**The Cosmic Microwave ** **Background in High **

**The Cosmic Microwave**

**Background in High**

**Definition**

**Definition**

**Gil Holder**

**Gil Holder**

**as part of:**

**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**

**– special timescale in the**

**universe for photons**

**Hot Big ** **Bang**

**Hot Big**

**Bang**

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

**The Cosmic ** **Microwave ** **Background**

**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**

**Isotropy**

### • Cosmic

### microwave background is

### remarkably isotropic

### • Unnaturally

### isotropic! +-3.5 mK scale

*WMAP science team*

**The Cosmic ** **Microwave ** **Background**

**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”: **

**Nothing too strange**

**within our “horizon”:**

**40 billion light years**

**40 billion light years**

### Image from WMAP

8

### Planck has higher resolution than WMAP

### 16 ^{o}

9
### WMAP 60 GHz Planck 143 GHz

**South Pole ** **Telescope**

**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 Polarization**

### • CMB fluctuations are relatively strongly

### polarized (~10%)

^{14}

### Q

### U

**E-modes/B-modes**

**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**

**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

^{−1}

### . 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

_{l}

### , which decreases as a function of wavenumber.

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

^{2}

### C

_{l}

### ∼ 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

^{−1}

### . 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

_{l}

### , which decreases as a function of wavenumber.

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

^{2}

### C

_{l}

### ∼ 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**

**B modes**

*Bunn*

**E-modes/B-modes**

**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**

**E modes**

### • **Lensing of CMB is **

**Lensing of CMB is**

**much more obvious in ** **polarization! **

**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 **

**only**

**upper **

**upper**

**limits **

**limits**

**on B **

**on B**

**mode **

**mode**

**power**

**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/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**

**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

^{−1}

### . 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

_{l}

### , which decreases as a function of wavenumber.

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

^{2}

### C

_{l}

### ∼ 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).

### 2. Pure and ambiguous modes

^{◦}

### angle, as shown in Figure 1.

^{−1}

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

_{l}

### , which decreases as a function of wavenumber.

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

^{2}

### C

_{l}

### ∼ constant.)

**B modes**

**B modes**

*Bunn*

**Cosmic Microwave ** **Background **

**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

^{23}

**WMAP** **(all sky)**

**South Pole Telescope** **(total 2500 sq deg)**

### 8

^{o}

### SPT-SZ Survey (completed)

Final survey depths of:

**- 100 GHz: < 40 uK**CMB-arcmin
**- 150 GHz: < 18 uK**CMB-arcmin
**- 220 GHz: < 80 uK**CMB-arcmin

**2500 square degrees**

**Gravity at work**

**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**

**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*

^{15 }

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**

**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**

**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**

**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

500### (z=0.6) > ~3x10

^{14}

### M

sol### /h

70### 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*

_{L}^{}

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

_{L}### *!!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**

**Characterizing Dark**

**Energy**

### from Dark Energy Task Force report **w=-0.9**

**w=-0.9**

**w=-1**

**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 σ_{8} 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
σ_{8} = 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 σ_{8}, 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 H_{0}
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− M^{500} 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} − M^{500} 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**

**Constraints on Dark Energy**

**w=−0.973 ± 0.063**

### SPT-clusters: −1.087 ± 0.363

**Benson et al 2011**

**Benson et al 2011**

**Gravitational deflection**

**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**

**Lensing simplified**

### • gravitational potentials

### distort shapes by stretching, squeezing,

### shearing

### Gravity

**Lensing simplified**

**Lensing simplified**

### • gravitational potentials

### distort shapes by stretching, squeezing,

### shearing

### Gravity

**Lensing simplified**

**Lensing simplified**

### • where gravity

### stretches, gradients become smaller

### • where gravity compresses,

### gradients are larger

### • shear changes **direction**

**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**

**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**

**SPT Lensing Mass Map**

**+-0.05 color bar**

**+-0.05 color bar**

**(noise ~0.01)**

**(noise ~0.01)**

### Planck

### (all-sky)

### SPT

### (2500 sq deg)

**CMB Lensing Power Spectrum **

**CMB Lensing Power Spectrum**

### • ^{well }

### measured with

### Planck, SPT, ACT

### Planck XVII 2013

**Massive Neutrinos in ** **Cosmology**

**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}

*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*

^{45}

### 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**

**SPT Lensing Mass Map**

**+-0.05 color bar**

**+-0.05 color bar**

**(noise ~0.01)**

**(noise ~0.01)**

**Cosmic Infrared **

**Cosmic Infrared**

**Background Traces Mass**

**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 deg^{2} of the total 100 deg^{2} 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)^{−1.25}Jy^{2}/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
Eˆ^{150} 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 χ^{2}/dof of 3.5/4
and a corresponding probability-to-exceed (PTE) of 48%.

**measured E modes** **estimated ** **predicted B**

**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 deg^{2} of the total 100 deg^{2} 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)^{−1.25}Jy^{2}/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
Eˆ^{150} 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 χ^{2}/dof of 3.5/4
and a corresponding probability-to-exceed (PTE) of 48%.

**measured E modes** **estimated ** **predicted B**

**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 χ^{2}/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)*

^{EB}

### X T

^{cib}