Expanding the Science Case for Baryon Acoustic Oscillations with Galaxy Redshift Surveys

Expanding the science case for Baryon Acoustic Oscillations with galaxy redshift surveys

Florian Beutler

25 March, 2019

Royal Society University Research Fellow Outline of the talk

1 Introduction & Motivation Galaxy redshift surveys Baryon Acoustic Oscillations (BAO)

2 Testing inﬂation with primordial features (Physical Review Research, 1, 2019)

3 Neutrinos in the phase of the BAO (Nature Physics, 15, 465, 2019)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 2 What is a galaxy redshift survey?

Measure the position of galaxies (redshift + RA, DEC). The CMB tells us a lot about the initial conditions for today’s distribution of matter. How the initial density ﬂuctuations in the CMB evolved from redshift z 1100 to today depends on Ωm, ΩΛ, H0 etc. ∼

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 3 From a point distribution to a power spectrum

Overdensity-ﬁeld:

ρ x ρ δ x ( ) − ( ) ρ

Two-point function:

homogeneity ( isotropy ξ r r x + r x ξ δ δ anisotropy ( ) ∫ 1 ( ) h ( ) ( )i r d r ξ` 1 µ ξ , µ ` µ ( ) − ( )L ( ) ...and in Fourier-space: ¹ P k 4π i ` r2drξ r j kr `( ) (− ) `( ) `( )

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 4 Why should you care?

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 5 The BOSS galaxy survey

Third version of the Sloan Digital Sky Survey (SDSS-III) Spectroscopic survey optimized for the measurement of Baryon Acoustic Oscillations (BAO) The galaxy sample includes 1 100 000 galaxy redshifts in the range 0.2 < z < 0.75 The eﬀective volume is 6 Gpc3 ∼ 1000 ﬁbres/redshifts per pointing

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 6 The DESI galaxy survey

Mayall 4m telescope at Kitt Peak, Arizona 5000 ﬁbres/pointing Will observe 3 types of galaxies (LRGs/ELGs/QSOs) + BGS 30 - 40 million galaxies in total z < 1.8 with galaxies and z < 3.5 with Ly-α forrest

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 7 The ESA Euclid mission

Launch scheduled for summer 2022 L2 point → Space-based weak lensing + gal. clustering survey over 15 000 deg2 30 million emission line galaxies over the redshift range 0.7 to 2.0 Slitless spectroscopy (grism)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 8 What are Baryon Acoustic Oscillations?

For the ﬁrst 380 000 years the evolution eq. of baryon and photon perturbations can be written as

2 2 2 δb c δb Φ Ü γ − s ∇ γ ∇ with the plane wave solution

δb A cos krs + φ γ ( ) Preferred distance scale between galaxies as a relic of sound waves in the early Universe. This signal is present at low redshift and detectable in ξ r /P k on very large scales. ( ) ( ) hhhhhhcredit: Martin White

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 9 What are Baryon Acoustic Oscillations?

rs zd 147.34 0.64 Mpc 0.43% ( ) ± ( )

Planck collaboration Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 10 Baryon Acoustic Oscillations in BOSS

1.08 0.2 < z < 0.51.08 1.06 1.06

1.04 1.04

(k) 1.02 1.02

smooth 1 1

0.98 0.98 P(k)/P 0.96 0.96

0.94 0.94

0.92 pre•recon0.92 post•recon 0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 1.08 0.4 < z < 0.61.08 1.06 1.06

1.04 1.04 (k) 1.02 1.02

smooth 1 1

0.98 0.98

P(k)/P 0.96 0.96

0.94 0.94

0.92 pre•recon0.92 post•recon 0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 1.08 0.5 < z < 0.751.08 1.06 1.06

1.04 1.04

(k) 1.02 1.02

smooth 1 1

0.98 0.98 P(k)/P 0.96 0.96

0.94 0.94

0.92 pre•recon0.92 post•recon 0.05 0.1 0.15 0.2 0.250.3 0.3 0.05 0.1 0.15 0.2 0.25 0.3 k [h Mpc•1] k [h Mpc•1] Beutler et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 11 Baryon Acoustic Oscillations in BOSS

¹ z cdz0 DA z ( ) H z 0 ( 0) p 3 2 H z H0 Ωm 1 + z + ΩΛ + Ωk 1 + z ( ) ( ) ( )

Beutler et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 12 Baryon Acoustic Oscillations in BOSS

ﬁd DV z 0.38 r rs 1476 15 Mpc 1.0% ( ) s / ± ( ) ﬁd DV z 0.61 r rs 2146 19 Mpc 0.9% ( ) s / ± ( ) 1 3 2 2 cz / DV z 1 + z D z ( ) ( ) A( )H z ( )

Beutler et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 13 Planck+SN:

Ωk 0.025 0.012 ± w 1.01 0.11 − ± Planck+SN+BAO:

Ωk 0.0003 0.0027 ± w 1.05 0.08 − ± Alam et al. (2017)

Baryon Acoustic Oscillations in BOSS

The BAO signal is located on very large scales and can be captured (mostly) with a linear model. In BOSS we used an agnostic broadband marginalisation using a set of polynomial terms and density ﬁeld reconstruction to boost the signal. Due to BAO we now know the distance to z 0.38 and z 0.61 with 1% uncertainty... better than our knowledge of H0. ∼

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 14 Baryon Acoustic Oscillations in BOSS

Ωk 0.025 0.012 ± w 1.01 0.11 − ± Planck+SN+BAO:

Ωk 0.0003 0.0027 ± w 1.05 0.08 − ± Alam et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 14 Theoretical systematics of BAO

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 15 Theoretical systematics of BAO

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 16 Looking into the (near) future

1.100 SDSS NOW 1.075 BOSS Ly- WiggleZ 1.050 ) z ( 1.025 k c n a l P V 1.000 D / ) z

( 0.975 V D BOSS 0.950 6dFGS 0.925 Measure of expansion history

0.900 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 redshift z

1 3 2 2 cz / DV z 1 + z D z ( ) ( ) A( )H z ( ) Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 17 Looking into the (near) future

1.100 SDSS SOON 1.075 BOSS Ly- WiggleZ 1.050 DESI (2020) ) z ( 1.025 k c n a l P V 1.000 D / ) z

( 0.975 V D BOSS Euclid (2022) 0.950 6dFGS 0.925 Measure of expansion history

0.900 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 redshift z

1 3 2 2 cz / DV z 1 + z D z ( ) ( ) A( )H z ( ) Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 18 Outline of the talk

1 Introduction & Motivation Galaxy redshift surveys Baryon Acoustic Oscillations (BAO)

2 Testing inﬂation with primordial features (Physical Review Research, 1, 2019)

3 Neutrinos in the phase of the BAO (Nature Physics, 15, 465, 2019)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 19 Inﬂation in one plot

Baumann (2009)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 20 Testing inﬂation through primordial features

No features 9 10− × 0.05 2.4 0.00 0

2.2 ζ, HZ spectrum ) /P ) k 0.05 ( 2.0 k −

( 0.05 P ζ

1.8 P ∆ 0.00 1.6 Inﬂationary spectrum 0.05 3 2 1 0 1 − 10− 10− 10− 10 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 1 k [Mpc− ] k [Mpc− ]

2 2 ns 1 2π 2π As k − Pζ,0 k ζ,0 k ( ) k3 P ( ) k3 k ∗

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 21 Testing inﬂation through primordial features

Linear features 10 9 ω = 271.8 Mpc × − lin 0.05 2.4 0.00 0

2.2 ζ, HZ spectrum ) /P ) k 0.05 ( 2.0 k −

( 0.05 P ζ

1.8 P ∆ 0.00 1.6 Linear feature Inﬂationary spectrum 0.05 3 2 1 0 1 − 10− 10− 10− 10 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 1 k [Mpc− ] k [Mpc− ]

∆Pζ k ( ) Alin sin ωlink + φlin P 0 k ζ, ( ) [Sharp Features] Starobinsky (1992) Adams, Cresswell & Easther (1997) ...

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 22 Testing inﬂation through primordial features

Logarithmic features 10 9 ω = 271.8 Mpc ω = 31.4 × − lin log 0.05 2.4 0.00 0

2.2 ζ, HZ spectrum ) /P ) k 0.05 ( 2.0 k −

( 0.05 P ζ

1.8 P ∆ Logarithmic feature 0.00 1.6 Linear feature Inﬂationary spectrum 0.05 3 2 1 0 1 − 10− 10− 10− 10 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 1 1 k [Mpc− ] k [Mpc− ]

∆Pζ k ( ) Alog sin ωloglog k k + φlog P 0 k ( / ∗) ζ, ( ) [Resonant features] Chen, Easther & Lim (2008) Silverstein & Westphal (2008) Flauger, McAllister, Pajer & Westphal (2010) ...

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 23 Feature damping

Linear Feature Logarithmic Feature

Damping factor of linear Damping factor of log features equal to BAO features approx. equal to damping for ωlin & 75 Mpc BAO damping for ωlog & 10

nw k2Σ2 2 h w w w lin,log i P k P k + e− nl/ P k α +P k + P k α δP k ( ) ( ) BAO( / ) lin,log( ) BAO( / ) ζ ( )

linear linear 0.10 nonlinear 0.10 nonlinear ) k

( 0.05 0.05 nw

/P 0.00 0.00 ) k (

w 0.05 0.05 − − P 0.10 0.10 − − 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 1 1 k [h Mpc− ] k [h Mpc− ]

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 24 Feature damping

Linear Feature Logarithmic Feature

Damping factor of linear Damping factor of log features equal to BAO features approx. equal to damping for ωlin & 75 Mpc BAO damping for ωlog & 10

nw k2Σ2 2 h w w w lin,log i P k P k +e− nl/ P k α + P k + P k α δP k ( ) ( ) BAO( / ) lin,log( ) BAO( / ) ζ ( )

linear linear 0.10 nonlinear 0.10 nonlinear ) k

( 0.05 0.05 nw

/P 0.00 0.00 ) k (

w 0.05 0.05 − − P 0.10 0.10 − − 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 1 1 k [h Mpc− ] k [h Mpc− ]

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 25 Feature constraints from BOSS DR12 and Planck

0.06 0.06 low-z BOSS low-z BOSS 0.05 high-z BOSS 0.05 high-z BOSS ) )

BOSS . BOSS . 0.04 0.04

0.03 0.03 (95% excl

(95% excl BOSS 0.02 0.02 lin log A 0.01 A 0.01

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog 0.06 0.06 Planck TT Planck TT 0.05 Planck TTTEEE 0.05 Planck TTTEEE ) )

BOSS . BOSS . 0.04 0.04 0.03 0.03 BOSS vs. Planck (95% excl (95% excl 0.02 0.02 lin log A 0.01 A 0.01

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog 0.03 0.03 BOSS BOSS BOSS+Planck TT BOSS+Planck TT ) )

BOSS+Planck TTTEEE . BOSS+Planck TTTEEE . 0.02 0.02 BOSS + Planck (95% excl (95% excl 0.01 0.01 lin log A A

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 26 Feature constraints from BOSS DR12 and Planck

0.06 0.06 low-z BOSS low-z BOSS 0.05 high-z BOSS 0.05 high-z BOSS ) )

BOSS . BOSS . 0.04 0.04

0.03 0.03 (95% excl

(95% excl BOSS 0.02 0.02 lin log A 0.01 A 0.01

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog 0.06 0.06 Planck TT Planck TT 0.05 Planck TTTEEE 0.05 Planck TTTEEE ) )

BOSS . BOSS . 0.04 0.04 0.03 0.03 BOSS vs. Planck (95% excl (95% excl 0.02 0.02 lin log A 0.01 A 0.01

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog 0.03 0.03 BOSS BOSS BOSS+Planck TT BOSS+Planck TT ) )

BOSS+Planck TTTEEE . BOSS+Planck TTTEEE . 0.02 0.02 BOSS + Planck (95% excl (95% excl 0.01 0.01 lin log A A

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 27 Feature constraints from BOSS DR12 and Planck

0.06 0.06 low-z BOSS low-z BOSS 0.05 high-z BOSS 0.05 high-z BOSS ) )

BOSS . BOSS . 0.04 0.04

0.03 0.03 (95% excl

(95% excl BOSS 0.02 0.02 lin log A 0.01 A 0.01

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog 0.06 0.06 Planck TT Planck TT 0.05 Planck TTTEEE 0.05 Planck TTTEEE ) )

BOSS . BOSS . 0.04 0.04 0.03 0.03 BOSS vs. Planck (95% excl (95% excl 0.02 0.02 lin log A 0.01 A 0.01

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog 0.03 0.03 BOSS BOSS BOSS+Planck TT BOSS+Planck TT ) )

BOSS+Planck TTTEEE . BOSS+Planck TTTEEE . 0.02 0.02 BOSS + Planck (95% excl (95% excl 0.01 0.01 lin log A A

0.00 0.00 200 400 600 800 20 40 60 80 ωlin [Mpc] ωlog

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 28 Forecasts for primordial feature constraints

LSS dominates on small frequencies, while the CMB can access higher frequencies→ DESI/Euclid are going to beat even CVL-CMB experiments →

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 29 Outline of the talk

1 Introduction & Motivation Galaxy redshift surveys Baryon Acoustic Oscillations (BAO)

2 Testing inﬂation with primordial features (Physical Review Research, 1, 2019)

3 Neutrinos in the phase of the BAO (Nature Physics, 15, 465, 2019)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 30 Motivation: Neutrinos in the phase of the BAO

一攀甀琀爀椀渀漀䈀䈀一搀攀挀漀甀瀀氀椀渀最儀䌀䐀倀吀爀攀栀攀愀琀椀渀最

猀瀀椀渀㴀

" 4 3 # 7 4 / ρ 1 + N ρ r 8 11 eﬀ γ

Baumann et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 31 Motivation: Neutrinos in the phase of the BAO

一攀甀琀爀椀渀漀䈀䈀一搀攀挀漀甀瀀氀椀渀最儀䌀䐀倀吀爀攀栀攀愀琀椀渀最

䔀砀挀氀甀搀攀搀戀礀倀氀愀渀挀欀

猀瀀椀渀㴀

N 3.04 0.18 (Planck) eﬀ ±

Baumann et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 32 Motivation: Neutrinos in the phase of the BAO

一攀甀琀爀椀渀漀䈀䈀一搀攀挀漀甀瀀氀椀渀最儀䌀䐀倀吀爀攀栀攀愀琀椀渀最

䔀砀挀氀甀搀攀搀戀礀倀氀愀渀挀欀

猀瀀椀渀㴀

σ N 0.030 (CMB-S4) ( eﬀ) σ N 0.027 (CMB-S4 + Euclid) ( eﬀ)

Baumann et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 33 Neutrinos in the CMB Spectrum

Current constraints are dominated by the damping of the power spectrum (degenerate with helium fraction).

Baumann et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 34 Phase shift detection in the CMB

The Phase shift has recently been detected in the temperature and polarisation CMB spectrum.

+1.1 Neﬀ 2.8 0.4 − Follin et al. (2015)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 35 Neutrinos in the BAO Spectrum

Baumann et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 36 Neutrinos in the BAO Spectrum

ﬁd k2σ2 2 O k O k α + β 1 f k r e− nl/ ( ) lin( / ( − ) ( )/ s )

Baumann et al. (2019)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 37 Neutrinos in the BAO Spectrum

ﬁd k2σ2 2 O k O k α + β 1 f k r e− nl/ ( ) lin( / ( − ) ( )/ s )

N β N with eff eff 4 3 ( ) fid 8 11 4 7 + N ( / ) / / eff Proof of principle! → Baumann et al. (2019)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 38 Neutrinos in the BAO Spectrum

Baumann et al. (2019)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 39 2 The BAO analysis pipeline is perfectly suited to extract primordial features to test inﬂation 3 Constraints on primordial features from LSS are already better than Planck for a large frequency range

4 The phase of the BAO carries information on Neff just as in the CMB first (low significance) detection in BOSS →

Summary

1.100 SDSS SOON 1.075 BOSS Ly- WiggleZ 1.050 DESI (2020) ) z ( 1.025 k c n a l P V 1.000 D / ) z

( 0.975 V D BOSS Euclid (2022) 0.950 6dFGS 0.925 Measure of expansion history

0.900 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 redshift z

1 The next generation of galaxy redshift surveys is just around the corner with BAO as a key science case →

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 40 4 The phase of the BAO carries information on Neff just as in the CMB first (low significance) detection in BOSS →

Summary

1.100 SDSS SOON 1.075 BOSS Ly- WiggleZ 1.050 DESI (2020) ) z ( 1.025 k c n a l P V 1.000 D / ) z

( 0.975 V D BOSS Euclid (2022) 0.950 6dFGS 0.925 Measure of expansion history

0.900 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 redshift z

1 The next generation of galaxy redshift surveys is just around the corner with BAO as a key science case → 2 The BAO analysis pipeline is perfectly suited to extract primordial features to test inﬂation 3 Constraints on primordial features from LSS are already better than Planck for a large frequency range

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 40 Summary

1.100 SDSS SOON 1.075 BOSS Ly- WiggleZ 1.050 DESI (2020) ) z ( 1.025 k c n a l P V 1.000 D / ) z

( 0.975 V D BOSS Euclid (2022) 0.950 6dFGS 0.925 Measure of expansion history

0.900 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 redshift z

4 The phase of the BAO carries information on Neff just as in the CMB first (low significance) detection in BOSS →

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 40 Baryon Acoustic Oscillations in BOSS

¹ z cdz0 DA z ( ) H z 0 ( 0) p 3 2 H z H0 Ωm 1 + z + ΩΛ + Ωk 1 + z ( ) ( ) ( ) 1 3 2 2 cz / DV z 1 + z D z ( ) ( ) A( )H z ( )

Beutler et al. (2017)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 41 Fitting the BAO

Start with linear P k and separate the broadband shape, Psm k , and the BAO feature O(lin) k . Include a damping of the BAO feature:( ) ( ) sm,lin sm h lin k2Σ2 2i P k P k 1 + O k α 1 e− nl/ ( ) ( ) ( ( / ) − ) Add broadband nuisance terms

a3 a4 a5 A k a1k + a2 + + + ( ) k k2 k3 Pﬁt k B2Psm,lin k α + A k ( ) ( / ) ( ) Marginalize to get α . L( )

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 42 Current constraints on Neﬀ Relic neutrinos make up 41% of the radiation density

" 4 3 # 7 4 / ρ 1 + N ρ r 8 11 eﬀ γ

CMB BBN

NCMB 3.04 0.18 NBBN 3.28 0.28 eﬀ ± eﬀ ±

Planck (2015), Cooke et al. (2015)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 43 Impact of the window function for features search

Beutler et al. (2019)

Florian Beutler Expanding the science case forBaryon Acoustic Oscillationswith galaxy redshift25 March, surveys 2019 44