Physicists Map Every Possible Fate of the Universe in Lovelock Gravity

A universe that never began with a Big Bang — that instead snaps into existence with finite density, a finite size, and an infinitely sharp burst of acceleration — is one of the striking possibilities mapped out in a new theoretical study published in The European Physical Journal C. Physicists A. V. Nikolaev, V. M. Zhuravlev and S. V. Chervon performed a complete qualitative analysis of cosmological evolution in Lovelock gravity, the most general extension of Einstein’s theory that keeps the field equations strictly second order. Working with a spatially flat Friedmann–Robertson–Walker universe filled with a perfect fluid obeying a barotropic equation of state, p = ωρ, the authors produced a full classification of every possible evolution scenario for spacetime dimensions from four to seven. Among the scenarios they identified is one they call the ‘Big Shock,’ which replaces the standard Big Bang in more than four dimensions.

Lovelock gravity occupies a special position in theoretical physics. In four dimensions it reduces exactly to general relativity, but in higher dimensions it admits additional polynomial curvature terms — the quadratic Gauss–Bonnet invariant, its cubic generalization, and higher orders — while preserving second-order field equations. This second-order property matters because it is what keeps the theory free of the ghost degrees of freedom that arise in generic higher-derivative theories: the Lovelock Lagrangian is the unique polynomial curvature invariant with this property, so the Ostrogradsky theorem, which predicts ghost instabilities in nondegenerate higher-derivative theories, does not apply to it. According to the authors, such modified theories play a key role in modern cosmological studies because their corrections can account for additional gravitational scalar fields, quantum effects and extra dimensions, which are especially important for describing the early universe and dark energy. The Einstein–Gauss–Bonnet theory, a special case of Lovelock gravity, can explain the accelerated expansion of the universe without invoking the ΛCDM model.

The technical heart of the paper is a dramatic simplification. In a spatially flat Friedmann geometry the Riemann tensor has only two independent components, which the authors denote f₁ = H² and f₂ = Ḣ + H², where H is the Hubble parameter. Using an approach based on these independent components, they compress the full system of generalized N-dimensional Friedmann equations into a single autonomous first-order equation for H. Its right-hand side is a ratio of two polynomials in H², written P₂(H²)/P₁(H²), whose coefficients are fixed by exactly three ingredients: the order n of the Lovelock polynomial, the number of spacetime dimensions N, and the coupling constants αᵢ that weight each curvature correction. Everything about the universe’s possible evolutions is then encoded in the real, positive roots of those two polynomials: the roots of the numerator polynomial P₂ determine the fixed points of the dynamics, while the roots of the denominator polynomial P₁ determine the singular points.

The qualitative analysis classifies what happens near each root. Each positive root hᵢ of P₂ generates a pair of fixed points, +hᵢ and −hᵢ, with opposite stability: linearizing the equation around such a point shows that if one is attractive, its mirror image is repulsive. Near an attractive fixed point with hᵢ > 0, the scale factor grows exponentially, approaching the de Sitter expansion regime, while the matter density dilutes to zero. Near an attractive fixed point with hᵢ < 0, the universe contracts exponentially, yet the matter density still tends to zero — a paradoxical behavior the authors trace directly to the structure of the Lovelock equations, since any real root of P₂ inevitably produces the two mirror fixed points. The point H = 0 requires separate treatment: it is a degenerate, nonhyperbolic fixed point where the dynamics becomes universal for all Lovelock models and reduces to the general-relativistic form, and trajectories approach it only asymptotically, never crossing it.

The roots sᵢ of the denominator polynomial P₁ play a different role. At the points H = ±sᵢ the right-hand side of the evolution equation has poles, so the rate of change of H diverges; these singular points act as impenetrable barriers that partition the Hubble axis into separate basins, each corresponding to a distinct cosmological history. A trajectory starting in one basin never leaves it. The authors also proved a sharp timing result: fixed points are reached only in infinite time, whereas singular points are reached in finite time. At a singular point the scale factor, the Hubble parameter, the density and the pressure all remain finite, but Ḣ blows up, so the scalar curvature diverges. In general relativity such a combination is impossible, because the field equations tie a divergence of Ḣ to a divergence in pressure. In the standard classification of finite-time singularities, these events behave as Type II, or ‘sudden,’ singularities, with the diverging pressure replaced by a degeneracy of the gravitational operator itself.

The most eye-catching scenario emerges only when the number of dimensions N exceeds four and some of the coupling constants are negative. The authors call it the ‘Big Shock.’ In this scenario the universe does not emerge from a state of infinite density and zero size, as in the standard Big Bang. Instead, it starts from a finite density and a finite scale factor — but with an infinite rate of change of the Hubble parameter. A universe that already exists experiences a sudden, infinitely sharp kick of acceleration and then settles into ordinary decelerating expansion, eventually approaching either a Friedmann-like state of zero density and infinite size or a de Sitter state of eternal exponential growth, depending on the basin. The authors argue that, from the standpoint of initial conditions, such a beginning looks more rational than a Big Bang, since it requires no infinities in the material content of the cosmos. The Big Shock is impossible in four-dimensional general relativity.

The analysis also uncovered ‘phantom intervals’ — ranges of the Hubble parameter in which the matter density required by the field equations becomes negative, which is physically impossible for an ordinary fluid. These intervals arise for n > 2 and N > 5 when the polynomial P₂ has positive real roots, forcing the Lovelock curvature corrections to demand more than the matter can supply. The authors show that trajectories with positive density can never enter these forbidden zones, because their boundaries are fixed points reachable only asymptotically; the central basin around H = 0 is always free of phantom intervals, and if all coupling constants are non-negative the intervals are absent altogether. In the effective-fluid picture, where all Lovelock corrections are moved to the matter side of the equations, the effective energy density remains non-negative everywhere. The authors note that realizing values of H from such forbidden ranges would require matter of a phantom type, which in principle opens a possibility of an observational test.

To make the general classification concrete, the paper works through three illustrative cases. For n = 1 and N = 4, Lovelock gravity reduces to general relativity, and the phase portrait contains only the degenerate fixed point H = 0, which is semi-stable: for one sign of the equation-of-state parameter it attracts expanding trajectories and repels contracting ones, and vice versa for the opposite sign. The expanding and contracting branches are dynamically disconnected. For n = 2 and N = 5, corresponding to Einstein–Gauss–Bonnet gravity with a negative coupling α₂ = −0.1, a pair of fixed points at H ≈ ±2.236 and a pair of singular points at H ≈ ±1.581 appear, generating six distinct evolution scenarios, two of which are Big Shock events differing in the sign of the initial acceleration kick. For n = 3 and N = 7, cubic Lovelock gravity, the polynomials acquire two positive roots each, the Hubble axis splits into ten basins, and the catalogue grows to include phantom-matter variants.

A further structural result concerns what determines the shape of the phase diagram. The skeleton of the phase line — the fixed points ±hᵢ, the singular barriers ±sᵢ, the basins between them and the phantom intervals — is fixed entirely by the geometry, that is, by n, N and the couplings αᵢ, and does not depend on the equation of state of the matter. If the barotropic parameter ω crosses the phantom divide at ω = −1 during cosmic history, the direction of motion along the same phase curve simply reverses for all H simultaneously: attracting fixed points become repelling and vice versa, and the trajectory retraces its path without ever leaving its basin. In models with singular points, a sufficiently long phantom phase inside the central basin drives the trajectory toward the singular point s₁ rather than to infinity, so that the Big Rip of general relativity is replaced by the Big Shock.

What emerges is a complete atlas of cosmic histories available to a flat universe in Lovelock gravity — Big Bang scenarios, Big Shock beginnings, de Sitter futures, contracting branches and phantom-matter exotics — each tied to specific regions of the theory’s parameter space. The authors emphasize that the work is purely theoretical: no datasets were generated or analysed, and the study does not claim that our universe actually began with a shock rather than a bang. Several questions remain open, including a detailed analysis of the Boulware–Deser ghost in the negative-coupling regime required for Big Shock scenarios, geodesic completeness near the new singularities, and the extension of the qualitative analysis to spatially curved geometries, where the dynamical system becomes two-dimensional in the variables (a, χ). The authors also note that scenarios with matter generated directly from geometry, with zero initial density and scale factor, appear in the higher-dimensional models and require further research. The Big Bang, the analysis shows, may be only one entry on a much longer list.

Subject of Research: Theoretical Physics / Cosmology

Article Title: Cosmological dynamics based on Lovelock’s gravity: qualitative analysis

Article References: Nikolaev, A. V., Zhuravlev, V. M., & Chervon, S. V. (2026). Cosmological dynamics based on Lovelock’s gravity: qualitative analysis. The European Physical Journal C, 86(10), Article 1209. https://doi.org/10.1140/epjc/s10052-026-16446-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16446-w

Keywords: Lovelock gravity, cosmology, modified gravity, Big Bang, Einstein-Gauss-Bonnet, singularities, higher dimensions, theoretical physics, Cosmological, dynamics, based, Lovelock