The new Planck polarization data at large angular scales

Transcripción

The new Planck polarization data at
large angular scales
Anna Mangilli
On behalf of the Planck Collaboration
!
Institut d’Astrophysique Spatiale
!
Paris Sud University
Instituto de Física Teórica (IFT), Universidad Autónoma de Madrid (UAM)
OUTLINE
The CMB polarization at large angular scales!
!
!
The challenge for Planck: systematics and statistics!
!
!
The new Planck HFI results!
!
The Planck Coll. A&A 2016: “Planck constraints on the reionization history” (arxiv:1605.03507)
!The Planck Coll. A&A 2016: “Improved large angular scale polarization data”(arxiv:1605.02985)
Anna Mangilli (IAS) - IFT, Universidad Autónoma de Madrid (UAM) - 20th June 2016
The CMB polarization
Polarization
Planck
CMB polarization signal: orders of magnitude weaker than temperature
E-modes
•
•
Electric type polarization field.
!
Generated by scalar density
perturbations.
B-modes
•
•
•
Magnetic type polarization field.
!
Can be generated only by
primordial tensor modes i.e.
primordial gravitational waves
!
Contribution from lensing
Generation of the CMB polarization
Planck Collaboration: Planck constraints on re
erage ionized fraction of the gas in the Universe rapidly increased until hydrogen became fully ionized. Empirical, anaDECOUPLING
lytic, and numerical models of the reionization process have
highlighted many pieces of the essential physics that led to the
birth to the ionized intergalactic medium (IGM) at late times
(Couchman & Rees 1986; Miralda-Escude & Ostriker 1990;
Meiksin & Madau 1993; Aghanim et al. 1996; Gruzinov & Hu
REIONIZATION
1998; Madau et al. 1999; Gnedin 2000; Barkana
& Loeb 2001;
Ciardi et al. 2003; Furlanetto et al. 2004; Pritchard et al. 2010;
Pandolfi et al. 2011; Mitra et al. 2011; Iliev et al. 2014). Such
studies provide predictions on the various reionization observables, including those associated with the CMB.
The most common physical quantity used to characterize
reionization is the Thomson scattering optical depth defined as
Z t0
Planck Collaboration: Reionisation
history
⌧(z) =
ne T cdt0 ,
(1)
fraction rea
rameterizat
tical depth
reionizatio
onset of th
constructio
full compl
mainly det
z. This is
logical pap
to z = 1.7
reionizatio
A reds
flexible des
process (e.
2015). A
by the co
t(z)
Thomson scattering
of star-for
diffuse gas in the Universe is mostly ionized up to a redshift Thomson scattering optical
depth
optical
depth:0
where ne is the number density of free electrons
time t , T probes suc
6 (Fan et al. 2006).
Z ⌘at
z_reio
0
is star
the Thomson
t(z) (Faisst et a
The cosmological evolution of galaxies, of the
formationscattering cross-section, t0 ⌧is =the timeantoday,d⌘,
e TequaRobertson
is the
time at of
redshift z, and we can use the Friedmann
and of the integrated luminosity of AGNs as
a function
0
tion
to
convert
dt
to
dz.
The
reionization
history
is conveniently choices of
shift is constrained by surveys of individual galaxies in the ulexpressed
terms where
of the nionized
(z) mialatora exp
number xdensity
ofe (z)/n
free Helectrons
co
e (z) ⌘ n
e is the fraction
iolet (HST), visible light (HST and large ground
basedinteleTheseatwo
where
n
(z)
is
the
hydrogen
number
density.
time
⌘,
is
the
Thomson
scattering
cross-section,
is t
H
T
pes), far-infrared and sub-millimeter wavelengths (space obadopt
here
In
this
study,
we
define
the
redshift
of
reionization,
z
⌘
factor
and
⌘
is
the
conformal
time
today.
The
reionisat
re
0
vatories ISO, SPITZER, HERSCHEL), and millimeter (with
Enhancement
of the E&B
which
reio
z
,
as
the
redshift
at
which
x
=
0.5
⇥
f
.
Here
the
normalizatory is conveniently
expressed in terms of the
ionised
% formation
e
und based millimeter telescopes SPT, ACT). 50
The
modes
at large
angular
we have
tion, f = 1 + fHe = x1e (z)
+ nHe
/n
, takes
into
account
electrons
in-scales:
H(z)/n
=
n
(z)
where
n
(z)
is
the
Hydrogen
numb
e
H
H
structures in the universe predicts that small masses detach
jected into the IGM by the first ionization
of helium (correspondREIONIZATION
BUMP
m the general expansion to collapse and virialize and if they sity.
25 eV),Autónoma
which is
assumed
to- 20th
happen
roughly
the beginning
same
the(UAM)
following,
defineatthe
and the
Mangillithe
(IAS)lowest
-ing
IFT, to
Universidad
deIn
Madrid
Junewe
2016
ch a high enough temperature Anna
to excite
electronic
The CMB E & B angular power spectra
CMB anisotropies:
l<20 (low-l)
Large scale reionization bump
CℓEE
τ
r
CℓBB
r = 0.1
lensing
τ, r
r = 0.03
r / Einflation
Scientific goals
Reionization history:
CℓEE
at large angular scales to constrain τ
Inflation:
CℓBB
at large and intermediate scales to constrain r
OUTLINE
!
!
!
!
!
The Planck satellite
➡ 9 frequency bands (7 polarized: 30GHz-353GHz)
➡ Two instruments: !
LFI: 30GHz, 44GHz, 70GHz
HFI: 100GHz, 143GHz, 217GHz
353GHz, 545GHz, 857GHz
Channels for CMB
characterization
Foregrounds
characterization
30GHz
44GHz
70GHz
100GHz
143GHz
217GHz
353GHz
545GHz
857GHz
Large scale polarization issues
• Planck detectors are sensitive to one polarization direction
Polarization reconstruction: detector combinations
!
• Mismatch between detectors will create spurious polarization signal
Major systematics in polarization at large angular scales: !
!
Intensity to Polarization leakage
LFI: negligible residuals with respect to noise, LFI 70GHz released
HFI has higher sensitivity, lower noise: residuals systematics
HFI 100GHz, 143GHz, 217GHz NOT used for the 2015 low-l analysis
NEW HFI results 2016
The challenge I: control of systematics
Measuring the large scale polarization is difficult!
Major HFI systematic residuals:
•
inter-calibration leakage
•
ADC non-linearity residuals
•
correlated 1/f noise residuals
•
foreground residuals
All these effects are important at low-l
E-Modes: noise&systematics
Planck Collaboration: Large angular scale data, reionization
Planck-HFI 2015
Signal
Planck-HFI NEW
Signal
Fig. 16. E-mode polarization pseudo
spectra,
plotted
as C` to emphasise
low
multipoles,
computed
the plotted
maps distributed
in the
Fig. 16. de
E-mode
polarization
pseudo
spectra,
as C` to empha
Anna Mangilli
(IAS)
- IFT, Universidad
Autónoma
Madrid
(UAM)
- 20th
June
2016 from
The
ogy
ethodology
challenge II: low-l data analysis
Statistical method(s) optimized to CMB analysis @ large angular scales
Polarized CMB from 70 GHz
Gaussian
in
the m=[T,Q,U]
maps,
with
CMB
Multivariate
GaussianPlanck
likelihood
in the
m=[T,Q,U]
maps,
with CMBin
So farlikelihood
(WMAP,
2013,
2015):
Gaussian
likelihood
noise
matrix M :matrix M :
signal covariance
plus noise covariance
map space
1. Low ell CMB polarization in Planck
2014 comes from 70 GHz.
2. Out of eight surveys, we exclude
Polarized CMB from 70
fromGHz
the dataset survey 2 and 4
ow-l Commander temperature map
because the exhibit unusual B mode
excess, presumably connected with
013, the low-l temperature likelihood is based on
sidelobe contamination.
M= CMB signal+noise covariance matrix
T,Q,U
maps are
of foreground
emission
and residual
are cleaned
of cleaned
foreground
emission and
residual
3. Templates (30 and 353) are built
U
1. Low ell CMB polarization in Planck
Q
systematics:
:
from full mission data.
eground-cleaned Commander map
like 2013, the 2014 map also incorporates the 9WMAP data and the 408 MHz Haslam
2014map
comes from
ore frequencies
better fg model
more clean sky
70 GHz.
4. Working resolution is nside = 16,
Out of eight surveys, we exclude
a. In2. T,
Commander
multiband
CMB
down
sampled
from
highsolution
resolution
T, Commander
multiband
CMB
solution
from the
dataset survey 2 and
4
noise weighting. No
because the exhibit unusual B through
mode
smoothing
is applied in polarization
In Q,U
polarized
CMB
isby
provided
70 GHz, after
excess,
presumably
connected
with
Q,U b.
polarized
CMB
is provided
Planck by
70 Planck
GHz, after
5. The analysis mask retains 47% of the Stokes Q
sidelobe contamination.
template
fitting for built
polarized
synchrotron
dust, based
sky.
3. Templates
(30 and 353) aresynchrotron
mplate fitting
for polarized
and dust, and
based
from full mission data.
on
Planck
30
and
353
GHz, and their polarization leakage
Planck 304.Problem:
and
353
GHz,
and
Working
resolution
is nside
= 16, their polarization leakage
noise covariance matrix reconstruction
down sampled from high resolution
corrections.
Preliminary
rections.
ee Wehus’ talk tomorrow for more details
Stokes Q
ysis chain:
erform component separation at 1 resolution
efine narrow 2–based processing mask to
emove obvious residuals
ll mask with a constrained Gaussian realization
Stokes U
mooth to 440’ FWHM, and repixelize at Nside=16
efine proper
side=256
2–based
confidence mask at
his year fsky = 0.93, which is up from 0.87 in 2013
owngrade mask, and apply to Nside=16 map
accuracy
through noise weighting. No
smoothing is applied in polarization
ange of different 2 thresholds considered; no systematic
ases or trends found in power spectrum until fsky 0.97
•
•
5. The analysis mask retains 47% of the
sky.
Stokes U
Can compromise parameter reconstruction in particular for the high
sensitivity of HFI channels!
Preliminary of noise bias/residual systematics
Difficult handling
Cross-spectra likelihood at large scales
[Mangilli, Plaszczynski, Tristram (MNRAS 483 2015)]
!
Use cross-spectra likelihood at large scales!
!
Noise bias removed. Exploit cross dataset informations!
Better handling of residual systematics/foregrounds!
Two solutions to solve for the non-Gaussianity of the estimator
distributions at low multipoles
•
Analytic approximation of the estimators: works for single-field and small mask
•
Modified Hamimeche&Lewis (2008) likelihood for cross-spectra (oHL)
Full temperature and polarization analysis
In eq.14,
f id
f id
@
fìd
B
C`TBBBB@ B `
`
C
` C
CCC
CÈB
A
10 + o BB
C`BB
`
-15
A
Cross-spectra likelihood at large scales
C1/2
f id
TB
C
is the square root of the C` matrix: `
CÈB
C`BB
used
Figure 6. The o↵
tensor
cross-spectra
and
-16
so [Mangilli,
that:0
Plaszczynski,1Tristram (MNRAS 483 2015)]
ol
∝
ol
ol
10
for a given fiducial model and the function (g[D(P)]) refers to the
(mode
E
T BC
BBBC T T C TLewis
2008):
A⇥B
A⇥B10-16
C
0
C
0transformation:
1
C
[X
(C
)]
!
[OX
]
=
[X
(O(C
))]
.
(18)
`
`
`
CCC likelihood
`
g `
g
`
X
`cross-spectra
T
TE g
T`B C
BBBC TB
B
• Modified Hamimeche&Lewis
p
C
0
10
15
(2008)
for
(oHL)
C
C
B
th
T
1 5
`
` C
C
BBB `B
C
2lnL
(C` ln(x)
|Ĉ` ) = 1)),
[Xg ]` [M f(16)
]``0 [Xg ]`0 .
B
TE
EECCC= sign(x
EB C
-14 (13)
l
g(x)
1)
(2(x
B
C
10
C
=
(15)
B
C
C
C
B
C
`
B
C
T
E
EE
EB
The
(oHL hereafter)
reads:
BBB new
CCC
`C` o↵set
C` = BBBC
(15)
0
C`` CCCH&L` likelihood
``
`
1/2
1/2
BBBapplied
CCC
consi
B
C
X
to
the
eigenvalues
of
the
matrix
P
=
C
Ĉ
C
,
where
A
data
@ T@B T BEB
A
EB
BB
mod1
mod
A⇥B
BB
A⇥B
T
Figure
6.
The
o↵se
1 0
Figure
6.
The
o↵set
functions
o
of
the o
C
C
C` C `C2lnL
C
`
The
[M
]
is
the
inverse
of
the
C
-covariance
matrix
that
allows
0
0
(C
|
Ĉ
)
=
[OX
]
[M
]
[OX
]
,
(19)
`
`
to
the
``
`
`
`
`
g
``
g
`
Cmod and Ĉdata are, respectively,
the matrices
` off the sampled C ` and
` f
cross-spectra and forcross-spectra
the six di↵erent combin
and
f
0
to
quantify
the
`
`
and
the
correlations
of
the
T,
E,
B
fields.
The
``
the
data.
given fiducial model and the function (g[D(P)]) refers to the
ven• fiducial
model and
the
function
toat the
vector
[Xg(g[D(P)])
]for
the H&Lrefers
transformed
C` angular
vector defined as: in ge
The
problem
is
that
and
large
` iscross-spectra
“Gaussianization”
formation:
The variable transformation g(x) is now modified for the cross-15
⇣ positive
⌘
scalesp
the P matrix is no longer guaranteed to be
definite. In
rmation:
10
1/2
1/2
T
tions
spectra
to
regularize
the
likelihood
around
zero
so
(16)
[X
]
=
vecp
C
U(g[D(P)])U
C
.
(14)
-14 that Eq.
g(x) = sign(x
1)
(2(x
ln(x)
1)),
(16)
g
`
f id
10
fact, as shown
of the cross- f id
p in Eq. (2) that describes the distribution
funct
now
reads: estimators
1/2
1/2can
-14
spectra
Ĉ
,
the
Ĉ
be
negative.
In
order
to
solve
g(x)
=
sign(x
1)
(2(x
ln(x)
1)),
(16)
`
data
ed to the eigenvalues of the matrix P = CmodInĈeq.14,
,1/2
where
10
data Cmod
C
is
the
square
root
of
the
C
matrix:
`
cial m
f id
for this issue, we propose a modification
of the H&L likelihood that
and Ĉdata are, respectively, the matrices of the
sampled
C` and 1/2
1/2
g(x)
!
sign(x)g(|x|).
(20)
Planc
consists
in
adding
an
e↵ective
o↵set
o
to
the
cross-spectra
on
the
to the eigenvalues of the matrix P = Cmod Ĉdata` Cmod , where
ata.
diagonal of the C` matrices
the positive definiteXYin order to ensure
early
0
1
-16
The
o↵set
function
o
can
be
derived
from
simulations.
We
TheĈproblem
is
that
for
cross-spectra
and
at
large
angular
nd
are,
respectively,
the
matrices
of
the
sampled
C
and
10
TE
T BC
` matrix. In particular,
data
` re-define
BBBC T Twe
C
ness
of
the
cross-spectra
P
each
C
C
C
8 guaranteed
Mangilli,
Plaszczynski,
Tristram
` -15
` C
lines
BBB ` ensuring
0
s
the
matrix
is
no
longer
to
be
positive
definite.
In
• P
C
estimate
the
o↵sets
from
the
MC
distributions
that
the
P
10
C
“Offset”
terms:
N
a.
eff (eq. 15) matrix
C` Covariance
matrix
as:
BBBcorrelations)
CCC
(l-l and T-E-B
T
E
EE
EB
C` =for
(15)
the
b
C
as shown in Eq. (2)
that describes
the distribution
of thedefinite
crossBBBC`moreCthan
C
`
` C
matrix
reconstructed
is
positive
99%
of
our
0
1
C
BB@ angular
CCA
he
problemĈìs, thethat
for
cross-spectra
and
at
large
that:
⌧ and the scalar
T
T
T
T
T
E
T
B
B
C
ra estimators
Ĉ
can
be
negative.
In
order
to
solve
B
C
of
tho↵
T
B
EB
BB
data
C
+
o
C
C
BBB `the o↵sets
CCCthe
simulations. Given⇣ that
are
to
C
distri`
` needed
`shift
C
C
C
Figure
7.
The
`
⌘⇣
⌘
` XY f id In
`
`
ing accordingly
t
-15
BBBXY tolikelihood
C
XY
f id that
he
P matrix
is noa longer
guaranteed
be
positive
definite.
XY A⇥B
XY
is issue,
we propose
modification
of
the
H&L
C
A⇥B[M A⇥B
10
EE 0 ) sim
EB
the
v
]``0 ` =field
h) (C
i MC , CCC (21)
2⌧ (so
cross-spectra
C` for
!f each
O(C
= BBB`T,) sim
(17)
`+negative
C`T EBC`to avoid
C`(C
oÈE C`0 Ceigenvalues
butions
E,
on
the
A
=
A
e
at
`
CCC
TT
s
`
B
sts
in
adding
an
e↵ective
o↵set
o
to
the
cross-spectra
on
the
`
B
for
a
given
fiducial
model
and
the
function
(g[D(P)])
refers
to
the
shown in Eq.P(2)
that
describes
the
of
the
crossB@ distribution
C
HFI⇥Planck-143
the⌧ fia
straints
of the
T
B
EB
BB
BB Athe C distriXY
XY
A⇥B
matrix,
the
o↵set
functions
depend
on
the
shape
of
`
C
C
C
+
o
where
C
⌘
(C
)
,
and
X,
Y
=
{T,
E,
B}.
onal of the C` matrices in order `to ensure
definite-`
`
`
`
transformation:
usedWe
in choose
the sime
` the positive
-16
estimators
Ĉ
,
the
Ĉ
can
be
negative.
In
order
to
solve
the (m
li
` P matrix.
data
bution
at each
`.it In
depend
on10report
the noise
Since
willparticular,
be we
useful
inthe
theo↵sets
following,
we also
the levelstensor modes
of the cross-spectra
In particular,
re-define
each
p
0
5
15
soequations
that:
the 10
set of tion
Planck
g(x)
=
sign(x
1)
(2(x
ln(x)
1)),
(16)
of
the
modified
oHL
likelihood
in
the
case
of
the
single
issue,
we
propose
a
modification
of
the
H&L
likelihood
that
(model 2) andl bla
atrix (eq. 15) as:of the maps involved in the cross-spectra and on the mask used.
each ` independe
Full temperature
andwe
polarization
A⇥BIn particular,
A⇥B analysis
field
approximation.
are
interested
in
applying
the
[Xof
)]` the
!
[OX
(O(C
))]
. more(18)
0 Ine↵ective
1g ]` = [Xgat
1/2
1/2 to de
g (C
àre
`
`
s in adding an
o↵set
o
to
cross-spectra
on
the
fact,
the
tails
the
C
distributions
each
`
negative
ent
methods,
sett
`
`
applied
to
eigenvalues
of the- matrix
PEE=A⇥B
Cmod Ĉdata Cmod , where
EE
T E(IAS)
B Autónoma
BBBC T T + method
CCC cross-spectra
Mangilli
- IFT, Universidad
de Madrid (UAM)C
20th June
2016 )
polarization
EE-only
⌘
(C
oT T Anna to
Cthe
C Tthe
A⇥B
`
`
Cross-spectra oHL: τ estimation
0.4
0.2
0.0
0.00
Large-scale
CMB
cross-spectra
likelih
0.02
0.04
0.06
0.08
0.10
0.12
τ
τ posterior from realistic MC simulations, different noise levels, l=[2,20]
1.0
0.8
1.0
WMAPxPlanck-70
WMAPxPlanck-100
WMAPxPlanck-143
Planck-70xPlanck-100
0.8
0.6
Unbiasedness
0.6
fsky=0.8
fsky=0.5 Cross-spectra
0.4
0.4
0.2
0.0
0.00
0.2
0.02
0.04
0.06
τ
0.08
0.10
Optimality
WMAPxPlanck-70
WMAPxPlanck-100
WMAPxPlanck-143
100x143
0.6
0.4
0.0
0.00
( f sky = 0.8)
WMAP⇥Planck-70
0.0108
WMAP⇥Planck-100
0.0075
WMAP⇥Planck-143
0.0079
⌧
(
Planck-70⇥Planck-100
0.0069
Large-scale
CMB cross-spectra
like
0.02
0.04
0.06
0.08
τ
0.10
0.0065
0.12
estimation at better than 1010% s
0.2
0.0
0.00
0.12
⌧
0.0049for ⌧-r and ⌧
Table
3. Results of the joint constraints
Figure 11. Validationwith
of the
The plots
show
theoHL
full multi-fields
temperaturelikelihood.
and polarization
oHL
like
that the oHL likelihood
computed
combining
the simulations
T, E and B fields
acmodel
used in
the
is theand
⇤CDM
Bestfiducial
constraints
expected
counting for both multipole
results
with ⌧ fand
0.078correlations
and ln(1010gives
(A s ) funbiased
For r
id =fields
id ) = 3.09.
from
HFI
100GHzx143GHz
on the estimation ofTable
the
reionization
parameter
⌧. The
r f idoptical
= Results
0.1.depthoftothe
3.
joint constraints
for ⌧-r
and top
⌧-ln(
panel shows the ⌧ posterior
six di↵erent
when
20% of
with the for
fullthe
temperature
and cross-spectra
polarization oHL
likelihood.
the sky is masked (cial
f sky model
= 0.8),
while
thesimulations
bottom panel
⌧shows
r the results
used
in the
is the
⇤CDM
Planck
10
for a bigger mask with
= 0.5.
The Adashed
line and
refers
value
⌧ = f sky
0.078,
ln(10
r to
= the
0.1.input
The results
s ) = 3.09
⌧
r
⌧
best
f
it
best
f
it
Comparison
pixel-based
⌧ f id = 0.078 used
inPlanck-100⇥Planck-143
the simulations. with the
cross-spectra
simulations with f
0.0772727
0.00492517
0.0971194
approach:
compatible
error
bars
10 )
⌧ ⌧ln(10
s A )
ln(10A10
1.0
0.8
Table 2. Results on the estimation of the ⌧ parameter w
perature and polarization oHL likelihood. The fiducial m
WMAPxPlanck-70
WMAPxPlanck-100
simulation is the Planck 2015 ⇤CDM best fit with ⌧ f id =
WMAPxPlanck-143
shows the comparison between the estimates obtained w
sky cuts: . the small mask with with f sky = 0.8 and a
f sky = 0.5.
0.02
0.04
0.06
τ
0.08
0.10
0.12
⌧best f it
(ln(10 A s ))best
⌧
⌧best f it
(ln(1010 fAits ))best(ln(
⌧
f it
0.0770
0.0050
3.0632
0.0
0.0775020
0.00501646
3.08478
⌧ r
⌧best f it
rbest f it
⌧
Figure11.
11.Validation
Comparison
posterior
distributions
of the
⌧ parameter
Figure
of of
thethe
oHL
multi-fields
likelihood.
The
plots showobfor current
and 0.0049
future CMB experiments.
We0.0
Annathe
Mangilli
(IAS) - IFT,and
Universidad
Autónoma
de Madrid
(UAM)
0.0777
0.0703
tained with
full temperature
polarization
oHL likelihood
(blue)
and - 20th June 2016
Cross-spectra oHL: τ-r estimation
l=[2,20], full temperature and polarization oHL likelihood!
MC simulations Planck 100x143 with correlated noise
0.10
1.0
0.8
0.09
0.8
0.6
0.08
0.6
τ
1.0
Mangilli et al. 2015
0.4
0.07
0.2
0.06
0.2
0.0
0.05
0.0
0.1
0.4
0.2
0.3
0.4
r
⌧-r) and (⌧-A s ). The plot shows
the joint constraints obtained with the full temperature and
OUTLINE
!
!
!
!
!
The new Planck-HFI results
•
Large scale Polarization l=[4-20] + lmin=2,3!
!
•
E-modes 100GHzx143GHz cross spectra (PCL, Xpol)!
Planck Collaboration: Planck constraints on reionization history
!
•
Sky fraction: 50% + validation 60%!
!
•
multipoles. The idea behind this approach is that the noise can be
considered as uncorrelated between maps and that systematics
will be considerably reduced in cross-correlation compared to
auto-correlation.
At low multipoles and for incomplete sky coverage, the
C` s are not Gaussian distributed and are correlated between multipoles. lollipop uses the approximation presented in Hamimeche & Lewis (2008), modified as described in
Mangilli et al. (2015) to apply to cross-power spectra. The idea
is to apply a change of variable C` ! X` so that the new variable X` is Gaussian. Similarly to Hamimeche & Lewis (2008),
we define
0
1
q
e` + O` CC q
BBB C
f
CCA C f + O` ,
X` = C` + O` gB@
(A.1)
`
C ` + O`
p
e` are the measured crosswhere g(x) = 2(x ln(x) 1), C
power spectra, C` are the power-spectra of the model to evaluate,
C`f is a fiducial model, and O` are the o↵sets needed in the case
of cross-spectra. For multi-dimensional CMB modes (i.e., T , E,
and B), C` is a 3 ⇥ 3 matrix of power-spectra:
0 TT TE TB 1
C
C CC
BBB C
C
B
ET
EE
B
C
C EB CCCC ,
C` = BB@ C
(A.2)
A
C BT C BE C BB `
Polarization foreground cleaning !
!
Fig. A.1. Galactic mask used for the lollipop likelihood, covering 50 % of the sky.
Planck frequencies corrected for polarization leakage:
!
•
-
30GHz for polarized synchrotron
-
353GHz for polarized dust
e` C 1/2 .
and the g function is applied to the eigenvalues of C` 1/2C
`
In the case of auto-spectra, the o↵sets are replaced by the
noise bias e↵ectively present in the measured power-spectra. For
cross-power spectra, the noise bias is null and here we use the
e↵ective o↵sets defined from the C` noise variance:
r
2
C` ⌘
O` .
(A.3)
2` + 1
Cross-spectra based likelihood analysis oHL (Mangilli
et al. MNRAS 2015)
The distribution of the new variable X can be approximated Fig. A.2. Distribution of the peak value of the posterior distrias Gaussian, with a covariance given by the covariance of the bution for optical depth from end-to-end simulations including
e` is then
noise, systematic e↵ects, Galactic dust signal, and CMB with the
C` s. The likelihood function of the C` given the data C
fiducial
value June
of ⌧ = 2016
0.06.
X
Anna Mangilli (IAS) - IFT, Universidad
Autónoma
de Madrid (UAM)
- 20th
T
1
e
0
Planck HFI 100GHzx143GHz E-modes at low-l:
!
Planck Collabo
first τ constraint from CMB polarization data alone
z = 0.5), for the various data combinations are:
Planck, Collaboration: Planck constraintslolli
on reion
⌧ = 0.053+0.014
0.016
⌧ = 0.058+0.012
0.012 ,
14
lollipop+Plan
Départemen
Quai E. Ans
15
⌧
,
lollipop+PlanckTT+len
Departamen
38206 La La
16
⌧
,
lollipop+PlanckTT+
Departamen
s/n, Oviedo,
17
We can see an improvement of the posterior
width
Department
Nijmegen,
P
temperature anisotropy data to the lollipop
lik
18
Department
comes from the fact that the temperature anisotrop
Columbia, 6
other ⇤CDM parameters, in particular theCanada
normal
19
initial power spectrum As , and its spectralDepartment
index, n
College of
ing also helps to reduce the degeneracy with
As , L
California,
20
Department
rid of the tension with the phenomenological
lensi
London
WC
AL when using PlanckTT only (see Planck
Colla
21
Department
2016), even if the impact on the error barsBrighton
is smal
BN
22
Department
the posteriors in Fig. 6 with the constraints
from Pla
Helsinki, He
= 0.058+0.011
0.012
= 0.054+0.012
0.013
(see figure 45 in Planck Collaboration XI 2016) s
asured crossel to evaluate,
ed in the case
des (i.e., T , E,
(A.2)
e` C 1/2 .
C` 1/2C
`
placed by the
er-spectra. For
re we use the
e:
(A.3)
approximated
ariance of the
e` is then
ata C
(A.4)
ed via Monte
-field approxbased only on
We use a conwith a Galactic
mplitude meaodized using a
seudo-C` estio polarisation)
and 143 GHz
rst two multicontamination
boration XLVI
14
Département de Physique Théorique, Université de Genève,
Quai E. Ansermet,1211 Genève 4, Switzerland
15
Departamento de Astrofı́sica, Universidad de La Laguna (ULL)
38206 La Laguna, Tenerife, Spain
16
Departamento de Fı́sica, Universidad de Oviedo, Avda. Calvo So
s/n, Oviedo, Spain
17
Department of Astrophysics/IMAPP, Radboud Univer
Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
18
Department of Physics & Astronomy, University of Bri
Columbia, 6224 Agricultural Road, Vancouver, British Colum
Canada
19
Department of Physics and Astronomy, Dana and David Dorn
College of Letter, Arts and Sciences, University of South
California, Los Angeles, CA 90089, U.S.A.
20
Department of Physics and Astronomy, University College Lond
London WC1E 6BT, U.K.
21
Department of Physics and Astronomy, University of Sus
Brighton BN1 9QH, U.K.
22
Department of Physics, Gustaf Hällströmin katu 2a, Universit
Helsinki, Helsinki, Finland
Fig. A.2. Distribution of the peak value of the posterior distri23
Department of Physics, Princeton University, Princeton, New Jer
bution for optical depth from end-to-end simulations including
U.S.A.
noise, systematic e↵ects, Galactic dust signal, and CMB with the
24
Department of Physics, University of California, Berke
fiducial value of ⌧ = 0.06.
California, U.S.A.
25
Department of Physics, University of California, One Shi
Avenue, Davis, California, U.S.A.
To validate the choice of multipole and the stability of the re26
Department of Physics, University of California, Santa Barb
sult on ⌧, we performed several consistency checks on the Planck
California, U.S.A.
27
data. Among them, we varied the minimum multipole used (from
Department of Physics, University of Illinois at Urbana-Champa
` = 2 to ` = 4) and allowed for larger sky coverage (increasing
1110 West Green Street, Urbana, Illinois, U.S.A.
28
to 60 % of the sky). The results are summarized in Fig. A.3.
Dipartimento di Fisica e Astronomia G. Galilei, Università d
Studi di Padova, via Marzolo 8, 35131 Padova, Italy
29
Dipartimento di Fisica e Astronomia, Alma Mater Studior
Appendix B: Impact on ⇤CDM parameters
Università degli Studi di Bologna, Viale Berti Pichat 6/2, I-401
Bologna, Italy
In addition to the restricted parameter set shown in Fig. 6,
30
Dipartimento di Fisica e Scienze della Terra, Università di Ferr
we describe here the impact of the lollipop likelihood on
Via Saragat 1, 44122 Ferrara, Italy
31
⇤CDM parameters in general. Figure B.1 compares results from
Dipartimento di Fisica, Università La Sapienza, P. le A. Mor
lollipop+PlanckTT with the lowP+PlanckTT 2015. The new
Roma, Italy
32
low-`
polarization
are sufficiently
that they
Dipartimento di Fisica, Università degli Studi di Milano,
Fig. A.3.
Posterior results
distributions
for opticalpowerful
depth, showing
the
Celoria, 16, Milano, Italy
break
between
ns choices
and ⌧. The
contours
⌧ and
e↵ectthe
of degeneracy
changing two
of the
made
in ourforanalysis.
33
Anna likelihood
Mangilli (IAS)
- IFT, Universidad
Autónoma
(UAM) - 20th di
June
2016Università di Roma Tor Vergata, Via d
Fisica,
As , where the lollipop
dominates
the constraint,
arede MadridDipartimento
End-to-end simulations
Sky fraction
(2014) and Planck Collaboration XIII (2016); however, some
tension still remains.
+0.014
⌧
=
0.053
0.016 , with
Combining with VHL data gives compatible results,
consistent error bars. The slight shift toward lower+0.012
⌧ value (by
⌧ = 0.058
0.012 ,alone
) is related
to the fact that the PlanckTT
likelihood
Combination0.3
of low-l
HFI with:
pushes towards higher ⌧ values (see Planck Collaboration
+0.011 XIII
⌧ = 0.058 0.012
,
1. +Planck 2016),
TT/lensing
(2015)
while the
addition of New
VHL data helps to some
extent in
+0.012
reducing
the tension on ⌧experiments
between high-`(ACT
and low-`
polarization.
2. +Very High-l
ground-based
& SPT)
cosmology corresp
Planck low-l polarization + other data
4.2. Kinetic Sunya
lolli
The Thomson sca
lollipop+Pla
trons induces seco
reionization proce
⌧ = 0.054 0.013 ,
thelollipop+
e↵ect of photo
velocity, which is
We can see an improvement
theItpost
kSZ of
e↵ect.
is co
neous”
temperature anisotropy data
to kSZ
the e↵ect
lol
(e.g., Ostriker & V
comes from the fact that the
temperatur
neous)
reionizatio
other ⇤CDM parameters,during
in particular
the process
bubbles
initial power spectrum As ,ionized
and its
specta
nents can be desc
ing also helps to reduce the
degenerac
computed
analytic
rid of the
tension
phenomenolo
Collaborati
Planck
2015 with thePlanck
homogeneous
AL when using PlanckTTononly
(see Pls
In the followin
2016), even if the impact given
on the
by error b
the posteriors in Fig. 6 with the constrai
(see figure 45 in Planck Collaboration
where D` = `(` +
deed, the polarization likelihood
kSZ” standisforsuffi
“h
breaks the degeneracy between
and
spectively.nsFor
the
template
power sp
⇤CDM parameters is small,
typically
Fig. 5. Posterior
distribution
forAutónoma
⌧ fromdethe
various
combinations
Anna Mangilli (IAS)
- IFT, Universidad
Madrid
(UAM) - 20th
June 2016
with a simulation
mum level allowed by the luminosity density constraints at redshifts zagreement
< 9, then the currently
observed
galaxy population at
Better
with
astrophysical
MUV < 17 seems to be sufficient to comply with all the observational constraints without the need for high-redshift (z = 10–
15) galaxies.
WMAP9, Planck 2015
data
The scientific results that we present today are a produc
the Planck Collaboration, including individuals from m
than 100 scientific institutes in Europe, the USA and Cana
optical depth & redshift
WMAP
Planck NEW
Bouwens et al. (2015)
Robertson et al. (2015)
Ishigaki et al. (2015)
•
integrated optical depth for the symmetric model (tanh, δz = 0.5).
•
models from Bouwens et al. (2015), Robertson et al. (2015), Ishigaki
et al. (2015), using high redshift galaxy UV and IR flux and/or direct
measurements.
Planc
projec
Europea
Agenc
instru
provide
scie
Consort
by ESA
stat
partic
lead co
France a
w
contri
from
(USA
tele
refle
provid
collab
between
a sci
Consor
and fu
Den
Fig. 17. Reioniza
eterization comp
piled by Bouwens
of ionized fractio
limits. The dark a
95 % allowed inte
Although the
Fig. 16. Evolution of the integrated
optical depth for the tanh above, understand
preliminary
M. Tristram
Japan, Dec.
2015
B-mode from Space, Kavli IPMU
in
the Universe i
functional Anna
form
(with
z
=
0.5,
blue
shaded
area).
The
Mangilli (IAS) - IFT, Universidad Autónoma de Madrid (UAM) - 20th June 2016
redshift
andusing
duration
of
thevanishing
EoR,
finding
are plotted in Fig. 11. In such a model, the
end of
reionization
C
be
modelled
a tanh
(Lewis
2008):
z <function
10.2
(95
%ionized
CL,
,Ontimes
sults
fractionat3 low
xprior)
early
strongly
depends
on theunform
constraint
redshift.
the òt
e at
strongly depends on the constraint at low redshift. On the other
hand,
theredshifts.
constraints on
z only
slightly depend
the
lo
can
hand, the constraints on z only slightly depend on the lowat
low
When
calculating
theone↵ect
"
!#
z
<
4.6
(95
%
CL).
(13)
redshift
prior.
This
results
shows
that
the
Universe
is
ionized
vely
z <f 6.8+0.9
(95y % ytoCL,
prior
zend width
> 6)to. the(i.etr
redshift prior. This results
shows that the Universe is ionized at
+1.0
re
necessary
a
non-zero
regive
less
than
the
10
%
level
above
z
=
9.4
±
1.2.
z
=
8.5
(uniform
prior)
,
(14)
z
=
8.0
(uniform
prior)
,
less than the 10 % level above z = 9.4 ± 1.2.
re
xe (z) re=
1 + tanh
,
(2)
1.1
1.1
aints
be modelled
20
2
y using a tanh function (LewisPla
+0.9
5.2. Redshift-asymmetric parameterization
" .
5.2. Redshift-asymmetric
parameterization
tio
anck
z
=
8.5
(prior zend >f 6)
zre = 8.8+0.9
(prior
z
>
6)
.
(15)
re
end
y
yre
0.9
0.9
We
now
explore
more
complex
reionization
histories
us
3
3/2
1/2
Fig. 10.
Posterior
distributions
(in
blue)
of
z
and
z
for
a
bee
reionization
histories
using
x
(z)
=
1
+
tanh
Fig. 10. Posterior distributions (in blue) of z and z for a We now explore more complex
e
where
y
=
(1
+
z)
and
y
=
(1
+
z)
z.
The
key
parameters
of x (z) described
VHL
redshift-symmetric
2
y
of x (z)parameterization
described in using the CMB likelihoods2 the redshift-asymmetric parameterization
redshift-symmetric parameterization using the CMB likelihoods the redshift-asymmetric parameterization
tail
Sect.
3.
In
the
same
manner
as
in
Sect.
5.1,
also
examine
in
polarization
and
temperature
(lollipop+PlanckTT).
The
are
also thus
examinezre
the, which measures the redshift at which the ionized
in polarization and temperature (lollipop+PlanckTT). The Sect. 3. In the same manner as in Sect. 5.1,
additional constraint from the Gu
This
redshift
is
lower
than
the
values
derived
previously
green
contours
and
lines
show
the distribution after imposing e↵ect of imposing the 3/2
ude.
e↵ect
of
imposing
the
additional
constraint
from
the
Gunn3 pa- 1/2(H
green contours and lines show the distribution after imposing
fraction
reaches
half
its
maximum
and
a
width
z.
The
tanh
Peterson
e↵ect.
where
y
=
(1
+
z)
and
y
=
z.
the
additional
prior
z
>
6.
Peterson
e↵ect.
2 (1 + z)
the additional prior z > 6.
The
distributions
of
the
two
parameters,
z
and
z
mea,
10
ACT
SPT
pos- from WMAP-9 data, in combination
The distributions of the twowith
parameters,
z and z and
, are
are
thus
zre12.
, us
which
measures
theopredshift
rameterization
of
the EoR transition
allows
to compute
the
plotted
in
Fig.
With
the
redshift-asymmetric
parameteri
plotted in Fig. 12. With the redshift-asymmetric parameteriza4 aon
reaches
wiz
in
The10.3
posterior
is shown
in
Fig.a10
after fraction
tion, we
obtain
z redshift-symmetric
=half
10.4its maximum
(imposing the and
prior
tical
depth
of
(1)
for
one-stage
almost
The
posterior
distribution
of
z
is
shown
in
Fig.
10
after
tion, we obtain zz re = 10.4
(imposing
the distribution
prior
on
z of
), z Eq.
(Hinshaw
et
al.
2013),
namely
=
±
1.1.
It
is
of marginalizing
6
marginalizing
overionized
z, with
and without the additional constraint rameterization
which disfavours anyofmajor
to the ionized
fract
the contribution
EoR
transition
allows
u
over z, with and without the additional constraint which disfavours any major contribution
to the
fraction
pan
reionization
transition,
where
the
redshift
interval
between
the
>
z
>
6.
This
suggests
that
the
reionization
process
occurred
at
from
sources
that
could
form
as
early
as
z
15.
⇠
>
z > 6. This
suggests
that
the
reionization
process
occurred
at
from
sources
that
could
form
as
early
as
z
15.
also lower than the value zre = 11.1 ±⇠ onset
1.1 ofderived
in
tical depth of Eq. (1) for a one-stage almost
ten
the reionization processreionization
and its half
completion
transition,
where is
the(by
redshift i
Planck Collaboration XVI (2014), based on Planck
2013
data
construction)
between
half completion
9 equal to the interval
onset
of the reionization
processand
and itsple
ha
(H
full completion. In this parameterization,
depth between
is
construction) the
equaloptical
to the interval
and the WMAP-9 polarization likelihood.
red
full
completion.
In with
this parameterization,
11.
Posterior
distributions
on the
mainly determined
by zFig.
almost
degenerate
the width
re and
Although the uncertainty is now smaller,
this
new
determined
by2015
zre andcosmoalmost degen
sul
z. This is the model used
in themainly
Planck
2013
and
reionization,
i.e.,
z
and
z
,
using
the
end
beg
z. This is the model used in the Planck pro
201
with forthe
TT) reionization redshift value is entirely consistent
logical papers,
which
we have
fixed
z
=
0.5
(corresponding
rameterization
without
(blue)
and
with
logical papers, for which we have fixed z (g
=
to
z
=
1.73).
In
this
case,
we
usually
talk
about
“instantaneous”
Planck
2015
results
(Planck
Collaboration
XIII
2016)
for
DM
to z = 1.73). In this case, we usually talk ab
+1.8
reionization.
Planck Collaboration:
Reionization history
= 9.9 1.6 orreionization.
in combinah z PlanckTT+lowP alone, zre Planck
A redshift-asymmetric
parameterization
A redshift-asymmetric
parameterization
is a better,
more 4.
Collaboration:
history
+1.3 Reionization
In
addition
to
the
posteriors
for zre a
s (in tion with other data sets, width
zPlanck
8.8While
(specifically
for
flexible
description
of
numerical
simulation
re z=Collaboration:
flexible
description
of
numerical
simulations
of
the
reionization
= 0.5.
there
is
still
some
debate
on
whether
heof
redshift
(Lewis
et
al.
2006).
The
1.2 Planck constraints on reionization history
Planck
Collaboration:
Planck
constraints
on
reionization
history
symmetric
parameterization,
the
distri
process
(e.g.,
Ahn
et
al.
2012;
Park
et
al.
2
Planck
Collaboration:
Reionization
history
n:
Planck
constraints
on reionization
process
(e.g.,
al. 2006).
2012;
Park
et
al.for2013;
Douspisexotic
et al.uselium
could
be
inhomogeneous
and
extended
(and
ful
investigating
reioni
the
PlanckTT+lowP+lensing+BAO)
estimated
with
z fixed
toAhn
0.5.
Re
dth
z=
0.5. While
there ishistory
still some debate
onreionization
whether heof
redshift
(Lewis
etetal.
They
should
be
particularly
2015).
A
function with zthis (i.e.,
behaviour
beginning
ofof
reionization,
zin99
thusextended
have an early
start, ful
Worseck
etAal.function
2014),
we
have
checked
reionization
Cen
(2003).
Howeve
end redshift2015).
with
this
behaviour
is
also
suggested
These
values
are
within
0.4
the
results
for
the
m
reionization
could
be
inhomogeneous
and
(and
for
investigating
exotic
reionization
histories,
e.g.,
double
by
the
constraints
from
ionizing
backgro
d in The
constraint
from
lollipop+PlanckTT
when
fixing
z
to
0.5
width
z
=
0.5.
While
there
is
still
some
debate
on
whether
heof
redshift
(Lewis
et
al.
2006).
They
should
be
particularly
u
redshift
can
break
this
degeneracy.
This
is
illustrated
in
Fig.
10,
redshift can break this degeneracy. Thisthat
is illustrated
in helium
Fig. 10,reionization
varying the
redshift between
2.5
and
4.5
10)
polarization
anisotropies
are
are
plotted
in
Fig.
11.
In
such
a
model,
symmetric
model.
For
the
duration
of
the
EoR,
the
upper
limits
by
the
constraints
from
ionizing
background
measurements
pre
<
of star-forming
galaxies
and
+1.0
lium
reionization
could
be inhomogeneous
and
extended
(and
ful
for investigating
exotic
reionization
histories,
e.g.,
dou
us have an
early
start,
Worseck
etthe
al.
2014),ofchanges
we
have
checked
reionization
Cen(in
(2003).
However,
thevalue
CMB
large-scale
(`low-re
where
we
show
green)
the
results
of
the
same
analysis
withfrom
⇠
show
(in
green)
results
the
same
analysis
with
most
iswhere
zre =we
8.2
.
This
slightly
lower
value
(compared
to
the
one
the
total
optical
depth
by
less
than
1
%.
all
of
the
optical
depth,
z are
of
star-forming
galaxies
and
from
low-redshift
line-of-sight
probes
such
as
quasars,
Lyman-↵
emitte
1.2
kS
strongly
depends
on
the
constraint
at
lo
thus have
an early
start, Worseck
et al.
2014),
we
haveonchecked
reionization
Cenare
(2003).
However,
the
CMB
large-scale
(`w
an
additional
prior
z
>
6.
In
this
case,
we
find
z
<
1.3
at
95
%
at varying the
helium
reionization
redshift
between
2.5
and
4.5
10)
polarization
anisotropies
mainly
sensitive
to
the
overend
an additional
prior zletting
> 6.varying
Inthe
thisreionization
case, we find
z <width
1.3redshift
atand
95
%
tude
the
reionization
bump
inovI
simplest
most
widely-used
deend that
(Faisst
et al. of
2014;
Chornock
et al.
probes
asparameterizations
quasars,
Lyman-↵
emitters,
or
-ray
bursts
the heliumThe
reionization
between
2.5such
and
4.5
10) polarization
anisotropies
are
mainly
sensitive
to2014;
the
over
obtained
when
be
free)
is
explained
ele
CL,
which
corresponds
to
a
reionization
duration
(z
z
)
of
hand,
the
constraints
on
z
only
sligh
beg
end
beg
anges the total
optical
depth
by
less
than
1
%.
all
value
of
the
optical
depth,
which
determines
the
ampliCL, which correspondschanges
to a reionization
duration
zend
of1 %. transition between
Fig.
have
the
im
scribes
the(zEoR
as
a)step-like
essentially
beg
Robertson
et3).
al.We
2015;
Bouwens
et al.
2015
zvalue
<an10.2
(95
% CL,
unform
prior)
,estimated
(18)
the total optical
depth
by less
than
all Chornock
of the
optical
depth,
which
determines
the
amp
(Faisst
et
al.
2014;
et
al.
2014;
Ishigaki
et
al.
2015;
3
onal
by theand
shape
the degeneracy
surface.
Allowing
larger
duredshift
prior.
This
results
shows
that
th(
ferent
models
(tanh
andspectrum
power
tude
of
the
reionization
bump
in
the
EE
power
spectrum
(seelaw
vanishing
ionized
fraction
xefor
at early
times,
to a of
value
of
unity
The simplest
mostofwidely-used
parameterizations
dechoices
of
redshift-asymmetric
parameteri
tude
the
reionization
bump
in
the >
EE
power
The simplest
and
most widely-used
parameterizations
deRobertson
et
al.
2015;
Bouwens
et
al.
2015).
The
two
simplest
z
<
6.8
(95
%
CL,
prior
z
6)
.
(19)
z
<
4.6
(95
%
CL).
(13)
end
EE ofofredshift
EE
zscribes
< 4.6
% as
CL).
(13)
found
di↵erences
less
tha
exponential
functions
(D
atavalue
low
When
calculating
the e↵ect
on
anisotropies
itmial
istheorand
Fig.
3).less
We have
estimated
the
impact
on Cthe
for
the
two
d
the(95
EoR
step-like
transition
between
an
essentially
low theration
keeping
the
same
of
⌧
pushes
towards
higher
than
10
%
level
above
z
=
9.4
±
Fig.
3).
We
have
estimated
the
impact
on
C
for
two
difribes
EoR aswhen
a step-like
transition
between
anredshifts.
essentially
`
4.1
choices
of
redshift-asymmetric
parameterizations
are
polyno`
3
EoR: z
beg
and duration
re
re
e
e
end
end
re
respondables, including those associated with the CMB.
ing toThe
25 eV),
isnassume
d ltoquantit
happen
the same
most which
commo
physica
y roughly
used to at
charact
erize
time
as
hydrog
We define
the beginn
and the
reionization isen
thereioniz
Thomsation.
on scatteri
ng optical
depthing
defined
end of the EoR by the redshifts zbeg ⌘ z10 % and zend ⌘ z99 % as
at
Z t0
which xe = 0.1⇥ f and 0.99⇥ f , respectively.
0 The duration of the
2,
(1)
EoR is then defined as⌧(z)z =
= z10 %ne zT99cdt
% . Moreover, to ensure
t(z)
that the Universe is fully reioniz
ed at low redshift, we impose
where
the
conditi
ne is
onthe
thatnumbe
the EoR
r density
is comple
of free
ted electro
beforenstheatsecond
time t0 ,he-T
is thereioniz
lium
Thoms
on scatteri
ation
phase ng
(corres
cross-s
pondin
ection,
g tot054
is eV),
the time
noting
today,
thatt(z)
it
isiscommo
the time
nlyatassume
redshif
d tthat
z, and
quasars
we can
are necessa
use the ryFriedm
to produc
ann eequathe
tion photon
hard
to conver
s needed
t dt to dz.
to ionize
The reioniz
heliumation
. To history
be explici
is conven
t about iently
how
express
we
treat ed
theinlowest
terms redshif
of the tsionized
we assume
fraction
thatxethe
(z) full
⌘ nereioniz
(z)/nHa(z)
where
tion
of helium
nH (z) ishappen
the hydrog
s fairly
en numbe
sharplyr density
at zHe =. 3.5 (Becker et al.
2011),
In followi
this study,
ng a we
transiti
define
on the
of hyperb
redshifolic
t oftangen
reioniz
t ation,
shape zwith
re ⌘
z502% , as the redshift at which xe = 0.5 ⇥ f . Here the normalizais that tinelectro
practice
we
tion,The
f =reason
1 + this
fHe =is not
1 +defined
nHe /nHsymmet
, takes rically
into accoun
ns
inhave
tighter
ntsby
on the
the end
reioniza
on the
beginni
ng.
jected
intoconstrai
the IGM
firstofionizat
iontion
ofthan
helium
(corres
ponding to 25 eV), which is assumed to happen roughly at the same
time as hydrogen reionization. We define the beginning and the
end of the EoR by the redshifts zbeg ⌘ z10 % and zend ⌘ z99 % at
which xe = 0.1⇥ f and 0.99⇥ f , respectively. The duration of the
EoR is then defined as z = z10 % z99 % .2 Moreover, to ensure
that the Universe is fully reionized at low redshift, we impose
the condition that the EoR is completed before the second helium reionization phase (corresponding to 54 eV), noting that it
is commonly assumed that quasars are necessary to produce the
hard photons needed to ionize helium. To be explicit about how
we treat the lowest redshifts we assume that the full reionization of helium happens fairly sharply at zHe = 3.5 (Becker et al.
2011), following a transition of hyperbolic tangent shape with
+1.9
1.6
end
beg
re
beg
end
end
beg
The reason this is not defined symmetrically is that in practice we
have tighter constraints on the end of reionization than on the beginning.
end
end
beg
+1.9
1.6
beg
e
2
These
parameterizations
in ⌧fact
double
reionization
model,
Fig.
ferent
models
andatwo
power
law)
having
the⌧are
same
= ve
0
necessary
atimes,
non-zero
width
the(tanh
transition,
and it(tanh
can
ionizedto
fraction
xetoat
early
to a value
oftounity
ferent
models
and
power
law)
having
the
same
=
0.06
nishing ionized fraction3 xe atvanishing
early times,
a value
ofgive
unity
mial
or
exponential
functions
of
redshift
(Douspis
et
al.
2015).
EE
adopt here
a is
power
defined
two
param
and found di↵erences
of
than
4%
for `given
<by 10.
at low redshifts. When
calculating
the e↵ect
anisotropies
it is2008):
C` less
quitelaw
weak,
theEven
actu
be modelled
using
a tanhonfunction
(Lewis
Th
and
found
di↵erences
of
less
than
4
%
for
`
<
10.
Even
for
low redshifts. When calculating
the
e↵ect
on
anisotropies
it
is
These
two
parameterizations
are
in
fact
very
similar,
and
we
which
reionization
ends
(z
);
and
the
expo
a
double
reionization
model,
Fig.
4
shows
that
the
impact
end
necessary to give a non-zero width to the transition,
and
it
can
cannot
be
distinguished
relative
to
"
!#
tro
5.2.
Redshift-asymmetric
parameteriz
EE
we
have
af adopt
double
model,
Fig.
4(i.e.,
shows
that
impact
cessary to give a non-zero width
to theusing
transition,
and it(Lewis
can 2008):
quite weak,
given
theeven
actual
measured
value
ofon⌧, a
herereionization
by
two
parameters:
the
atexperim
be modelled
a tanh function
8 the
ya power
yre Claw
for
a redshift
full-sky
`9 isdefined
>
of
EE
x
(z)
=
1
+
tanh
,
(2)
>
f ✓not of
forspre
zm<
e
cannot
be
distinguished
relative
to the
cosmic
variance
>
which
reionization
ends
(z
);
and
the
exponent
↵.
Specifically
C
is
quite
weak,
given
the
actual
measured
value
⌧,
and
modelled using a tanhSymmetric
function (Lewis
2008):
"
!#
Planck
data
do
allow
for
<
end
model:
Asymmetric
model:
◆
2
y
`
↵
xemore
=and
Werelative
now
explore
complex
reion
Fig.
13.
Posterior
distributions
for
z(z)rein
using
the
at tr
f (in blue)
y of
yre zwe have
>
zz
z checked
(i.e.,
even
for
a full-sky
experiment).
We
also
>
Fig.
Posterior
distributions
and
z
for
a
>
tion
of
x
redshift
bins.
Princ
re
cannot
be
distinguished
to
the
cosmic
variance
spread
f
for
z>
e
x
(z)
=
1
+
tanh
,
(2)
:
Fig.
11.10.
Posterior
distributions
on
the
end
and
beginning
of
8
e
z
z
"
!#
Planck
data
do
not
allow
for
model-independent
reconstr
sca
2
y
the
redshift-asymmetric
parameterizatio
asymmetric
parameterization
without
(blue)
with
(g
>
1/2
been
proposed
as
anand
explicit
appr
>
redshift-symmetric
the
CMB
f x✓parameters
for
z
<
z
,
f i.e., zend andyparameterization
where
y = (1 using
+ z)3/2 and
ypa= 32even
(1likelihoods
+ z)for
z.a The
key
reionization,
zbegy, reusing
the redshift-symmetric
(i.e.,
full-sky
experiment).
We
also
checked
that
end
>
tion
of
in
redshift
bins.
Principal
component
analysis
h
<
◆↵thethe
e
dit
x
(z)
=
1
+
tanh
,
(2)
In
following,
we
fix
z
=
20,
the
red
tails
of
the
reionization
history
i
e
early
Sect.
3.
In
same
manner
as
in
Sec
x
(z)
=
(3)
prior
z
>
6.
>
are
thus
z
,
which
measures
the
redshift
at
which
the
ionized
z
z
rameterization
(blue)
prior
> 6.
3/2 (green)
1/2 zendPlanck
early
re3the
in polarization
and
temperature
been>
proposed
as an
explicit
approach
to tryreconstructo capture the
data end
doeThe
not
allow
for
model-independent
where
y = (1
+ z)with
and y =(lollipop+PlanckTT).
z. The
key parameters
2 without
yand
>
f
for
z
>
z
.
the&
the
first
emitting
sources
form,
and at which
:
2 (1 + z)
end
(Hu
&
Holder
2003;
Mortonson
z
z
early
end
fraction
reaches
half
its
maximum
and
a
width
z.
The
tanh
patails
of
the
reionization
history
in
a
small
set
of
paramet
e↵ect
of
imposing
the
additional
con
are
thus
z
,
which
measures
the
redshift
at
which
the
ionized
tionafter
of xeimposing
in redshift bins. Principal
component
analysis
green contours and relines show
the distribution
xe (z) tomethods
the
ionized
fraction
left
overhas
from
siz
are
generally
considered
rameterization
of
the
EoR
transition
allows
us
to
compute
the
op(Hu
&
Holder
2003;
Mortonson
&
Hu
2008).
Although
tht
fraction
reaches
half
its
maximum
and
a
width
z.
The
tanh
pa3
3/2
1/2
Peterson
e↵ect.
checked
that
our
results
are
not
sensitive
to
been
proposed
as
an
explicit
approach
to
try
to
capture
the
de(IAS)
- IFT,
Universidad
Autónoma
Madrid
(UAM)
20thzare
June
2016
4 considered
additional
6.
thedefollowing,
we -fix
=
20,
the
redshift
which they
here y = (1 + z)theand
y = (1prior
+Anna
z) zMangilli
key
parameters
tial
in fact
based
on
a description
of
endz.>The
early
methods
generally
to bearound
non-parametric,
tical
depth
of Eq. (1) for
aIn
one-stage
almost
redshift-symmetric
early
early
end
Improved and lower τ: impact on parameters
nck constraints on reionization history
2015
New
Lowl HFI
(4)
(5)
(6)
(7)
ding
This
o fix
f the
enstting
meter
XIII
aring
lone
2015
New
Lowl HFI
e show only ⌧the
results 0.016 ,
= 0.053
lollipop
; > 6.
(4)In this case, we find z <
an additional
prior zend
nds to approximatively+0.012
+1.0
Conclusions
CL,
which(uniform
corresponds
to(5)
a ,reionization duration
(zb
z
=
8.5
prior)
(14)
⌧
=
0.058
,
lollipop+PlanckTT
;
re
1.1
y looking at constraints 0.012
+0.9
ic models using
z
=
8.8
. (95 % CL).(15)
⌧ = Planck
0.058+0.011
,
lollipop+PlanckTT+lensing
; >
(6)
z <6)
4.6
re
0.9 (prior zend
0.012
we introduce the VHL+0.012
⌧
=
0.054
,
lollipop+PlanckTT+VHL
.
(7)
0.013
This
redshift
is
lower
than
the
values
derived
om the
• kSZ
Aamplitude.
significantly lower value for the reionization optical
depth:previously
from WMAP-9
data,
in combination
with ACT and SPT
ts that follow
fromsee
posWe
can
an
improvement
of
the
posterior
width
when
adding
- is consistent with a fully reionized Universe at z ∼ 6
al.the
2013),
namely
zre = This
10.3 ± 1.1. It is
pleted at temperature
a redshift of anisotropy
6 (Hinshaw
dataet to
lollipop
likelihood.
- 6.
is infrom
better
with
astrophysical
constraints
!
alsothat
lower
thanrecent
the value
zre =help
11.1
± 1.1 derived
in
rior zend >
comes
theagreement
fact
the temperature
anisotropies
to fix
- disfavors
Planck
Collaboration
(2014),
basedofon
Planck 2013
data
reionization
tailXVI
and
complicated
reionization
histories!
other
⇤CDM high-z
parameters,
in particular
the normalization
the
and the
WMAP-9
polarization
likelihood.
- makes
the
quest
of
B-modes
at
low-l
more
challenging!
initial
power
spectrum
A
,
and
its
spectral
index,
n
.
CMB lenss
s
on
Although
the uncertainty
is now
smaller, this new
ing also helps to reduce
the degeneracy
with As , while
getting
•
τ constraint:
tighter
constraints
cosmological
reionization
redshift
value lensing
is on
entirely
consistent parameters
with the
emperature
(PlanckTT)
ridImproved
of the tension
with the phenomenological
parameter
Planck
2015
results
(Planck
Collaboration
XIII 2016) for
σ8, Σm
s, n
s, using
ν
constraints
⇤CDM
ALAon
when
PlanckTT
only (see
Planck
Collaboration
XIII
+1.8
PlanckTT+lowP
alone,
z
=
9.9
or in combinare
dshift zre2016),
and width
z the impact on the error bars is small. Comparing
even
if
1.6
The Planck Coll. A&A 2016: “Planck constraints on the reionization history”
(arxiv:1605.03507)
+1.3
n. Figure
10posteriors
shows
(inin2016:
tion
other
data
sets,
zrePlanckTT
= 8.8alone
Fig.
6 with
the
constraints
from
aints
on
the
kSZ
amplitude
at `“Improved
= 3000
using
Thethe
Planck
Coll.
A&A
large
angular
scale
polarization
data”(arxiv:1605.02985)
1.2 (specifically for
arginalization
over the
PlanckTT+lowP+lensing+BAO)
estimated
fixed
0.5.
anckTT+VHL
likelihoods.
three Collaboration
cases corre(see figure
45 inThe
Planck
XI 2016) shows
that with
in- zFig.
6. to
Constraint
ent kSZAs
templates.
eters.
discussed
in The constraint
lollipop+PlanckTT
when
z
to
0.5
deed,
the polarization
likelihoodfrom
is sufficiently
powerful that
it fixing
ogy from PlanckT
+1.0
anisotropies
are the
almost
is zre =between
8.2 1.2 .nThis
value
(compared to the one
breaks
degeneracy
⌧. Thelower
impact
on other
s andslightly
likelihood from P
What’s
next:!
nction. We
thus recover
obtained
when
letting the
reionization
width
be free) is explained
⇤CDM
parameters
is
small,
typically
below
0.3
(as
shown
likelihood. The to
sis.
The
data
presented
here
provide
the
best
conmposingmore
an
byAppendix
thedata
shape
of
the
degeneracy
surface. are
Allowing
for
larger
du• additional
Further
Planck
improvement
at
map-making
level
on
going
explicitly
in
B).
The
largest
changes
for
the bottom panels
on the kSZ power and is a factor of 2 lower than
on
fraction
at
very
low
ration
when
keeping
the
same
value
of
⌧
pushes
towards
higher
•
⌧
and
A
,
where
the
lollipop
likelihood
dominates
the
conPrimordial
B-modes
at
low-ell:
s
ted in George et al. (2015). Our limit is certainly
parameter
shiftspretowards
smaller val- The
with the straint.
homogeneous
kSZ template,
8which
improved
S/N, lensing
is not a slightly
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not
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1 needed:
. This
in
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to help
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that the
onal kSZ power
patchy
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Universidad
with George et al. (2015),Anna
weMangilli
find(IAS)
the- IFT,
total
kSZ Autónoma de Madrid (UAM) - 20th June 2016
Thank you!
CITA – ICAT
UNIVERSITÀ DEGLI STUDI
DI MILANO
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The new Planck polarization data at large angular scales

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