Ecole Rodolphe Cledassou 2026

 

Corentin Ravoux

LPCA, CNRS/IN2P3, Clermont-Ferrand

 

ACCEL² simulation

Clustering des galaxies observations 4/4 - The Lyman-α forest

Cosmological probes

You are here

CMB, BBN

21cm, high-z galaxies/QSO, CMB lensing

Galaxies clustering (BAO, RSD, AP), Lyman-α forest, voids, WL, SL, clusters, high-z SN, SZ, ISW, CIB

SN, peculiar velocities (TF, FP, SN), cosmography, clusters, local structures

+ GW, GRB, PTA, Astroparticles...

+ Emerging cosmic probes (arxiv:2201.07241)

The Story of Our Universe - NASA

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Outline

1. Lyman-α forest physics

2. Observations

3. Clustering with Lyman-α forest

    a. 3D correlation function

    b. 1D power spectrum

    c. Lyman-α tomography

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Outline

1. Lyman-α forest physics

2. Observations

3. Clustering with Lyman-α forest

    a. 3D correlation function

    b. 1D power spectrum

    c. Lyman-α tomography

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1. Lyman-α forest physics

 

Quasars as back light sources

  • Detected in 1963 at z=0.158, very bright point source
  • Quasar = supermassive black hole + accretion disk
  • Brightest objects in the Universe (up to 1000 x Milky Way)
  • Variable (month/year)
  • Radio loud/quiet
  • Observed up to z~7

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  • 1990-2000: Global classification of Active Galaxy Nucleus (AGN)
    • Seyfert galaxies, quasars, liners, blazars, radio galaxies...
  • High gravitational energy in the SMBH
  • Accretion disk, heating of matter by collisions

Quasars as back light sources

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The Lyman-α forest

Wavelength [Angstrom]

Flux

  • Lyman-α: transition of neutral Hydrogen to first excited state

  • Lines in quasar spectra at                                     caused by absorbers in the intergalactic medium (IGM) at

\lambda_{\mathrm{obs}} = (1 + z_{\mathrm{abs}}) \lambda_{\alpha}
z_{\mathrm{abs}}

DESI quasar at z =3.42

\lambda_{\alpha} = 121.567\ \mathrm{nm}

Ravoux et al. 2023

Not only one redshift ! info on the full spectrum

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Lyman-α physics

\tau_\alpha(z)=\int n_{\mathrm{HI}}(z) \sigma_\alpha(z) d z
  • Lyman-α absorption optical depth (degree of transparency of the IGM):

Density of neutral hydrogen

Cross section of Lyman-α transition

\lambda_{\alpha} = 121.567\ \mathrm{nm}
  • Fraction of transmitted flux:
\mathrm{F}(z)= \frac{f_{\mathrm{out}}(z)}{C(z)}=\exp \left(-\tau_\alpha(z)\right)

F

Quasar continuum

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Gunn-Peterson effect

  • Universe is expanding: Hubble velocity flow
  • The matter has peculiar velocity
\mathrm{v}=\mathrm{c}\left[\nu_\alpha / \nu-1\right] = c z
\tau_\alpha(\mathrm{v})=\left(\frac{\sqrt{\pi} e^2}{m_{\mathrm{e}}}\right)\left(\frac{1}{4 \pi \epsilon_0}\right) \lambda_\alpha f_\alpha \int n_{\mathrm{HI}}\left(\mathrm{v}^{\prime}\right) \frac{1}{b\left(\mathrm{v}^{\prime}\right) H\left(\mathrm{v}^{\prime}\right)} e^{-\left(\frac{\mathrm{v}-\mathrm{v}^{\prime}-\mathrm{v}_{\|}\left(\mathrm{v}^{\prime}\right)}{b\left(\mathrm{v}^{\prime}\right)}\right)^2} d \mathrm{v}^{\prime}
\mathrm{v}_{\|}

Velocity space:

  • Complete calculation gives:

Density of neutral hydrogen/baryon

Doppler parameter (depends on temperature)

Universe expansion

IGM peculiar velocity

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Fluctuating Gunn-Peterson approximation

  • IGM thermal equilibrium + highly photoionized + optically thin approximation + larges scales (> 2 Mpc):
\tau_\alpha(z) \propto \frac{(1+z)^6}{H(z)} \times (1+\eta(z))^{-1} \times T(z)^{-0.7} \times \Omega_{\mathrm{b}}^2\left(1+\delta_{\mathrm{b}}(z)\right)^\beta

Density of neutral hydrogen/baryon

Peculiar velocities

Universe expansion

Universe temperature

Lyman-α forest = non-linear tracer of neutral hydrogen in the IGM

  • At high redshift (z >2), fraction of neutral hydrogen

Lyman-α forest observable from the ground

x_{\mathrm{HI}} \sim 10^{-5} \longrightarrow \tau_{\alpha} \sim 1

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

  • Near the quasar:
    • Unknown continuum
    • Broad absorption line quasars (BAL)

Contaminants

  • Along the line-of-sight:
    • Metal absorptions in the IGM
    • Damped Lyman-α systems (DLA)
  • Near the telescope:
    • Atmospheric absorption and emission
    • Instrument noise
    • Spectrograph resolution

Adapted from A. Pontzen video

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Contaminants: Broad absorption line quasars

Adapted from A. Pontzen video

  • BAL quasars: Absorptions occuring inside the quasar (non-IGM = contaminant)
  • High velocity outflows from the quasar torus
  • Absorption from one or several clouds of gas
  • ~15 % of quasars

Capellupo et al. 2017

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Contaminants: Damped Lyman-α systems

  • DLA: passing of the QSO light in the circum-galactic medium
  • Oversaturated absorption impacts a large spectra portion = clustering contaminant
  • Present on 30-40% of the quasars

DESI DR2 DLAs

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Contaminants: Metal absorptions

Adapted from A. Pontzen video

  • Metals: Other elements present in the IGM (Si, C, N, O...)
  • Like hydrogen, every element has a forest associated to each line
  • Depends a lot on the IGM proportion of each elements
F = \left(F_{\mathrm{Ly}\alpha} + F_{\mathrm{Si_{IV}}} + F_{\mathrm{C_{IV}}} + ..... \right.

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Contaminants: Instrument

Adapted from A. Pontzen video

  • Atmosphere:
    • Absorptions reduces the signal in the UV
    • Emission lines bias the clustering
  • Noise:
    • Very various sources (Poisson, electronic, calibration images...)
    • Hard to model
  • Spectral resolution:
    • Ability to distinguish two consecutive lines
    • Smooth the thinnest lines in the signal
F = \left(F_{\mathrm{Ly}\alpha} + F_{\mathrm{Si_{IV}}} + F_{\mathrm{C_{IV}}} + ..... + N \right) \circledast W\left(\lambda, R\right)
R = \lambda / \Delta \lambda

Resolution power

Ravoux et al. 2023

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Lyman-α forest examples

?

?

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Outline

1. Lyman-α forest physics

2. Observations

3. Clustering with Lyman-α forest

    a. 3D correlation function

    b. 1D power spectrum

    c. Lyman-α tomography

 

2. Observations

 

 

 

 

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Intermediate resolution: eBOSS (2014 - 2019)

  • Multi-object spectrograph
  • 2.5 m telescope at Apache Point Observatory
  • 210 000 Lyman-α forests

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DESI (2021 - 2028)

  • Multi-object spectrograph
  • 4 m telescope at Kitt Peak Observatory
  • more than 1 million Lyman-α forests expected

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Intermediate resolution: DESI focal plane

  • 5 μm positioning in few sec. for 5000 targets
  • All fibers connected to 10 spectrographs

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Comparison of spectral resolution

  • DESI:

 

 

  • eBOSS:
R \sim 1500 - 2500
R \sim 2000 - 5000
R \sim 50000
  • KODIAQ (High res):

c.f.  Y. Copin "La spectrométrie IR"

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High resolution: VLT

  • Cross-dispersed echelle spectrograph: part of the spectra spreaded into diffraction order and separatated
  • UVES on VLT:

                                                     for one object

                                                     for 8 fibers

R \sim 80000 - 110000
R \sim 47000
  • Very detailed spectra on few quasars
  • Science cases: Astrophysics, quasar properties, deuterium abundance, fine structure constant
  • Examples: UVES, XSHOOTER (VLT), HIRES (Keck)

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Outline

1. Lyman-α forest physics

2. Observations

3. Clustering with Lyman-α forest

    a. 3D correlation function

    b. 1D power spectrum

    c. Lyman-α tomography

 

 

3. Clustering with Lyman-α forest

 

 

 

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Quasar continuum

  • Quasar continuum: unknown and different for each quasar
  • Flux contrast (normalization):

 

Wavelength [Angstrom]

Flux

\begin{aligned} \delta_F(\lambda) &= \frac{F}{\overline{F}} - 1 \\ &= \frac{f(\lambda)}{\bar{F}(\lambda) C_{\mathrm{q}}\left(\lambda, z_{\mathrm{q}}\right)}-1 \end{aligned}

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Quasar continuum fit

  • Several methods:
    • Using very high resolution quasar spectra/stack of spectra
    • Principal Component Analysis template decomposition
    • Measuring on hydro simulation
    • Brute force: Likelihood minimization of a common model
C_{\mathrm{q}}\left(\lambda, z_{\mathrm{q}}\right)=\left(a_{\mathrm{q}}+b_{\mathrm{q}} \Lambda\right) C\left(\lambda_{\mathrm{rf}}=\frac{\lambda}{\left(1+z_{\mathrm{q}}\right)}\right)
\mathcal{L}=-\sum_i \frac{\left[f_i-\bar{F}\left(\lambda_i\right) C_{\mathrm{q}}\left(\lambda_i, z_{\mathrm{q}}, a_{\mathrm{q}}, b_{\mathrm{q}}\right)\right]^2}{\sigma_{\mathrm{q}}^2\left(\lambda_i\right)}-\ln \left[\sigma_{\mathrm{q}}^2\left(\lambda_i\right)\right]

picca python package

Common continuum model

Flux noise renormalized

\delta_F(\lambda)

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Definition of observational statistics

Configuration space

Fourier space

  • 3D auto-correlation:
\begin{aligned} &P_\alpha(\vec{k}) \delta_{\mathrm{D}}^{(3)}\left(\vec{k}+\vec{k^{\prime}}\right) \\ & =\frac{1}{(2 \pi)^3}\left\langle\delta_F(\vec{k}) \delta_F\left(\vec{k^{\prime}}\right)\right\rangle \end{aligned}
  • 3D power spectrum:
  • 1D power spectrum:
P_{1 \mathrm{D}, \alpha}\left(k_{\|}\right)=\int \frac{d \vec{k_{\perp}}}{(2 \pi)^2} P_\alpha\left(\vec{k}_{\perp}, k_{\|}\right)
\xi_{\alpha}(\vec{r})=<\delta_F(\vec{x}) \delta_F(\vec{x}+\vec{r})>_x
\xi_{\alpha X}(\vec{r})=<\delta_F(\vec{x}) \delta_X(\vec{x}+\vec{r})>_x
  • 3D cross-correlation:
  • 1D auto-correlation (not very used)
\vec{r}
\vec{k}
\xi_{1\mathrm{D},\alpha}(r_{\|})= <\delta_F(x) \delta_X(x+r_{\|})>_x

X = quasar, void

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Lyman-α clustering

Observer

Quasar

Lyman-α forest

\vec{r_{\parallel}}
\vec{r_{\bot}}
\vec{r}
\theta
\mu = \cos \theta

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Lyman-α clustering

Correlation

1D power spectrum (P1D)

Quasar

Lyman-α forest

1D auto correlation

Fourier space

Configuration space

P_{1 \mathrm{D}, \alpha}\left(k_{\|}\right)=\int \frac{d \vec{k_{\perp}}}{(2 \pi)^2} P_\alpha\left(\vec{k}_{\perp}, k_{\|}\right)
\xi_{1\mathrm{D},\alpha}(r_{\|})= <\delta_F(x) \delta_X(x+r_{\|})>_x
\vec{r_{\parallel}}
\vec{r_{\bot}}
\vec{r}
\theta
\mu = \cos \theta

Correlation

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Lyman-α clustering

Correlation

Quasar

Lyman-α forest

\xi_{\alpha}(\vec{r})=<\delta_F(\vec{x}) \delta_F(\vec{x}+\vec{r})>_x
P_\alpha(\vec{k}) \delta_{\mathrm{D}}^{(3)}\left(\vec{k}+\vec{k^{\prime}}\right) =\frac{1}{(2 \pi)^3}\left\langle\delta_F(\vec{k}) \delta_F\left(\vec{k^{\prime}}\right)\right\rangle

3D power spectrum (P3D)

3D auto correlation

\vec{r_{\parallel}}
\vec{r_{\bot}}
\vec{r}
\theta
\mu = \cos \theta

Fourier space

Configuration space

Correlation

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Lyman-α clustering

Configuration space

3D cross correlation with quasars

Correlation

Quasar

Lyman-α forest

\vec{r_{\parallel}}
\vec{r_{\bot}}
\vec{r}
\theta
\mu = \cos \theta
\xi_{\alpha q}(\vec{r})=<\delta_F(\vec{x}) \delta_q(\vec{x}+\vec{r})>_x

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Lyman-α clustering

Configuration space

Quasar

Lyman-α forest

\vec{r_{\parallel}}
\vec{r_{\bot}}
\vec{r}
\theta
\mu = \cos \theta

Interpolation

Interpolation

Lyman-α tomography

Interpolation algorithm to construct a 3D map

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Outline

1. Lyman-α forest physics

2. Observations

3. Clustering with Lyman-α forest

    a. 3D correlation function

    b. 1D power spectrum

    c. Lyman-α tomography

 

 

 

    a. 3D correlation function

 

 

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Baryon acoustic oscillations

  • BAO: Sound waves in the primordial Universe imprint a characteristic scale in the density distribution
  • Standard ruler in cosmology to measure distances
  • Observed reduced variables:
  •                           = Sound horizon at the drag epoch (baryon/photon decouple)
\begin{aligned} \alpha_{\|} & = (1 / H(z) r_d) /\left(1 / H(z) r_d\right)_{\text {fiducial }} \\ \alpha_{\perp} & = (D_M(z) / r_d) /\left(D_M(z) / r_d\right)_{\text {fiducial }} \end{aligned}
r_{\mathrm{d}} \sim 100~\mathrm{Mpc}/h

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Correlation estimator

  • Need to discretize the statistic:
\vec{r_{\bot}}
\vec{r_{\parallel}}
\vec{r}
\vec{r}

Bin A

\xi_{\alpha}(\vec{r})=<\delta_F(\vec{x}) \delta_F(\vec{x}+\vec{r})>_x
  • Estimator for auto-correlation:
\hat{\xi}_{\alpha}(A)=\frac{\sum_{(i, j) \in A} w_i w_j \delta_{F}^i \delta_{F}^j}{\sum_{(i, j) \in A} w_i w_j}
  • Run over Lyman-alpha forest quasars, can be computationally expensive
  • Similar estimator with point-like contrast for quasar, voids, galaxies...

= pixel weights from continuum fitting

w_i
\delta_{F}^i
\delta_{F}^j

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BAO with auto-correlation

3D auto correlation

  • Measurement of α parameters with the BAO scale

DESI DR2 2025

du Mas des Bourboux et al. 2020

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BAO with auto-correlation

BAO scale

3D auto correlation

  • Measurement of α parameters with the BAO scale

DESI DR2 2025

du Mas des Bourboux et al. 2020

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BAO with auto-correlation

3D auto correlation

  • Contaminants: Continuum fitting error, DLA, BAL, metals

?

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BAO with auto-correlation

3D auto correlation

  • Contaminants: Continuum fitting error, DLA, BAL, metals

SiII doublet

SiIII singulet

\delta_{F,\mathrm{Ly}\alpha}\delta_{F,\mathrm{Si}_{\mathrm{II}}}
\delta_{F,\mathrm{Ly}\alpha}\delta_{F,\mathrm{Si}_{\mathrm{III}}}

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BAO with quasar cross-correlation

3D cross correlation with quasars

  • Other measurement with additional information

BAO scale

DESI DR2 2025

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Modeling: theoretical model

P_{XY}(k, \mu)=b_X b_Y\left(1+\beta_X \mu^2\right)\left(1+\beta_Y \mu^2\right) P_{\mathrm{L}}(k,\mu)
  • Model for correlation between tracer X and Y:

Linear perturbation theory (Kaiser)

Tracer biases

RSD parameters

Linear power spectrum

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Modeling: theoretical model

P_{XY}(k, \mu)=b_X b_Y\left(1+\beta_X \mu^2\right)\left(1+\beta_Y \mu^2\right) P_{\mathrm{QL}}(k,\mu)
  • Model for correlation between tracer X and Y:

Power spectrum modified

P_{\mathrm{QL}}(k,\mu, z)=P_{\mathrm{sm}}(k)+\exp \left[-\frac{k_{\|}^2 \Sigma_{\|}^2+k_{\perp}^2 \Sigma_{\perp}^2}{2}\right] P_{\text {peak }}(k)

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Modeling: theoretical model

P_{XY}(k, \mu)=b_X b_Y\left(1+\beta_X \mu^2\right)\left(1+\beta_Y \mu^2\right) P_{\mathrm{QL}}(k,\mu)
  • Model for correlation between tracer X and Y:

Power spectrum modified

P_{\mathrm{QL}}(k,\mu, z)=P_{\mathrm{sm}}(k)+\exp \left[-\frac{k_{\|}^2 \Sigma_{\|}^2+k_{\perp}^2 \Sigma_{\perp}^2}{2}\right] P_{\text {peak }}(k)

Smooth component (no BAO)

BAO damping due to non-linear clustering

BAO component (wiggles)

k_{\|}=k_{\|}^{\mathrm{fid}} / \alpha_{\|}
k_{\bot}=k_{\bot}^{\mathrm{fid}} / \alpha_{\bot}
  • Parameters introduced by rescaling:

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Modeling: theoretical model

P_{XY}(k, \mu)=b_X b_Y\left(1+\beta_X \mu^2\right)\left(1+\beta_Y \mu^2\right) P_{\mathrm{QL}}(k,\mu) F_{\mathrm{NL}}(k,\mu) G(k,\mu)
  • Model for correlation between tracer X and Y:

Correction for binning

Non-linear Lyman-α clustering correction

D_1(k, \mu)=\exp \left\{\left[q_1 \Delta^2(k)+q_2 \Delta^4(k)\right]\left[1-\left(\frac{k}{k_v}\right)^{a_v} \mu^{b_v}\right]-\left(\frac{k}{k_p}\right)^2\right\}

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Modeling: theoretical model

P_{XY}(k, \mu)=b_X b_Y\left(1+\beta_X \mu^2\right)\left(1+\beta_Y \mu^2\right) P_{\mathrm{QL}}(k,\mu) F_{\mathrm{NL}}(k,\mu) G(k,\mu)
  • Model for correlation between tracer X and Y:

Correction for binning

Non-linear Lyman-α clustering correction

D_1(k, \mu)=\exp \left\{\left[q_1 \Delta^2(k)+q_2 \Delta^4(k)\right]\left[1-\left(\frac{k}{k_v}\right)^{a_v} \mu^{b_v}\right]-\left(\frac{k}{k_p}\right)^2\right\}

Non-linear power spectrum correction

Jeans smoothing (isotrop) 

Thermal broadening (directional)

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Modeling: theoretical model

P_{XY}(k, \mu)=b_X b_Y\left(1+\beta_X \mu^2\right)\left(1+\beta_Y \mu^2\right) P_{\mathrm{QL}}(k,\mu) F_{\mathrm{NL}}(k,\mu) G(k,\mu)
  • Model for correlation between tracer X and Y:

+ Full modeling of each contaminants (metals, DLA, BAL...)

FFT

\xi_{\mathrm{Ly}\alpha \times \mathrm{Ly}\alpha}
\xi_{\mathrm{Ly}\alpha \times QSO}

Templates for fit

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

BAO fit results

  • Auto-correlation and cross-correlation with quasars
  • Covariance matrix estimation for the fit:
    • Subsampling
    • Jackknife
    • Estimator on data
    • Theoretical

DESI DR2 2025

Used as a high-redshift BAO point for constraints on Hubble constant, neutrino mass, dark energy

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

BAO fit results

High complementarity with "low-z" galaxy clustering measurements

eBOSS DR16 2021

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Next steps in 3D clustering

Full shape

EFT modeling

  • High-z full-shape measurement with growth rate constraints

Cuceu et al. 2021

\begin{aligned} P^{1-\operatorname{loop}}(k, \mu) & =\left(b_1-b_\eta f \mu^2\right)^2 P_{\operatorname{lin}}(k) \\ & +2 \int_{\mathbf{q}} K_2^2(\mathbf{q}, \mathbf{k}-\mathbf{q}) P_{\operatorname{lin}}(|\mathbf{k}-\mathbf{q}|) P_{\operatorname{lin}}(q) \\ & +6 K_1(\mathbf{k}) P_{\operatorname{lin}}(k) \int_{\mathbf{q}} K_3(\mathbf{k},-\mathbf{q}, \mathbf{q}) P_{\operatorname{lin}}(q) \\ & -2\left(c_0+c_2 \mu^2+c_4 \mu^4\right) k^2 P_{\operatorname{lin}}(k) . \end{aligned}
  • New EFT models for exact treatment of non-linear clustering corrections
\begin{aligned} & \tilde{K}_1(\mathbf{k})=\left(-\tau_0 b_1^\tau\right)+\left(-\tau_0\right) f \mu^2 \\ & \tilde{K}_2\left(\mathbf{k}_1, \mathbf{k}_2\right)=\frac{\left(-\tau_0 b_2^\tau\right)}{2}+\left(-\tau_0 b_{\mathcal{G}_2}^\tau\right)\left(\frac{\left(\mathbf{k}_1 \cdot \mathbf{k}_2\right)^2}{k_1^2 k_2^2}-1\right)+\left(-\tau_0 b_1^\tau\right) F_2\left(\mathbf{k}_1, \mathbf{k}_2\right)-\tau_0 f \mu^2 G_2\left(\mathbf{k}_1, \mathbf{k}_2\right) \\ & -\tau_0 \frac{f \mu k}{2}\left(\frac{\mu_1}{k_1}\left(b_1^\tau+f \mu_2^2\right)+\frac{\mu_2}{k_2}\left(b_1^\tau+f \mu_1^2\right)\right)+\frac{\tau_0^2}{2}\left(\left(b_1^\tau\right)^2+b_1^\tau f\left(\mu_1^2+\mu_2^2\right)+f^2 \mu_1^2 \mu_2^2\right) \\ & \tilde{K}_3\left(\mathbf{k}_1, \mathbf{k}_2, \mathbf{k}_3\right)=-\tau_0 Z_3\left(\mathbf{q}_1, \mathbf{q}_2, \mathbf{q}_3\right)+\tau_0^2\left[Z_2\left(\mathbf{q}_1, \mathbf{q}_2\right) Z_1\left(\mathbf{q}_3\right)\right]_{\text {symm. }}-\frac{\tau_0^3}{3!} Z_1\left(\mathbf{q}_1\right) Z_1\left(\mathbf{q}_2\right) Z_1\left(\mathbf{q}_3\right) \end{aligned}

Ivanov 2024

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Outline

1. Lyman-α forest physics

2. Observations

3. Clustering with Lyman-α forest

    a. 3D correlation function

    b. 1D power spectrum

    c. Lyman-α tomography

 

 

 

     

    b. 1D power spectrum

 

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Small scales

Large scales

1D power spectrum (P1D)

  • Correlations along individual lines-of-sight

  • Sensitive to small-scale matter clustering and IGM thermal state

  • Probe the small-scale matter power spectrum

 

 

 

\Omega_{\mathrm{m}}, \sigma_8, n_s
A_{\mathrm{Ly\alpha}}, n_{\mathrm{Ly\alpha}}

Amplitude and slope at

z=3\ \&\ k=1\ h/ \mathrm{Mpc}

Chabanier et al. 2019

P1D range

Wavenumber k [Mpc^-1]

Matter power spectrum

One-dimensional power spectrum

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Neutrino masses

\sum m_{\nu}

P1D range

CMB range

  • Oscillations experiments measure mass differences

 

  • Cosmology:
    • Neutrinos impact structure formation by smoothing matter power spectrum at small scales
    • Effect depends on the sum of neutrino masses

P1D constrain neutrino masses

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

  • Warm dark matter:
    • Thermal relics from cosmic microwave background
    • Mass-dependent power spectrum cut-off on small scales
  • Other models:
    • Fuzzy dark matter
    • Self-interacting dark matter
    • Primordial black holes

Dark matter properties

P1D range

Baur et al. 2015

P1D constrain dark matter models

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

P1D estimator

  • P1D measured using Fast Fourrier Transform estimator
\begin{aligned} P_{1\mathrm{D},\alpha}(k) = & \left(\left\langle\left[P_{\mathrm{raw}}(k)-P_{\mathrm {pipeline}}(k) - \alpha \right] \cdot \mathbf{R}^{-2}(k) \right\rangle - P_{\mathrm{SB1,m}}(k) \right) \\ & \cdot A_{\mathrm{line}}(z,k) \cdot A_{\mathrm{dla}}(z,k) \cdot A_{\mathrm{cont}}(z,k) \cdot A_{\mathrm{res}}(k) \end{aligned}
= \left\langle|\delta_F(k)|^2\right\rangle
\big \rangle
\delta_F(k)

Without correction, Fourier transform of flux contrasts and averaging for all quasars

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

P1D estimator

  • P1D measured using Fast Fourrier Transform estimator
\begin{aligned} P_{1\mathrm{D},\alpha}(k) = & \left(\left\langle\left[P_{\mathrm{raw}}(k)-P_{\mathrm {pipeline}}(k) - \alpha \right] \cdot \mathbf{R}^{-2}(k) \right\rangle - P_{\mathrm{SB1,m}}(k) \right) \\ & \cdot A_{\mathrm{line}}(z,k) \cdot A_{\mathrm{dla}}(z,k) \cdot A_{\mathrm{cont}}(z,k) \cdot A_{\mathrm{res}}(k) \end{aligned}

Correction of the noise estimation

Spectroscopic resolution correction

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

P1D estimator

  • P1D measured using Fast Fourrier Transform estimator
\begin{aligned} P_{1\mathrm{D},\alpha}(k) = & \left(\left\langle\left[P_{\mathrm{raw}}(k)-P_{\mathrm {pipeline}}(k) - \alpha \right] \cdot \mathbf{R}^{-2}(k) \right\rangle - P_{\mathrm{SB1,m}}(k) \right) \\ & \cdot A_{\mathrm{line}}(z,k) \cdot A_{\mathrm{dla}}(z,k) \cdot A_{\mathrm{cont}}(z,k) \cdot A_{\mathrm{res}}(k) \end{aligned}

Measurement of  absorption from other IGM elements

Correction of the noise estimation

Spectroscopic resolution correction

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

P1D estimator

  • P1D measured using Fast Fourrier Transform estimator
\begin{aligned} P_{1\mathrm{D},\alpha}(k) = & \left(\left\langle\left[P_{\mathrm{raw}}(k)-P_{\mathrm {pipeline}}(k) - \alpha \right] \cdot \mathbf{R}^{-2}(k) \right\rangle - P_{\mathrm{SB1,m}}(k) \right) \\ & \cdot A_{\mathrm{line}}(z,k) \cdot A_{\mathrm{dla}}(z,k) \cdot A_{\mathrm{cont}}(z,k) \cdot A_{\mathrm{res}}(k) \end{aligned}

Masking of atmospheric lines

Masking of Damped Lyman-α systems

Correction of continuum fitting error

Correction of the noise estimation

Spectroscopic resolution correction

Measurement of  absorption from other IGM elements

Correction of the resolution estimation

All corrections derived from simulation/models on simulations

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

P1D measurements on intermediate resolution

P1D

Wavenumber k [Ang^-1]

  • DESI DR1 measurement
  • Large number of quasars (159132) with rather noisy spectra = small statistical error
  • Error bars accounting for systematic and statistical uncertainties

Ravoux et al. 2025

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

P1D measurements on high resolution

  • KODIAQ + SQUAD + XQ-100 measurement
  • Very few quasars (538)
  • Goes to very small scales = more constraining power for dark matter models

Karacaylı et al. 2022

P1D

Wavenumber k [s/km]

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Modeling: High-resolution simulations

Walther et al. 2021

  • Interpretation of P1D measurement with high-resolution hydrodynamical simulations (very expensive)

 

  • Machine learning methods:
    • Interpolation between simulations
    • Covers cosmological parameter space evenly

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Modeling: emulators

  • Two simulation grids built:
    • Nyx (full hydro)
    • MP-Gadget (SPH particles)
  • Gaussian processes
    emulators with adapted
    cosmological/IGM parameters
    • Interpolation between simulations
    • Covers cosmological parameter space evenly
    • Include IGM thermal state marginalization

Cabayol-Garcia  et al. 2023

Walther et al. 2021

Walther et al. 2025

Pedersen et al. 2023

Pedersen et al. 2021

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

DESI DR1 constraints: Primordial power spectrum

\begin{aligned} & \log \Delta_{\mathcal{R}}^2(k)=\log A_{\mathrm{s}}+\left(n_{\mathrm{s}}-1\right) \log \left(k / k_{\mathrm{s}}\right) \\ \end{aligned}
  • Expressing primordial power spectrum amplitude (     or     ) and index (    ) at the scale and redshift of P1D: bijective transformation
  • Large error bar from conservative systematics and emulator errors 
  • In agreement with eBOSS
A_\mathrm{s}
n_\mathrm{s}
\sigma_8

Chaves-Montero et al. 2025

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

DESI DR1 constraints: Inflation

\begin{aligned} & \log \Delta_{\mathcal{R}}^2(k)=\log A_{\mathrm{s}}+\left(n_{\mathrm{s}}-1\right) \log \left(k / k_{\mathrm{s}}\right) \\ & +\frac{\alpha_{\mathrm{s}}}{2} \log \left(k / k_{\mathrm{s}}\right)^2+\frac{\beta_{\mathrm{s}}}{6} \log \left(k / k_{\mathrm{s}}\right)^3 \end{aligned}
  • Relaxing slope variation for primordial power spectrum = constraints on inflation models
  • Improve Planck constraints

Chaves-Montero et al. 2025

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

DESI DR1 constraints: Neutrinos

  • Large constraints by comparing probes of large scales (CMB) with small scale ones (P1D)
  • Does not improve much the DESI DR2 BAO constraints but:
    • P1D constraints from LSS not evolution (BAO)

Chaves-Montero et al. 2025

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

DESI DR3 forecasts

\sigma\left(\sum m_{\nu}\right) = 0.03~\mathrm{eV}
  • Constraints on WDM improved by a factor 1.6
  • IGM thermal parameters by a factor 2.
  • Neutrino mass, in association with BAO and CMB:

 Valluri et al. 2022

Credits to A. Font-Ribera

Preliminary

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Future of Fourier space measurements

3D power spectrum

Bispectrum

  • 3-point correlation function in Fourier space = more information from cosmic web
  • 3D information in Fourier space
  • Made complicated by the intrinsic anisotropy of the Lyman-α forest
  • 3-point correlation function in Fourier space = more information from cosmic web

 de la Cruz et al. 2024

Abdul Karim et al. 2023

+ Work on P1D interpretation with EFTofLSS theories (Karacaylı et al. 2026)

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Outline

1. Lyman-α forest physics

2. Observations

3. Clustering with Lyman-α forest

    a. 3D correlation function

    b. 1D power spectrum

    c. Lyman-α tomography

 

 

 

   

   

    c. Lyman-α tomography

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Lyman-α tomography

  • 3D map of Lyman-α absorption from 1D spectra

 

  • Goal: Map the cosmic web at high redshift

 

  • Require large sample of Lyman-α with high angulardensity

Lyman-α tomography

Ravoux et al. 2020

Horowitz et al. 2021

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Interpolation algorithms

  • Wiener filter ~ Noise-dependent interpolation of forests
\delta_F^{\mathrm{rec}}=\mathbf{C}_{\mathrm{MD}} \cdot\left(\mathbf{C}_{\mathrm{DD}}+\mathbf{N}\right)^{-1} \cdot \delta_F
\mathbf{C}\left(\mathbf{r}_1, \mathbf{r}_2\right)=\sigma_F^2 \exp \left[-\frac{\left(\Delta r_{\|}\right)^2}{2 L_{\|}^2}\right] \exp \left[-\frac{\left(\Delta r_{\perp}\right)^2}{2 L_{\perp}^2}\right]

Map (output)

Forests (input)

Noise

Map-data and data-data covariance, modeled with Gaussian kernels

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Interpolation algorithm data

  • Forward modeling ~ Evolving density field from simulation to reproduce data (expensive and hard for large maps)

Horowitz et al. 2020

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Example: wiener filter map on eBOSS data

Lyman-α forest

Quasar

Reconstructed Lyman-α  contrast

Void

Ravoux et al. 2020

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Gives a 3D map tracing matter at redshift z > 2

Redshift

Redshift

Ravoux et al. 2020

Example: wiener filter map on eBOSS data

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Applications

  • Identification of proto-cluster candidates:
    • Densest regions of the map with density threshold
    • Criteria on the number of crossed lines-of-sight

 

  • High redshift cosmic voids:
    • Adapted for void grid finders
    • Void can be used to probe the dynamic of the cosmic web

Ravoux et al. 2020

Ravoux et al. 2020

cf. M. Aubert "Le clustering des galaxies : les vides"

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Lyman-α void cross correlation

  • Use distance between void centers and flux contrast
  • Same estimator as cross-correlation with quasars

eBOSS measurement

Ravoux et al. 2022

3D cross correlation with voids

Quasar

Lyman-α forest

Void

x

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Linear void model

\color{red}\vec{u} = \color{black} - \frac{1}{3} \frac{f H}{1+z} \overline{\delta}(r) \color{green} \vec{r}
  • Velocity field around void center:
\color{red}\vec{u}
\color{green} \vec{r}

Velocity

Position

  • Due to velocities, the void shape is deformed.
  • Measuring void shape = constraints on the cosmic web dynamics

Density

Cosmic web dynamics parameter

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Example: eBOSS data

  • Use of a spherical decomposition to measure deformation of voids
  • Effect of velocities seen in eBOSS data

 

Hexadecapole

r x Monopole

Quadrupole

View of velocity
flow around voids at z > 2

\beta = \frac{b_{\eta} f}{b}= 0.52 \pm 0.05

Ravoux et al. 2022

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

Tutorials

P1D

BAO

Tomography

Understand Lyman-α forest and manipulate a small tomographic map

Understand DESI data and compute a Fast Fourrier Transform

Build an estimator of 3D correlation functions used in BAO measurement

Ecole Rodolphe Cledassou 2025 - galaxy clustering: observation 4/4                                                                                                                                  

+ Some suggestions for projects if you want to go further

Ecole Rodolphe Cledassou 2026 - galaxy clustering: observation 4/4                                                                                                                                  

\vec{r}
\frac{\phi}{2}
\frac{\phi}{2}
r_{\bot}
r_{\parallel}

Earth

Forests