh, Renata; Linde, Andrei; Vercnocke, Bert
 Physical Review D, Vol. 90, Issue 4
Rescuing quadratic inflation
journal, February 2014
 Ellis, John; Fairbairn, Malcolm; Sueiro, Maria
 Journal of Cosmology and Astroparticle Physics, Vol. 2014, Issue 02
Extranatural Inflation
journal, June 2003
 ArkaniHamed, Nima; Cheng, HsinChia; Creminelli, Paolo
 Physical Review Letters, Vol. 90, Issue 22
Recent progress in the inflationary universe scenario
journal, January 1985
 Linde, A. D.
 Nuclear Physics B, Vol. 252
Detection of $B$ Mode Polarization at Degree Angular Scales by BICEP2
journal, June 2014
 Ade, P. A. R.; Aikin, R. W.; Barkats, D.
 Physical Review Letters, Vol. 112, Issue 24
A natural measure on the set of all universes
journal, February 1987
 Gibbons, G. W.; Hawking, S. W.; Stewart, J. M.
 Nuclear Physics B, Vol. 281, Issue 34
Natural inflation: Particle physics models, powerlaw spectra for largescale structure, and constraints from the Cosmic Background Explorer
journal, January 1993
 Adams, Fred C.; Bond, J. Richard; Freese, Katherine
 Physical Review D, Vol. 47, Issue 2
UltravioletProtected Inflation
journal, April 2011
 Germani, Cristiano; Kehagias, Alex
 Physical Review Letters, Vol. 106, Issue 16
Natural inflation with pseudo NambuGoldstone bosons
journal, December 1990
 Freese, Katherine; Frieman, Joshua A.; Olinto, Angela V.
 Physical Review Letters, Vol. 65, Issue 26
Difficulties with Inflationary Cosmology
journal, December 1989
 Penrose, Roger
 Annals of the New York Academy of Sciences, Vol. 571, Issue 1 Texas Symposi
Measure problem in cosmology
journal, March 2008
 Gibbons, G. W.; Turok, Neil
 Physical Review D, Vol. 77, Issue 6
Chaotic inflation
journal, March 1988
 Madsen, Mark S.; Coles, Peter
 Nuclear Physics B, Vol. 298, Issue 4
Works referencing / citing this record:
Global dynamics and inflationary center manifold and slowroll approximants
journal, January 2015
 Alho, Artur; Uggla, Claes
 Journal of Mathematical Physics, Vol. 56, Issue 1