On relativistic effects and large scale cosmology In collaboration - - PowerPoint PPT Presentation

On relativistic effects and large scale cosmology

In collaboration with Enea Di Dio (INAF, Trieste) Matteo Viel (SISSA, Trieste) Carlo Baccigalupi (SISSA, Trieste) Ruth Durrer (Université de Genève) Eleonora Villa (SISSA, Trieste)

Francesca Lepori

Cosmology at large and small scales

Universitá di Cagliari (7 marzo 2017)

1

Introduction

Analysis of CMB anisotropies started the era of PRECISION COSMOLOGY

Credit: ESA and the Planck Collaboration.

◮ CMB offers a 2D map of the universe

− → LSS will provide 3D map of the distribution of galaxies: potentially richer information

Francesca Lepori | @Casteddu

2

Power spectrum in Fourier space NOT DIRECTLY OBSERVABLE

Credit: A. Challinor, A. Lewis (2011)

→ Assume a cosmology to convert observed redshift and angles into length scales. → Theoretical predictions are gauge-dependent.

Francesca Lepori | @Casteddu

3

Observable in Galaxy Surveys

Credit: M. Blanton and the Sloan Digital Sky Survey.

Which coordinates do we observe?

◮ Redshift z ◮ Direction of incoming photons n = (θ, φ)

dΩ dz

Francesca Lepori | @Casteddu

4

Galaxy Number Count

∆obs(n, z) = N(n, z) − ¯ N(z) ¯ N(z)

◮ Relate the observable to the local density of galaxies

N(n, z) = ρ(n, z) · V (n, z), ¯ N(z) = ¯ ρ(z) · ¯ V (z) ¯ ρ(z) ≈ ¯ ρ(¯ z) + ∂z ¯ ρ · δz

◮ In the linear regime

∆obs(n, z) = δg − 3 1 + z δz(n, z) + δV (n, z) ¯ V

• C. Bonvin, R. Durrer - What galaxy surveys really measure (2011)
• A. Challinor, A. Lewis - Linear power spectrum of observed source number counts (2011)

Francesca Lepori | @Casteddu

5

Physical interpretation of δz

The distance we measure between us and the bin depends on the motion of the galaxies inside the bin dΩ dz dΩ dz − → If in a redshift bin the galaxies are moving towards us with the same velocities, it will appear closer

Francesca Lepori | @Casteddu

6

Physical interpretation of δV

The direction of the incoming light is perturbed by the presence of intervening matter: fluctuation in the observed solid angle dΩ dz dΩ dz δθ

Francesca Lepori | @Casteddu

7

Relation with velocity and metric perturbation

∆obs(n, z) = ∆g + 1 H(z)∂r(V · n)

Standard terms

+(5 s(m∗, z)

Magnification bias!

− 2) r(z) r(z) − r 2r(z)r ∆Ω(Φ + Ψ)dr +

• H′

H2 + 2 − 5s(m∗, z) rH + 5s(m∗, z)

• (V · n) − 3HV

+(5s − 2)Φ + (1 + 5s)Ψ + 1 HΦ′ + 2 − 5s r(z) r(z) dr(Φ + Ψ)+ +

• H′

H2 + 2 − 5s r(z)H + 5s

• Ψ +

r(z) dr(Φ′ + Ψ′)

• Hierarchy −

→ ∼ ∆g ∼ H

k ∆g

∼

• H

k

2 D

Francesca Lepori | @Casteddu

8

Do we care about relativistic effects?

• 1. Neglecting relativistic corrections correction may bias the analysis

Credit: Cardona et al. (2016)

Francesca Lepori | @Casteddu

A model independent method for the Alcock Paczyński test

In collaboration with Enea Di Dio, Matteo Viel, Carlo Baccigalupi, Ruth Durrer

Based on arxiv:1606.03114 - Published in JCAP

9

Introduction - The Alcock Paczyński test Purely geometric test of cosmic expansion

• C. Alcock, B. Paczyński - An evolution free test for non-zero cosmological constant (1979)

zmean OBS ∆z θ θ =

L (1+z)DA(z)

∆z = L H(z)

The BAO feature in the galaxy correlation function can provide the scale L!

Francesca Lepori | @Casteddu

10

Baryon Acoustic Oscillations

In the early universe baryons are coupled to photons: single photon-baryon fluid GRAVITY

Credit: Wayne Hu.

PHOTON PRESSURE

Credit: Wayne Hu.

Sound waves which propagates until recombination imprint a feature in the correlation function at the sound horizon: ≈ 150Mpc

Francesca Lepori | @Casteddu

11

BAO feature in the correlation function

Evolution of an overdense region at a single point: dark matter, baryons, photons.

◮ The coupled fluid of photons and

baryons move away from the dark matter at the centre (top panels)

◮ After recombination, photons

continue propagating, baryons stay were they are (middle panels)

◮ Baryons fall into the gravitational

potential generated by Dark Matter and viceversa. They share the same distribution (bottom panels)

D., Eisenstein, H. Seo, M. White - On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter (2007)

Francesca Lepori | @Casteddu

12

Current and future BAO measurements

Sound horizon at the drag epoch is measured by Planck at the 0.3% level

− → Use BAO scale as a standard ruler to test distance-redshift relation and map the expansion history

BAOs are one of the main target of future generation of galaxy surveys

BOSS

Current measurements

◮ From BOSS (the Baryon

Oscillation Spectroscopic Survey)

1% precision

DESI & Euclid

Next generation of surveys

◮ Euclid Spacecraft,

launch planned for 2020

◮ DESI (Dark Energy Spectroscopic

Instrument), starting in 2018

0.1% precision

Francesca Lepori | @Casteddu

13

Overview & Notation

OVERVIEW

• 1. Test a model independent method for the AP test

(Montanari and Durrer, 2012)

• 2. Systematic effects due to lensing and galaxy bias
• 3. Window function shift
• 4. Statistical uncertainties (shot-noise and cosmic variance)

NOTATION ∆obs(n, z) = ∆g + ∆RSD + ∆κ + ✟✟ ❍❍ ∆rel ,

◮ Redshift space distortion

∆RSD(n, z) = 1 H(z)∂r(V · n) +

• H′

H2 + 2 rH

• (V · n) − 3HV

◮ Lensing Convergence

∆κ = (5 [s(m∗, z) =0 − 2) r(z) r(z) − r 2r(z)r ∆Ω(Φ + Ψ)dr

Francesca Lepori | @Casteddu

14

Methodology

• 1. Compute the angular power spectrum

◮ Run classgal for our fiducial cosmology (Planck + external data)

• E. Di Dio, F. Montanari, J. Lesgourgues, R. Durrer (2013)
• 2. Compute the correlation function in both radial and transverse

direction from the redshift dependent angular power spectra

• 3. Use a phenomenological model for the correlation function

ξ(x) = A · e−(x−xFIT)2/2σ2 +

N

• n=0

Kn · xn,

• 4. Estimate the peak position ∆zF IT and θFIT
• 5. Compare FAP = ∆zFIT/θFIT to its expected value

FAP = (1 + z)H(z)DA(z)

Francesca Lepori | @Casteddu

15

Results: RSD & Lensing

z = 0.7 Transverse

2.5 3.0 3.5 4.0

θ[deg]

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

ξ(θ)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

Radial

0.035 0.040 0.045 0.050 0.055 0.060 0.065

∆z

−0.006 −0.004 −0.002 0.000 0.002

ξ(∆z)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

Francesca Lepori | @Casteddu

16

RSD & Lensing

z = 1 Transverse

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

θ[deg]

0.000 0.001 0.002 0.003 0.004 0.005 0.006

ξ(θ)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

Radial

0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080

∆z

−0.006 −0.004 −0.002 0.000 0.002

ξ(∆z)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

Francesca Lepori | @Casteddu

17

RSD & Lensing

z = 1.5 Transverse

1.4 1.6 1.8 2.0 2.2 2.4

θ[deg]

0.000 0.001 0.002 0.003 0.004

ξ(θ)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

Radial

0.06 0.07 0.08 0.09 0.10

∆z

−0.004 −0.003 −0.002 −0.001 0.000 0.001 0.002

ξ(∆z)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

Francesca Lepori | @Casteddu

18

RSD & Lensing

z = 2.0 Transverse

1.2 1.4 1.6 1.8 2.0

θ[deg]

−0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

ξ(θ)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

Radial

0.07 0.08 0.09 0.10 0.11 0.12 0.13

∆z

−0.003 −0.002 −0.001 0.000 0.001

ξ(∆z)

DENSITY DENSITY + RSD DENSITY + RSD + LENS CLASSgal FIT FIT - GAUSSIAN

70 80 90 100 110 120 130

r[Mpc/h]

ONLY DENSITY: shift in the estimated peak position ≈ 0.5% in both directions plus LENSING: change in the amplitude of radial correlations, but the peak positions is not affected in both directions

Francesca Lepori | @Casteddu

19

Consistency Test

0.5 1.0 1.5 2.0

zmean

0.02 0.01 0.00 0.01 0.02 0.03

F AP(z)/F(z) − 1

DENSITY DENSITY + RSD DENSITY + RSD + LENSING

The shaded region shows the errors computed from the resolution in ∆z and θ

When ONLY DENSITY: ∼ 1% offset between the expected and computed value of FAP LENSING does not affect the AP test

Francesca Lepori | @Casteddu

20

Window function

− → Need spectroscopic resolution for radial BAO − → Maximize S/N choosing optimal redshift binning in the transverse Example: Euclid Survey - z = 1

0.000 0.004 0.008

ξ(θ)

σz = 0. 005(1 + zmean) σz = 0. 01(1 + zmean) σz = 0. 02(1 + zmean) σz = 0. 03(1 + zmean)

1.5 2.0 2.5 3.0 3.5

θ[deg]

10-4 10-3

σξ(θ)

1.5 2.0 2.5 3.0 3.5

θ[deg]

2 4 6 8 10

(S/N)θ

σz = 0. 005(1 + zmean) σz = 0. 01(1 + zmean) σz = 0. 02(1 + zmean) σz = 0. 03(1 + zmean)

Optimized window function for 3 spectroscopic survey Euclid SKA DESI σz = 0.02(1 + zmean)

Francesca Lepori | @Casteddu

21

σz = 0.02(1 + zmean)

OFFSET BETWEEN 3% AND 5%

Model for the AP function (Montanari and Durrer, 2012) FAP (zmean) = ∆zBAO θBAO

• 1 − γ ·
• σz

∆zBAO 2 We estimate γ by minimizing χ2 =

• i
• FAP (zi) − Fth(zi)

∆FAP (zi) 2 Bias γ EUCLID √1 + z 0.166 SKA c4 exp (c5z) 0.161 DESI - ELGs 0.84/D(z) 0.151 DESI - LRGs 1.7/D(z) 0.154

0.01 0.01 0.03 0.05 EUCLID 0.01 0.01 0.03 0.05 SKA 0.01 0.01 0.03 0.05 DESI - ELGs 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.01 0.01 0.03 0.05 DESI - LRGs

FAP (z)/F(z)−1 zmean Francesca Lepori | @Casteddu

22

Impact of the shot-noise

EUCLID-LIKE SURVEY

0.8 1.0 1.2 1.4 1.6 1.8 2.0

zmean

0.10 0.05 0.00 0.05 0.10

∆FAP(z)/FAP(z)

Systematics + cosmic variance + shot-noise Systematics + cosmic variance Systematics

Statistical error is dominated by cosmic variance at low redshift Shot-noise is the major contributor at high redshift ERRORS ON PARAMETERS

0.2 0.3 0.4

∆Ωm

Planck Errorbars BOSS Errorbars

0.15 0.00 0.15

∆Ωk

0.8 1.0 1.2 1.4 1.6 1.8 2.0

zmean

3 2 1 1

∆w

Light shaded region − → Error from one measurement Dark shaded region − → Error from 10 independent measurements

Francesca Lepori | @Casteddu

23

Conclusions

◮ We tested the possibility of performing the AP test in a model

independent way

◮ Gravitational lensing does not affect the estimation of the BAO

peak position

◮ The parametrization introduces an extra 1% error due to

uncertainty on galaxy bias

◮ We computed the correction to the projection effect,

introduced by a window function in the transverse direction

◮ The shot-noise has a significant impact on the error

◮ Cosmic variance dominates the statistical error at low z ◮ The shot-noise is the major contribution at high z Francesca Lepori | @Casteddu

Optimal survey for relativistic distortions in the cross-correlation 21cm - galaxies

In collaboration with Enea Di Dio, Eleonora Villa, Matteo Viel

WORK IN PROGRESS

24

Introduction

◮ Anisotropy induced by peculiar velocities

Credit: A. J. S. Hamilton, 1997

◮ Asymmetry induced by relativistic distortions

Example of Gravitational Redshift

Credit: C. Bonvin et al., 2014

→ Imaginary part of the Fourier space power spectrum (Patrick McDonald, 2009) → Dipole of the correlation function (C. Bonvin et al., 2014)

Francesca Lepori | @Casteddu

25

Multiple expansion of the CF and P(k)

◮ Relation correlation function - power spectrum multiples

ξAB

ℓ

(d, z1, z2) = (−i)ℓ k2dk 2π2 P AB

ℓ

(k, z1, z2)jℓ(k d),

◮ where

P AB

1

= (−i)

• 2

rH + ˙ H H2

• (bB − bA)

+

• 1 −

1 rH

• [3(sA − sB) + 5(bBsA − bAsB)]
• f H

k Pm(k) P AB

3

= (2i)

• 1 −

1 rH

• (sB − sA)f2 H

k Pm(k)

• S

N

• = f(bias, magnification biases, evolution bias, volume of the survey...)

NOT ALL INDEPENDENT

Francesca Lepori | @Casteddu

26

Intensity mapping in a nutshell

Intensity mapping

Measure the integrated radio emission from unresolved gas clouds

Many planned low-z 21 cm IM experiment

(the Canadian Hydrogen Intensity Mapping Experiment(CHIME), Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX) , BAO from Integrated Neutral Gas Observations (BINGO), Square Kilometer Array (SKA-mid)...)

Observed quantity: ∆Tb(z, n) ≡ fluctuation in the 21 cm brightness temperature

◮ Computed at first order in perturbation theory (A. Hall et al., 2013) ◮ Equivalent to the number count, with NO LENSING: → sHI = 0.4

Francesca Lepori | @Casteddu

27

Modeling HI distribution Halo based method

Villaescusa-Navarro et al., 2014

◮ Abundance of neutral hydrogen inside halos

ΩHI(z) = 1 ρ0

c

∞ n(M, z)MHI(M, z)dM, n(M, z) → Tinker halo mass function at z (Tinker et al., 2008) MHI(M, z) → average HI mass inside a halo of mass M at redshift z

◮ Shot-Noise (From Castorina and Villaescusa-Navarro, 2016)

PSN(z) =

• 1

ρ0

cΩHI(z)

2 ∞ n(M, z)M 2

HI(z)dM, ◮ HI bias

bHI(z) = 1 ρ0

cΩHI(z)

∞ n(M, z)b(M, z)MHI(z)dM,

Francesca Lepori | @Casteddu

28

Modeling HI distribution

MHI(M, z) = C · (1 − Yp) Ωb Ωm exp

• −(Mmin/M)
• · M α

(From Castorina and Villaescusa-Navarro, 2016)

108 109 1010 1011 1012

Mmin[M ⊙/h]

0.5 1.0 1.5 2.0 2.5

bHI z =0.15

α =0.6 α =0.75 α =1.0 α =1.2 MOD 1 MOD 2 HALOs

108 109 1010 1011 1012

Mmin[M ⊙/h]

10-1 100 101 102 103 104 105

PSN[(Mpc/h)3 ] z =0.15

α =0.6 α =0.75 α =1.0 α =1.2 MOD 1 MOD 2 HALOs Francesca Lepori | @Casteddu

29

Bias VS shot-noise

10-1 100 101 102 103 104 105

PSN[(Mpc/h)3 ]

0.5 1.0 1.5 2.0

bHI z =0.15

α =0.6 α =0.75 α =1.0 α =1.2 MOD 1 MOD 2 HALOs

Francesca Lepori | @Casteddu

30

Modeling Galaxies

We model the average number of galaxies within a halo of mass M as Nav(M) =      if M ≤ Mmin A ·

• M

Mmin

α , if M > Mmin Bias vs Shot-Noise

10-4 10-3 10-2 10-1 100 101 102 103 104 105

1/ng[(Mpc/h)3 ]

1.0 1.5 2.0

bgal z =0.15

A =0.1 A ≈0.8 A =10 α =0.6 α =0.8 α =1.0 Francesca Lepori | @Casteddu

31

Preliminary S/N analysis in Fourier space

MOD 1

Mmin = 108 and αHI = 0.6 − → bHI = 0.67, PSN ∼ 1(Mpc/h)3

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

bgal

2 4 6 8 10 12

S/N fsky =0.1, zmax =0.3

sgal =0.1 sgal =0.3 sgal =0.4 sgal =0.5 sgal =0.7

MOD 2

Mmin = 1010, αHI = 0.75 − →, bHI = 0.96, PSN ∼ 100(Mpc/h)3

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

bgal

2 4 6 8 10 12

S/N fsky =0.1, zmax =0.3

sgal =0.1 sgal =0.3 sgal =0.4 sgal =0.5 sgal =0.7 Francesca Lepori | @Casteddu

32

Preliminary S/N analysis in Fourier space

NO INTERF. NOISE

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

bgal

2 4 6 8 10 12

S/N fsky =0.1, zmax =0.3

sgal =0.1 sgal =0.3 sgal =0.4 sgal =0.5 sgal =0.7

+ INTERF. NOISE

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

bgal

2 4 6 8 10 12

S/N fsky =0.1, zmax =0.3

sgal =0.1 sgal =0.3 sgal =0.4 sgal =0.5 sgal =0.7 Francesca Lepori | @Casteddu

33

Work in Progress

◮ Generalize the analysis to the configuration space, including full

covariance and interferometer noise

◮ Modeling the luminosity function for specific galaxy surveys ◮ ... ◮ ...

Thanks for your attention!

Francesca Lepori | @Casteddu

34

Correlation function and angular power spectrum

Compare the galaxy number count in different bins

◮ Correlation function

ξ(θ, z1, z2) = ∆obs(n1, z1)∆obs(n2, z2) , cos θ ≡ n1 · n2

◮ Angular power spectrum

∆obs(n, z) =

• ℓm

aℓm(z)Yℓm(n), aℓm =

• dΩnY ∗

ℓm∆obs(n, z)

Cℓ(z1, z2) = aℓm(z1)aℓm(z2)

Francesca Lepori | @Casteddu

35

Transverse and Radial correlation function

◮ Compute ξ(θ, z1, z2) from Cℓ(z1, z2)

ξ(θ, z1, z2) = 1 4π

∞

• ℓ=0

(2ℓ + 1)Cℓ(z1, z2)Pℓ(cos θ),

◮ Transverse correlation function

ξ⊥(θ, zmean) = 1 4π

∞

• ℓ=0

(2ℓ + 1)Cℓ(zmean, zmean)Pℓ(cos θ)

◮ Radial correlation function

ξ(∆z, zmean) = 1 4π

∞

• ℓ=0

(2ℓ+1)Cℓ

• zmean − ∆z

2 , zmean + ∆z 2

• Pℓ(1)

Francesca Lepori | @Casteddu