CMB B-mode polarization from Thomson scattering in the local universe

0.26 MB·English

by  Christopher M. Hirata

CMB B-mode polarization from Thomson scattering in the local universe
Christopher M. Hirata,1,∗ Abraham Loeb,2 and Niayesh Afshordi2
1Department of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA
2Harvard-Smithsonian Center for Astrophysics, MS-51, 60 Garden Street, Cambridge, MA 02138, USA
(Dated: January 10, 2005)
The polarization of thecosmic microwavebackground (CMB) is widely recognized as apotential
source of information about primordial gravitational waves. The gravitational wave contribution
can be separated from the dominant CMB polarization created by density perturbations at the
times of recombination and reionization because it generates both E and B polarization modes,
whereas the density perturbations create only E polarization. The limits of our ability to measure
gravitationalwavesarethusdeterminedbystatisticalandsystematicerrorsfromCMBexperiments,
foregrounds, and nonlinear evolution effects such as gravitational lensing of the CMB. Usually it
is assumed that most foregrounds can be removed because of their frequency dependence,however
5
ThomsonscatteringoftheCMBquadrupolebyelectronsintheGalaxyornearbystructuresshares
0
0 the blackbody frequency dependenceof the CMB. If the optical depth from these nearby electrons
2 is anisotropic, the polarization generated can include B modes even if no tensor perturbations are
present. We estimate this effect for the Galactic disk and nearby extragalactic structures, and
n find that it contributes to the B polarization at the level of ∼ (1–2)×10−4µK per logarithmic
a
interval in multipole ℓ for ℓ<30. This is well below the detectability level even for a future CMB
J
polarization satellite and hence is negligible. Depending on its structure and extent, the Galactic
0
coronamaybeasourceofB-modescomparabletotheresiduallarge-scalelensingB-modeafterthe
1
latter has been cleaned using lensing reconstruction techniques. For an extremely ambitious post-
PlanckCMBexperiment,ThomsonscatteringintheGalacticcoronaisthusapotentialcontaminant
1
of the gravitational wave signal; conversely, if the other foregrounds can be cleaned out, such an
v
experiment might beable to constrain models of the corona.
7
6
PACSnumbers: 98.80.Es,95.30.Gv,98.35.Gi
1
1
0
I. INTRODUCTION the ℓ < 100 temperature power spectrum of the CMB,
5
andthenondetectionofthisfeatureintheWilkinsonMi-
0
/ Recent observations of the cosmic microwave back- crowave Anisotropy Probe (WMAP) data sets an upper
h
limit of T/S < 0.7 [7]. However the use of the temper-
p ground(CMB)anisotropieshaveconfirmedseveralofthe
ature power spectrum to constrain T/S suffers from the
- predictions of the simplest inflationary models [1, 2, 3].
o large cosmic variance error bars on the low CMB mul-
They have (in combination with other datasets) shown
r tipoles, which impose a fundamental limit on measure-
st that the universe is close to spatially flat, with Ωtot = ments of T/S from CTT. In addition, these measure-
a 1.02 0.02 [4]; that the perturbations in the CMB ℓ
± ments are model-dependent in the sense that modifica-
: are close to Gaussian [5]; that the temperature and E-
v tions to the scalar power spectrum can produce excess
polarization power spectra match those expected from
i
X adiabatic initial conditions [6]; and that the perturba- power at ℓ<100 without invoking tensors (for example,
T/S ispartiallydegeneratewiththerunningofthespec-
r tions are close to scale-invariantwith ns 1 and αs 0
a [4]. Thus there is now interest in testing≈the last of≈the tralindexαs inthecurrentdata[8]). Analternativeisto
usetheCMBpolarization,whichcanbedecomposedinto
major predictions ofinflation: the existence ofa roughly
E and B modes [9, 10]. Since the scalar density fluctua-
scale-invariantbackgroundoftensor(gravitationalwave)
tionsofmultipoleℓhaveparity( 1)ℓ,whereastheE and
perturbations. The amplitude of the tensor perturba-
−
B polarization modes have parity ( 1)ℓ and ( 1)ℓ re-
tions is typically described with the ratio of the temper-
− − −
ature quadrupoles in the CMB T/S Ctensor/Cscalar. spectively,it followsthat the scalarscanproduce only E
≡ 2 2 polarization in linear perturbation theory. However ten-
The value of T/S depends on the specific inflationary
modelthroughtherelationT/S =V /(3.7 1016 GeV)4, sor perturbations consist of both positive and negative
∗
× parity modes for every value of ℓ 2, so the tensors can
where V is the energy density at the time during infla-
∗ ≥
tion when observable scales in the CMB (10−4 < k < contribute to both E and B. The measurement of T/S
10−2h Mpc−1) exited the horizon. A detection of the through B-mode polarization thus is not limited by the
cosmicvarianceassociatedwiththescalars,andisnotde-
tensor perturbations would bolster confidence in infla-
generatewithanyfeaturesinthe scalarpowerspectrum;
tion, as well as providing a measurement of V .
∗
insteaditislimitedbytheobservations,foregrounds,and
If T/S is large, then the tensor perturbations add to
nonlinear processes that can create B-mode polarization
from scalar initial conditions.
The dominant nonlinear process that generates B-
∗Electronicaddress: [email protected] mode polarization in the CMB is gravitational lensing
2
of the E polarization generated during recombination. and
On the large angular scales relevant for tensor B-mode
sCeBarBches,2t.h4elen1s0i−n6gµBK-m2;otdheisapwpoeualrdsasestwahitdeetneocitsaebwiliitthy E2m(rnˆ)= Y2Em†(nˆ′)P(nˆ′)|rnˆd2nˆ′ (3)
limℓ it o≈f (T/S×) = 6 10−5 for a full-sky experiment Z
if τ = 0.17 andmi(nT/S)m×in = 2×10−5 if τ = 0.07. (The amreenttsheofttehmepCeMraBtuartepaonsditiopnolranˆrizraeltaiotinveqtuoatdhreupooblseermveor-,
optical depth to reionization τ is relevant since this de-
and
terminestheamplitudeoflarge-scalepolarization.) How-
ever, lensing results in higher-ordercorrelationsbetween dτ(r,nˆ)
g(r,nˆ)=e−τ(r,nˆ) (4)
theEandBpolarizationmodesthatallowoneto“clean” dr
out the lensing B-mode [11, 12, 13]. The amount by
whichonecancleantheB-modeusingtheiterativetech- isthe Thomsonvisibilityfunction, whichdepends onthe
niqueofRef.[14]wascomputedbyRef.[15]asafunction opticaldepth τ(r,nˆ)to distance r; we havetriviallygen-
of the detector noise and beam size; for the most sen- eralizedfromRef. [16]toallowτ to dependonthe direc-
sitive experiment considered (0.25 µK arcmin noise and tion nˆ. Here Yℓm are the spherical harmonics and YℓEm
2arcminbeam),theB-modepolarizationcanbecleaned are the tensor spherical harmonics. (We normalize these
downtoCBB 5.8 10−8µK2, whichreduces(T/S) to YE†YE (nˆ)d2nˆ = 1 over the whole sphere, with
by a factorℓof≈40. × min the innℓemr prℓmoduct P†P = Q∗Q +U∗U ; this is con-
R 1 2 1 2 1 2
The polarized foregrounds contain both Galactic and sistent with Ref. [9] and with CMBFast [17] but differs
extragalacticcomponents. Theextragalacticforegrounds by a factor of √2 from Ref. [10].) If we break the line-
such as point sources are approximately white noise and of-sightintegralinto a piece containedwithin our galaxy
hence are largest on small angular scales, whereas the (r < R = 100 kpc) and a piece external to the galaxy
Galactic synchrotron and dust emission are expected to (r >R), we find to first order in τ(R,nˆ):
have a significant polarization at low ℓ. These can be at
leastpartiallycleanedusing their frequency dependence, P(nˆ)=[1 τ(R,nˆ)]P (nˆ)+P (nˆ), (5)
cosmic gal
−
whichdiffersfromtheblackbodysignatureexpectedfrom
tensors. However it is also possible to generate polariza- where Pcosmic(nˆ) is the polarization that we compute if
tion via Thomson scattering of the CMB quadrupole by we neglect the Galactic Thomson scattering and
free electrons in the Galaxy or other nearby structures.
While this polarization signal is extremely small and is √6 2
P (nˆ)= τ(R,nˆ) (T √6E )YE (nˆ),
well below the predicted level of the synchrotronand/or gal −10 2m− 2m 2m
m=−2
dust foregrounds at all frequencies, it cannot be cleaned X
(6)
using frequency information since it is a blackbody sig-
(We have assumed that the cosmic quadrupoles
nal. It can contain both E and B polarization if the
T (rnˆ) √6E (rnˆ) at position rnˆ inside the Galaxy
distribution of scattering electrons is anisotropic, hence 2m − 2m
can be replaced by their values at the observer, T
it is a potential foreground for tensor B-mode searches. 2m −
√6E , which is a good approximation since the cos-
The primary purpose of this paper is to investigate this 2m
mic quadrupole is dominated by perturbation modes
Thomson polarization from the local universe.
with wavenumber k R−1.) The optical depth is
≪
related to the electron distribution through the result
dτ/dr = n σ /(1+z) where n is the electron density
e T e
II. GALACTIC CONTRIBUTION TO andσ =6.65 10−25cm2 istheThomsoncrosssection.
T
POLARIZATION If τ(R,nˆ) we×re independent of nˆ, i.e. if the Galactic
optical depth were the same in all directions, then the
A. Basic model only effect of the Thomson scattering would be to (i)
suppressthe CMB powerspectrum because ofthe factor
The polarization Stokes parameters P = (Q,U) pro- of 1 τ(R,nˆ), and (ii) generate a small amount of ℓ=2
ducedbyThomsonscatteringaremosteasilydetermined E-m−ode through Pgal. However if τ(R,nˆ) is anisotropic
in the line-of-sight formalism [16] as with fluctuations δτ/τ 1, which is the case in the real
Galaxy we inhabit, the∼n P becomes a mixture of E
gal
√6 ∞ 2 andB modesthatis genericallyoforderτ times thecos-
P(nˆ)= g(r,nˆ) [T (rnˆ) √6E (rnˆ)] mic temperature+polarization quadrupole. In the rest
2m 2m
−10 −
Z0 mX=−2 of this paper, we will neglect E2m compared to T2m in
YE (nˆ)dr, (1) Eq. (6), since even for the WMAP τ = 0.17 cosmology
× 2m we have CEE =0.08µK2, whereas the observedtemper-
2
ature quadrupole in the map of Tegmark et al. [18] is
where
CTT =200µK2.
2
Since this temperature quadrupole has already been
T2m(rnˆ)= Y2∗m(nˆ′)T(nˆ′)|rnˆd2nˆ′ (2) measured, we can compute the induced polarization if
Z
3
we have a map of the Galactic electron distribution. In be comparable to the disk contribution. If instead the
this paper we compute the polarization P using the baryons are uniformly distributed in a ball, the radius
gal
modelofCordes&Lazio[19]fortheGalacticdistribution and optical depth are related to the number density by
of free electrons. This model is based on pulsar disper-
sion (DM) and scattering (SM) measures as well as SMs M /1011M 1/3
b ⊙
R = 195kpc
of other Galactic and extragalactic radio sources, and is n /10−4cm−3
(cid:18) H (cid:19)
appropriatefor the electroncontentofthe Galactic disk;
M 1/3 n 2/3
thepredictedopticaldepthinthedirectionofthe Galac- τ = 4.6 10−5 b H .(8)
tsiocmpeowlehsaitsl4ar×ge1r0−th5a(nNtohretho)potric5al×d1e0p−th5 (oSfout3h.)4. Th1i0s−i5s × (cid:18)1011M⊙(cid:19) (cid:16)10−4cm−3(cid:17)
∼ × One of the available constraints on the optical depth
predicted by the older Taylor & Cordes model [20].
through the corona comes from the dispersion measures
TheinputdatatotheCordes&Laziomodelissparsely
(DMs)ofpulsarsobservedintheLarge(LMC)andSmall
sampled, witha typicaldensity of pulsarsathigh Galac-
(SMC) Magellanic Clouds. The Thomson optical depth
tic latitude (b > 20 deg) of n 6sr−1, and small-scale
| | ∼ and the DM along a line of sight are both proportional
structurein the electrondensity mapisthus expected to
to n dr, hence they are related by the equation τ =
be lost [21]. As a crude estimate, we note that the num- e
2.05 10−6DMwheretheDMisinunitsofcm−3pc. The
ber of modes per steradian is ∼ ℓ2max/4π; setting this obsRe×rved DM in the Magellanic Cloud pulsars is greater
equal to n implies that structures smaller than ℓ 9
max ∼ thanthatpredictedbytheGalacticthickdiskmodel[19]
shouldbe lost. (The 14 non-pulsarSMs at highGalactic
by16–91cm−3 pcdependingonthepulsar[21]. However
latitude do not significantly alter this conclusion.) The
muchofthisexcessDMmaybeinternaltotheMagellanic
effective ℓ at high Galactic latitude may be even less
max Clouds and not part of the Galactic corona. Cordes &
thanthisbecausesomeofthepulsarsliewithinthethick
Lazio [21] argue for this interpretation since the DMs of
diskandtheirlinesofsightdonotprobestructurebehind
pulsars not in the Magellanic Clouds are only fit if the
them. Fortunately, this loss of resolution will not have
electrondensityfallsoffbeyond 1kpcfromtheplaneof
a large influence on the results of this paper because we ∼
the Galaxy (see their Sec. 3.1.1). This is also consistent
will find that the extragalactic contribution dominates
with the factor of 5 disparity among DM excesses in
the Galactic thick disk for ℓ 5. ∼
≥ the LMC along lines of sight separated by only a few
degrees. In any case the contribution of the Galactic
corona along the line of sight to the LMC at a distance
B. Milky Way corona
of50kpcis limited to DM 16cm−3 pc orτ 3 10−5.
≤ ≤ ×
Ifthisupperlimitistypicaloflinesofsightto50kpcand
TheCordes&LaziomodeldoesnotincludetheGalac-
we use Eq. (7) with M =1.2 1011M and R=50kpc
tic corona component. Indeed, very few constraints on b × ⊙
as an upper limit on the optical depth beyond 50kpc,
the electron density in the corona are available, which
we find that the optical depth through the Milky Way
is unfortunate because the corona may contain a signif-
corona is at most 3 10−4. If one instead assumes the
icant fraction of the Milky Way’s baryons [22]. Models ×
uniform ball model (Eq. 8) for the corona and assigns
of the Milky Way suggest a baryonic mass in the bulge
a baryonic mass M = 1.2 1011M , the DM to the
amnadssdiisskesctoimmaptoendeantts∼of1∼0125M×⊙10[1203M].⊙F,owrthheerecausrrtehnetvlyirfiaa-l aLnMdCτp<rov1id.1es a1n0−upb4.perThliimsi×tupopfenrHlim<⊙it3.2is×si1g0n−ifi4ccamnt−ly3
svuogregdesbtsartyhoantfraanctaiodndiftibo=naΩl b/Ω1m.2=01.10711±M0.01of[7“]m, tihssis- strengthenedi×fthecoronadensityisassumedtodecrease
∼ × ⊙ outwards as a power-law more gradual than 1/r since
ing” baryons should either be present inside the Milky ∝
this places the bulk of the baryons even farther away
Way’s virial radius or else must have been ejected. (In-
(if the power law is steeper than 1/r the optical depth
deed, Ref. [23] argued that at least half of the baryons
diverges at small r, and would be cut off by either some
aremissingeventhoughthey assumedf =0.1. Increas-
b core radius or our finite distance from the center of the
ing f to 0.17only makesthe problemworse.) Assuming
b Galaxy).
that a mass M of baryons were ionized and are present
b ThereareotherconstraintsonthedensityoftheGalac-
in ashell atradiusR, they presenta meanopticaldepth
tic corona. If the infall velocities of high-velocity clouds
(1 Y/2)σ M (M /1011M ) areassumedtobelessthantheirterminalvelocitydueto
τ¯= − T b =5.9 10−5 b ⊙ , (7) drag in the Galactic corona, the density of the corona is
4πm R2 × (R/100kpc)2
p limited to n < 3 10−4cm−3 at 100kpc [24]. Mod-
H
×
where Y = 0.24 is the Helium abundance, m is the els of the kinematics [25] and Hα emission [26] from
p
proton mass, and we have neglected the distance from the Magellanic Stream argue for n 10−4cm−3 and
H
theEarthtothecenteroftheGalaxyincomparisonwith 5 10−5cm−3 respectively at radiu∼s 50–65kpc (but
∼ ×
R. The virial radius for the Milky Way is estimated see criticism of these models arguing that survivalof the
as r 260kpc [23], so that if a large fraction of the stream then requires n < 10−5cm−3 [27]). Interaction
vir H
∼
missing baryons are significantly inside the virial radius ofhigh-velocitycloudswithaGalacticcoronaisalsosug-
then the corona contribution to the optical depth could gested in order to produce some of the O vi detected in
4
region in Eq. 2. Unlike the case of an all-sky map where
Contamination of all-sky power spectrum
Eq. (9) is the only “reasonable” (rotationally invariant)
2K) 10-5 Lensing (uncleaned) measurementoftheB-modepowerspectrum,ananalysis
µ Lensing (cleaned) ofthe B-mode powerspectrum ona cut sky requiresthe
π2 ( choice of an estimator. We choose the quadratic estima-
BB /+1)CL 10-6 otofrl,ewnghtihch2iNnv,olwvheserteheNveicstotrhexn=um(Qbe1r,..o.Qf Npi,xUe1ls,....UTNhe)
L
um L( Tensors (T/S=10-5) covariance matrix C=hxx†i is then written as
ctr 10-7
pe ℓmax ℓ
wer s Galactic Thomson Cij = Nij + CℓEEYℓEm(nˆi)YℓEm†(nˆj)
Po 10-8 Xℓ=2mX=−ℓh
2 5 10 20 50 +CℓBBYℓBm(nˆi)YℓBm†(nˆj)
Multipole, L
= N + p C , i (10)
ij α αij
FIG.1: ThecontributionstotheB-modepolarization power
spectrum. The white-noise (Cℓ ∝ ℓ2) lines are the lensing Xα
signal before cleaning (top line) and after cleaning assuming where N is the noise covariance matrix, p are the pa-
α
iterative lensing reconstruction with 0.25 µK arcmin noise rametersofthecovariancematrix,andC arethepower
α
and 2 arcmin beam (bottom line). The dashed line shows
spectrum templates. Our objective is to estimate the
the tensor B-mode spectrum for T/S = 10−5. The Galactic
parameters p . The quadratic estimator determines the
Thomson scattering contribution is shown by the solid line α
parameters using the relation
with points; note that due to the approximate parity sym-
metry of the Galactic electron distribution, the odd ℓ modes
pˆ = [F−1] q ,
havemuch more power than even ℓ. α αβ β
1 1
q = x†C−1C C−1x Tr C−1C C−1N ,
α 2 0 α 0 − 2 0 α 0
absorption [28, 29]. If the higher values of nH [25, 26] F = 1Tr C−1C C−1C ; (cid:0) (cid:1)(11)
are correct, and the corona remains this dense out to αβ 2 0 α 0 β
radii>100kpc,thenitispossiblethatthe opticaldepth (cid:0) (cid:1)
through the halo may be 10−4. here F is the Fisher matrix and C0 is a positive defi-
∼ nite Hermitian weighting matrix. The quadratic estima-
InordertoassesstheB-modesproducedinthecorona,
tor method is unbiased regardless of the choice of C in
we must remember that it is the fluctuations in optical 0
Eq.(11), but is only optimal if C is the true covariance
depth δτ, rather than the mean optical depth τ¯, that 0
matrix. (C controls the relative weighting of modes.)
produces B-modes. We will return to this point in the 0
In our case, we will choose C to contain the noise due
discussion, and for now we focus on the smooth Cordes 0
tolensingaftertheiterativecleaning,sothatitequalsC
& Laziomodel[19]for the knownpopulationofGalactic
if N is white noise with 0.83µK arcmin and CBB = 0,
electrons. ij ℓ
and CEE noise in Eq. (10) since the CMB E-mode
ℓ ≫
power spectrum is large.
To understand the relationship between this method
III. POWER SPECTRUM
and the E/B decomposition of Ref. [30], note that as
CEE , any polarization mode x that has non-zero
We next analyze the B-mode power spectrum of the inℓner→pro∞duct with an E mode has x†C x . Also,
scatteredradiation. Inthecaseofanall-skyexperiment, 0 → ∞
any mode that is orthogonal to all E modes – i.e. that
one cansimply do the sphericalharmonicdecomposition
is a “pure B-mode” in the terminology of Ref. [30] –
has C x = Nx and so x is an eigenvector of C−1 with
1 ℓ 2 0 0
CBB(scat)= YB†(nˆ)P(nˆ)d2nˆ , (9) eigenvalue given by the inverse noise variance per pixel.
ℓ 2ℓ+1 ℓm Thus the quadratic estimator procedure Eq. (11) can be
m=−ℓ(cid:12)Z (cid:12)
X (cid:12) (cid:12) thought of as a method for projecting out the pure E-
(cid:12) (cid:12)
This is shown in Fig. 1, alo(cid:12)ng with the lensing(cid:12) contri- modes and ambiguous modes, and then computing the
bution, and the gravitational waves for T/S = 10−5 as power spectrum; the Fisher matrix F serves also as the
computed by CMBFast [17]. matrix of window functions for the quadratic estimators
A realistic low-ℓ B-mode experiment will exclude the q . (This same type of projectionhas been suggestedby
α
GalacticPlanesincethisiswheretheforegroundcontam- Ref. [31].)
ination is the worst. We would therefore like to under- Numerically,wehaveevaluatedEq.(11)usingaweight
stand the B-mode contamination on the cut sky. Here with CEE equal to 100 times the noise level for 2 ℓ
ℓ ≤ ≤
we will considercutting out the regionwithin 10 degrees ℓ = 80. We use the implementation of the quadratic
max
of the Galactic Plane; note that we are cutting this re- estimatorsdescribed in Ref. [32] (with no preconditioner
gionoutofthe finalmapP ,notoutofthe integration for the C−1 operations) and the HEALPix resolution
gal 0
5
(of order 1 kpc) is comparable to (and in some cases,
B-modes from Thomson scattering with Galactic Plane cut
greaterthan)the distancetotheforeground-emittingre-
10-6 CCBLBL BB <>00 gions. SinceWMAPreportsthetemperatureobservedin
adirectionnˆ butcannotspecifythedistancetotheemit-
2πµ2 (K) 10-7 ttthionecgamrlceicugrlioaowtne,aPvoengeafoln.reeWegdreoswuaninldlamcdaadnkitebiotehndeaelacasossmusumpmopspteitdoioninnhtieonreaotr“hdlaoetr-
BB /+1)CL 10-8 cthael”scizoentorfibthuetioGnalfaroxmy)naenadrbay“(gdloisbtaaln”cceosnmtruibchutlieosns ftrhoamn
L
m, L( distances of order the size of the Galaxy. The local con-
ctru 10-9 tributionwe assumeto be plane-parallel;inthis casethe
e quadrupole is independent of the position of the scatter-
p
s
er ing electron and we may use the observedquadrupole at
w
Po 10-10 Earth. Theglobalcontributiontothephotonquadrupole
atthepositionofthescatteringelectroncanalsobetaken
to equal the foreground quadrupole observed at Earth,
10-11 since the distance to the scattering electron (< 1 kpc)
2 3 4 5 10 15 20 is small compared to the size of the Galaxy. B∼eing non-
Multipole, L
axisymmetric, the global contribution can produce both
FIG.2: TheB-modepowerspectrumonacutsky(with|b|< EandBpolarization. Withinthissetofassumptions,we
10degreesremoved)fromThomsonscatteringintheGalactic can calculate the B-modes from Thomson scattering of
disk,computed with Eq.(11). The dottedline is thecleaned Galacticforegroundemissionbyrepeatingtheprocedure
lensing signal assuming iterative lensing reconstruction with ofSec.III withthe uncleanedWMAP maps. Since these
0.25µKarcminnoiseand2arcminbeam,andthedashedline
maps contain both foregrounds and CMB, the result –
is the tensor contribution for T/S = 10−5. The points show
shown in Fig. 3 – is the B-mode power spectrum from
the polarization band powers from Thomson scattering (the
Thomson scattering of both the CMB and the Galactic
open squares represent negative band powers). Some power
foreground emission.
may be missing at ℓ ≥ 10 due to the sparse sampling of the
pulsars used to construct theelectron density model.
V. EXTRAGALACTIC CONTRIBUTION
5 pixelization system [33], which has 12 288 pixels. In
Fig.2wehaveshownthepowerspectrumofthe B-mode In addition to the Galactic electrons, the CMB
polarizationobtained if we accept only the 10 112 pixels quadrupole can also be scattered off of nearby struc-
at Galactic latitude b > 10 degrees. We have plugged tures in the universe, generating B-mode polarization.
the resulting polariza|ti|on data vector x of length 20 224 In this section, we begin by computing the B polariza-
into Eq.(11), leavingout the Tr(C−01CαC−01N) term in tion from structures out to a distance of 80 h−1 Mpc
qα since the Thomson scattering map does not include using a model for the local electron distribution. We
the noise. The resulting CBB(scat) represent the addi- then calculate the expected CBB statistically from the
ℓ ℓ
tive bias to the B-mode power spectrum estimators due power spectrum. The former method has the advantage
to the Galactic Thomson scattering component. (These of correspondingapproximately to the actual realization
arenegativeforℓ=4,5;notethatonacutskyitispossi- of the electron distribution in the universe, whereas the
bleforthepowerspectrumestimatortobecomenegative latter method is better for addressing the contribution
even without noise subtraction.) from larger distances.
IV. POLARIZATION FROM SCATTERED A. Constrained realization method
GALACTIC EMISSION
Our first method for estimating the local extragalac-
In additionto scatteringthe CMB quadrupole, Galac- tic contribution to the B-mode polarization is to use
tic electrons can scatter the microwaveforeground emis- the “constrained realization” N-body simulations from
sion emitted by the Galaxy itself. In order to compute Mathis et al. [34] and assume that the electron den-
thiseffect,oneneedsnotonlyamodelforthedistribution sity traces the dark matter. These simulations have
of electrons but also for the foregroundemission, so that constrained initial conditions intended to reproduce the
one can calculate the quadrupole in Eq. (6). While the structuresobservedinthelocaluniverseouttoadistance
foreground emission at frequencies from 23–94 GHz has of 80h−1 Mpc andonscaleslargerthan5h−1 Mpc (on
been mapped out by WMAP, the quadrupole T (rnˆ) sm∼aller scales, the initial conditions are filled in with a
2m
at the scattering electron may differ from the locally ob- Gaussian random field with appropriate power). Mathis
served quadrupole because the distance to the electron etal. [34]simulatedbothaΛCDMandEinstein-deSitter
6
(a) B-modes from Thomson scattering in K-band (b) B-modes from Thomson scattering in W-band
CCBLBL BB <>00 CCBLBL BB ><00
10-5 10-5
2K) 2K)
µ µ
π ( π (
2 2
BB /CL 10-6 BBC /L 10-6
1) 1)
+ +
L L
L( L(
m, 10-7 m, 10-7
u u
ctr ctr
e e
p p
s s
wer 10-8 wer 10-8
o o
P P
10-9 10-9
2 3 4 5 10 15 20 2 3 4 5 10 15 20
Multipole, L Multipole, L
FIG. 3: The B-mode power spectrum on a cut sky (with |b| < 10 degrees removed) from Thomson scattering, in (a) the K
(23 GHz) band and (b) the W (94 GHz) band. This figure is the same as Fig. 2 except that we have replaced the cosmic
quadrupolewiththerawKandWbandquadrupolesobservedbyWMAP(nofrequencycleaning). Somepowermaybemissing
at ℓ≥10 dueto the sparse sampling of thepulsars used to construct theelectron density model.
cosmology;herewehaveusedtheformer(withΩ =0.3, wavenumber,
m
H = 70 km/s/Mpc, and σ = 0.9) as it corresponds
0 8
most closely with the currently favored cosmologicalpa- 4πGρ¯(1+δ) 104K
rameters. kJ = =4.2hMpc−1 (1+δ),
The perturbation to the optical depth due to nearby p cs s Tg
(14)
extragalactic structures is calculated according to
where c is the sound speed, T is the gas tempera-
s g
ture, and we have assumed the sound speed relation
n¯ (z)
δτ =σ e δ dr, (12) c2 =5k T/3µwith mean mass per particle µ=0.59m ,
T 1+z e s B p
Z appropriate for ionized gas with Y = 0.24. The inter-
galacticmediumisphotoionizedsoweexpectT 104K
where δe is the fractional electron density perturbation; and hence in regions of average density all sgtr≥uctures
here we assume δ = δ, which should be valid except on
e smaller than k−1 0.2h−1 Mpc should be washed out if
very small scales where the baryons are segregated from J ≥
weconsiderδ insteadofδ. Inthissenseourestimateus-
the dark matter. We have written this as an integral e
ing smoothing at 0.1h−1 Mpc is conservativeand should
overthecomovingdistancer. Themeanelectrondensity
overestimateCBB. Mostofthesmall-scalepowerishow-
assuming complete ionization is ℓ
ever in overdense regions where 1+δ 1, so depending
≫
on T in these regions the smoothing scale may be dif-
3Ω H2(1 Y/2) g
n¯ = b 0 − (1+z)3 =2.2 10−7(1+z)3cm−3, ferent from the typical value of k−1. We find that the
e 8πGm × J
p ℓ<10modesareonlyslightlyaffected(by<20%)ifwe
(13)
increase the smoothing length to 0.5h−1 Mpc, indicat-
where G is Newton’s gravitational constant and the
ingthatthe resultsarenotdominatedby the small-scale
baryon density Ω h2 is from WMAP [4]. Here we have
b modes.
neglectedredshiftevolutionofn¯ (z)/(1+z)andδ since
e e Once δτ maps are constructed, these can be fed
the simulation goes out to a distance of 80h−1 Mpc
through Eq. (6) to compute the polarization signal. The
(z = 0.027); n¯ (z)/(1 + z) is only 5.4% greater at
e power spectrum on a cut sky (retaining the b > 10 de-
z =0.027thantoday,whileδe isslightlyless(by1.3%in grees region) can then be determined from |a|quadratic
thelinearregime). Equation(12)cannotbeuseddirectly
estimator, as was done in Sec. III.
because the dark matter distribution in the simulation
is represented by individual particles; we have therefore
smoothed with a Gaussian of 1σ width 0.1h−1 Mpc in
eachdimension. ThePoissonnoiseatthesmoothingscale B. Power spectrum method
k =10h/Mpcis4%ofthelinearmatterpowerspectrum.
Physically, the free electrons cannot trace the dark OursecondapproachtocomputingCBB istoestimate
ℓ
matter down to arbitrarily small scales, and one would thepowerspectrumoftheelectrondensityperturbations,
expect smoothing on scales shorter than the Jeans and then convert this via Eq. (6) into a power spectrum
7
for the polarization. This approach is purely statistical by only 5% for ℓ=1 and 2% for ℓ=2.
and does not require any particular realization of the TogofromEq.(19)totheB-modepowerspectrum,we
electron distribution. Its principal disadvantage is that notethatfromEq.(6),themultipolesofthepolarization
the sky τ(nˆ) may lookvery differentfrom differentparts from the local universe are
of the universe, and thus the local value of Cττ may
deviate substantially from the global average coℓmputed Bloc,ℓm ≡ YℓBm†(nˆ)P(nˆ)d2nˆ
here. Z
recTtihoenpnˆeritsurgbivaetnionbytoEtqh.e(T12h)o.mWsoenboepgtiincawlidtehptthheincadsie- = −√106 Imℓℓm′′′m′′T2m′τℓ′′m′′, (21)
whereδ isaFourierwave,δ (r,nˆ)=eirk·nˆ−ǫr,whereǫis m′Xℓ′′m′′
e e
where we have used the mode coupling integral
aninfinitesimalparameterthatforcesconvergenceofthe
integralatinfinity. Wethenhaveδτ(nˆ)=n¯ σ /(ǫ ikµ)
where µ=kˆ nˆ is the angle between the waeveTvecto−rand Imℓℓm′′′m′′ = YℓBm†(nˆ)Y2Em′(nˆ)Yℓ′′m′′(nˆ)d2nˆ. (22)
· Z
the line of sight. Taking the spherical harmonic trans-
Expressingthetensorsphericalharmonicsintermsofthe
form, we find
spin-weighted spherical harmonics,
2π 1 1
δτℓm = dφ dµYℓm(nˆ)τ(nˆ) YℓEm Q = YℓBm U = 2(Yℓ2m+Yℓ−m2) and
Z0 Z−1
= iδm0n¯eσT 4π(2ℓ+1) 1 Pℓ(µ)dµ, (15) (cid:2)YℓEm(cid:3)U = −(cid:2) YℓB(cid:3)m Q = 21i(Yℓ2m−Yℓ−m2), (23)
2k µ+iǫ
p Z−1 gives(cid:2) (cid:3) (cid:2) (cid:3)
where P is a Legendre polynomial and we have taken k
to be aloℓng the z-axis. If we define the coefficients Imℓℓm′′′m′′ = 21i (Yℓ−m2∗Y22m′ −Yℓ2m∗Y2−m2′)Yℓ′′m′′d2nˆ
Z
1 Pℓ(µ)dµ 5(2ℓ+1)(2ℓ′′+1)
aℓ = , (16) = i( 1)m
Z−1 µ+iǫ − − r 4π
ℓ ℓ′′ 2 ℓ ℓ′′ 2
then using the recursion relation [35] (24)
× 2 0 2 m m′′ m′
(cid:18) − (cid:19)(cid:18)− (cid:19)
an1dℓPnℓo(µti)ng=(t2hℓa−t 1f)oµrPℓℓ−1(>µ)−0(ℓor−th1o)gPoℓn−a2l(iµty), imp(l1i7es) f(ℓofo′r′rℓℓ′′′ℓ′=−>ℓℓ=2±t01heoarn3d±j2Is,ymℓℓmIm′m′ℓbℓ′mm′o′′l′′ms=′v′a=n0i0sohtv)ha.enrHiwsehirseeeswbbyye sphyaamvriemtyu,estferodyr
P (µ)dµ = 0, we find a = (ℓ 1)a /ℓ. It is
−1 ℓ ℓ − − ℓ−2 t|he−thr|ee spherical harmonic integral [16, 36, 37]. Now
then easy to show by induction from the initial values
R the power spectrum of the Thomson-scattered radiation
a = iπ and a =2 that
0 1
− is obtained by taking the mean square value of Eq. (21):
−2
aℓ =2−ℓiℓ−1πΓ(ℓ+1) Γ 2ℓ +1 . (18) CℓBB(loc) = 530 |Imℓℓm′′′m′′|2h|T2m′τℓ′′m′′|2i
(cid:20) (cid:18) (cid:19)(cid:21) m′ℓ′′m′′
X
toTaoragnendeormalifizeeldfr,owme tnheeedcatsoeinocfoahesrienngtlelyFionutergierratmeotdhee = 530C2TT |Imℓℓm′′′m′′|2Cℓτ′τ′, (25)
power spectrum Cττ over all Fourier modes: m′Xℓ′′m′′
ℓ
where we may drop cross-terms between different values
Cℓττ = ℓm=−2ℓℓ|+δτℓ1m(kˆ)|2∆2δe(k)dkk oreflmat′e,dℓ′a′,nadnwdemh′′asviencaesstuhmeeddifftehraenttτminultthipeolloecsaalruenuinvceorsre-
Z P is independent of the CMB quadrupole. (The value of
π3[Γ(ℓ+1)]2(n¯ σ )2 dk
= e T k−2∆2 (k) ; (19) the sum cannot depend on m because of rotationalsym-
4ℓ Γ[(ℓ/2)+1] 4 δe k metry.) We may do the summation over m′ and m′′ by
{ } Z
applying the 3j symbol orthogonality relations, yielding
inthelastlinewehavesubstitutedtheresultofEq.(18).
(for ℓ′′ =ℓ 1)
(The summation over m is rotationally invariant and so ±
izt-ddioreecstnioont.)maTttheirstihsastliwgehthlyavsemaaslsluermethdaknttohebeLiinmbtheer |Imℓℓm′′′m′′|2 = 5(2ℓ4′′π+1) 2ℓ ℓ0′′ 22 2. (26)
approximation m′m′′ (cid:18) − (cid:19)
X
Using the tabulated values of this specific form of the 3j
2π2
Cττ = (n¯ σ )2 r−2 ∆2 (k) dr symbol [36] gives the final result
ℓ,Limber e T k3 δe |k=(ℓ+1/2)/r
Z 3CTT[(ℓ+2)Cττ +(ℓ 1)Cττ ]
= 2πℓ2(+n¯e1σ/T2)2 k−2∆2δe(k)dkk (20) CℓBB(loc)= 2 80π(ℓ2−ℓ1+1) − ℓ+1 . (27)
Z
8
C. Results statistically using its covariance matrix C(2):
q
T/S
∆(T/S) = h i
We can evaluate Eq. (27) given any power spectrum
F
∆2 (k), which here we will assume to be the nonlinear T/S,T/S
δe 1
matter power spectrum. We have computed the spec- = (F )−1 x(1)†C−1C C−1x(1)
T/S,T/S 2 0 T/S 0
trum using the Eisenstein & Hu [38] transfer function
1 h
and the Peacock & Dodds [39] nonlinear mapping. The + Tr(C−1C C−1C(2)) . (28)
resultsfromthiscalculationareshowninthethicklinein 2 0 T/S 0
i
Fig.4(a). Forcomparison,wehavealsoshown(thinline)
This equation yields ∆(T/S) = 1.0 10−7 for the tem-
tahtekrmeisnu=lts(oℓb+t1a/in2e)d/rbmyaxc,uwtthinegreorffmtahxe=in8te0ghr−a1liMnpEcqi.s(t1h9e) plate with optical depth τ = 0.17,×and 3.6×10−7 for
τ =0.07. This is negligible, since even for the optimistic
radius to which we integrate in the constrained realiza-
(0.25µKarcmin noise, 2 arcmin beam) experiment, the
tion method. Within the context of the Limber approx-
Fisher matrix uncertainties in T/S are 6.2 10−7 and
imation, this measures the amount of this power that is 1.7 10−6 for τ =0.17 and 0.07 respectively×.
recovered by the constrained realization method. The ×
difference between the thick and thin curves in Fig. 4(a)
represents the contribution to CBB coming from struc-
ℓ
tures farther away than 80h−1 Mpc; this is 50% of the VI. DISCUSSION
power at ℓ = 10, with nearby structures dominating at
lower ℓ and more distant structures at higher ℓ. We have considered the contribution to the CMB B-
mode polarization from Thomson scattering in the local
Alsoshownisthepowerspectrumfromtheconstrained
universe, which is a potential contaminant to the gravi-
realization method (points in Fig. 4a). We can see from
tational wave signal. Our findings are that:
the figure that the power spectrum CBB obtained from
ℓ
this method is greater than the predicted result out to
1. The Thomson scattering signal from the Galactic
a distance of 80h−1 Mpc (thin line). This indicates that
disk and from nearby extragalactic structures is
the nearby universe is more inhomogeneous than aver-
well below the weak lensing B-modes even after
age, which is not entirely surprising since we live in an
the lensing signal is “cleaned” out using a lensing
overdense region of the universe and it is these regions
reconstruction from a very optimistic experiment.
that contribute most of the small-scale power.
This signal is believed to be negligible.
Since the Mathis et al. [34] simulations provide an
actual optical depth map, it is possible to construct an 2. The Galactic disk also Thomson-scatters the fore-
actual realization for the combined Thomson scattering ground radiation emitted by the Milky Way. At
signal from the Milky Way’s disk and from nearby ex- the low multipoles, this radiation can exceed the
tragalactic structures. We have computed the power cleaned lensing signal; in the K band it is roughly
spectrum CBB of this realization with the quadratic equal to the uncleaned lensing signal at ℓ = 2–
ℓ
estimator, and added to it the “missing” power from 3. However this signal is strongly frequency-
r >80h−1 Mpc; the result is shown in Fig. 4(b). This is dependent, and inprinciple canbe removedby the
ourfinalresultforthefrequency-independent,Thomson- same techniques used to clean other Galactic fore-
induced B-mode power spectrum from the Galactic disk grounds.
and nearby extragalactic structures (but excluding the
corona). Thecontributionisroughly(1–2) 10−4µKper 3. ThomsonscatteringwithinthehotGalacticcorona
× makes an unknown contribution to the B-modes,
logarithmic intervalin ℓ, which is well below the cleaned
depending on the flucutations in the optical depth
lensing signal even for an optimistic experiment.
through the corona. The contribution can be
We can be more quantitative about the effect on the roughly estimated based on Eq. (27):
estimation of the tensor signal T/S by using the ten-
sor signal CT/S ≡ dC/d(T/S) as a template in the ℓ(ℓ+1)CBB 3C2TT∆2 2.4µK2∆2; (29)
quadratic estimator instead of the band powers. In 2π ℓ ∼ 80π τ ∼ τ
this case we can get the spurious contribution to T/S
fromtheequation∆(T/S)=q /F ,whereq this can exceed the cleaned lensing signal if the
T/S T/S,T/S T/S
and F are computed using Eq. (11) with C . largeangularscalefluctuationsintheopticaldepth
T/S,T/S T/S
The map x to be inserted into the equation for q is through the corona are ∆ 2 10−4. Since a
T/S τ
≥ ×
the sum of the known Galactic disk and extragalactic mean optical depth through the corona of up to
(r < 80h−1Mpc) signals and the unknown extragalac- τ¯ O(10−4) appears to be reasonable, and there
tic signal from r > 80h−1Mpc, which we denote x(1) are∼only very limited constraints on the angular
and x(2) respectively. We can put into the estimator the distribution of free electrons in the corona, fluctu-
actual value of x(1), whereas we must account for x(2) ations of ∆ 2 10−4 cannot yet be ruled out.
τ
≥ ×
9
(a) B-modes from nearby extragalactic structure (b) Total (Galactic + extragalactic) B-modes
Power spectrum Local scattering
2µK) 10-6 Power spectrum (to M80a thh-is1M ept ca)l 2µK) 10-6 TenCsloerasn (eTd/ Sle=n1s0in-5g)
π ( π (
2 2
BBC /L BBC /L
+1) 10-7 +1) 10-7
L L
L( L(
m, m,
u u
pectr 10-8 pectr 10-8
er s er s
w w
o o
P P
10-9 10-9
2 3 4 5 10 15 20 30 2 3 4 5 10 15 20 30
Multipole, L Multipole, L
FIG. 4: The contribution from nearby extragalactic structures to the B-modes. Panel (a) shows the power spectrum for
the constrained realization method of Sec. VA (points), the power spectrum method of Sec. VB (thick line), and the power
spectrum method with the integration cut off at rmax = 80h−1 Mpc (thin line). Panel (b) shows the total B-mode from
local scattering (points), constructed by taking the optical depth map from the Galaxy and nearby structures, computing the
resulting CℓBB using the quadratic estimator, and then adding the estimated power from r > 80h−1 Mpc. For reference, the
cleaned-lensing and tensor curvesfrom Fig. 2 are shown.
Of these, the scattering from the Galactic corona is of periments, however, if a frequency-independent B-mode
the most interest. In principle, detection of this signal signalwere observedclose to the detectability threshold,
wouldprovideevidence thatthe coronacontainssomeof then the interpretation as a gravitational wave source
the missing baryons; non-detection of the B-mode sig- would require that Thomson scattering in the local uni-
nal would provide an upper limit on the anisotropy of verse be robustly ruled out as the source. This would
the corona. However the practical difficulties should not require significant improvement in our understanding of
be understated: reaching the “cleaned lensing” limit of the Galactic corona.
CBB = 5.8 10−8µK2 requires a detector sensitivity of
ℓ ×
0.25µKarcmin,two ordersof magnitude better thanthe
upcomingPlancksatellite[40]. Itisalsopossiblethatthe
Acknowledgments
frequency structure of the polarized foregrounds is suffi-
ciently complicated that extraction of small large-angle
B-mode signals becomes impractical. Finally it is pos- We thank Joseph Taylor and Bob Benjamin for use-
sible that tensor perturbations, or some other source of ful conversations about the Galactic electron density,
frequency-independent B-modes such as cosmic strings, and Nikhil Padmanabhan for his assistance with the
are presentandswampthe Galactic Thomsonscattering E/B decomposition software. C.H. is supported by
signal. NASA NGT5-50383. This work was supported in part
From the perspective of CMB gravitational wave by NSF grants AST-0204514, AST-0071019 and NASA
searches,theGalacticThomsonscatteringsignalappears grantNAG5-13292(forA.L.). Someoftheresultsinthis
to be negligible for all except the most futuristic exper- paper have been derived using the HEALPix [33] pack-
iments. Part of the reason for this is the low value of age. We acknowledge the use of the Legacy Archive for
the local quadrupole CTT, which is only 20% of the Microwave Background Data Analysis (LAMBDA) [41].
2 ∼
ΛCDMexpectedvalue;ifΛCDMisindeedcorrect,weare Support for LAMBDA is provided by the NASA Office
in this respect simply lucky. For the very futuristic ex- of Space Science.
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