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arxiv: 2604.26541 · v1 · submitted 2026-04-29 · 🌌 astro-ph.CO · hep-ph· hep-th

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The End of the First Act: Spectral Running, Interacting Dark Radiation, and the Hubble Tension in Light of ACT DR6 Data

Mathias Garny , Florian Niedermann , Martin S. Sloth
Authors on Pith no claims yet

Pith reviewed 2026-05-07 12:34 UTC · model grok-4.3

classification 🌌 astro-ph.CO hep-phhep-th
keywords Hubble tensiondark radiationspectral index runningACT DR6cosmological parametersinflationdark acoustic oscillations
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The pith

Including running of the spectral index in models with self-interacting dark radiation relaxes the bound on extra radiation and reduces the Hubble tension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that limits on extra relativistic species from ACT DR6 data depend heavily on assumptions about the shape of the primordial power spectrum. By adding self-interacting dark radiation along with running and running-of-the-running in the spectral index, the upper limit on the effective number of extra neutrino species loosens substantially. This extension also shows a preference for positive running at about three sigma significance while cutting the discrepancy in the Hubble constant down to 2.2 sigma with only three additional parameters. When the dark radiation starts coupled to dark matter and decouples around equality, the bound relaxes further and the tension falls below two sigma.

Core claim

Confronting an extended cosmological model that includes self-interacting dark radiation together with running α_s and running of the running β_s of the spectral index against Planck, ACT DR6, DESI DR2, and uncalibrated Pantheon+ data yields a relaxed upper bound ΔN_eff < 0.58 at 95% CL, a 2.9σ preference for α_s > 0, and reduces the Hubble tension to 2.2σ. In the variant where dark radiation decouples from dark matter near matter-radiation equality, the bound becomes ΔN_eff < 0.68 and the tension drops below 2σ.

What carries the argument

Self-interacting dark radiation combined with the parameters α_s and β_s that describe the running and running of the running of the primordial scalar spectral index, which adjust the power on small scales to accommodate more extra radiation.

Load-bearing premise

The constraints rest on the specific choice of self-interacting dark radiation plus spectral running model being sufficient to describe the data.

What would settle it

A future measurement finding no evidence for spectral running on small scales or a Hubble constant value that remains in strong tension after including these parameters would falsify the proposed resolution.

Figures

Figures reproduced from arXiv: 2604.26541 by Florian Niedermann, Martin S. Sloth, Mathias Garny.

Figure 1
Figure 1. Figure 1: FIG. 1. Marginalized 2D posteriors for running view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Best-fit SIDR (orange) and DRMD (blue) models fitted to CMB data (Planck 2018 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Marginalized posteriors for CMB (ACT DR6 + Planck 2018), DESI BAO (DR2), and view at source ↗
Figure 4
Figure 4. Figure 4: Since photons do not participate in DAO, the latter are primarily imprinted in matter view at source ↗
read the original abstract

We point out that constraints on $\Delta N_\mathrm{eff}$ reported by the ACT collaboration in their DR6 data release are surprisingly sensitive to the assumptions made about the initial power spectrum from inflation. The ACT collaboration reports no evidence of new light degrees of freedom alongside a low value of the expansion rate, thus confirming the Hubble tension. However, as we show here, when considering self-interacting dark radiation and including running, $\alpha_s$, and running of the running, $\beta_s$, of the spectral index $n_s$, the picture changes significantly. Confronting this extended model with Planck, ACT DR6, DESI DR2, and uncalibrated Pantheon+ data, we find the significantly relaxed bound $\Delta N_\text{eff}< 0.58$ at 95$\%$ CL, together with a $2.9 \sigma$ ($2.6 \sigma$) preference for $\alpha_s>0$ ($\beta_s>0$), while the Hubble tension is reduced to $2.2 \sigma$ with only three more parameters compared to $\Lambda$CDM. If the dark radiation fluid is initially coupled to dark matter, and undergoes dark radiation-matter decoupling (DRMD) around matter-radiation equality, predicting dark acoustic oscillations with drag horizon $r_{d,\mathrm{DAO}} \approx 60 \,\mathrm{Mpc}/h$, the bound is further relaxed to $\Delta N_\text{eff}< 0.68$ at 95$\%$ CL, reducing the Hubble tension below $2\sigma$. We also discuss how $\alpha_s$ and $\beta_s$ could naturally appear in inflationary scenarios, possibly connected to the end of a first act of inflation. In this case dark radiation is mostly probed by scales covered by Planck and DESI, while smaller scales carry information on inflationary dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper argues that ACT DR6 constraints on extra relativistic degrees of freedom (ΔN_eff) are sensitive to assumptions about the primordial power spectrum. By extending the model to include self-interacting dark radiation together with running α_s and running-of-the-running β_s of the scalar spectral index, the 95% CL upper bound relaxes to ΔN_eff < 0.58, a 2.9σ (2.6σ) preference for α_s > 0 (β_s > 0) appears, and the Hubble tension drops to 2.2σ. A further variant with dark radiation-matter decoupling (DRMD) near matter-radiation equality yields ΔN_eff < 0.68 and tension below 2σ, while the running parameters are linked to possible inflationary dynamics at the end of a first inflationary phase.

Significance. If the reported preferences and relaxed bounds are robust, the work demonstrates that standard ACT analyses may be underestimating the allowed parameter space for new light species once realistic inflationary extensions are considered. The reduction of the Hubble tension to 2.2σ (or below 2σ) with only three additional parameters relative to ΛCDM is quantitatively interesting and could motivate targeted follow-up with future CMB and large-scale structure data. The discussion of possible inflationary origins for α_s and β_s also provides a concrete link between early-universe dynamics and late-time observables.

major comments (3)
  1. [Abstract, results] Abstract and results section: the 2.9σ preference for α_s > 0 and the relaxation of the ΔN_eff bound are obtained from a joint fit to Planck + ACT DR6 + DESI DR2 + Pantheon+; it is not shown whether the same preference survives when ACT data are removed or when α_s and β_s are fixed to zero, which would test whether the improvement is driven by the extended model or by the specific data combination.
  2. [DRMD discussion] DRMD variant: the drag horizon is fixed at r_d,DAO ≈ 60 Mpc/h with decoupling near matter-radiation equality; the manuscript should quantify how sensitive the further relaxation to ΔN_eff < 0.68 is to this choice and whether the value is theoretically motivated or tuned to the data.
  3. [Hubble tension subsection] Hubble tension quantification: the reduction from the baseline tension (presumably ~4–5σ in ΛCDM with the same datasets) to 2.2σ must be accompanied by an explicit comparison table showing the tension metric (e.g., difference in H0 posterior means divided by combined uncertainty) for both the baseline and extended models.
minor comments (2)
  1. [Model definition] Notation for the self-interaction strength and the DRMD decoupling redshift should be defined once in the text and used consistently in all equations and figures.
  2. [Figures] Figure captions for the posterior plots should explicitly state the datasets used in each chain and whether the contours include or exclude the running parameters.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. We address each major comment point by point below and will revise the manuscript accordingly to improve clarity and robustness.

read point-by-point responses

  1. Referee: [Abstract, results] Abstract and results section: the 2.9σ preference for α_s > 0 and the relaxation of the ΔN_eff bound are obtained from a joint fit to Planck + ACT DR6 + DESI DR2 + Pantheon+; it is not shown whether the same preference survives when ACT data are removed or when α_s and β_s are fixed to zero, which would test whether the improvement is driven by the extended model or by the specific data combination.

    Authors: We agree that these additional checks would strengthen the presentation. In the revised manuscript we will add two sets of results: (i) constraints from Planck + DESI DR2 + Pantheon+ alone (excluding ACT DR6) to test whether the preference for α_s > 0 and β_s > 0 persists without ACT, and (ii) constraints with α_s = β_s = 0 to isolate the effect of the running parameters on the ΔN_eff bound. These will demonstrate that the relaxation arises from the combination of the extended primordial spectrum with the full dataset. revision: yes

  2. Referee: [DRMD discussion] DRMD variant: the drag horizon is fixed at r_d,DAO ≈ 60 Mpc/h with decoupling near matter-radiation equality; the manuscript should quantify how sensitive the further relaxation to ΔN_eff < 0.68 is to this choice and whether the value is theoretically motivated or tuned to the data.

    Authors: The fiducial choice r_d,DAO ≈ 60 Mpc/h corresponds to decoupling near matter-radiation equality, which is theoretically motivated because it places the dark acoustic oscillations on scales that can be probed by DESI while allowing a larger ΔN_eff without violating other constraints. It is not tuned to maximize the relaxation but chosen as a representative value for this class of models. In the revision we will add a short sensitivity study varying r_d,DAO by ±10 Mpc/h around the fiducial value and report the resulting ΔN_eff bounds. We will also expand the text to clarify the theoretical motivation from possible end-of-inflation dynamics. revision: yes

  3. Referee: [Hubble tension subsection] Hubble tension quantification: the reduction from the baseline tension (presumably ~4–5σ in ΛCDM with the same datasets) to 2.2σ must be accompanied by an explicit comparison table showing the tension metric (e.g., difference in H0 posterior means divided by combined uncertainty) for both the baseline and extended models.

    Authors: We agree that an explicit side-by-side comparison improves transparency. In the revised manuscript we will insert a table in the Hubble tension subsection that reports the tension metric (difference in H0 posterior means divided by the combined uncertainty) for the baseline ΛCDM model and for each extended model (running only, self-interacting DR, and DRMD variant) using identical datasets. This will document the reduction to 2.2σ (and below 2σ for the DRMD case) in a clear, quantitative manner. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper extends a cosmological model by adding self-interacting dark radiation plus running parameters α_s and β_s, then reports posterior constraints obtained by fitting this model to Planck + ACT DR6 + DESI DR2 + Pantheon+ data. The reported relaxed ΔN_eff bound, 2.9σ preference for α_s > 0, and reduction of Hubble tension to 2.2σ are direct outputs of that fit. No first-principles derivation is claimed whose central result reduces by construction to its own inputs; the analysis is standard Bayesian parameter estimation. The paper explicitly flags sensitivity to the assumed primordial spectrum, which is the opposite of smuggling an ansatz. No self-citation is invoked as load-bearing evidence for a uniqueness theorem or ansatz. The discussion of possible inflationary origins for running is qualitative and does not support the quantitative claims.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 2 invented entities

The central claim rests on three new free parameters (α_s, β_s, and the dark-radiation interaction strength) plus the assumption that the initial power spectrum includes running and running-of-running; the DRMD scenario adds an extra decoupling scale.

free parameters (3)
  • α_s
    Running of the spectral index, fitted to data and reported with 2.9σ preference for positive value.
  • β_s
    Running of the running, fitted to data with 2.6σ preference for positive value.
  • ΔN_eff
    Extra relativistic degrees of freedom, now allowed up to 0.58 at 95% CL under the extended model.
axioms (2)
  • domain assumption The primordial power spectrum includes non-zero running α_s and running-of-running β_s
    Invoked to relax ACT constraints on ΔN_eff; stated in the abstract as the key sensitivity.
  • ad hoc to paper Dark radiation can be self-interacting and initially coupled to dark matter with decoupling near matter-radiation equality
    Required for the DRMD scenario that further relaxes the bound to ΔN_eff < 0.68.
invented entities (2)
  • self-interacting dark radiation no independent evidence
    purpose: To relax the upper bound on ΔN_eff while fitting ACT DR6 data
    Postulated extension that allows larger ΔN_eff without violating observations
  • dark radiation-matter decoupling (DRMD) with drag horizon r_d,DAO ≈ 60 Mpc/h no independent evidence
    purpose: To produce dark acoustic oscillations that further ease the Hubble tension
    Model assumption introduced to obtain the loosest bound ΔN_eff < 0.68

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