Cosmology with positive and negative exponential potentials
Open accessAbstract
We present a phase-plane analysis of cosmologies containing a scalar field ϕ with an exponential potential V ∝ exp(−λ κ ϕ) where κ 2 = 8π G and V may be positive or negative. We show that power-law kinetic–potential scaling solutions only exist for sufficiently flat (λ 2 < 6) positive potentials or steep (λ 2 > 6) negative potentials. The latter correspond to a class of ever-expanding cosmologies with negative potential. However, we show that these expanding solutions with a negative potential are unstable in the presence of ordinary matter, spatial curvature or anisotropic shear, and generic solutions always recollapse to a singularity. Power-law kinetic–potential scaling solutions are the late-time attractor in a collapsing universe for steep negative potentials (the ekpyrotic scenario) and stable against matter, curvature or shear perturbations. Otherwise kinetic-dominated solutions are the attractor during collapse (the pre-big-bang scenario) and are only marginally stable with respect to anisotropic shear.
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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 [24]
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Cites
Cosmological consequences of a rolling homogeneous scalar field
Ratra, B., & Peebles, P. J. E. (1988) · Physical Review D · doi:10.1103/physrevd.37.3406 · Cited by 4,310 · see citing works
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Ekpyrotic and Cyclic Cosmology
Lehners, J.-L. (2008) · Physics Reports · doi:10.1016/j.physrep.2008.06.001 · Cited by 381 · see citing works
Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology
Stingley, J. (2026) · General Relativity and Gravitation · doi:10.1007/s10714-026-03547-w
- Authors
- Heard, I. P. C., & Wands, D.
- Affiliations
- University of Portsmouth (GB)
- ORCID
- Wands, David
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2002-10-15
- Volume
- 19
- Issue
- 21
- Pages
- 5435-5447
- ISSN
- 0264-9381 (ISSN-L)
- OpenAlex
- W2113073233
- Cited by
- 168
- References
- 1
- Citations by year
- 2026: 6 · 2025: 6 · 2024: 6 · 2023: 23 · 2022: 2 · 2021: 5 · 2020: 3 · 2019: 7 · 2018: 8 · 2017: 6 · 2016: 3 · 2015: 5 · 2014: 8 · 2013: 9 · 2012: 4
- FWCI
- 6.98
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Engineering and Physical Sciences Research Council
- 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 › Advanced Differential Geometry Research
- Keywords
- cosmology, exponential potential, negative potential, scalar field, phase plane analysis, pre-big bang scenario, recollapse, singularities, spatial curvature, matter perturbations, attractor solutions
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