The Cosmic Microwave Background in High Definition

The Cosmic Microwave Background in High

Definition

Gil Holder

as part of:

SPT collaboration

Outline

• the cosmic microwave background (CMB) – temperature & polarization fluctuations

• Sunyaev-Zeldovich effect – galaxy clusters

• CMB gravitational lensing – chasing neutrino masses

• first detection of “B-modes”

2

• Expanding => cooling

• At earlier times, the universe was hotter

• when atoms formed, universe became

transparent to photons

– special timescale in the universe for photons

Hot Big Bang

http://www.astro.ucla.edu/~wright/BBhistory.html

The Cosmic Microwave Background

CMB according to COBE

(Bennett et al 1996)

Image from COBE science team: http://lambda.gsfc.nasa.gov/product/cobe/

Isotropy

• Cosmic

microwave background is

remarkably isotropic

• Unnaturally

isotropic! +-3.5 mK scale

WMAP science team

The Cosmic Microwave Background

CMB according to COBE

(Bennett et al 1996)

Image from COBE science team: http://lambda.gsfc.nasa.gov/product/cobe/

Nothing too strange within our “horizon”:

40 billion light years

Image from WMAP

8

Planck has higher resolution than WMAP

16 ^o

WMAP 60 GHz Planck 143 GHz

South Pole Telescope

10 m mm-wave (3 different wavelengths) telescope

at the south pole

• extremely dry

• very stable

• good support

photo by Keith Vanderlinde

photo by Dana Hrubes

Chicago Colorado UC Berkeley Case Western

McGill Harvard UC Davis Munich +++

The evolution of SPT cameras

2007-2011: SPT 960 detectors

2012-2015: SPTpol

~1600 detectors 2016: SPT-3G

~15,200 detectors

Now with polarization!

slide from S Hoover

12

SPT has higher resolution than Planck

Planck 143 GHz Planck+SPT

4 ^o

CMB Angular Power Spectrum

CMB Polarization

• CMB fluctuations are relatively strongly

polarized (~10%)

Q

U

E-modes/B-modes

• E-modes vary spatially parallel or perpedicular to polarization direction

• B-modes vary spatially at 45 degrees

• CMB

• scalar perturbations only generate *only* E

E modes

(a) (b)

Figure 1. A pure E Fourier mode (a), and a pure B mode (b).

example, consider the original COBE detection: although the key science was contained in the two-point correlation function and power spectrum estimates, the actual real-space maps were invaluable in convincing the world of the validity and importance of the results.)

Consideration of issues related to E/B separation is important in experiment design and optimization as well. For example, the ambiguity in E/B separation significantly alters the optimal tradeoff between sky coverage and noise per pixel in a degree-scale B mode experiment [6].

2. Pure and ambiguous modes

The E/B decomposition is easiest to understand in Fourier space. For any given wavevector k, define a coordinate system (x, y) with the x axis parallel to k, and compute the Stokes parameters Q, U . An E mode contains only Q, while a B mode contains only U . In other words, in an E mode, the polarization direction is always parallel or perpendicular to the wavevector, while in a B mode it always makes a 45

angle, as shown in Figure 1.

In a map that covers a finite portion of the sky, of course, the Fourier transform cannot be determined with infinite k-space resolution. According to the Heisenberg uncertainty principle, if the observed region has size L, an estimate of an individual Fourier mode with wavevector q will be a weighted average of true Fourier modes k in a region around q of width |k − q| ∼ L

. These Fourier modes will all point in slightly different directions, spanning a range of angles

∼ qL. Since the mapping between (Q, U) and (E, B) depends on the angle of the wavevector, we expect the amount of E/B mixing to be of order qL. In particular, this means that the largest scales probed by a given experiment will always have nearly complete E/B mixing. This is unfortunate, since the largest modes probed are generally the ones with highest signal-to-noise ratio. Typically, the noise variance is about the same in all Fourier modes detected by a given experiment, while the signal variance scales as C

, which decreases as a function of wavenumber.

(Remember, even a “flat” power spectrum is one with l

C

∼ constant.)

One way to quantify the amount of information lost in a given experimental setup is to decompose the observed map into a set of orthogonal modes consisting of pure E modes, pure B modes, and ambiguous modes [7]. A pure E mode is orthogonal to all B modes, which means that any power detected in such a mode is guaranteed to come from the E power spectrum.

(a) (b)

Figure 1. A pure E Fourier mode (a), and a pure B mode (b).

example, consider the original COBE detection: although the key science was contained in the two-point correlation function and power spectrum estimates, the actual real-space maps were invaluable in convincing the world of the validity and importance of the results.)

Consideration of issues related to E/B separation is important in experiment design and optimization as well. For example, the ambiguity in E/B separation significantly alters the optimal tradeoff between sky coverage and noise per pixel in a degree-scale B mode experiment [6].

2. Pure and ambiguous modes

The E/B decomposition is easiest to understand in Fourier space. For any given wavevector k, define a coordinate system (x, y) with the x axis parallel to k, and compute the Stokes parameters Q, U . An E mode contains only Q, while a B mode contains only U . In other words, in an E mode, the polarization direction is always parallel or perpendicular to the wavevector, while in a B mode it always makes a 45

angle, as shown in Figure 1.

In a map that covers a finite portion of the sky, of course, the Fourier transform cannot be determined with infinite k-space resolution. According to the Heisenberg uncertainty principle, if the observed region has size L, an estimate of an individual Fourier mode with wavevector q will be a weighted average of true Fourier modes k in a region around q of width |k − q| ∼ L

. These Fourier modes will all point in slightly different directions, spanning a range of angles

∼ qL. Since the mapping between (Q, U) and (E, B) depends on the angle of the wavevector, we expect the amount of E/B mixing to be of order qL. In particular, this means that the largest scales probed by a given experiment will always have nearly complete E/B mixing. This is unfortunate, since the largest modes probed are generally the ones with highest signal-to-noise ratio. Typically, the noise variance is about the same in all Fourier modes detected by a given experiment, while the signal variance scales as C

, which decreases as a function of wavenumber.

(Remember, even a “flat” power spectrum is one with l

C

∼ constant.)

One way to quantify the amount of information lost in a given experimental setup is to decompose the observed map into a set of orthogonal modes consisting of pure E modes, pure B modes, and ambiguous modes [7]. A pure E mode is orthogonal to all B modes, which means that any power detected in such a mode is guaranteed to come from the E power spectrum.

B modes

Bunn

E-modes/B-modes

• E-modes vary spatially parallel or perpedicular to polarization direction

• B-modes vary spatially at 45 degrees

• CMB

• scalar perturbations only generate *only* E

Image of positive kx/positive ky Fourier transform of a 10x10 deg chunk of

Stokes Q CMB map [simulated; nothing clever done to it]

E modes

• Lensing of CMB is

much more obvious in polarization!

k x

k y

E-mode polarization of ra23h30, dec -55 field (150 GHz)

E

CMB Polarization Angular Power Spectrum

Barkats et al (BICEP)

only

upper

limits

on B

mode

power

Two Expected Sources of B Modes

Gravitational Radiation in Early Universe

(amplitude unknown!) Gravitational lensing of

E modes (amplitude

well-predicted, but no

measured B modes

until later in talk)

E-modes/B-modes

• E-modes vary spatially parallel or perpedicular to polarization direction

• B-modes vary spatially at 45 degrees

• scalar perturbations only generate *only* E

E modes

(a) (b)

Figure 1. A pure E Fourier mode (a), and a pure B mode (b).

example, consider the original COBE detection: although the key science was contained in the two-point correlation function and power spectrum estimates, the actual real-space maps were invaluable in convincing the world of the validity and importance of the results.)

Consideration of issues related to E/B separation is important in experiment design and optimization as well. For example, the ambiguity in E/B separation significantly alters the optimal tradeoff between sky coverage and noise per pixel in a degree-scale B mode experiment [6].

2. Pure and ambiguous modes

The E/B decomposition is easiest to understand in Fourier space. For any given wavevector k, define a coordinate system (x, y) with the x axis parallel to k, and compute the Stokes parameters Q, U . An E mode contains only Q, while a B mode contains only U . In other words, in an E mode, the polarization direction is always parallel or perpendicular to the wavevector, while in a B mode it always makes a 45

angle, as shown in Figure 1.

In a map that covers a finite portion of the sky, of course, the Fourier transform cannot be determined with infinite k-space resolution. According to the Heisenberg uncertainty principle, if the observed region has size L, an estimate of an individual Fourier mode with wavevector q will be a weighted average of true Fourier modes k in a region around q of width |k − q| ∼ L

. These Fourier modes will all point in slightly different directions, spanning a range of angles

∼ qL. Since the mapping between (Q, U) and (E, B) depends on the angle of the wavevector, we expect the amount of E/B mixing to be of order qL. In particular, this means that the largest scales probed by a given experiment will always have nearly complete E/B mixing. This is unfortunate, since the largest modes probed are generally the ones with highest signal-to-noise ratio. Typically, the noise variance is about the same in all Fourier modes detected by a given experiment, while the signal variance scales as C

, which decreases as a function of wavenumber.

(Remember, even a “flat” power spectrum is one with l

C

∼ constant.)

One way to quantify the amount of information lost in a given experimental setup is to decompose the observed map into a set of orthogonal modes consisting of pure E modes, pure B modes, and ambiguous modes [7]. A pure E mode is orthogonal to all B modes, which means that any power detected in such a mode is guaranteed to come from the E power spectrum.

(a) (b)

Figure 1. A pure E Fourier mode (a), and a pure B mode (b).

example, consider the original COBE detection: although the key science was contained in the two-point correlation function and power spectrum estimates, the actual real-space maps were invaluable in convincing the world of the validity and importance of the results.)

Consideration of issues related to E/B separation is important in experiment design and optimization as well. For example, the ambiguity in E/B separation significantly alters the optimal tradeoff between sky coverage and noise per pixel in a degree-scale B mode experiment [6].

2. Pure and ambiguous modes

The E/B decomposition is easiest to understand in Fourier space. For any given wavevector k, define a coordinate system (x, y) with the x axis parallel to k, and compute the Stokes parameters Q, U . An E mode contains only Q, while a B mode contains only U . In other words, in an E mode, the polarization direction is always parallel or perpendicular to the wavevector, while in a B mode it always makes a 45

angle, as shown in Figure 1.

In a map that covers a finite portion of the sky, of course, the Fourier transform cannot be determined with infinite k-space resolution. According to the Heisenberg uncertainty principle, if the observed region has size L, an estimate of an individual Fourier mode with wavevector q will be a weighted average of true Fourier modes k in a region around q of width |k − q| ∼ L

. These Fourier modes will all point in slightly different directions, spanning a range of angles

∼ qL. Since the mapping between (Q, U) and (E, B) depends on the angle of the wavevector, we expect the amount of E/B mixing to be of order qL. In particular, this means that the largest scales probed by a given experiment will always have nearly complete E/B mixing. This is unfortunate, since the largest modes probed are generally the ones with highest signal-to-noise ratio. Typically, the noise variance is about the same in all Fourier modes detected by a given experiment, while the signal variance scales as C

, which decreases as a function of wavenumber.

(Remember, even a “flat” power spectrum is one with l

C

∼ constant.)

One way to quantify the amount of information lost in a given experimental setup is to decompose the observed map into a set of orthogonal modes consisting of pure E modes, pure B modes, and ambiguous modes [7]. A pure E mode is orthogonal to all B modes, which means that any power detected in such a mode is guaranteed to come from the E power spectrum.

B modes

Bunn

Cosmic Microwave Background

• acoustic scale (in cm) set by physics

unrelated to dark energy

– angular scale depends on

expansion history

• provides

normalization of

fluctuation amplitude

at z~1100

WMAP (all sky)

South Pole Telescope (total 2500 sq deg)

8

SPT-SZ Survey (completed)

Final survey depths of:

- 100 GHz: < 40 uKCMB-arcmin - 150 GHz: < 18 uKCMB-arcmin - 220 GHz: < 80 uKCMB-arcmin

2500 square degrees

Gravity at work

simulations carried out by the Virgo Supercomputing Consortium using computers based at Computing Centre of the Max-Planck Society in Garching and at the Edinburgh Parallel Computing Centre. The data are publicly available at www.mpa-garching.mpg.de/NumCos

t=400 000 yrs t=20 million yrs t=500 million yrs t=13.7 billion yrs

simulated density contrast at different times

1 billion light years

Zoom in on 2 mm map

~ 4 deg ² of SPT data

18 uK sensitivity at observing wavelength of 2 mm roughly same resolution as your eye

Image by Will High in recent paper by Williamson et al

One of the heaviest objects in the universe

>10 ¹⁵ solar masses

Fig. 16.— SPT-CL J0438-5419, also known as ACT-CL J0438-5419, at zrs = 0.45. Blanco/MOSAIC-II irg images are shown in the optical/infrared panel.

Fig. 17.— SPT-CL J0549-6204 at zrs = 0.32. Blanco/MOSAIC-II irg images are shown in the optical/infrared panel.

patch of

isolated

cosmic fog

Thermal Sunyaev-Zel’dovich Effect

CMB Hot

electrons

CMB+ ν

I

Optical depth: τ ~ 0.01

Fractional energy gain per scatter: ~ 0.01

Typical cluster signal: ~500 uK

SPT Cluster Images

0658-5556 (z=0.30) (Bullet)

2344-4243 (z=0.62)

2106-5844 (z=1.13) 2337-5942 (z=0.78)

SZ

IR-Optical

12 ’ 6 ’

Williamson et al 2011, arXiv:1101.1290

SPT Cluster Sample Properties

• Optically confirmed >300 clusters, ~80% newly discovered

• High redshift: < z> = 0.55 and ~20-25% of clusters at z > 0.8

• Optical measurements also confirm ~95% purity at S/N = 5

• Mass threshold flat/falling w/ redshift: M

(z=0.6) > ~3x10

M

/h

Redshift Histogram SZ Mass vs Redshift

! "#

$%&&'()*!+,!-./0!12%&/3/3'4!5%&!-'!$'%42('67!/0'!8'(%53/9!.,!/034!(':%/3.&!5%&!-'!50'5;'67!

permitting a test of the underlying GR theory*!

!

<3=2('!+>?@!3::24/(%/'4!/0'!',,'5/!.,!6%(;!'&'(=9!.&!/0'!634/%&5'?('6403,/!%&6!=(.A/0?

('6403,/!(':%/3.&47!03=0:3=0/3&=!/0'!&''6!,.(!B'(5'&/?:'8':!B('5343.&!3&!/0'4'!12%&/3/3'4!3,!

A'!%('!/.!5.&4/(%3&!/0'!6%(;?'&'(=9!'12%/3.&!.,!4/%/'!/.!%-.2/!C*#!%552(%59*!

!

!!! !

Fig. VI-2: The primary observables for dark-energy – the distance-redshift relation DDz)!

and the growth-redshift relation gDz) – are plotted vs. redshift for three cosmological models. The green curve is an open-Universe model with no dark energy at all. The black curve is the “concordance” / CDM model, which is flat and has a cosmological constant, i.e., w  . This model is consistent with all reliable present-day data. The red curve is a dark-energy model with w , for which other parameters have been adjusted to match WMAP data. At left one sees that dark-energy models are easily

distinguished from non-dark-energy models. At right, we plot the ratios of each model to the / CDM model, and it is apparent that distinguishing the w  model from / CDM requires percent-level precision on the diagnostic quantities.!

Four Astrophysical Approaches to Dark Energy Measurements

1. Type Ia Supernovae

E9B'!+%!42B'(&.8%'!%('!-':3'8'6!/.!-'!/0'!'FB:.438'!6343&/'=(%/3.&4!.,!A03/'?6A%(,!4/%(4!

/0%/!%55('/'!$%/'(3%:!/.!'F5''6!/0'!4/%-3:3/9!:3$3/!.,!#*G!4.:%(!$%44'4!6'(38'6!-9!

H0%&6(%4';0%(*!!!I'5%24'!/0'!$%44'4!.,!/0'4'!.-J'5/4!%('!&'%(:9!%::!/0'!4%$'7!/0'3(!

'FB:.43.&4!%('!'FB'5/'6!/.!4'(8'!%4!4/%&6%(6!5%&6:'4!.,!;&.A&!:2$3&.43/9!L7!3&!A0350!

5%4'!/0'!(':%/3.&!f = LKGSd

!5%&!-'!24'6!/.!3&,'(!/0'!:2$3&.43/9!634/%&5'!d

*!!LB'5/(%:!

:3&'4!3&!/0'!42B'(&.8%!:3=0/!$%9!-'!24'6!/.!36'&/3,9!/0'!('6403,/7!%4!5%&!4B'5/(%:!,'%/2('4!

.,!/0'!=%:%F9!0.4/3&=!/0'!'FB:.43.&*!!

!

Characterizing Dark Energy

from Dark Energy Task Force report w=-0.9

w=-1

33 10

Fig. 4.— Constraints on the Ωm, σ8, and w parameters using the SPT and CMB+BAO+SNe data sets assuming a wCDM cosmology.

Contours show the 68% and 95% confidence regions for the CMB+BAO+SNe (gray, dashed), and CMB+BAO+SNe+SPT (green, solid) data sets. The SPT data most significantly improves the constraints on σ8 and w, improving the constraints on each, by factors of 1.4 and 1.3, respectively.

Figure 4, we show the constraints of the combined CMB+BAO+SNe data set, before and after including the SPT cluster data. The SPT data most significantly improves the constraints on σ₈ and w; reducing their 2-d likelihood area by a factor of ∼1.8. In Table 6, we give the marginalized constraints for several param- eters before and after the inclusion of the SPT data.

The combined constraints are w = −0.973 ± 0.063 and σ₈ = 0.793 ± 0.028, a factor of 1.3 and 1.4 improvement, respectively, over the constraints without clusters. The combined data set also constrains Ω_m = 0.273 ± 0.015 and h = 0.697 ± 0.018. These constraints are consistent with previous cluster-based results from Vikhlinin et al.

(2009b); Mantz et al. (2010c); Rozo et al. (2010), which used X-ray and optical selected samples of typically lower redshift clusters. The sensitivity of the SPT cluster data to the amplitude of structure, or σ₈, is primarily what gives it the ability to break degeneracies with the distance-relation based constraints from the BAO and SNe data sets. We note the slight tension with the H₀ constraints from (Riess et al. 2011) of h = 0.738 ± 0.024.

While this tension is not significant, it helps to intuitively

explain some constraints on neutrino mass in Section 5.2, so we make note of it here.

5.1.2. ζ− M⁵⁰⁰ Constraints

Given the work of V10 and other cluster results (e.g.

Vikhlinin et al. 2009b; Mantz et al. 2010c; Rozo et al.

2010), we expect the cluster mass-calibration to be the dominant systematic uncertainty limiting our results. In Figure 5, we show the constraints on ASZ and σ8. The SPT cluster data has a significant degeneracy between the constraints on ASZ and σ8. From the SPT data, we constrain the fractional uncertainty, δA_SZ/ASZ, to be 27%, which is effectively only constrained by the 14 clusters that have both X-ray and SZ measurements.

This constraint is not significantly better than the un- certainty in the simulation based prior of 30%. With enough X-ray observations, we expect the ζ − M500 cal- ibration to track the Y_X − M⁵⁰⁰ relation, because the latter is currently better observationally constrained. In this limit, we would expect a fractional uncertainty on A_SZ of B_SZ(δA_X/A_X) ∼ 14%. In Section 4, for a ΛCDM cosmology, we nearly achieved this limit, constraining

Constraints on Dark Energy

w=−0.973 ± 0.063

SPT-clusters: −1.087 ± 0.363

Benson et al 2011

Gravitational deflection

• CMB is a unique source for lensing

• Gaussian, with well-understood power spectrum (contains all info)

• At redshift which is (a) unique, (b) known, and (c) highest

T ^L (ˆn) = T ^U (ˆn + ∇φ(ˆn))

CMB Lensing

∇φ(ˆn) = −2

� χ

0

dχ χ _� − χ

χ _� χ ∇ ⊥ Φ(χˆn, χ),

Broad kernel, peaks at z ~ 2 In WL limit, add many

deflections along line of sight

Photons get shifted ^{n ˆ}

T

n + ∇φ ˆ

Lensing simplified

• gravitational potentials

distort shapes by stretching, squeezing,

shearing

Gravity

Lensing simplified

• gravitational potentials

distort shapes by stretching, squeezing,

shearing

Gravity

Lensing simplified

• where gravity

stretches, gradients become smaller

• where gravity compresses,

gradients are larger

• shear changes direction

Gravity

• We extract ϕ by taking a suitable average over CMB multipoles

separated by a distance L

• We use the standard Hu quadratic estimator.

Mode Coupling from Lensing

T ^L (ˆn) = T ^U (ˆn + ∇φ(ˆn))

= T ^U (ˆn) + ∇T ^U (ˆn) · ∇φ(ˆn) + O(φ ² ),

• Non-gaussian mode coupling for l ₁ �= −l ² :

l x

l y

L

l CMB1

l CMB2

SPT Lensing Mass Map

+-0.05 color bar

(noise ~0.01)

Planck

(all-sky)

SPT

(2500 sq deg)

CMB Lensing Power Spectrum

• ^well

measured with

Planck, SPT, ACT

Planck XVII 2013

Massive Neutrinos in Cosmology

–Below free-streaming scale, neutrinos act like radiation

• drag on growth

–Above free-streaming scale, neutrinos act like matter

€

Ω _ν ≈ (m _i

i

∑ /0.1 eV ) 0.0022 h _0.7 ⁻²

Neutrinos & CMB Lensing

44

Neutrino masses

• Perturbations are washed out on

scales smaller than neutrino free-

streaming scale

• current upper bounds from CMB are WMAP: m nu < 1.3 eV ; WMAP+BAO+H 0 : m nu <

0.56 eV

d ∼ T ^ν /m _ν × 1/H

Neutrino masses

• Perturbations are washed out on

scales smaller than neutrino free-

streaming scale

• current upper bounds from CMB are WMAP: m nu < 1.3 eV ; WMAP+BAO+H 0 : m nu <

0.56 eV

d ∼ T ^ν /m _ν × 1/H

• Peak at l=40 (k eq =[300 Mpc] ^-1 at z = 2): coherent over degree scales

• RMS deflection angle is only ~2.7’

Upper limits on neutrino masses

• CMB experiments closing in on

interesting

neutrino mass range

• CMB lensing adds new information

– forecast ~0.05 eV sensitivity in ~4

yrs Planck collaboration 2013

Not everything makes total sense

• combining all cosmological

information leads to a preference for a high neutrino

mass and some form of new light particle in the

universe

Wyman et al 2013

SPT Lensing Mass Map

+-0.05 color bar

(noise ~0.01)

Cosmic Infrared

Background Traces Mass

48 SPT TT Lensing map 100 sq deg Herschel 500 um

Two Expected Sources of B Modes

Gravitational Radiation in Early Universe

(amplitude unknown!) Gravitational lensing of

E modes (amplitude

well-predicted, but no

measured B modes

until later in talk)

E-mode polarization of ra23h30, dec -55 field (150 GHz)

E

Predicting B-Modes

3

FIG. 1: (Left panel): Wiener-filtered E-mode polarization measured by SPTpol at 150 GHz. (Center panel): Wiener-filtered CMB lensing potential inferred from CIB fluctuations measured by Herschel at 500 µm. (Right panel): Gravitational lensing B-mode estimate synthesized using Eq. (1). The lower left corner of each panel indicates the blue(-)/red(+) color scale.

[23] onboard the Herschel space observatory [24] as a tracer of the CMB lensing potential φ. The CIB has been established as a well-matched tracer of the lens- ing potential [22, 25, 26] and currently provides a higher signal-to-noise estimate of φ than is available with CMB lens reconstruction. Its use in cross-correlation with the SPTpol data also makes our measurement less sensitive to instrumental systematic effects [27]. We focus on the Herschel 500 µm map, which has the best overlap with the CMB lensing kernel [22].

Post-Map Analysis: We obtain Fourier-domain CMB temperature and polarization modes using a Wiener filter (e.g. [28] and refs. therein), derived by maximizing the likelihood of the observed I, Q, and U maps as a function of the fields T (�l), E(�l), and B(�l). The filter simultaneously deconvolves the two- dimensional transfer function due to beam, TOD filter- ing, and map pixelization while down-weighting modes that are “noisy” due to either atmospheric fluctuations, extragalactic foreground power, or instrumental noise.

We place a prior on the CMB auto-spectra, using the best-fit cosmological model given by [29]. We use a sim- ple model for the extragalactic foreground power in tem- perature [19]. We use jackknife difference maps to deter- mine a combined atmosphere+instrument noise model, following [30]. We set the noise level to infinity for any pixels within 5^� of sources detected at > 5σ in [31]. We extend this mask to 10^� for all sources with flux greater than 50 mJy, as well as galaxy clusters detected using the Sunyaev-Zel’dovich effect in [32]. These cuts remove approximately 5 deg² of the total 100 deg² survey area.

We remove spatial modes close to the scan direction with an �_x < 400 cut, as well as all modes with l > 3000. For these cuts, our estimated beam and filter map transfer functions are within 20% of unity for every unmasked mode (and accounted for in our analysis in any case).

The Wiener filter naturally separates E and B con-

tributions, although in principle this separation depends on the priors placed on their power spectra. To check that we have successfully separated E and B, we also form a simpler estimate using the χ_B formalism advo- cated in [33]. This uses numerical derivatives to estimate a field χ_B(�x) which is proportional to B in harmonic space. This approach cleanly separates E and B, al- though it can be somewhat noisier due to mode-mixing induced by point source masking. We therefore do not mask point sources when applying the χB estimator.

We obtain Wiener-filtered estimates ˆφ^CIB of the lensing potential from the Herschel 500 µm maps by applying an apodized mask, Fourier transforming, and then multiply- ing by C^CIB-φ

l (C^CIB-^CIB

l C_l^φφ)⁻¹. We limit our analysis to modes l ≥ 150 of the CIB maps. We model the power spectrum of the CIB following [34], with C^CIB-^CIB

l =

3500(l/3000)^−1.25Jy²/sr. We model the cross-spectrum C^CIB-^φ

l between the CIB fluctuations and the lensing po- tential using the SSED model of [35], which places the peak of the CIB emissivity at redshift zc = 2 with a broad redshift kernel of width σ_z = 2. We choose a linear bias parameter for this model to agree with the results of [22, 26]. More realistic multi-frequency CIB models are available (for example, [36]); however, we only require a reasonable template. The detection significance is inde- pendent of errors in the amplitude of the assumed CIB-φ correlation.

Results: In Fig. 1, we plot Wiener-filtered estimates Eˆ¹⁵⁰ and ˆφ^CIB using the CMB measured by SPTpol at 150 GHz and the CIB fluctuations traced by Herschel. In addition, we plot our estimate of the lensing B modes, Bˆ^lens, obtained by applying Eq. (1) to these measure- ments. In Fig. 2 we show the cross-spectrum between this lensing B-mode estimate and the B modes mea- sured directly by SPTpol. The data points are a good fit to the expected cross-correlation, with a χ²/dof of 3.5/4 and a corresponding probability-to-exceed (PTE) of 48%.

measured E modes estimated  predicted B

Hanson, Hoover, Crites et al 2013

Many Ways of Predicting B-Modes

3

FIG. 1: (Left panel): Wiener-filtered E-mode polarization measured by SPTpol at 150 GHz. (Center panel): Wiener-filtered CMB lensing potential inferred from CIB fluctuations measured by Herschel at 500 µm. (Right panel): Gravitational lensing B-mode estimate synthesized using Eq. (1). The lower left corner of each panel indicates the blue(-)/red(+) color scale.

[23] onboard the Herschel space observatory [24] as a tracer of the CMB lensing potential φ. The CIB has been established as a well-matched tracer of the lens- ing potential [22, 25, 26] and currently provides a higher signal-to-noise estimate of φ than is available with CMB lens reconstruction. Its use in cross-correlation with the SPTpol data also makes our measurement less sensitive to instrumental systematic effects [27]. We focus on the Herschel 500 µm map, which has the best overlap with the CMB lensing kernel [22].

Post-Map Analysis: We obtain Fourier-domain CMB temperature and polarization modes using a Wiener filter (e.g. [28] and refs. therein), derived by maximizing the likelihood of the observed I, Q, and U maps as a function of the fields T (�l), E(�l), and B(�l). The filter simultaneously deconvolves the two- dimensional transfer function due to beam, TOD filter- ing, and map pixelization while down-weighting modes that are “noisy” due to either atmospheric fluctuations, extragalactic foreground power, or instrumental noise.

We place a prior on the CMB auto-spectra, using the best-fit cosmological model given by [29]. We use a sim- ple model for the extragalactic foreground power in tem- perature [19]. We use jackknife difference maps to deter- mine a combined atmosphere+instrument noise model, following [30]. We set the noise level to infinity for any pixels within 5^� of sources detected at > 5σ in [31]. We extend this mask to 10^� for all sources with flux greater than 50 mJy, as well as galaxy clusters detected using the Sunyaev-Zel’dovich effect in [32]. These cuts remove approximately 5 deg² of the total 100 deg² survey area.

We remove spatial modes close to the scan direction with an �_x < 400 cut, as well as all modes with l > 3000. For these cuts, our estimated beam and filter map transfer functions are within 20% of unity for every unmasked mode (and accounted for in our analysis in any case).

The Wiener filter naturally separates E and B con-

tributions, although in principle this separation depends on the priors placed on their power spectra. To check that we have successfully separated E and B, we also form a simpler estimate using the χ_B formalism advo- cated in [33]. This uses numerical derivatives to estimate a field χ_B(�x) which is proportional to B in harmonic space. This approach cleanly separates E and B, al- though it can be somewhat noisier due to mode-mixing induced by point source masking. We therefore do not mask point sources when applying the χB estimator.

We obtain Wiener-filtered estimates ˆφ^CIB of the lensing potential from the Herschel 500 µm maps by applying an apodized mask, Fourier transforming, and then multiply- ing by C^CIB-φ

l (C^CIB-^CIB

l C_l^φφ)⁻¹. We limit our analysis to modes l ≥ 150 of the CIB maps. We model the power spectrum of the CIB following [34], with C^CIB-^CIB

l =

3500(l/3000)^−1.25Jy²/sr. We model the cross-spectrum C^CIB-^φ

l between the CIB fluctuations and the lensing po- tential using the SSED model of [35], which places the peak of the CIB emissivity at redshift zc = 2 with a broad redshift kernel of width σ_z = 2. We choose a linear bias parameter for this model to agree with the results of [22, 26]. More realistic multi-frequency CIB models are available (for example, [36]); however, we only require a reasonable template. The detection significance is inde- pendent of errors in the amplitude of the assumed CIB-φ correlation.

Results: In Fig. 1, we plot Wiener-filtered estimates Eˆ¹⁵⁰ and ˆφ^CIB using the CMB measured by SPTpol at 150 GHz and the CIB fluctuations traced by Herschel. In addition, we plot our estimate of the lensing B modes, Bˆ^lens, obtained by applying Eq. (1) to these measure- ments. In Fig. 2 we show the cross-spectrum between this lensing B-mode estimate and the B modes mea- sured directly by SPTpol. The data points are a good fit to the expected cross-correlation, with a χ²/dof of 3.5/4 and a corresponding probability-to-exceed (PTE) of 48%.

measured E modes estimated  predicted B

E 150 GHz E 90 GHz

E from Temperature

 CIB

 TT

 EE

 TE

( Spitzer cat)

X B 150 GHz X B 90 GHz

Hanson, Hoover, Crites et al 2013

4

FIG. 2: (Black, center bars): Cross-correlation of the lens- ing B modes measured by SPTpol at 150 GHz with lensing B modes inferred from CIB fluctuations measured by Herschel and E modes measured by SPTpol at 150 GHz; as shown in Fig. 1. (Green, left-offset bars): Same as black, but using E modes measured at 95 GHz, testing both foreground contam- ination and instrumental systematics. (Orange, right-offset bars): Same as black, but with B modes obtained using the χB procedure described in the text rather than our fiducial Wiener filter. (Gray bars): Curl-mode null test as described in the text. (Dashed black curve): Lensing B-mode power spectrum in the fiducial cosmological model.

We determine the uncertainty and normalization of the cross-spectrum estimate using an ensemble of simulated, lensed CMB+noise maps and simulated Herschel maps.

We obtain comparable uncertainties if we replace any of the three fields involved in this procedure with observed data rather than a simulation, and the normalization we determine for each bin is within 15% of an analytical prediction based on approximating the Wiener filtering procedure as diagonal in Fourier space.

In addition to the cross-correlation Eφ×B, it is also interesting to take a “lensing perspective” and rear- range the fields to measure the correlation EB ×φ. In this approach, we perform a quadratic “EB” lens re- construction [13] to estimate the lensing potential ˆφ_EB, which we then cross-correlate with CIB fluctuations. The observed cross-spectrum can be compared to previous temperature-based lens reconstruction results [22, 26].

This cross-correlation is plotted in Fig. 3. Again, the shape of the cross-correlation which we observe is in good agreement with the fiducial model, with a χ²/dof of 2.2/4 and a PTE of 70%.

Both the Eφ×B and EB ×φ cross-spectra discussed above are probing the three-point correlation function (or bispectrum) between E, B, and φ that is induced by lensing. We assess the overall significance of the measure- ment by constructing a minimum-variance estimator for the amplitude ˆA of this bispectrum, normalized to have

FIG. 3: “Lensing view” of the EBφ correlation plotted in Fig. 2, in which we cross-correlate an EB lens reconstruc- tion from SPTpol data with CIB intensity fluctuations mea- sured by Herschel. Left green, center black, and right or- ange bars are as described in Fig. 2. Previous analyses using temperature-based lens reconstruction from Planck [26] and SPT-SZ [22] are shown with boxes. The results of [26] are at a nominal wavelength of 550 µm, which we scale to 500 µm with a factor of 1.22 [37]. The dashed black curve gives our fiducial model for C^CIB-φ

l as described in the text.

a value of unity for the fiducial cosmology+CIB model (analogous to the analyses of [38, 39] for the T T φ bis- pectrum). This estimator can be written as a weighted sum over either of the two cross-spectra already dis- cussed. Use of ˆA removes an arbitrary choice between the “lensing” or “B-mode” perspectives, as both are sim- ply collapsed faces of the EBφ bispectrum. Relative to our fiducial model, we measure a bispectrum amplitude A = 1.092ˆ ± 0.141, non-zero at approximately 7.7σ.

We have tested that this result is insensitive to analy- sis choices. Replacement of the B modes obtained using the baseline Wiener filter with those determined using the χ_B estimator causes a shift of 0.2σ. Our standard B-mode estimate incorporates a mask to exclude bright point sources, while the χB estimate does not. The good agreement between them indicates the insensitivity of po- larization lensing measurements to point-source contam- ination. If we change the scan direction cut from lx< 400 to 200 or 600, the measured amplitude shifts are less than 1.2σ, consistent with the root-mean-squared (RMS) shifts seen in simulations. If we repeat the analysis with- out correcting for I → Q, U leakage, the measured ampli- tude shifts by less than 0.1σ. A similar shift is found if we rotate the map polarization vectors by one degree to mimic an error in the average PSB angle.

We have produced estimates of ˆB^lens using alterna- tive estimators of E. When we replace the E modes measured at 150 GHz with those measured at 95 GHz, we measure an amplitude ˆA = 1.225± 0.164, indicating

First Detection of B-Modes

(predicted B) X (measured B) 

X T

Hanson, Hoover, Crites et al 2013

not zero at 7.7

Summary & Outlook

• high resolution CMB maps give new information about the universe

• gravitational lensing of the CMB a powerful new probe

– lensing of temperature fluctuations now a mature field

– lensing of polarization fluctuations just measured for first time

• B-mode polarization anisotropy has now been detected

– next up: B modes from early universe!