<figure><img src="image_20.png" /><figcaption>**Figure 19.** Left: Comparison between the measurement performed here and that of the eBOSS data (Chabanier et al. 2019). For this comparison, all the analysis was performed in velocity units using the conversion given by equation (26). Both normalized $P_{1D,\alpha}$ are shown in the top panel (our measurement with points and the eBOSS measurement with shaded coloured areas). The ratio between DESI and eBOSS measurements is shown in the bottom panel. The striped grey area in the bottom panel shows the centered error bar of the ratio averaged over all shown redshift bins. We remove high-redshift bins for clarity. A description of the tests used to explain the difference between DESI and eBOSS can be found in Appendix D. (right) Same comparison with the high-resolution measurement obtained using the combination of KODIAQ, SQUAD, and XQ-100 surveys (Karaçaylı et al. 2022). In this case, our $P_{1D,\alpha}$ measurement and the error bars associated are rebinned to the wavenumber binning of the high-resolution measurement. For clarity, only the four first redshift bins are shown in the bottom panel of each figure, along with the error on the ratio for $z = 2.8$ centred to unity.</figcaption></figure>

and Walther et al. (2021) will be employed with next DESI $P_{1D,\alpha}$ measurement to obtain constraints on cosmological and intergalactic medium parameters.

## ACKNOWLEDGEMENTS

The authors acknowledge support from Agence National de la Recherche (grant ANR-16-CE31-0021). The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University – A*MIDEX, a French ‘Investissements d’Avenir’ programme (grant AMX-20-CE-02–DARKUNI).

This material is based upon work supported by the US Department of Energy (DOE), Office of Science, Office of High-Energy Physics, under Contract No. DE-AC02-05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Science and Technology of Mexico (CONACYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U.S. National Science Foundation, the U.S. Department of Energy, or any of the listed funding agencies.

The authors are honoured to be permitted to conduct scientific research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation.

## DATA AVAILABILITY

The DESI spectra from SV1 and SV3 will be publicly available in the Early Data Release (EDR) in 2023. We added the first 2 months of the main survey to improve our statistics. The main survey spectra will be made publicly available as part of Year 1 data release in the future. All the data points of the figures in this article are made available according to the data management policy of DESI (Ravoux 2023). All the outputs and plots of this article are generated with picca (8.1.2) and p1desi https://github.com/coretinravoux/p1desi (1.0.0).

## REFERENCES

* Abareshi B. et al., 2022, Overview of the Instrumentation for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2205.10939

* Ahumada R. et al., 2020, ApJS, 249, 3

* Alexander D. M. et al., 2022, The DESI Survey Validation: Results from Visual Inspection of the Quasar Survey Spectra. Available at: http://arxiv.org/abs/2208.08517

* Armengaud E., Palanque-Delabrouille N., Yèche C., Marsh D. J. E., Baur J., 2017, MNRAS, 471, 4606

* Bajtlik S., Duncan R. C., Ostriker J. P., 1988, ApJ, 327, 570

* Baur J., Palanque-Delabrouille N., Yèche C., Boyarsky A., Ruchayskiy O., Armengaud É., Lesgourgues J., 2017, J. Cosmol. Astropart. Phys., 2017, 013

* Baur J., Palanque-Delabrouille N., Yèche C., Magneville C., Viel M., 2016, J. Cosmol. Astropart. Phys., 2016, 012

* Bautista J. E. et al., 2017, Astron. Astrophys., 603, A12

* Blanton M. R. et al., 2017, AJ, 154, 28

* Boera E., Becker G. D., Bolton J. S., Nasir F., 2019, ApJ, 872, 101

* Bolton A. S., Schlegel D. J., 2010, Publ. Astron. Soc. Pac., 122, 248

* Bolton J. S., Puchwein E., Sijacki D., Haehnelt M. G., Kim T.-S., Meiksin A., Regan J. A., Viel M., 2017, MNRAS, 464, 897

* Borde A., Palanque-Delabrouille N., Rossi G., Viel M., Bolton J., Yèche C., LeGoff J.-M., Rich J., 2014, J. Cosmol. Astropart. Phys., 2014, 005
<figure><img src="image_21.png" /><figcaption>**Figure 19.** Left: Comparison between the measurement performed here and that of the eBOSS data (Chabanier et al. 2019). For this comparison, all the analysis was performed in velocity units using the conversion given by equation (26). Both normalized $P_{1D,\alpha}$ are shown in the top panel (our measurement with points and the eBOSS measurement with shaded coloured areas). The ratio between DESI and eBOSS measurements is shown in the bottom panel. The striped grey area in the bottom panel shows the centered error bar of the ratio averaged over all shown redshift bins. We remove high-redshift bins for clarity. A description of the tests used to explain the difference between DESI and eBOSS can be found in Appendix D. (right) Same comparison with the high-resolution measurement obtained using the combination of KODIAQ, SQUAD, and XQ-100 surveys (Karaçaylı et al. 2022). In this case, our $P_{1D,\alpha}$ measurement and the error bars associated are rebinned to the wavenumber binning of the high-resolution measurement. For clarity, only the four first redshift bins are shown in the bottom panel of each figure, along with the error on the ratio for $z = 2.8$ centred to unity.</figcaption></figure>

and Walther et al. (2021) will be employed with next DESI $P_{1D,\alpha}$ measurement to obtain constraints on cosmological and intergalactic medium parameters.

## ACKNOWLEDGEMENTS

The authors acknowledge support from Agence National de la Recherche (grant ANR-16-CE31-0021). The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University – A*MIDEX, a French ‘Investissements d’Avenir’ programme (grant AMX-20-CE-02–DARKUNI).

This material is based upon work supported by the US Department of Energy (DOE), Office of Science, Office of High-Energy Physics, under Contract No. DE-AC02-05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Science and Technology of Mexico (CONACYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U.S. National Science Foundation, the U.S. Department of Energy, or any of the listed funding agencies.

The authors are honoured to be permitted to conduct scientific research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation.

## DATA AVAILABILITY

The DESI spectra from SV1 and SV3 will be publicly available in the Early Data Release (EDR) in 2023. We added the first 2 months of the main survey to improve our statistics. The main survey spectra will be made publicly available as part of Year 1 data release in the future. All the data points of the figures in this article are made available according to the data management policy of DESI (Ravoux 2023). All the outputs and plots of this article are generated with picca (8.1.2) and p1desi https://github.com/coretinravoux/p1desi (1.0.0).

## REFERENCES

* Abareshi B. et al., 2022, Overview of the Instrumentation for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2205.10939

* Ahumada R. et al., 2020, ApJS, 249, 3

* Alexander D. M. et al., 2022, The DESI Survey Validation: Results from Visual Inspection of the Quasar Survey Spectra. Available at: http://arxiv.org/abs/2208.08517

* Armengaud E., Palanque-Delabrouille N., Yèche C., Marsh D. J. E., Baur J., 2017, MNRAS, 471, 4606

* Bajtlik S., Duncan R. C., Ostriker J. P., 1988, ApJ, 327, 570

* Baur J., Palanque-Delabrouille N., Yèche C., Boyarsky A., Ruchayskiy O., Armengaud É., Lesgourgues J., 2017, J. Cosmol. Astropart. Phys., 2017, 013

* Baur J., Palanque-Delabrouille N., Yèche C., Magneville C., Viel M., 2016, J. Cosmol. Astropart. Phys., 2016, 012

* Bautista J. E. et al., 2017, Astron. Astrophys., 603, A12

* Blanton M. R. et al., 2017, AJ, 154, 28

* Boera E., Becker G. D., Bolton J. S., Nasir F., 2019, ApJ, 872, 101

* Bolton A. S., Schlegel D. J., 2010, Publ. Astron. Soc. Pac., 122, 248

* Bolton J. S., Puchwein E., Sijacki D., Haehnelt M. G., Kim T.-S., Meiksin A., Regan J. A., Viel M., 2017, MNRAS, 464, 897

* Borde A., Palanque-Delabrouille N., Rossi G., Viel M., Bolton J., Yèche C., LeGoff J.-M., Rich J., 2014, J. Cosmol. Astropart. Phys., 2014, 005
Brodzeller A. et al., 2023, Performance of the Quasar Spectral Templates for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/ abs/2305.10426

Busca N., Balland C., 2018, QuasarNET: Human-level spectral classification and redshifting with Deep Neural Networks. Available at: http://arxiv.org/ abs/1808.09955

Chabanier S. et al., 2019, J. Cosmol. Astropart. Phys., 2019, 017

Chabanier S. et al., 2021, ApJS, 258, 18

Chabanier S. et al., 2022, Modeling the Lyman-α forest with Eulerian and SPH hydrodynamical methods. Available at: http://arxiv.org/abs/2207.05023

Chaussidon E. et al., 2022, Target Selection and Validation of DESI Quasars. Available at: http://arxiv.org/abs/2208.08511

Croft R. A. C., Weinberg D. H., Bolte M., Burles S., Hernquist L., Katz N., Kirkman D., Tytler D., 2002, ApJ, 581, 20

Croft R. A. C., Weinberg D. H., Katz N., Hernquist L., 1998, ApJ, 495, 44

Dawson K. S. et al., 2016, AJ, 151, 44

Day A., Tytler D., Kambalur B., 2019, MNRAS, 489, 2536

DESI, 2016a, The DESI Experiment Part I: Science,Targeting, and Survey Design. Available at: http://arxiv.org/abs/1611.00036

DESI, 2016b, The DESI Experiment Part II: Instrument Design. Available at: http://arxiv.org/abs/1611.00037

DESI, 2023a, Validation of the Scientific Program for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2306.06307

DESI, 2023b, The Early Data Release of the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2306.06308

Dey A. et al., 2019, ApJ, 157, 168

du Mas des Bourboux H. et al., 2020, ApJ, 901, 153

du Mas des Bourboux H. et al., 2021, Astrophysics Source Code Library, record ascl:2106.018

Farr J., Font-Ribera A., Pontzen A., 2020, JCAP, 2020, 015

Gaikwad P., Srikanand R., Haehnelt M. G., Choudhury T. R., 2021, MNRAS, 506, 4389

Gunn J. E. et al., 2006, AJ, 131, 2332

Gunn J. E., Peterson B. A., 1965, ApJ, 142, 1633

Guy J. et al., 2022, The Spectroscopic Data Processing Pipeline for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2209 .14482

Ho M.-F., Bird S., Garnett R., 2021, MNRAS, 507, 704

Iršič V. et al., 2016, MNRAS, 466, 4332

Iršič V., Viel M., Haehnelt M. G., Bolton J. S., Becker G. D., 2017, Phys. Rev. Lett., 119, 031302

Karaçaylı N. G. et al., 2022, MNRAS, 509, 2842

Karaçaylı N. G. et al., 2023a, Optimal 1D Ly α Forest Power Spectrum Estimation – III. DESI Early Data. Available at: http://arxiv.org/abs/2306 .06316

Karaçaylı N. G. et al., 2023b, MNRAS, 522, 5980

Karaçaylı N. G., Font-Ribera A., Padmanabhan N., 2020, MNRAS, 497, 4742

Khaire V. et al., 2019, MNRAS, 486, 769

Kim T.-S., Viel M., Haehnelt M. G., Carswell R. F., Cristiani S., 2004, MNRAS, 347, 355

Kramida A., Ralchenko Yu., Reader J., ASD NIST, 2021, NIST Atomic Spec- tra Database (ver. 5.9), National Institute of Standards and Technology. Available at: https://physics.nist.gov/asd

Lee K.-G. et al., 2013, AJ, 145, 69

Lesgourgues J., Pastor S., 2006, Phys. Rep., 429, 307

Lesgourgues J., Pastor S., 2012, Adv. High Energy Phys., 2012, 1

Levi M. et al., 2013, The DESI Experiment, a whitepaper for Snowmass 2013. Available at: https://arxiv.org/abs/1308.0847

Lopez S. et al., 2016, A&A, 594, A91

Lukić Z., Stark C., Nugent P., White M., Meiksin A., Almgren A., 2015, MNRAS, 446, 3697

Lynds R., 1971, ApJ, 164, L73

McDonald P. et al., 2006, ApJS, 163, 80

McDonald P., Miralda-Escude J., Rauch M., Sargent W. L. W., Barlow T. A., Cen R., Ostriker J. P., 2000, ApJ, 543, 1

McDonald P., Seljak U., Cen R., Bode P., Ostriker J. P., 2005, MNRAS, 360, 1471

McQuinn M., 2016, Annu. Rev. Astron. Astrophys., 54, 313

Meiksin A. A., 2009, Rev. Mod. Phys., 81, 1405

Miller T. N. et al., 2023, The Optical Corrector for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2306.06310

Murphy M. T., Kacprzak G. G., Savorgnan G. A. D., Carswell R. F., 2018, MNRAS, 482, 3458

Myers A. D. et al., 2022, The Target Selection Pipeline for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2208.08518

O'Meara J. M. et al., 2015, AJ, 150, 111

O'Meara J. M., Lehner N., Howk J. C., Prochaska J. X., Fox A. J., Peeples M. S., Tumlinson J., O'Shea B. W., 2017, AJ, 154, 114

Palanque-Delabrouille N. et al., 2013, A&A, 559, A85

Palanque-Delabrouille N. et al., 2015, J. Cosmol. Astropart. Phys., 2015, 011

Palanque-Delabrouille N., Yèche C., Schöneberg N., Lesgourgues J., Walther M., Chabanier S., Armengaud E., 2020, J. Cosmol. Astropart. Phys., 2020, 038

Parks D., Prochaska J. X., Dong S., Cai Z., 2017, MNRAS, 476, 1151

Pedersen C., Font-Ribera A., Gnedin N. Y., 2023, ApJ, 944, 223

Pedersen C., Font-Ribera A., Kitching T. D., McDonald P., Bird S., Slosar A., Rogers K. K., Pontzen A., 2020a, JCAP, 2020, 025

Pedersen C., Font-Ribera A., Rogers K. K., McDonald P., Peiris H. V., Pontzen A., Slosar A., 2020b, JCAP, 2021, 033

Pieri M. M. et al., 2014, MNRAS, 441, 1718

Pontzen A. et al., 2008, MNRAS

Puchwein E. et al., 2023, MNRAS, 519, 6162

Ramírez-Pérez C. et al., 2023, The Lyman-alpha forest catalog from the Dark Energy Spectroscopic Instrument Early Data Release. Available at: http://arxiv.org/abs/2306.06312

Ravoux C., 2023, The Dark Energy Spectroscopic Instrument: One- Dimensional Power Spectrum from First Lyman-alpha Forest Samples with Fast Fourier Transform. preprint (arXiv:2306.06311)

Rogers K. K., Bird S., Peiris H. V., Pontzen A., Font-Ribera A., Leistedt B., 2017, MNRAS, 474, 3032

Schlafly E. F. et al., 2023, Survey Operations for the Dark Energy Spectro- scopic Instrument. Available at: http://arxiv.org/abs/2306.06309

Seljak U., Slosar A., McDonald P., 2006, J. Cosmol. Astropart. Phys., 2006, 014

Silber J. H. et al., 2022, The Robotic Multi-Object Focal Plane System of the Dark Energy Spectroscopic Instrument (DESI). Available at: http: //arxiv.org/abs/2205.09014

Smee S. A. et al., 2013, AJ, 146, 32

Valluri M. et al., 2022, Snowmass2021 Cosmic Frontier White Paper: Prospects for obtaining Dark Matter Constraints with DESI. Available at: https://arxiv.org/abs/2203.07491

Viel M., Becker G. D., Bolton J. S., Haehnelt M. G., 2013, Phys. Rev. D, 88, 043502

Viel M., Becker G. D., Bolton J. S., Haehnelt M. G., Rauch M., Sargent W. L. W., 2008, Phys. Rev. Lett., 100, 041304

Viel M., Lesgourgues J., Haehnelt M. G., Matarrese S., Riotto A., 2005, Phys. Rev. D, 71, 063534

Walther M., Armengaud E., Ravoux C., Palanque-Delabrouille N., Yèche C., Lukić Z., 2021, J. Cosmol. Astropart. Phys., 2021, 059

Walther M., Hennawi J. F., Hiss H., Oñorbe J., Lee K.-G., Rorai A., O'Meara J., 2018, ApJ, 852, 22

Yang L., Zheng Z., Bourboux H. d. M. d., Dawson K., Pieri M. M., Rossi G., Schneider D. P., de la Macorra A., 2022, ApJ, 935, 121

Yèche C. et al., 2020, Res. Notes Am. Astron. Soc., 4, 179

Yèche C., Palanque-Delabrouille N., Baur J., BourBoux H. d. M. d., 2017, J. Cosmol. Astropart. Phys., 2017, 047

Zou H. et al., 2017, PASP, 129, 064101

APPENDIX A: DESI DATA SET COMPARISON

SV1, SV3, and M2 are very different data sets: they have different target selections and exposure times and were collected for different states of the DESI instrument. From a P<sub>ID,α</sub> point of view, the noise properties of SV1 are a potential issue. While our initial goal was to analyse the full data sample available, we choose first to compare
Brodzeller A. et al., 2023, Performance of the Quasar Spectral Templates for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/ abs/2305.10426

Busca N., Balland C., 2018, QuasarNET: Human-level spectral classification and redshifting with Deep Neural Networks. Available at: http://arxiv.org/ abs/1808.09955

Chabanier S. et al., 2019, J. Cosmol. Astropart. Phys., 2019, 017

Chabanier S. et al., 2021, ApJS, 258, 18

Chabanier S. et al., 2022, Modeling the Lyman-α forest with Eulerian and SPH hydrodynamical methods. Available at: http://arxiv.org/abs/2207.05023

Chaussidon E. et al., 2022, Target Selection and Validation of DESI Quasars. Available at: http://arxiv.org/abs/2208.08511

Croft R. A. C., Weinberg D. H., Bolte M., Burles S., Hernquist L., Katz N., Kirkman D., Tytler D., 2002, ApJ, 581, 20

Croft R. A. C., Weinberg D. H., Katz N., Hernquist L., 1998, ApJ, 495, 44

Dawson K. S. et al., 2016, AJ, 151, 44

Day A., Tytler D., Kambalur B., 2019, MNRAS, 489, 2536

DESI, 2016a, The DESI Experiment Part I: Science,Targeting, and Survey Design. Available at: http://arxiv.org/abs/1611.00036

DESI, 2016b, The DESI Experiment Part II: Instrument Design. Available at: http://arxiv.org/abs/1611.00037

DESI, 2023a, Validation of the Scientific Program for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2306.06307

DESI, 2023b, The Early Data Release of the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2306.06308

Dey A. et al., 2019, ApJ, 157, 168

du Mas des Bourboux H. et al., 2020, ApJ, 901, 153

du Mas des Bourboux H. et al., 2021, Astrophysics Source Code Library, record ascl:2106.018

Farr J., Font-Ribera A., Pontzen A., 2020, JCAP, 2020, 015

Gaikwad P., Srikanand R., Haehnelt M. G., Choudhury T. R., 2021, MNRAS, 506, 4389

Gunn J. E. et al., 2006, AJ, 131, 2332

Gunn J. E., Peterson B. A., 1965, ApJ, 142, 1633

Guy J. et al., 2022, The Spectroscopic Data Processing Pipeline for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2209 .14482

Ho M.-F., Bird S., Garnett R., 2021, MNRAS, 507, 704

Iršič V. et al., 2016, MNRAS, 466, 4332

Iršič V., Viel M., Haehnelt M. G., Bolton J. S., Becker G. D., 2017, Phys. Rev. Lett., 119, 031302

Karaçaylı N. G. et al., 2022, MNRAS, 509, 2842

Karaçaylı N. G. et al., 2023a, Optimal 1D Ly α Forest Power Spectrum Estimation – III. DESI Early Data. Available at: http://arxiv.org/abs/2306 .06316

Karaçaylı N. G. et al., 2023b, MNRAS, 522, 5980

Karaçaylı N. G., Font-Ribera A., Padmanabhan N., 2020, MNRAS, 497, 4742

Khaire V. et al., 2019, MNRAS, 486, 769

Kim T.-S., Viel M., Haehnelt M. G., Carswell R. F., Cristiani S., 2004, MNRAS, 347, 355

Kramida A., Ralchenko Yu., Reader J., ASD NIST, 2021, NIST Atomic Spec- tra Database (ver. 5.9), National Institute of Standards and Technology. Available at: https://physics.nist.gov/asd

Lee K.-G. et al., 2013, AJ, 145, 69

Lesgourgues J., Pastor S., 2006, Phys. Rep., 429, 307

Lesgourgues J., Pastor S., 2012, Adv. High Energy Phys., 2012, 1

Levi M. et al., 2013, The DESI Experiment, a whitepaper for Snowmass 2013. Available at: https://arxiv.org/abs/1308.0847

Lopez S. et al., 2016, A&A, 594, A91

Lukić Z., Stark C., Nugent P., White M., Meiksin A., Almgren A., 2015, MNRAS, 446, 3697

Lynds R., 1971, ApJ, 164, L73

McDonald P. et al., 2006, ApJS, 163, 80

McDonald P., Miralda-Escude J., Rauch M., Sargent W. L. W., Barlow T. A., Cen R., Ostriker J. P., 2000, ApJ, 543, 1

McDonald P., Seljak U., Cen R., Bode P., Ostriker J. P., 2005, MNRAS, 360, 1471

McQuinn M., 2016, Annu. Rev. Astron. Astrophys., 54, 313

Meiksin A. A., 2009, Rev. Mod. Phys., 81, 1405

Miller T. N. et al., 2023, The Optical Corrector for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2306.06310

Murphy M. T., Kacprzak G. G., Savorgnan G. A. D., Carswell R. F., 2018, MNRAS, 482, 3458

Myers A. D. et al., 2022, The Target Selection Pipeline for the Dark Energy Spectroscopic Instrument. Available at: http://arxiv.org/abs/2208.08518

O'Meara J. M. et al., 2015, AJ, 150, 111

O'Meara J. M., Lehner N., Howk J. C., Prochaska J. X., Fox A. J., Peeples M. S., Tumlinson J., O'Shea B. W., 2017, AJ, 154, 114

Palanque-Delabrouille N. et al., 2013, A&A, 559, A85

Palanque-Delabrouille N. et al., 2015, J. Cosmol. Astropart. Phys., 2015, 011

Palanque-Delabrouille N., Yèche C., Schöneberg N., Lesgourgues J., Walther M., Chabanier S., Armengaud E., 2020, J. Cosmol. Astropart. Phys., 2020, 038

Parks D., Prochaska J. X., Dong S., Cai Z., 2017, MNRAS, 476, 1151

Pedersen C., Font-Ribera A., Gnedin N. Y., 2023, ApJ, 944, 223

Pedersen C., Font-Ribera A., Kitching T. D., McDonald P., Bird S., Slosar A., Rogers K. K., Pontzen A., 2020a, JCAP, 2020, 025

Pedersen C., Font-Ribera A., Rogers K. K., McDonald P., Peiris H. V., Pontzen A., Slosar A., 2020b, JCAP, 2021, 033

Pieri M. M. et al., 2014, MNRAS, 441, 1718

Pontzen A. et al., 2008, MNRAS

Puchwein E. et al., 2023, MNRAS, 519, 6162

Ramírez-Pérez C. et al., 2023, The Lyman-alpha forest catalog from the Dark Energy Spectroscopic Instrument Early Data Release. Available at: http://arxiv.org/abs/2306.06312

Ravoux C., 2023, The Dark Energy Spectroscopic Instrument: One- Dimensional Power Spectrum from First Lyman-alpha Forest Samples with Fast Fourier Transform. preprint (arXiv:2306.06311)

Rogers K. K., Bird S., Peiris H. V., Pontzen A., Font-Ribera A., Leistedt B., 2017, MNRAS, 474, 3032

Schlafly E. F. et al., 2023, Survey Operations for the Dark Energy Spectro- scopic Instrument. Available at: http://arxiv.org/abs/2306.06309

Seljak U., Slosar A., McDonald P., 2006, J. Cosmol. Astropart. Phys., 2006, 014

Silber J. H. et al., 2022, The Robotic Multi-Object Focal Plane System of the Dark Energy Spectroscopic Instrument (DESI). Available at: http: //arxiv.org/abs/2205.09014

Smee S. A. et al., 2013, AJ, 146, 32

Valluri M. et al., 2022, Snowmass2021 Cosmic Frontier White Paper: Prospects for obtaining Dark Matter Constraints with DESI. Available at: https://arxiv.org/abs/2203.07491

Viel M., Becker G. D., Bolton J. S., Haehnelt M. G., 2013, Phys. Rev. D, 88, 043502

Viel M., Becker G. D., Bolton J. S., Haehnelt M. G., Rauch M., Sargent W. L. W., 2008, Phys. Rev. Lett., 100, 041304

Viel M., Lesgourgues J., Haehnelt M. G., Matarrese S., Riotto A., 2005, Phys. Rev. D, 71, 063534

Walther M., Armengaud E., Ravoux C., Palanque-Delabrouille N., Yèche C., Lukić Z., 2021, J. Cosmol. Astropart. Phys., 2021, 059

Walther M., Hennawi J. F., Hiss H., Oñorbe J., Lee K.-G., Rorai A., O'Meara J., 2018, ApJ, 852, 22

Yang L., Zheng Z., Bourboux H. d. M. d., Dawson K., Pieri M. M., Rossi G., Schneider D. P., de la Macorra A., 2022, ApJ, 935, 121

Yèche C. et al., 2020, Res. Notes Am. Astron. Soc., 4, 179

Yèche C., Palanque-Delabrouille N., Baur J., BourBoux H. d. M. d., 2017, J. Cosmol. Astropart. Phys., 2017, 047

Zou H. et al., 2017, PASP, 129, 064101

APPENDIX A: DESI DATA SET COMPARISON

SV1, SV3, and M2 are very different data sets: they have different target selections and exposure times and were collected for different states of the DESI instrument. From a P<sub>ID,α</sub> point of view, the noise properties of SV1 are a potential issue. While our initial goal was to analyse the full data sample available, we choose first to compare
<figure><img src="image_21.png" /><figcaption>**Figure A1.** Ratio of the $P_{1D,α}$, measured with the same parameters, between SV1 and SV3 (top), and M2 and SV3 (bottom), for four redshift bins.</figcaption></figure>

the measured $P_{1D,α}$ on the separate SV1, SV3, and M2 data sets. Fig. A1 shows their respective ratios, on the four redshift bins with largest statistics. It appears that the measurement of $P_{1D,α}$ on SV1 is biased compared to the other two data sets. In particular, we believe that the difference at $k \gtrsim 1.0 \AA^{-1}$ is due to an imperfection in the noise correction presented in Section 4.2. Consequently, we decide to remove the SV1 data set in this study to remain conservative.

## APPENDIX B: SNR WEIGHTING

For this $P_{1D,α}$ measurement, we keep all Ly α subforests available in the measurement sample, independently of their $\overline{SNR}$, unlike what was done in eBOSS (Chabanier et al. 2019), reminding that only one $\overline{SNR}$ cut is applied at earlier stages of the analysis, during the continuum fitting procedure as described in Section 3.1.

Individual Ly α power spectra, falling into the same wavenumber bin, do not have the same dispersion which varies as function of the $\overline{SNR}$. In our analysis, we account for this effect at the last step of our FFT estimator pipeline, by weighting each of the Ly α subforests by a $\overline{SNR}$ dependent factor, while averaging over the full measurement sample.

First, for each Ly α subforest (i), the variance of individual $P_{1D,α,i}$ is fitted according to the following:

$$ \sigma^2(P_{1D,\alpha,i}) = \frac{a}{(\overline{SNR}_i - 1)^2} + b, \qquad (B1) $$

where $\overline{SNR}_i$ is the mean signal-to-noise ratio of the Ly α sub-forest (i), defined in equation (6).

Hence, the weighting factor is

$$ W_i = \frac{1}{\sigma^2(P_{1D,\alpha,i})}. \qquad (B2) $$

The employed fitting model works well empirically with the measured $\sigma^2$ ($P_{1D,α,i}$). Also according to equation (B1), $W_i$ tends to 0 as $\overline{SNR}_i$ tends to 1, consistently with the applied $\overline{SNR}$ cut = 1.

Tests for this $\overline{SNR}$ weighting method were done on DESI-LITE mocks that mimic the SV1 + SV3 data set described in Section 2, and are specifically designed for $P_{1D,α}$ measurement. A comparison

<figure><img src="image_22.png" /><figcaption>**Figure B1.** Comparison between measured $P_{1D,α}$ and DESI-LITE mocks truth power spectrum, for the redshift bin $z = 3.8$, for both $P_{1D,α}$ measured with eBOSS $\overline{SNR}$ cut method, and the $\overline{SNR}$ weighting method.</figcaption></figure>

between measured $P_{1D,α}$ and mocks truth power spectrum is represented in Fig. B1, for both $P_{1D,α}$ measured with eBOSS $\overline{SNR}$ cut method, and our $\overline{SNR}$ 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 Section 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:

(i) **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 modelling the CCD.

(ii) **Overscan:** The overscan measures the bulk offset, i.e. the average level of all CCD pixels. It is used to remove small variations in the bias. Overscan suppression introduces noise.

(iii) **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.

(iv) **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 $\sigma_{pip}$ is then obtained.
<figure><img src="image_22.png" /><figcaption>**Figure A1.** Ratio of the $P_{1D,α}$, measured with the same parameters, between SV1 and SV3 (top), and M2 and SV3 (bottom), for four redshift bins.</figcaption></figure>

the measured $P_{1D,α}$ on the separate SV1, SV3, and M2 data sets. Fig. A1 shows their respective ratios, on the four redshift bins with largest statistics. It appears that the measurement of $P_{1D,α}$ on SV1 is biased compared to the other two data sets. In particular, we believe that the difference at $k \gtrsim 1.0 \text{ Å}^{-1}$ is due to an imperfection in the noise correction presented in Section 4.2. Consequently, we decide to remove the SV1 data set in this study to remain conservative.

## APPENDIX B: SNR WEIGHTING

For this $P_{1D,α}$ measurement, we keep all Ly α subforests available in the measurement sample, independently of their $\overline{SNR}$, unlike what was done in eBOSS (Chabanier et al. 2019), reminding that only one $\overline{SNR}$ cut is applied at earlier stages of the analysis, during the continuum fitting procedure as described in Section 3.1.

Individual Ly α power spectra, falling into the same wavenumber bin, do not have the same dispersion which varies as function of the $\overline{SNR}$. In our analysis, we account for this effect at the last step of our FFT estimator pipeline, by weighting each of the Ly α subforests by a $\overline{SNR}$ dependent factor, while averaging over the full measurement sample.

First, for each Ly α subforest (i), the variance of individual $P_{1D,α,i}$ is fitted according to the following:

$$ \sigma^2(P_{1D,\alpha,i}) = \frac{a}{(\overline{SNR}_i - 1)^2} + b, \quad (B1) $$

where $\overline{SNR}_i$ is the mean signal-to-noise ratio of the Ly α sub-forest (i), defined in equation (6).

Hence, the weighting factor is

$$ W_i = \frac{1}{\sigma^2(P_{1D,\alpha,i})}. \quad (B2) $$

The employed fitting model works well empirically with the measured $\sigma^2(P_{1D,α,i})$. Also according to equation (B1), $W_i$ tends to 0 as $\overline{SNR}_i$ tends to 1, consistently with the applied $\overline{SNR}$ cut = 1.

Tests for this $\overline{SNR}$ weighting method were done on DESI-LITE mocks that mimic the SV1 + SV3 data set described in Section 2, and are specifically designed for $P_{1D,α}$ measurement. A comparison

<figure><img src="image_23.png" /><figcaption>**Figure B1.** Comparison between measured $P_{1D,α}$ and DESI-LITE mocks truth power spectrum, for the redshift bin $z = 3.8$, for both $P_{1D,α}$ measured with eBOSS $\overline{SNR}$ cut method, and the $\overline{SNR}$ weighting method.</figcaption></figure>

between measured $P_{1D,α}$ and mocks truth power spectrum is represented in Fig. B1, for both $P_{1D,α}$ measured with eBOSS $\overline{SNR}$ cut method, and our $\overline{SNR}$ 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 Section 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:

(i) **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 modelling the CCD.

(ii) **Overscan:** The overscan measures the bulk offset, i.e. the average level of all CCD pixels. It is used to remove small variations in the bias. Overscan suppression introduces noise.

(iii) **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.

(iv) **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 $\sigma_{pip}$ is then obtained.
The noise power spectrum estimated from pipeline ($P_{pipeline}$) is computed from the standard deviation $\sigma_{\delta_F}$ linked to $\sigma_{pip}$ by equation (5). For each unmasked spectrum pixel, a contrast $\delta_{pipeline}$ is generated following a normal probability distribution such that:

$$ \delta_{pipeline}(\lambda) \leftarrow \mathcal{N}(0, \sigma_{\delta_F}(\lambda)). \quad (C1) $$

This procedure is repeated $N_G$ times ($N_G = 2500$) to obtain a converged noise power spectrum. For each quasar, the associated noise power spectrum is the average of the $N_G$ noise contrasts after Fourier transformation:

$$ P_{pipeline}(k) = \left\langle \left| \delta_{pipeline} \right|^2 \right\rangle_{N_G}. \quad (C2) $$

## 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 j by separating half of its exposures in the even category ($N_{even}$ exposures) and the other half in the odd category ($N_{odd}$ exposures) such that

$$ \Delta f_j = \frac{1}{2} \left( \frac{\sum_{k=1}^{N_{even}} (\sigma_{pip,k})^{-2} f_k}{\sum_{k=1}^{N_{even}} (\sigma_{pip,k})^{-2}} - \frac{\sum_{k=1}^{N_{odd}} (\sigma_{pip,k})^{-2} f_k}{\sum_{k=1}^{N_{odd}} (\sigma_{pip,k})^{-2}} \right), \quad (C3) $$

where $\sigma_{pip,k}^2$ is the pipeline noise of the exposure k for quasar j. In the case where the total number of exposures is even, $N_{even} = N_{odd}$. The standard deviation of $\Delta f_j$ can be calculated from the variances of individual exposures:

$$ \sigma_{\Delta f_j} = \frac{1}{2} \sqrt{ \frac{1}{\sum_{k=1}^{N_{even}} (\sigma_{pip,k})^{-2}} + \frac{1}{\sum_{k=1}^{N_{odd}} (\sigma_{pip,k})^{-2}} } \quad (C4) $$

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 $P_{noise}$, the variance of $\Delta f_j$ must be equal to that of the coadded flux defined by

$$ f_j = \frac{\sum_k (\sigma_{pip,k})^{-2} f_k}{\sum_k (\sigma_{pip,k})^{-2}}, \quad (C5) $$

To obtain the same variance, we multiply $\Delta f_j$ by the ratio $\sigma_{f_j}/\sigma_{\Delta f_j}$:

$$ \Delta f_j^{\text{corr}} = 2 \frac{\frac{1}{\sqrt{\sum_{k=1}^{N_{\text{tot}}} (\sigma_{\text{pip},k})^{-2}}}}{\sqrt{\frac{1}{\sum_{k=1}^{N_{\text{even}}} (\sigma_{\text{pip},k})^{-2}} + \frac{1}{\sum_{k=1}^{N_{\text{odd}}} (\sigma_{\text{pip},k})^{-2}}}} \Delta f_j, \quad (C6) $$

where $N_{tot}$ is the total number of exposure for the quasar j ($N_{tot} = N_{even} + N_{odd}$).

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, $\sigma_{\Delta f_j}$ 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) and Chabanier et al. (2019). Our new estimator corrects the variance for any exposure time. It is

<figure><img src="image_23.png" /><figcaption>Figure C1. Asymptotic ratios $\beta$ 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 $\beta$. Stars represent $\beta$ as derived from the asymptotic differences in Fig. 8. Second-order polynomial fits are shown only for representation.</figcaption></figure>

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 (Bourboux et al. 2021). We obtain an estimator of $P_{noise}$ called difference power spectrum and noted $P_{diff}$, such that

$$ P_{\text{diff}}(k) = \left| \mathcal{F} \left[ \frac{\Delta f_j^{\text{corr}}(\lambda)}{\bar{F}(\lambda)C_q(\lambda)} - 1 \right] \right|^2. \quad (C7) $$

## C3 Additional considerations on noise estimation

Fig. C1 shows the asymptotic ratios $\beta$ for the three data sets. We remark that for SV3 and M2, the SNRdependence of $\beta$ is much more pronounced than that of $\alpha$. 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 $\beta$ values derived from the $\alpha$ of Fig. 8, using the mean value of $P_{noise}$ for all redshift. They are shown as stars in Fig. C1, and exhibit similar trends to the direct $\beta$ computation, which corroborates that the missing noise is additive. Consequently, we decide to correct $P_{pipeline}$ using an additive term $\alpha$ ($P_{noise} = P_{pipeline} + \alpha$).

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 $\alpha$ band. The $\alpha$ 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 $\alpha$ band. This is likely due to the use of different quasar populations employed for side-band and Ly $\alpha$ measurements. Indeed, the DESI observation strategy is different for low redshift quasars (used for side-band study) and Ly $\alpha$ quasars. On average, the number of exposures is larger for Ly $\alpha$ than low-redshift quasars. Consequently, and in accordance with our previous interpretations, the misestimation of noise is larger for Ly $\alpha$ 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.
The noise power spectrum estimated from pipeline ($P_{pipeline}$) is computed from the standard deviation $\sigma_{\delta_F}$ linked to $\sigma_{pip}$ by equation (5). For each unmasked spectrum pixel, a contrast $\delta_{pipeline}$ is generated following a normal probability distribution such that:

$$ \delta_{pipeline}(\lambda) \leftarrow \mathcal{N}(0, \sigma_{\delta_F}(\lambda)). \quad (C1) $$

This procedure is repeated $N_G$ times ($N_G = 2500$) to obtain a converged noise power spectrum. For each quasar, the associated noise power spectrum is the average of the $N_G$ noise contrasts after Fourier transformation:

$$ P_{pipeline}(k) = \left\langle |\delta_{pipeline}|^2 \right\rangle_{N_G}. \quad (C2) $$

## 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 j by separating half of its exposures in the even category ($N_{even}$ exposures) and the other half in the odd category ($N_{odd}$ exposures) such that

$$ \Delta f_j = \frac{1}{2} \left( \frac{\sum_{k=1}^{N_{even}} (\sigma_{pip,k})^{-2} f_k}{\sum_{k=1}^{N_{even}} (\sigma_{pip,k})^{-2}} - \frac{\sum_{k=1}^{N_{odd}} (\sigma_{pip,k})^{-2} f_k}{\sum_{k=1}^{N_{odd}} (\sigma_{pip,k})^{-2}} \right), \quad (C3) $$

where $\sigma_{pip,k}^2$ is the pipeline noise of the exposure k for quasar j. In the case where the total number of exposures is even, $N_{even} = N_{odd}$. The standard deviation of $\Delta f_j$ can be calculated from the variances of individual exposures:

$$ \sigma_{\Delta f_j} = \frac{1}{2} \sqrt{ \frac{1}{\sum_{k=1}^{N_{even}} (\sigma_{pip,k})^{-2}} + \frac{1}{\sum_{k=1}^{N_{odd}} (\sigma_{pip,k})^{-2}} } \quad (C4) $$

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 $P_{noise}$, the variance of $\Delta f_j$ must be equal to that of the coadded flux defined by

$$ f_j = \frac{\sum_k (\sigma_{pip,k})^{-2} f_k}{\sum_k (\sigma_{pip,k})^{-2}}, \quad (C5) $$

To obtain the same variance, we multiply $\Delta f_j$ by the ratio $\sigma_{f_j}/\sigma_{\Delta f_j}$:

$$ \Delta f_j^{\text{corr}} = 2 \frac{\frac{1}{\sqrt{\sum_{k=1}^{N_{\text{tot}}} (\sigma_{\text{pip},k})^{-2}}}}{\sqrt{\frac{1}{\sum_{k=1}^{N_{\text{even}}} (\sigma_{\text{pip},k})^{-2}} + \frac{1}{\sum_{k=1}^{N_{\text{odd}}} (\sigma_{\text{pip},k})^{-2}}}} \Delta f_j, \quad (C6) $$

where $N_{tot}$ is the total number of exposure for the quasar j ($N_{tot} = N_{even} + N_{odd}$).

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, $\sigma_{\Delta f_j}$ 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) and Chabanier et al. (2019). Our new estimator corrects the variance for any exposure time. It is

<figure><img src="image_24.png" /><figcaption>Figure C1. Asymptotic ratios $\beta$ 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 $\beta$. Stars represent $\beta$ as derived from the asymptotic differences in Fig. 8. Second-order polynomial fits are shown only for representation.</figcaption></figure>

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 (Bourboux et al. 2021). We obtain an estimator of $P_{noise}$ called difference power spectrum and noted $P_{diff}$, such that

$$ P_{\text{diff}}(k) = \left| \mathcal{F} \left[ \frac{\Delta f_j^{\text{corr}}(\lambda)}{\bar{F}(\lambda)C_q(\lambda)} - 1 \right] \right|^2. \quad (C7) $$

## C3 Additional considerations on noise estimation

Fig. C1 shows the asymptotic ratios $\beta$ for the three data sets. We remark that for SV3 and M2, the SNRdependence of $\beta$ is much more pronounced than that of $\alpha$. 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 $\beta$ values derived from the $\alpha$ of Fig. 8, using the mean value of $P_{noise}$ for all redshift. They are shown as stars in Fig. C1, and exhibit similar trends to the direct $\beta$ computation, which corroborates that the missing noise is additive. Consequently, we decide to correct $P_{pipeline}$ using an additive term $\alpha$ ($P_{noise} = P_{pipeline} + \alpha$).

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 $\alpha$ band. The $\alpha$ 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 $\alpha$ band. This is likely due to the use of different quasar populations employed for side-band and Ly $\alpha$ measurements. Indeed, the DESI observation strategy is different for low redshift quasars (used for side-band study) and Ly $\alpha$ quasars. On average, the number of exposures is larger for Ly $\alpha$ than low-redshift quasars. Consequently, and in accordance with our previous interpretations, the misestimation of noise is larger for Ly $\alpha$ 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.
<figure><img src="image_24.png" /><figcaption>**Figure C2.** Asymptotic differences $\alpha$ for SV1(blue), SV3(yellow), and M2(green) data sets in the side-band regions SB1 (crosses) and SB2 (triangles), for the pipeline noise. The continuous lines are fits of the $\alpha$ values, whose parameters are given in Table 3.</figcaption></figure>

## APPENDIX D: COMPARISON WITH EBOSS MEASUREMENT ON LARGE SCALES

We performed a series of tests to investigate the discrepancy at large scales ($k < 0.01$ km s⁻<fnref n="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 Section 3.1, Fourier transform and averaging in Section 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 Section 3.1. For eBOSS, this pipeline step was performed separately on subforests instead of the total Ly $\alpha$ forest. We checked that performing our continuum fitting on subforest does not modify the large-scales level of $P_{1D,\alpha}$. 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 $P_{1D,\alpha}$ at large scales. We also applied non-constant weights in equation (4), removed the forcing to zero of the Ly $\alpha$ 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 per cent discrepancy visible on the eBOSS comparison. Furthermore, the corrections we derived from the mocks in Section 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 per cent) 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 catalogue, we created a common set of quasars and DLAs by merging eBOSS and DESI catalogues. 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 catalogue used by varying confidence levels, or column density $N_{HI}$ to include sub-DLAs. Masking and correcting the damping wings for $\log(N_{HI}) < 20.3$ systems decreases the discrepancy only for the very first wavenumber bins ($k < 0.004$ km s⁻<fnref n="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.

<footnote n="1">¹Aix Marseille Univ, CNRS/IN2P3, CPPM, F-13288 Marseille, France</footnote>

<footnote n="2">²IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France</footnote>

<footnote n="3">³Excellence Cluster ORIGINS, Boltzmannstrasse 2, D-85748 Garching, Germany</footnote>

<footnote n="4">⁴University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, D-81677 München, Germany</footnote>

<footnote n="5">⁵Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA</footnote>

<footnote n="6">⁶Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA</footnote>

<footnote n="7">⁷Department of Astronomy, The Ohio State University, 4055 McPherson Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA</footnote>

<footnote n="8">⁸The Ohio State University, Columbus, 43210 OH, USA</footnote>

<footnote n="9">⁹Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA</footnote>

<footnote n="10">¹⁰Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA</footnote>

<footnote n="11">¹¹Departamento de Física, Universidad de Guanajuato – DCI, C.P. 37150, Leon, Guanajuato, México</footnote>

<footnote n="12">¹²Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, UK</footnote>

<footnote n="13">¹³Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, E-08193 Bellaterra (Barcelona), Spain</footnote>

<footnote n="14">¹⁴Department of Physics and Astronomy, The University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA</footnote>

<footnote n="15">¹⁵Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México</footnote>

<footnote n="16">¹⁶Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio Ip, CP 111711, Bogotá, Colombia</footnote>

<footnote n="17">¹⁷Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio H, CP 111711 Bogotá, Colombia</footnote>

<footnote n="18">¹⁸Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582. Colonia Crédito Constructor, Del. Benito Juárez C.P. 03940, México D.F. México</footnote>

<footnote n="19">¹⁹Kavli Institute for Cosmology, Department of Physics, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="20">²⁰Department of Physics, Southern Methodist University, 3215 Daniel Ave., Dallas, TX 75205, USA</footnote>

<footnote n="21">²¹Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), F-75005 Paris, France</footnote>

<footnote n="22">²²NSF's NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA</footnote>

<footnote n="23">²³Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companies, 23, E-08010 Barcelona, Spain</footnote>

<footnote n="24">²⁴Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA</footnote>

<footnote n="25">²⁵Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK</footnote>

<footnote n="26">²⁶Department of Physics & Astronomy, University of Wyoming, 1000 E. University, Dept. 3905, Laramie, WY 82071, USA</footnote>
<figure><img src="image_25.png" /><figcaption>**Figure C2.** Asymptotic differences $\alpha$ for SV1(blue), SV3(yellow), and M2(green) data sets in the side-band regions SB1 (crosses) and SB2 (triangles), for the pipeline noise. The continuous lines are fits of the $\alpha$ values, whose parameters are given in Table 3.</figcaption></figure>

## APPENDIX D: COMPARISON WITH EBOSS MEASUREMENT ON LARGE SCALES

We performed a series of tests to investigate the discrepancy at large scales ($k < 0.01$ km s⁻<fnref n="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 Section 3.1, Fourier transform and averaging in Section 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 Section 3.1. For eBOSS, this pipeline step was performed separately on subforests instead of the total Ly $\alpha$ forest. We checked that performing our continuum fitting on subforest does not modify the large-scales level of $P_{1D,\alpha}$. 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 $P_{1D,\alpha}$ at large scales. We also applied non-constant weights in equation (4), removed the forcing to zero of the Ly $\alpha$ 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 per cent discrepancy visible on the eBOSS comparison. Furthermore, the corrections we derived from the mocks in Section 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 per cent) 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 catalogue, we created a common set of quasars and DLAs by merging eBOSS and DESI catalogues. 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 catalogue used by varying confidence levels, or column density $N_{HI}$ to include sub-DLAs. Masking and correcting the damping wings for $\log(N_{HI}) < 20.3$ systems decreases the discrepancy only for the very first wavenumber bins ($k < 0.004$ km s⁻<fnref n="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.

<footnote n="1">¹Aix Marseille Univ, CNRS/IN2P3, CPPM, F-13288 Marseille, France</footnote>

<footnote n="2">²IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France</footnote>

<footnote n="3">³Excellence Cluster ORIGINS, Boltzmannstrasse 2, D-85748 Garching, Germany</footnote>

<footnote n="4">⁴University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, D-81677 München, Germany</footnote>

<footnote n="5">⁵Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA</footnote>

<footnote n="6">⁶Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA</footnote>

<footnote n="7">⁷Department of Astronomy, The Ohio State University, 4055 McPherson Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA</footnote>

<footnote n="8">⁸The Ohio State University, Columbus, 43210 OH, USA</footnote>

<footnote n="9">⁹Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA</footnote>

<footnote n="10">¹⁰Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA</footnote>

<footnote n="11">¹¹Departamento de Física, Universidad de Guanajuato – DCI, C.P. 37150, Leon, Guanajuato, México</footnote>

<footnote n="12">¹²Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, UK</footnote>

<footnote n="13">¹³Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, E-08193 Bellaterra (Barcelona), Spain</footnote>

<footnote n="14">¹⁴Department of Physics and Astronomy, The University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA</footnote>

<footnote n="15">¹⁵Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México</footnote>

<footnote n="16">¹⁶Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio Ip, CP 111711, Bogotá, Colombia</footnote>

<footnote n="17">¹⁷Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio H, CP 111711 Bogotá, Colombia</footnote>

<footnote n="18">¹⁸Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582. Colonia Crédito Constructor, Del. Benito Juárez C.P. 03940, México D.F. México</footnote>

<footnote n="19">¹⁹Kavli Institute for Cosmology, Department of Physics, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK</footnote>

<footnote n="20">²⁰Department of Physics, Southern Methodist University, 3215 Daniel Ave., Dallas, TX 75205, USA</footnote>

<footnote n="21">²¹Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), F-75005 Paris, France</footnote>

<footnote n="22">²²NSF's NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA</footnote>

<footnote n="23">²³Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companies, 23, E-08010 Barcelona, Spain</footnote>

<footnote n="24">²⁴Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA</footnote>

<footnote n="25">²⁵Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK</footnote>

<footnote n="26">²⁶Department of Physics & Astronomy, University of Wyoming, 1000 E. University, Dept. 3905, Laramie, WY 82071, USA</footnote>
This paper has been typeset from a TeX/Latex file prepared by the author.

<footnote n="27"><sup>27</sup>National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd., Chaoyang District, Beijing 100012, P.R. China</footnote>

<footnote n="28"><sup>28</sup>Instituto Avanzado de Cosmología A. C. San Marcos 11 – Atenas 202. Magdalena Contreras, 10720. Ciudad de México, México</footnote>

<footnote n="29"><sup>29</sup>Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada</footnote>

<footnote n="30"><sup>30</sup>Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada</footnote>

<footnote n="31"><sup>31</sup>Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada</footnote>

<footnote n="32"><sup>32</sup>Aix Marseille Univ, CNRS, CNES, LAM, F-13013 Marseille, France</footnote>

<footnote n="33"><sup>33</sup>Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA</footnote>

<footnote n="34"><sup>34</sup>University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA</footnote>

<footnote n="35"><sup>35</sup>Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain</footnote>

<footnote n="36"><sup>36</sup>Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea</footnote>

<footnote n="37"><sup>37</sup>CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain</footnote>

<footnote n="38"><sup>38</sup>Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="39"><sup>39</sup>University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="40"><sup>40</sup>Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA</footnote>

<footnote n="41"><sup>41</sup>Instituto de Astrofísica de Canarias, C/ Vía Láctea, s/n, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="42"><sup>42</sup>Universidad de La Laguna, Dept. de Astrofísica, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="43"><sup>43</sup>Department of Astronomy, Tsinghua University, 30 Shuangqing Road, Haidian District, Beijing, 100190, China</footnote>
This paper has been typeset from a TeX/Latex file prepared by the author.

<footnote n="27"><sup>27</sup>National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd., Chaoyang District, Beijing 100012, P.R. China</footnote>

<footnote n="28"><sup>28</sup>Instituto Avanzado de Cosmología A. C. San Marcos 11 – Atenas 202. Magdalena Contreras, 10720. Ciudad de México, México</footnote>

<footnote n="29"><sup>29</sup>Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada</footnote>

<footnote n="30"><sup>30</sup>Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada</footnote>

<footnote n="31"><sup>31</sup>Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada</footnote>

<footnote n="32"><sup>32</sup>Aix Marseille Univ, CNRS, CNES, LAM, F-13013 Marseille, France</footnote>

<footnote n="33"><sup>33</sup>Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA</footnote>

<footnote n="34"><sup>34</sup>University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA</footnote>

<footnote n="35"><sup>35</sup>Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain</footnote>

<footnote n="36"><sup>36</sup>Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea</footnote>

<footnote n="37"><sup>37</sup>CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain</footnote>

<footnote n="38"><sup>38</sup>Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="39"><sup>39</sup>University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="40"><sup>40</sup>Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA</footnote>

<footnote n="41"><sup>41</sup>Instituto de Astrofísica de Canarias, C/ Vía Láctea, s/n, E-38205 La Laguna, Tenerife, Spain</footnote>

<footnote n="42"><sup>42</sup>Universidad de La Laguna, Dept. de Astrofísica, E-38206 La Laguna, Tenerife, Spain</footnote>

<footnote n="43"><sup>43</sup>Department of Astronomy, Tsinghua University, 30 Shuangqing Road, Haidian District, Beijing, 100190, China</footnote>