We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann–Lemaître–Robertson–Walker universe. The existence and properties of separatrices are investigated in the framework of the Hamilton–Jacobi formalism, where the main quantity is the Hubble parameter considered as a function of the inflaton field. A wide class of inflaton models that have separatrix solutions (and include many of the most physically relevant potentials) is introduced, and the properties of the corresponding separatrices are investigated, in particular, asymptotic inflationary stages, leading approximations to the separatrices, and full asymptotic expansions thereof. We also prove an optimal growth criterion for potentials that do not have separatrices.
REFERENCES
1.
A. A.
Starobinsky
, “A new type of isotropic cosmological models without singularity
,” Phys. Lett. B
91
, 99
(1980
).2.
A. H.
Guth
, “Inflationary universe: A possible solution to the horizon and flatness problems
,” Phys. Rev. D
23
, 347
(1981
).3.
A. D.
Linde
, “A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems
,” Phys. Lett. B
108
, 389
(1982
).4.
V.
Mukhanov
, Physical Foundations of Cosmology
(Cambridge University Press
, 2005
).5.
6.
V. A.
Belinskii
, L. P.
Grishchuk
, Y. B.
Zel’dovich
, and I. M.
Khalatnikov
, “Inflationary stages in cosmological models with a scalar field
,” Sov. Phys. JETP
62
, 195
(1985
).7.
A. R.
Liddle
, P.
Parsons
, and J. D.
Barrow
, “Formalizing the slow-roll approximation in inflation
,” Phys. Rev. D
50
, 7222
(1994
).8.
G. N.
Remmen
and S. M.
Carroll
, “Attractor solutions in scalar-field cosmology
,” Phys. Rev. D
88
, 083518
(2013
).9.
M.
Hurley
, “Attractors: Persistence, and density of their basins
,” Trans. Am. Math. Soc.
269
, 247
(1982
).10.
E. J.
Copeland
, A. R.
Liddle
, and D.
Wands
, “Exponential potentials and cosmological scaling solutions
,” Phys. Rev. D
57
, 4686
(1998
).11.
S.
Foster
, “Scalar field cosmological models with hard potential walls
,” arXiv:gr-qc/9806113 (1998
).12.
C.
Uggla
, “Global cosmological dynamics for the scalar field representation of the modified Chaplygin gas
,” Phys. Rev. D
88
, 064040
(2013
).13.
N.
Frusciante
, M.
Raveri
, and A.
Silvestri
, “Effective field theory of dark energy: A dynamical analysis
,” J. Cosmol. Astropart. Phys.
2014
, 026
.14.
N.
Roy
and N.
Banerjee
, “Quintessence scalar field: A dynamical systems study
,” Eur. Phys. J. Plus
129
, 162
(2014
).15.
N.
Tamanini
, “Dynamics of cosmological scalar fields
,” Phys. Rev. D
89
, 083521
(2014
).16.
A.
Paliathanasis
, M.
Tsamparlis
, S.
Basilakos
, and J. D.
Barrow
, “Dynamical analysis in scalar field cosmology
,” Phys. Rev. D
91
, 123535
(2015
).17.
R.
García-Salcedo
, T.
Gonzalez
, F. A.
Horta-Rangel
, I.
Quiros
, and D.
Sanchez-Guzmán
, “Introduction to the application of dynamical systems theory in the study of the dynamics of cosmological models of dark energy
,” Eur. J. Phys.
36
, 025008
(2015
).18.
A.
Alho
and C.
Uggla
, “Global dynamics and inflationary center manifold and slow-roll approximants
,” J. Math. Phys.
56
, 012502
(2015
).19.
A.
Alho
and C.
Uggla
, “Scalar field deformations of ΛCDM cosmology
,” Phys. Rev. D
92
, 103502
(2015
).20.
A.
Alho
and C.
Uggla
, “Inflationary α-attractor cosmology: A global dynamical systems perspective
,” Phys. Rev. D
95
, 083517
(2017
).21.
S.
Bahamonde
, C. G.
Böhmer
, S.
Carloni
, E. J.
Copeland
, W.
Fang
, and N.
Tamanini
, “Dynamical systems applied to cosmology: Dark energy and modified gravity
,” Phys. Rep.
775-777
, 1
(2018
).22.
D. S.
Salopek
and J. R.
Bond
, “Nonlinear evolution of long-wavelength metric fluctuations in inflationary models
,” Phys. Rev. D
42
, 3936
(1990
).23.
W.
Handley
, S.
Brechet
, A.
Lasenby
, and M.
Hobson
, “Kinetic initial conditions for inflation
,” Phys. Rev. D
89
, 063505
(2014
).24.
25.
G.
Teschl
, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics
(American Mathematical Society
, 2012
), Vol. 140.26.
N.
Onuchic
, “Invariance properties in the theory of ordinary differential equations with applications to stability problems
,” SIAM J. Control
9
, 97
(1971
).27.
N.
Onuchic
, “Invariance and stability for ordinary differential equations
,” J. Math. Anal. Appl.
63
, 9
(1978
).28.
S.
Murakami
, “Asymptotic behavior of solutions of ordinary differential equations
,” Tohoku Math. J.
34
, 559
(1982
).29.
C.
Destri
, H. J.
de Vega
, and N. G.
Sanchez
, “Pre-inflationary and inflationary fast-roll eras and their signatures in the low CMB multipoles
,” Phys. Rev. D
81
, 063520
(2010
).30.
W.
Handley
, A.
Lasenby
, and M.
Hobson
, “Logolinear series expansions with applications to primordial cosmology
,” Phys. Rev. D
99
, 123512
(2019
).31.
D. H.
Lyth
and A. R.
Liddle
, The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure
(Cambridge University Press
, 2009
).32.
J. J.
Halliwell
, “Scalar fields in cosmology with an exponential potential
,” Phys. Lett. B
185
, 341
(1987
).33.
J.
Martin
, C.
Ringeval
, and V.
Vennin
, “Encyclopaedia inflationaris
,” Phys. Dark Univ.
5-6
, 75
(2014
).34.
G.
Álvarez
and H. J.
Silverstone
, “A new method to sum divergent power series: Educated match
,” J. Phys. Commun.
1
, 025005
(2017
).35.
Handbook of Mathematical Functions
, edited by M.
Abramowitz
and I. A.
Stegun
(Dover
, 1972
).36.
R.
Kallosh
and A.
Linde
, “Universality class in conformal inflation
,” J. Cosmol. Astropart. Phys.
2013
, 002
.37.
S.
Ferrara
, R.
Kallosh
, A.
Linde
, and M.
Porrati
, “Minimal supergravity models of inflation
,” Phys. Rev. D
88
, 085038
(2013
).38.
R.
Kallosh
, A.
Linde
, and D.
Roest
, “Superconformal inflationary α-attractors
,” J. High Energy Phys.
2013
, 1311
.39.
B.
Whitt
, “Fourth order gravity as general relativity plus matter
,” Phys. Lett. B
145
, 176
(1984
).40.
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