Ecole Rodolphe Cledassou 2025

 

Corentin Ravoux

LPCA, CNRS/IN2P3, Clermont-Ferrand

 

ACCEL² simulation

Le clustering des galaxies observation 4/4: The Lyman-α forest

Cosmological context

How did the Universe evolve to its current state?
What are the fundamental constituents of our Universe?
How is matter distributed in the Universe?

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The Story of Our Universe - NASA

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Universe expansion/composition

Large scale structures

Chabanier et al. 2019

Walther et al. 2021

Adapted from Dijkstra 2017

  • Initial density perturbations grows to form the cosmic web
  • Distribution of matter characterized by matter power spectrum
  • Components of the Universe (dark energy, dark matter, baryons, neutrinos, photons) modify the Universe expansion (Only 5% known)
\mathrm{flat}\ \Lambda\mathrm{CDM} = \left\{\Omega_{\mathrm{b}} ; \Omega_{\mathrm{cdm}} ; H_0 ; \tau ; A_{\mathrm{s}} ; n_{\mathrm{s}}\right\}

Cosmological probes

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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

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 2025 - 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 2025 - galaxy clustering: observation 4/4                                                                                                                                  

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}

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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

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

Quasar continuum

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:

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

Density of neutral hydrogen/baryon

Doppler parameter (depends on temperature)

Universe expansion

IGM peculiar velocity

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 2025 - 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

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

Contaminants: Broad absorption line quasars

Adapted from A. Pontzen video

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  • 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

Contaminants: Damped Lyman-α systems

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

  • 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

Cooke et al. 2017

Contaminants: Metal absorptions

Adapted from A. Pontzen video

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

  • 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.

Contaminants: Instrument

Adapted from A. Pontzen video

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

  • 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

Lyman-α forest examples

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

 

2. Observations

 

 

 

 

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

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R \sim 50000
  • KODIAQ (High res):

c.f.  Y. Copin "La spectrométrie IR"

High resolution: VLT

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  • 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)

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 2025 - galaxy clustering: observation 4/4                                                                                                                                  

 

 

3. Clustering with Lyman-α forest

 

 

 

Quasar continuum

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

 

Wavelength [Angstrom]

Flux

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\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}

Quasar continuum fit

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  • 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)

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

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

Quasar

Lyman-α forest

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

Lyman-α clustering

Correlation

1D power spectrum (P1D)

Quasar

Lyman-α forest

1D auto correlation

Fourier space

Configuration space

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

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

Lyman-α clustering

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

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

Lyman-α clustering

Configuration space

3D cross correlation with quasars

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

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

Lyman-α clustering

Configuration space

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

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

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 2025 - galaxy clustering: observation 4/4                                                                                                                                  

 

 

 

    a. 3D correlation function

 

 

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}

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r_{\mathrm{d}} \sim 100~\mathrm{Mpc}/h

Correlation estimator

  • Need to discretize the statistic:

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\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

BAO with auto-correlation

3D auto correlation

  • Measurement of α parameters with the BAO scale

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DESI DR2 2025

du Mas des Bourboux et al. 2020

BAO with auto-correlation

BAO scale

3D auto correlation

  • Measurement of α parameters with the BAO scale

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

DESI DR2 2025

du Mas des Bourboux et al. 2020

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

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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}}}

BAO with quasar cross-correlation

3D cross correlation with quasars

  • Other measurement with additional information

BAO scale

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DESI DR2 2025

Modeling: theoretical model

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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

Modeling: theoretical model

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

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)

Modeling: theoretical model

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

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:

Modeling: theoretical model

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

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\}

Modeling: theoretical model

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

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)

Modeling: theoretical model

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

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

BAO fit results

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  • 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

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BAO fit results

High complementarity with "low-z" galaxy clustering measurements

eBOSS DR16 2021

Ecole Rodolphe Cledassou 2025 - 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

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 2025 - galaxy clustering: observation 4/4                                                                                                                                  

 

 

 

     

    b. 1D power spectrum

 

Small scales

Large scales

1D power spectrum (P1D)

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

  • 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

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 2025 - 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 2025 - 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

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

\delta_F(k)

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

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}

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

Correction of the noise estimation

Spectroscopic resolution correction

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

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Correction of the noise estimation

Spectroscopic resolution correction

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

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

All corrections derived from simulation/models on simulations

P1D measurements on intermediate resolution

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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

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

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

P1D

Wavenumber k [s/km]

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 2025 - galaxy clustering: observation 4/4                                                                                                                                  

Current and future constraints

Palanque-Delabrouille et al. 2020

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  • Neutrino mass  (P1D +CMB):
\sum m_{\nu} < 0.11~\mathrm{eV}
m_{\mathrm{X}} > 5.3~\mathrm{keV}
  • Warm dark matter model:
  • Other constraints: Fuzzy dark matter, sterile neutrinos, running of the primordial power spectrum

at 95% C.L.

at 95% C.L.

Current and future constraints

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

\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

Future of Fourier space measurements

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

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

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 2025 - galaxy clustering: observation 4/4                                                                                                                                  

 

 

 

   

   

    c. Lyman-α tomography

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

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

Ravoux et al. 2020

Horowitz et al. 2021

Interpolation algorithms

  • Wiener filter ~ Noise-dependent interpolation of forests

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\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

Interpolation algorithm data

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

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

Horowitz et al. 2020

Example: wiener filter map on eBOSS data

Lyman-α forest

Quasar

Reconstructed Lyman-α  contrast

Void

Ravoux et al. 2020

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

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

Redshift

Redshift

Ravoux et al. 2020

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

Example: wiener filter map on eBOSS data

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

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

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

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

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

Quasar

Lyman-α forest

Void

x

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

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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

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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

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

Earth

Forests