A softly broken symmetry offers a route across the phantom divide

The cosmological constant has a fixed equation of state, w = pressure/energy density = −1. Some combinations of distance measurements have instead motivated models in which the effective dark-energy equation of state evolves from below −1 at earlier times to above it more recently. Crossing that boundary, known as the phantom divide, is difficult for a single minimally coupled scalar field in ordinary general relativity.

In arXiv preprint 2611.07035v1, Calderón and Linder ask whether a simple modified-gravity theory can make the crossing while remaining consistent with the standard early expansion history. Their proposed softly broken shift symmetric gravity, or SBSS, produces a representative crossing near redshift z ≈ 0.5 and an effective w today of roughly −0.8. These are model-dependent fits, not a direct detection of a crossing.

A symmetry that relaxes over time

SBSS belongs to the Horndeski family of scalar–tensor theories. Its action combines the usual Einstein gravity term with a noncanonical scalar kinetic term and a kinetic-gravity-braiding term. Braiding couples the scalar’s dynamics to the expansion of spacetime. The authors assume gravitational waves travel at the speed of light and keep the ordinary gravitational coupling to matter minimal. They introduce neither a cosmological constant nor a scalar potential to drive acceleration.

The construction starts from shift symmetry: the field can be displaced by a constant without changing the theory if the action depends only on its derivatives. Exact symmetry is relaxed by multiplying both scalar terms by exponential functions of the field. Those factors evolve very slowly at early times, making the symmetry approximate when the universe is young, but become consequential later. The authors argue that this offers substantial protection against high-energy quantum corrections; it is a motivation for the ansatz, not a complete solution of the cosmological naturalness problem.

For simplicity, the authors choose power laws in the kinetic variable X, with exponents n and m. Early-time charge conservation and the desired balance between kinetic and braiding contributions relate the two exponents. In the matter era, viable histories approach an attractor that largely erases their dependence on the initial braiding fraction. This helps make the late-time predictions economical without making the underlying functional choices unique.

Comparing models with the sky

The analysis combines DESI DR2 baryon acoustic oscillation distances, Union3.1 Type Ia supernova distances, and Planck 2018 CMB power spectra. The authors study two principal cases, fixing n to 1/8 or 1/4. In each, the dark-energy sector has two effective adjustable parameters beyond the standard cosmological model, matching the count in the comparison model, w₀wₐCDM, a widely used phenomenological parametrization of evolving dark energy.

Relative to w₀wₐCDM, the best-fit total χ² improves by 3.02 for n = 1/8 and 3.33 for n = 1/4. Most of that improvement comes from the CMB contribution, while the supernova contribution is slightly worse in both cases. The reported log Bayesian evidence differences are approximately +3, favoring these SBSS cases under the adopted priors. Evidence depends on prior ranges and parameterization, however, and neither the χ² difference nor the evidence establishes that modified gravity is the correct explanation. The fits also assume massless neutrinos.

The authors discard histories with ghosts or negative scalar sound-speed squared between their initial epoch and today. This is a stability check for the cosmological histories considered, not proof that every realization of the theory is healthy under all circumstances. Their analysis also explores other parameter choices, including allowing n to vary, so the fixed-exponent results should not be read as an exhaustive search over scalar–tensor gravity.

The next discriminant: structure

A similar expansion history does not guarantee similar growth of cosmic structure. In this model, the effective gravitational coupling affecting matter and light can differ modestly from Newton’s constant. The paper predicts a modest increase in the growth observable fσ₈ relative to w₀wₐCDM and examines the integrated Sachs–Wolfe–galaxy correlation kernel.

Those predictions are a reason to test the proposal, not an additional fit result: the main parameter constraints do not include separate structure-growth data. The authors argue that the model looks promising against growth-related considerations, but a full comparison with those observations remains necessary. The work is a preprint, and its conclusions remain contingent on its chosen datasets, priors, stability criteria and action. Its principal contribution is a concrete, testable example of a phantom-divide crossing without inserting a cosmological constant or potential by hand.