One-dimensional Flux Power Spectrum

• The one-dimensional flux power spectrum is defined as the Fourier transform of Lyman-α forest flux fluctuations normalized by the mean transmission, summarizing small-scale intergalactic absorption.
• It is derived from quasar absorption spectra using methods like FFT and QMLE, providing key constraints on the thermal state, reionization, and amplitude of the matter power spectrum.
• The estimator carefully accounts for contaminants and systematics such as high column density absorbers, instrumental resolution, and noise to ensure robust cosmological inference.

The one-dimensional flux power spectrum, usually denoted $P_{1\mathrm D}$ or $P_F(k)$, is the line-of-sight power spectrum of Lyman-$\alpha$ forest transmitted-flux fluctuations measured from quasar absorption spectra. In practice it is defined in velocity-space coordinates along individual sightlines, averaged over many forests or synthetic skewers, and used as a compact summary of small-scale intergalactic absorption structure. Across the literature, $P_{1\mathrm D}$ is central to constraints on the thermal state and reionization history of the intergalactic medium (IGM), the amplitude and slope of the underlying matter power spectrum, neutrino free streaming, and non-cold-dark-matter scenarios [(Palanque-Delabrouille et al., 2013); (Chabanier et al., 2018); (Karaçaylı et al., 12 May 2025)]. The quantity is not defined identically in all analyses: most observational and simulation papers work with a mean-normalized flux-contrast field, while some studies use the Fourier power of the transmitted flux itself normalized by the mean flux, without explicitly introducing $\delta_F$ (Iršič et al., 2017, Chabanier et al., 2018, Mishra et al., 2021).

1. Definitions and normalization conventions

In the standard Lyman-$\alpha$ forest convention, the transmitted flux fraction is built from the observed flux $f$ and the quasar continuum $C$, so that $F=f/C$, and the fluctuation field is defined by

$\delta_F=\frac{F}{\bar F}-1.$

This definition is explicit in the XQ-100 analysis, where $P_F(k)$0 is measured in each redshift bin and the 1D power spectrum is obtained from the Fourier transform of $P_F(k)$1 (Iršič et al., 2017). Closely related conventions appear in BOSS DR9, where

$P_F(k)$2

and in SDSS DR14, where

$P_F(k)$3

with $P_F(k)$4 and $P_F(k)$5 [(Palanque-Delabrouille et al., 2013); (Chabanier et al., 2018)]. DESI DR1 likewise uses

$P_F(k)$6

in its QMLE measurement, and the FFT-based DESI DR1 estimator adopts the same flux-contrast logic after continuum fitting and mean-transmission normalization (Karaçaylı et al., 12 May 2025, Ravoux et al., 14 May 2025).

The corresponding 1D power estimator is usually the squared Fourier amplitude of this line-of-sight field. A representative explicit form is

$P_F(k)$7

with $P_F(k)$8 when the discrete Fourier transform is performed over a spectrum of velocity-space length $P_F(k)$9 (Coughlin et al., 2018). In SDSS DR14 the raw power is written as

$\alpha$0

and the final estimator subtracts noise and divides by the instrumental window function before averaging over forest chunks (Chabanier et al., 2018). BOSS DR9 expresses the same object through the line-of-sight covariance or, equivalently, through Fourier power of the transmitted-flux fluctuation field (Palanque-Delabrouille et al., 2013).

Not all papers adopt the contrast-field convention. In the CROC reionization study, the operative definition is

$\alpha$1

where $\alpha$2 is the Fourier transform of the simulated transmitted flux and averaging is over 1000 lines of sight. No mean-subtracted $\alpha$3 field is introduced there, and no Fourier-normalization convention is specified (Mishra et al., 2021). That difference is substantive: the statistic is a mean-flux-normalized power of the Fourier-transformed transmitted flux field itself, rather than explicitly the power spectrum of a contrast field.

A separate but related convention appears in theoretical work that starts from a three-dimensional field and defines dimensionless 1D power. One modified-gravity study writes the 1D flux-decrement power as

$\alpha$4

with logarithmic power $\alpha$5 (Brax et al., 2018). A recent phenomenological mapping between 3D matter power and 1D flux power instead uses

$\alpha$6

explicitly emphasizing that the observable is a projection over all transverse modes (Ridkokasha et al., 20 May 2025).

2. Estimation from spectra and simulated skewers

The measurement pipeline begins with quasar spectra, continuum estimation, forest selection, and conversion to a line-of-sight fluctuation field in velocity space. BOSS DR9 defines the forest in the quasar rest frame by $\alpha$7, rebins spectra on a uniform $\alpha$8 grid with $\alpha$9, splits forests into sub-sectors of limited redshift extent, and estimates $P_{1\mathrm D}$0 with both a Fourier-transform estimator and a maximum-likelihood estimator (Palanque-Delabrouille et al., 2013). SDSS DR14 uses the same rest-frame forest interval, divides each forest into three consecutive non-overlapping subregions so that each chunk spans at most $P_{1\mathrm D}$1, and measures the power in 35 $P_{1\mathrm D}$2-bins over $P_{1\mathrm D}$3 to $P_{1\mathrm D}$4 (Chabanier et al., 2018).

The standard FFT-based measurement writes the observed raw power as a combination of astrophysical signal, metal contamination, instrumental smoothing, and noise. In SDSS DR14,

$P_{1\mathrm D}$5

where

$P_{1\mathrm D}$6

The final estimator is

$P_{1\mathrm D}$7

with the sideband subtraction removing uncorrelated metal power and residual spectroscopic-pipeline contamination (Chabanier et al., 2018).

DESI DR1 extends this framework in two directions. The FFT-based measurement uses observed spectra rebinned linearly in wavelength, converts to the flux fluctuation field

$P_{1\mathrm D}$8

and estimates

$P_{1\mathrm D}$9

where $\delta_F$0 is derived from the DESI resolution matrix averaged over the sub-forest (Ravoux et al., 14 May 2025). The QMLE analysis instead works directly with the covariance of the pixel data and estimates band powers through inverse-covariance weighting, with the combined estimator

$\delta_F$1

thereby deconvolving survey-window effects from masking, continuum marginalization, and irregular sampling (Karaçaylı et al., 12 May 2025).

Mock spectra and simulation skewers follow analogous logic. XQ-100 divides each continuum-normalized spectrum into redshift chunks, computes the Fourier transform of $\delta_F$2, deconvolves instrumental smoothing and pixelization in Fourier space, subtracts white noise, and then subtracts a metal power term $\delta_F$3 so that the Ly$\delta_F$4-only quantity is

$\delta_F$5

(Iršič et al., 2017). The dark-energy simulation study gives a fully explicit synthetic-spectrum pipeline, beginning with $\delta_F$6, forming $\delta_F$7, then

$\delta_F$8

and finally

$\delta_F$9

from FFTW3-transformed skewers in velocity space (Coughlin et al., 2018).

3. Major survey measurements and simulation data sets

The observational development of $\alpha$0 has proceeded from BOSS-era surveys to DESI, with high-resolution surveys supplying complementary small-scale information. BOSS DR9 measured the 1D transmitted-flux power spectrum from 13,821 selected quasar spectra over 12 redshift bins from $\alpha$1 to $\alpha$2 and scales $\alpha$3 to $\alpha$4, using both Fourier and maximum-likelihood estimators (Palanque-Delabrouille et al., 2013). SDSS DR14 then extended the measurement to 43,751 high-quality quasar spectra and 94,558 forest chunks, covering 13 redshift bins from $\alpha$5 to 4.6 and $\alpha$6 to $\alpha$7, with statistical uncertainties reduced by about a factor of two relative to the earlier DR9 result (Chabanier et al., 2018).

High-resolution spectroscopy probes smaller scales than BOSS-like surveys. The XQ-100 Legacy Survey measured the Ly$\alpha$8 flux power spectrum from 100 quasar spectra over $\alpha$9 to $f$0 and $f$1 to $f$2, with total errors comparable to BOSS in the overlap region and more than 50% smaller for $f$3 and $f$4 (Iršič et al., 2017). A later high-resolution cosmology analysis with XQ100 and KODIAQ-SQUAD used $f$5–4.2 and $f$6–0.064 $f$7 for XQ100, and $f$8–4.2 and $f$9–0.065 $C$0 for KODIAQ-SQUAD, explicitly targeting the small-scale regime inaccessible to eBOSS-like samples (Ho et al., 22 Sep 2025).

DESI has shifted the field from precision measurement to very-high-statistics precision measurement. The early DESI FFT analysis used 26,330 quasar spectra at $C$1, yielding 73,839 sub-forests over $C$2 to $C$3 and $C$4 (Ravoux et al., 2023). The DESI DR1 QMLE measurement later used 314,241 quasars in the Ly$C$5 rest-frame region $C$6–$C$7 $C$8, and the abstract characterizes the DR1 sample as over 300,000 Ly$C$9 quasars, larger than eBOSS by a factor of 1.7 (Karaçaylı et al., 12 May 2025). The DR1 QMLE and companion FFT analyses are consistent with each other and are described as the most precise $F=f/C$0 measurements to date (Karaçaylı et al., 12 May 2025, Ravoux et al., 14 May 2025).

Simulation papers use the same statistic in different regimes. The CROC reionization study measures a 1D line-of-sight flux power spectrum from 7 realizations of $F=f/C$1 boxes and 1000 genuinely random lines of sight per realization over $F=f/C$2 (Mishra et al., 2021). The time-dependent-dark-energy study uses dark-matter-only simulations in a $F=f/C$3 box with $F=f/C$4 particles, extracts 1152 synthetic spectra per model at each redshift, and evaluates $F=f/C$5 at $F=f/C$6 (Coughlin et al., 2018).

4. Physical content: scales, shape, amplitude, and cosmological sensitivity

At fixed redshift, $F=f/C$7 generally decreases toward high $F=f/C$8, owing to thermal broadening, pressure smoothing, and instrumental resolution; at fixed $F=f/C$9, the amplitude generally increases with redshift over the observed Lyman-$\delta_F=\frac{F}{\bar F}-1.$0 range (Iršič et al., 2017, Chabanier et al., 2018). In the XQ-100 measurement, for example, $\delta_F=\frac{F}{\bar F}-1.$1 increases with redshift and decreases with $\delta_F=\frac{F}{\bar F}-1.$2, while the DR14 measurement likewise shows larger amplitude at higher redshift and a falloff toward high $\delta_F=\frac{F}{\bar F}-1.$3 (Iršič et al., 2017, Chabanier et al., 2018). These trends encode both cosmology and IGM microphysics.

The statistic is sensitive to several distinct physical effects. Measurement papers explicitly emphasize sensitivity to the IGM temperature and thermal history on Mpc and sub-Mpc scales, and to suppression of small-scale clustering from free-streaming particles on smaller scales (Chabanier et al., 2018, Karaçaylı et al., 12 May 2025). The dark-energy simulation study shows that, within observationally allowed CPL models, the changes induced in the Ly$\delta_F=\frac{F}{\bar F}-1.$4 flux power spectrum are only marginal and not statistically distinguishable from intrinsic line-of-sight variance, although an extreme $\delta_F=\frac{F}{\bar F}-1.$5 model remains weakly distinguishable when identical sightlines are used (Coughlin et al., 2018). Modified-gravity modeling finds that the 1D projection smoothes scale-dependent deviations, making $\delta_F=\frac{F}{\bar F}-1.$6 relatively uncompetitive for $\delta_F=\frac{F}{\bar F}-1.$7 models but potentially more informative for K-mouflage, whose deviations are broader and more nearly scale independent (Brax et al., 2018).

Reionization studies highlight a different decomposition of information. In CROC, the shape of the 1D flux power spectrum is comparatively stable, while the amplitude evolves strongly with time and is almost perfectly correlated with reionization timing (Mishra et al., 2021). The authors summarize this by stating that the shape at large scales $\delta_F=\frac{F}{\bar F}-1.$8 varies little, at the level of 10%, with the redshift of the measurement or the redshift of overlap, except at the highest redshifts. By contrast, the amplitude $\delta_F=\frac{F}{\bar F}-1.$9 evolves rapidly, and the redshift shift required to align $P_F(k)$00 across realizations correlates approximately linearly with the overlap-redshift proxy defined by $P_F(k)$01 with $P_F(k)$02 (Mishra et al., 2021). This suggests that the amplitude history of $P_F(k)$03, rather than detailed scale dependence, can be a useful summary of end-of-reionization information in that simulation suite.

The connection to the underlying three-dimensional matter field is intrinsically nontrivial. A recent phenomenological correspondence paper emphasizes three complications simultaneously: the flux power is a 1D projection, traces neutral hydrogen subject to pressure, and is a nonlinear function of the local matter density, with thermal broadening and redshift-space distortions adding further scale dependence (Ridkokasha et al., 20 May 2025). Its practical formula,

$P_F(k)$04

shows why small-scale suppression in 3D influences the entire 1D spectrum. The same work argues that warm-dark-matter free streaming and baryonic filtering combine approximately through

$P_F(k)$05

so that the 1D cutoff is degenerate between dark-matter physics and thermal-history smoothing (Ridkokasha et al., 20 May 2025).

5. Contaminants, nuisance modeling, and cosmological inference

Precision use of $P_F(k)$06 depends on controlling contaminants whose scale dependence can masquerade as cosmological structure. High column density absorbers are a canonical example. Illustris-based simulations show that absorbers with $P_F(k)$07 produce broad damping wings that bias the 1D flux power spectrum away from the absorber center itself, especially on large scales, and that the effect depends strongly on column density and redshift (Rogers et al., 2017). That work replaces single-template HCD corrections with redshift-dependent, column-density-resolved multiplicative templates for LLSs, sub-DLAs, small DLAs, and large DLAs, specifically to model residual contamination after clipping or masking (Rogers et al., 2017).

Continuum fitting, sky-line masking, DLA masking, spectral resolution, and noise estimation enter the modern survey measurements as explicit systematic-error terms. DR14 identifies eight systematic sources—continuum determination, noise estimation, spectrograph resolution, sideband power estimation, sky-line masking, DLA masking, incompleteness of the DLA catalog, and incompleteness of the BAL catalog—and provides per-bin systematic contributions alongside the measured power (Chabanier et al., 2018). DESI DR1 QMLE adds correlated and uncorrelated noise-systematic modes, a continuum correction derived from mocks, pixel-by-pixel resolution-matrix treatment, and a cross-exposure estimator that removes the need to model pipeline noise bias (Karaçaylı et al., 12 May 2025). The companion DESI DR1 FFT analysis uses cross-exposure power at low redshift, mock-calibrated multiplicative corrections for line masking, DLA masking, BAL masking, continuum fitting, and residual resolution mismatch, and an explicit covariance estimator for the measurement (Ravoux et al., 14 May 2025).

These nuisance treatments directly affect cosmological interpretation. A recent high-resolution analysis with the PRIYA emulator finds that $P_F(k)$08 at $P_F(k)$09 is especially sensitive to Lyman-limit-system contamination and thermal history, while lower $P_F(k)$10 carries more robust cosmological information (Ho et al., 22 Sep 2025). XQ100 alone gives $P_F(k)$11 constraints consistent with earlier eBOSS DR14 and Planck results, whereas KODIAQ-SQUAD favors substantially higher $P_F(k)$12, which the authors attribute to selection bias toward high-column-density absorbers and to residual LLS contamination rather than to a genuine cosmological shift (Ho et al., 22 Sep 2025).

In current neutrino-mass analyses, $P_F(k)$13 often enters through compressed likelihoods rather than raw spectral estimators. A recent frequentist overview uses SDSS/eBOSS Lyman-$P_F(k)$14 P1D information through the Taylor and Lyssa compressed likelihoods on $P_F(k)$15, rather than rebuilding the flux-power likelihood from spectra (Chebat et al., 16 Jul 2025). In that framework, adding Lyman-$P_F(k)$16 P1D to Planck PR4 and DESI DR1 BAO improves the constraining power from $P_F(k)$17 meV to 50 meV or 48 meV depending on the compression, and the combination DESI full-shape + BBN + eBOSS P1D yields a CMB-independent limit $P_F(k)$18 eV at 95% C.L. for the Lyssa implementation (Chebat et al., 16 Jul 2025). This underscores a broader point: by the mid-2020s, $P_F(k)$19 had become both a direct observational data product and a compressed small-scale-structure likelihood.

6. Methodological tensions, synthetic spectra, and related line-of-sight observables

A persistent issue in the $P_F(k)$20 literature is methodology dependence. The CROC reionization study reports only $P_F(k)$21 large-scale shape variation with reionization timing, in “surprising disagreement” with Wu et al. (2019), whose AREPO-RT simulations found systematic large-scale shape deviations of order 40% (Mishra et al., 2021). Because the two calculations have comparable box sizes, mass resolution, radiation frequency coverage, and moment-based radiative transfer, the discrepancy is left unresolved. A plausible implication is that shape sensitivity of $P_F(k)$22 to reionization timing was not yet numerically robust across simulation methodologies at that stage.

Estimator dependence has been more favorably resolved. BOSS DR9 developed both Fourier and maximum-likelihood measurements and found good agreement between them over all 12 redshift bins and the full $P_F(k)$23-range (Palanque-Delabrouille et al., 2013). DESI DR1 repeats this pattern at much higher precision: the QMLE and FFT measurements are consistent with each other, while the QMLE cross-exposure estimator and the FFT cross-exposure construction both show that exposure cross-correlation is a powerful route to suppressing pipeline-noise systematics (Karaçaylı et al., 12 May 2025, Ravoux et al., 14 May 2025).

Synthetic spectra are increasingly used not just for covariance estimation but for $P_F(k)$24-focused validation. A recent lognormal mock framework generates one-dimensional Lyman-$P_F(k)$25 forest spectra by tuning the Gaussian correlation function so that the resulting spectra recover the target mean flux and $P_F(k)$26, achieving sub-percent accuracy in mean flux and percent-level accuracy in $P_F(k)$27 across the DESI EDR redshift range $P_F(k)$28 (Herbold et al., 4 Sep 2025). The method is explicitly intended for estimator validation and systematics studies, rather than for replacing hydrodynamical simulations in precision inference (Herbold et al., 4 Sep 2025).

Finally, the term “1D power spectrum” now extends beyond transmitted-flux statistics in the Ly$P_F(k)$29 forest. In the 21 cm forest literature, the analogous line-of-sight observable is typically a 1D power spectrum of an absorption-related brightness-temperature field, not of transmitted flux. One 21 cm halo-model paper defines

$P_F(k)$30

for a field proportional to $P_F(k)$31 rather than $P_F(k)$32 (Shao et al., 2024). A later 21 cm forest forecast instead Fourier transforms the line-of-sight brightness temperature $P_F(k)$33 and defines

$P_F(k)$34

using suppression of the 1D power amplitude as a probe of dark-matter annihilation, decay, and primordial-black-hole heating (Zhao et al., 6 Sep 2025). These developments are directly analogous in geometry but not in field definition; they clarify that “one-dimensional power spectrum” is a broader line-of-sight concept, while “one-dimensional flux power spectrum” remains most precisely associated with Lyman-$P_F(k)$35 transmitted-flux statistics.