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arxiv: 2605.18615 · v1 · pith:7E7VYIR7new · submitted 2026-05-18 · 🌌 astro-ph.CO

Primordial power spectrum reconstructions from BOSS + eBOSS

Guillermo Mart\'inez-Somonte , H\'ector Gil-Mar\'in , Airam Marcos-Caballero , Enrique Mart\'inez-Gonz\'alez This is my paper

Pith reviewed 2026-05-20 09:07 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords primordial power spectrumBOSS surveyeBOSS surveynon-parametric reconstructionprimordial featuresspectral indexgalaxy clusteringinflationary models
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The pith

BOSS and eBOSS galaxy clustering data reconstruct the primordial power spectrum as a quasi-scale-invariant power law with no detected features.

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

The paper uses a non-parametric Bayesian approach to reconstruct the primordial curvature power spectrum P_R(k) directly from the observed galaxy power spectra in the BOSS DR12 LRG and eBOSS DR16 QSO catalogs. Knots are placed in the logarithmic plane of wavenumber and power, and a separate seven-parameter model isolates galaxy bias effects, redshift-space distortions, and baryon acoustic oscillation wiggles from the underlying matter power spectrum template. This separation permits reliable reconstruction out to nonlinear scales of k = 0.3 h Mpc^{-1}. Application to the real data, validated first on mock catalogs and injected-feature templates, yields consistent results across redshift slices and sky regions. The reconstruction prefers a simple power-law form whose spectral index matches the value measured by the Planck satellite.

Core claim

The analysis reconstructs P_R(k) by linearly interpolating N knots in the {log k, log P_R(k)} plane while fitting a seven-parameter model that describes the galaxy power spectrum after separating any power-law deviations and BAO contributions from the matter power spectrum template. Validation on MD-Patchy and EZmock catalogs recovers the input power law, and tests on primordial-feature templates confirm the method's ability to detect deviations if present. Both individual and combined BOSS plus eBOSS measurements are consistent with a quasi-scale-invariant power law and show no evidence for additional features, producing a robust constraint n_s = 0.976 ± 0.021.

What carries the argument

Non-parametric reconstruction via linear interpolation of N knots in the {log k, log P_R(k)} plane, combined with a seven-parameter parametric model Θ_model that isolates power-law deviations and BAO wiggles from the matter power spectrum template.

If this is right

  • The reconstructed spectrum is consistent across separate redshift bins and northern and southern galactic caps.
  • No evidence appears for primordial features in P_R(k) up to k = 0.3 h Mpc^{-1}.
  • A quasi-scale-invariant power law remains the preferred model, matching the expectation of most slow-roll inflationary scenarios.
  • The spectral index is constrained to n_s = 0.976 ± 0.021, compatible with the Planck measurement.
  • The method successfully recovers the input power law from representative mock catalogs of the BOSS and eBOSS samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The absence of features strengthens the case for single-field slow-roll inflation over models that predict oscillatory or scale-dependent deviations at these wavenumbers.
  • Applying the same knot-based reconstruction to future surveys such as DESI or Euclid could tighten the limits on any residual non-power-law behavior.
  • If the parametric separation of BAO and bias effects remains accurate, the approach offers a route to model-independent tests of inflation using only late-time large-scale structure.

Load-bearing premise

The seven-parameter model can accurately separate power-law deviations and BAO contributions from the underlying matter power spectrum template at non-linear scales.

What would settle it

Re-running the identical knot reconstruction on an independent, higher-precision dataset or on mocks that contain known oscillatory features and recovering a statistically significant departure from the recovered power-law shape would falsify the claim of no primordial features.

read the original abstract

We reconstruct the primordial power spectrum $P_{\mathcal{R}}(k)$ from the BOSS DR 12 LRG and eBOSS DR 16 QSO catalogs with a non-parametric Bayesian method. The $P_{\mathcal{R}}(k)$ is reconstructed by linearly interpolating $N$ knots in the $\{ \log k, \log P_{\mathcal{R}}(k) \}$ plane. We use a parametric model to describe the galaxy power spectra of the BOSS+eBOSS catalogs, assuming any power-law deviations and BAO contributions separately from the matter power spectrum template, composed of seven parameters $\Theta_{\text{model}}$. This parametric model enables us to reconstruct $P_{\mathcal{R}}(k)$ at non-linear scales, reaching $k = 0.3 \text{ h} \text{ Mpc}^{-1}$. The method is validated by applying it to different Primordial Features (PF) templates and by recovering the input power law of \textsc{MD-Patchy} and \textsc{EZmock} mock catalogs, representative of the BOSS and eBOSS data. These mocks provide additional information on $\Theta_{\text{model}}$, enabling us to impose Gaussian correlated priors on $\Theta_{\text{model}}$. This prior set allows us to reconstruct $P_{\mathcal{R}}(k)$ more precisely and to alleviate the degeneracies between the model and knot parameters. The results for both individual and combined $z$-bins and galactic caps of the BOSS and eBOSS catalogs are consistent, showing no evidence of the presence of PF in $P_{\mathcal{R}}(k)$ and pointing to a quasi-scale-invariant power law as the preferred model for $P_{\mathcal{R}}(k)$, as predicted by most slow-roll inflationary models. With a different prior set that relaxes the Gaussian constraints on $\Theta_{\text{model}}$ and imposes Planck-based priors on the extreme knots, the results also favor the power law. From the knot reconstructions, we robustly constrain the spectral index $n_s = 0.976 \pm 0.021$, compatible with the Planck value.

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

2 major / 2 minor

Summary. The manuscript presents a non-parametric Bayesian reconstruction of the primordial curvature power spectrum P_R(k) from BOSS DR12 LRG and eBOSS DR16 QSO catalogs. P_R(k) is obtained by linear interpolation among N knots in the {log k, log P_R(k)} plane. A seven-parameter parametric model Θ_model is used to describe the observed galaxy power spectra, isolating power-law deviations and BAO wiggles from a fixed matter template; this permits reconstruction out to k = 0.3 h Mpc^{-1}. The approach is validated on injected primordial-feature templates and on MD-Patchy/EZmock catalogs, from which Gaussian correlated priors on Θ_model are derived to reduce degeneracies. Results are reported for individual and combined redshift bins and galactic caps; all are consistent with a featureless quasi-scale-invariant power law, yielding the constraint n_s = 0.976 ± 0.021.

Significance. If robust, the work extends direct constraints on P_R(k) into the mildly non-linear regime using galaxy clustering, providing a valuable cross-check on slow-roll inflation beyond CMB scales. Explicit credit is due for the mock-based validation (recovery of input power-law spectra) and for the reported consistency across multiple independent subsamples of BOSS and eBOSS data.

major comments (2)
  1. [§3.2] §3.2 (definition of Θ_model): the seven-parameter parametric form is asserted to fully capture scale-dependent galaxy bias, RSD, and higher-order corrections up to k = 0.3 h Mpc^{-1}, yet the mock validation in §5 recovers the input spectrum only when the same parametric assumption is imposed on the mocks. This leaves open the possibility that unmodeled k-dependent bias or stochasticity could be absorbed into the knot amplitudes, directly affecting the central claim of no primordial features.
  2. [§4.1] §4.1 (prior construction): the Gaussian correlated priors on Θ_model are derived from the identical class of mocks used for validation. While this reduces degeneracies, it ties the reconstruction to the mock assumptions; the paper does not quantify how the posterior on the knots changes when these priors are replaced by broader or Planck-informed alternatives beyond the single test mentioned in the abstract.
minor comments (2)
  1. [§2] The number of knots N and the precise knot placement in log k are not stated explicitly in the main text; a short table or equation would improve reproducibility.
  2. [Figure 5] Figure captions for the reconstructed P_R(k) should indicate the exact k-range used for the n_s fit and whether the quoted uncertainty includes marginalization over Θ_model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for the constructive comments, which help clarify the robustness of our non-parametric reconstruction. We address each major comment below.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (definition of Θ_model): the seven-parameter parametric form is asserted to fully capture scale-dependent galaxy bias, RSD, and higher-order corrections up to k = 0.3 h Mpc^{-1}, yet the mock validation in §5 recovers the input spectrum only when the same parametric assumption is imposed on the mocks. This leaves open the possibility that unmodeled k-dependent bias or stochasticity could be absorbed into the knot amplitudes, directly affecting the central claim of no primordial features.

    Authors: We thank the referee for raising this important caveat. The seven-parameter Θ_model is constructed to isolate the dominant contributions from scale-dependent bias, RSD, and BAO wiggles relative to a fixed matter template, thereby enabling reconstruction at mildly non-linear scales. While it is correct that the mock validation assumes the same parametric form, the MD-Patchy and EZmock catalogs are specifically calibrated to reproduce the observed BOSS and eBOSS clustering statistics. In addition, the manuscript already reports results under a relaxed prior set on Θ_model (with Planck-informed priors on the extreme knots), which yields consistent conclusions with no evidence for primordial features. In the revised version we will expand §5 with an explicit discussion of possible residual systematics and their potential leakage into the knot amplitudes. revision: partial

  2. Referee: [§4.1] §4.1 (prior construction): the Gaussian correlated priors on Θ_model are derived from the identical class of mocks used for validation. While this reduces degeneracies, it ties the reconstruction to the mock assumptions; the paper does not quantify how the posterior on the knots changes when these priors are replaced by broader or Planck-informed alternatives beyond the single test mentioned in the abstract.

    Authors: We appreciate the referee's point on prior dependence. As stated in the abstract and detailed in the text, we have already performed a test replacing the mock-derived Gaussian priors with a relaxed set that incorporates Planck-based priors on the extreme knots. This alternative prior produces a reconstruction that remains consistent with a quasi-scale-invariant power law and shows no evidence for primordial features. To provide the requested quantification, we will add a direct comparison of the knot posteriors (including shifts in n_s and feature amplitudes) under the two prior choices in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the reconstruction method

full rationale

The paper presents a Bayesian reconstruction of the primordial power spectrum from real BOSS and eBOSS galaxy clustering data using knot interpolation and a 7-parameter parametric model for the galaxy power spectrum. While Gaussian priors on the model parameters are informed by mock catalogs to reduce degeneracies, the primary results—no evidence for primordial features and a constrained spectral index—are obtained by fitting to the actual observational data. Validation on mocks demonstrates recovery of input power laws under the model assumptions but does not make the data analysis circular, as the mocks serve as an external benchmark rather than defining the output by construction. The derivation chain remains self-contained with independent content from the survey data.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard cosmological modeling assumptions plus the specific choice of priors and interpolation scheme; no new particles or forces are introduced.

free parameters (2)
  • N (number of knots)
    Number of interpolation points chosen to balance flexibility against data constraints; value not stated in abstract.
  • Θ_model (seven parameters)
    Parameters describing galaxy bias, BAO, and power-law deviations; fitted with Gaussian correlated priors derived from mocks.
axioms (2)
  • domain assumption Linear interpolation between knots in the {log k, log P_R(k)} plane sufficiently captures any deviations from a power law.
    Central modeling choice for the non-parametric reconstruction.
  • domain assumption The MD-Patchy and EZmock catalogs accurately represent the statistical properties of the BOSS and eBOSS data for the purpose of setting priors on Θ_model.
    Used to impose Gaussian correlated priors that alleviate degeneracies.

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Reference graph

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