Flexknot Dark Energy Reconstruction

• Flexknot dark energy is a nonparametric reconstruction framework that represents w(a) as a linear spline between free-moving knots, capturing localized features missed by standard models.
• The approach employs Bayesian model comparison across varying knot numbers to assess how well the data support deviations from ΛCDM with refined dynamical features.
• Analyses combining DESI BAO and supernova data reveal a W-shaped posterior in w(a), though alternative models like a low-/high-redshift magnitude offset can sometimes provide a better explanation.

Flexknot dark energy is a nonparametric reconstruction framework for the dark-energy equation of state $w(a)$ in which $w$ is represented as a linear spline between free-moving knots in scale factor, with Bayesian model comparison used to infer how much functional complexity the data support. In the 2025 literature, the framework was used to test whether late-time geometric data favor departures from $\Lambda$CDM that are too structured to be captured by $w$CDM or Chevallier–Polarski–Linder (CPL), especially in combinations of DESI BAO with Pantheon+ or DES5Y supernovae (Ormondroyd et al., 11 Mar 2025, Ormondroyd et al., 21 Mar 2025). It was subsequently reinterpreted in a model-selection setting where a low-/high-redshift supernova magnitude offset outperformed the earlier flexknot explanation for the DES-5Y + DESI signal, substantially weakening the case that those data require dynamical dark energy (Ormondroyd et al., 16 Sep 2025).

1. Conceptual role and motivation

Flexknot dark energy was introduced to address a specific limitation of standard low-dimensional dark-energy parameterizations: if the data prefer localized or alternating structure in $w(a)$, then $\Lambda$CDM, constant-$w$ $w$CDM, and the single-slope CPL form can miss that structure or distort it. The reconstruction therefore treats $w(a)$ as a free-form object rather than imposing a rigid global shape, while still remaining simple enough for Bayesian evidence calculations (Ormondroyd et al., 11 Mar 2025).

The motivating empirical context was the appearance of nontrivial reconstructed structure when DESI BAO data were combined with either Pantheon+ or DES5Y supernovae. In that setting the functional posterior for $w(a)$ exhibited a “W-shaped” structure, with alternating quintessence-like and phantom-like excursions. The authors argued that this pattern was generally too complex to be captured by $w$0CDM or CPL, even when the overall Bayesian evidence still favored $w$1CDM in some data combinations (Ormondroyd et al., 11 Mar 2025).

The framework also acquired a second role: not only as a reconstruction tool, but as a benchmark for testing whether an apparent preference for evolving dark energy is genuinely cosmological or instead a symptom of supernova systematics. In that later use, flexknot models were compared directly against a phenomenological low-redshift supernova magnitude offset, and the offset model was preferred for DES-5Y + DESI BAO (Ormondroyd et al., 16 Sep 2025).

2. Spline construction and model space

The flexknot model represents $w$2 as a linear spline in $w$3-space between movable nodes. The parameter vector is written as

$w$4

with endpoint scale factors fixed at

$w$5

and interior knot locations drawn as ordered variables,

$w$6

Between adjacent knots, the reconstruction is piecewise linear: $w$7 This makes $w$8 continuous on $w$9, but allows derivative jumps at the knot locations (Ormondroyd et al., 11 Mar 2025, Ormondroyd et al., 21 Mar 2025).

A key feature is that standard dark-energy models appear as special cases. The flexknot family is therefore nested relative to conventional late-time parameterizations.

Model	Flexknot identification	Characterization
$\Lambda$0CDM	reference model	$\Lambda$1
$\Lambda$2CDM	$\Lambda$3	constant $\Lambda$4
CPL	$\Lambda$5	$\Lambda$6
Higher-order flexknots	$\Lambda$7	multiple linear segments with free break locations

In the DR1 and DR2 flexknot reconstruction papers, the knot amplitudes were restricted to

$\Lambda$8

which excludes positive-pressure dark energy and allows $\Lambda$9 crossing within the reconstructed posterior. In the later supernova-offset comparison paper, the corresponding flexknot analysis used

$w$0

so the exact prior volume is analysis-dependent (Ormondroyd et al., 11 Mar 2025, Ormondroyd et al., 16 Sep 2025).

The knot number itself is part of the model space. The sampled range was

$w$1

and early tests up to thirty knots were reported not to materially change the reconstructed shape, so $w$2 was adopted as sufficient in the original nonparametric study (Ormondroyd et al., 11 Mar 2025).

3. Cosmological implementation and Bayesian inference

The reconstruction is embedded in a late-time background-expansion analysis. In the original nonparametric study the background was taken to be spatially flat, with

$w$3

and

$w$4

Equivalently,

$w$5

The reconstruction is built directly for

$w$6

Because each segment is linear in $w$7, the dark-energy evolution can be partly integrated analytically on each interval, while the remaining distance integrals are evaluated numerically (Ormondroyd et al., 11 Mar 2025).

The observational inputs were deliberately restricted to late-time geometric probes. BAO-only analyses sampled

$w$8

while supernova-only analyses used

$w$9

with $w(a)$0 analytically marginalized in practice for the supernova likelihood. The matter-density prior was

$w(a)$1

For supernovae, the absolute magnitude $w(a)$2 was also analytically marginalized in the original flexknot implementation (Ormondroyd et al., 11 Mar 2025).

Model comparison is central to the framework. Separate nested-sampling runs are performed for fixed knot number $w(a)$3, and then combined by evidence weighting. If $w(a)$4 denotes the evidence for the model with $w(a)$5 knots and $w(a)$6 the model prior, then

$w(a)$7

With a uniform prior over knot number,

$w(a)$8

This same evidence weighting is used for model-averaged expectations and for model-marginalized tension measures (Ormondroyd et al., 11 Mar 2025).

The original reconstruction studies used PolyChord nested sampling. In the later offset-comparison paper, the likelihoods were also implemented in JAX; BlackJAX nested slice sampling was used for $w(a)$9CDM and CPL, while PolyChord was retained for the more multimodal flexknot runs. That paper additionally tested Nested Bridge Sampling with Sequential Monte Carlo as an alternative method for Bayes-factor evaluation in nested models (Ormondroyd et al., 16 Sep 2025).

4. Reconstructions from DESI DR1, Pantheon+, and DES5Y

In the DR1-era flexknot analysis, the main scientific result was the appearance of a W-shaped posterior structure in $\Lambda$0 when DESI BAO were combined with either Pantheon+ or DES5Y. The common local maximum appeared around

$\Lambda$1

while the higher-redshift DESI-driven feature was associated especially with the LRG region around

$\Lambda$2

The lower-redshift feature was driven by the supernova data, and the higher-redshift feature by DESI BAO (Ormondroyd et al., 11 Mar 2025).

For DESI + Pantheon+, the W-shape was reported as the clearest and most complex, with “around ten knots or more” needed to reproduce it faithfully. Even so, the fully marginalized Bayesian evidence still favored $\Lambda$3CDM over the flexknot family: $\Lambda$4 The conclusion was therefore not a detection of dynamical dark energy, but rather the presence of structured preferences not well represented by $\Lambda$5CDM or CPL (Ormondroyd et al., 11 Mar 2025).

For DESI + DES5Y, the overall evidence relative to $\Lambda$6CDM was nearly neutral,

$\Lambda$7

but the evidence as a function of knot number was more suggestive: the $\Lambda$8 models each individually exceeded $\Lambda$9CDM before the larger-$w$0 Occam penalty erased that advantage in the model-averaged comparison. DES5Y alone was the only individual dataset in that paper with positive evidence for flexknot evolution relative to $w$1CDM,

$w$2

By contrast, Pantheon+ alone favored $w$3CDM,

$w$4

and DESI alone also disfavored the full flexknot family,

$w$5

These results established flexknots as a sensitive probe of late-time structure in $w$6, but also showed that claims of dynamical dark energy depended strongly on how model complexity was marginalized (Ormondroyd et al., 11 Mar 2025).

The same work also emphasized inter-dataset consistency. DESI and Pantheon+ were found to become broadly compatible once enough flexibility was allowed, whereas DESI and DES5Y remained substantially more discrepant. The least tension for DESI and DES5Y occurred around $w$7, and allowing more knots worsened the disagreement because the two datasets preferred incompatible behavior near $w$8 (Ormondroyd et al., 11 Mar 2025).

5. DESI DR2 and the change in evidence across model complexity

The DR2 update retained the same basic flexknot philosophy but replaced DESI DR1 BAO with DESI DR2. The qualitative structure of the reconstructed $w$9 remained similar, including a transition around

$w$0

but the posterior band became tighter, the transition was less pronounced in the broad reconstruction, and it moved to slightly higher redshift (Ormondroyd et al., 21 Mar 2025).

The shift in Bayesian support was selective rather than uniform. For DESI DR2 alone, support for high-knot models was reduced relative to DR1, while the simpler dynamical extensions became more competitive with $w$1CDM. The paper states that there was now a “very slight preference” for $w$2CDM over $w$3CDM, and that the evidence for the $w$4 case corresponding to CPL also slightly increased relative to $w$5CDM. Evidence for models with more than four knots was reduced relative to DR1 (Ormondroyd et al., 21 Mar 2025).

For DESI DR2 + Pantheon+, the conclusions remained broadly similar to the DR1 case, but with a systematic tendency for high-knot models to be less favored. By contrast, DESI DR2 + DES5Y strengthened the case for dynamical models: the evidence for all $w$6 models increased significantly compared with DR1, and CPL remained the preferred model. In the abstract and detailed summary, all such models were reported as preferred over $w$7CDM in that combination, with CPL favored by a Bayes factor of $w$8 relative to $w$9CDM (Ormondroyd et al., 21 Mar 2025).

The DR2 paper also revisited dataset concordance via $w(a)$0. DESI and Pantheon+ remained consistent, albeit with slightly increased tension from DR1 to DR2. DESI and DES5Y became more consistent in DR2, with $w(a)$1 for all but the $w(a)$2CDM and $w(a)$3CDM cases. This reinforced the interpretation that dynamical dark-energy models were being rewarded not only for fitting the combined data, but also for reducing the mismatch between DESI and DES5Y (Ormondroyd et al., 21 Mar 2025).

6. Supernova-offset reinterpretation and the status of the flexknot signal

The most consequential reinterpretation of flexknot dark energy came in the later comparison between dynamical dark energy and a supernova systematic model. That paper asked whether the earlier preference for non-$w(a)$4 behavior in DES-5Y + DESI is better explained by flexknot dark energy or by a constant magnitude offset between the low-redshift non-DES anchor supernovae and the DES high-redshift sample (Ormondroyd et al., 16 Sep 2025).

The phenomenological offset model modifies the supernova residual vector as

$w(a)$5

where $w(a)$6 is a binary mask equal to 1 for supernovae not from DES itself and 0 for DES supernovae. Setting $w(a)$7 recovers the standard DES-5Y likelihood. Because $w(a)$8 is also free, only the relative offset between the low-$w(a)$9 and high-$w(a)$0 subsets matters (Ormondroyd et al., 16 Sep 2025).

For DES-5Y alone, evidence for the offset was weak. For DES-5Y + DESI BAO, however, the offset within $w(a)$1CDM became strongly preferred, while the same parameter was not preferred within CPL or flexknot cosmologies. The key DES-5Y + DESI BAO results are:

Cosmological model	$w(a)$2	log Bayes factor for adding offset
$w(a)$3CDM	$w(a)$4	$w(a)$5
CPL	$w(a)$6	$w(a)$7
flexknot	$w(a)$8	$w(a)$9

These values imply that if cosmology is restricted to $w$00CDM, the data prefer a low-$w$01/high-$w$02 supernova shift of roughly $w$03 mag; but if dynamical dark energy is allowed through CPL or flexknots, the cosmology can absorb the effect and the explicit offset is no longer needed. The decisive model-selection question is then which overall explanation has higher evidence. The answer reported was that $w$04CDM with the offset outperforms no-offset flexknot dark energy and other dynamical alternatives, especially for DES-5Y + DESI BAO (Ormondroyd et al., 16 Sep 2025).

The paper further reported that the inferred $w$05CDM offset was centered near $w$06, consistent with the earlier George et al. claim of a low-redshift DES-5Y shift, and that allowing the offset significantly reduced the DES-5Y–DESI tension. It also used Nested Bridge Sampling with Sequential Monte Carlo as a cross-check. For the most important case, $w$07CDM with DES-5Y + DESI, the bridge-sampling estimate of the Bayes factor,

$w$08

agreed closely with the nested-sampling result,

$w$09

supporting the numerical robustness of the offset preference (Ormondroyd et al., 16 Sep 2025).

Within the model space explicitly tested, the earlier evidence for flexknot dark energy in DES-5Y + DESI was therefore overtaken by a simpler supernova-systematic explanation. The paper’s final conclusion was that the systematic is a better model than dynamical dark energy in that setting (Ormondroyd et al., 16 Sep 2025).

7. Interpretation, caveats, and broader theoretical context

Flexknot dark energy is a phenomenological reconstruction rather than a microphysical model. Its spline segments should not be interpreted literally as the Lagrangian dynamics of a dark-energy sector; rather, they expose structure in the expansion history that any successful physical model would have to reproduce. This is one reason the evidence is sensitive to priors over knot number and parameter volume: the framework is deliberately broad, and Bayesian model averaging can favor $w$10CDM even when specific low-$w$11 flexknot models outperform it (Ormondroyd et al., 11 Mar 2025, Ormondroyd et al., 21 Mar 2025).

The supernova-offset reinterpretation adds a second caveat. The offset model was explicitly described as simple and phenomenological: adjusting a subset of apparent magnitudes post bias correction does not constitute a sensible supernova catalogue, and the result is not a claim that a specific pipeline bug has been identified. The conclusion is instead a Bayesian statement within a simplified model space: once that low-/high-redshift offset is admitted, flexknot dark energy is no longer the preferred explanation of the DES-5Y + DESI signal (Ormondroyd et al., 16 Sep 2025).

Broader theory also cautions against reading a reconstructed $w$12 as direct evidence for dark-energy microphysics. In interacting dark-energy models, a true constant non-phantom equation of state can be inferred as an evolving effective $w$13 if the dark-sector interaction is ignored. The exact and approximate expressions derived for $w$14 show how an apparent low-redshift phantom excursion can arise from a misassigned matter density rather than intrinsic dark-energy evolution (Avelino et al., 2012). A plausible implication is that flexible reconstructions, including flexknots, can absorb missing dark-sector physics into the inferred shape of $w$15.

A second caution comes from inhomogeneous-light-cone arguments. A nonperturbative comparison between the idealized FLRW past light cone and the physical light cone has been used to define an effective redshift-dependent correction

$w$16

with characteristic magnitude

$w$17

at scales where cosmological expansion couples to local virialized structure dynamics. In that interpretation, an inferred departure from constant $w$18 could reflect observer- and scale-dependent light-cone effects rather than a fundamental evolving fluid (Carfora et al., 2024). This suggests that flexknot reconstructions are best viewed as high-fidelity summaries of what late-time distance data prefer, not as uniquely identifying the ontology behind those preferences.

Taken together, the 2025 flexknot literature establishes three points. First, free-form knot-based reconstructions can reveal structured features in $w$19 that standard one- and two-parameter models compress or miss. Second, the Bayesian support for those features is highly dependent on data combination, model prior, and treatment of systematics. Third, the most prominent apparent evidence for flexknot dark energy in DES-5Y + DESI was substantially weakened once a plausible supernova offset model was included, leaving flexknots as a powerful diagnostic of late-time structure in the data, but not, at present, a robust demonstration of dynamical dark energy (Ormondroyd et al., 11 Mar 2025, Ormondroyd et al., 21 Mar 2025, Ormondroyd et al., 16 Sep 2025).