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The ACCEL2 Project: Precision Measurements of EFT Parameters and BAO Peak Shifts for the Lyman-$\alpha$ Forest

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read One-loop EFT puts the Lyman-alpha BAO shift near 0.3 percent.

desk verdict Useful calibration paper with an inflated abstract and a shift error budget that omits truncation sensitivity; the central result is plausible and deserves peer review after revisions. read the letter →

arxiv 2412.06892 v1 pith:QF3MBB4N submitted 2024-12-09 astro-ph.CO

Roger de Belsunce , Shi-Fan Chen , Mikhail M. Ivanov , Corentin Ravoux , Solene Chabanier , Jean Sexton , Zarija Lukic This is my paper · ORCID
classification astro-ph.CO PACS 98.80.-k
keywords Lyman-alphaforesteffectivefieldtheoryoflarge-scalestructurebaryonacousticoscillationsBAOshiftfluxpowerspectrumhydrodynamicsimulationsbiasparametersDESI
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that the one-loop effective field theory of large-scale structure (EFT), calibrated on the ACCEL2 hydrodynamic simulations, describes the three-dimensional Lyman-$\alpha$ flux power spectrum accurately enough to serve as the theoretical backbone for baryon acoustic oscillation (BAO) measurements: below 2 percent on small scales up to $k=2\,h\,\mathrm{Mpc}^{-1}$, and at the 5–10 percent level on large scales. From the fitted bias parameters, the paper derives an analytic prediction for how nonlinear clustering shifts the apparent BAO scale. At $z=2$, the radial and transverse dilation parameters shift by $\Delta\alpha_\parallel=-0.20\pm0.09\%$ and $\Delta\alpha_\perp=-0.31\pm0.11\%$, corresponding to $\Delta\alpha_{\rm iso}=-0.28\pm0.09\%$ and $\Delta\alpha_{\rm ap}=0.11\pm0.07\%$. If these numbers hold, current and future Lyman-$\alpha$ BAO analyses can attach a small, quantifiable theory error budget instead of relying on unquantified systematics. The paper also estimates the shift in the Lyman-$\alpha$–quasar cross-correlation at roughly $-0.1\%$, with similar values for two different quasar bias models.

What carries the argument

The load-bearing object is the one-loop EFT flux power spectrum $P^{\rm th}(k,\mu)=P_{\rm tree}+P_{\rm 1\text{-}loop}+P_{\rm ct}+P_{\rm st}$, with redshift-space kernels from the rotationally invariant bias expansion and IR resummation of the BAO wiggle. The BAO shift is carried by the mode-coupling integral $P^{(22)}$, whose dominant piece is $\langle\Psi\delta\rangle\xi'(x)$: long-wavelength displacements contract the BAO sphere around overdensities where the forest signal is suppressed, producing an analytic shift proportional to $\sigma_d^2\,k\,P_w'(k)$ with a bias-dependent prefactor. A Fisher formalism converts this shift into predicted offsets in $\alpha_\parallel$ and $\alpha_\perp$ for a survey of DESI-like volume.

What would settle it

Compute a two-loop Lyman-alpha flux power spectrum at the ACCEL2 cosmology, or perform a field-level comparison on the same simulation boxes, and refit the one-loop model: if the best-fit bias parameters move by more than their quoted uncertainties, the derived BAO-shift budget is not reliable. A second check is to fit the same model to an independent hydrodynamic simulation with different intergalactic-medium thermal physics and see whether the 0.1–0.3 percent shift values persist.

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Extended reading notes

Core claim

The paper establishes that a one-loop EFT model for the Lyman-alpha forest—built from the bias expansion invariant under rotations around the line of sight, with counterterms, stochastic terms, and infrared resummation—fits the ACCEL2 simulated flux power spectrum to sub-2% accuracy on small scales and about 10% on large scales across five snapshots at redshifts $z=2$ to $4$. Using the same fitted parameters, the paper derives a closed-form expression for the nonlinear BAO shift and finds that nonlinearities displace the BAO scale by only a few tenths of a percent, with the sign and size set by the quadratic bias coefficients. This result does not support earlier simulation-based claims of a percent-level redshift-space BAO shift, and it supplies both informative priors and a concrete error budget for full-shape and compressed analyses of the forest.

Load-bearing premise

The one-loop EFT is assumed to remain controlled up to $k_{\max}=2\,h\,\mathrm{Mpc}^{-1}$, even though the loop-to-tree ratio crosses unity near $k\sim1.5\,h\,\mathrm{Mpc}^{-1}$, so the fitted bias parameters and the BAO shift built from them could absorb uncontrolled higher-order corrections.

Editorial extensions

If this is right

  • DESI Lyman-alpha BAO analyses can add a theory error term of order 0.1–0.3 percent to their covariance, comparable to or below the current statistical precision.
  • The measured bias parameters provide informative priors for full-shape analyses of the three-dimensional Lyman-alpha power spectrum.
  • The isotropic shift of about $-0.3\%$ is far smaller than the percent-level shift suggested by some simulation-based studies, so the BAO scale in the forest is largely robust to nonlinear modeling.
  • The Lyman-alpha–quasar cross-correlation shift is only about $-0.1\%$, so cross-correlation BAO measurements need only a modest theory correction.
  • The same EFT framework carries over to future surveys and to field-level analyses, where the bias parameters can be checked against simulation output.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the small BAO shift relies on an accidental suppression of tree-level terms, a two-loop computation or field-level fit could break the cancellation and enlarge the shift beyond the quoted budget.
  • The cross-correlation estimates use quasar bias parameters measured at an effective redshift near $z\simeq1.48$, while the DESI forest measurement sits near $z\simeq2.33$; a bias measurement at the matching redshift could move the cross-correlation shift by an amount comparable to its size.
  • The quoted uncertainties on the shifts include only EFT parameter posteriors; they omit the spread across quasar bias models and across choices of $k_{\max}$, so the true cross-correlation error budget may be larger.
  • A clean test would be to predict the one-dimensional flux power spectrum or the flux bispectrum from the same bias parameters; either observable would exercise the model in a regime that the BAO-shift calculation does not directly use.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper calibrates the one-loop effective field theory (EFT) power spectrum for the Lyman-α forest, as developed in Ref. [49], against the three-dimensional flux power spectrum measured from the ACCEL2 hydrodynamic simulations at five redshifts z = 2.0–4.0. An 18-parameter model (linear, quadratic, and cubic biases; counterterms; stochastic terms) is fitted with Gaussian priors and analytic marginalization over the linear parameters, and the authors report fit residuals at the 2% level for k ≳ 1 h/Mpc and 5–10% on large scales (k ≲ 0.5 h/Mpc). Using the MCMC chains, the fitted bias parameters are propagated through an analytic expression (Eq. A1) and a Fisher forecast to predict non-linear BAO peak shifts: Δα∥ = −0.20 ± 0.09% and Δα⊥ = −0.31 ± 0.11% at z = 2.0 (Eq. 19), corresponding to Δαiso = −0.28 ± 0.09% and Δαap = 0.11 ± 0.07%, with analogous numbers at z = 2.6 and for the Lyα–quasar cross-correlation using eBOSS quasar biases and analytic/simulation-based bias relations. The paper presents these numbers as a theoretical error budget for DESI Lyα BAO measurements and as informative priors for full-shape analyses.

Significance. The paper is a timely and useful contribution if the quoted error budget can be made robust. Its strengths include: the first precision calibration of the one-loop Lyα-forest EFT with full posterior distributions for the bias set; an analytic extension of the galaxy BAO-shift formalism of Ref. [71] to the anisotropic Lyα forest and to the Lyα–quasar cross-correlation (Appendix A); a transparent, quantified disagreement with the percent-level simulation-based shift of Ref. [72]; and a commendably explicit statement of its own limitations (possible phenomenological status of the model at kmax = 2, resolution differences between the two boxes, and the mismatched effective redshift of the eBOSS quasar biases). The concern about circularity raised in the stress test does not, in my reading, land: the BAO shift is never fitted to the simulations; it is propagated from power-spectrum-calibrated parameters through an independent analytic formula. However, the central deliverable is an error budget, and the omission of a truncation-error term, together with the ambiguity over which chains (minimal vs. stochastic) produce Eq.

major comments (4)
  1. [Sec. 6.2, Eq. (19), Fig. 7] The quoted error bars in Eqs. (19)–(22) propagate only the MCMC parameter covariance and do not include a truncation-error term, although the one-loop expansion is not manifestly controlled at the fitting scale kmax = 2 h/Mpc: Fig. 2 shows the one-loop-to-tree ratio crossing unity near k ≈ 1.5 h/Mpc at z = 2, and Sec. 6.1 explicitly leaves open the possibility that the model is only phenomenological there. This is directly relevant to the error budget: Fig. 7 shows that the radial shift changes sign as kmax is increased from 0.5 to 2 h/Mpc, a variation far larger than the quoted ±0.09% uncertainty. I request that a systematic truncation error (e.g., from varying kmax, or from a two-loop estimate) be added to Eq. (19), or that the quoted budget be restricted to scales on which the loop expansion is controlled.
  2. [Abstract; Sec. 6.1, Fig. 1] The abstract states that the EFT model "fits the data with an accuracy of below 2 percent up to a wavenumber of k = 2 h/Mpc." The body (Sec. 6.1 and Fig. 1) instead reports residuals of 5–10% on large scales (k ≲ 0.5 h/Mpc), with sub-2% accuracy only for k ≳ 1 h/Mpc. The body is honest about this scale dependence, but the abstract overstates the accuracy and should be rephrased to describe the scale-dependent residuals, e.g., "below 2% on small scales (k > 1 h/Mpc) and 5–10% on large scales."
  3. [Sec. 6.2, Table II, Fig. 3] It is not specified which MCMC chains were used to produce the baseline BAO shifts in Eqs. (19)–(22): the minimal model or the model with stochastic terms ("+st."). This matters because Table II shows the stochastic terms are detected at ≫5σ at z = 2.0–3.0, and the minimal versus +st. best fits differ substantially in exactly the quadratic biases entering Eq. (A1) (e.g., at z = 2, bη2 = −0.35 versus −1.18 and bG2 = 0.05 versus −0.17). Since Eq. (A1) is linear in these parameters, the predicted BAO shift will differ by more than the quoted uncertainty between the two model choices. Please state explicitly which configuration was used for Eqs. (19)–(22) and report the shift from both configurations as a robustness check.
  4. [Sec. 6.1, discussion after Fig. 2] The claim that the BAO shift estimates "will not depend on a particular point of view on the role of the one-loop corrections" is too strong. The fitted quadratic and cubic biases used in Eq. (A1) are obtained from fits at kmax = 2 h/Mpc, where the one-loop-to-tree ratio exceeds unity and the stochastic/counterterm contributions are large; any mis-absorption of two-loop UV physics into those biases propagates directly into Eq. (19) through the analytic relation in Eq. (A1). A concrete, feasible test would be to compute the shift using only fits with kmax ≤ 1 h/Mpc, where the loop expansion is controlled, and to compare the result with Eq. (19); this would place the "phenomenological versus controlled" debate on a quantitative footing for the BAO-shift claim.
minor comments (6)
  1. [Sec. 2] There are typos in the resolution units: "resolution down to 100 h kpc−1" should read "100 h−1 kpc," and similarly "25 hkpc−1" should read "25 h−1 kpc" (the correct form appears elsewhere in the paper).
  2. [Sec. 3, Eq. (7)] The symbol kNL is used in Eq. (7) but is not defined at first use; please define it explicitly as the nonlinear scale (e.g., the scale where the dimensionless power spectrum is of order unity).
  3. [Sec. 6.2, Eqs. (24)–(30)] The eBOSS quasar biases used for the cross-correlation shifts are measured at z_eff = 1.48, while the Lyα parameters are at z = 2.0; the authors appropriately caution that the result "should only be taken indicatively," but the headline cross-correlation numbers (Eqs. 27–30) could be misinterpreted. A direct statement of the implied systematic from the redshift mismatch, or a matched-redshift estimate, would remove ambiguity.
  4. [Sec. 4, Eq. (13)] The likelihood uses a diagonal Gaussian covariance derived from the number of Fourier modes per bin. Given the very small quoted parameter uncertainties (e.g., b1 to 0.003 in Table II), a short test of sensitivity to off-diagonal or non-Gaussian covariance (e.g., a split-box or jackknife estimate) would increase confidence that the uncertainties are not underestimated.
  5. [Fig. 3 and Sec. 6.2] The figure legend distinguishes "kmax = 2.0 h/Mpc," "kmax = 3.0 h/Mpc," and "kmax = 3.0 h/Mpc + st.," which implies the baseline kmax = 2 chains do not include stochastic terms. This is also relevant to the major comment about which chains produce Eq. (19); a consistent labeling or an explicit statement in Sec. 6.2 would help the reader.
  6. [Throughout] The simulation name is written inconsistently as "ACCEL2" and "ACCL2"; please unify.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the BAO shift is a derived functional of independently fitted EFT parameters, not a refit of the shift; minor same-author citations are present but not load-bearing.

full rationale

The derivation chain is not circular. The EFT parameters in Eq. (2)-(7) are calibrated to the ACCEL2 3D flux power spectrum through the chi-squared in Eq. (13); this is a fit to simulation data, not an input defined in terms of the later BAO-shift output. The BAO shift is computed from Eq. (A1), an analytic one-loop functional of the fitted quadratic and cubic bias parameters, with the Fisher formalism taken from Ref. [71]. The shift itself is never fitted, and no equation in the paper sets the shift equal to a fit parameter by construction. Varying kmax changes the fitted biases and therefore the shift (Fig. 7), which is error propagation rather than tautology. Same-author references appear in the model construction ([49], [50]) and in the BAO-shift formalism ([71]), but the model is tested against independent hydrodynamic simulation power spectra and Eq. (A1) is derived in the appendix; no uniqueness theorem or ansatz is imported to force the result. The paper explicitly acknowledges the main limitation in Sec. 6.1: Fig. 2 shows the one-loop-to-tree ratio crossing unity near k~1.5 h/Mpc at z=2, and the text states that the model may need to be treated as phenomenological beyond k~1 h/Mpc, deferring a definitive two-loop or field-level check. That is a truncation/correctness risk for the quoted error budget, not a circularity: the fit quality and the shift estimate are not definitions of one another. The cross-correlation shift uses eBOSS quasar biases at zeff=1.48 and external bias relations from Refs. [76] and [110], with the paper itself cautioning that these values are only indicative. Overall, no step reduces by construction to its own input; the only ground for a nonzero score is the presence of minor self-citations in the theoretical machinery, none of which is load-bearing.

Assumptions & free parameters 18 free parameters · 8 assumptions · 0 invented entities

The central claim rests on 18 fitted EFT bias, counterterm, and stochastic parameters, plus assumptions that the one-loop expansion is controlled at kmax=2, that the diagonal Gaussian covariance is adequate, that ACCEL2 simulations represent the forest, that the Ref. [71] Fisher mapping is valid, and that external quasar bias relations approximate quasar clustering. No new particles or forces are introduced.

free parameters (18)
  • b1 = -0.0739 +/- 0.0051 (z=2.0, extended)
    Linear density bias; sampled EFT parameter fitted to ACCEL2 P(k,mu).
  • b_eta = 0.1245 +0.0129/-0.0140 (z=2.0, extended)
    Linear velocity gradient bias; sampled EFT parameter.
  • b2 = 0.0504 +0.0970/-0.0734 (z=2.0, extended)
    Quadratic density bias; enters the BAO shift formula Eq. A1.
  • b_G2 = -0.1684 +0.1039/-0.0765 (z=2.0, extended)
    Tidal quadratic bias; enters Eq. A1.
  • b_eta2 = -1.1780 +0.2389/-0.2645 (z=2.0, extended)
    Quadratic velocity bias; sampled EFT parameter.
  • b_delta_eta = -0.4730 +0.1421/-0.1702 (z=2.0, extended)
    Density-velocity bias; sampled EFT parameter.
  • b_(KK)_parallel = 0.7508 +0.1979/-0.1782 (z=2.0, extended)
    Line-of-sight operator bias; sampled EFT parameter.
  • b_Pi[2]_parallel = 0.1464 +/- 0.0988 (z=2.0, extended)
    Second-order line-of-sight bias; sampled EFT parameter.
  • b_Pi[3]_parallel = 0.5863 +/- 0.0393 (z=2.0, extended)
    Cubic EFT term; analytically marginalized and recovered a posteriori.
  • b_(K Pi[2])_parallel = -1.0424 +/- 0.0943 (z=2.0, extended)
    Cubic EFT term; analytically marginalized.
  • b_delta Pi[2]_parallel = 1.6968 +/- 0.0685 (z=2.0, extended)
    Cubic EFT term; analytically marginalized.
  • b_eta Pi[2]_parallel = 5.1990 +/- 0.1273 (z=2.0, extended)
    Cubic EFT term; analytically marginalized.
  • c0 = -0.1557 +/- 0.0072 (z=2.0, extended), h^2/Mpc^2
    Counterterm; analytically marginalized.
  • c2 = 0.2554 +/- 0.0086 (z=2.0, extended), h^2/Mpc^2
    Counterterm; analytically marginalized.
  • c4 = -0.1675 +/- 0.0055 (z=2.0, extended), h^2/Mpc^2
    Counterterm; analytically marginalized.
  • P_shot = 0.1562 +/- 0.0107 (z=2.0, extended)
    Stochastic shot noise term; analytically marginalized.
  • a0 = -0.1156 +/- 0.0101 (z=2.0, extended)
    Stochastic k^2 term; analytically marginalized.
  • a2 = -0.0409 +/- 0.0330 (z=2.0, extended)
    Stochastic k^2 mu^2 term; analytically marginalized.
assumptions (8)
  • domain assumption The one-loop EFT Lyman-alpha power spectrum (Eqs. 2-7) with kernels from Ref. [49] correctly describes the flux power spectrum at kmax=2 h/Mpc.
    Sec. 3 and Fig. 2: the loop-to-tree ratio reaches unity near k=1.5, and the paper defers a two-loop or field-level validation.
  • domain assumption IR resummation of long-wavelength displacements is required and correctly implemented.
    Sec. 3 invokes Refs. [83-88]; no independent verification is provided in this paper.
  • domain assumption A diagonal Gaussian covariance based on Fourier mode counts, Eq. (13), is a sufficient description of the ACCEL2 power spectrum errors.
    Sec. 4; off-diagonal and non-Gaussian covariance contributions are neglected.
  • domain assumption The ACCEL2/Nyx simulations provide unbiased realizations of the Lyman-alpha forest EFT bias parameters at each snapshot.
    Sec. 2; a single hydro code and IGM model are used, and resolution effects are not fully spliced for the large box.
  • domain assumption The Fisher formalism of Ref. [71] correctly maps the analytic P_shift to shifts in alpha_parallel and alpha_perp for BAO fits.
    Sec. 5 and Eq. (18); assumes a specific template fitting procedure and a DESI-like covariance.
  • domain assumption External quasar bias values and relations (Refs. [75,76,110,119]) approximate the nonlinear quasar clustering at the redshifts of interest.
    Sec. 6.2; eBOSS values are at z_eff=1.48 rather than z=2, and the authors call the result indicative.
  • domain assumption Setting b_Gamma3=0 is harmless because it is degenerate with b_G2.
    Footnote 10; a modeling choice not directly tested in this work.
  • domain assumption Stochastic shot-noise terms P_shot, a0, a2 are physically small at high redshift and can be included as Wilson coefficients.
    Sec. 3 naturalness estimates and the redshift dependence of detections; not proven from first principles.

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Pith. "Pith review of The ACCEL2 Project: Precision Measurements of EFT Parameters and BAO Peak Shifts for the Lyman-$\alpha$ Forest." pith.science (2026). https://pith.science/paper/QF3MBB4N

@misc{pith2026241206892,
  author       = {Pith},
  title        = {Pith review of: The ACCEL2 Project: Precision Measurements of EFT Parameters and BAO Peak Shifts for the Lyman-$\alpha$ Forest},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QF3MBB4N}},
  note         = {Machine review of arXiv:2412.06892}
}
read the original abstract

We present precision measurements of the bias parameters of the one-loop power spectrum model of the Lyman-alpha (Lya) forest, derived within the effective field theory of large-scale structure (EFT). We fit our model to the three-dimensional flux power spectrum measured from the ACCEL2 hydrodynamic simulations. The EFT model fits the data with an accuracy of below 2 percent up to a wavenumber of k = 2 h/Mpc. Further, we analytically derive how non-linearities in the three-dimensional clustering of the Lya forest introduce biases in measurements of the Baryon Acoustic Oscillations (BAO) scaling parameters in radial and transverse directions. From our EFT parameter measurements, we obtain a theoretical error budget of -0.2 (-0.3) percent for the radial (transverse) parameters at redshift two. This corresponds to a shift of -0.3 (0.1) percent for the isotropic (anisotropic) distance measurements. We provide an estimate for the shift of the BAO peak for Lya-quasar cross-correlation measurements assuming analytical and simulation-based scaling relations for the non-linear quasar bias parameters resulting in a shift of -0.2 (-0.1) percent for the radial (transverse) dilation parameters, respectively. This analysis emphasizes the robustness of Lya forest BAO measurements to the theory modeling. We provide informative priors and an error budget for measuring the BAO feature -- a key science driver of the currently observing Dark Energy Spectroscopic Instrument (DESI). Our work paves the way for full-shape cosmological analyses of Lya forest data from DESI and upcoming surveys such as the Prime Focus Spectrograph, WEAVE-QSO, and 4MOST.

Figures

Figures reproduced from arXiv: 2412.06892 by the authors.

Figure 1
Figure 1. FIG. 1. The best-fit EFT model in four angular bins [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Bias parameters and counterterms fit to the ACCEL [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figures from the paper (5 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Size of the one-loop corrections obtained by compar [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Bias relations assumed for the Ly- [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Percentage of shift parameter sampled from the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Fisher forecasts for the BAO shift in radial ( [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Percentage of BAO shift parameter for the Ly- [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

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    ACCEL 2 SIMULA TIONS In the present work, we perform precision measure- ments of the EFT bias parameters on a suite of ACCEL2 (ACCELerated expansion of the universe with ACCEL- erated computing on GPUs) hydrodynamical simulations of the Ly- α forest [63]. We use the ACCEL 2 simulation with the highest resolution with a box of length L = 160h−1Mpc with N =...

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    THEORETICAL MODEL One of the key advantages of the Ly- α forest is that it probes the Universe at intermediate redshifts (2 ≤ z ≤ 4), with access to many more linear modes than, e.g., galaxy surveys. In the following, we briefly summarize the theoretical model of the Ly- α forest one-loop (auto) power spectrum in the EFT framework [49]. The model consists...

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