Evolution to a smooth universe in an ekpyrotic contracting phase withw>1
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A period of slow contraction with equation of state $w>1$, known as an ekpyrotic phase, has been shown to flatten and smooth the universe if it begins the phase with small perturbations. In this paper, we explore how robust and powerful the ekpyrotic smoothing mechanism is by beginning with highly inhomogeneous and anisotropic initial conditions and numerically solving for the subsequent evolution of the universe. Our studies, based on a universe with gravity plus a scalar field with a negative exponential potential, show that some regions become homogeneous and isotropic while others exhibit inhomogeneous and anisotropic behavior in which the scalar field behaves like a fluid with $w=1$. We find that the ekpyrotic smoothing mechanism is robust in the sense that the ratio of the proper volume of the smooth to nonsmooth region grows exponentially fast along time slices of constant mean curvature.
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On the contracting branch, the field rolls down the steep negative potential into an ekpyrotic phase with wφ≫1; this super-stiff phase drives the homogeneous shear fraction Σ to zero exponentially, so the isotropic solution is a strong attractor.
“On the contracting branch (H < 0) the field rolls down the steep negative part of the potential (V < 0) into an ekpyrotic phase with w φ ≫ 1.As shown in Sec.IV A, this super-stiff phase drives the homogeneous shear fraction Σ to zero exponentially, so the isotropic solution is a strong attractor [27,34].”
From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [34]
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Experimental — accuracy not yet measuredA steep negative potential with wφ ≫ 1 makes the ekpyrotic regime a strong homogeneous attractor, with Σ ∝ e^{(q-2)N} → 0 along the contracting branch.
“By contrast, as shown in Sec.IV A, a steep negative potential with w φ ≫ 1 makes the ekpyrotic regime a strong homogeneous attractor [27,34]: Σ ∝ e (q-2)N → 0 along the contracting branch.”
From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [34]
The Results (pp. 5–7) link the steep negative exponential potential and w ≫ 1 to smoothing and isotropization during contraction, but report that some regions remain anisotropic. The paper does not state the claimed decay law Σ ∝ e^{(q−2)N}; its displayed formulas are undecoded in our copy. It therefore supports the qualitative behavior, not the specific law or an unqualified attractor claim. (LLM, conf=0.86)
Experimental — accuracy not yet measuredDuring ekpyrotic contraction with wφ>1, the shear energy density ratio ρσ/ρφ ∝ a^{3(1+wφ)-6} → 0 as a→0, so the anisotropic contribution becomes negligible and the ekpyrotic phase acts as a strong attractor towards isotropy.
“Thus the anisotropic contribution becomes negligible during ekpyrotic contraction, and at the homogeneous level the ekpyrotic phase acts as a strong attractor towards isotropy, suppressing the growth of shear anisotropy.”
From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [34]
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Ijjas, A., & Steinhardt, P. J. (2019) · Physics Letters B · doi:10.1016/j.physletb.2019.06.056 · Cited by 145 · see citing works
Self-citationAnisotropic pressures at ultrastiff singularities and the stability of cyclic universes
Barrow, J. D., & Yamamoto, K. (2010) · Physical Review D · doi:10.1103/physrevd.82.063516 · Cited by 34 · 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
- Garfinkle, D., Lim, W. C., Pretorius, F., & Steinhardt, P. J.
- Affiliations
- Oakland University (US)
- Princeton University (US)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2008-10-27
- Volume
- 78
- Issue
- 8
- ISSN
- 0556-2821 (ISSN-L)
- 1550-7998
- 1550-2368
- OpenAlex
- W2073009683
- Cited by
- 85
- References
- 0
- Citations by year
- 2025: 4 · 2024: 6 · 2023: 6 · 2022: 5 · 2021: 11 · 2020: 7 · 2019: 8 · 2018: 4 · 2017: 5 · 2016: 4 · 2015: 2 · 2014: 2 · 2013: 2 · 2012: 6
- FWCI
- 2.53
- 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)
- Max Planck Institute for Plasma Physics (submittedVersion)
- Max Planck Digital Library (submittedVersion, other-oa)
- MPG.PuRe (Max Planck Society) (submittedVersion, other-oa)
- 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 › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- ekpyrotic cosmology, contracting universe, scalar field, inhomogeneous initial conditions, structure formation
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