Exponential potentials and cosmological scaling solutions
Open accessAbstract
We present a phase-plane analysis of cosmologies containing a baryotropic fluid with an equation of state ${p}_{\ensuremath{\gamma}}=(\ensuremath{\gamma}\ensuremath{-}1){\ensuremath{\rho}}_{\ensuremath{\gamma}},$ plus a scalar field $\ensuremath{\varphi}$ with an exponential potential $V\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\ensuremath{\lambda}\ensuremath{\kappa}\ensuremath{\varphi})$ where ${\ensuremath{\kappa}}^{2}=8\ensuremath{\pi}G$. In addition to the well-known inflationary solutions for ${\ensuremath{\lambda}}^{2}<2$, there exist scaling solutions when ${\ensuremath{\lambda}}^{2}>3\ensuremath{\gamma}$ in which the scalar field energy density tracks that of the baryotropic fluid (which for example might be radiation or dust). We show that the scaling solutions are the unique late-time attractors whenever they exist. The fluid-dominated solutions, where $V(\ensuremath{\varphi})/{\ensuremath{\rho}}_{\ensuremath{\gamma}}\ensuremath{\rightarrow}0$ at late times, are always unstable (except for the cosmological constant case $\ensuremath{\gamma}=0)$. The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to ${\ensuremath{\lambda}}^{2}>20$. We show that standard inflation models are unable to solve this ``relic density'' problem.
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For λ2>3γm, the scalar tracks the fluid with wφ=wm and Ωφ=3γm/λ2; this scaling point is stable in the (x,y) sector but, once curvature is included, the curvature direction is unstable for γm≥1, so in the full six-dimensional space it is a saddle.
“For λ 2 > 3γ m , the scalar tracks the fluid with w φ = w m and Ω φ = 3γ m /λ 2 [23,24].This point is stable in the (x, y) sector but, once curvature is included, the curvature direction is unstable for γ m ≥ 1 (the GR result of [26]).In the full six-dimensional space the scaling point is a saddle.”
From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [23]
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Experimental — accuracy not yet measuredThe extended EC model reduces consistently to the CLW system in the flat, isotropic, torsionless case, and the numerics are benchmarked against this baseline.
“confirming that the extended EC model reduces consistently to CLW in the flat, isotropic, torsionless case and that the numerics are benchmarked against this baseline.”
From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [23]
Section II (p. 1) presents the CLW autonomous system for a spatially flat FRW universe, supporting its identification as the baseline model. The paper does not examine an extended EC model or establish that it reduces to CLW, and it does not report benchmarking the citing authors’ numerics against that baseline. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThe eigenvalues at the scaling point are μ± = -[3(2 - γ_m)/4][1 ± sqrt(1 - 8γ_m(λ^2 - 3γ_m)/(λ^2(2 - γ_m)))], μΣ = (3/2)(γ_m - 2), μk = 3γ_m - 2, and μs = 3(γ_m - 2).
“The two non-trivial eigenvalues in the scalar-matter subspace are [23]”
From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [23]
The Appendix (p. 5) identifies two eigenvalues, m±, for the scaling solution, supporting the citation’s reference to two eigenvalues in the scalar–matter phase plane. Their formula is not decoded in our copy, so the stated expression for μ± cannot be checked. The citation is attached to the sentence about those two eigenvalues, not to the additional μΣ, μk, and μs values. (LLM, conf=0.82)
Experimental — accuracy not yet measuredIn the CLW limit, the fixed points and their stability properties coincide with the original CLW analysis: the scalar-field dominated point is stable for λ^2 < 3γ_m and drives accelerated expansion when λ^2 < 2; the scaling solution exists and is stable for λ^2 > 3γ_m; and the kinetic ± points are always unstable.
“In this limit the fixed points and their stability properties coincide with the original CLW analysis.The scalar-field dominated point is stable for λ 2 < 3γ m and drives accelerated expansion when λ 2 < 2; the scaling (tracking) solution exists and is stable for λ 2 > 3γ m ; and the kinetic ± points are always unstable.”
From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [23]
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“In particular, quintessence models governed by JCAP09 ( 2024 ) 073 exponential potentials [12–15], V = V 0 e − λϕ , are especially interesting as they are ubiquitous at the boundaries of moduli spaces in string theory, which are regions that are parametrically under good theoretical control.”
Jacobi stability analysis of dynamical systems—applications in gravitation and cosmology
Boehmer, C., Harko, T., & Sabău, S. (2010) · doi:10.4310/atmp.2012.v16.n4.a2 · Cited by 105
“Let us consider a dynamically evolving scalar field with exponential potential V = V0 exp(−λκφ) to model dark energy in a spatially flat FriedmannRobertson–Walker (FRW) universe [9].”
BackgroundMethodDynamics of quintessence models of dark energy with exponential coupling to dark matter
González, T., Leon, G., & Quiros, I. (2006) · doi:10.1088/0264-9381/23/9/025 · Cited by 104
Disformal scalar fields and the dark sector of the universe
Zumalacárregui, M., Koivisto, T., Mota, D., & Ruiz-Lapuente, P. (2010) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2010/05/038 · Cited by 103 · see citing works
UNIFIED DARK MATTER IN SCALAR FIELD COSMOLOGIES
Bertacca, D., Matarrese, S., & Pietroni, M. (2007) · Modern Physics Letters A · doi:10.1142/s0217732307025893 · Cited by 100 · see citing works
Unification of inflation and dark energy à la quintessential inflation
Hossain, M. W., Myrzakulov, R., Sami, M., & Saridakis, E. (2014) · International Journal of Modern Physics D · doi:10.1142/s0218271815300141 · Cited by 98 · see citing works
“low roll to hold till the end of inflation. In case of single-field inflation taking place in 4-dimensional spacetime and braneworld cosmology, respectively we have H2 ≃ V 3M2 Pl ; H2 ≃ 1 6M2 Pl V2 λb , (78) 20 M. W. Hossain, R. Myrzakulov, M. Sami and E. N. Saridakis whereλb isthe branetension.Usingthen the slow-rollequ…”
BackgroundMethodUnified Dark Matter Scalar Field Models
Bertacca, D., Bartolo, N., & Matarrese, S. (2010) · Advances in Astronomy · doi:10.1155/2010/904379 · Cited by 87 · see citing works
“Indeed, this asymptotic form, in the presence of an extra radiation component, allows to recover one of the stable nodes obtained in [118] for quintessence fields with exponential potentials, where the scalar field mimics a pressureless fluid.”
BackgroundNon-Gaussianities from ekpyrotic collapse with multiple fields
Koyama, K., Mizuno, S., Vernizzi, F., & Wands, D. (2007) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2007/11/024 · Cited by 98 · see citing works
Self-citationTowards inflation and dark energy cosmologies from modified Gauss–Bonnet theory
Neupane, I. P., & Carter, B. (2005) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2006/06/004 · Cited by 95 · see citing works
Dynamical analysis of anisotropic scalar-field cosmologies for a wide range of potentials
Fadragas, C. R., Leon, G., & Saridakis, E. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/31/7/075018 · Cited by 93 · see citing works
“In the case of exponential potential this point becomes the most important one, since it is both stable and compatible with observations [76].”
BackgroundCosmological Constraints on the Sign-Changeable Interactions
Wei, H. (2010) · Communications in Theoretical Physics · doi:10.1088/0253-6102/56/5/29 · Cited by 93 · see citing works
THEORETICAL MODELS OF DARK ENERGY
Yoo, J., & Watanabe, Y. (2012) · International Journal of Modern Physics D · doi:10.1142/s0218271812300029 · Cited by 232 · see citing works
“We will use the following dimensionless variables [118, 3, 4]”
MethodThe quintom model with O(N) symmetry
Setare, M., & Saridakis, E. (2008) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2008/09/026 · Cited by 92 · see citing works
Selected topics in scalar–tensor theories and beyond
Quiros, I. (2019) · International Journal of Modern Physics D · doi:10.1142/s021827181930012x · Cited by 91 · see citing works
Cosmographic Constraints and Cosmic Fluids
Capozziello, S., Laurentis, M. D., Luongo, O., & Ruggeri, A. C. (2013) · Galaxies · doi:10.3390/galaxies1030216 · Cited by 107 · see citing works
“Tighter bounds of the observable universe have been therefore measured, definitively showing that a possible new ingredient [26–28], dubbed dark energy (DE), is actually required in order to accelerate the universe today.”
BackgroundDark degeneracy and interacting cosmic components
Aviles, A., & Cervantes-Cota, J. (2011) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.84.083515 · Cited by 91 · see citing works
POWER-LAWS f(R) THEORIES ARE COSMOLOGICALLY UNACCEPTABLE
Amendola, L., Polarski, D., & Tsujikawa, S. (2006) · International Journal of Modern Physics D · doi:10.1142/s0218271807010936 · Cited by 91 · see citing works
Phantom damping of matter perturbations
Amendola, L., Tsujikawa, S., & Sami, M. (2005) · Physics Letters B · doi:10.1016/j.physletb.2005.10.065 · Cited by 91 · see citing works
Phantom cosmic dynamics: Tracking attractor and cosmic doomsday
Hao, J.-G., & Li, X.-Z. (2003) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.70.043529 · Cited by 91 · see citing works
Precision gravity tests and the Einstein Equivalence Principle
Tino, G., Cacciapuoti, L., Capozziello, S., Lambiase, G., & Sorrentino, F. (2020) · Progress in Particle and Nuclear Physics · doi:10.1016/j.ppnp.2020.103772 · Cited by 90 · see citing works
“A candidate for DE is quintessence (the energy density associated a scalar field that evolves slowly in time) [162–166].”
BackgroundRiding on irrelevant operators
de Rham, C., & Ribeiro, R. H. (2014) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2014/11/016 · Cited by 89 · see citing works
Instant preheating in quintessential inflation with α-attractors
Dimopoulos, K., Wood, L., & Owen, C. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.97.063525 · Cited by 88 · see citing works
“It has been shown that, when the background density becomes comparable to the field’s residual potential density V (φF ), the field unfreezes and briefly oscillates about the attractor before settling on the attractor solution [45].”
BackgroundEnergy transfer in multi field inflation and cosmological perturbations
Ashoorioon, A., Krause, A., & Turzyński, K. (2008) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2009/02/014 · Cited by 88 · see citing works
Cosmological behavior in extended nonlinear massive gravity
Leon, G., Saavedra, J., & Saridakis, E. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/30/13/135001 · Cited by 85 · see citing works
“2, the curve Q5 contains the quintessencelike critical points presented in table 4, which are obtained in standard quintessence too in flat [79] or non-flat geometries [85].”
BackgroundInfluential
- Authors
- Copeland, E. J., Liddle, A. R., & Wands, D.
- Affiliations
- University of Sussex (GB)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 1998-04-15
- Volume
- 57
- Issue
- 8
- Pages
- 4686-4690
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W2140950522
- Cited by
- 1,468
- References
- 12
- Citations by year
- 2026: 64 · 2025: 84 · 2024: 89 · 2023: 82 · 2022: 80 · 2021: 64 · 2020: 62 · 2019: 59 · 2018: 49 · 2017: 43 · 2016: 63 · 2015: 50 · 2014: 46 · 2013: 52 · 2012: 51
- FWCI
- 21.77
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- Sussex Research Online (University of Sussex) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Figshare (submittedVersion, cc-by)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Galaxies: Formation, Evolution, Phenomena
- Keywords
- exponential potential, phase plane analysis, scalar field, equation of state, inflation (cosmology), late-time attractors, nucleosynthesis
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