A&A 607, A95 (2017)
DOI:
10.1051/0004-6361/201629504
c
ESO 2017
Astronomy
&
Astrophysics
Planck intermediate results
LI. Features in the cosmic microwave background temperature
power spectrum and shifts in cosmological parameters
Planck Collaboration: N. Aghanim
51
, Y. Akrami
53, 55
, M. Ashdown
62, 6
, J. Aumont
51
, C. Baccigalupi
75
, M. Ballardini
29, 43, 46
, A. J. Banday
88, 9
,
R. B. Barreiro
57
, N. Bartolo
28, 58
, S. Basak
81
, K. Benabed
52, 87
, M. Bersanelli
32, 44
, P. Bielewicz
72, 9, 75
, A. Bonaldi
60
, L. Bonavera
16
, J. R. Bond
8
,
J. Borrill
12, 85
, F. R. Bouchet
52, 83
, C. Burigana
43, 30, 46
, E. Calabrese
78
, J.-F. Cardoso
65, 1, 52
, A. Challinor
54, 62, 11
, H. C. Chiang
23, 7
,
L. P. L. Colombo
20, 59
, C. Combet
66
, B. P. Crill
59, 10
, A. Curto
57, 6, 62
, F. Cuttaia
43
, P. de Bernardis
31
, A. de Rosa
43
, G. de Zotti
41, 75
, J. Delabrouille
1
,
E. Di Valentino
52, 83
, C. Dickinson
60
, J. M. Diego
57
, O. Doré
59, 10
, A. Ducout
52, 50
, X. Dupac
35
, S. Dusini
58
, G. Efstathiou
62, 54
, F. Elsner
70
,
T. A. Enßlin
70
, H. K. Eriksen
55
, Y. Fantaye
2, 18
, F. Finelli
43, 46
, F. Forastieri
30, 47
, M. Frailis
42
, E. Franceschi
43
, A. Frolov
82
, S. Galeotta
42
,
S. Galli
61,
⋆
, K. Ganga
1
, R. T. Génova-Santos
56, 15
, M. Gerbino
86, 73, 31
, J. González-Nuevo
16, 57
, K. M. Górski
59, 90
, S. Gratton
62, 54
,
A. Gruppuso
43, 46
, J. E. Gudmundsson
86, 23
, D. Herranz
57
, E. Hivon
52, 87
, Z. Huang
79
, A. H. Jaffe
50
, W. C. Jones
23
, E. Keihänen
22
, R. Keskitalo
12
,
K. Kiiveri
22, 40
, J. Kim
70
, T. S. Kisner
68
, L. Knox
25
, N. Krachmalnicoff
75
, M. Kunz
14, 51, 2
, H. Kurki-Suonio
22, 40
, G. Lagache
5, 51
, J.-M. Lamarre
64
,
A. Lasenby
6, 62
, M. Lattanzi
30, 47
, C. R. Lawrence
59
, M. Le Jeune
1
, F. Levrier
64
, A. Lewis
21
, M. Liguori
28, 58
, P. B. Lilje
55
, M. Lilley
52, 83
,
V. Lindholm
22, 40
, M. López-Caniego
35
, P. M. Lubin
26
, Y.-Z. Ma
60, 77, 74
, J. F. Macías-Pérez
66
, G. Maggio
42
, D. Maino
32, 44
, N. Mandolesi
43, 30
,
A. Mangilli
51, 63
, M. Maris
42
, P. G. Martin
8
, E. Martínez-González
57
, S. Matarrese
28, 58, 37
, N. Mauri
46
, J. D. McEwen
71
, P. R. Meinhold
26
,
A. Mennella
32, 44
, M. Migliaccio
3, 48
, M. Millea
25, 84, 52, ⋆
, M.-A. Miville-Deschênes
51, 8
, D. Molinari
30, 43, 47
, A. Moneti
52
, L. Montier
88, 9
,
G. Morgante
43
, A. Moss
80
, A. Narimani
19
, P. Natoli
30, 3, 47
, C. A. Oxborrow
13
, L. Pagano
51
, D. Paoletti
43, 46
, B. Partridge
39
, G. Patanchon
1
,
L. Patrizii
46
, V. Pettorino
38
, F. Piacentini
31
, L. Polastri
30, 47
, G. Polenta
4
, J.-L. Puget
51
, J. P. Rachen
17
, B. Racine
55
, M. Reinecke
70
,
M. Remazeilles
60, 51, 1
, A. Renzi
75, 49
, G. Rocha
59, 10
, M. Rossetti
32, 44
, G. Roudier
1, 64, 59
, J. A. Rubiño-Martín
56, 15
, B. Ruiz-Granados
89
, L. Salvati
51
,
M. Sandri
43
, M. Savelainen
22, 40, 69
, D. Scott
19
, C. Sirignano
28, 58
, G. Sirri
46
, L. Stanco
58
, A.-S. Suur-Uski
22, 40
, J. A. Tauber
36
, D. Tavagnacco
42, 33
,
M. Tenti
45
, L. Toffolatti
16, 57, 43
, M. Tomasi
32, 44
, M. Tristram
63
, T. Trombetti
43, 30, 46
, J. Valiviita
22, 40
, F. Van Tent
67
, P. Vielva
57
, F. Villa
43
,
N. Vittorio
34
, B. D. Wandelt
52, 87, 27
, I. K. Wehus
59, 55
, M. White
24
, A. Zacchei
42
, and A. Zonca
76
(Affiliations can be found after the references)
Received 8 August 2016 / Accepted 10 September 2017
ABSTRACT
The six parameters of the standard ΛCDM model have best-fit values derived from the Planck temperature power spectrum that are shifted
somewhat from the best-fit values derived from WMAP data. These shifts are driven by features in the Planck temperature power spectrum at
angular scales that had never before been measured to cosmic-variance level precision. We have investigated these shifts to determine whether
they are within the range of expectation and to understand their origin in the data. Taking our parameter set to be the optical depth of the reionized
intergalactic medium τ, the baryon density ω
b
, the matter density ω
m
, the angular size of the sound horizon θ
∗
, the spectral index of the primordial
power spectrum, n
s
, and A
s
e
−2τ
(where A
s
is the amplitude of the primordial power spectrum), we have examined the change in best-fit values
between a WMAP-like large angular-scale data set (with multipole moment ℓ < 800 in the Planck temperature power spectrum) and an all angular-
scale data set (ℓ < 2500 Planck temperature power spectrum), each with a prior on τ of 0.07 ± 0.02. We find that the shifts, in units of the 1σ
expected dispersion for each parameter, are {∆τ, ∆A
s
e
−2τ
, ∆n
s
, ∆ω
m
, ∆ω
b
, ∆θ
∗
} = {−1.7, −2.2, 1.2, −2.0, 1.1, 0.9}, with a χ
2
value of 8.0. We find
that this χ
2
value is exceeded in 15% of our simulated data sets, and that a parameter deviates by more than 2.2σ in 9% of simulated data sets,
meaning that the shifts are not unusually large. Comparing ℓ < 800 instead to ℓ > 800, or splitting at a different multipole, yields similar results.
We examined the ℓ < 800 model residuals in the ℓ > 800 power spectrum data and find that the features there that drive these shifts are a set of
oscillations across a broad range of angular scales. Although they partly appear similar to the effects of enhanced gravitational lensing, the shifts
in ΛCDM parameters that arise in response to these features correspond to model spectrum changes that are predominantly due to non-lensing
effects; the only exception is τ, which, at fixed A
s
e
−2τ
, affects the ℓ > 800 temperature power spectrum solely through the associated change in
A
s
and the impact of that on the lensing potential power spectrum. We also ask, “what is it about the power spectrum at ℓ < 800 that leads to
somewhat different best-fit parameters than come from the full ℓ range?” We find that if we discard the data at ℓ < 30, where there is a roughly
2σ downward fluctuation in power relative to the model that best fits the full ℓ range, the ℓ < 800 best-fit parameters shift significantly towards the
ℓ < 2500 best-fit parameters. In contrast, including ℓ < 30, this previously noted “low-ℓ deficit” drives n
s
up and impacts parameters correlated with
n
s
, such as ω
m
and H
0
. As expected, the ℓ < 30 data have a much greater impact on the ℓ < 800 best fit than on the ℓ < 2500 best fit. So although the
shifts are not very significant, we find that they can be understood through the combined effects of an oscillatory-like set of high-ℓ residuals and
the deficit in low-ℓ power, excursions consistent with sample variance that happen to map onto changes in cosmological parameters. Finally, we
examine agreement between Planck TT data and two other CMB data sets, namely the Planck lensing reconstruction and the TT power spectrum
measured by the South Pole Telescope, again finding a lack of convincing evidence of any significant deviations in parameters, suggesting that
current CMB data sets give an internally consistent picture of the ΛCDM model.
Key words. cosmology: observations – cosmic background radiation – cosmological parameters – cosmology: theory
⋆
Corresponding authors: Silvia Galli, e-mail: gallis@iap.fr;
Marius Millea, e-mail: millea@iap.fr
Article published by EDP Sciences A95, page 1 of 27
A&A 607, A95 (2017)
1. Introduction
Probably the most important high-level result from the Planck
satellite
1
(Planck Collaboration I 2016) is the good agreement
of the statistical properties of the cosmic microwave background
anisotropies (CMB) with the predictions of the six-parameter
standard ΛCDM cosmological model (Planck Collaboration XV
2014; Planck Collaboration XVI 2014; Planck Collaboration XI
2016; Planck Collaboration XIII 2016). This agreement is quite
remarkable, given the very significant increase in precision of
the Planck measurements over those of prior experiments. The
continuing success of the ΛCDM model has deepened the moti-
vation for attempts to understand why the Universe is so well-
described as having emerged from Gaussian adiabatic initial
conditions with a particular mix of baryons, cold dark matter
(CDM), and a cosmological constant (Λ).
Since the main message from Planck, and indeed from the
Wilkinson Microwave Anisotropy Probe (WMAP;
Bennett et al.
2013) before it, has been the continued success of the six-
parameter ΛCDM model, attention naturally turns to precise
details of the values of the best-fit parameters of the model.
Many cosmologists have focused on the parameter shifts with
respect to the best-fit values preferred by pre-Planck data. Com-
pared to the WMAP data, for example, Planck data prefer a
somewhat slower expansion rate, higher dark matter density, and
higher matter power spectrum amplitude, as discussed in several
Planck Collaboration papers (
Planck Collaboration XV 2014;
Planck Collaboration XVI 2014; Planck Collaboration XI 2016;
Planck Collaboration XIII 2016), as well as in Addison et al.
(2016). These shifts in parameters have increased the degree of
tension between CMB-derived values and those determined from
some other astrophysical data sets, and have thereby motivated
discussion of extensions to the standard cosmological model
(e.g.
Verde et al. 2013; Marra et al. 2013; Efstathiou 2014;
Wyman et al. 2014; Beutler et al. 2014; MacCrann et al. 2015;
Seehars et al. 2016; Hildebrandt et al. 2016). However, none of
these extensions are strongly supported by the Planck data them-
selves (e.g. see discussion in Planck Collaboration XIII 2016).
Despite the interest that the shifts in best-fit parameters has
generated, there has not yet been an identification of the particu-
lar aspects of the Planck data, and their differences from WMAP
data, that give rise to the shifts. The main goal of this paper is
to identify the aspects of the data that lead to the shifts, and to
understand the physics that drives ΛCDM parameters to respond
to these differences in the way they do. We chose to pursue this
goal with analysis that is entirely internal to the Planck data.
In carrying out this Planck-based analysis, we still shed light
on the WMAP-to-Planck parameter shifts, because when we re-
strict ourselves to modes that WMAP measures at high signal-to-
noise ratio, the WMAP and Planck temperature maps agree well
(e.g.
Kovács et al. 2013; Planck Collaboration XXXI 2014). The
qualitatively new attribute of the Planck data that leads to the pa-
rameter shifts is the high-precision measurement of the temper-
ature power spectrum in the 600
<
∼
ℓ
<
∼
2000 range
2
. Restricting
1
Planck (
http://www.esa.int/Planck) is a project of the Euro-
pean Space Agency (ESA) with instruments provided by two scientific
consortia funded by ESA member states and led by Principal Investi-
gators from France and Italy, telescope reflectors provided through a
collaboration between ESA and a scientific consortium led and funded
by Denmark, and additional contributions from NASA (USA).
2
Although the South Pole Telescope and Atacama Cosmology Tele-
scope had already measured the CMB T T power spectrum over this
multipole range (e.g.
Story et al. 2013; Das et al. 2014), Planck’s dra-
matically increased sky coverage leads to a much more precise power
spectrum determination.
our analysis to be internal to Planck has the advantage of sim-
plicity, without altering the main conclusions.
We also investigated the consistency of the differences in
parameters inferred from different multipole ranges with ex-
pectations, given the ΛCDM model and our understanding of
the sources of error. The consistency of such parameter shifts
has been previously studied in
Planck Collaboration XI (2016),
Couchot et al. (2015), and Addison et al. (2016). In studying the
consistency of parameters inferred from ℓ < 1000 with those in-
ferred from ℓ > 1000
Addison et al. (2016) claim to find signifi-
cant evidence for internal inconsistencies in the Planck data. Our
analysis improves upon theirs in several ways, mainly through
our use of simulations to account for covariances between the
pair of data sets being compared, as well as the “look elsewhere
effect”, and the departure of the true distribution of the shift
statistics away from a χ
2
distribution.
Much has already been demonstrated about the robustness
of the Planck parameter results to data processing, data se-
lection, foreground removal, and instrument modelling choices
Planck Collaboration XI (2016). We will not revisit all of that
here. However, having identified the power spectrum features
that are causing the shifts in cosmological parameters, we show
that these features are all present in multiple individual fre-
quency channels, as one would expect from the previous studies.
The features in the data therefore appear to be cosmological in
origin.
The Planck polarization maps, and the T E and EE polar-
ization power spectra determinations they enable, are also new
aspects of the Planck data. These new data are in agreement with
the T T results and point to similar shifts away from the WMAP
parameters (
Planck Collaboration XIII 2016), although with less
statistical weight. In order to focus on the primary driver of the
parameter shifts, namely the temperature power spectrum, we
have ignored polarization data except for the constraint on the
value of the optical depth τ coming from polarization at the
largest angular scales, which in practice we folded in with a prior
on τ.
Our primary analysis is of the shift in best-fit cosmologi-
cal parameters as determined from: (1) a prior on the value of
τ (as a proxy for low-ℓ polarization data) and PlanckTT
3
data re-
stricted to ℓ < 800
4
; and (2) the same τ prior and the full ℓ-range
(ℓ < 2500) of PlanckTT data. Taking the former data set as a
proxy for WMAP, these are the parameter shifts that have been
of great interest to the community. There is of course a degree of
arbitrariness in the particular choice of ℓ = 800 for defining the
low-ℓ data set. One might argue for a lower ℓ, based on the fact
that the WMAP temperature maps reach a signal-to-noise ratio
of unity by ℓ ≃ 600, and thus above 600 the power spectrum er-
ror bars are at least twice as large as the Planck ones. However,
we explicitly selected ℓ = 800 for our primary analysis because
it splits the weight on ΛCDM parameters coming from Planck
3
In common with other Planck papers, we use PlanckTT to refer to the
full Planck temperature-only C
T T
ℓ
likelihood. We often omit the “TT”
when also specifying a multipole range, for example by Planck ℓ < 800
we mean PlanckTT ℓ < 800.
4
To avoid unnecessary detail, we write ℓ
max
of 800, 1000, and 2500,
even though the true ℓ
max
values are 796, 996, and 2509 (since this is
where the nearest data bins happen to fall). For brevity, the implied
ℓ
min
is always two unless otherwise stated, for example ℓ < 800 means
2 ≤ ℓ < 800.
A95, page 2 of
27
Planck Collaboration: Parameter shifts
so that half is from ℓ < 800 and half is from ℓ > 800
5
. Address-
ing the parameter shifts from ℓ < 800 versus ℓ > 800 is a related
and interesting issue, and while our main focus is on the com-
parison of the full-ℓ results to those from ℓ < 800, we computed
and showed the low-ℓ versus high-ℓ results as well. Additionally,
as described in Appendix A, we performed an exhaustive search
over many different choices for the multipole at which to split
the data.
In addition to the high-ℓ Planck temperature data, in-
ferences of the reionization optical depth obtained from
the low-ℓ Planck polarization data also have an impor-
tant impact on the determination of the other cosmolog-
ical parameters. The parameter shifts that have been dis-
cussed in the literature to date have generally assumed
a constraint on τ coming from Planck LFI polarization
data (
Planck Collaboration XI 2016; Planck Collaboration XIII
2016). During the writing of this paper, new and tighter
constraints on τ were released using improved Planck
HFI polarization data (Planck Collaboration Int. XLVI 2016;
Planck Collaboration Int. XLVII 2016). These are consistent
with the previous ones, shrinking the error by approximately a
factor of two and moving the best fit to slightly lower values of
τ. To make our work more easily comparable to previous discus-
sions, and because the impact of this updated constraint is not
very large, we have chosen to write the main body of this paper
assuming the old τ prior. This also allows us to more cleanly iso-
late and discuss separately the impact of the new prior, which we
do in a later section of this paper.
Our focus here is on the results from Planck, and so an
in-depth study comparing the Planck results with those from
other cosmological data sets is beyond our scope. Neverthe-
less, there do exist claims of internal inconsistencies in CMB
data (
Addison et al. 2016; Riess et al. 2016), with the parameter
shifts we discuss here playing an important role, since they serve
to drive the PlanckTT best fits away from those of the two other
CMB data sets, namely the Planck measurements of the φφ lens-
ing potential power spectrum (Planck Collaboration XVII 2014;
Planck Collaboration XV 2016) and the South Pole Telescope
(SPT) measurement of the TT damping tail (Story et al. 2013).
Thus, we also briefly examine whether there is any evidence of
discrepancies that are not just internal to the PlanckTT data, but
also when comparing with these other two probes.
The features we identify that are driving the changes in pa-
rameters are approximately oscillatory in nature, a part of them
with a frequency and phasing such that they could be caused by
a smoothing of the power spectrum, of the sort that is generated
by gravitational lensing. We thus investigate the role of lensing
in the parameter shifts. The impact of lensing in PlanckTT pa-
rameter estimates has previously been investigated via use of the
parameter “A
L
” that artificially scales the lensing power spec-
trum (as discussed on p. 28 of
Planck Collaboration XVI 2014;
and p. 24 of
Planck Collaboration XIII 2016). Here we introduce
a new method that more directly elucidates the impact of lensing
on cosmological parameter determination.
Given that we regard the ℓ < 2500 Planck data as provid-
ing a better determination of the cosmological parameters than
the ℓ < 800 Planck data, it is natural to turn our primary ques-
tion around and ask: what is it about the ℓ < 800 data that
makes the inferred parameter values differ from the full ℓ-range
parameters? Addressing this question, we find that the deficit
5
More precisely, the product of eigenvalues of the two Fisher informa-
tion matrices (see e.g. Schervish 1996, for a definition) – one for ℓ < 800
and the other for ℓ > 800 – is approximately equal at this multipole split.
in low-multipole power at ℓ
<
∼
30, the “low-ℓ deficit”
6
, plays a
significant role in driving the ℓ < 800 parameters away from the
results coming from the full ℓ-range.
The paper is organized as follows. Section
2 introduces the
shifts seen in parameters between using Planck ℓ < 800 data and
full-ℓ data. Section 3 describes the extent to which the observed
shifts are consistent with expectations; we make some simplify-
ing assumptions in our analysis and justify their use here. Sec-
tion 4 represents a pedagogical summary of the physical effects
underlying the various parameter shifts. We then turn to a more
detailed characterization of the parameter shifts and their origin.
The most elementary, unornamented description of the shifts is
presented in Sect.
5.1, followed by a discussion of the effects of
gravitational lensing in Sect. 5.2 and the role of the low-ℓ deficit
in Sect. 5.3. In Sect. 5.4 we consider whether there might be sys-
tematic effects significantly impacting the parameter shifts and
in Sect. 5.5 we add a discussion of the effect of changing the τ
prior. Finally, we comment on some differences with respect to
other CMB experiments in Sect. 6 and conclude in Sect. 7.
Throughout we work within the context of the six-parameter,
vacuum-dominated, cold dark matter (ΛCDM) model. This
model is based upon a spatially flat, expanding Universe whose
dynamics are governed by general relativity and dominated by
cold dark matter and a cosmological constant (Λ). We shall
assume that the primordial fluctuations have Gaussian statis-
tics, with a power-law power spectrum of adiabatic fluctuations.
Within that framework the usual set of cosmological parameters
used in CMB studies is: ω
b
≡ Ω
b
h
2
, the physical baryon density;
ω
c
≡ Ω
c
h
2
, the physical density of cold dark matter (or ω
m
for
baryons plus cold dark matter plus neutrinos); θ
∗
, the ratio of
sound horizon to angular diameter distance to the last-scattering
surface; A
s
, the amplitude of the (scalar) initial power spectrum;
n
s
, the power-law slope of those initial perturbations; and τ,
the optical depth to Thomson scattering through the reionized
intergalactic medium. Here the Hubble constant is expressed
as H
0
= 100 h km s
−1
Mpc
−1
. In more detail, we follow the pre-
cise definitions used in Planck Collaboration XVI (2014) and
Planck Collaboration XIII (2016).
Parameter constraints for our simulations and comparison
to data use the publicly available CosmoSlik package (Millea
2017), and the full simulation pipeline code will be released
publicly pending acceptance of this work. Other parameter con-
straints are determined using the Markov chain Monte Carlo
package cosmomc (
Lewis & Bridle 2002), with a convergence
diagnostic based on the Gelman and Rubin statistic performed on
four chains. Theoretical power spectra are calculated with CAMB
(
Lewis et al. 2000).
2. Parameters from low-ℓ versus full-ℓ
Planck data
Figure
1 compares the constraints on six parameters of the base-
ΛCDM model from the PlanckTT+τprior data for ℓ < 2500 with
those using only the data at ℓ < 800. We have imposed a specific
prior on the optical depth, τ = 0.07 ± 0.02, as a proxy for the
Planck LFI low-ℓ polarization data, in order to make it easier to
compare the constraints, and to restrict our investigation to the
T T power spectrum only. As mentioned before, we will discuss
the impact of the newer HFI polarization results in Sect. 5.5. The
6
This is the same feature that has sometimes previously been called
the “low-ℓ anomaly”. We choose to use the name “low-ℓ deficit”
throughout this work to avoid ambiguity with other large scale “anoma-
lies” and because it is more appropriate for a feature of only moderate
significance. See Sect.
5.3 for further discussion.
A95, page 3 of 27
A&A 607, A95 (2017)
0.12 0.13 0.14 0.15
ω
m
0.021 0.022 0.023 0.024
ω
b
1.035 1.040 1.045
100θ
∗
0.03 0.06 0.09 0.12
τ
0.93 0.96 0.99 1.02
n
s
1.74 1.80 1.86 1.92
10
9
A
s
e
−2τ
2.96 3.04 3.12 3.20
ln(10
10
A
s
)
64 68 72 76
H
0
0.70 0.75 0.80 0.85
σ
8
Fig. 1. Cosmological parameter constraints from PlanckTT+τprior for
the full multipole range (orange) and for ℓ < 800 (blue) – see the text for
the definitions of the parameters. We note that the constraints are gener-
ally in good agreement, with the full Planck data providing tighter lim-
its on the parameters; however, the best-fit values certainly do shift. It is
these shifts that we seek to explain in this paper. A prior τ = 0.07 ± 0.02
has been used here as a proxy for the effect of the low-ℓ polarization
data (with the impact of a different prior discussed later). As a compari-
son, we also show results for WMAP T T data combined with the same
prior on τ (grey).
constraints shown are one-dimensional marginal posterior distri-
butions of the cosmological parameters given the data, obtained
using the cosmomc code (
Lewis & Bridle 2002), as described in
Sect. 1, and applying exactly the same priors and assumptions for
the Planck likelihoods as detailed in Planck Collaboration XIII
(2016).
We see that the constraints from the full data set are tighter
than those from using only ℓ < 800, and that the peaks of the
distributions
7
are slightly shifted. It is these shifts that we seek
to explain in the later sections. Figure 1 also shows constraints
from the WMAP T T spectrum. As already mentioned, these
constraints are qualitatively very similar to those from Planck
ℓ < 800, although not exactly the same, since WMAP reaches
the cosmic variance limit closer to ℓ = 600. Nevertheless, as was
already shown by Kovács et al. (2013), Larson et al. (2015), the
CMB maps themselves agree very well, and thus the small differ-
ences in parameter inferences (the largest of which is a roughly
1σ difference in θ
∗
) are presumably due to small differences in
sky coverage and WMAP instrumental noise. We see that the
dominant source of parameter shifts between Planck and WMAP
is the new information contained in the ℓ > 800 modes, and that
7
We loosely refer here to the “peaks of the distributions”. In the next
sections, we will more carefully specify whether we quantify the shifts
in terms of difference in the best-fit values (i.e., the maximum of the
full-dimensional posterior distribution of the parameters) or in terms of
the marginalized means. Choosing one or the other should not signif-
icantly change our conclusions, since the posterior distributions of the
parameters are nearly Gaussian, and therefore these two quantities are
very close to each other.
by discussing parameter shifts internal to Planck we are also di-
rectly addressing the differences between WMAP and Planck.
Figure
1 shows the shifts for some additional derived pa-
rameters, as well as the basic six-parameter set. In particular,
one can choose to use the conventional cosmological param-
eter H
0
, rather than the CMB parameter θ
∗
, as part of a six-
parameter set. Of course neither choice is unique, and we could
have also focused on other derived quantities in addition to six
that span the space; for the amplitude, we have presented re-
sults for the usual choice A
s
, but added panels for the alterna-
tive choices A
s
e
−2τ
(which will be important later in this paper)
and σ
8
(the rms density variation in spheres of size 8 h
−1
Mpc
in linear theory at z = 0). The shifts shown in Fig.
1 are fairly
representative of the sorts of shifts that have already been dis-
cussed in previous papers (e.g. Planck Collaboration XVI 2014;
Planck Collaboration XI 2016; Addison et al. 2016), despite dif-
ferent choices of τ prior and ℓ ranges.
To simplify the analysis as much as possible, throughout
most of this paper we will choose our parametrization of the
six degrees of freedom in the ΛCDM model so that we reduce
the correlations between parameters, and also so that our choice
maps onto the physically meaningful effects that will be de-
scribed in Sect.
4. While a choice of six parameters satisfying
both criteria is not possible, we have settled on θ
∗
, ω
m
, ω
b
, n
s
,
A
s
e
−2τ
, and τ. Most of these choices are standard, but two are
not the same as those focused on in most CMB papers: we have
chosen ω
m
instead of ω
c
, because the former governs the size
of the horizon at the epoch of matter-radiation equality, which
controls both the potential-envelope effect and the amplitude of
gravitational lensing (see Sect.
4); and we have chosen to use
A
s
e
−2τ
in place of A
s
, because the former is much more pre-
cisely determined and much less correlated with τ. Physically,
this arises because at angular scales smaller than those that sub-
tend the horizon at the epoch of reionization (ℓ ≃ 10) the primary
impact of τ is to suppress power by e
−2τ
(again, see Sect.
4).
As a consequence of this last fact, the temperature power
spectrum places a much tighter constraint on the combination
A
s
e
−2τ
than it does on τ or A
s
. Due to the strong correlation be-
tween these two parameters, any extra information on one will
then also translate into a constraint on the other. For this rea-
son, a change in the prior we use on τ will be mirrored by a
change in A
s
, given a fixed A
s
e
−2τ
combination. Conversely, the
extra information one obtains on A
s
from the smoothing of the
small-scale power spectrum due to gravitational lensing will be
mirrored by a change in the recovered value of τ (and this will
be important, as we will show later). As a result, since we will
mainly focus on the shifts of A
s
e
−2τ
and τ, we will often inter-
pret changes in the value of τ as a proxy for changes in A
s
(at
fixed A
s
e
−2τ
), and thus for the level of lensing observed in the
data (see Sect. 5.2).
3. Comparison of parameter shifts
with expectations
In light of the shifts in parameters described in the previous sec-
tion, we would of course like to know whether they are large
enough to indicate a failure of the ΛCDM model or the presence
of systematic errors in the data, or if they can be explained sim-
ply as an expected statistical fluctuation arising from instrumen-
tal noise and sample variance. The aim of this section is to give
a precise determination based on simulations, in particular one
that avoids several approximations used by previous analyses.
One of the first attempts to quantify the shifts was per-
formed in Appendix A of
Planck Collaboration XVI (2014),
A95, page 4 of 27
Planck Collaboration: Parameter shifts
and was based on a set of Gaussian simulations. More re-
cent studies using the Planck 2015 data have generally com-
pared posteriors of disjoint sets of Planck multipole ranges
(e.g.
Planck Collaboration XI 2016; Addison et al. 2016). There,
the posterior distribution of the parameters shifts given the
data is P( ¯p
(1)
− ¯p
(2)
|d), with ¯p
α
being the vector of parameter-
marginalized means estimated from the multipole range α = 1, 2.
This posterior distribution is assumed to be a Gaussian with zero
mean and covariance Σ = C
(1)
+ C
(2)
, where C
(α)
are the param-
eter posterior covariances of the two data sets and both ¯p
α
and
C
(α)
are estimated from MCMC runs. Therefore, there it is as-
sumed that, if one excludes from the parameter vector the optical
depth τ for which prior information goes into both sets, the re-
maining five cosmological parameters are independent random
variables. Additionally, to quantify the overall shift in parame-
ters, a χ
2
statistic is computed,
χ
2
= ( ¯p
(1)
− ¯p
(2)
)Σ
−1
( ¯p
(1)
− ¯p
(2)
). (1)
The probability to exceed χ
2
is then calculated assuming that it
has a χ
2
distribution with degrees of freedom equal to the num-
ber of parameters (usually five since τ is ignored).
There are assumptions, both explicit and implicit, in previ-
ous analyses which we avoid with our procedure. We take into
account the covariance in the parameter errors from one data set
to the next, and do not assume that the parameter errors are nor-
mally distributed. Additionally our procedure allows us to in-
clude τ in the set of compared parameters. As we will see, our
more exact procedure shows that consistency is somewhat better
than would have appeared to be the case otherwise.
3.1. General outline of the procedure
We schematically outline here the steps of the procedure that we
apply, with more details being provided in the following section.
First, we choose to quantify the shifts between parameters
estimated from different multipole ranges as differences in best-
fit values ˜p, that is, the values that maximize their posterior dis-
tributions, rather than differences in the mean values ¯p of their
marginal distributions. We adopt this choice because best-fit val-
ues are much faster to compute (they are determined with a min-
imizer algorithm, while the means require full MCMC chains).
We justify this choice by the fact that the posterior distributions
of cosmological parameters in the ΛCDM model are very closely
Gaussian, so that their means and maxima are very similar. Fur-
thermore, we will consistently compare the shifts in best-fit pa-
rameters measured from the data with their probability distribu-
tion estimated from the simulations. Therefore we are confident
that this choice should not affect our final results.
Next, we wish to determine the probability distribution of
the parameter shifts given the data, that is, P( ˜p
(1)
− ˜p
(2)
|d). Since
when estimating ˜p
1,2
we use the same Gaussian prior on τ, ˜p
(1)
and ˜p
(2)
are correlated. Therefore, we use simulations to numeri-
cally build this distribution. The idea is to draw simulations from
the Planck likelihoods P(d| ˜p
fid
), where ˜p
fid
is a fiducial model.
For each of these simulations, we estimate the best-fit parame-
ters ˜p
1,2
i
for each of the multipole ranges considered. This allows
us to build the probability distribution of the shifts in parameters
given a fiducial model, P( ˜p
(1)
− ˜p
(2)
| ˜p
fid
).
The fiducial model we use is the best-fit (the maximum of
the posterior distribution) ΛCDM model for the full ℓ = 2–2500
PlanckTT data, with τ fixed to 0.07, and the Planck calibration
parameter, y
P
, fixed to one (see details, for example about treat-
ment of foregrounds, in the next section; y
P
is a map-level rescal-
ing of the data as defined in
Planck Collaboration XI (2016)).
More explicitly, we use {A
s
e
−2τ
, n
s
, ω
m
, ω
b
, θ
∗
, τ, y
P
} =
{1.886, 0.959, 0.1438, 0.02206, 1.04062, 0.07, 1}. The rea-
son for fixing τ and the calibration in obtaining the fiducial
model is that for the analysis of each simulation, priors on these
two parameters are applied, centred on 0.07 and 1, respectively;
if our fiducial model had different values, the distribution
of best-fits across simulations for those and all correlated
parameters would be biased from their fiducial values, and
one would need to recentre the distributions; our procedure is
more straightforward and clearer to interpret. In any case, our
analysis is not very sensitive to the exact fiducial values and
we have checked that for a slightly different fiducial model
with τ = 0.055, the significance levels of the shifts given in
Sect.
3.3 change by <0.1σ
8
. This allows us to take the final step,
which assumes that the distribution of the shifts in parameters
is weakly dependent on the fiducial model in the range allowed
by its probability distribution given the data, P( ˜p
fid
|d), so that
we can estimate the posterior distribution of the parameter
differences given the data from
P( ˜p
(1)
− ˜p
(2)
|d) =
Z
P( ˜p
(1)
− ˜p
(2)
| ˜p
fid
)P( ˜p
fid
|d)d ˜p
fid
, (2)
∼ P( ˜p
(1)
− ˜p
(2)
|d, ˜p
fid
). (3)
In fact, the uncertainty on the fiducial model estimated from the
data, encoded in P( ˜p
fid
|d), is small (at the percent level for most
of the parameters), and we explicitly checked in the τ = 0.055
case that its value does not change our results. Moreover, since
we are interested in the distribution of the differences of the
parameter best-fits, and not in the absolute values of the best-
fits themselves, we expect that this difference essentially only
depends on the scatter of the data as described by the Planck like-
lihood from which we generate the simulations. Since this like-
lihood is assumed to be weakly dependent on the fiducial model,
again roughly in the range allowed by P( ˜p
fid
|d), we expect the
distribution of the differences to have a weak dependence on the
fiducial model.
3.2. Detailed description of the simulations
We now turn to describe these simulations in more detail. The
goal of these simulations is to be as consistent as possible with
the approximations made in the real analysis (as opposed to,
for example, the suite of end-to-end simulations described in
Planck Collaboration XI 2016, which aim to simulate system-
atics not directly accounted for by the real likelihood). In this
sense, our simulations are a self-consistency check of Planck
data and likelihood products. We will now describe these sim-
ulations in more detail.
For each simulation, we draw a realization of the data
independently at ℓ < 30 and at ℓ > 30
9
. At ℓ < 30 we draw
realizations directly at the map level, whereas for ℓ > 30
we use the plik_lite CMB covariance (described in
Planck Collaboration XI 2016) to draw power spectrum realiza-
tions. For both ℓ < 30 and ℓ > 30, each realization is drawn as-
suming a fiducial model.
For ℓ > 30, we draw a random Gaussian sample from the
plik_lite covariance and add it to the fiducial model. This,
along with the covariance itself, forms the simulated likelihood.
8
In Sect.
5.5 we discuss changing the prior on τ, rather than changing
its fiducial value, which does affect the significance levels somewhat.
9
We thus ignore ℓ-to-ℓ correlations across this multipole, consistent
with what is assumed in the real likelihood (
Planck Collaboration XI
2016
).
A95, page 5 of 27