The Dark Energy Spectroscopic Instrument: One-dimensional power spectrum from first Lyman-$\alpha$ forest samples with Fast Fourier Transform

We present the one-dimensional Lyman-$\alpha$ forest power spectrum measurement using the first data provided by the Dark Energy Spectroscopic Instrument (DESI). The data sample comprises $26,330$ quasar spectra, at redshift $z>2.1$, contained in the DESI Early Data Release and the first two months of the main survey. We employ a Fast Fourier Transform (FFT) estimator and compare the resulting power spectrum to an alternative likelihood-based method in a companion paper. We investigate methodological and instrumental contaminants associated to the new DESI instrument, applying techniques similar to previous Sloan Digital Sky Survey (SDSS) measurements. We use synthetic data based on log-normal approximation to validate and correct our measurement. We compare our resulting power spectrum with previous SDSS and high-resolution measurements. With relatively small number statistics, we successfully perform the FFT measurement, which is already competitive in terms of the scale range. At the end of the DESI survey, we expect a five times larger Lyman-$\alpha$ forest sample than SDSS, providing an unprecedented precise one-dimensional power spectrum measurement.


INTRODUCTION
The Lyman- (Ly) forest can be observed from the ground in the optical spectra of distant quasars at redshift between the end phase of reionization ( ∼ 6) and the peak of galaxy formation  ∼ 2. The Ly forest consists of a series of Ly absorption lines caused by intervening neutral hydrogen located at various redshifts between the quasar and the observer.The Ly forest is a powerful probe of the underlying matter density field at redshift  > 2, together with the astrophysical state of the intergalactic medium (Gunn & Peterson 1965;Lynds 1971;Meiksin 2009;McQuinn 2016).
In particular, the small-scale distribution of neutral hydrogen (∼ Mpc) is imprinted in the fluctuations of the Ly forest along the line-of-sight that can be accessed by measuring the one-dimensional Ly forest power spectrum (denoted  1D,  ).This measurement is sensitive to the amplitude and slope of the matter power spectrum at redshift  > 2. The impact of cosmological parameters on  1D,  can only be accurately predicted using hydrodynamical simulations (Borde et al. 2014;Bolton et al. 2017;Walther et al. 2021;Puchwein et al. 2023).The realization of those simulations is made arduous by the large dynamic range needed to model the Ly forest adequately (Lukić et al. 2015;Chabanier et al. 2022).Fitting data measurements with those simulation predictions provides constraints on the cosmological parameters  8 ,  s , and Ω m , as well as on parameters describing the thermal properties of the intergalactic medium.In particular, the simulations described in Walther et al. (2021) are able to predict  1D,  with sufficient accuracy (at the 1% level) when compared to expected uncertainties from the Dark Energy Spectroscopic Instrument (DESI) measurement.
Due to its sensitivity to the matter fluctuations at small scales, measurements of  1D,  can constrain physics beyond the Standard Model, such as the mass of neutrinos, the mass of warm dark matter candidates, or a possible running of the spectral index due to primordial inflation physics.First,  1D,  is well suited to constrain the sum of neutrino masses which damps the matter power spectrum at small scales (Lesgourgues & Pastor 2006, 2012).Stringent constraints are obtained by coupling  1D,  with hydrodynamical simulations, and by combining it with the Cosmic Microwave Background (CMB) (Seljak et al. 2006;Palanque-Delabrouille et al. 2015;Yèche et al. 2017;Palanque-Delabrouille et al. 2020).Secondly, several studies combined high-and moderate-resolution  1D,  measurements to obtain constraints on the warm dark matter mass (Viel et al. 2005(Viel et al. , 2008(Viel et al. , 2013;;Baur et al. 2016;Yèche et al. 2017;Baur et al. 2017;Palanque-Delabrouille et al. 2020).The hydrodynamical simulations used in those study either directly model neutrinos as particles or using a rescaling of the matter power spectrum to account for neutrinos (Pedersen et al. 2020a(Pedersen et al. ,b, 2023)).Finally, other exotic dark matter models such as fuzzy dark matter (Iršič et al. 2017;Armengaud et al. 2017) can also be constrained using  1D,  measurement.
Moderate-resolution surveys provide large numbers of Ly forests, which yield smaller statistical uncertainties on  1D,  estimates.However, the resolution of such spectrographs limits the reach of very small scales.The  1D,  measurement with moderate-resolution spectrograph was first performed on a small sample of the Sloan Digital Sky Survey (SDSS) data in McDonald et al. (2006).Subsequently, the increase of the Ly forest statistic has largely improved this measurement with the Baryon Oscillation Spectroscopic Survey (BOSS) in Palanque-Delabrouille et al. (2013) and the extended Baryon Oscillation Spectroscopic Survey (eBOSS) in Chabanier et al. (2019) using 43, 751 quasar spectra.
Several methods can be used to measure  1D,  from Ly forest samples.The most straightforward relies on the fast Fourier transform (FFT) and was applied in BOSS and eBOSS (Palanque-Delabrouille et al. 2013;Chabanier et al. 2019) analyses.The one-dimensional power spectrum can also be measured with configuration space estimators such as the quadratic maximum likelihood estimator (QMLE).This method has already been applied to moderate resolution observations (McDonald et al. 2006;Palanque-Delabrouille et al. 2013) and more recently on high-resolution data in Karaçaylı et al. (2022).The QMLE method is applied to the same data than used in the Table 1.Spectral range and effective resolving power ( = Δ/) for each channel of the DESI spectrographs (Abareshi et al. 2022).

Channel Spectral range (Å) Resolving power
Blue (B) 3600 -5930 2,000 -3,200 Red (R) 100 Near Infrared (Z) 7470 -9800 4,100 -5,100 present paper and presented in a companion paper (Karaçaylı et al. 2023a).The FFT method yields a straightforward calculation of  1D,  and offers more control over the different calculation steps.Conversely, the more-complex QMLE estimation is not sensitive to gaps in the quasar spectra.The results between the two methods are presented in the companion paper and are in good agreement.FFT and QMLE results agree at 1% level precision up to half the Nyquist frequency.
The purpose of this work is to compute  1D,  from the first DESI data, following the same methodology as in the latest eBOSS measurement in Chabanier et al. (2019).Using the same method facilitate the comparison between eBOSS and DESI.The  1D,  is sensitive to instrumental properties such as noise and spectral resolution.As the telescopes used and the data are very different, it is essential to characterize the DESI instrument.We improve the algorithms and methodology used in Chabanier et al. (2019) to account for systematic and instrumental differences between eBOSS and DESI.In particular, due to the spectral resolution improvement of DESI, our measurement allows accessing smaller scales than eBOSS.
The outline of this paper is as follows: Sec. 2 describes the DESI instrument and data processing used to perform this  1D,  measurement.The  1D,  pipeline is presented in Sec. 3, and the characterization of the DESI instrument in Sec. 4. We generate synthetic data to validate and correct our measurement in Sec. 5.The treatment of statistical and systematic uncertainties for the  1D,  measurement is given in Sec. 6.Finally, we present our measurement on DESI data, as well as a comparison to previous measurements in Sec. 7, and conclude in Sec. 8.

INSTRUMENT AND DATA DESCRIPTION
DESI has as objective to measure the spectra of 40 million galaxies and quasars in a footprint of 14, 000 deg2 over 5 years (Levi et al. 2013;DESI 2016a;Abareshi et al. 2022).This project aims to continue the cosmic mapping efforts started by SDSS, while drastically increasing its constraining power on the ΛCDM model and its possible extensions.
We first focus on the description of the data used for our measurement.In the following, we describe the data starting from the instrument (DESI), its associated spectroscopic pipeline, the different data acquisition phases, and the input catalogs of our study.

DESI instrument
The DESI instrument is mounted on the Mayall telescope, located on the Kitt Peak National Observatory (KPNO) in the Tohono O'odham Nation.The Mayall telescope is a reflective prime-focus telescope with a 4-meter diameter primary mirror.The DESI instrument (DESI 2016b;Miller et al. 2023) receives photons through an optical corrector designed to increase the field of view to 7.5 deg 2 on the focal plane.The focal plane system, composed of 5, 000 robotically controlled fibers, can quickly modify its configuration to aim at the targeted objects on a specific footprint (Silber et al. 2022).An optical fiber system redirects the light of the observed objects to a separate climate-controlled enclosure containing 10 spectrographs.Each spectrograph comprises 3 CCD cameras whose properties are given in the Tab. 1.In comparison to SDSS spectrographs, the effective resolving power (Δ/) improved by at least a factor of two.

DESI spectroscopic pipeline
The high complexity of the DESI survey induces the need for advanced software pipelines and products, including the imaging from the DESI Legacy Imaging Surveys (Zou et al. 2017;Dey et al. 2019;Schlegel et al. 2023), a pipeline to select the targets to observe (Myers et al. 2022), a pipeline to assign fibers (Raichoor et al. 2023), a pipeline to parse the survey and to optimize the observation strategy (Schlafly et al. 2023), and an extensive spectroscopic reduction pipeline (Guy et al. 2022).
This spectroscopic pipeline, called desispec1 , transforms the raw CCD images into spectra, and is detailed in Guy et al. (2022).Before extracting the spectra, the images are subtracted by dark and bias calibration frames to remove expected background sources, and to estimate the associated readout noise (noise estimation details are given in appendix C).The non-uniform CCD pixel response is corrected using a dedicated flat-field slit on the spectrograph and the CCD over-scan is removed.A dedicated software detects and flags cosmic rays or defective CCD pixels.
The spectral extraction is performed using the "spectroperfectionism" method (Bolton & Schlegel 2010), an optimal spectroscopic extraction that correctly models complex 2D point spread functions.This method provides the encoding for each fiber and each wavelength of the non-Gaussian instrument resolution into a resolution matrix (noted R in next sections) used to compute  1D,  .
Once the spectrum of each fiber has been extracted, several postprocessing steps allow removing a variety of further observational effects.The non-uniform response of individual fibers as a function of wavelength is corrected with flat-field frames by observing a white screen attached to the telescope dome and illuminated with a LED array.For all exposures, some fibers are dedicated to observing the sky.The so-called sky spectra associated with these fibers provide the sky level and the intensity of atmospheric emission lines and are subtracted to the spectra associated with targets (sky subtraction).The transmission defaults of the atmosphere and telescope as a whole are corrected by the observation of calibrated star spectra.This step converts CCD units (number of electrons) to observed flux units.Finally, the spectrum of an object is obtained by coadding its different exposures.The resulting spectrum is expressed separately into the three spectrographs bands described in Tab. 1.
All the software of the pipeline employed for the analysis of DESI data are listed in the repository desihub 2 .In particular, the spectra analyzed in this article have been processed with the fuji3 version of the spectroscopic pipeline.

DESI data
The spectroscopic pipeline described previously was used to analyze data obtained over different periods.In this article, we use three data sets from the first observations of DESI described in Tab. 2, which includes a total of 112, 414 quasar spectra whose redshifts are between 2.0 and 5.0.The first two data sets 'Target Selection Validation' (SV1) (DESI 2023a) and 'One-Percent Survey' (SV3) are part of the Early Data Release (EDR) whose complete description is given in DESI (2023b).'Target Selection Validation' was conducted from December 2020 to March 2021 and includes a large number of exposures (up to 16) for the same targets.The objective of this survey was among others, to study extensively the survey performance as a function of instrumental depth and to build visual inspection truth tables.The SV1 data set also includes 'Secondary Tiles' as detailed in DESI (2023b).Following the completion of 'Target Selection Validation' observations, the 'One-Percent Survey' phase was dedicated more specifically to the evaluation of the survey design.The number of exposures is similar to the main survey at its end (between 4 and 5 for each Ly quasars) and the goal was to determine the best strategy to cover the sky while limiting fiber loss.
The main DESI survey started in June 2021 and in this article we also use the first two months of data, named DESI-M2 (and noted M2 in this paper for conciseness), which is not present in the EDR but will be included in the Data Release 1.In the M2 data set, most quasars have only one exposure.While all three data sets are studied in this article, in the end we removed SV1 due to its different noise properties (see appendix A).The final measurement is computed on SV3+M2 data set only.

Input catalogs
The input catalogs used to compute  1D,  were obtained by applying specific procedures to the three data sets previously described.
The targeting of the quasars used in our study (Yèche et al. 2020;Chaussidon et al. 2022) was verified with visual inspection of subsets of early observations (Alexander et al. 2022).For quasars, the DESI pipeline categorizes the observed spectra and estimates their redshifts using the redrock spectral template fitting software4 (Bailey et al. 2023;Brodzeller et al. 2023).In order to optimize the completeness of the quasar catalog while keeping a high purity, additionally a broad Mgii line finder mgii_afterburner5 and a machine learning classifier applying deep convolutional neural networks QuasarNP6 (Busca & Balland 2018;Farr et al. 2020) are run after redrock.Both post-processing programs are run on all objects targeted by DESI as detailed in Chaussidon et al. (2022).
An example of a quasar spectrum at redshift  = 3.42 is given in Fig. 1.This spectrum, with a particularly high flux, is part of the SV3 data set which contains spectra with exposure time equivalent to the end of the DESI survey.The redshifts and effective exposure times for each data set considered in this paper are shown in Fig. 2 and 3 respectively.The effective exposure time accounts for nightly observing conditions by normalizing the real exposure time to a reference with airmass 1, zero galactic extinction, a 1.1 ′′ seeing (FWHM), and zenith dark sky (Guy et al. 2022).The nominal exposure time of one exposure is defined to 1000 .The large differences in term of exposure time emphasize the need to treat the data sets differently, at least for noise properties.SV1 and SV3 contains a small number of forests but with heterogeneous exposure times, in opposition to M2, which contains many quasars only observed once.
Broad absorption line (BAL) quasars are specific quasars whose spectra exhibit consistent blueshifted absorptions associated with many spectral features.They are identified using the baltools7 software.It consists of a  2 minimizer algorithm that looks for blueshifted Civ or Siiv absorptions in an unabsorbed quasar model.The fit is performed for rest-frame wavelengths between 1, 260 and 2, 400 Å.A quasar is considered a BAL type if its spectrum exhibits a region between Civ and Siiv emission lines, with at least 10% flux decrement below the continuum and a width greater than 2, 000 km•s −1 .BAL quasars spectra (4.18% in the total data set) are removed in the analysis performed in this work.
Damped Lyman- absorbers (DLA) are regions within a quasar spectrum that show over-saturated absorption with prominent Lorentzian wings as the quasar flux intersects the dense, circumgalactic medium of an intervening (proto-)galaxy.They are a subclass of high-column density systems and are a significant contaminant of the Ly forest signal, particularly because of their wings and the additional contamination by circumgalactic metal absorption lines (McDonald et al. 2005).The correct modeling of such systems in simulations has been proven to be particularly complicated (Pontzen et al. 2008).
We use a catalog resulting from the combination of a convolutional neural network (CNN) algorithm desi-dlas8 (Parks et al. 2017;Zou et al. 2023) and a Gaussian process (GP) algorithm 9 (Ho et al. 2021).desi-dlas is trained with SDSS spectra to identify candidate high-column density objects for rest-frame wavelengths between 900 and 1, 346 Å.It returns locations of high-column density systems in the spectra, as well as their Hi column density and a confidence parameter.The GP finder provides similar output using the same training set and a Bayesian model selection.We only consider the high-column density objects with column density  Hi > 10 20.3 cm −2 (DLAs).In accordance with recommendations from Parks et al. (2017);Ho et al. (2021), we consider CNN confidence level higher than 0.2 as valid DLA detections when the ratio between the quasar continuum and the noise is higher than 3.We  Finally, the number of BAL quasars characterized by a balnicity index higher than zero, and not used in the  1D,  computation pipeline, is shown with a dotted line.
take a confidence level limit of 0.3 when this ratio is lower than 3.For the GP model, a 0.9 minimal confidence level is applied.In the case when absorbers are detected by the two models, the combined DLA catalog uses  Hi values and DLA redshifts from GP model.Although DLAs by themselves constitute tracers of the matter distribution, they have an extended impact on the observed spectra.They increase the correlations of neighbor spectrum pixels, thus artificially increasing  1D,  level.Therefore, we choose to mask the core of DLA regions of the spectra by fixing the transmitted flux fraction to its mean value for spectrum pixels where the DLAinduced absorption is larger than 20 %.In addition, the absorption in the Lorentzian damping wings that remain after the cut is corrected with a Voigt profile following Bautista et al. (2017); Chabanier et al. (2021).
Finally, we use a catalog of masks to account for atmospheric and Galactic emission lines, which has been adapted to DESI resolution 10 .The creation of this catalog is detailed in Sec.4.4.

ONE-DIMENSIONAL POWER SPECTRUM ESTIMATION
The  1D,  estimator is build using the data product described previously in two phases.First, the fitting of the continuum of quasars is used to convert the absolute received flux to a normalized quantity   .Secondly,  1D,  is computed by employing a Fast Fourier Transform and by averaging the product of this transformation for all the selected Ly forests.

Continuum fitting
A standard normalized quantity used in the calculation of correlations and power spectra is the flux contrast   of the Ly forest, defined as where  is the transmitted flux fraction, and  () is its average value, the mean transmission of the intervening IGM.Note that for the purpose of this work, we do not need to know the individual quasar continua   , but only the product  q (,  q ) ().This software also merges the quasar spectra over different bands to obtain  over the all wavelength range.The continuum of each quasar is modeled as the product of a universal continuum  common to all quasars, and a first-order polynomial term in wavelength: where  q and  q are quasar-dependent constants.In previous studies (Palanque-Delabrouille et al. 2013;Chabanier et al. 2019),  q = 0 was assumed for all quasars, i.e.only a wavelength-independent normalization factor was taken into account.We add an additional linear wavelength-dependent term to account for the diversity of quasars after verifying that this change does not impact the mean level of our  1D,  measurement.The  q and  q parameters for each quasar are determined along with  by maximizing the following log-likelihood: where the sum is run over all the spectrum pixels of the quasar q, and  q is the standard deviation estimator of the flux  .
In contrast to analyses of the large-scale 3d correlation function such as du Mas des Bourboux et al. (2020), we want all spectrum pixels to contribute equally to the continuum fitting and the  1D,  computation, as the opposite could bias  1D,  .Therefore, we impose noise-independent weights in the continuum fitting procedure: 11 https://github.com/igmhub/picca This procedure is the same as in the previous  1D,  analyses based on BOSS/eBOSS data (Palanque-Delabrouille et al. 2013;Chabanier et al. 2019).The standard deviation associated with   at the end of the continuum fitting procedure is defined by where  pip,q is the noise provided by the DESI spectroscopic pipeline detailed in appendix C. The universal continuum  and the  q and  q parameters are computed iteratively.In particular,  is estimated from the average of all spectra, i.e. in a non-parametric way.During the entire fitting procedure, spectrum pixels that are masked due to the presence of a DLA or an atmospheric line are not considered in the fit.We use 7 iterations and have verified that the continuum fits are converged at this point.
The noise level of a Ly forest is characterized by defining the average signal-to-noise ratio SNR in the Ly forest region: Only Ly forests with a SNR larger than 1 are used in the continuum fitting procedure.This procedure is also restricted to the observed redshift range 3, 600 <  < 7, 600 Å, to avoid the shorter wavelength range where a large fraction of the quasar spectra is absorbed by the atmosphere.We also select the rest-frame wavelength in the range 1, 050 <  rf < 1, 180 Å, so that the measured contrasts are dominated by the Ly forest.In particular, we try to avoid the Ly singlet and the Ovi doublet emission regions respectively located at  Ly = 1, 025.72 Å and  Ovi = (1031.912,1037.613)Å in the rest-frame.
The cut   < 1, 180 Å facilitates the continuum fitting procedure and mitigates most of the proximity effect: close to a quasar, the neutral hydrogen fraction is indeed influenced by the quasar's UV radiation in addition to the extragalactic UV radiation background (Bajtlik et al. 1988).
As detailed in Sec.2.2, the quasar spectra are linearly binned in observed wavelength with Δ pix = 0.8 Å.Note that when converting to rest-frame wavelength  rf the pixel size will be redshift dependent.For the continuum fitting process, we thus need to rebin our spectra to a uniform grid in  rf .As the quasar continuum is relatively smooth and to avoid noisy continuum fits for analyses of relatively small data sets, we chose a grid for the common continuum  that is 10 times coarser than the lowest redshift quasar pixels considered, i.e.: By taking  min = 2.0 as the lowest redshift, we obtain Δ pix,rf = 2.67 Å.With increasing size of the dataset, such a rebinning could be relaxed in future DESI measurements.At the end of the continuum fitting procedure, the stacking of all the Ly contrasts is forced to be equal to zero to avoid introducing flux calibration errors.

Fast Fourier transform power spectrum estimator
Conceptually, the   quantity can be separated into different contributions in the Ly forest region: where  astro corresponds to the fluctuations caused by all the elements of the intergalactic medium (including Ly), and  noise corresponds to noise fluctuations.Considering instrumental effects, the true underlying flux contrast is modified on its way through the instrument in multiple ways.First, photons traverse the spectrograph leading to the output flux being convolved with the spectrograph line spread function  (, R, Δ pix ) which depends on the resolution matrix presented in Sec.2.2.The  term also account for the signal pixelization as the photons are counted into CCD pixels of size Δ pix .Finally, we need to account for noise sourcing from the processes of photon counting and readout.In total, we can write the measured flux contrast   as We assume that the impact of the noise and the resolution term  are decorrelated, i.e., that the noise contrast is not affected by the instrumental effects of pixelization and resolution.Verifying this assumption is beyond the scope of this paper.
The  astro contrast can be decomposed between the Ly signal and the one from all other elements of the intergalactic medium called metals.The contribution of those metal absorptions to   can be decomposed into two parts.On the one hand, there are absorption lines with rest-frame wavelength  ≫  Ly = 1, 215.67 Å.Those lines can be independently observed redwards of the Ly forest, using specific rest-frame spectral regions called side-bands SB1 (1, 270 <  rf < 1, 380 Å), and SB2 (1, 410 <  rf < 1, 520 Å), as shown in Fig. 1.We group the absorption of all those metals in a contrast noted  metals .
On the other hand, there are absorption lines with rest-frame wavelength  ≲  Ly , such as Siii and Siiii elements ( Siii = 1, 190 and 1, 193 Å, and  Siiii = 1, 206.50 Å).They cannot be observed independently of the Ly forest, but will show absorption that is correlated with the Ly absorption and will lead to an oscillatory feature in the estimated  1D,  .For those lines, we adopt the same approach developed in McDonald et al. (2006) and subsequently used in other analyses.We leave the features inside our power spectrum estimates to be corrected by fitting an additional oscillation during parameter inference.We note  Ly the contrast containing the Ly forest and the effect of those latter metals.Finally, the flux contrast can be expressed: The one-dimensional Ly power spectrum ( 1D,  ) can be estimated from this decomposition by applying a fast Fourier transform (FFT) algorithm on   of each Ly forest.This method was applied in previous measurements (Croft et al. 1998;Palanque-Delabrouille et al. 2013;Chabanier et al. 2019).
The FFT estimator is implemented in picca 11 (du Mas des Bourboux et al. 2021).For each Ly forest, the raw power spectrum12 is defined from the Fourier transform of   (): where  D is the one-dimensional Dirac distribution.Applying a Fourier transform on equation ( 10), the raw power spectrum is expressed by In this decomposition,  Ly ,  metals , and  noise are the power spectra respectively associated to the contrasts  Ly ,  metals , and  noise with the same definition as in equation ( 11).We assumed here that the Siii and Siiii power spectra are negligible and that all cross-correlation terms between the contrasts are null.The only nonneglected cross-term is  Ly−Siii/Siiii () = 2  Ly () Siii/Siiii () , which corresponds to the correlated absorptions of Ly with either Siii or Siiii.The oscillations induced by this term have a wavenumber 2/( Ly −  Siii/Siiii ) when "" is expressed in Å −1 .
The FFT estimator for the one-dimensional power spectrum is computed as an average over all available Ly forests in the measurement sample.It is designed to match the sum of Ly and Ly−Siii/Siiii power spectra in equation ( 12), so that we define where ⟨.⟩ denotes the average over all the Ly forests used for  1D,  calculation.From equation ( 12), the estimator of  1D,  is defined by The  1D,  measurement is split in different redshift bins to take into account its evolution.The Ly forest is split into sub-forests which correspond to consecutive and non-overlapping sub-regions of equal length.This procedure also reduces the correlations between the different redshift bins.We chose to cut Ly forests into three subforests whose rest-frame wavelength boundaries are  rf = 1, 093.3 and 1, 136.6 Å, so that the length of each sub-forest is  sub = 43.3Å.With this sub-forest separation, a single Ly forest can contribute to up to 3 different redshift bins in the  1D,  measurement.The subforest splitting constrains the minimal accessible observed wavenumber to  min = 2/( sub (1 +  min )) = 0.0453 Å −1 , by taking the minimal redshift used.Each sub-forest spans at most Δ = 0.2 and we choose the same Δ to define the redshift binning for  1D,  .
For observed wavelength  ≲ 3, 700 Å, the noise level is high in comparison to the spectra because of atmospheric absorptions.To minimize the impact of this noise, we remove the spectrum pixels for which the observed wavelength is lower than 3, 750 Å, which corresponds to Ly absorbers located at  = 2.085.In the future, with a dedicated study to control the noise at shorter wavelength, the  1D,  analysis can be extended to redshift  ∼ 2.
In accordance with the eBOSS study (Chabanier et al. 2019), we remove sub-forests shorter than 75 spectrum pixels due to a cut in the UV region or to the presence of a large DLA.We also do not consider the Ly sub-forests with more than 120 masked spectrum pixels.
Unlike the analysis in Chabanier et al. (2019), we do not apply a second redshift dependent SNR cut for the averaging of  1D,  .Instead, we develop and test a SNR weighting scheme, as detailed in the appendix B. This procedure is used for all the article except in Sec.4.2.1 where the impact of the SNR cut is investigated.(Gunn et al. 2006;Smee et al. 2013;Dawson et al. 2016;Blanton et al. 2017) in the 16  ℎ Data Release (DR16) (Ahumada et al. 2020) of the extended Baryon Oscillation Spectroscopic Survey (eBOSS).The SDSS name associated to this quasar is 142903.03-014519.3.The DESI spectrum is obtained after 3 individual exposures for a total exposure time of 2, 300 seconds.The eBOSS spectrum have 11 exposures for a total of 6, 300 seconds.The large-scale absorption structures are similar but due to its improved spectroscopic resolution, the DESI spectrum clearly shows more details at small scales.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 (blue points) using Ly forest from SV3+M2 data set.This resolution correction is weakly dependent on the redshift range (shown by shaded dashed black lines along the blue points).Only one shaded black line is above the blue points ( = 3.8).All the others are at the same level as the blue points.The mean value over all redshift bins is shown with the points.The shaded area represents the regime for which the associated impact of resolution and pixelization removes more than 80% of the power spectrum.This criteria is chosen to define the maximal wavenumber of our  1D,  measurement, shown with a vertical black line.

DESI INSTRUMENTAL CHARACTERIZATION
As the DESI instrument is new, we first focus the analysis on characterizing the instrumental effects on our  1D,  measurement.In particular, we describe the impact of spectral resolution, instrumental noise, metal power spectrum, and atmospheric emission lines in the following.

Resolution correction modeling
As mentioned in Sec.2.2, DESI spectra are linearly binned in observed wavelength, so that the natural unit for wavenumbers is Å −1 .The maximal measurable wavenumber follows the Nyquist-Shannon limit: For SDSS  1D,  measurements (Croft et al. 1998;McDonald et al. 2006;Palanque-Delabrouille et al. 2013;Chabanier et al. 2019), the pixelization is logarithmically binned in observed wavelength, making it suitable to express the power spectrum in Hubble velocity unit , because Δ ∝ Δ/ = Δ log().In eBOSS (Chabanier et al. 2019), the resolution correction is modeled by a Gaussian function , where Δ is the spectral resolution in velocity units.
The spectroscopic resolution of DESI is improved with respect to the SDSS spectrographs.On the DESI blue band (see Tab. 1) where most of the Ly forest are observed, the effective resolving power  = /Δ ranges from 2, 000 to 3, 200 (Abareshi et al. 2022).In comparison, SDSS spectrographs had a 1, 500 <  < 2, 300 in its blue band (3, 600 <  < 6, 350 Å).This improved resolution brings the opportunity to probe the clustering of matter at smaller scale by measuring the small fluctuations in the Ly forest, as illustrated in Fig. 4.
As described in Sec.2.2, the DESI spectrograph resolution is entirely characterized by the resolution matrix R (Guy et al. 2022).In opposition to SDSS, the resolution matrix also accounts for the pixelization of the signal.Consequently, we choose to express the resolution correction function  in equation ( 12) directly as the Fourier transform of the resolution matrix that we note R().
Fig. 5 shows the average correction due to resolution and pixelization for the SV3+M2 data set.This correction indicates that resolution and pixelization suppress more than 95 % of the signal at  >  res,95 = 2.73 Å −1 , and 98 % at  >  res,98 = 3.15 Å −1 .This observation drives the maximal wavenumber that of the  1D,  measurement.We choose the conservative value  max = 2 Å −1 , for which the average resolution corrections is equal to 80%.We will extend this conservative limit in future studies after a full characterization of resolution on CCD pixel-level simulations.

Validation with CCD image simulations
We use CCD image simulations of the DESI instrument to verify our resolution modeling described previously.This method is also used for the QMLE estimation of  1D,  in a companion paper (Karaçaylı et al. 2023a).Those simulations are built using the desisim13 package developed to model and validate the spectroscopic extraction pipeline presented in Sec.2.2 and detailed in Guy et al. (2022).We use the desisim package to produce realistic realizations of twodimensional spectroscopic images.Those images simulate various instrumental effects such as the different sources of noise detailed in appendix C, gain and bias of the CCD amplifiers, the throughput of each spectrograph, point spread function (PSF) of each fiber, and sky emission.
Using the desisim package, we transform Ly transmissions that follows a given input  1D,  into realistic Ly forests with apparent magnitudes representative of DESI quasar targets and noise that is representative of a single, 1000 s exposure in nominal conditions.We simulate 45,000 Ly forest spectra located over ten DESI tiles and process this simulated dataset with the spectroscopic pipeline.Since we only want to see the impact of spectral resolution, the true imposed noise level is used to reduce the data.
Two sets of Ly contrasts   are generated from this simulated quasar sample.The first set, called RAW, is produced directly from Ly transmissions.This type of realization does not contain the effect of spectral resolution.In parallel, we create Ly contrasts from the full CCD image simulations, noted CCD, but using the true imposed continuum for each quasar to only see the impact of resolution.Finally, we run the  1D,  FFT pipeline presented in Sec.3.2 with our resolution modeling on both Ly contrast sets.
The ratio between  1D,  obtained from RAW and CCD sets is shown in Fig. 6.As expected, the main difference between those measurements resides in the smallest scales ( > 1.0 Å −1 ).This ratio is not redshift dependent, as indicated by the light black curves in the background.We only consider the average overall redshifts.We checked that applying an additional pixelization correction sinc 2 (0.5Δ pix ) similarly to eBOSS increases a lot the discrepancy between RAW and CCD power spectrum; thus confirming that the resolution matrix accounts the pixelization, at least partly.We derive a correction by fitting a second-order polynomial function to the following averaged ratio: This term is directly multiplied to our  1D,  measurement to account for the miss-estimation of the resolution correction.

Comparison of noise estimators with high-wavenumber data
The  1D,  measurement is significantly impacted by the noise power spectrum at small scales.Thus, it is necessary to obtain an accurate estimate of this component to correct for it.
The noise power spectrum is estimated either directly from the pipeline noise, or by using an exposure difference method.A de-  tailed description of the obtained noise power spectrum estimators, respectively noted  pipeline and  diff , can be found in appendix C. Additionally, the noise power spectrum level is determined by taking advantage of the combined effect of the resolution and the pixelization shown in Fig. 5, as well as the Ly thermal broadening.Those effects erase essentially all "signal" power and thus at large wavenumbers, equation ( 12) simplifies into  raw () ≃  noise ().The difference (or ratio) of  raw and  noise on the largest k-bins accessible can be used to validate the noise estimator and correct it empirically.We define the following asymptotic difference and ratio by averaging those quantities at large wavenumbers.We decide to use the criteria  >  res,98 where  res,98 is defined as the wavenumber for which the resolution and pixelization suppress more than 98% of the signal: Fig. 7 shows the measurement of  and  on SV3 data set with SNR > 3 (with SNR defined by equation 6), using the pipeline noise to compute  noise .Considering the observed statistical fluctuations, we notice that the asymptotic behavior of power spectra at high wavenumber enables a good measurement of  (respectively ), whose value is close to 0 (respectively 1).Additionally, we verified  that the variation of  and  as a function of redshift is small in comparison to the statistical fluctuations of the ratio and difference.

Characterization on DESI data sets
We compute the  and  coefficients for the pipeline ( pipeline ) and difference ( diff ) noise estimators on SV1, SV3, and M2 data sets, while varying the applied SNR cut.Fig. 8 shows the measured  values on the Ly forest regions for the different data sets and noise estimators.As previously stated,  should be equal to zero when the noise is perfectly estimated.The values of  are small compared to the absolute level of the noise power spectrum shown in Fig. 7.The SV1 data set exhibits a SNR cut dependence which is not present for M2 and SV3.We choose to make a data set-dependent correction to remove this residual noise.The miss-estimated noise is higher for data sets with a larger number of exposures (such as SV1).We think this originates from unaccounted common sources of noise coming from the statistical uncertainties in the CCD calibration data (dark current, pixel flat field), which explain why this effect increases with the number of exposures.The dependence in SNR can be explained by the fact that this effect is amplified when we consider noisier spectra.
In Fig. 8, the exposure difference noise estimator  diff is shown for SV1 and SV3.In M2, quasars are observed with one, or a small number of exposures; thus,  diff can not be reliably computed.For the SV1 and SV3 observations, the difference noise power spectrum exhibits the same trend as the pipeline noise estimate.How-ever,  diff consistently underestimates the noise level compared to  pipeline .We think that this under-estimation is due to the fact that  diff is not accounting for all the common sources of noise between exposures.As a consequence, we only consider the  pipeline estimation from now on.
In appendix C3, we perform additional studies to characterize the additivity of the miss-estimated noise and its behavior for different spectra regions.Those regions, called side-bands, are used in the next section for metal power spectrum estimation.Tab. 3 summarizes the corrections to the pipeline noise we computed for the different data sets.For SV3 and M2, we choose to apply a constant additive correction ().For SV1, given the observed dependence of  as a function of the minimal SNR cut, we fit it with a power-law.

Side-band power spectrum
Following previous  1D,  studies (McDonald et al. 2006;Palanque-Delabrouille et al. 2013;Chabanier et al. 2019), special spectrum regions, called side-bands and that are devoid of Ly absorption, are used to statistically estimate the power spectrum components  metals caused by metal absorptions in the Ly forest.The resulting signal from the side-bands contains information about the abundance, temperature and clustering of metals in the intergalactic medium.In our study, we aim at creating a model to closely reproduce the sideband power spectrum ( SB ) so that we can statistically subtract it in the measurement of  1D,  in equation ( 14).
We define the side-bands SB1 (1, 270 <  rf < 1, 380 Å) and SB2 (1, 410 <  rf < 1, 520 Å).In both side-bands, the fraction of transmitted flux contrast can be expressed similarly to equation ( 10): The  metals ()| SB contrast contains all the fluctuations caused by metals with rest-frame absorption wavelength higher than 1, 380 Å for SB1, and 1, 520 Å for SB2.Similarly to the calculation of  1D,  , the side-band power spectrum writes: The main difference between both side-bands is that SB1 contains Siiv absorption, which is not present in SB2.Consequently, we use the side-band power spectrum of SB1 to estimate  metals in equation ( 14), and the SB2 power spectrum as a consistency check.The side-band power spectrum is computed at the same observed wavelength range as  1D,  .Because of the higher rest-frame absorption wavelength, the quasars employed to calculate the side-band power spectrum are at a lower redshift than the sample used for Ly.In particular, quasars at  < 2.0 are employed to calculate  SB in the lowest redshift bins of  1D,  .However, the metals in the intergalactic medium which produces the absorptions responsible for the sideband power spectrum ( SB ) are at the same redshift as those which produced the metal power spectrum ( metals ) in the  1D,  calculation for Ly.Consequently, the redshift dependence of metal absorptions is correctly taken into account.
We note that the method we use to remove metal contribution is not perfect.In particular, the blending of metals with Ly emission pointed out in Day et al. (2019) is not fully accounted here.This second-order effect should be included in future studies for which the precision level will significantly improve.
The measurement of the side-band power spectrum using the In top panel of Fig. 9, the average side-band power is lower for the SB2 than SB1, as expected by the addition of Siiv absorptions.
In the eBOSS measurement (Chabanier et al. 2019), a sixth-degree polynomial is used to fit the shape of the side-band power.For our measurement, we exploit the stacked  SB profile to design a more physically motivated model.
A complete list of metals present in the Ly forest and that impact  1D,  is given in Pieri et al. (2014); Yang et al. ( 2022) and the strongest absorptions are from Siiii, Siii, Siiv, and Civ.
The emission peaks, and consequently absorption peaks, of Siiv and Civ are actually two doublets.Their rest-frame wavelength given by NIST (Kramida et al. 2021) are  Siiv a = 1, 393.76 Å,  Siiv b = 1, 402.77 Å,  Civ a = 1, 548.202 Å, and  Civ b = 1, 550.774 Å.The presence of an absorption doublet in the side-band creates a peak in the one-dimensional correlation function, which translates into an oscillatory pattern in the power spectrum, whose periodicity depends on the doublet separation.This effect is studied more in detail on the same dataset in Karaçaylı et al. (2023b) to determine cosmic ion abundance.In the top panel of Fig. 9, both side-band power spectra display a large oscillation caused by the Civ doublet.As expected, the SB1 power spectrum shows an additional oscillation induced by Siiv doublet absorptions.These considerations lead us to model  SB as the sum of a power-law including all-metal contributions and oscillations due to Siiv and Civ doublets: Oscillations induced by a doublet have a frequency characterized by the rest-frame wavenumber  rest, = 2/  where   is the doublet separation in Å.We choose to use damped sinusoidal functions to model the doublet oscillations as follows: where   is a free parameter with a uniform prior centered around  rest, .
The result of the fit on the redshift-averaged  SB , taking into account the oscillations of Civ and Siiv for SB1, and only Civ for SB2, is shown in the top panel of Fig. 9.
The SB1 fitted function is used to derive the redshift dependence of the side-band power spectrum, shown in the bottom panel of Fig. 9.For each redshift bin, we fit a product between the global SB1 fitted function (expressed in the observational wavelength frame), and a first-order polynomial.As  SB may also include other uncorrelated contaminations besides metals, we do not seek to interpret the fitted values for each power spectrum.In particular, we note that the   are systematically shifted in comparison to their doublet oscillation frequency  rest, .For SB1, the fitted values are  Civ = 3.32 Å −1 and  Siiv = 0.812 Å −1 whereas rest-frame wavenumber are  rest,Civ = 2.44 Å −1 and  rest,Siiv = 0.697 Å −1 .For SB2, we obtain  Civ = 3.23 Å −1 .We think that this effect might be due to the blended impact of all metals present in the intergalactic medium.Thus, it is necessary to vary   parameters to closely fit our data.
This  SB measurement already represents a clear improvement with respect to that of BOSS (Palanque-Delabrouille et al. 2013), and eBOSS (Chabanier et al. 2019): by eye, Siiv and Civ induced oscillatory patterns are seen even for individual redshift bins, and  SB is essentially a decreasing function of wavenumber, even at high wavenumber.This indicates an improvement in the noise modeling.

Atmospheric and Galactic emission lines
Atmospheric emission lines are corrected from DESI spectra by the spectral extraction pipeline as described in Sec.2.2.The average of 15,000 sky spectra on exposures with optimal observing conditions, noted  sky , is shown in Fig. 10.
The noise of spectrum pixels associated with intense atmospheric lines is strongly increased.It induces additional oscillations in the Ly contrasts and increases the level of  1D,  .We need to correct this effect as those atmospheric lines are not linked to IGM physics.We choose to mask the major atmospheric lines as in previous measurements (Palanque-Delabrouille et al. 2013;Chabanier et al. 2019).The catalog of lines in these studies14 was adapted to the spectral resolution of the SDSS instrument.The improved resolution of DESI makes it possible to reduce the masking size for narrow atmospheric lines, decreasing the impact of masking on the  1D,  measurement We develop an algorithm similar to Lee et al. (2013)   atmospheric line catalog adapted to the DESI instrument.A median smoothing M (  ) of spectral width   = 160 Å is applied on the average sky spectrum.Atmospheric lines are selected when the average sky spectrum is larger than the product of the smoothed sky flux, M (  ) ⊛  sky , by a threshold Λ l .In Fig. 10, the dashed red line represents this product for Λ l = 2.5 and   = 160 Å.
A second threshold Λ w = 1.2 defines the width of atmospheric lines.The upper and lower wavelength limits of an atmospheric line are defined as the first wavelengths on each side whose average sky spectrum is lower than Λ w ×M (  ) ⊛  sky .To remain conservative and prevent numerical effects potentially caused by masking at a spectrum pixel position, the line widths are increased by 1 Å on each side.In this atmospheric line catalog, we also add the galactic absorption lines, which correspond to relatively broad absorptions made by dust in the Milky Way.We take the same lines as eBOSS: Caii H and K lines at 3, 968 and 3, 933 Å, and the NaD doublet at 5, 893 Å.The DESI atmospheric emission line catalog built from this procedure is available online 10 .
We verify that the produced atmospheric line catalog correctly masks the DESI noise.For this purpose, we compute an average noise by stacking the pipeline noise  pip of 125, 477 objects categorized by redrock as quasars or luminous red galaxies on SV3 observations.This average noise with the DESI atmospheric line catalog such that (Λ l = 2.5, Λ w = 1.2) is shown in the bottom panel of Fig. 11.A zoom on an atmospheric line is shown in the top panels of Fig. 11 with eBOSS and DESI masks.The eBOSS mask is too wide for DESI stacked noise, which highlights the benefit of creating a new catalog.After a visual inspection of most atmospheric lines, we validate that all the spectrum pixels showing an increase in noise are masked by setting the width threshold to Λ w = 1.2.
Comparing the average noise (Fig. 11) to the average sky flux (Fig. 10), there is a consistency between atmospheric emission lines and observed peaks in the pipeline noise.The feature at 4, 360 Å is an exception, as it appears wide and relatively high in the DESI noise and not in the average sky flux.Its wavelength is inside a known transmission dip around 4, 400 Å due to an issue with DESI's spectrograph collimator coating (Guy et al. 2022).For this specific line, we take the same value as the eBOSS catalog and force the algorithm to consider it as an atmospheric line even if it does not pass the Λ l requirement.
A comparison of the percentage of Ly forest masked for eBOSS and DESI catalog, as a function of redshift, is given in Fig. 12.To remain conservative, we chose the value of Λ l = 2.5 to obtain a catalog of atmospheric emission lines with a spectral length masked similar to the eBOSS catalog.A complete study on synthetic data, out of this paper's scope, will be done to decrease the length masked, and consequently the impact on  1D,  .

SYNTHETIC DATA CORRECTIONS
Synthetic data (otherwise called mocks) are generated to characterize the impact on  1D,  of continuum fitting, spectral resolution, noise modeling, and spectrum pixel masking.From this, we derive empirical corrections of these effects and apply them to the data measurement.

Synthetic data sets
We generated a set of DESI-Lite (Karaçaylı et al. 2020) mocks, specifically designed for  1D,  .The full description of these mocks is given in Karaçaylı et al. (2023a).The DESI-Lite software produces uncorrelated Ly forests that mimic the redshift and noise distribution of the SV3+M2 dataset.Ten independent realizations are generated with different initial conditions.For each realization, a random catalog of DLAs is created to follow the redshift and column density distribution of the latest eBOSS catalog (Chabanier et al. 2021).
The quickquasars software 15 (Herrera-Alcantar et al. 2023), included in the desisim package 13 , transforms Ly transmission into spectra with observational and astrophysical contaminants.For each realization, two sets of spectra are generated by imprinting DLAs according to the catalog aforementioned or not.
We run 5 different  1D,  variations to study the impact of different Ly contaminants: • TRUECONT: The true quasar continuum imposed by quickquasars is applied instead of the continuum fitting procedure described in Sec.3.1.In comparison to RAW mocks, this realization is impacted by finite noise and resolution.
• CONT: The Ly contrasts are calculated using the pipeline detailed in Sec.3.1.This type of mocks includes the impact of continuum fitting.
• DLAm: Realization for which the DLAs are not added to forests at the quickquasars stage, though we mask spectrum pixels as if they were present.The objective of this kind of mocks is to characterize the impact of DLA masking.
• LINEm: Similarly to DLAm but masking the atmospheric emission lines catalog built in Sec.4.4.
• DLA: Realization for which the DLA are applied to the spectra without masking them.The objective of this mocks is to measure the impact of DLA to compute a DLA completeness systematic error in Sec. 6.For all the mocks, we take the same procedure as for the FFT calculation on observational data.In particular, the same SNR-weighting is applied, and the number of sub-forest for each realization is around 81, 500 with a small statistical variation between realizations.This is slightly larger than the number of sub-forest of the data sample given in Sec. 7.
In the next sections, all the results are shown for the combination of ten independent realizations.To decrease the error bars, we also performed a linear rebin that provides a wavenumber binning three times coarser than what we used for observational data.The error bars of the presented ratios are computed in quadrature.

Continuum fitting correction
The continuum fitting procedure defined in Sec.3.1 systematically distort the measured  q (,  q ) () term by suppressing large-scale modes, and may bias the  1D,  measurement.This is a well-known effect in BAO measurements (du Mas des Bourboux et al. 2020).To create a correction which contains this effect, we compare the mock computed using the true continuum (TRUECONT) with the one which follows the standard continuum fitting procedure (CONT): This correction is shown for the combination of ten DESI-Lite mocks in Fig. 13.We use a second-order polynomial function to fit this correction and apply it to the  1D,  measurement.
This correction differs in amplitude compared to the eBOSS measurement (Chabanier et al. 2019).As for eBOSS, the one-dimensional power spectrum with continuum fitting is higher than that measured with the true continuum.However, in our case, the impact is much smaller than eBOSS, for which this ratio was near 4% (without using a first-order polynomial function in the continuum fitting).Furthermore, we do not have a large-scale impact as significant as eBOSS.

Spectrum pixel masking
For both DLA and atmospheric line masking, we remove some data points from the measured spectra.This does not impact studies performed on real-space spectra, such as the continuum fitting or the quadratic maximum likelihood estimator for  1D,  (Karaçaylı et al. 2022).On the other hand, the FFT calculation requires that spectrum pixels are equally-spaced.Consequently, when computing the Fourier transform, we impose a value of   = 0 (equivalent to mean transmitted flux fraction value for ) and infinite standard deviation to the masked spectrum pixels.This masking introduces a -dependent bias, which we need to quantify.
In order to determine and correct this bias in our  1D,  measurement, we compare mocks for which DLAs or atmospheric lines are masked (respectively DLAm and LINEm) with mocks where no masking is applied (CONT).On those two mocks, the DLAs and atmospheric emission lines are not imposed on spectra.We want to derive only the corrections of masking in order to apply them on data.The coefficients used for both masking corrections are defined as the ratio between the unmasked and the masked power spectra: dla (, ) =  1D, ,CONT (, )  1D, ,DLAm (, ) . (23)

Atmospheric emission lines
The correction induced by the DESI atmospheric line mask, as defined in Sec.4.4, is shown in Fig. 14.We verified that, for all redshifts where the masked Ly forest length of DESI is close to eBOSS, the impact of masking is lower in the DESI case.It indicates that applying thinner masks to our measurement mitigates the impact of masking.
As expected, the correction roughly scales with the number of masked spectrum pixels.The effect of masking is a relatively smooth function of wavenumber, and its main impact is at low wavenumber.The most impacted redshift bins are  = 2.2 (Caii galactic absorption lines),  = 2.6 (lines at 4, 360 Å in the transmission dip), and at high redshift for which many atmospheric lines need to be masked.As shown in Fig. 12, redshifts  = 2.4, 2.8, 3.0, 3.2 have no masks applied, and only a few for  = 3.4.It is also in agreement with the level of corrections.The impact of atmospheric line masking is qualitatively in agreement with the eBOSS results in Chabanier et al. (2019).We choose to model  line (, ) by a second-order polynomial fit and use this correction in the final calculation of  1D,  .

DLA masking
DLAs are added at random locations in the Ly forest during the creation of the mocks.For this study, we do not attempt to characterize the completeness of the DLA finder applied to the data, and we use a "truth" DLA catalog for masking.
We mask the "truth" catalog with the same parameters as the DLA data catalog, i.e., for  Hi > 10 20.3 cm −2 .The correction induced by the masking,  dla (, ), is represented in Fig. 15.As it was already seen in the eBOSS measurement (Chabanier et al. 2019), the DLA masking has a small impact compare to atmospheric emission lines.This is due to the random distribution of DLAs and the smaller masking in terms of Ly forest length.As the impact is very similar for all wavenumbers, we apply a k-independent correction  dla (, ) =  dla (), whose amplitude is 0.5 % on average.

UNCERTAINTY ESTIMATION
The statistical uncertainty of our averaged  1D,  measurement, noted  stat , is obtained during the SNR weighting scheme presented in appendix B. For each (, ) bin, a binned histogram of standard deviation as a function of SNR is derived.Fitting this histogram provides a function  , (SNR) that is used to define the statistical uncertainty: MNRAS 000, 1-23 (2023) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 where  is the fitted function, and the  index runs over the SNR bins chosen.
The obtained statistical uncertainties are shown in Fig. 16.Despite using a SNR-dependent weighting in the  1D,  calculation compared to a redshift-dependent SNR cut as eBOSS, we find similar trends as in Chabanier et al. (2019).The statistical uncertainty depends mainly on the number of sub-forests for each redshift bin, so that  stat is an increasing function of redshift.This error bar depends also on the power spectrum level, as pointed out by the right panel of Fig. 16, for which low redshift bins are not separated.In the case of  ∼ 2.2−2.4,at small scales ( ≳ 1.5 Å −1 ), the statistical uncertainties are crossing each other.It is caused by the large noise increase in the blue spectral band due to atmospheric absorptions.
Looking at wavenumber dependence,  stat increases as a function of  for small scales ( ≳ 1.5 Å −1 ).This is due to the resolution correction, which effectively increases the rms of individual  1D,  .At large scales ( ≲ 1 Å −1 ),  stat is a decreasing function of , mainly due to the decrease in Fourier modes available to compute  1D,  .
In our study, we characterized the impact of several instrumental and astrophysical contaminants.From this extensive study, we associate systematic errors, noted  syst , to our  1D,  measurement.Fig. 17 shows the systematic uncertainties for different redshift bins and their relative values with respect to statistical errors.Similarly to Chabanier et al. (2019), we made conservative choices to define those uncertainties: • Noise estimation: As presented in Sec.4.2, the pipeline noise is corrected using the  corrective term, which depends on the data set considered.We assign a systematic uncertainty equal to 30 % of the average  for each redshift bin.
• Resolution: We fit the average resolution correction given in (2022) (Fig. 30), indicate that its fractional change is less than 1 % over all spectrographs.We therefore assign a conservative systematic uncertainty  Δ = 1 %Δ.Using the abovementioned simplified resolution model, this translates into a  1D,  uncertainty equal to 2 2 Δ Δ •  1D,  ().
• Resolution correction: We apply a correction to the resolution modeling as presented in Sec.4.1, by multiplying  res to  1D,  .We add an associated systematic error defined as 30% of this correction.
• Side-band: The fitted side-band power spectrum  SB1,m measured in Sec.4.3 is subtracted to  1D,  to account essentially for metal absorptions in the Ly forest region.We associate to this correction a systematic uncertainty equal to the statistical errors of the measured SB1 power spectrum.This is a conservative choice, as the modeling performed in Sec.4.3 closely reproduces  SB .
• Spectrum pixel masking: The impact of masking DLAs and atmospheric emission lines on the  1D,  measurement was determined with synthetic data in Sec.5.3.Spectrum pixel masking is corrected by multiplying  line (, ) •  dla (, ) to the  1D,  estimator.We define the systematic error associated with each masking as 30 % of this correction.
• DLA completeness: Using the synthetic data described in Sec. 5, we derived the impact of DLA on the one-dimensional powerspectrum as the ratio between mock with DLA (DLA) and without (CONT).We fit this ratio with an adapted function provided by Rogers et al. (2017).As detailed previously in Sec.2.4, our DLA catalog of data results from the combination of two finders.The trend of this ratio is reported on the penultimate panel of Fig. 17.In Chabanier et al. (2021), the authors perform a full study on eBOSS data and provide the completeness of the CNN finder.The completeness of this finder is higher than 85% for log( Hi ) > 20.3.To be conservative, we choose to associate an uncertainty of 15% of the total impact of DLAs on  1D,  to the incompleteness of our catalog.We stress that this uncertainty is over-estimated, as the CNN finder has a higher completeness for DLAs with higher column density and since we are using an additional GP algorithm.
For all the corrections we applied on our  1D,  measurement, the choice of 30% in the associated systematic uncertainties is motivated by the fact that we consider a shift randomly ranging between no correction and 100% of the correction.It is described by a uniform distribution between 0 and 1.The standard deviation of the distribution, equal to 0.30, quantifies the spread among the possible values, leading to a systematic uncertainty equal to 30% of the correction.
Opposite to eBOSS (Chabanier et al. 2019), we chose to not ac-0.000.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00  count for the incompleteness of the BAL catalog in our analysis, as it is one of the weaker contaminants.The study of BAL catalog completeness will be performed on further studies.
The general trends in Fig. 17 are similar to those of the eBOSS measurement (Chabanier et al. 2019).Given our limited statistics, most of the systematic errors are smaller than the statistical uncertainties for all redshift bins and all scales.However, this is expected to change for future DESI measurements, which will offer unprecedented statistics, thus reducing the statistical errors.
There is room for improvement for the major source of systematic uncertainties presented above.The noise modeling can be improved by understanding and correcting the source of unaccounted noise.Regarding the resolution modeling, the mathematical model and its verification with the relatively new CCD mocks presented in Sec.4.1 can be improved with additional tests and larger datasets.Decreasing statistical error on the side-band power spectrum will directly reduce the associated systematic error.Concerning the spectrum pixel masking, especially for the atmospheric lines, thinner masks can be applied considering the improvement of atmospheric emission line subtraction in DESI (see Guy et al. (2022)).Furthermore, a more complex but analytical mathematical correction could be derived for this regular masking.For the DLA completeness, a more advanced study as  et al. (2017), the impact and correction of objects with lower column density (sub-DLAs, Lyman limit systems...) should be accounted for in a future, more developed study.Finally, the systematics are defined from a simplified assumption of uncertainty propagation, along with conservative choices.We plan to improve the modeling of each systematic and obtain more reliable uncertainties by modeling them directly into large samples of synthetic data.
We choose to remove the SV1 data set from this measurement due to considerations on the noise power spectrum shown in Sec.4.2 and measurements performed in the appendix A.
The  1D,  measurement is done using the pipeline and parameters presented in Sec. 3. Considering all the corrections defined in the previous section, the final  1D,  estimator is defined by (25) Fig. 18 presents the normalized  1D,  measurement such that Δ 2 1D,  =   1D,  /.This observable is shown for 9 redshift bins ranging from 2.2 to 3.8, for wavenumbers 0.145 ≤  ≤ 2 Å −1 , and using a total of 73, 839 sub-forests extracted from 26, 330 quasar spectra.The represented error bars are the statistical and systematic uncertainties added in quadrature.The details of sub-forest properties for each redshift bin are given in Tab. 4.

Comparison with other measurements
We perform a comparison with past measurements.The comparison of the DESI  1D,  with the last moderate-resolution measurement (eBOSS survey Chabanier et al. (2019)) is shown in Fig. 19 (left).We also compare our measurement with the last high-resolution measurement obtained using the combination of KODIAQ, SQUAD, and XQ-100 surveys (Karaçaylı et al. 2022) in Fig. 19 (right).
Both moderate-and high-resolution measurements were expressed in velocity units (s•km −1 ).The conversion between this unit system and the one used for DESI (Å −1 ) is defined by: This conversion is performed at the stage of the FFT estimation by converting the terms in equation ( 14) in velocity units before performing the ensemble average to compute  1D,  .We found that doing this conversion on the averaged  1D,  significantly shift  1D,  because of the redshift term in the equation ( 26).This is caused by the non-uniform redshift distribution in each z-bins, pointed out by Fig. 2. Similarly, we converted all corrections terms appearing in equation ( 25) in velocity units.For the comparison with eBOSS measurement, we compute  1D,  in velocity units with the same binning.For the high-resolution measurement, we rebin our DESI measurement to the same wavenumber binning, and account this rebinning in the calculation of error bars, to obtain a fair comparison.
The comparison with the eBOSS measurement in Fig. 19 (left) yields a ∼ 15% difference at small wavenumber ( < 0.01 km•s −1 ).To investigate this discrepancy, we performed a detailed investigation by varying most of the parameters of our analysis, which are susceptible to impact small scales.The discussion concerning those tests is detailed in appendix D. On the other scales (0.01 km•s −1 <  < 0.02 km•s −1 ), our measurement agrees with eBOSS considering the error bars.
The major improvement lays at large wavenumber ( > 0.015 km•s −1 ) where the improved DESI resolution and noise modeling allows us reaching much smaller scales than eBOSS, especially for high redshifts.We are able to conservatively reach the wavenumber  max = 2.0 Å −1 for all redshift bins.In comparison, the eBOSS measurement (Chabanier et al. 2019) achieved maximal wavenumber of  max = 1.54 Å −1 for  = 2.2 and  max = 1.03 Å −1 for  = 3.8.At large wavenumbers, the eBOSS measurement is highly contaminated by noise and resolution.We consider that our measurement is more suitable to probe those scales, considering the improvement in resolution and noise estimation.
We compare our measurement with the high-resolution measurements in Fig. 19 (right).This measurement (Karaçaylı et al. 2022) is performed with a statistically smaller sample of quasars but with a very high spectral resolution (5, 000 ≲  ≲ 60, 000) and SNR.
The high-resolution  1D,  measurements thus allow reaching very small scales with large error bars.Our agreement shows a 20% disagreement at small scales ( > 0.02 km•s −1 ), mostly for the lower redshifts measured.It indicates that there is still room for improvement in our measurement's noise and resolution modeling.We note that considering error bars, both measurements agree on intermediate scales.

CONCLUSION AND PROSPECTS
We performed the first measurement of the one-dimensional power spectrum ( 1D,  ) with DESI data.The main objective of this paper is to carefully characterize the different contaminants of  1D,  with regard to DESI instrument.In particular, we modeled the noise and spectral resolution of DESI.In comparison to the previous eBOSS measurement (Chabanier et al. 2019), we improved the analysis of side-band power spectrum and atmospheric emission lines.
We used adapted synthetic data to correct the impact of spectrum pixel masking, continuum fitting and spectral resolution modeling.We performed a complete review of the systematic uncertainties linked to the  1D,  pipeline and compared the DESI measurement with previous moderate-and high-resolution measurements.We find a relatively good agreement, except for a slight difference at large scales with the eBOSS measurement, partially due to the different residual correction we apply.Our estimation of  1D,  is also compared with the QMLE method in a companion paper Karaçaylı et al. (2023a).Measurements with FFT and QMLE methods agree at 1% level precision up to half the Nyquist frequency.
The DESI spectral resolution is approximately two times better than SDSS.Consequently, our  1D,  measurement is of high scientific interest to probe the small scales of the intergalactic medium.However, the data sets we exploited remains inferior to eBOSS in terms of statistics.If we apply the same SNR cut as in Chabanier et al. (2019), our sample contains 17, 333 sub-forests compared to 94, 558 for eBOSS.However, we expect future DESI data sets to provide high increase of statistics (up to 1 million Ly forest, thus almost 3 milion sub-forests).This unprecedented dataset will allow obtaining a sub-percent precision measurement.The resulting  1D,  measurement will provide stringent constraints on the sum of neutrinos masses, warm dark matter models, and on the parameters of the intergalactic medium (DESI 2016a;Valluri et al. 2022).
We plan to improve our analysis to keep the level of systematic error close to the statistical one.First, applying stricter constraints on a larger Ly forest sample will be beneficial to reduce systematical uncertainties.Furthermore, we also plan to improve the treatment of contaminants presented in Sec. 6.In particular, we plan to test extensively the resolution and noise estimations on pixel-level simulations of the CCD camera.

APPENDIX B: SNR WEIGHTING
For this  1D,  measurement, we keep all Ly sub-forests available in the measurement sample, independently of their SNR, unlike what was done in eBOSS (Chabanier et al. 2019), reminding that only one SNR cut is applied at earlier stages of the analysis, during the continuum fitting procedure as described in Sec.3.1.Individual Ly power spectra, falling into the same wavenumber bin, do not have the same dispersion which varies as function of the SNR.In our analysis, we account for this effect at the last step of our FFT estimator pipeline, by weighting each of the Ly sub-forests by a SNR dependent factor, while averaging over the full measurement sample.
First, for each Ly sub-forest (i), the variance of individual  1D,  , is fitted according to the following: where SNR  is the mean signal to noise ratio of the Ly sub-forest (i), defined in equation ( 6).Hence, the weighting factor is: The employed fitting model works well empirically with the measured  2 ( 1D, ,i ).Also according to equation (B1), W  tends to 0 as SNR  tends to 1, consistently with the applied SNR cut = 1.
Tests for this SNR weighting method were done on DESI-Lite mocks that mimic the SV1+SV3 data set described in Sec. 2, and are specifically designed for  1D,  measurement.A comparison between measured  1D,  and mocks truth power spectrum is represented in Fig. B1, for both  1D,  measured with eBOSS SNR cut method, and our SNR weighting method.
Fig. B1 shows that we have an improvement compared to the eBOSS method, especially at large wavenumber and redshift values, where we are mostly limited by the statistics, as well as at low wavenumber and redshift values, while for the eBOSS measurement, there was no possible optimization at both small and large wavenumber ranges at the same time.

APPENDIX C: DETAILS ON NOISE ESTIMATION C1 Pipeline noise
The noise associated to each spectrum is computed with the spectroscopic pipeline presented in Sec.2.2.It is modeled as the addition of effects at the CCD scale.Each contribution is calculated by measuring the spatial variance on the CCD image from the associated noise source.It is assumed that this noise comes from the following four sources: • Poisson noise: Measuring photons with a CCD is a statistical process.It creates a noise source which is directly linked to the input flux, and particularly dominant for low fluxes.For DESI, this noise is estimated by modeling the CCD.
• Over-scan: The over-scan measures the bulk offset, i.e., the average level of all CCD pixels.It is used to remove small variations in the bias.Over-scan suppression introduces noise.
• Bias: Noise due to the response of the CCD to a minimal exposition time.It emerges from parasite electron or CCD pixel defects.The master bias estimates this noise.
• Dark current: Readout noise due to the thermal motion of the atoms composing CCD material which induces charge deposit.Dark current is estimated using the master dark.In DESI, the modeling of noise is improved to account for Poisson noise in the dark frames.
All those noise sources have all been corrected for their dependence on the CCD position.By adding these four terms, we obtain a CCD noise estimator which is propagated to the spectra by the "spectroperfectionism" formalism.A pipeline noise estimator  pip is then obtained.
The noise power spectrum estimated from pipeline ( pipeline ) is computed from the standard deviation    linked to  pip by equation (5).For each unmasked spectrum pixel, a contrast  pipeline is generated following a normal probability distribution such that: This procedure is repeated  G times ( G = 2500) to obtain a converged noise power spectrum.For each quasar, the associated noise power spectrum is the average of the  G noise contrasts after Fourier transformation:

C2 Exposure difference noise
Another noise estimation can be done using the difference between exposures of the same quasar, when several exposures are available for the same object.The difference between exposures removes the physical signal, leaving only the fluctuations due to noise.We implemented a noise power spectrum estimator using this principle.We define the difference coadd of a quasar of index  by separating half of its exposures in the even category ( even exposures) and the other half in the odd category ( odd exposures) such that: where  2 pip,k is the pipeline noise of the exposure  for quasar .In the case where the total number of exposures is even,  even =  odd .
The standard deviation of Δ   can be calculated from the variances of individual exposures: This difference coadd is unbiased, i.e., of zero average, whatever the values of the sum of the inverse variance for both exposure populations.Finally, this estimator does not necessarily need an even total number of exposures.
To derive an estimator of  noise , the variance of Δ   must be equal to that of the coadded flux defined by: To obtain the same variance, we multiply Δ   by the ratio    / Δ   : where  tot is the total number of exposure for the quasar  ( tot =  even +  odd ).
In SDSS analysis (McDonald et al. 2006;Palanque-Delabrouille et al. 2013;Chabanier et al. 2021), the variance of all exposures for a given object was considered equal.In this case,  Δ   can be simplified, and a correction was applied only in the case of an odd number of exposures.In the case of a constant exposure variance, the difference coadd in equation (C6) is equal to the one derived in Palanque-Delabrouille et al. (2013); Chabanier et al. (2019).Our new estimator corrects the variance for any exposure time.It is essential in the case of DESI first data, for which the exposure times can be very variable compared to SDSS.
The exposure difference coadd is computed in picca 11 (du Mas des Bourboux et al. 2021).We obtain an estimator of  noise called difference power spectrum and noted  diff , such that: (C7)

C3 Additional considerations on noise estimation
Fig. C1 shows the asymptotic ratios  for the three data sets.We remark that for SV3 and M2, the SNR dependence of  is much more pronounced than that of .The absolute noise level is the main parameter which varies when changing the minimal SNR cut.It indicates that the residual noise source is additive rather than multiplicative.To support this hypothesis, we computed the  values derived from the  of Fig. 8, using the mean value of  noise for all redshift.They are shown as stars in Fig. C1, and exhibit similar trends to the direct  computation, which corroborates that the missing noise is additive.Consequently, we decide to correct  pipeline using an additive term  ( noise =  pipeline + ).
The same noise study is performed on side-band regions SB1 and SB2 for which the astrophysical signal, i.e. absorption from intergalactic elements, is much lower than the Ly band.The  values are shown in Fig. C2.For side-bands, the overall missing noise level exhibits similar trends as a function of the minimal SNR cut, but is lower than for the Ly band.This is likely due to the use of different quasar populations employed for side-band and Ly measurements.Indeed, the DESI observation strategy is different for low redshift quasars (used for side-band study) and Ly quasars.On average, the number of exposures is larger for Ly than low-redshift quasars.Consequently, and in accordance with our previous interpretations, the misestimation of noise is larger for Ly measurement than side-bands.The difference might also be due to the much lower astrophysical signal in the side-bands, allowing an improved estimation of asymptotic noise level.As the results for SB1 and SB2 are very similar, we decide to apply the same correction for these two bands.

APPENDIX D: COMPARISON WITH EBOSS MEASUREMENT ON LARGE SCALES
We performed a series of tests to investigate the discrepancy at large scales ( < 0.01 km•s −1 ) between our measurement and the eBOSS measurement in Chabanier et al. (2019) as seen in Fig. 19 (left).We first focused on reproducing the eBOSS measurement with the picca software used in our analysis and the eBOSS parameters.Starting from SDSS spectra, we applied the pipeline used in Chabanier et al. (2019), and successively replaced each step (continuum fitting in Sec.3.1, Fourier transform and averaging in Sec.3.2) by the new picca software.We assessed that the version used in our analysis could reproduce the eBOSS measurement without noticeable bias at all scales.
Compared to the eBOSS measurement, some parameters are changed for the continuum fitting presented in Sec.3.1.For eBOSS, this pipeline step was performed separately on sub-forests instead of the total Ly forest.We checked that performing our continuum fitting on sub-forest does not modify the large-scales level of  1D,  .Additionally, we have performed the following changes in the continuum analysis.We removed the smoothing of the common continuum in equation 7, which was not used in Chabanier et al. (2019).We changed the polynomial order in equation 2 to zero as in eBOSS.Additionally, we tested to modify parameters of the continuum fitting procedure, which were used as eBOSS but could potentially change the level of  1D,  at large scales.We also applied non-constant weights in equation 4, removed the forcing to zero of the Ly contrast stack, or changed the observed wavelength to a smaller range.The conclusion of those continuum fitting tests is that none of those effects could be responsible for the ∼ 15% discrepancy visible on the eBOSS comparison.Furthermore, the corrections we derived from the mocks in Sec. 5 are different than the eBOSS corrections and these differences could explain the disagreement between the two measurements.To check the impact of corrections, we computed an eBOSS measurement without any corrections, and we realized the comparison in Fig. 19 (left), adding the corrections successively.This test yields that part of the disagreement (between 4 and 6 %) is due to the continuum fitting correction, which is different from eBOSS (see Fig. 7 in Chabanier et al. (2019)).
Another effect that could impact the largest scale is the possible impurity and incompleteness of the DLA catalog.In order to eliminate effects due to differences in the DLA catalog, we created a common set of quasars and DLAs by merging eBOSS and DESI catalogs.We found that the results from this common set of quasars are not significantly different, which indicates that missing DLAs cannot fully explain the disagreement.Finally, we varied the DLA and BAL catalog used by varying confidence levels, or column density  Hi to include sub-DLAs.Masking and correcting the damping wings for log( Hi ) < 20.3 systems decreases the discrepancy only for the very first wavenumber bins ( < 0.004 km•s −1 ).
To conclude, the large-scale disagreement between our measurement and eBOSS (Chabanier et al. 2019) cannot be fully explained for the moment.The continuum correction is responsible for a portion of this discrepancy.A complete study on DLA completeness or a comparison at the spectrum level between eBOSS and DESI will be performed in future studies to investigate in detail this discrepancy This paper has been typeset from a T E X/L A T E X file prepared by the author.

Figure 2 .
Figure 2. Histogram of the quasar redshift (with  > 2.0) whose spectra are observed in the SV1, SV3, and M2 data sets.The histogram of the sum of data sets is shown in dashed line.Finally, the number of BAL quasars characterized by a balnicity index higher than zero, and not used in the  1D,  computation pipeline, is shown with a dotted line.

Figure 3 .
Figure 3. Normalized histogram of the effective exposure time in the Ly forest region for the quasar spectra in the SV1, SV3, and M2 data sets.As mentioned in Sec.2.3, there is a wide disparity of exposure time for the three data sets.As a reference, the nominal time of one DESI exposure is set to 1200 s.

Figure 4 .
Figure 4. Illustration of the resolution improvement between eBOSS and DESI.The DESI spectrum of the quasar represented in Fig. 1 is zoomed on a region of the Ly forest on the top panel.The spectrum of the same quasar obtained by the Sloan Digital Sky Survey-IV (SDSS-IV)(Gunn et al. 2006;Smee et al. 2013;Dawson et al. 2016;Blanton et al. 2017) in the 16  ℎ Data Release (DR16)(Ahumada et al. 2020) of the extended Baryon Oscillation Spectroscopic Survey (eBOSS).The SDSS name associated to this quasar is 142903.03-014519.3.The DESI spectrum is obtained after 3 individual exposures for a total exposure time of 2, 300 seconds.The eBOSS spectrum have 11 exposures for a total of 6, 300 seconds.The large-scale absorption structures are similar but due to its improved spectroscopic resolution, the DESI spectrum clearly shows more details at small scales.

Figure 6 .
Figure6.Ratios between the power spectrum obtained directly from Ly transmissions (RAW) and the one derived from the CCD image simulations (CCD), for different redshift bins (light black) and averaged over all redshifts bins (blue points with error bars).Each power spectrum is re-binned by a factor 3 to reduce error bars.A second order polynomial is fitted to the ratio (blue continuous line).It is used to correct the miss-estimation of resolution in the  1D,  measurement.

Figure 8 .
Figure8.Asymptotic differences  between the noise and raw power spectra for SV1 (blue), SV3 (yellow), and M2 (green) data sets, as a function of the minimal SNR cut.This difference is measured for both  noise estimators from the pipeline ( pipeline , points) and from exposure differences ( diff , stars).The continuous lines are fits of the  values for pipeline noise, whose parameters are given in Tab. 3.

Figure 11 .
Figure 11.Comparison between the atmospheric emission line mask catalogs used for eBOSS 14 and the new DESI catalog we designed in our study.(bottom) Average of the DESI pipeline noise for 125,477 objects categorized by redrock as quasars or luminous red galaxies on SV3 observations (blue curve).The DESI (Λ l = 2.5, Λ w = 1.2) line catalog is shown on top with purple vertical lines.(top) Zoom on a specific atmospheric emission line on the DESI stacked noise (blue curve).The mask used on this specific line for DESI is shown in the left panel in purple and for eBOSS in orange on the right panel.The DESI mask decreases the masked length in accordance with stacked DESI noise.

Figure 13 .
Figure13.Ratios between the power spectrum obtained using true continuum (TRUECONT) and the one derived with our pipeline (CONT) on the combination of 10 mocks.Each power spectrum is re-binned by a factor 3 to reduce error bars.Fitting functions are represented by continuous lines, and used to correct the  1D,  measurement.For clarity, we artificially offset the points corresponding to different redshifts.

Figure 14 .Figure 15 .
Figure14.Ratio between the unmasked (CONT) and masked (LINEm) power spectra (equation (22)) for atmospheric line masking on the combination of the 10 mocks.The DESI (Λ l = 2.5, Λ w = 1.2) atmospheric line catalog is used.Each power spectrum is re-binned by a factor 3 to reduce error bars.Second-order polynomial functions are employed to fit the corrections in each redshift bin.For clarity, we artificially offset the points corresponding to different redshifts.

Figure 16 .
Figure 16.Statistical uncertainties (left) and its relative value with respect to  1D,  (right) of the DESI SV3+M2 measurement in Å, as a function of wavenumber.

Figure 18 .
Figure18.Normalized one-dimensional Ly forest power spectrum (Δ 1D,  ( )) using the SV3+M2 data set, for redshift bins from  = 2.2 to  = 3.8.All the corrections given in equation (25) are applied to perform this measurement.As an illustration, wavenumbers in velocity space for different redshifts are represented at the top of the figure.Error bars are systematic and statistical uncertainties added in quadrature.

Figure A1 .
Figure A1.Ratio of the  1D,  , measured with the same parameters, between SV1 and SV3 (top), and M2 and SV3 (bottom), for four redshift bins.

Figure C1 .Figure C2 .
Figure C1.Asymptotic ratios  between the noise and raw power spectra for SV1 (blue), SV3 (yellow), and M2 (green) data sets, as a function of the minimal SNR cut, for the pipeline noise.Points give the direct estimation of .Stars represent  as derived from the asymptotic differences in Fig.8.Second-order polynomial fits are shown only for representation.

Table 2 .
Summary of the DESI data sets used in this study, their associated acronyms, a subjective description, and the total number of quasar spectra whose redshifts are between 2.0 and 5.0.

Table 3 .
Additive corrections applied to the pipeline noise for different spectral regions and data sets.An SNR dependence is included in the case of SV1 only.The same parameters are used for both SB1 and SB2.
Average of the sky spectra of 15,000 sky fibers with optimal observation conditions (speed > 105, effective exposure time > 1, 100 sec, seeing < 1.05 deg, and airmass < 1.3).These sky spectra originate from three exposures in the SV1 and SV3 data sets.The different spectral bands of DESI are represented (B in blue, R in orange, and Z in green).The median smoothing of this average sky spectrum multiplied by a threshold Λ l = 2.5, shown as a dashed red line, is used to select atmospheric emission lines we want to mask (light purple lines).The line located at 4, 360 Å was added manually, considering its large impact on noise seen in Fig.11. obs[Å]

Table 4 .
Number of sub-forest, average redshift, and signal-to-noise ratio for each redshift bins in the final data set sample used in this measurement.
preparation du Mas desBourboux H., et al., 2020, The Astrophysical Journal, 901, 153  du Mas des Bourboux H., et al., 2021, Astrophysics Source Code Library, p.  1D,  point of view, the noise properties of SV1 are a potential issue.While our initial goal was to analyze the full data sample available, we choose first to compare the measured  1D,  on the separate SV1, SV3, and M2 data sets.Comparison between measured  1D,  and DESI-Lite mocks truth power spectrum, for the redshift bin z = 3.8, for both  1D,  measured with eBOSS SNR cut method, and the SNR weighting method.
Fig.A1shows their respective ratios, on the four redshift bins with largest statistics.It appears that the measurement of  1D,  on SV1 is biased compared to the other two data sets.In particular, we believe that the difference at  ≳ 1.0 Å −1 is due to an imperfection in the noise correction presented in Sec.4.2.Consequently, we decide to remove the SV1 data set in this study to remain conservative.