Introduction to galaxy clustering

Arnaud de Mattia

CEA Saclay, Irfu/DPhP

Goals

broad overview of the domain
keys to easily read and criticize clustering analysis papers
to get a bit deeper: Percival 2013, Percival 2018

Past and current surveys
Clustering observables
Spectroscopic surveys and systematics
Large-scale structure formation
BAO and RSD theory models
Current constraints
Other clustering analyses

Goals

broad overview of the domain
keys to easily read and criticize clustering analysis papers
to get a bit deeper: Percival 2013, Percival 2018

Goals

broad overview of the domain
keys to easily read and criticize clustering analysis papers
to get a bit deeper: Percival 2013, Percival 2018

Past and current surveys
Clustering observables
Spectroscopic surveys and systematics
Large-scale structure formation
BAO and RSD theory models
Current constraints
Other clustering analyses

1924, Mount Wilson: Edwin Hubble shows the existence of
galaxies (previoulsy nebulae) using cepheids

Cataloguing galaxies

1924, Mount Wilson: Edwin Hubble shows the existence of
galaxies (previously nebulae) using cepheids
early imaging surveys carried out with photometric plates:
- 1949 - 1958: Palomar Observatory Sky Survey (POSS I)
- 1974 - 1999: UKST sky surveys, Siding Spring, Australia
- 1985 - 2000: POSS II

Cataloguing galaxies

galaxy catalogs:
- 1961 - 1968: Zwicky Catalog of galaxies based on POSS I
- 1967: Lick catalog of 1M galaxies
numerization:
- 1994: Digitized Sky Survey, scanning of POSS I/II and UKST: 102 CD-roms
- 1997 - 2001: 2MASS (Arizona, USA, and Chile) 3-band photometric survey

1924, Mount Wilson: Edwin Hubble shows the existence of
galaxies (previously nebulae) using cepheids
early imaging surveys carried out with photometric plates:
- 1949 - 1958: Palomar Observatory Sky Survey (POSS I)
- 1974 - 1999: UKST sky surveys, Siding Spring, Australia
- 1985 - 2000: POSS II

How to go 3D?

R.A.

Dec.

R.A.

Dec.

With spectroscopic surveys!

More specifically:

broad-band photometric surveys: $\sigma(z) \sim 0.03 (1 + z)$
spectroscopic surveys: $\sigma(z) \sim 0.0001 (1 + z)$

CfA redshift surveys

Mesurement of galaxy redshifts using a spectrograph.

redshifts of galaxies from updated Zwicky catalogs
1977 - 1982: CfA I, 1100 redshifts, structures formed by galaxies
1985 - 1995: CfA II, 18k redshifts, Great Wall (Geller and Huchra 1989) at $z \simeq 0.03$

CfA redshift surveys

Mesurement of galaxy redshifts using a spectrograph.

redshifts of galaxies from updated Zwicky catalogs
1977 - 1982: CfA I, 1100 redshifts, structures formed by galaxies
1985 - 1995: CfA II, 18k redshifts, Great Wall (Geller and Huchra 1989) at $z \simeq 0.03$

First cosmological constraints

Vogeley et al. 1992 with CfA: inconsistency with CDM model at 99% (propose oCDM)
Cole et al. 1994 with IRAS (infrared space telescope) data (15k redshifts): first measurement of anisotropy ($\beta$ parameter) w.r.t. line-of-sight (due to redshift-space distortions)

First cosmological constraints

Vogeley et al. 1992 with CfA: inconsistency with CDM model at 99% (propose oCDM)
Cole et al. 1994 with IRAS (infrared space telescope) data (15k redshifts): first measurement of anisotropy ($\beta$ parameter) w.r.t. line-of-sight (due to redshift-space distortions)

Since the 90's, multi-object spectroscopy

How to speed up redshift surveys? Many spectra at once!
Demonstrators in the 80's (MEDUSA, FOCAP)
Las Campanas Redshift Survey (1991 - 1996)
- 2.5 m Dupont Telescope at LCO, Chile, 50 fibers
- 26k galaxies at $z < 0.2$ over $700\;\mathrm{deg}^2$

CfA

Since the 90's, multi-object spectroscopy

How to speed up redshift surveys? Many spectra at once!
Demonstrators in the 80's (MEDUSA, FOCAP)
Las Campanas Redshift Survey (1991 - 1996)
- 2.5 m Dupont Telescope at LCO, Chile, 50 fibers
- 26k galaxies at $z < 0.2$ over $700\;\mathrm{deg}^2$

CfA

2dFGRS

2dF Galaxy Redshift Survey (1997 - 2002)
- Anglo-Australian 3.9-m telescope (AAT), 400 fibers
- 221k galaxies at $z < 0.3$ over $1800\;\mathrm{deg}^2$

2dFGRS

single robotic positioner

Lewis et al. 2002

SDSS I-II

SDSS-I and II (2000 - 2008)
- 2.5-m telescope at Apache Point Observatory, 500 fibers
- Main Galaxy Sample (MGS), 800k galaxies at $z < 0.25$ over $8000\;\mathrm{deg}^2$

SDSS I-II

credits: LBL

credits: Thomas Nash

2dFGRS, SDSS: first BAO detection!

In 2005, first evidence for Baryon Acoustic Oscillations
Standard ruler to probe the expansion history of the Universe

SDSS - 47k LRG $0.16 < z < 0.47$
Eisenstein et al. 2005

2dFGRS

Cole et al. 2005

WiggleZ, BOSS/eBOSS: higher z

WiggleZ

Blake et al., 2011

WiggleZ (2006-2011): upated 2dFGRS facility, 240k star-forming galaxies at intermediate redshifts ($z \sim 0.6$)
SDSS-III BOSS (2009-2014): 1.5M Luminous Red Galaxies and Quasars $(0.3 < z < 0.7)$
SDSS-IV eBOSS (2014-2020): +800k LRG, Emission Line Galaxies and Quasars up to $z \sim 2$

WiggleZ, BOSS/eBOSS: higher z

WiggleZ

Blake et al., 2011

WiggleZ (2006-2011): upated 2dFGRS facility, 240k star-forming galaxies at intermediate redshifts ($z \sim 0.6$)
SDSS-III BOSS (2009-2014): 1.5M Luminous Red Galaxies and Quasars $(0.3 < z < 0.7)$
SDSS-IV eBOSS (2014-2020): +800k LRG, Emission Line Galaxies and Quasars up to $z \sim 2$

Large surveys

made possible by multi-object spectroscopy
high redshift (z > 0.5) surveys since 2005

Moore's law for spectroscopic surveys; Schlegel et al. 2022

Euclid

10 years = $10 \times $

Stage IV experiments: Euclid

2023 - 2029: 35M H$\alpha$ emitters at $0.7 < z < 1.8$ over $14 000 \; \mathrm{deg}²$
Slitless spectroscopy with NISP: disperses the entire field-of-view:

all spectra at once, no moving parts
no need for target selection
but difficult to deal with spectra overlap, noise subtraction, etc.

NISP instrument. Euclid consortium

ESA

Stage IV experiments: DESI

2021 - 2025: 40M redshifts at $0 < z < 3$ over $14 000 \; \mathrm{deg}²$

Mayall Telescope at Kitt Peak, AZ

5000 robotically-positioned spectroscopic fibers

robotic positioners

Taken from Zhao et al. (2020)

DESI

Credit: NSF

Taken from Zhao et al. (2020)

DESI focal plane

DESI cosmological constraints

Measuring dark energy

$\Lambda$

2024

2025

DESI cosmological constraints

Measuring dark energy

$\Lambda$

Testing general relativity

Taken from Zhao et al. (2020)

Other redshift surveys

HETDEX (2017-2023): 1M (Lyman-$\alpha$ emitting) galaxies, untargeted survey (R ∼ 800)
4MOST (2024-2029): 25M spectra $15,000\;\mathrm{deg}^2$, Paranal Observatory, Chile
Nancy-Grace-Roman / WFIRST (2025-2030): 20M H$\alpha$ emitters (R = 70 − 140 and R = 450 − 850), slitless spectroscopy

Past and current surveys: take-aways

Catalogs of galaxies have been built starting the 1920's (Hubble, Zwicky, etc.)
Digitalization in the 1970's
Redshift surveys as early as 1980's (CfA) revealed cosmic structures (walls, filaments). But it was slow...
In the 90's, first evidence for $\Omega_\mathrm{m} < 1$
>90's, multi-object spectroscopy allowed 100-1000 spectra to be measured at once: a revolution in survey speed!
Big names: 2dFGRS, SDSS, DESI, Euclid...
First evidence for BAO in 2005 (WiggleZ, SDSS), much progress since then!
With DESI and Euclid: pin down dark energy, test modified gravity, primordial non-Gaussianity, etc.

how to extract cosmological information from our survey data?

Past and current surveys: take-aways

Catalogs of galaxies have been built starting the 1920's (Hubble, Zwicky, etc.)
Digitalization in the 1970's
Redshift surveys as early as 1980's (CfA) revealed cosmic structures (walls, filaments). But it was slow...
In the 90's, first evidence for $\Omega_\mathrm{m} < 1$
>90's, multi-object spectroscopy allowed 100-1000 spectra to be measured at once: a revolution in survey speed!
Big names: 2dFGRS, SDSS, DESI, Euclid...
First evidence for BAO in 2005 (WiggleZ, SDSS), much progress since then!
With DESI and Euclid: pin down dark energy, test modified gravity, primordial non-Gaussianity, etc.

how to extract cosmological information from our survey data?

Outline

Past and current surveys
Clustering observables
Spectroscopic surveys and systematics
Large-scale structure formation
BAO and RSD theory models
Current constraints
Other clustering analyses

What do we measure?

We measure angular positions (right ascension (R.A.), declination
(Dec.)) and redshifts ($z$) of $\mathcal{O}(10^6)$ galaxies.

What to do with this data?

SDSS data. Credits: EPFL

An example: ...

An example: Paris metro & bus stations

Credit to Etienne Burtin for the idea!

bois de Boulogne

bois de Vincennes

Counting pairs

edges = np.linspace(0., 2., 51)
# positions.shape = (N, 2)
# positions[:, 0] is x, positions[:, 1] is y

# Count pairs of points within a distance range
def pair_count_2d(positions, edges):
    counts = np.zeros(len(edges) - 1)
    for i in range(positions.shape[0]):
        for j in range(i + 1, positions.shape[0]):
            dx = positions[i, 0] - positions[j, 0]
            dy = positions[i, 1] - positions[j, 1]
            dist2 = dx * dx + dy * dy
            # Only count if within the maximum distance
            if dist2 < edges[-1]**2:
                # Find the index in the edges array
                idx = int((np.sqrt(dist2) - edges[0])\
                          / (edges[-1] - edges[0])\
                          * len(counts))
                counts[idx] += 1
    return counts

...a bit hard to interpret! Is the trend consistent with what one would expect is stations were distributed uniformly?

Randoms

Let's just generate some uniformly-distributed "randoms"

Bonus question: what is ~the slope of this curve?

Randoms

Let's imprint the footprint of Paris!

Randoms

DD: data pair counts

RR: randoms pair counts

Randoms

DD: data pair counts

RR: randoms pair counts

clustered stations

characteristic scale of 0.4 km

A bit of formalism

$n_\mathrm{g}(\mathbf{x}) = \bar{n}(\mathbf{x})\left[1 + \delta_\mathrm{g}(\mathbf{x}) \right]$ $\delta_\mathrm{g}$ density contrast

Density of galaxies

A bit of formalism

$n_\mathrm{g}(\mathbf{x}) = \bar{n}(\mathbf{x})\left[1 + \delta_\mathrm{g}(\mathbf{x}) \right]$ $\delta_\mathrm{g}$ density contrast

Density of galaxies

Probability to find:

one galaxy in $dV_1$: $dP_1 = \langle n_\mathrm{g}(\mathbf{x}_1) dV_1 \rangle = \bar{n}(\mathbf{x}_1) dV_1$

A bit of formalism

Density of galaxies

Probability to find:

one galaxy in $dV_1$: $dP_1 = \langle n_\mathrm{g}(\mathbf{x}_1) dV_1 \rangle = \bar{n}(\mathbf{x}_1) dV_1$
two galaxies in $dV_1$ and $dV_2$:

dP_{12} = \left\langle n_\mathrm{g}(\mathbf{x}_1) n_\mathrm{g}(\mathbf{x}_2) \right\rangle dV_1 dV_2 = \bar{n}(\mathbf{x}_1) \bar{n}(\mathbf{x}_2) \left[ 1 + \xi_\mathrm{gg}(\mathbf{x}_1, \mathbf{x}_2) \right] dV_1 dV_2

$\xi_\mathrm{gg}(\mathbf{s}) = \left\langle \delta_\mathrm{g}(\mathbf{x}_1) \delta_\mathrm{g}(\mathbf{x}_1 + \mathbf{s}) \right\rangle$

Covariance of the density contrast as a function of separation $\mathbf{s}$

Galaxy correlation function

Independent of position assuming spatial homogeneity.

$n_\mathrm{g}(\mathbf{x}) = \bar{n}(\mathbf{x})\left[1 + \delta_\mathrm{g}(\mathbf{x}) \right]$ $\delta_\mathrm{g}$ density contrast

Fiducial coordinates

Wait! What is $\mathbf{x}$? I thought that in the catalog we had R.A., Dec., z?

Fiducial coordinates

We use a fiducial cosmology to convert $z$ to distance

\orange{D_c(z)} = \int_0^z \frac{c dz^\prime}{\green{H(z^\prime)}} \qquad \text{with} \qquad \green{H(z)} = \purple{H_0} \sqrt{\blue{\Omega_\mathrm{m}} (1 + z)^3 + 1 - \Omega_\mathrm{m}}

Distance in $\mathrm{Mpc}/h$ units: only need to assume a fiducial $\Omega_\mathrm{m}$

Two angles on the sky (R.A., Dec.), and distance

$\Rightarrow$ fiducial cartesian comoving coordinates $\mathbf{x}$

Note for later: include this in the theory model!

comoving radial distance

Hubble rate

matter density

Hubble parameter $H_0 = 100\;h\;\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$

Fiducial coordinates

We use a fiducial cosmology to convert $z$ to distance

Distance in $\mathrm{Mpc}/h$ units: only need to assume a fiducial $\Omega_\mathrm{m}$

Two angles on the sky (R.A., Dec.), and distance

$\Rightarrow$ fiducial cartesian comoving coordinates $\mathbf{x}$

Note for later: include this in the theory model!

comoving radial distance

Hubble rate

matter density

Hubble parameter $H_0 = 100\;h\;\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$

\orange{D_c(z)} = \int_0^z \frac{c dz^\prime}{\green{H(z^\prime)}} \qquad \text{with} \qquad \green{H(z)} = \purple{H_0} \sqrt{\blue{\Omega_\mathrm{m}} (1 + z)^3 + 1 - \Omega_\mathrm{m}}

(Isotropic) correlation function

separation between galaxies

correlation function

excess probability that 2 galaxies are close

$<0$ as $\int d^3s \xi(s) = 0$

excess probability that 2 galaxies are close

Power spectrum

Fourier transform of the density contrast $\delta_\mathrm{g}(\mathbf{x})$

\delta_\mathrm{g}(\mathbf{k}) = \int d^3x \, \delta_\mathrm{g}(\mathbf{x}) \, e^{i \mathbf{k} \cdot \mathbf{x}}

$(2\pi)^3 \delta_D^{(3)}(\mathbf{k} + \mathbf{k}') P_\mathrm{gg}(\mathbf{k}) = \langle \delta_\mathrm{g}(\mathbf{k}) \delta_\mathrm{g}(\mathbf{k}') \rangle$

Galaxy power spectrum

Dirac $\delta_D^{(3)}(\mathbf{k} + \mathbf{k}')$ comes from homogeneity

Power spectrum

Fourier transform of the density contrast $\delta_g(\mathbf{x})$

Galaxy power spectrum

Dirac $\delta_D^{(3)}(\mathbf{k} + \mathbf{k}')$ comes from homogeneity
$\xi_\mathrm{gg}(\mathbf{s})$ and $P_\mathrm{gg}(\mathbf{k})$ are Fourier transform pairs:

Early time/large scales, $\delta$ follows Gaussian statistics: fully described by 2-point function.

P_\mathrm{gg}(\mathbf{k}) = \int d^3 s \, \xi_\mathrm{gg}(\mathbf{s}) \, e^{i \mathbf{k} \cdot \mathbf{s}}

\delta_\mathrm{g}(\mathbf{k}) = \int d^3x \, \delta_\mathrm{g}(\mathbf{x}) \, e^{i \mathbf{k} \cdot \mathbf{x}}

$(2\pi)^3 \delta_D^{(3)}(\mathbf{k} + \mathbf{k}') P_\mathrm{gg}(\mathbf{k}) = \langle \delta_\mathrm{g}(\mathbf{k}) \delta_\mathrm{g}(\mathbf{k}') \rangle$

(Isotropic) power spectrum

power spectrum

wavenumber

small scales

large scales

Anisotropic clustering

In practice, the clustering amplitude does not only depend on the separation $|\mathbf{s}|$ or wavenumber $|\mathbf{k}|$...

But also on the direction of $\mathbf{s}$ and $\mathbf{k}$

Q: Wait, isn't the Universe homogeneous and isotropic?

Anisotropic clustering

In practice, the clustering amplitude does not only depend on the separation $|\mathbf{s}|$ or wavenumber $|\mathbf{k}|$...

But also on the direction of $\mathbf{s}$ and $\mathbf{k}$

Q: Wait, isn't the Universe homogeneous and isotropic?

In practice, the clustering amplitude does not only depend on the separation $|\mathbf{s}|$ or wavenumber $|\mathbf{k}|$...

But also on the direction of $\mathbf{s}$ and $\mathbf{k}$

direction of a galaxy = line-of-sight

Anisotropic clustering

In practice, the clustering amplitude does not only depend on the separation $|\mathbf{s}|$ or wavenumber $|\mathbf{k}|$...

But also on the direction of $\mathbf{s}$ and $\mathbf{k}$

"midpoint" line-of-sight

$\mathbf{s} = \mathbf{x}_2 - \mathbf{x}_1$ separation

We usually call $\mu = \hat{\mathbf{s}} \cdot \hat{\mathbf{\eta}}$ the cosine angle between the separation vector $\mathbf{s}$ and the line-of-sight $\hat{\mathbf{\eta}}$

Similarly: $\mu = \hat{\mathbf{k}} \cdot \hat{\mathbf{\eta}}$

$\hat{\eta} = \widehat{\mathbf{x}_1 + \mathbf{x}_2}$

$\mathbf{x}_1$

$\mathbf{x}_2$

$\mu$

Anisotropic clustering

In practice, rather than binning in $\mu$, we prefer to measure Legendre multipoles $\xi_\ell(s)$ and $P_\ell(k)$, typically $0 \leq \ell \leq 4$

\xi(s, \mu) = \sum_{\ell = 0}^{+\infty} \xi_{\ell}(s) \orange{\mathcal{L}_{\ell}(\mu)} \quad \& \quad P(k, \mu) = \sum_{\ell = 0}^{+\infty} P_{\ell}(k) \orange{\mathcal{L}_{\ell}(\mu)}

\xi_{\ell}(s) = \frac{2 \ell + 1}{2} \int_{-1}^{1}d\mu \xi(s, \mu) \orange{\mathcal{L}_{\ell}(\mu)} \quad \& \quad P_{\ell}(k) = \frac{2 \ell + 1}{2} \int_{-1}^{1}d\mu P(k, \mu) \orange{\mathcal{L}_{\ell}(\mu)}

Legendre polynomial

Anisotropic clustering

In practice, rather than binning in $\mu$, we prefer to measure Legendre multipoles $\xi_\ell(s)$ and $P_\ell(k)$, typically $0 \leq \ell \leq 4$

\xi(s, \mu) = \sum_{\ell = 0}^{+\infty} \xi_{\ell}(s) \orange{\mathcal{L}_{\ell}(\mu)} \quad \& \quad P(k, \mu) = \sum_{\ell = 0}^{+\infty} P_{\ell}(k) \orange{\mathcal{L}_{\ell}(\mu)}

Then one can show that $\xi_\ell(s)$ and $P_\ell(k)$ are related through a Hankel transform:

P_\ell(k) = 4\pi i^\ell \int s^2 ds \orange{j_\ell(k s)} \xi_\ell(s) \quad \& \quad \xi_\ell(s) = \frac{(-i)^\ell}{2\pi^2} \int k^2 dk \orange{j_\ell(k s)} P_\ell(k)

spherical Bessel function

Anisotropic clustering

How to estimate $\xi_\ell(s)$ and $P_\ell(k)$ from galaxy catalogs?

Clustering catalogs

Catalog of particles (R.A., Dec., $z$) that randomly sample the survey selection function $\bar{n}$ (i.e. where we carried out observations).

Usually $>20\times$ denser than the data catalogs, to reduce sampling noise.

"Randoms"

Catalog of galaxies (R.A., Dec., $z$, optionally weights)

"Data"

Correlation function estimators

Let $XY(\mathbf{s})$ be the (normalized, weighted) number of pairs of objects from catalogs $X, Y$ as a function of separation $\mathrm{s}$

$\hat{\xi}(\mathbf{s}) = \frac{DD(\mathbf{s})}{RR(\mathbf{s})} − 1$ minimally biased but large variance

Natural estimator

$\hat{\xi}(\mathbf{s}) = \frac{DD(\mathbf{s})}{DR(\mathbf{s})} − 1$ biased and not minimal variance

Davis and Peebles 1983 estimator

$\hat{\xi}(\mathbf{s}) = \frac{DD(\mathbf{s}) RR(\mathbf{s})}{DR(\mathbf{s})^2} − 1$ minimal variance but biased

Hamilton 1993 estimator

Correlation function estimators

$\hat{\xi}(\mathbf{s}) = \frac{DD(\mathbf{s})}{RR(\mathbf{s})} − 1$ minimally biased but large variance

Natural estimator

$\hat{\xi}(\mathbf{s}) = \frac{DD(\mathbf{s})}{DR(\mathbf{s})} − 1$ biased and not minimal variance

Davis and Peebles 1983 estimator

$\hat{\xi}(\mathbf{s}) = \frac{DD(\mathbf{s}) RR(\mathbf{s})}{DR(\mathbf{s})^2} − 1$ minimal variance but biased

Hamilton 1993 estimator

$\hat{\xi}(\mathbf{s}) = \frac{DD(\mathbf{s}) - 2DR(\mathbf{s}) + RR(\mathbf{s})}{RR(\mathbf{s})}$ minimally biased, minimal variance

Landy-Szalay 1993 estimator

Let $XY(\mathbf{s})$ be the (normalized, weighted) number of pairs of objects from catalogs $X, Y$ as a function of separation $\mathrm{s}$

Power spectrum estimator

$F(\mathbf{x}) = n_\mathrm{g}(\mathbf{x}) − \bar{n}(\mathbf{x})$

Density fluctuations

Yamamoto 2006 estimator

\hat{P}_{\ell}(k) = \frac{2\ell + 1}{A N_k} \sum_{\mathbf{k} \in k-\mathrm{bin}} F_{\ell}^{\star}(\mathbf{k}) F_{0}(\mathbf{k}) - P_{\ell}^{\mathrm{noise}}(k)

F_{\ell}(\mathbf{k}) = \sum_{\mathbf{x}} e^{i \mathbf{k} \cdot \mathbf{x}} F(\mathbf{x}) \mathcal{L}_\ell(\mathbf{\hat{k}} \cdot \mathbf{\hat{x}})

$P_\ell^\mathrm{noise}(k) \simeq \delta_{\ell 0}^{K} \bar{n}^{-1}$: shot noise due to finite number of galaxies

number of $\mathbf{k}$-modes in the bin

A = \sum_\mathbf{x} \bar{n}^{2}(\mathbf{x})

normalisation

computed with Fast Fourier Transforms

$F$ painted on a mesh ($\Rightarrow$ aliasing effects)

Power spectrum estimator

$F(\mathbf{x}) = n_\mathrm{g}(\mathbf{x}) − \bar{n}(\mathbf{x})$

Density fluctuations

Yamamoto 2006 estimator

\hat{P}_{\ell}(k) = \frac{2\ell + 1}{A N_k} \sum_{\mathbf{k} \in k-\mathrm{bin}} F_{\ell}^{\star}(\mathbf{k}) F_{0}(\mathbf{k}) - P_{\ell}^{\mathrm{noise}}(k)

F_{\ell}(\mathbf{k}) = \sum_{\mathbf{x}} V_\mathrm{cell} e^{i \mathbf{k} \cdot \mathbf{x}} F(\mathbf{x}) \mathcal{L}_\ell(\mathbf{\hat{k}} \cdot \mathbf{\hat{x}})

$P_\ell^\mathrm{noise}(k) \simeq \delta_{\ell 0}^{K} \bar{n}^{-1}$: shot noise due to finite number of galaxies

number of $\mathbf{k}$-modes in the bin

A = \sum_\mathbf{x} V_\mathrm{cell} \bar{n}^{2}(\mathbf{x})

normalisation

computed with Fast Fourier Transforms

Window function effect

Survey has finite size: window function effect

correlation function: already corrected for by the $RR(s, \mu)$ term in the denominator
power spectrum: to be deconvolved, or included in power spectrum model: matrix multiplication (e.g. Beutler and McDonald 2021) $W_{\alpha\beta} = d\langle \hat{P}_\alpha \rangle / d P_\beta$
other effects: wide-angle, integral constraints (Beutler et al. 2019; de Mattia and Ruhlmann-Kleider 2019)
"alternative": Optimal Quadratic Estimator (e.g. Philcox et al. 2024)

For a $6\; \mathrm{Gpc}/h$ box

Measurement covariance

Power spectrum covariance is, using Wick’s theorem (Gaussian field):

\sigma(P(k))^2 \simeq \frac{2 (2\pi)^{3}}{A^{2}\Delta V_{k}} \int d^{3} x \bar{n}(\mathbf{x})^{4} \left[P(\mathbf{k}) + \frac{1}{\bar{n}(\mathbf{x})}\right]^{2}

minimizing variance: FKP (Feldman et al. 1994 ) weights
$w_\mathrm{FKP} = 1/ [1 + \bar{n}(z)P_0)]$ applied to galaxies (and randoms)

e.g. Grieb et al. 2015

Measurement covariance

Power spectrum covariance is, using Wick’s theorem (Gaussian field):

covariance matrix typically estimated from mocks: fast simulations of the galaxy density field, including survey selection function
noise in the covariance matrix: noise in the cosmological measurement. Artificially enlarge error bars (Hartlap et al. 2007; Percival et al. 2014; Percival et al. 2022)
accurate analytic estimations are developed (e.g. Lin et al. 2018, Wadekar et al. 2019)

\sigma(P(k))^2 \simeq \frac{2 (2\pi)^{3}}{A^{2}\Delta V_{k}} \int d^{3} x \bar{n}(\mathbf{x})^{4} \left[P(\mathbf{k}) + \frac{1}{\bar{n}(\mathbf{x})}\right]^{2}

minimizing variance: FKP (Feldman et al. 1994 ) weights
$w_\mathrm{FKP} = 1/ [1 + \bar{n}(z)P_0)]$ applied to galaxies (and randoms)

e.g. Grieb et al. 2015

Measurement covariance

In the uniform $\bar{n}$ limit:

larger survey volume: $\mathrm{error} \propto 1/\sqrt{\blue{V_s}}$
higher density: $\mathrm{error} \propto 1/\orange{\bar{n}}$ when in the shot-noise dominated regime $\bar{n} \ll 1/P_0$

Two leverages to minimize variance (= higher measurement precision):

Credit: DESI

\sigma(P(k))^2 \simeq \frac{2 (2\pi)^{3}}{\Delta V_{k} \blue{V_{s}}} \left[P(k) + \frac{1}{\orange{\bar{n}}}\right]^{2}

Clustering observables: take-aways

The clustering of galaxies can be probed through the correlation function $\xi(\mathbf{s})$ or power spectrum $P(\mathbf{k})$ (FT pair)
Contain all information on $\delta_\mathrm{g}$ if it is Gaussian (large scales)
Anisotropic measurements: dependence on the (cosine) angle w.r.t. the line-of-sight $\mu$; expansion in multipoles
Various estimators exist; beware of the effect of the survey selection function!
Measurement uncertainty is $\propto 1/\sqrt{V}$ and, in the shot noise dominated regime $1/\bar{n}$ $\Rightarrow$ $\bar{n} P = 1$ criterion

how to carry out galaxy redshift surveys?

(disclaimer: with a focus on DESI!)

Clustering observables: take-aways

The clustering of galaxies can be probed through the correlation function $\xi(\mathbf{s})$ or power spectrum $P(\mathbf{k})$ (FT pair)
Contain all information on $\delta_\mathrm{g}$ if it is Gaussian (large scales)
Anisotropic measurements: dependence on the (cosine) angle w.r.t. the line-of-sight $\mu$; expansion in multipoles
Various estimators exist; beware of the effect of the survey selection function!
Measurement uncertainty is $\propto 1/\sqrt{V}$ and, in the shot noise dominated regime $\propto 1/\bar{n}$ $\Rightarrow$ $\bar{n} P = 1$ criterion

how to carry out galaxy redshift surveys?

(disclaimer: with a focus on DESI!)

Outline

Past and current surveys
Clustering observables
Spectroscopic surveys and systematics
Large-scale structure formation
BAO and RSD theory models
Current constraints
Other clustering analyses

DESI Y5 galaxy samples

Bright Galaxies: 14M (SDSS: 600k)

0 < z < 0.4

LRG: 8M (SDSS: 1M)

0.4 < z < 1.1

ELG: 16M (SDSS: 200k)

0.6 < z < 1.6

QSO: 3M (SDSS: 500k)

Lya $1.8 < z$

Tracers $0.8 < z < 2.1$

Y5 (DR1-DR2-DR3) $\sim 40$M galaxy redshifts!

$z = 0.4$

$z = 0.8$

$z = 0$

$z = 1.6$

$z = 2.0$

$z = 3.0$

Spectroscopic galaxy surveys

imaging surveys (2014 - 2019) + WISE (IR)

target selection

spectroscopic observations

spectra and redshift measurements

two steps: photometry and spectroscopy ⇒ selection effects
catalog of angular positions $\mathrm{R.A.}, \mathrm{Dec.}$ and redshifts $z$

Survey selection function

specify the survey selection function $\bar{n}$ $\Rightarrow$ account for systematic effects due to photometry/spectroscopy

Expected density without clustering = angular & radial footprint

Survey selection function $\bar{n}$

survey selection function $\bar{n}$

Taken from DESI Collaboration et al. 2024

Photometry

photometric surveys: images of the sky, taken with different filters
mainly characterized by their depth (magnitude corresponding to given probability of source detection), seeing ($\in$ size of PSF)
pipeline for source detection and source fitting (flux, shape, etc.)
Legacy Surveys: https://www.legacysurvey.org/viewer

Taken from Zhao et al. (2020)

From left to right: data, model, residual. From Dey et al. (2019) (DECaLS DR8).

Target selection

to obtain objects of certain class (luminous red galaxies, emission line galaxies, quasars...) in a redshift range
typically with cuts in colour and magnitude
usually, tradeoff between target density and purity

Taken from Zhao et al. (2020)

c) high-z

b) star / low-z rejection

d) [OII]

Left: taken from Raichoor et al. (2022). Right: taken from DESI Collaboration et al. (2016).

Photometric systematics

veto masks in selection function n̄ for bright objects, (variable) stars, bad pixels...
target density varies with observational conditions: depth,
seeing, galaxy extinction, star density... to be modelled

Taken from Zhao et al. (2020)

Left: masks on a legacypipe $0.25^\circ × 0.25^\circ$ brick.

Taken from Raichoor et al. (2020).

Fiber assignment

Fiber-fed spectrographs: fibers cannot be too close (e.g. 62'′ for SDSS) / "fixed density of fibers"
- SDSS-I < IV: hand-plugged fibers
- 2dFGRS, WiggleZ: fibers positioned by a robot
- SDSS V, DESI: robotic positioner for each fiber

Taken from Zhao et al. (2020)

Credit: SDSS

Credit: DESI

Fiber assignment

density-dependent effect: correlates with clustering

Taken from Zhao et al. (2020)

Individual galaxy weights not sufficient:

pairwise inverse probability weights (PIP) (e.g. Bianchi and Percival 2017): rerun fiber assignment many times with different random seeds
$\theta$-cut: remove all small scale angular pairs (e.g. Pinon et al. 2024)

$0.05^\circ \simeq$ positioner patrol diameter

Spectroscopic measurements

Taken from Zhao et al. (2020)

grims disperse light onto the focal plane
reduction of 2D traces into 1D
spectrum fit with a basis of archetypes / PCA templates
criterion to select reliable redshifts

wavelength

fiber number

$z = 2.1$ QSO

$z = 0.9$ ELG

Ly$\alpha$

CIV

CIII

[OII] doublet at $3727 \AA$ up to $z = 1.6$

[OII]

Ly$\alpha$ at $1216 \AA$ down to $z = 2.0$

Redshift failures

Taken from Zhao et al. (2020)

redshift efficiency involves the response of the telescope,
spectrograph and redshift determination pipeline
may vary with spectroscopic observing conditions / instrument
corrected by a weight

Taken from Krolewski et al. 2024

Spectroscopic surveys: take-aways

Taken from Zhao et al. (2020)

$\bar{n}$ varies due to photometry and spectroscopy:
- angular photometric systematics
- fibre assignment
- redshift failures
Understanding $\bar{n}$ is key to reliable clustering measurements
Effects of systematics tested on fast simulations: mocks

Spectroscopic surveys: take-aways

Taken from Zhao et al. (2020)

$\bar{n}$ varies due to photometry and spectroscopy:
- angular photometric systematics
- fibre assignment
- redshift failures
Understanding $\bar{n}$ is key to reliable clustering measurements
Effects of systematics tested on fast simulations: mocks

Taken from Zhao et al. (2020)

Outline

Past and current surveys
Clustering observables
Spectroscopic surveys and systematics
Large-scale structure formation
BAO and RSD theory models
Current constraints
Other clustering analyses

Galaxy - matter relation

How do galaxies "populate" the (dark) matter density field?

\Omega_\mathrm{cdm}

\Omega_\mathrm{b}

credit: ESA

galaxies:

14% of $\Omega_\mathrm{b}$

3% of $\Omega_\mathrm{m}$

Galaxy bias

galaxy formation in two steps (White and Rees 1978):
- dark matter forms halos = gravitationally-bound structures
- gas cools down, baryons aggregate into galaxies
galaxies trace the density field, in large overdensities → bias

Galaxy bias

galaxy formation in two steps (White and Rees 1978):
- dark matter forms halos = gravitationally-bound structures
- gas cools down, baryons aggregate into galaxies
galaxies trace the density field, in large overdensities → bias

on large scales

halo model (e.g. Seljak 2000):
- galaxies ⇔ DM halos with halo occupation distribution
- DM halos ⇔ DM field with halo bias (Press and Schechter 1974, Bardeen et al. 1986, Sheth and Tormen 1999)

\delta_\mathrm{g} = \orange{b_1} \delta + \epsilon \Rightarrow P_\mathrm{gg}(k) = \orange{b_1}^2 P_\mathrm{\delta\delta}(k) + N_\epsilon

linear bias parameter

Poisson noise

Galaxy bias

galaxy formation in two steps (White and Rees 1978):
- dark matter forms halos = gravitationally-bound structures
- gas cools down, baryons aggregate into galaxies
galaxies trace the density field, in large overdensities → bias

bias expansion (e.g. McDonald and Roy 2009):

\delta_\mathrm{g} = \orange{b_1} \delta + \orange{b_2} \delta^2 + \orange{b_s} s^2 + \orange{b_\nabla} \nabla \delta + \cdots

second order bias

shear bias

non-local bias

shear bias

s^2 = s_{ij} s_{ji} \text{ with } s_{ij} = \left[\nabla_i \nabla_j \nabla^{-2} - \delta_{ij}^K / 3\right] \delta

Structure formation

matter is described as a collisionless fluid that evolves only through gravitation, in an expanding Universe
Gravity:

First two moments of the Vlasov-Poisson equation:

\frac{\partial f}{\partial \eta} + \frac{\mathbf{p}}{ma} \cdot \frac{\partial f}{\partial \mathbf{x}} - am \nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{p}} = 0

\Delta \Phi = a^{2} 4\pi G_{N} \bar{\rho} \, \delta = \frac{3}{2} a^2 H^{2} \Omega_\mathrm{m} \, \delta

\delta^\prime + \nabla_i \left[(1 + \delta) u_i \right] = 0

continuity equation

Euler equation

u_i^\prime + a H u_i + u_j \nabla_j u_i = - \nabla_i \Phi - \frac{1}{\rho} \nabla_j (\rho \sigma_{ij})

anisotropic stress tensor, sourced by multi-streams / shell-crossing

conformal time derivative $d\eta = dt / a$

velocity

Poisson equation

gravitational potential

Perturbation theory: first order

Linear order Eulerian PT

First order Lagrangian PT = Zeldovich approximation

\frac{\partial f}{\partial \eta} + \frac{\mathbf{p}}{ma} \cdot \frac{\partial f}{\partial \mathbf{x}} - am \nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{p}} = 0

\delta^{\prime\prime} + a H \delta^\prime - \frac{3}{2} \Omega_\mathrm{m} \delta = 0

\orange{D_{+}(a) = a}

linear growth factor

\theta = \nabla_i u_i = \orange{f} a H \delta

velocity divergence

\orange{f = \frac{d \ln D}{d \ln a}} \simeq \Omega_\mathrm{m}^{0.55}

logarithmic growth rate

\mathbf{x}(\eta) = \mathbf{q} + \mathbf{\Psi}(\mathbf{q}, \eta)

\mathbf{\Psi}(\mathbf{k}, \eta) = -\frac{i \mathbf{k}}{k^2} D_+(\eta) \delta_0(\mathbf{k})

\delta = D_{+} \delta_0

decreasing mode, and growing mode

Zeldovich displacement

Lagrangian picture

Perturbation theory: first order

Linear order Eulerian PT

First order Lagrangian PT = Zeldovich approximation

\frac{\partial f}{\partial \eta} + \frac{\mathbf{p}}{ma} \cdot \frac{\partial f}{\partial \mathbf{x}} - am \nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{p}} = 0

\delta^{\prime\prime} + a H \delta^\prime - \frac{3}{2} \Omega_\mathrm{m} \delta = 0

\orange{D_{+}(a) = a}

linear growth factor

\theta = \nabla_i u_i = \orange{f} a H \delta

velocity divergence

\orange{f = \frac{d \ln D}{d \ln a}} \simeq \Omega_\mathrm{m}^{0.55}

logarithmic growth rate

\mathbf{x}(\eta) = \mathbf{q} + \mathbf{\Psi}(\mathbf{q}, \eta)

\mathbf{\Psi}(\mathbf{k}, \eta) = \frac{i \mathbf{k}}{k^2} D_+(\eta) \delta_0(\mathbf{k})

\delta = D_{+} \delta_0

decreasing mode, and growing mode

Zeldovich displacement

Lagrangian picture

initial positions

Perturbation theory: higher order

Write $\delta$ and $\theta$ as an expansion:

\frac{\partial f}{\partial \eta} + \frac{\mathbf{p}}{ma} \cdot \frac{\partial f}{\partial \mathbf{x}} - am \nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{p}} = 0

\delta(\mathbf{k}) = D_+ \delta^{(1)}(\mathbf{k}) + \cdots + D_+^n \delta^{(n)}(\mathbf{k}) + \cdots

\delta^{(n)}(\mathbf{k}) = \int \frac{d^3 q_1 \cdots d^3 q_n}{(2\pi)^{3(n - 1)}} \delta_D(\mathbf{k} - \mathbf{q}_1 - \cdots - \mathbf{q}_n) \orange{F^{(n)}(\mathbf{q}_1, \cdots, \mathbf{q}_n)} \delta_0(\mathbf{q}_1) \cdots \delta_0(\mathbf{q}_n)

perturbation theory kernels

(geometrical functions of $\mathbf{q}$'s that can be computed recursively)

Diagrammatic representation, up to order 3:

$\mathbf{q}_1$

$\mathbf{q}_2$

$\mathbf{q}_1$

$\mathbf{q}_2$

$\mathbf{q}_3$

$\mathbf{k}$

Using Wick's theorem, compute any $n$-point statistics, e.g. $P_{\delta\delta}(\mathbf{k})$ at 1-loop

Taken from Crocce & Scoccimarro

Perturbation theory: higher order

Write $\delta$ and $\theta$ as an expansion:

\frac{\partial f}{\partial \eta} + \frac{\mathbf{p}}{ma} \cdot \frac{\partial f}{\partial \mathbf{x}} - am \nabla \Phi \cdot \frac{\partial f}{\partial \mathbf{p}} = 0

\delta(\mathbf{k}) = D_+ \delta^{(1)}(\mathbf{k}) + \cdots + D_+^n \delta^{(n)}(\mathbf{k}) + \cdots

perturbation theory kernels

(geometrical functions of $\mathbf{q}$'s that can be computed recursively)

Diagrammatic representation, up to order 3:

$\mathbf{q}_1$

$\mathbf{q}_2$

$\mathbf{q}_1$

$\mathbf{q}_2$

$\mathbf{q}_3$

$\mathbf{k}$

Using Wick's theorem, compute any $n$-point statistics, e.g. $P_{\delta\delta}(\mathbf{k})$ at 1-loop

Taken from Crocce & Scoccimarro

\langle \delta_{a_1}(\mathbf{k}_1) \cdots \delta_{a_{2n}}(\mathbf{k}_{2n}) \rangle = \sum_{\text{all pair associations}} \prod_{\text{pairs } (i,j) \in p} \langle \delta_{a_i}(\mathbf{k}_i)\, \delta_{a_j}(\mathbf{k}_j) \rangle

Taken from Zhao et al. (2020)

N-body simulations

solve numerically the Vlasov-Poisson equations for the dark matter fluid by sampling the phase-space with particles

N-body simulations

Credit: The AbacusSummit Team

Dark matter particles virialize into dark matter halos
Halo finders (Friend-of-Friend, Spherical Overdensity, Rockstar...)

Adapted from Hadzhiyska et al. 2021

Taken from Zhao et al. (2020)

Halo occupation distribution

specify the probability to find $N$ galaxies in a halo of mass $M$

Halo occupation distribution

Right: HOD measured on the outputs of two semi-analytical models (GALFORM and LGALAXIES) run on the Millennium simulation. Taken from Contreras et al. (2013).

split between central and satellite galaxies
also sample galaxy velocities
many extensions (assembly bias = dependence on other properties (e.g. local density or shear)

Taken from Zhao et al. (2020)

halos

emission line galaxies

Credit: Mathilde Pinon

Halo occupation distribution

LSS formation: take-aways

linear matter power spectrum

LSS formation: take-aways

evolved matter power spectrum ($z = 0.8$)

LSS formation: take-aways

galaxy power spectrum ($z = 0.8$)

Interlude: the primordial power spectrum

The initial power spectrum (of density fluctuations after the inflaction) is:

P_{\delta\delta}^\mathrm{ini}(k) \propto \orange{A_s} \left(\frac{k}{k_\mathrm{pivot}}\right)^{\orange{n_\mathrm{s}}}

scalar index

amplitude

Early Universe is dominated by radiation:
- $k \ll H a \equiv \mathcal{H}$, $\delta$ grows as $a$ (scale factor)
- $k \gg \mathcal{H}$ (Jeans scale), growth suppressed by radiation pressure: $\delta$ constant or logarithmically growing
At $z_\mathrm{eq}$, matter - radiation equality: $\delta$ grows as $a$

$\Rightarrow$ characteristic "equality" scale $k_\mathrm{eq} = \mathcal{H}_\mathrm{eq}$

Encoded in the transfer function

P_{\delta\delta}(k) = \orange{T(k)}^2 P_{\delta\delta}^\mathrm{ini}(k)

\orange{T(k) \propto \left\lbrace \begin{align*} 1 & \qquad k \ll k_\mathrm{eq}\\ (k_\mathrm{eq} / k)^2 & \qquad k \gg k_\mathrm{eq} \end{align*} \right.}

Interlude: the primordial power spectrum

The initial power spectrum (of density fluctuations after the inflaction) is:

P_{\delta\delta}^\mathrm{ini}(k) \propto \orange{A_s} \left(\frac{k}{k_\mathrm{pivot}}\right)^{\orange{n_\mathrm{s}}}

scalar index

amplitude

P_{\delta\delta}(k) = \orange{T(k)}^2 P_{\delta\delta}^\mathrm{ini}(k)

Early Universe is dominated by radiation:
- $k \ll H a \equiv \mathcal{H}$, $\delta$ grows as $ \mathcal{H}^{-2} $
- $k \gg \mathcal{H}$ (Jeans scale), growth suppressed by radiation pressure: $\delta$ constant or logarithmically growing
After $z_\mathrm{eq}$, matter - radiation equality: $\delta$ grows as $ a \propto \mathcal{H}^{-2} $

$\Rightarrow$ characteristic "equality" scale $k_\mathrm{eq} = \mathcal{H}_\mathrm{eq}$

Encoded in the transfer function

Interlude: the primordial power spectrum

The initial power spectrum (of density fluctuations after the inflaction) is:

P_{\delta\delta}^\mathrm{ini}(k) \propto \orange{A_s} \left(\frac{k}{k_\mathrm{pivot}}\right)^{\orange{n_\mathrm{s}}}

scalar index

amplitude

Early Universe is dominated by radiation:
- $k \ll H a \equiv \mathcal{H}$, $\delta$ grows as $ \mathcal{H}^{-2} $
- $k \gg \mathcal{H}$ (Jeans scale), growth suppressed by radiation pressure: $\delta$ constant or logarithmically growing
After $z_\mathrm{eq}$, matter - radiation equality: $\delta$ grows as $ a \propto \mathcal{H}^{-2} $

$\Rightarrow$ characteristic "equality" scale $k_\mathrm{eq} = \mathcal{H}_\mathrm{eq}$

Encoded in the transfer function

P_{\delta\delta}(k) = \orange{T(k)}^2 P_{\delta\delta}^\mathrm{ini}(k)

\orange{T(k) \propto \left\lbrace \begin{align*} 1 & \qquad k \ll k_\mathrm{eq}\\ (k_\mathrm{eq} / k)^2 & \qquad k \gg k_\mathrm{eq} \end{align*} \right.}

Interlude: the primordial power spectrum

Features

peak

oscillations

What are the noticeable features in $\xi_\mathrm{gg}(s)$ or $P_\mathrm{gg}(k)$?

Baryon acoustic oscillations (BAO)

Sound waves in primordial plasma

At recombination ($z \simeq 1100$)

plasma changes to optically thin
baryons decouple from photons
sound wave stalls after travelling $r_\mathrm{d}$

Sound horizon scale at the drag epoch

$r_\mathrm{d} \simeq 150\; \mathrm{Mpc}$

Baryon acoustic oscillations (BAO)

CMB ($z \simeq 1100$)

At recombination ($z \simeq 1100$)

plasma changes to optically thin
baryons decouple from photons
sound wave stalls after travelling $r_\mathrm{d}$

Sound horizon scale at the drag epoch

$r_\mathrm{d} \simeq 150\; \mathrm{Mpc}$

Sound waves in primordial plasma

Baryon acoustic oscillations (BAO)

CMB ($z \simeq 1100$)

Baryon acoustic oscillations (BAO)

Thanks to Julian Bautista!

BAO

Credits: CAASTRO, https://www.youtube.com/watch?v=jpXuYc-wzk4

Baryon acoustic oscillations (BAO)

distribution of galaxies (cartoonish)

\theta_\mathrm{BAO} = r_\mathrm{d} / D_\mathrm{M}(z)

transverse comoving distance

sound horizon $r_\mathrm{d}$

Let's measure:

angle on the sky (transverse to the line-of-sight): $\theta_\mathrm{BAO} = \orange{r_\mathrm{d}}/\green{D_\mathrm{M}(z)}$
$\Delta z$ (along the line-of-sight): $ \Delta z_\mathrm{BAO} = r_\mathrm{d} / D_\mathrm{H}(z) = \green{H(z)} \orange{r_\mathrm{d}} / c $

Baryon acoustic oscillations (BAO)

Let's measure:

angle on the sky (transverse to the line-of-sight): $\theta_\mathrm{BAO} = \orange{r_\mathrm{d}}/\green{D_\mathrm{M}(z)}$
$\Delta z$ (along the line-of-sight): $ \Delta z_\mathrm{BAO} = \orange{r_\mathrm{d}} / \green{D_\mathrm{H}(z)} = \green{H(z)} \orange{r_\mathrm{d}} / c $

distribution of galaxies (cartoonish)

\Delta z_\mathrm{BAO} = r_\mathrm{d} / D_\mathrm{H}(z)

Hubble distance $c/H(z)$

sound horizon $r_\mathrm{d}$

Baryon acoustic oscillations (BAO)

Let's measure:

angle on the sky (transverse to the line-of-sight): $\theta_\mathrm{BAO} = \orange{r_\mathrm{d}}/\green{D_\mathrm{M}(z)}$
$\Delta z$ (along the line-of-sight): $ \Delta z_\mathrm{BAO} = \orange{r_\mathrm{d}} / \green{D_\mathrm{H}(z)} = \green{H(z)} \orange{r_\mathrm{d}} / c $
at multiple redshifts $z$

Probes the expansion history ($\green{D_\mathrm{M}, D_H}$), hence the energy content (e.g. dark energy)

Absolute size at $z = 0$: $H_0 \orange{r_\mathrm{d}}$

z_1

z_2

z_3

Baryon acoustic oscillations (BAO)

correlation function

\alpha_\mathrm{iso} \propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d}

BAO peak

line-of-sight

monopole

isotropic

comoving transverse distance

Hubble distance $c/H(z)$

sound horizon (standard ruler)

Baryon acoustic oscillations (BAO)

isotropic

anisotropic

\propto D_{\mathrm{M}}(z) / D_\mathrm{H}(z)

BAO peak

line-of-sight

monopole

quadrupole

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d}

line-of-sight

Q: What can make the BAO look anisotropic?

Alcock-Paczynski effect

R.A., Dec., $z$ $\Rightarrow$ $\mathbf{x}$ with the true cosmology

Alcock-Paczynski effect

$\propto q_\parallel = D_\mathrm{H}^\mathrm{fid}(z) / D_\mathrm{H}(z) $

R.A., Dec., $z$ $\Rightarrow$ $\mathbf{x}$ with wrong (fiducial) cosmology

$\propto q_\perp = D_\mathrm{M}^\mathrm{fid}(z) / D_\mathrm{M}(z) $

Alcock-Paczynski effect

In the theory:

\left\lbrace \begin{align*} &q_\parallel = D_\mathrm{H}(z) / D_\mathrm{H}^\mathrm{fid}(z)\\ &q_\perp = D_\mathrm{M}(z) / D_\mathrm{M}^\mathrm{fid}(z) \end{align*} \right .

P^\mathrm{fid}(k_\parallel, k_\perp) = 1 / (\orange{q_\parallel q_\perp}^2) P(k_\parallel / \orange{q_\parallel}, k_\perp / \orange{q_\perp})

\Rightarrow P^\mathrm{fid}(k, \mu) = 1 / (\orange{q_\parallel q_\perp}^2) P(k^\prime, \mu^\prime)

rescaled in fiducial coordinates

\left\lbrace \begin{align*} k^\prime &= \frac{k}{\orange{q_\perp}} \left[ 1 + \mu^2 \left( \frac{\orange{q_\perp}^2}{\orange{q_\parallel}^2} - 1 \right) \right]^{1/2}\\ \mu^\prime &= \frac{\mu \orange{q_\perp}}{\orange{q_\parallel}} \left[ 1 + \mu^2 \left( \frac{\orange{q_\perp}^2}{\orange{q_\parallel}^2} - 1 \right) \right]^{-1/2} \end{align*}\right .

$\propto q_\parallel$

$\propto q_\perp $

Features

non-zero quadrupole!

What are the noticeable features in $\xi_\mathrm{gg}(s)$ or $P_\mathrm{gg}(k)$?

non-zero quadrupole!

Q: where do you think it (mainly) comes from?

Redshift space distortions (RSD)

observed redshifts ($z_\mathrm{obs}$) =

Hubble flow ($\blue{z_\mathrm{cosmo}}$)
+ peculiar velocities ($\orange{u_z/c}$)
+ (relativistic terms)

redshift-space positions ($\mathbf{s}$) =

real space position ($\blue{\mathbf{r}}$)

+ RSD shift ($\orange{u_z/H\mathbf{\hat{z}}}$)

Redshift space distortions (RSD)

observed redshifts ($z_\mathrm{obs}$) =

Hubble flow ($\blue{z_\mathrm{cosmo}}$)
+ peculiar velocities ($\orange{u_z/c}$)
+ (relativistic terms)

redshift-space positions ($\mathbf{s}$) =

real space position ($\blue{\mathbf{r}}$)

+ RSD shift ($\orange{u_z/H\mathbf{\hat{z}}}$)

u_z \propto f\sigma_8

Redshift space distortions (RSD)

observed redshifts ($z_\mathrm{obs}$) =

Hubble flow ($\blue{z_\mathrm{cosmo}}$)
+ peculiar velocities ($\orange{u_z/c}$)
+ (relativistic terms)

redshift-space positions ($\mathbf{s}$) =

real space position ($\blue{\mathbf{r}}$)

+ RSD shift ($\orange{u_z/H\mathbf{\hat{z}}}$)

$s = D_\mathrm{c}(z_\mathrm{obs})$

u_z \propto f\sigma_8

Redshift space distortions (RSD)

real-space

Credit: Mathilde Pinon

Redshift space distortions (RSD)

redshift-space

Credit: Mathilde Pinon

Redshift space distortions (RSD)

galaxy positions in redshift space: $\mathbf{s} = \mathbf{r} - v_z \hat{z}$ with $v_z = -\frac{\mathbf{u} \cdot \hat{z}}{H}$

mass conservation:

$[1 + \delta_s(\mathbf{s})] d^3s = [1 + \delta_r(\mathbf{r})] d^3r \implies \delta_s(\mathbf{s}) = \left[ 1 + \delta_r(\mathbf{r}) \right] \left| \frac{d^3 s}{d^3 r} \right|^{-1} - 1 $

power spectrum in redshift space:

Redshift space distortions (RSD)

galaxy positions in redshift space: $\mathbf{s} = \mathbf{r} - v_z \hat{z}$ with $v_z = -\frac{\mathbf{u} \cdot \hat{z}}{H}$

mass conservation:

$[1 + \delta_s(\mathbf{s})] d^3s = [1 + \delta_r(\mathbf{r})] d^3r \implies \delta_s(\mathbf{s}) = \left[ 1 + \delta_r(\mathbf{r}) \right] \left| \frac{d^3 s}{d^3 r} \right|^{-1} - 1 $

power spectrum in redshift space:

P_s(\mathbf{k}) = \int d^3x e^{i \mathbf{k} \cdot \mathbf{x}} \left\langle \textcolor{red}{e^{-i k_\mu \Delta v_z}} \textcolor{blue}{\left[ \delta_r(\mathbf{r}) + \partial_z v_z(\mathbf{r}) \right] \left[ \delta_r(\mathbf{r} + \mathbf{x}) + \partial_z v_z(\mathbf{r} + \mathbf{x}) \right]} \right\rangle

Redshift space distortions (RSD)

galaxy positions in redshift space: $\mathbf{s} = \mathbf{r} - v_z \hat{z}$ with $v_z = -\frac{\mathbf{u} \cdot \hat{z}}{H}$

mass conservation:

$[1 + \delta_s(\mathbf{s})] d^3s = [1 + \delta_r(\mathbf{r})] d^3r \implies \delta_s(\mathbf{s}) = \left[ 1 + \delta_r(\mathbf{r}) \right] \left| \frac{d^3 s}{d^3 r} \right|^{-1} - 1 $

power spectrum in redshift space:

Kaiser: $\delta_r + \partial_z v_z \rightarrow (b_1 + f \mu^2)\delta$ in linear theory, enhancement on large scales

Finger-of-God: $e^{-ik_\mu \Delta v_z}$ damping on scales $\lesssim 3\, \mathrm{Mpc}$

Kaiser

Redshift space distortions (RSD)

galaxy positions in redshift space: $\mathbf{s} = \mathbf{r} - v_z \hat{z}$ with $v_z = -\frac{\mathbf{u} \cdot \hat{z}}{H}$

mass conservation:

$[1 + \delta_s(\mathbf{s})] d^3s = [1 + \delta_r(\mathbf{r})] d^3r \implies \delta_s(\mathbf{s}) = \left[ 1 + \delta_r(\mathbf{r}) \right] \left| \frac{d^3 s}{d^3 r} \right|^{-1} - 1 $

power spectrum in redshift space:

Kaiser: $\delta_r + \partial_z v_z \rightarrow (b_1 + f \mu^2)\delta$ in linear theory, enhancement on large scales

Finger-of-God: $e^{-ik_\mu \Delta v_z}$ damping on scales $\lesssim 3\, \mathrm{Mpc}$

Finger-of-God

Measurement of $f\sigma_8$

$P_s(k, \mu) = (b_1 + f \mu^2)^2 P_{\delta\delta}(k) = b_1^2 (1 + \beta \mu^2)^2 P_{\delta\delta}(k)$

Kaiser model (= linear order)

with $\beta = f / b_1$. Equivalently:

P_s(k, \mu) = \left(\beta^{-1} + \mu^2\right)^2 P_{\theta\theta}(k)

(for historical reasons) at a pivot point of $8\;\mathrm{Mpc}/h$
$= f \sigma_8$ with $f = \frac{d \ln D}{d \ln a} \simeq \Omega_\mathrm{m}^{0.55}$ within ΛCDM
probe matter density $\Omega_\mathrm{m}$ / test of general relativity

Typically bias is marginalised over:

effectively measure the (amplitude of) the velocity divergence power spectrum $P_{\theta\theta}(k)$

BAO and RSD: take-aways

RSD

BAO

Anisotropic correlation function or power spectrum of galaxies. Sensitive to:

RSD: $f\sigma_8$ ⇒ energy content, test of general relativity
BAO: $D_M / r_\mathrm{d} , D_H /r_\mathrm{d}$ ⇒ energy content
more generally: primordial power spectrum (inflation, matter-radiation equality), neutrinos, etc.

Taken from Zhao et al. (2020)

Outline

Past and current surveys
Clustering observables
Spectroscopic surveys and systematics
Large-scale structure formation
BAO and RSD theory models
Current constraints
Other clustering analyses

Taken from Zhao et al. (2020)

BAO model

measure only the position of the BAO peak: robust
split fiducial power spectrum $P(k)$ into no-wiggle $P_\mathrm{nw}(k)$ and wiggles $P_\mathrm{w}(k) = P(k) - P_\mathrm{nw}(k)$
marginalize over the shape: broadband parameters (polynomials)
adjust position of wiggles $P_\mathrm{w}(k^\prime)$

Chen, Howlett et al. 2024

P_\mathrm{gg}(k, \mu) = \green{\mathcal{B}(k, \mu) P_\mathrm{nw}(k)} + \mathcal{C}(k^\prime, \mu^\prime) \red{P_\mathrm{w}(k^\prime)} + \blue{\mathcal{D}(k, \mu)}

"no-wiggle Kaiser"

correlation function = Hankel transform(power spectrum)

\left\lbrace \begin{align*} k^\prime &= \frac{k \orange{\alpha_\mathrm{ap}}^{1/3}}{\orange{\alpha_\mathrm{iso}}} \left[ 1 + \mu^2 \left( \orange{\alpha_\mathrm{ap}}^{-2} - 1 \right) \right]^{1/2}\\ \mu^\prime &= \frac{\mu}{\orange{\alpha_\mathrm{ap}}} \left[ 1 + \mu^2 \left( \orange{\alpha_\mathrm{ap}}^{-2} - 1 \right) \right]^{-1/2} \end{align*} \right.

Non-linear structure growth and peculiar velocities blur and shrink (slightly) the ruler

Eisenstein et al. 2008, Padmanabhan et al. 2012

BAO reconstruction

reconstruction

Estimates Zeldovich displacements from observed field and moves galaxies back: refurbishes the ruler (improves precision and accuracy)

reconstruction

BAO reconstruction

Credit: DESI

Taken from Zhao et al. (2020)

Full Shape models

Unbiased measurement of amplitude $f\sigma_8$ ⇒ accurate model for the full shape power spectrum.

Various approaches (% accuracy at $z = 1$):

WiggleZ (Blake, 2010): Halofit $P_\mathrm{m}(k)$ + Kaiser + FoG
in BOSS/eBOSS (< 2020): perturbation theory models:
- power spectrum: standard/regularized PT (RPT, RegPT Taruya et al. 2012), RSD (Taruya et al. 2010), bias expansion (McDonald and Roy 2009) ($k < 0.2 \; h/\mathrm{Mpc}$)
- correlation function: Gaussian streaming model (Reid and White 2011) ($s > 30 \; \mathrm{Mpc}/h$)

Taken from Zhao et al. (2020)

Full Shape models

Unbiased measurement of amplitude $f\sigma_8$ ⇒ accurate model for the full shape power spectrum.

Various approaches (% accuracy at $z = 1$):

WiggleZ (Blake, 2010): Halofit $P_\mathrm{m}(k)$ + Kaiser + FoG
in BOSS/eBOSS (< 2020): perturbation theory models:
- power spectrum: standard/regularized PT (RPT, RegPT Taruya et al. 2012), RSD (Taruya et al. 2010), bias expansion (McDonald and Roy 2009) ($k < 0.2 \; h/\mathrm{Mpc}$)
- correlation function: Gaussian streaming model (Reid and White 2011) ($s > 30 \; \mathrm{Mpc}/h$)
in DESI, Euclid, effective field theory: small-scale-sourced counterterm to regularize loop integrals (pybird, CLASS-PT, velocileptors, folps...) ($k < 0.25 \; h/\mathrm{Mpc}$)

Taken from Zhao et al. (2020)

EFT models

P_{s, g}(k, \mu) = \blue{P^\mathrm{PT}(k, \mu)} + \orange{(b + f\mu^2) (b \alpha_0 + f\alpha_2 \mu^2 + f\alpha_4 \mu^4) k^2 P_{s, b_1^2}(k)} + \purple{\mathrm{SN}_0 + \mathrm{SN}_2 k^2 \mu^2 + \mathrm{SN}_4 k^4 \mu^4}

perturbation theory term = $f(P_\mathrm{lin}, f)$

linear and quasi-linear physics

counter-terms contribution

truncation of perturbative series

stochastic-terms contribution

small-scale galaxy physics

The Effective Field Theory in a nutshell

perturbation theory model + counter-terms and stochastic terms
dependence on cosmology into $P_\mathrm{lin}(k)$, $f$ and Alcock-Paczynski transform ($\mathrm{R.A.}, \mathrm{Dec.}, z \Rightarrow \text{fiducial coordinates}$)

Taken from Zhao et al. (2020)

Full Shape models

Unbiased measurement of amplitude $f\sigma_8$ ⇒ accurate model for the full shape power spectrum.

Various approaches (% accuracy at $z = 1$):

WiggleZ (Blake, 2010): Halofit $P_\mathrm{m}(k)$ + Kaiser + FoG
in BOSS/eBOSS (< 2020): perturbation theory models:
- power spectrum: standard/regularized PT (RPT, RegPT Taruya et al. 2012), RSD (Taruya et al. 2010), bias expansion (McDonald and Roy 2009) ($k < 0.2 \; h/\mathrm{Mpc}$)
- correlation function: Gaussian streaming model (Reid and White 2011) ($s > 30 \; \mathrm{Mpc}/h$)
in DESI, Euclid, effective field theory: small-scale-sourced counterterm to regularize loop integrals (pybird, CLASS-PT, velocileptors, folps...) ($k < 0.25 \; h/\mathrm{Mpc}$)
hybrid PT/HOD models, e.g. Hand et al. 2017 ($k < 0.4 \; h/\mathrm{Mpc}$)
simulation-based models, e.g. SimBig Lemos et al. 2023

Taken from Zhao et al. (2020)

Mock challenge

Test the theoretical model accuracy against simulations (mocks)

Mock challenge

Taken from Findlay et al. 2024

fitted cosmological parameters

HOD-variations for each tracer (conformity, assembly bias, etc.)

Theory models: take-aways

Taken from Zhao et al. (2020)

BAO models: fit the position of the BAO feature, marginalize over the shape of the power spectrum ("broadband")
Full Shape models: full shape of the power spectrum, linear and quasi-linear scales. State-of-the-art of standard analyses: EFT models
tested with N-body mock challenge

Likelihood

Taken from Zhao et al. (2020)

We usually assume a Gaussian likelihood

\ln{p(\mathbf{d} | \mathbf{\theta})} \propto -\frac{1}{2} (\mathbf{d} - \orange{\mathbf{t}}(\red{\mathbf{\theta}}))^T \green{\mathrm{C}}^{-1} (\mathbf{d} - \orange{\mathbf{t}}(\red{\mathbf{\theta}}))

theory model

data vector

($P_\ell(k)$ or $\xi_\ell(s)$)

parameters

covariance matrix

Full Shape fit: $A_\mathrm{s} \text{ or }\sigma_8, \omega_\mathrm{cdm}, h, n_\mathrm{s}$
BAO fit: $\alpha_\mathrm{iso}, \alpha_\mathrm{ap}$

+ bias or "nuisance" parameters

analytic or based on fast simulations

We sample the posterior $p(\red{\mathbf{\theta}} | \mathbf{d}) \propto p(\mathbf{d} | \red{\mathbf{\theta}}) \red{p(\mathbf{\theta})}$

prior

In a nutshell

Taken from Zhao et al. (2020)

galaxy catalog

galaxy power spectrum (or correlation function)

cosmological constraints

compression = "we measure specific features"

e.g. BAO model $\Rightarrow$ $\alpha_\mathrm{iso}, \alpha_\mathrm{ap}$

"variance of the density field as a function of scale"

Full Shape

Taken from Zhao et al. (2020)

Outline

Past and current surveys
Clustering observables
Spectroscopic surveys and systematics
Large-scale structure formation
BAO and RSD theory models
Current constraints
Other clustering analyses

Taken from Zhao et al. (2020)

DESI DR1 BAO

6dFGRS

SDSS (MGS)

SDSS (BOSS/eBOSS)

WiggleZ

\alpha_\mathrm{iso} \propto D_\mathrm{V}(z) / r_\mathrm{d} = (z D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d}

\alpha_\mathrm{AP}^{-1} \propto D_\mathrm{M}(z) / D_\mathrm{H}(z)

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

DESI DR2 BAO

\alpha_\mathrm{iso} \propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

\orange{D_c(z)} = \int_0^z \frac{c dz^\prime}{\green{H(z^\prime)}} \qquad \text{with} \qquad \green{H(z)} = \purple{H_0} \sqrt{\blue{\Omega_\mathrm{m}} (1 + z)^3 + 1 - \Omega_\mathrm{m}}

Reminder:

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

\propto \alpha_\mathrm{AP}^{-1} = (D_{\mathrm{M}}(z) / D_\mathrm{H}(z)) = f(\Omega_\mathrm{m})

DESI DR2 BAO

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

DESI DR2 BAO

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

DESI DR2 BAO

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

DESI DR2 BAO

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

DESI DR2 BAO

\propto (D_{\mathrm{M}}^{2}(z) D_\mathrm{H}(z))^{1/3} / r_\mathrm{d} = f(\Omega_\mathrm{m}) / (H_0 r_\mathrm{d})

DESI DR2 BAO

Consistent with each other,

and complementary

\underbrace{\begin{align*} \Omega_\mathrm{m} &= 0.2975 \pm 0.0086 & \mathbf{(3.0\%)} \\ H_{0} r_\mathrm{d} &= (101.54 \pm 0.73) \, [100 \, \mathrm{km} \, \mathrm{s}^{-1}] & \mathbf{(0.7\%)} \end{align*}}_{\textstyle \text{\color{black}{DESI}}}

DESI BAO prefers slightly higher $H_0r_\mathrm{d}$, slightly lower $\Omega_\mathrm{m}$ than Planck (and ACT DR6 and SPT-3G)
$2.3 \sigma$ discrepancy with primary CMB¹ + CMB lensing²

1. Planck PR4 CamSpec

2. Planck PR4 + ACT DR6 lensing

DESI DR2 BAO

BAO constrains $\Omega_\mathrm{m}$, $h \times r_d(\Omega_\mathrm{b}h^2, \Omega_\mathrm{m}h^2 - \Omega_\nu h^2)$
Calibrating BAO relative distance measurements using BBN $\Omega_\mathrm{b} h^2$

\underbrace{ H_0 = (68.51 \pm 0.58) \, {\rm km\,s^{-1}\,Mpc^{-1}} }_{\textstyle \color{darkblue}{\text{DESI} + \text{BBN}}}

\underbrace{ H_0 = (68.45 \pm 0.47) \, {\rm km\,s^{-1}\,Mpc^{-1}} }_{\textstyle \color{orange}{\text{DESI} + \theta_\ast + \text{BBN}}}

Adding very precise CMB acoustic angular scale

In $4.5\sigma$ tension with SH0ES (Breuval+24) (independently of the CMB)

DESI DR2 BAO

Dark energy fluid

No strong preference for dark energy evolution: $1.7\sigma$ from DESI data alone

$\Lambda$

p / \rho = w = w_0 + w_a (1 - a)

pressure

density

CPL

DESI DR2 BAO

Combining all DESI + CMB + SN

\underbrace{ w_{0} = -0.838 \pm 0.055 \qquad w_{a} = -0.62^{+0.22}_{-0.19} }_{\textstyle \textcolor{blue}{\text{DESI + CMB + Pantheon+} \; \implies \; 2.8\sigma}}

\underbrace{ w_{0} = -0.667 \pm 0.088 \qquad w_{a} = -1.09^{+0.31}_{-0.27} }_{\textstyle \color{orange}{\text{DESI + CMB + Union3} \; \implies \; 3.8\sigma}}

\underbrace{ w_{0} = -0.752 \pm 0.057 \qquad w_{a} = -0.86^{+0.23}_{-0.20} }_{\textstyle \color{green}{\text{DESI + CMB + DES-SN5YR} \; \implies \; 4.2\sigma}}

Internal CMB degeneracies limiting precision on the sum of neutrino masses

Broken by BAO, which favors low $\Omega_\mathrm{m}$

\sum m_\nu < 0.064 \, \mathrm{eV} \; (95\%, \text{DESI + CMB})

DESI DR2 BAO

Taken from Zhao et al. (2020)

DESI DR1 Full Shape

DESI DR2 Full Shape results are not yet published! Come back next year ;) In the meantime, let's use DR1!

Taken from Zhao et al. (2020)

DESI DR1 Full Shape + BAO

$\omega_\mathrm{b}$: BBN, $n_\mathrm{s} \sim \mathcal{G}(0.9649, 0.042^2)$

Taken from Zhao et al. (2020)

DESI DR1 Full Shape + BAO

$\omega_\mathrm{b}$: BBN, $n_\mathrm{s} \sim \mathcal{G}(0.9649, 0.042^2)$

Taken from Zhao et al. (2020)

DESI DR1 Full Shape + BAO

$\omega_\mathrm{b}$: BBN, $n_\mathrm{s} \sim \mathcal{G}(0.9649, 0.042^2)$

Taken from Zhao et al. (2020)

DESI DR1 Full Shape + BAO

$\omega_\mathrm{b}$: BBN, $n_\mathrm{s} \sim \mathcal{G}(0.9649, 0.042^2)$

Taken from Zhao et al. (2020)

DESI DR1 Full Shape + BAO

$\omega_\mathrm{b}$: BBN, $n_\mathrm{s} \sim \mathcal{G}(0.9649, 0.042^2)$

Taken from Zhao et al. (2020)

DESI DR1 Full Shape + BAO

$\omega_\mathrm{b}$: BBN, $n_\mathrm{s} \sim \mathcal{G}(0.9649, 0.042^2)$

Taken from Zhao et al. (2020)

Consistency with SDSS
In agreement with CMB
Weak lensing prefers lower $S_8$, but still consistent
FS measurement competitive with weak lensing

$S_8 = \sigma_8 (\Omega_\mathrm{m} / 0.3)^{0.5}$ best constrained by weak lensing surveys

\underbrace{ S_8 = 0.836 \pm 0.035 }_{\textstyle \text{\color{blue}{DESI + BBN + $n_{s10}$}}}

DESI DR1 Full Shape + BAO

Taken from Zhao et al. (2020)

Modified gravity constraints

In general relativity, $\green{\mu(a, k)} = \green{\Sigma(a, k)} = 1$

To test GR, introduce $\green{\mu_0, \Sigma_0}$

\mu(a) = 1 + \frac{\Omega_\Lambda(a)}{\Omega_\Lambda} \green{\mu_0}, \Sigma(a) = 1 + \frac{\Omega_\Lambda(a)}{\Omega_\Lambda} \green{\Sigma_0}

Perturbed FLRW metric

$ds^2=a(\tau)^2[-(1+2\orange{\Psi})d\tau^2+(1-2\orange{\Phi})\delta_{ij}dx^i dx^j]$

At late times:

(mass) $k^2\orange{\Psi} = -4\pi G a^2 \green{\mu(a,k)} \blue{\sum_i\rho_i\Delta_i}$

(light) $k^2(\orange{\Phi} + \orange{\Psi})=-8\pi G a^2 \green{\Sigma(a,k)} \blue{\sum_i\rho_i\Delta_i}$

gravitational potentials

density perturbations

Taken from Zhao et al. (2020)

$\Sigma_0$ constrained by

- CMB (ISW and lensing)

- galaxy lensing

\underbrace{\begin{align*} \mu_0 &= 0.04\pm 0.22 \\ \Sigma_0 &= 0.045\pm 0.046 \end{align*}}_{\textstyle \text{\color{purple}{DESI + CMB-nl + DESY3 ($3\times 2$-pt)}}}

\underbrace{ \mu_0 = 0.11^{+0.45}_{-0.54} }_{\textstyle \text{\color{blue}{DESI + BBN + $n_{\mathrm{s}10}$}}}

compared to CMB-nl + DESY3 (3x2pt) only: $\sigma(\mu_0) / 2.5$, $\sigma(\Sigma_0) / 2$

DESI constrains

Modified gravity constraints

Taken from Zhao et al. (2020)

Cosmological constraints - take aways

BAO

constrains distances / the Hubble rate $\Rightarrow$ energy content
compared to Planck: low $\Omega_\mathrm{m}$, high $H_{0}$
hint of dynamical dark energy (depending on SN dataset)

Adding Full Shape

probes structure growth
$\sigma_8, S_8$ consistent with Planck
modified gravity parameter $\mu_0$ consistent with GR

Primordial non-Gaussianity (left by inflation):

scale-dependent bias $\propto k^{-2}$
consistent with 0: $f_\mathrm{NL}^\mathrm{loc} = -3.6^{+9.0}_{-9.1}$ (Chaussidon et al. 2024)

Taken from Zhao et al. (2020)

Cosmological constraints - take aways

BAO

constrains distances / the Hubble rate $\Rightarrow$ energy content
compared to Planck: low $\Omega_\mathrm{m}$, high $H_{0}$
hint of dynamical dark energy (depending on SN dataset)

Adding Full Shape

probes structure growth
$\sigma_8, S_8$ consistent with Planck
modified gravity parameter $\mu_0$ consistent with GR

Primordial non-Gaussianity (left by inflation):

scale-dependent bias $\propto k^{-2}$
consistent with 0: $f_\mathrm{NL}^\mathrm{loc} = -3.6^{+9.0}_{-9.1}$ (Chaussidon et al. 2024)

Taken from Zhao et al. (2020)

Other clustering analyses

higher order correlation functions (3-pt, 4-pt...): e.g. Slepian et al. 2017, Hou et al. 2022, Philcox et al. 2022
alternative clustering statistics: e.g. density-split correlation function, 1D PDF, Wavelet Scattering Transforms, neural compression, e.g. Paillas et al. 2023, Beyond-2pt collaboration, Lemos et al. 2023
field-level inference of the galaxy density, e.g. Lavaux et al. 2019
cross-correlations: galaxy clustering x galaxy weak lensing, e.g. Chen et al. 2024, galaxy clustering x CMB weak lensing S ailer et al. 2024
continuous tracer Ly$\alpha$: probe HI density along line-of-sight ⇒ BAO D ESI 2025, P1D Ravoux et al. 2025
photometric surveys: BAO with DES Collaboration et al. 2025, Vera Rubin (LSST)

LSS formation galaxy bias

truth

samples

Hugo Simon 2025

Taken from Zhao et al. (2020)

Other clustering analyses

higher order correlation functions (3-pt, 4-pt...): e.g. Slepian et al. 2017, Hou et al. 2022, Philcox et al. 2022
alternative clustering statistics: e.g. density-split correlation function, 1D PDF, Wavelet Scattering Transforms, neural compression, e.g. Paillas et al. 2023, Beyond-2pt collaboration, Lemos et al. 2023
field-level inference of the galaxy density, e.g. Lavaux et al. 2019
cross-correlations: galaxy clustering x galaxy weak lensing, e.g. Chen et al. 2024, galaxy clustering x CMB weak lensing S ailer et al. 2024
continuous tracer Ly$\alpha$: probe HI density along line-of-sight ⇒ BAO D ESI 2025, P1D Ravoux et al. 2025
photometric surveys: BAO with DES Collaboration et al. 2025, Vera Rubin (LSST)

Introduction_to_galaxy_clustering_2025

By Arnaud De Mattia

Introduction_to_galaxy_clustering_2025