In an illustrative analysis, a new dark-energy method identifies redshift 0.098 as the optimum for testing a departure from the cosmological-constant value. The reported one-dimensional significance at that point is 2.78 standard deviations. The paper asks which redshift most effectively tests whether the equation of state, written as w, differs from the reference value w = −1, and how that point relates to the pivot and crossing redshifts.
Why the error bar is not enough
That is a different goal from simply finding the smallest error bar. In the paper’s terminology, the pivot redshift is the minimum-variance projection of the equation of state. The proposed optimum instead selects the redshift where the squared distance from w = −1 is largest relative to the associated uncertainty. One point is chosen for precision; the other is chosen for standardized separation from the cosmological-constant boundary.
Within the generalized CPL framework, a model for how w changes with redshift, the authors derive analytic relationships among the optimal, pivot and crossing descriptions. They use the reported CPL parameters, their posterior variances and covariance, which captures how the fitted parameters vary together, to evaluate the EOS at different redshifts. The target is the location where the standardized departure from w = −1 is greatest, so the method combines the size of the offset with the uncertainty attached to it.
One data combination, one result
For the worked example, the inputs are CPL constraints from DESI DR2 BAO, the independent DES Year-6 BAO measurement without overlap, and the recalibrated DES-Dovekie Type-Ia supernova compilation. The reported CPL values are w0 = −0.840 with uncertainty components of +0.065 and −0.076, and wa = −0.52 ± 0.49.
For those inputs, the three characteristic redshifts are ordered as zopt = 0.098, zp = 0.139 and zc = 0.444. The optimum comes first, followed by the pivot and then the crossing point. In this example, the preferred test point is therefore at lower redshift than either of the other two.
At the pivot, the reported EOS is wp = −0.903 ± 0.038. At the optimum, it is wopt = −0.887 ± 0.041. The optimum’s central value lies farther from −1, while its reported uncertainty is slightly larger. The comparison illustrates the method’s trade-off: the pivot has the tighter EOS estimate, but the optimum is farther from the reference after uncertainty is taken into account.
A different target from the pivot
The optimized statistic is R(aopt) = 7.71. The corresponding reported one-dimensional significance is nσ = 2.78, compared with nσ = 2.56 at the pivot. The optimum thus has the larger standardized separation, despite its wider error bar.
The paper treats the gap between the two redshifts as a geometric consequence of asking different questions. In the CPL parameter space, the pivot minimizes the projected EOS variance, while the optimum maximizes statistical separation from the cosmological-constant boundary relative to uncertainty. For a positive-definite observational covariance matrix, the optimum and pivot are generically distinct. Exact coincidence is limited to a singular covariance matrix.
A result tied to its model
The numerical result is not a universal redshift. It depends on the dataset combination and generally on the dark-energy parameterization, although the optimization principle itself is presented as more general. The document is an arXiv version 1 preprint dated 25 Aug 2026, so the low-redshift value is best read as an illustrative application of the formalism, not as a fixed target for every cosmological analysis.
Paper data and sources
Original title: The optimal redshift for dark energy I: formalism and interpretation
Authors: Mustapha Ishak, Travis Seth Rippentrop, Kristian Gonzalez
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text