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

History
Observables and cosmological information
Spectroscopic surveys and systematics
Standard (RSD and BAO) analyses
Current constraints
On-going and future surveys
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

History
Observables and cosmological information
Spectroscopic surveys and systematics
Standard (RSD and BAO) analyses
Current constraints
On-going and future surveys
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

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 (previoulsy nebulae) using cepheids
early imaging surveys carried out with photographic plates:
- 1949 - 1958: Palomar Observatory Sky Survey (POSS I)
- 1974 - 1999: UKST sky surveys, Siding Spring, Australia
- 1985 - 2000: POSS II

Cataloguing galaxies

1924, Mount Wilson: Edwin Hubble shows the existence of
galaxies (previoulsy nebulae) using cepheids
early imaging surveys carried out with photographic plates:
- 1949 - 1958: Palomar Observatory Sky Survey (POSS I)
- 1974 - 1999: UKST sky surveys, Siding Spring, Australia
- 1985 - 2000: POSS II
galaxy catalogs:
- 1961 - 1968: Zwicky Catalog of galaxies based on POSS I
- 1967: Lick catalog of 1 M 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

How to go 3D?

R.A.

Dec.

R.A.

Dec.

With spectroscopic surveys!

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

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, 18 000 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)

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)

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

Outline

History
Observables and cosmological information
Spectroscopic surveys and systematics
Standard (RSD and BAO) analyses
Current constraints
On-going and future surveys
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?

An example: Paris metro & bus stations

Credits: Etienne Burtin

Counting pairs

Randoms

Randoms

Credits: Etienne Burtin

A bit of formalism

\(n_g(\mathbf{x}) = \bar{n}(\mathbf{x})\left[1 + \delta_g(\mathbf{x}) \right]\) \(\delta_g\) density contrast

Density of galaxies

Probability to find:

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

A bit of formalism

\(n_g(\mathbf{x}) = \bar{n}(\mathbf{x})\left[1 + \delta_g(\mathbf{x}) \right]\) \(\delta_g\) density contrast

Density of galaxies

Probability to find:

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

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

\(\xi_{gg}(\mathbf{s}) = \left\langle \delta_g(\mathbf{x}_1) \delta_g(\mathbf{x}_1 + \mathbf{s}) \right\rangle\)

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

Galaxy correlation function

Independent of position assuming spatial homogeneity.

Correlation function

SDSS eBOSS DR16 LRG correlation function

Power spectrum

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

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

\((2\pi)^3 \delta_D^{(3)}(\mathbf{k} + \mathbf{k}') P_{gg}(\mathbf{k}) = \langle \delta_g(\mathbf{k}) \delta_g(\mathbf{k}') \rangle\)

Galaxy power spectrum

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

Power spectrum

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

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

\((2\pi)^3 \delta_D^{(3)}(\mathbf{k} + \mathbf{k}') P_{gg}(\mathbf{k}) = \langle \delta_g(\mathbf{k}) \delta_g(\mathbf{k}') \rangle\)

Galaxy power spectrum

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

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

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

Power spectrum

SDSS eBOSS DR16 LRG power spectrum

Galaxy - matter relation

How are \(\xi_{gg}(\mathbf{s})\) and \(P_{gg}(\mathbf{k})\) related to the (theory) matter power spectrum?

Galaxy bias

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

perturbative expansion; at linear order \(\delta_g = b_1 \delta\), \(\delta\) matter field

\Rightarrow \xi_{gg,r}(\mathbf{s}) = b_1^2 \xi_{\delta\delta}(\mathbf{s}) \qquad P_{gg,r}(\mathbf{k}) = b_1^2 P_{\delta\delta}(\mathbf{k})

bottom-up approach (e.g. Seljak 2000):
- galaxies ⇔ DM halos with halo occupation distribution
- DM halos ⇔ DM field with halo bias (Sheth and Tormen 1999)

Features

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

Features

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

peak

oscillations

Baryon acoustic oscillations (BAO)

before photon decoupling (CMB emission), acoustic waves propagated in a plasma of baryons and photons

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

Baryon acoustic oscillations (BAO)

before photon decoupling (CMB emission), acoustic waves propagated in a plasma of baryons and photons
baryon acoustic oscillations (BAO) imprinted in the CMB...

Credits: Esa & Planck

Baryon acoustic oscillations (BAO)

...and in the matter distribution at a fixed comoving scale: sound horizon at the drag epoch \(r_\mathrm{d}\)

Baryon acoustic oscillations (BAO)

...and in the matter distribution at a fixed comoving scale: sound horizon at the drag epoch \(r_\mathrm{d}\)

= standard ruler

Baryon acoustic oscillations (BAO)

In practice: catalog of angular positions and redshifts, so we constrain:

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

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

transverse comoving distance

sound horizon \(r_d\)

Baryon acoustic oscillations (BAO)

In practice: catalog of angular positions and redshifts, so we constrain:

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

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

Hubble distance

sound horizon \(r_d\)

Baryon acoustic oscillations (BAO)

In practice: catalog of angular positions and redshifts, so we constrain:

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

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

Absolute size at \(z = 0\): \(H_0 \green{r_d}\)

z_1

z_2

z_3

Features

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

non-zero quadrupole

Redshift space distortions (RSD)

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

Hubble flow (\(z_\mathrm{cosmo}\))
+ peculiar velocities (\(u_z/(ac)\))
+ (relativistic terms)

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

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

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

Redshift space distortions (RSD)

dependence in \(\mu = \cos \theta\) cosine angle to the line-of-sight
Legendre expansion in multipoles (usually truncated at \(\ell = 4\)):

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

Redshift space distortions (RSD)

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

mass conservation:

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

Redshift space distortions (RSD)

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

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

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 \(\lesssim 3\, \mathrm{Mpc}\)

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

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 \(f = \frac{d \ln D}{d \ln a} \simeq \Omega_m^{0.55}\) within ΛCDM
probe matter density \(\Omega_m\) / test of general relativity

Typically bias is marginalised over:

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

Summary

Measurement of the 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.

Standard clustering analyses

Estimators

How to compute \(\xi_{gg}\) or \(P_{gg}\) from a galaxy catalog?

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 \(\mathrm{s}\)

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 \(\mathrm{s}\)

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

Natural estimator

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

Davis and Peebles 1983 estimator

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

Hamilton 1993 estimator

Correlation function estimators

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

Natural estimator

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

Davis and Peebles 1983 estimator

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

Hamilton 1993 estimator

\(\hat{\xi}_g(\mathbf{s}) = \frac{DD(\mathbf{s}) - 2DR(\mathbf{s}) + RR(\mathbf{s})}{RR(\mathbf{s})}\) 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

\(F(\mathbf{x}) = n_g(\mathbf{x}) − \bar{n}(\mathbf{x})\)

Density fluctuations

Yamamoto 2006 estimator

\hat{P}_{\ell}(k_{\mu}) = \frac{2\ell+1}{A V_{k_{\mu}}} \int_{V_{k_{\mu}}} d^{3}k F_{\ell}^{\star}(\mathbf{k}) F_{0}(\mathbf{k}) - P_{\ell}^{\mathrm{noise}}(k_{\mu})

with:

F_{\ell}(\mathbf{k}) = \int d^{3}x e^{i \mathbf{k} \cdot \mathbf{x}} F(\mathbf{x}) \mathcal{L}_\ell(\mathbf{\hat{k}} \cdot \mathbf{\hat{x}}) \qquad A = \int d^{3} x \bar{n}^{2}(\mathbf{x})

Power spectrum estimator

\(F(\mathbf{x}) = n_g(\mathbf{x}) − \bar{n}(\mathbf{x})\)

Density fluctuations

Yamamoto 2006 estimator

\hat{P}_{\ell}(k_{\mu}) = \frac{2\ell+1}{A V_{k_{\mu}}} \int_{V_{k_{\mu}}} d^{3}k F_{\ell}^{\star}(\mathbf{k}) F_{0}(\mathbf{k}) - P_{\ell}^{\mathrm{noise}}(k_{\mu})

with:

F_{\ell}(\mathbf{k}) = \int d^{3}x e^{i \mathbf{k} \cdot \mathbf{x}} F(\mathbf{x}) \mathcal{L}_\ell(\mathbf{\hat{k}} \cdot \mathbf{\hat{x}}) \qquad A = \int d^{3} x \bar{n}^{2}(\mathbf{x})

\(F_\ell(\mathbf{k})\) ⇒ simple Fourier transforms
\(F\) (galaxies, randoms) painted on a mesh for FFT
aliasing effects
shotnoise \(P_\ell^\mathrm{noise}(k_\mu) \simeq \delta_{\ell 0}^{K} \bar{n}^{-1}\) : subtracted as galaxies assumed to be a ∼ Poisson sampling of the dark matter field

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)

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

Measurement covariance

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

\begin{align*} \mathrm{cov}_{\ell \ell^{\prime}}(k) \simeq \, & 2 (2\pi)^{3} \frac{(2\ell + 1)(2\ell^{\prime} + 1)}{A^{2}\Delta V_{k}} \\ &\times \int d^{3} x \bar{n}(\mathbf{x})^{4} \left[P(k,\mu) + \frac{1}{\bar{n}(\mathbf{x})}\right]^{2} \mathcal{L}_{\ell}(\mu) \mathcal{L}_{\ell^{\prime}}(\mu) \end{align*}

with \(\mu = \mathbf{\hat{k}} \cdot \mathbf{\hat{x}}\)

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

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

with \(\mu = \mathbf{\hat{k}} \cdot \mathbf{\hat{x}}\)

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

Grieb et al. 2015

Measurement covariance

In the uniform \(\bar{n}\) limit:

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

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

Credit: DESI

\mathrm{cov}_{\ell\ell^{\prime}}(k) \simeq 2 (2\pi)^{3} \frac{(2\ell + 1)(2\ell^{\prime} + 1)}{\Delta V_{k} \textcolor{blue}{V_{s}}} \int d\mu \left[P(k,\mu) + \frac{1}{\textcolor{red}{\bar{n}}}\right]^{2} \mathcal{L}_{\ell}(\mu) \mathcal{L}_{\ell^{\prime}}(\mu)

Outline

History
Observables and cosmological information
Spectroscopic surveys and systematics
Standard (RSD and BAO) analyses
Current constraints
On-going and future surveys
Other clustering analyses

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 \(\bar{n}\) ⇒ account for systematic effects due to photometry/spectroscopy

Taken from Zhao et al. (2020)

Expected density without clustering = angular & radial footprint

Survey selection function

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 (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

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 collisions

Fiber-fed spectrographs: fibers cannot be too close (e.g. 62'′ for SDSS) / "fixed density of fibers"
- SDSS-I < IV: hand-plugged fibers
- 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 collisions

density-dependent effect: correlates with clustering

Taken from Zhao et al. (2020)

Pinon et al. 2024

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

Taken from Zhu et al. 2015

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 Raichoor et al. 2020

Summary

Taken from Zhao et al. (2020)

\(\bar{n}\) varies due to photometry and spectroscopy:

- angular photometric systematics

- fibre collisions

- redshift failures

Understanding \(\bar{n}\) is key to reliable clustering measurements.
Effects of systematics tested on fast simulations: mocks.

Survey selection function

Taken from Zhao et al. (2020)

Outline

History
Observables and cosmological information
Spectroscopic surveys and systematics
Standard (RSD and BAO) analyses
Current constraints
On-going and future surveys
Other clustering analyses

Taken from Zhao et al. (2020)

Alcock-Paczynski effect

angular positions \((\mathrm{R.A.}, \mathrm{Dec.})\) and redshifts \(z\) of galaxies converted into Cartesian coordinates assuming a fiducial cosmology (\(\mathrm{fid}\))

In the model, wavevectors in the true cosmology ⇒ fiducial cosmology, multiply by:

q_{\parallel} = \frac{D_H(z)}{D_H^{\text{fid}}(z)} \quad q_{\perp} = \frac{D_M(z)}{D_M^{\text{fid}}(z)}

Taken from Zhao et al. (2020)

BAO model

only measuring 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 \(P_\mathrm{w}(k^\prime = k / \alpha)\)

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)} + \orange{\mathcal{D}(k, \mu)}

"no-wiggle Kaiser"

\alpha \simeq \frac{r_{\text{drag}}^{\text{fid}}/D_V^{\text{fid}}}{r_{\text{drag}}/D_V} = \alpha_{\parallel}^{1/3} \alpha_{\perp}^{2/3}

Taken from Zhao et al. (2020)

BAO model

only measuring 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 \(P_\mathrm{w}(k^\prime = k / \alpha)\)
correlation function: Hankel transform of \(P_{gg}(k)\)

\alpha \simeq \frac{r_{\text{drag}}^{\text{fid}}/D_V^{\text{fid}}}{r_{\text{drag}}/D_V} = \alpha_{\parallel}^{1/3} \alpha_{\perp}^{2/3}

Taken from Zhao et al. (2020)

BAO model

If enough S/N: also fit quadrupole to measure anisotropic BAO \(\alpha_\parallel / \alpha_\perp = D_\mathrm{H} / D_\mathrm{M}\)

\(P_{gg}\)

\(\xi_{gg}\)

\(P_{gg}\)

\propto D_{V} / r_\mathrm{d}

\propto D_{\mathrm{H}} / D_\mathrm{M}

monopole

quadrupole

monopole

quadrupole

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

Eisenstein et al. 2008, Padmanabhan et al. 2012

BAO reconstruction

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

reconstruction

BAO reconstruction

Taken from Zhao et al. (2020)

RSD 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 \(P_m(k)\) + 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)

RSD models

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

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

for DESI, Euclid, effective field theory: small scale sourced counterterm to regularize loop integrals (pybird, CLASS-PT, velocileptors, folps...) (\(k < 0.25 \; h/\mathrm{Mpc}\))

\begin{align*} P^{\text{EFT}}_s(k, \mu) &= P_{\delta\delta}(k) + 2f_0 \mu^2 P_{\delta\theta}(k) + f_0^2 \mu^4 P_{\theta\theta}(k) + A^{\text{TNS}}(k,\mu) + D(k,\mu) \\ &+ \left( \alpha_0 + \alpha_2 \mu^2 + \alpha_4 \mu^4 \right) k^2 P_{\text{lin}}(k) \\ &+ P_{\text{shot}} \left[ \alpha_0^{\text{shot}} + \alpha_2^{\text{shot}} (k \mu)^2 \right] \end{align*}

counterterms

stochastic terms

SPT

Maus et al. 2024

Taken from Zhao et al. (2020)

RSD models

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

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

for DESI, Euclid, effective field theory: small scale sourced counterterm to regularize loop integrals (pybird, CLASS-PT, velocileptors, folps...) (\(k < 0.25 \; h/\mathrm{Mpc}\))

also 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

counterterms

stochastic terms

SPT

Taken from Zhao et al. (2020)

Testing models: N-body mock challenge

Snapshot of the OuterRim simulation at \(z = 0\). Taken from Heitmann et al. (2019).

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

N-body simulations

\(10,240^3\) DM particles, \(1.85 \times 10^9 M_\odot/h\)
particles clustered in halos, found with a Friend-of-Friend algorithm

OuterRim simulation

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

usually split between central and satellite galaxies
typically tuned to reproduce semi-analytic models of galaxy formation (SAM) and observational data

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).

other approach: sub-halo abundance matching (SHAM)

Taken from Zhao et al. (2020)

Mock challenge

Test the theoretical model accuracy against simulations (mocks)

Mock challenge

vary HOD shape, satellite fraction, positions and velocities, probability laws...

e.g.: ELG in halo outskirts,
infalling velocities

Taken from Orsi and Angulo (2018)

Summary

Taken from Zhao et al. (2020)

BAO models: marginalize over the shape of the power spectrum ("broadband")
RSD models: full shape of the power spectrum up to midly non-linear scales
tested with N-body mock challenge

Theory models

Taken from Zhao et al. (2020)

Outline

History
Observables and cosmological information
Spectroscopic surveys and systematics
Standard (RSD and BAO) analyses
Current constraints
On-going and future surveys
Other clustering analyses

Taken from Zhao et al. (2020)

BAO

6dFGRS

SDSS (MGS)

SDSS (BOSS/eBOSS)

WiggleZ

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

DESI Y1 BAO measurements

BAO (DESI)

DESI Y1 BAO measurements

BAO (DESI)

DESI Y1 BAO measurements

BAO (DESI)

DESI Y1 BAO measurements

BAO (DESI)

DESI Y1 BAO measurements

Consistent with each other,

and complementary

\underbrace{\begin{align*} \Omega_\mathrm{m} &= 0.295 \pm 0.015 & \mathbf{(5.1\%)} \\ H_{0} r_\mathrm{d} &= (101.8 \pm 1.3) \, [100 \, \mathrm{km} \, \mathrm{s}^{-1}] & \mathbf{(1.3\%)} \end{align*}}_{\textstyle \text{\color{black}{DESI}}}

BAO (DESI)

BAO constraints (DESI, SDSS) consistent with CMB (primary and lensing Planck Collaboration, 2018, ACT Collaboratio n, 2023, Carron, Mirmelstein, Lewis, 2022)

\underbrace{ \Omega_\mathrm{m} = 0.3069 \pm 0.0050 \; \mathbf{(1.6\%)} }_{\textstyle \text{\green{DESI + CMB}}}

BAO (DESI)

BAO constrains \( r_\mathrm{d}(\blue{\Omega_\mathrm{m}} h^{2}, \orange{\Omega_\mathrm{b} h^{2}}) h \)
\( \blue{\Omega_\mathrm{m}} \) constrained by BAO at different \(z\)
\(\orange{\Omega_\mathrm{b}h^2}\) can be constrained by light element abundance from Big Bang Nucleosynthesis (BBN): Schöneberg et al., 2024

\(\implies\) constraints on \(h\) i.e. \(H_0 = 100 h \; \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\)

BAO (DESI)

BAO constrains \( r_\mathrm{d}(\blue{\Omega_\mathrm{m}} h^{2}, \orange{\Omega_\mathrm{b} h^{2}}) h \)
\( \blue{\Omega_\mathrm{m}} \) constrained by BAO at different \(z\)
\(\orange{\Omega_\mathrm{b}h^2}\) can be constrained by light element abundance from Big Bang Nucleosynthesis (BBN): Schöneberg et al., 2024

\(\implies\) constraints on \(h\) i.e. \(H_0 = 100 h \; \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\)

\underbrace{ H_0 = (68.53 \pm 0.80) \, {\rm km\,s^{-1}\,Mpc^{-1}} }_{\textstyle \color{darkblue}{\text{DESI} + \text{BBN}}}

SDSS / DESI consistency
In agreement with CMB
In \(3.7 \sigma\) tension with SH0ES

BAO (DESI)

BAO + CMB measurements favor a flat Universe

\underbrace{ \Omega_\mathrm{m} = 0.0024 \pm 0.0016 \; \mathbf{(0.16\%)} }_{\textstyle \text{\green{DESI + CMB}}}

BAO (DESI)

Dark Energy fluid, pressure \(p\), density \(\rho\), Equation of State parameter \(w = p / \rho\)

BAO (DESI)

Dark Energy fluid, pressure \(p\), density \(\rho\), Equation of State parameter \(w = p / \rho\)

\underbrace{\begin{align*} \Omega_\mathrm{m} &= 0.3095 \pm 0.0065 & \mathbf{(2.1\%)} \\ w &= -0.997 \pm 0.025 & \mathbf{(2.5\%)} \end{align*}}_{\textstyle \text{\color{orange}{DESI + CMB + Pantheon+}}}

Assuming a constant EoS, DESI BAO fully compatible with a cosmological constant...

\underbrace{\begin{align*} \Omega_\mathrm{m} &= 0.293 \pm 0.015 & \mathbf{(5.1\%)} \\ w &= -0.99^{+0.15}_{-0.13} & \mathbf{(15\%)} \end{align*}}_{\textstyle \text{\color{darkblue}{DESI}}}

Constant EoS parameter \(w\)

BAO (DESI)

Dark Energy fluid, pressure \(p\), density \(\rho\), Equation of State parameter \(w = p / \rho\)

\underbrace{ w_{0} = -0.827 \pm 0.063 \qquad w_{a} = -0.75^{+0.29}_{-0.25} }_{\textstyle \blue{\text{DESI + CMB + Pantheon+} \; \implies \; 2.5\sigma}}

\underbrace{ w_{0} = -0.64 \pm 0.11\qquad w_{a} = -1.27^{+0.40}_{-0.34} }_{\textstyle \color{orange}{\text{DESI + CMB + Union3} \; \implies \; 3.5\sigma}}

\underbrace{ w_{0} = -0.727 \pm 0.067 \qquad w_{a} = -1.05^{+0.31}_{-0.27} }_{\textstyle \color{green}{\text{DESI + CMB + DES-SN5YR} \; \implies \; 3.9\sigma}}

Varying EoS

w(z) = w_{0} + \frac{z}{1 + z} w_{a} \qquad \text{(CPL)}

BAO (DESI)

Internal CMB degeneracies limiting precision on the sum of neutrino masses

Broken by BAO, especially through \(H_{0}\)

Low preferred value of \(H_{0}\) yields

\(\sum m_\nu < 0.072 \, \mathrm{eV} \; (95\%, \color{green}{\text{DESI + CMB})}\)

Limit relaxed for extensions to \(\Lambda\mathrm{CDM}\)

\(\sum m_\nu < 0.195 \, \mathrm{eV}\) for \(w_0w_a\mathrm{CDM}\)

Taken from Zhao et al. (2020)

RSD (SDSS)

DESI RSD results are not yet published! Come back next year ;)

In the meantime, let's use SDSS!

Taken from Zhao et al. (2020)

RSD (SDSS)

RSD constrain structure growth, competitive with weak lensing

Taken from Zhao et al. (2020)

RSD (SDSS)

RSD constrain structure growth, competitive with weak lensing

With DESI Y1...

Taken from Zhao et al. (2020)

RSD (SDSS)

RSD constrain structure growth, competitive with weak lensing
RSD constrain deviations to GR e.g.

complementary to weak lensing and CMB lensing

\begin{align*} k^{2} \Phi &= - 4 \pi G_\mathrm{N} a^{2} \left(1+\mu \right) \bar{\rho} \delta \\ k^{2} \left(\Phi + \Psi\right) &= - 8 \pi G_\mathrm{N} a^{2} \left(1+\Sigma\right) \bar{\rho} \delta \end{align*}

Taken from Zhao et al. (2020)

Outline

History
Observables and cosmological information
Spectroscopic surveys and systematics
Standard (RSD and BAO) analyses
Current constraints
On-going and future surveys
Other clustering analyses

Taken from Zhao et al. (2020)

Euclid (spectro)

\(15 000 \; \mathrm{deg}²\), 50M H\(\alpha\) emitters between \(0.9 < z < 1.8\)
slitless spectroscopy with NISP, R = \(\lambda / \Delta \lambda\) = 380
just launched!

NISP instrument. Euclid consortium

Taken from Zhao et al. (2020)

DESI

Mayall Telescope at Kitt Peak, AZ

focal plane 5000 fibers

wide-field corrector

6 lenses, FoV \(\sim 8~\mathrm{deg}^{2}\)

4 m mirror

fiber view camera

ten 3-channel spectrographs

49 m, 10-cable fiber run

Taken from Zhao et al. (2020)

DESI

Credit: NSF

Taken from Zhao et al. (2020)

DESI focal plane

Taken from Zhao et al. (2020)

DESI Y5 galaxy samples

Bright Galaxies: 14M

0 < z < 0.4

LRG: 8M

0.4 < z < 0.8

ELG: 16M

0.6 < z < 1.6

QSO: 3M

Lya \(1.8 < z\)

Tracers \(0.8 < z < 2.1\)

\(z = 0.4\)

\(z = 0.8\)

\(z = 0\)

\(z = 1.6\)

\(z = 2.0\)

\(z = 3.0\)

Y5 \(\sim 40\)M galaxy redshifts!

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 \(15,000\;\mathrm{deg}^2\)
Nancy-Grace-Roman / WFIRST (2025-2030): 20M H\(\alpha\) emitters (R = 70 − 140 and R = 450 − 850), slitless spectroscopy

Taken from Zhao et al. (2020)

Other clustering analyses

small scales: HOD + cosmology fitting, e.g. Chapman et al. 2021
large scales: primordial non-Gaussianity \(f_\mathrm{NL}\), scale-dependent \(\propto 1/k^2\) bias, e.g. Mueller et al. 2021
higher order correlation functions (3-pt, 4-pt...): e.g. Slepian et al. 2017
alternative clustering statistics: e.g. marked correlation function, density-split correlation function, e.g. Paillas et al. 2023
field-level inference of the galaxy: density Lavaux et al. 2019
cross-correlations: galaxy clustering x galaxy weak lensing DES Collaboration 2021, galaxy clustering x CMB weak lensing Kitanidis and White 2021
continuous tracer Ly\(\alpha\): probe HI density along line-of-sight ⇒ BAO du Mas des Bourboux et al. 2020, tomographic maps Ravoux et al. 2020
photometric surveys: BAO with DES Collaboration et al. 2021, Vera Rubin (LSST)

Taken from Zhao et al. (2020)

Summary

Future (on-going) surveys increase statistics by a factor ×20!

Challenge for treatment of observational and theoretical systematics
Development of new techniques to better exploit information: cross-correlation, higher order or alternative statistics, forward modelling of the density field