Enhancing constraints on modified gravity and inflation with - - PowerPoint PPT Presentation

Enhancing constraints

• n modified gravity and inflation

with multi-tracer cosmological surveys

arXiv:1505.04106 (MNRAS 2016) 1403.5237 (J-PAS white paper) 1302.5444 (MNRAS 2013) 1108.5449 (MNRAS 2012)

• L. Raul Abramo

Instituto de Fisica & LabCosmos Universidade de São Paulo

http://www.iag.usp.br/labcosmos/en/

Galaxy surveys are evolving

We used to live in an

era of shot (“counts”) noise

We are now in the age of cosmic variance and systematics

[Finding galaxies was the limiting factor] [Volume and control are the limiting factors]

SDSS

Surveys of large-scale structure are now limited by cosmic variance and systematics

However, up to any given redshift there is only a finite volume. Moreover, we are reaching closer to the limits of the observable Universe!

Anderson et al. [BOSS] 1312.4877

P(k)/Pdewig(k)

Shot noise:

finite number of counts of the tracers

• f the underlying density field

(Poisson statistics)

Cosmic variance:

finite volume inside which we can estimate the amplitudes and phases of the (Gaussian) random modes of the density field

Pg(~ k) ' b2

g Pm(~

k) + 1 ¯ ng Clustering in units of shot noise ¯ ng Pg(~ k) ' ¯ ng b2

g Pm(~

k) + 1

SNR noise

Fisher information of galaxy surveys

0.1 0.5 1.0 5.0 10.0 50.0 0.01 0.02 0.05 0.10 0.20 0.50 1.00 L

shot noise Cosmic Variance

Pg

$$$ more galaxies

Signal/Noise: Clustering strength

• f galaxy type “g” in

redshift space

happiness Information ∼ ✓ Pg 1 + Pg ◆2 Pg(~ x,~ k) ≡ ¯ ng(~ x) ⇥ bg(z, k) + f(z) µ2

k

⇤2 G2(z) Pm(k)

Feldman, Kaiser & Peacock 1994 (FKP) Tegmark et al. 1997, 1998

Fisher information in phase space

On each unit volume of phase space there is a certain amount of information about the clustering strength: F[log Pg] × ∆Vx ∆Vk (2π)3 = 1 2 ✓ Pg 1 + Pg ◆2 × ∆Vx ∆Vk (2π)3 The precision (SNR) with which we can estimate the clustering strength is: k x

phase space volume = .

∆V

P2

g

σ2(Pg) = F[log Pg] × ∆V = 1 2 ✓ Pg 1 + Pg ◆2 ∆V

Bandpower z-slice/volume

Fisher information/ (phase space volume) FKP 1994 Hamilton 2005

• R. A. 2012

F = 1 2 ✓ signal signal + noise ◆2 ≤ 1 2

The Universe has many different types of galaxies, halos, etc…

Clustering in position space Clustering in Fourier space

bias bias

Let’s say we have several (α = 1,2, ... N) different types of tracers of large-scale

• structure. E.g. : α=1=LRGs , α=2=ELGs , α=3=quasars , etc.

Multi-tracer Fisher information matrix

R.A. 2012 R.A. & Katie Leonard 2013

Pα(k, µk; z) = nα(z) ⇥ bα(z) + f(z) µ2

k

⇤2 P(k; z) Fα β = F(log Pα, log Pβ) = 1 4  δαβ Pα P 1 + P + Pα Pβ (1 − P) (1 + P)2

• ,

P = X

α

Pα µk = k|| k

The Fisher matrix for the N clustering strengths (power spectra) is:

Multi-tracer Fisher information

= ⇒ 1 4   

P2

1 (1−P)

(1+P)2 + P1P 1+P P1P2(1−P) (1+P)2 P1P2(1−P) (1+P)2 P2

2 (1−P)

(1+P)2 + P2P 1+P

  

= ⇒ 1 4        

P2

1 (1−P)

(1+P)2 + P1P 1+P P1P2(1−P) (1+P)2 P1P3(1−P) (1+P)2 P1P2(1−P) (1+P)2 P2

2 (1−P)

(1+P)2 + P2P 1+P P2P3(1−P) (1+P)2 P1P3(1−P) (1+P)2 P2P3(1−P) (1+P)2 P2

3 (1−P)

(1+P)2 + P3P 1+P

       

F[log Pα, log Pβ] = 1 4 PαPβ(1 − P) (1 + P)2 + δαβ PαP 1 + P

• F = 1

2 ✓ P 1 + P ◆2

OK… but are we in fact gaining any information by splitting galaxies into types,

• r are we just “shuffling around” the information?

Multi-tracer technique: Seljak 2008 McDonald & Seljak 2008 Gil-Marín et al. 2011 Hamaus et al. 2011,2012 Cai & Bernstein 2011

Fisher matrix: R.A. 2012 R.A. & K. Leonard 2013

1 tracer 2 tracers 3 tracers

Multi-tracer Fisher information

Yes, we gain information

> In fact, with multiple tracers the Fisher information is unbounded! We can diagonalize the multi-tracer Fisher matrix by changing variables: ⇒ (hyper) spherical coordinates!

           P1 P2 P3            =            x2 y2 z2            ⇐ ⇒            r2 tan2 θ tan2 φ            =            P

P3 P1+P2 P2 P1

           Relative

clustering strengths

Total

clustering strength

P1 → x2 P3 → z2 P2 → y2

In “spherical" coordinates (i.e., using the total clustering strength and the relative clustering strengths) the Fisher matrix becomes diagonal! E.g.: three species of tracers

FSph = 8 > > < > > :

1 2

⇣

P 1+P

⌘2

1 4 P2 1+P sin2 θ cos2 θ 1 4 P2 1+P sin2 θ sin2 φ cos2 φ

9 > > = > > ;

Relative clusterings: information ~ ☛ unbounded ☛ extra information! Total clustering: < 1/2

P = X

α

¯ nαb2

αPm

Multi-tracer Fisher information matrix

Why?

Cosmic variance is only inherited through the spectrum

By comparing the clustering between different tracers of large-scale structure (e.g.: LRGs, ELGs, etc.), we can measure with arbitrary accuracy* the physical parameters that distinguish the different clustering strengths:

P2 = n2 (b2 + f µ2

k)2 P(k; z)

P1 = n1 (b1 + f µ2

k)2 P(k; z)

P1 P2 = n1 (b1 + f µ2

k)2

n2 (b2 + f µ2

k)2

Cosmic variance does not apply:

• *bias *RSDs
• *PNGs *HOD

The key: high number densities of distinct types of tracers (red galaxies, blue galaxies, emission-line galaxies, quasars, etc.)

Seljak 2008 McDonald & Seljak 2008

0.1 0.5 1.0 5.0 10.0 50.0 100.0 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00

1

1 , 2

P2 = 0.5 P2 = 1 P2 = 2 P2 = 4 P2 = 8

Total clustering Relative clustering

FP = P2 σ2(P) = 1 2 P2 (1 + P)2 FR = (P2/P1)2 σ2(P2/P1) = 1 4 P1 P2 1 + P

FR FP

Simplest example: two types of tracers of large-scale structure

Where the hell are we going to get all those galaxies — with decent redshifts??

JPCam

• factory acceptance (Nov’14)
• J-PAS

J-PAS

1/5 of full sky (8500 deg2) ~3 mags > SDSS 𝜏z ~ 0.003(1+z) survey starts in Q1 2017 !

Benítez et al., 1403.5237 Benítez et al. 2016 (to appear)

2017 2018 2019 2020 2021 2022 2023 J-PAS J-PAS J-PAS J-PAS1 J-PAS J-PAS2

DESI (?) DESI DESI DESI

Euclid Euclid Euclid Euclid

Massive & deep multi-tracer survey with J-PAS

0.5 1.0 1.5 2.0 2.5 3.0 0.1 1 10 100

z PR , PE , PQ

ELGs LRGs QSOs

@ k=0.1 h/Mpc

* GAMA - Blake et al., MNRAS 2013 : P1 >10 for z<0.25 * Radio galaxies & SKA - Ferramacho et al. 2014, Camera et al. 2015, … * 21cm intensity mapping - Bull, Ferreira, Patel & Santos 2015 * SKA + optical surveys - Fonseca, Camera, Santos & Maartens 2015 * DESI, Euclid…

cosmic variance shot noise

0.2 0.4 0.6 0.8 1.0 1.2 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 z f 0.2 0.4 0.6 0.8 1.0 1.2 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 z f 0.2 0.4 0.6 0.8 1.0 1.2 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 z f 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 z

• marg. ΣHfΣ8 L ê HfΣ8 L

0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 z

• marg. ΣHfΣ8 L ê HfΣ8 L

0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 z

• marg. ΣHfΣ8 L ê HfΣ8 L

LRGs ELGs Combined — 𝚳CDM

• - 𝜹 ± 20%

— LRGs — ELGs — Combined

No prior on bias Weak (~20%) prior on bias Strong (~5%) prior on bias

Application: RSDs in J-PAS

Marginalized* errors on matter growth rate

* Marginalized 7 "global" cosmological parameters (𝜵m, h, etc.) + 5 parameters on each redshift slice

Pg = ng (bg + f µ2

k)2 P(k; z)

f = d ln G d ln a ' Ωγ

m

~modified gravity (𝜹GR ≅ 0.55)

J-PAS forecast for constraint on 𝛿: 𝜏(𝛿)=0.025

R.A. & Leonard 2013 Benítez et al. 2014

J-PAS

J-PAS constraints on local non-Gaussianity parameter fNL

ELGs only LRGs only QSOs only Total

Information from relative clustering can improve constraints

• n fNL by ~5 at low-z!

0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 Relat.êTotal

0.5 1.0 1.5 2.0 2.5 3.0

• 5
• 4
• 3
• 2
• 1

z Log10@ FHfNLL D

Pg = ng (bg + f µ2

k)2 P(k; z)

F(θ) = 1 σ2

c(θ)

bg → bg + ∆bg(fNL, k)

J-PAS

R.A. & Leonard 2013 Benítez et al. 2014

0.5 1.0 2.0 1 5 10 50 100 500 z

• cumul. marg. ΣHfNLL

0.5 1.0 2.0 1 5 10 50 100 500 z

• cumul. marg. ΣHfNLL

0.5 1.0 2.0 1 5 10 50 100 500 z

• cumul. marg. ΣHfNLL

— LRGs — ELGs — quasars — Combined

fNL is almost unaffected by marginalization w.r.t. bias The k-dependence of 𝜠bNL~ fNL x k -2 helps break the degeneracy Cumulative uncertainty on fNL when the redshift slices are combined

No prior on bias Weak (~20%) prior on bias Strong (~5%) prior on bias 1𝜏 : ~2

WARNING: this is Fisherology — not robust w.r.t. systematics

Planck: 𝜏(fNL)~5 arXiv: 1502.01592

J-PAS

R.A. & Leonard 2013 Benítez et al. 2014

How to do it in practice

Lucas ➭ UPenn Arthur ➭ UCL

Types and positions

• f galaxies

P(k), BAOs RSDs NGs, etc.

From catalogs to the power spectrum

Fourier analysis of galaxy surveys

Given a galaxy catalog ng(x), the optimal estimator for the spectrum (FKP) is:

δng(x) ¯ ng = δg(x) − → fg(x) = wF KP (x) δg(x)

The FKP weights express the best compromise between cosmic variance and shot noise:

wF KP = 1 1 + Pg ¯ ng Bg Bg = bg(z) + f(z)µ2

k + . . .

Feldman, Kaiser & Peacock, 1994 (FKP)

The FKP estimator is optimal — it is unbiased, and it saturates the Cramér-Rao bound:

Cov[ ˆ Pg(ki), ˆ Pg(kj)] → [Fisher]−1

The estimated spectrum for a Fourier bin ki (the bandpower) is then:

ˆ Pg(ki) = 1 N h| ˜ fg|2iki ˜ fg(k) = FFt[fg(x)]

Fourier analysis of multi-tracer surveys

Given any number of galaxy catalogs n𝞶(x), the weighted fields are:

R.A., L. Secco & A. Loureiro, MNRAS 2016

The multi-tracer weights are:

wµν =  δµν − Pµ 1 + P

• ¯

nνBν δµ(x) − → fµ(x) = X

ν

wµν δν(x)

The estimated auto-spectra are:

ˆ Pµ(ki) = X

ν

N −1

µν h| ˜

fν|2iki

These estimators are optimal: their covariance is the inverse of the Fisher matrix!

Cov[ ˆ Pµ, ˆ Pν] = [Fµν]−1

Testing and validating the multi-tracer estimators

Case ¯ n1 (h3 Mpc−3) b1 ¯ n2 (h3 Mpc−3) b2 A

• 1. 10−2

1.0

• 1. 10−2

1.2 B

• 1. 10−2

1.0

• 1. 10−5

1.2 C

• 1. 10−5

1.0

• 1. 10−5

1.2

* Volume = (1280 h-1 Mpc)3 = (128 x 10 h-1 Mpc)3 * 103 lognormal realizations * Planck-vanilla fiducial parameters

Theoretical spectra

P1(k) P2(k)

✓ Unbiased

R.A., L. Secco & A. Loureiro, MNRAS 2016

Testing and validating the multi-tracer estimators

Case ¯ n1 (h3 Mpc−3) b1 ¯ n2 (h3 Mpc−3) b2 A

• 1. 10−2

1.0

• 1. 10−2

1.2 B

• 1. 10−2

1.0

• 1. 10−5

1.2 C

• 1. 10−5

1.0

• 1. 10−5

1.2

✓ Unbiased ✓ Optimal

tracer 2 (multi-tracer) tracer 2 (FKP) tracer 1 (multi-tracer) tracer 1 (FKP)

• - - /- - - Theory (FKP)

—— /—— Theory (multi-tracer)

Framework to combine data from all surveys to fully exploit the science: DE, MoG, inflation, …

• Emission line galaxies
• LRGs
• Ly-breaks
• Ly-𝛽 forest
• Quasars/AGNs
• HI intensity maps
• Sub-mm galaxies
• Galaxies ⇢ halos
• …

The End