Table Of ContentUnveiling ν secrets with cosmological data: neutrino masses and mass hierarchy
Sunny Vagnozzi,1,∗ Elena Giusarma,2,3,4,† Olga Mena,5,‡ Katherine
Freese,1,6 Martina Gerbino,1 Shirley Ho,2,3,4 and Massimiliano Lattanzi7,8
1The Oskar Klein Centre for Cosmoparticle Physics,
Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden
2McWilliams Center for Cosmology, Department of Physics,
Carnegie Mellon University, Pittsburgh, PA 15213, USA
3Lawrence Berkeley National Laboratory (LBNL),
Physics Division, Berkeley, CA 94720-8153, USA
4Berkeley Center for Cosmological Physics, University of California, Berkeley, CA 94720, USA
5Instituto de F´ısica Corpuscolar (IFIC), Universidad de Valencia-CSIC, E-46980, Valencia, Spain
6Michigan Center for Theoretical Physics, Department of Physics,
University of Michigan, Ann Arbor, MI 48109, USA
7Dipartimento di Fisica e Scienze della Terra, Universita` di Ferrara, I-44122 Ferrara, Italy
8Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Ferrara, I-44122 Ferrara, Italy
(Dated: January 31, 2017)
7 Usingsomeofthelatestcosmologicaldatasetspubliclyavailable, wederivethestrongestbounds
1 in the literature on the sum of the three active neutrino masses, M , within the assumption of
ν
0
a background flat ΛCDM cosmology. In the most conservative scheme, combining Planck cosmic
2
microwave background (CMB) temperature anisotropies and baryon acoustic oscillations (BAO)
n data, as well as the up-to-date constraint on the optical depth to reionization (τ), the tightest
a 95% confidence level (C.L.) upper bound we find is M < 0.151 eV. The addition of Planck high-
ν
J (cid:96) polarization data, which however might still be contaminated by systematics, further tightens
7 the bound to Mν < 0.118 eV. Further improvements are possible when a less conservative prior
2 on the Hubble parameter [H0 = (73.02±1.79) km s−1Mpc−1], which is in slight tension with
CMB-only determinations, is added, bringing down the 95% C.L. upper limit to ∼ 0.09 eV. The
] three aforementioned combinations exclude values of M larger than the minimal value allowed
O ν
in the inverted hierarchical (IH) mass ordering, 0.0986 eV, at ∼ 82% C.L., ∼ 91% C.L., and
C ∼96% C.L. respectively, thus reducing the volume of parameter space allowed within the inverted
hierarchyscenario. Apropermodelcomparisontreatmentaccountingforvolumeeffectsshowsthat
.
h the same combinations exclude the IH at ∼ 64% C.L., ∼ 71% C.L., and ∼ 77% C.L. respectively.
p Furthermore,wetestthestabilityoftheboundsagainstthedistributionofthetotalmassM among
ν
-
thethreemasseigenstates,findingthatthesearerelativelystableagainstthechoiceofdistributing
o
r thetotalmassamongthree(theusualapproximation)oronemasseigenstate. Finally,wecompare
t the constraining power of measurements of the full-shape galaxy power spectrum versus the BAO
s
a signature, from the BOSS survey. Even though the latest BOSS full shape measurements cover a
[ larger volume and benefit from smaller error bars compared to previous similar measurements, the
analysismethodcommonlyadoptedresultsintheirconstrainingpowerstillbeinglesspowerfulthan
1
that of the extracted BAO signal. Our work uses only cosmological data; imposing the constraint
v
M > 0.06eV from oscillations data would raise the quoted upper bounds by O(0.1σ) and would
2 ν
not affect our conclusions.
7
1
8
I. Introduction Global fits [10–12] to oscillation measurements have
0
determinedwithunprecedentedaccuracyfivemixingpa-
.
1 The discovery of neutrino oscillations, awarded with rameters,namely,sin2θ12,sin2θ13,sin2θ23,aswellasthe
0 the 2015 Nobel Prize in Physics [1], has robustly estab- two mass-squared splittings governing the solar and the
7
lished the fact that neutrinos are massive [2–9]. The atmospheric transitions. The solar mass-squared split-
1
: results from oscillation experiments can therefore be ting is given by ∆m221 ≡ m22 − m21 (cid:39) 7.6 × 10−5 eV2.
v successfully explained assuming that the three neutrino Because of matter effects in the Sun, we know that the
i
X flavour eigenstates (ν , ν , ν ) are quantum superposi- mass eigenstate with the larger electron neutrino frac-
e µ τ
r tionsofthreemasseigenstates(ν1,ν2,ν3). Inanalogyto tion is the one with the smallest mass. We identify the
a thequarksector,flavourandmasseigenstatesarerelated lighter state with “1” and the heavier state (which has
viaamixingmatrixparametrizedbythreemixingangles a smaller electron neutrino fraction) with “2”. Conse-
(θ , θ , θ ) and a CP-violating phase δ . quently, thesolarmass-squaredsplittingispositive. The
12 13 23 CP
atmospheric mass-squared splitting is instead given by
|∆m2 | ≡ |m2 −m2| (cid:39) 2.5×10−3eV2. Since the sign
31 3 1
ofthe largest mass-squaredsplitting |∆m2 | remains un-
∗ [email protected] 31
known, there are two possibilities for the mass order-
† [email protected]
‡ omena@ific.uv.es ing: the normal hierarchy (NH, ∆m231 > 0, with m1 <
2
m < m ) and the inverted hierarchy (IH, ∆m2 < 0, [m = M /3, with i = 1,2,3, which we will later refer
2 3 31 νi ν
and m < m < m ). Other unknowns in the neutrino to as 3deg, see Eq.(3)]. Strictly speaking, this is a valid
3 1 2
sector are the presence of CP-violation effects (i.e. the approximation as long as the mass of the lightest eigen-
value of δ ), the θ octant, the Dirac versus Majorana state, m ≡m [m ] in the case of NH [IH], satisfies:
CP 23 light 1 3
neutrino nature, and, finally, the absolute neutrino mass
scale, see Ref. [13] for a recent review on unknowns of mlight (cid:29)|mi−mj| , ∀i,j =1,2,3. (2)
the neutrino sector.
The approximation might fail in capturing the exact be-
Cosmology can address two out of the above five un-
haviour of massive neutrinos when M ∼ M , where
knowns: the absolute mass scale and the mass ordering. ν ν,min
(cid:112) (cid:112) (cid:112)
M = ∆m2 + ∆m2 (cid:39) 0.06 eV[= ∆m2 +
Throughbackgroundeffects,cosmologyistozeroth-order ν,min 21 31 31
(cid:112)
sensitive to the absolute neutrino mass scale, that is, to ∆m2 +∆m2 (cid:39) 0.1 eV] is the minimal mass allowed
31 21
the quantity: byoscillationmeasurementsintheNH[IH]scenario[10–
12], see the Appendix for detailed discussions. Further-
Mν ≡mν1 +mν2 +mν3, (1) more, it has been argued that the ability to reach a
robust upper bound on the total neutrino mass below
where m denotes the mass of the ith neutrino mass
eigenstateνi. Indeed, the tightest current bounds on the Mν,min = 0.1 eV would imply having discarded at some
statisticalsignificancetheinvertedhierarchyscenario. In
neutrino mass scale come from cosmological probes, see
this case, one has to check that the conclusion is robust
for instance [14–21]. More subtle perturbation effects
against the underlying assumptions about the neutrino
make cosmology in principle sensitive to the mass hier-
mass spectrum, as well as provide a rigorous statistical
archy as well (see e.g. [22–27] for comprehensive reviews
treatment of the preference for one hierarchy over the
ontheimpactofnonzeroneutrinomassesoncosmology),
other [35–37].
although not with current datasets.
For all the reasons discussed above, it is timely to in-
As light massive particles, relic neutrinos are relativis-
vestigate the dependence of the cosmological limits on
tic in the early Universe and contribute to the radiation
the total neutrino mass M under the assumption of dif-
energydensity. However,whentheyturnnon-relativistic ν
ferent neutrino mass spectra. In particular, we shall im-
atlatetimes,theirenergydensitycontributestothetotal
prove recent results (see e.g. [21]), focusing also here on
matter density. Thus, relic neutrinos leave a character-
the usual scenario of three massive degenerate neutrinos
istic imprint on cosmological observables, altering both
(which we refer to as 3deg), as well as on the approxi-
the background evolution and the spectra of matter per-
mate scheme with a single massive neutrino carrying the
turbations and Cosmic Microwave Background (CMB)
totalmassM togetherwithtwomasslessspecies(which
anisotropies (see [22–27] as well as the recent [28] for ν
we refer to as 1mass). That is, we consider the following
a detailed review on massive neutrinos in cosmology, in
twoschemesforthedistributionofthetotalmassamong
light of both current and future datasets).
the three eigenstates:
Among all the physical effects of massive neutrinos on
the different cosmological observables, on which we shall M
m =m =m = ν (3deg),
commentonlater,thereisoneofparticularrelevance,re- 1 2 3 3
lated to the neutrino-induced suppression of the growth m =m =0,m =M (1mass). (3)
1 2 3 ν
ofstructuresatsmallscales. Duetotheircollisionlessna-
ture, after the non-relativistic transition, neutrinos free- Themotivationforthe1mass choiceistwofold: i)itis
stream on scales smaller than the distance traveled dur- the usual approximation adopted when performing cos-
ing a Hubble time (k > k , where k is the neutrino mological analyses with the total neutrino mass fixed to
fs fs
free-streaming scale), washing out perturbations below M = 0.06 eV, in order to mimic the minimal mass
ν,min
this characteristic scale. This effect is translated into a scenario in the case of the NH (m = 0 eV, m (cid:28) m ),
1 2 3
lack of power at small scales in the power spectrum of and ii) it might provide a better description of the un-
matter perturbations, as well as in a modification of the derlying neutrino mass ordering in the M < 0.1 eV
ν
gravitational lensing potential sourced by cosmological mass region, in which m ∼ m (cid:28) m , although a
1 2 3
structures, which in turn alters the shape of the spec- complete assessment goes beyond the scope of our work.
trum of the primary CMB anisotropies by the bending The 3deg choice is instead that adopted by the vast ma-
of the CMB photons [29]. jority of works when M is allowed to vary. This in-
ν
Cosmologicalprobesareprimarilysensitivetothesum cludes the Planck collaboration, which recently obtained
ofthethreeactiveneutrinomassesM . Theexactdistri- M <0.234 eV at 95% C.L. [39] through a combination
ν ν
butionofthetotalmassamongthethreemasseigenstates of temperature and low-(cid:96) polarization anisotropy mea-
induces sub-percent effects on the different cosmological surements, within the assumption of a flat ΛCDM+M
ν
observables, which are below the sensitivities of ongo- cosmology.
ing and near future experiments [30–34]. As a result, Our focus, however, will be on results obtained within
cosmological constraints on M are usually obtained by the 3deg scheme. The choice is dictated by the observa-
ν
making the assumption of a fully degenerate mass spec- tion that the impact of the NH and IH mass splittings
trum, with the three neutrinos sharing the total mass on cosmological data is tiny if one compares the 3deg
3
approximation to the corresponding NH and IH models themostconstraining95%C.L.boundsonthesumofthe
withthesamevalueofthetotalmassM . However, this neutrino masses using a combination of CMB (tempera-
ν
does not necessarily hold when the comparison is made tureandpolarization),BAO,andotherexternaldatasets.
between 3deg and 1mass, because the latter always has The bounds in Tab. VII have been obtained using also
twopureradiationcomponentsthroughoutthewholeex- small-scale CMB polarization data which may be con-
pansionhistoryand,inparticular,atthepresenttime(on taminated by systematics, yet we present the results as
the other hand, NH and IH can have at most one pure they are useful for comparing to previous work. Finally
radiation component at present time, a situation which Tab.IXpresentsexclusionlimitsontheInvertedHierar-
occurs in the minimal mass scenario and thus only for chy neutrino mass ordering, which is excluded at about
one specific point in parameter space). The extra mass- 70% C.L. statistical significance.
less component(s) present in the 1mass case, but not for
the NH and IH (one extra component if the NH and IH
happen to correspond to the minimal mass scenario, two II. Analysis method
extra components otherwise), are known to have a non-
negligible impact on cosmological observables, in partic-
In the following we shall provide a careful description
ulartheCMBanisotropyspectrum[21,28]. Nonetheless,
ofthestatisticalmethodsemployedinordertoobtainthe
for the reasons we thoroughly explained above, it is still
bounds on the sum of the three active neutrino masses
very useful to compare the results obtained in the 3deg
we show in Sec. IV, as well as caveats to our analyses.
case against those obtained in the 1mass one, but focus-
Furthermore,weprovideabriefdescriptionofthestatis-
ing on results obtained within the 3deg approximation,
tical method adopted to quantify the exclusion limits on
see the Appendix for further discussions.
the IH from our bounds on M . For more details on the
ν
For each of the two neutrino mass spectra described latter, we refer the reader to [35] where this method was
above, we present the constraints in light of the most re- originally described.
cent cosmological data publicly available. In particular,
we make use of i) measurements of the temperature and
polarization anisotropies of the CMB as reported by the A. Bounds on the total neutrino mass
Planck satellite in the 2015 data release; ii) a selection
of recent baryon acoustic oscillations (BAO) measure-
In our work, we perform standard Bayesian inference
ments; iii) measurements of the galaxy power spectrum
(see e.g. [40, 41] for recent reviews) to derive constraints
as reported by the final data release from the SDSS-III
on the sum of the three active neutrino masses. That is,
BOSS collaboration; iv) local measurements of the Hub-
given a model described by the parameter vector θ, and
ble parameter (H ); v) the latest measurement of the
0 a set of data x, we derive posterior probabilities of the
optical depth to reionization (τ) coming from the analy-
parameters given the data, p(θ|x), according to:
sisofthehigh-frequencychannelsofthePlancksatellite,
andvi)clustercountsfromtheobservationofthethermal
p(θ|x)∝L(x|θ)p(θ), (4)
Sunyaev-Zeldovich (SZ) effect by the Planck satellite.
In addition to providing bounds on Mν, we also use where L(x|θ) is the likelihood function of the data
these bounds to provide a rigorous statistical treatment given the model parameters, and p(θ) denotes the data-
ofthepreferencefortheNHovertheIH.Wedosobyap- independent prior. We derive the posteriors using the
plying the simple but rigorous method proposed in [35], Markov Chain Monte Carlo (MCMC) sampler cosmomc
and evaluate both posterior odds for NH against IH, as with an efficient sampling method [42, 43]. To assess
well as the C.L. at which current datasets can exclude the convergence of the generated chains, we employ the
the IH. Gelman and Rubin statistics [44] R − 1, which we re-
The paper is organized as follows. In Sec. II, we de- quire to satisfy R−1 < 0.01 when the datasets do not
scribe our analysis methodology. In Sec. III we instead include SZ cluster counts, R−1 < 0.03 otherwise (this
provide a careful description of the datasets employed, choice is dictated by time and resource considerations:
complemented with a full explanation of the physical ef- runs involving SZ cluster counts are more computation-
fectsofmassiveneutrinosoneachofthem. Weshowcase allyexpensivethanthosethatdonotincludeSZclusters,
our main results in Sec. IV, with Sec. IVA in partic- to achieve the same convergence). In this way, the con-
ular devoted to an analysis of the relative constraining tribution from statistical fluctuations is roughly a few
power of shape power spectrum versus geometrical BAO percent the limits quoted. 1
measurements, whereas in Sec. IVB we provide a rigor-
ous quantification of the exclusion limits on the inverted
hierarchy from current datasets. Finally, we draw our
conclusions in Sec. V.
1 Notice that this is a very conservative requirement, as a con-
For the reader who wants to skip to the results: the vergence of 0.05 is typically more than sufficient for the explo-
most important results of this paper can be found in rationoftheposteriorofaparameterwhosedistributionisuni-
Tabs. VI, VII, IX. The first two of these tables present modal[45].
4
We work under the assumption of a background flat andwoulddifferifoneweretoconsiderextendedparam-
ΛCDM Universe, and thus consider the following seven- eter models, for instance scenarios in which the number
dimensional parameter vector: of relativistic degrees of freedom N and/or the dark
eff
energy equation of state w are allowed to vary, or if the
θ ≡{Ω h2,Ω h2,Θ ,τ,n ,log(1010A ),M }. (5)
b c s s s ν assumptionofflatnessisrelaxed. Forrecentrelatedstud-
Here, Ω h2 and Ω h2 denote the physical baryon and ies considering extensions to the minimal ΛCDM model
b c
dark matter energy densities respectively, Θ is the ratio we refer the reader to e.g. [47–68].
s
of the sound horizon to the angular diameter distance
at decoupling, τ indicates the optical depth to reioniza- Parameter Prior Name
tion,whereasthedetailsoftheprimordialdensityfluctu- Ω h2 [0.005,0.1]
b
ations are encoded in the amplitude (A ) and the spec- Ω h2 [0.01,0.99]
s c
tral index (n ) of its power spectrum at the pivot scale Θ [0.5,10]
s s
k = 0.05 h Mpc−1. Finally, the sum of the three neu- τ [0.01,0.8]
(cid:63)
trinomassesisdenotedbyM . Foralltheseparameters, 0.055 ± 0.009 τ0p055
ν
a uniform prior is assumed unless otherwise specified. ns [0.8,1.2]
Concerning M , we impose the requirement M ≥ 0. log(1010As) [2,4]
ν ν
M (eV) [0,3]
Thus, we ignore prior information from oscillation ex- ν
periments, which, as previously stated, set a lower limit H0 (km/s/Mpc) 20 → 100 (Implicit)
72.5 ± 2.5 H072p5
of M ∼ 0.06 eV[0.10 eV] for the NH [IH] mass or-
ν,min 73.02 ± 1.79 H073p02
dering. If we instead had chosen not to ignore prior in-
1−b [0.1,1.3]
formation from oscillation experiments, the result would
bHF [0,10]
be a slight shift of the center of mass of our posteriors
PHF [0,10000]
on M towards higher values. As a consequence of these
ν
shifts,the95%C.L.upperlimitswereportwouldalsobe TABLE I. Priors on cosmological and nuisance parameters
shifted to slightly higher values; conversely, the C.L. at considered in this work. Priors on a parameter p of the form
which we can exclude a given values of M greater than [A,B] are uniform within the range A<p<B, whereas pri-
ν
a given value would be slightly reduced. Nonetheless, in ors of the form A±B are Gaussian with central value and
thiswaywecanobtainanindependentupperlimitonM variancegivenbyAandB,respectively. Thefirstsevenrows
ν
fromcosmologyalone,whileatthesametimemakingthe refertothebasicparametervectorinEq.(5). H0 referstothe
Hubble parameter and is a derived parameter, whereas 1−b
least amount of assumptions. It also allows us to remain
istheclustermassbiasparameter,seeSec.IIIF.Theparam-
open to the possibility of cosmological models predicting
etersbHF andPHF arenuisanceparametersusedtomodelthe
a vanishing neutrino density today, or models where the
galaxy power spectrum, see Eq. (12).
effect of neutrino masses on cosmological observables is
hidden due to degeneracies with other parameters (see
e.g. [46, 47]). One can get a feeling for the size of the
shifts by comparing our results to those of [17], where a
B. Model comparison between mass hierarchies
prior M ≥0.06 eV was assumed. As we see, the size of
ν
the shifts is small, of O(0.1σ). We summarize the priors
As we discussed previously, several works have argued
on cosmological parameters, as well as some of the main
that reaching an upper bound on M of order 0.1eV
nuisance parameters, in Tab. I. ν
would imply having discarded the IH at some statistical
AlltheboundsonM reportedinSec.IVare95%C.L.
ν significance. While CL (the C.L. at which cosmol-
upper limits. For each dataset combination, we also re- 0.0986
ogyisabletoexcludevaluesofM largerthan0.0986eV)
port the C.L. at which values of M greater than the ν
ν doesprovideanindicationofboththeremainingamount
minimum mass in the IH scenario, 0.0986eV, are ex-
of parameter space available within the IH, as well as on
cluded (we only report these values if C.L. > 70%): we
the sensitivity of current datasets, it does not properly
refer to this value as CL . While this does not pro-
0.0986 quantify the statistical significance at which the IH is
videarobuststatisticalindicatoroftheC.L.atwhichwe
excluded. In order to quantify the exclusion limits on
can exclude the IH, it does provide an indication of how
the IH, a proper model comparison treatment, thus rig-
much the volume of parameter space available in the IH
orously taking into account volume effects, is required.
scenario is reduced to. Given the posterior distribution
Variousmethodswhichallowtheestimationoftheex-
of M , that is, p(M ), we calculate CL in the usual
ν ν 0.0986 clusion limits on the IH have been devised in the re-
way, by considering the ratio of the area subtended by
cent literature, see e.g. [35–37]. Here, we will briefly de-
the posterior for M <0.0986eV, over the total area:
ν scribethesimplebutrigorousmodelcomparisonmethod
(cid:82)0.0986dM p(M ) which we will use in our work, proposed by Hannestad
CL0.0986 = 0(cid:82)∞ ν ν . (6) and Schwetz in [35], and based on previous work in [38].
dM p(M )
0 ν ν The method allows the quantification of the statistical
The bounds we obtain on M depend more or less significance at which the IH can be discarded, given the
ν
strongly on our assumption of a standard ΛCDM model, cosmological bounds on M . We refer the reader to the
ν
5
original paper [35] for further details. decoupling r (z ), thus reducing the angular size
s dec
Let us again consider the likelihood function L of the of the sound horizon at decoupling Θ and shifting
s
data x given a set of cosmological parameters θ, the all the peaks to higher multipoles (cid:96)’s [26].
mass of the lightest neutrino m ≡m =m [m ] for
0 light 1 3 • By suppressing the structure growth on small
NH [IH], and the discrete parameter H representing the
scales,reducingthelensingpotentialandhencethe
masshierarchy,withH =N [I]forNH[IH]respectively:
smearing of the high-(cid:96) multipoles due to gravita-
L(x|θ,m ,H). Then, given the prior(s) on cosmological
0 tional lensing. 2
parameters p(θ), we define the likelihood marginalized
over cosmological parameters θ as: • Massive neutrinos will also lead to a small change
(cid:90) in the diffusion scale, which affects the photon dif-
L(D|m ,H)≡ dθ L(D|θ,m ,H)p(θ). (7) fusion pattern at high-(cid:96) multipoles [26].
0 0
• Finally,sincetheenhancementofthefirstpeakdue
Imposing an uniform prior m ≥0eV and assuming fac-
0 to the EISW depends, in principle, on the precise
torizable priors for the other cosmological parameters it
isnothardtoshowthat, asaconsequenceofBayes’the- epoch of transition to the non-relativistic regime
orem, the posterior probability of a mass hierarchy H of each neutrino species, that is, on the individual
given the data, pH ≡p(H|x), can be obtained as below: neutrino masses, future CMB-only measurements
such as those of [71–73, 77–86] could provide some
p = (8)
H
p(N)(cid:82)0∞dm0pL(H(D)|(cid:82)m0∞0,dNm)0+Lp((DI)|m(cid:82)00∞,Hdm)0 L(D|m0,I), Althouhginhtsaltlothuenarabvoevletehffeenctesumtrianyosmugagsesstortdheartinthge[2C6M].B3
is exquisitely sensitive to the neutrino mass, in practice,
where p(N) and p(I) denote priors on the NH and IH
the shape of the CMB anisotropy spectra is governed by
respectively, with p(N)+p(I) = 1. The posterior odds
several parameters, some of which are degenerate among
of NH against IH are then given by p /p , whereas the
N I themselves [87, 88]. We refer the reader to the dedicated
C.L. at which the IH is excluded, which we refer to as
study of Ref. [28] (see also [89]). 4
CL is given by:
IH
CL =1−p . (9)
IH I
Baseline combinations of datasets used, and
their definitions, I.
III. Datasets and their sensitivity to M
ν
Measurements of the CMB temperature, polarization,
Wepresentbelowadetaileddescriptionofthedatasets and cross-correlation spectra from the Planck 2015 data
used in our analyses and their modeling, discussing their
sensitivity to the sum of the active neutrino masses.
For clarity, all the denominations of the combinations 2 This is a promising route towards determining both the abso-
of datasets we consider are summarized in Tab. II. lute neutrino mass scale and the neutrino mass hierarchy, see
e.g. Ref. [69, 70], because it probes the matter distribution in
the linear regime at higher redshift, and because the unlensed
background is precisely understood. CMB lensing suffers from
A. Cosmic Microwave Background
systematics as well, although these tend to be of instrumental
originandhencedecreasewithhigherresolution. Infact,acom-
Massive neutrinos may, depending on the precise val- bination of CMB-S4 [71–73] lensing and DESI [74–76] BAO is
ues of their masses, turn non-relativistic around the expectedtoachieveanuncertaintyonMν of0.016eV[71].
3 In practice, the effect is below the level for all multipoles,
epoch of photon decoupling. Therefore neutrinos can hence well beyond the reach of Plan(cid:104)ck. The effect will be be-
leave an imprint on the CMB (both at the background lowthereachofground-basedStage-IIIexperimentssuchasAd-
and at the perturbation level) in, at least, five different vanced ACTPol [77, 78], SPT-3G [79], the Simons Array [80]
and the Simons Observatory [81]. It will most likely be below
ways, extensively explored in the literature [22–28]:
the reach of ground-based Stage-IV experiments such as CMB-
• Bydelayingtheepochofmatter-radiationequality, S4 [71–73], or next-generation satellites such as the proposed
LiteBIRD[82],COrE[83,84],andPIXIE[86].
massive neutrinos lead to an enhanced early inte-
4 ToassesstheimpactofmassiveneutrinosontheCMB,allchar-
grated Sachs-Wolfe (EISW) effect [26]. This effect
acteristictimes,scales,anddensityratiosgoverningtheshapeof
is due to the time-variation of gravitational poten- theCMBanisotropyspectrumshouldbekeptfixed,i.e. keeping
tials which occurs during the radiation-dominated, zeqandtheangulardiameterdistancetolast-scatteringdA(zdec)
butnotduringthematter-dominatedera,andleads fixed. This would result in: a decrease in the late integrated
Sachs-Wolfe(LISW)effect,whichhoweverispoorlyconstrained
toanenhancementofthefirstacousticpeakinpar-
owing to the fact that the relevant multipole range is cosmic
ticular. Traditionally this has been the most rele- variancelimited;amodestchangeinthediffusiondampingscale
vant neutrino mass signature in the CMB data. for Mν (cid:38)0.6 eV; and finally, a ∆C(cid:96)/C(cid:96) ∼−(Mν/0.1 eV)% de-
pletion of the amplitude of the C ’s for 20(cid:46)(cid:96)(cid:46)200, due to a
(cid:96)
• Because of the same delay as above, massive neu- smaller EISW effect, which also contains a sub- effect due to
trinos also decrease the comoving sound horizon at theindividualneutrinomasses,essentiallyimpos(cid:104)sibletodetect.
6
release [39, 90] are included. We consider a combina- ple be the most sensitive cosmological probe of massive
tion of the high-(cid:96) (30 ≤ (cid:96) ≤ 2508) TT likelihood, as neutrinos among those exploited here. Sub-eV neutri-
well as the low-(cid:96) (2 ≤ (cid:96) ≤ 29) TT likelihood based on nos behave as a hot dark matter component with large
the CMB maps recovered with Commander: we refer to thermal velocities, clustering only on scales below the
thiscombinationasPlanckTT.Wefurthermoreinclude neutrino free-streaming wavenumber k [24, 26]:
fs
the Planck polarization data in the low-(cid:96) (2 ≤ (cid:96) ≤ 29)
likelihood, referring to it as lowP. Our baseline model, (cid:18)M (cid:19)1/2
k (cid:39)0.018 Ω1/2 ν h Mpc−1. (10)
consisting of a combination of PlanckTT and lowP, is fs m 1eV
referred to as base.
In addition to the above, we also consider the high-(cid:96) Onscalesbelowthefree-streamingscale(or,correspond-
(30 ≤ (cid:96) ≤ 1996) EE and TE likelihood, which we refer ingly, for wavenumbers larger than the free-streaming
to as highP. In order to ease the comparison of our re- wavenumber), neutrinos cannot cluster as their thermal
sults to those previously presented in the literature, we velocity exceeds the escape velocity of the gravitational
shall add high-(cid:96) polarization measurements to our base- potentials on those scales. Conversely, on scales well
linemodelseparately,referringtothecombinationofbase above the free-streaming scale, neutrinos behave as cold
and highP as basepol. Due to possible residual system- dark matter after the transition to the non-relativistic
atics in the datasets, which are still being analyzed by regime. Massive neutrinos leave their imprint on the
the Planck collaboration, the results obtained here with galaxy power spectrum in several different ways:
theinclusionofhigh-(cid:96)polarizationmeasurementsshould
• For wavenumbers k > k , the power spectrum in
be regarded as less conservative and thus should be in- fs
thelinearperturbationregimeissubjecttoascale-
terpreted with more caution. For the purpose of clarity,
independent reduction by a factor of (1 − f )2,
we have summarized our nomenclature of datasets and ν
where f ≡ Ω /Ω is defined as the ratio of the
their combinations in Tab. II. ν ν m
energy content in neutrinos to that in matter [26].
Allthemeasurementsdescribedaboveareanalyzedby
means of the publicly available Planck likelihoods [91]. 5 • In addition, the power-spectrum for wavenumbers
When considering a prior on the optical depth to reion- k >k isfurthersubjecttoascale-dependentstep-
fs
ization τ we shall only consider the TT likelihood in like suppression, starting at k and saturating at
fs
the multipole range 2 ≤ (cid:96) ≤ 29. We do so for avoid- k ∼1 h Mpc−1. This suppression is due to the ab-
ing double-counting of information, see Sec. IIIE. Of sence of neutrino perturbations in the total matter
course,theselikelihoodsdependalsoonanumberofnui- power spectrum, ultimately due to the fact that
sance parameters, which should be (and are) marginal- neutrinos do not cluster on scales k > k . At
fs
ized over. These nuisance parameters describe, for in- k ∼ 1 h Mpc−1, the suppression reaches a con-
stance, residual foreground contamination, calibration, stant amplitude of ∆P(k)/P(k)(cid:39)−10f [26] (the
ν
and beam-leakage (see Refs. [39, 91]). amplitudeofthesuppressionisindependentofred-
CMBmeasurementshavebeencomplementedwithad- shift, however see the point below).
ditional probes which will help breaking the parameter
degeneracies discussed. These additional datasets in- • The growth rate of the dark matter perturbations
clude large-scale structure probes and direct measure- is reduced from δ ∝ a to δ ∝ a1−53fν, due to the
ments of the Hubble parameter, and will be described in absence of gravitational back-reaction effects from
what follows. We make the conservative choice of not free-streaming neutrinos. The redshift dependence
including lensing potential measurements, despite mea- of this suppression implies that this effect could be
suring M via lensing potential reconstruction is the ex- disentangled from that of a similar suppression in
ν
pected target of the next-generation CMB experiments. the primordial power spectrum by measuring the
This choice is dictated by the observation that lensing galaxy power spectrum at several redshifts, which
potential measurements via reconstruction through the amounts to measuring the time-dependence of the
temperature 4-point function are known to be in tension neutrino mass effect [26].
with the lensing amplitude as constrained by the CMB
• On very large scales (10−3 < k < 10−2), the mat-
power spectra through the A parameter [39] (see also
lens
ter power spectrum is enhanced by the presence of
[92–95] for relevant work).
massive neutrinos [96].
• As in the case of the EISW effect in the CMB,
B. Galaxy power spectrum
the step-like suppression in the matter power spec-
trum carries a non-trivial dependence on the indi-
Once CMB data is used to fix the other cosmological
vidual neutrino masses, as it depends on the time
parameters, the galaxy power spectrum could in princi-
of the transition to the non-relativistic regime for
eachneutrinomasseigenstate[30,33](k ∝m1/2),
fs νi
and thus is in principle extremely sensitive to the
5 www.cosmos.esa.int/web/planck/pla neutrinomasshierarchy. However,theeffectisvery
7
smallandveryhardtomeasure,evenwiththemost icaldatasets. Thereareseveralpromisingroutestowards
ambitiousnext-generationlarge-scalestructuresur- achievingthis,forinstancethroughCMBlensing,galaxy
veys [31, 32, 34]. 6 lensing, cross-correlations of the former with galaxy or
quasar clustering measurements, or higher order corre-
Notice that, in principle, once CMB data is used to
lators of the former datasets, see e.g. Refs. [100–108].
fix the other cosmological parameters, the galaxy power
A sensitivity on M of 0.023 eV has been forecasted
spectrum could be the most sensitive probe of neutrino ν
from a combination of Planck CMB measurements to-
masses. In practice, the potential of this dataset is lim-
gether with weak lensing shear auto-correlation, galaxy
ited by several effects. Galaxy surveys have access to a
auto-correlation,andgalaxy-shearcross-correlationfrom
region of k-space k < k < k where the step-like
min max Euclid [109], after marginalization over the bias, with
suppression effect is neither null nor maximal. The min-
the figure improving to 0.01 eV after including a weak
imum wavenumber accessible is limited both by signal-
lensing-selected cluster sample from Euclid [109–114].
to-noise ratio and by systematics effects, and is typically
Similar results are expected to be achieved for certain
of order k ∼ 10−2 h Mpc−1, meaning that the fourth
configurations of the proposed WFIRST survey [115]. It
effect outlined above is currently not appreciable. The
isworthconsideringthatthesensitivityofthesedatasets
maximum wavenumber accessible is instead limited by
wouldbesubstantiallyboostedbydeterminingthescale-
the reliability of the non-linear predictions for the mat-
dependent bias as discussed above.
ter power spectrum.
Galaxy clustering measurements are addressed by
At any given redshift, there exists a non-linear
means of the Sloan Digital Sky Survey III (SDSS-III;
wavenumber, above which the galaxy power spectrum
[116]) Baryon Oscillation Spectroscopic Survey (BOSS;
is only useful insofar as one is able to model non-linear
[117–119]) DR12 [120, 121]. The SDSS-III BOSS DR12
effects, redshift space distortions, and the possible scale-
CMASS sample covers an effective volume of V ≈
dependence of the bias (a factor relating the spatial dis- eff
7.4Gpc3[122]. Itcontains777202massivegalaxiesinthe
tribution of galaxies and the underlying dark matter
range 0.43 < z < 0.7, at an effective redshift z = 0.57
density field [97]) correctly. The non-linear wavenum-
(see footnote 7 for the definition of effective redshift),
ber depends not only on the redshift of the sample but
covering9376.09deg2 overthesky. Hereweconsiderthe
also on other characteristics of the sample itself (e.g.
spherically averaged power spectrum of this sample, as
whether the galaxies are more or less massive). At
measured by Gil-Mar´ın et al. in [123]. We refer to this
the present time, the non-linear wavenumber is approxi-
dataset as P(k). The measured galaxy power spectrum
matelyk =0.15hMpc−1,whereasforthegalaxysample
Pg consists of a convolution of the true galaxy power
we will consider (DR12 CMASS, at an effective redshift meas
spectrum Pg with a window function W(k ,k ), which
of z = 0.57, see footnote 7 for the definition of effective true i j
accounts for correlations between the measurements at
redshift) we will show that wavenumbers smaller than
different scales due to the finite size of the survey geom-
k =0.2hMpc−1 aresafeagainstlargenon-linearcorrec-
etry:
tions (see also Fig. 1 and Ref. [21]). 7
(cid:88)
The issue of the scale-dependent bias is indeed more Pg (k )= W(k ,k )Pg (k ) (11)
meas i i j true j
subtle than it might seem, given that neutrinos them-
j
selves induce a scale-dependent bias [98, 99]. A
Thus, at each step of the Monte Carlo, we need to con-
parametrization of the galaxy power spectrum in the
volve the theoretical galaxy power spectrum P at the
th
presence of massive neutrinos in terms of a scale-
givenpointintheparameterspacewiththewindowfunc-
independent bias and a shot-noise component [see
tion,beforecomparingitwiththemeasuredgalaxypower
Eq.(12)], which in itself adds two extra nuisance param-
spectrum and constructing the likelihood.
eters, may not capture all the relevant effects at play.
Followingpreviousworks[124,125],wemodelthethe-
Despite these difficulties, the galaxy power spectrum is
oretical galaxy power spectrum as:
still a very useful dataset as it helps breaking some of
P =b2 Pm (k,z)+Ps , (12)
the degeneracies present with CMB-only data, in partic- th HF HFν HF
ularbyimprovingthedeterminationofΩ h2 andn ,the where Pm denotes the matter power spectrum cal-
m s HFν
latter being slightly degenerate with M . Moreover, as culated at each step by the Boltzmann solver camb,
ν
we shall show in this paper, the galaxy power spectrum corrected for non-linear effects using the Halofit
represents a conservative dataset (see Sec. IVA). method [127, 128]. We make use of the modified ver-
Nonetheless, a great deal of effort is being invested sion of Halofit designed by [129] to improve the treat-
intodeterminingthescale-dependentbiasfromcosmolog- ment of non-linearities in the presence of massive neu-
trinos. In order to reduce the impact of non-linearities
we impose the conservative choice of considering a max-
imum wavenumber k = 0.2 h Mpc−1. As we show
6 Throughthesameeffect,thelensedCMBaswellasthelensing max
in Fig. 1, this region is safe against uncertainties due to
potentialpowerspectrumcouldalsobesensitivetotheneutrino
masshierarchy. non-linear evolution, and is also convenient for compari-
7 The effective redshift consists of the weighted mean redshift of son with other works which have adopted a similar max-
thegalaxiesofthesample,withtheweightsdescribedin[121]. imum wavenumber cutoff. The smallest wavenumber we
8
areconsideringisinsteadofk =0.03hMpc−1,andis BAO measurements contain geometrical information
min
determined by the control over systematics, which dom- in the sense that, as a “standard ruler” of known and
inate at smaller wavenumbers. The parameters b and measured length, theyallow for thedetermination ofthe
HF
Ps denotethescale-independentbiasandtheshotnoise angulardiameterdistancetotheredshiftofinterest, and
HF
contributions: the former reflects the fact that galaxies hence make it possible to map out the expansion history
arebiasedtracersoftheunderlyingdarkmatterdistribu- oftheUniverseafterthelastscattering. Inaddition,they
tion,whereasthelatterarisesfromthediscretepoint-like are affected by uncertainties due to the non-linear evolu-
nature of the galaxies as tracers of the dark matter. We tionofthematterdensityfieldtoalesserextentthanthe
impose flat priors in the range [0.1,10] and [0,10000] re- galaxy power spectrum, making them less prone to sys-
spectively for b and Ps . 8 tematic effects than the latter. An angle-averaged BAO
HF HF
measurement constrains the quantity D (z )/r (z ),
v eff s drag
where the dilation scale D at the effective redshift of
v
C. Baryon acoustic oscillations the survey z is a combination of the physical angular
eff
diameterdistanceD (z)andtheHubbleparameterH(z)
A
Priortotherecombinationepoch,photonsandbaryons (which control the radial and the tangential separations
in the early Universe behave as a tightly coupled fluid, within a given cosmology, respectively):
whose evolution is determined by the interplay between
(cid:20) (cid:21)1
cz 3
thegravitationalpullofpotentialwells,andtherestoring D (z)= (1+z)2D (z)2 . (15)
force due to the large pressure of the radiation compo- v A H(z)
nent. The resulting pressure waves which set up, before D quantifies the dilation in distances when the fiducial
v
freezing at recombination, imprint a characteristic scale cosmology ismodified. Thepower ofthe BAO technique
on the late-time matter clustering, in the form of a lo- residesonitsabilityofresolvingtheexistingdegeneracies
calized peak in the two-point correlation function, or a present when the CMB data alone is used, in particular
seriesofsmearedpeaksinthepowerspectrum. Thisscale insharpeningthedeterminationofΩ andoftheHubble
m
corresponds to the sound horizon at the drag epoch, de- parameter H , discarding the low values of H allowed
0 0
noted by rs(zdrag), where the drag epoch is defined as by the CMB data.
the time when baryons were released from the Compton Massive neutrinos affect both the low-redshift geom-
dragofphotons, seeRef.[136]. Then, rs(zdrag)takesthe etry and the growth of structure, and correspondingly
form: BAO measurements. If we increase M , while keeping
ν
rs(zdrag)=(cid:90) ∞ dz Hcs((zz)), (13) Ωinbchre2aasnedd.ΩTchh2erfiexfoerde,,tihneoerxdpearntsoioknereaptefixaetdeatrhlye tainmgeuslaisr
zdrag scale of the sound horizon at last scattering Θ (which is
s
where c (z) denotes the sound speed and is given by very well constrained by the CMB acoustic peak struc-
s
cs(z) = c/(cid:112)3(1+R), with R = 3ρb/4ρr being the ra- ture), it is necessary to decrease ΩΛ. As ΩΛ decreases,
tio of the baryon to photon momentum density. Finally, it is found that H(z) decreases for z < 1 [137]. It can
the baryon drag epoch zdrag is defined as the redshift be shown that an increase in Mν has a negligible effect
such that the baryon drag optical depth τdrag is equal to on rs(zdrag). Hence, we conclude that the main effect of
one: massive neutrinos on BAO measurements is to increase
D (z)/r (z )anddecreaseH ,asM isincreased(see
4Ω (cid:90) zdrag dησ x (z) v s drag 0 ν
τ (η )= r dz T e =1, (14) [137]).
drag drag 3Ω da 1+z
b 0
whereσ =6.65×10−29 m2 denotestheThomsoncross-
T Baseline combinations of datasets used, and
sectionandx (z)representsthefractionoffreeelectrons.
e their definitions, II.
In this work, we make use of BAO measurements ex-
8 Although in this simple model the bias and shot noise are as- tracted from a number of galaxy surveys. When us-
sumed to be scale-independent, there is no unique prescription ing BAO measurements in combination with the DR12
for the form of these quantities. In particular, concerning the CMASSP(k),weconsiderdatafromtheSix-degreeField
bias, several theoretically well-motivated scale-dependent func-
Galaxy Survey (6dFGS) [138], the WiggleZ survey [139],
tionalformsexistintheliterature(suchastheQmodelof[133],
and the DR11 LOWZ sample [140], as done in [21]. We
thatof[134],orthatof[135]motivatedbylocalprimordialnon-
Gaussianity). Itisbeyondthescopeofourpapertoexplorethe refer to the combination of these three BAO measure-
impact of different bias function choices on the neutrino mass ments as BAO. When combining BAO with the base
bounds. Instead, we simply note that it is not necessarily true CMB dataset and the DR12 CMASS P(k) measure-
that increasing the number of parameters governing the bias
ments, we refer to the combination as basePK. When
shape may result in broader constraints. Indeed, tighter con-
straints on Mν may arise in some of the bias parameterizations combining BAO with the basepol CMB dataset and the
withmorethanoneparameterinvolved,becausetheymighthave DR12 CMASS P(k) measurements, we refer to the com-
comparableeffectsonthepowerspectrum. bination as basepolPK. Recall that we have summa-
9
105
]
3
c
p 104
M
3
− Halofit Non Linear Matter Power, PHF
h
)[ Non Linear Matter Power, PNL
P(k 103 (PNL-PHF)/PHF
DR9 data
DR12 data
102
0.8 10-2 10-1
0.6
P 0.4
/
P 0.2
∆
0.0
0.2
10-2 10-1
k[h/Mpc]
FIG. 1. Top: Non-linear galaxy power spectrum computed using the Halofit method with the camb code [126] (red line) and
the Coyote emulator (blue line) [130–132] at z=0.57 for the ΛCDM best-fit parameters from Planck TT 2015 data. Green
triangledatapointsaretheclusteringmeasurementsfromtheBOSSDR12CMASSsample. Theerrorbarsarecomputedfrom
thediagonalelementsC ofthecovariancematrix. Forcomparisonwithpreviouswork[21],purplecirclesrepresentclustering
ii
measurementsfromtheBOSSDataRelease9(DR9)CMASSsample. Averyslightsuppressioninpoweronsmallscales(large
k) of the DR12 sample compared to the DR9 sample is visible. Note that the binning strategy adopted in DR9 and DR12 is
different. Bottom: Residuals with respect to the non-linear model with Halofit. The orange horizontal line indicates the k
range used in our analysis. As it is visually clear, the k range we choose is safe from large non-linear corrections.
rized our nomenclature of datasets (including baseline pleteness,weconsiderthiscaseaswell. Namely,wedrop
datasets) and their combinations in Tab. II. the DR12 CMASS power spectrum from our datasets,
The 6dFGS data consists of a measurement of replacing it with the DR11 CMASS BAO measurement.
r (z )/D (z) at z = 0.106 (as per the discussion This consists of a measurement of D (z )/r (z ) at
s drag V v eff s drag
above, r /D decreases as M is increased). The Wig- z =0.57.
s V ν eff
gleZdatainsteadconsistofmeasurementsoftheacoustic
parameterA(z)atthreeredshifts: z =0.44,z =0.6,and
z =0.73, where the acoustic parameter is defined as: Baseline combinations of datasets used, and
√ their definitions, III.
100D (z) Ω h2
A(z)= v m . (16)
cz
WerefertothecombinationofthefourBAOmeasure-
Given the effect of M on D (z), A(z) will increase as ments (6dFGS, WiggleZ, DR11 LOWZ, DR11 CMASS)
ν v
M increases. Finally, the DR11 LOWZ data consists as BAOFULL. We instead refer to the combination of
ν
of a measurement of D (z)/r (z ) (which increases as the base CMB and the BAOFULL datasets with the
v s drag
M is increased) at z =0.32. nomenclature baseBAO. When high − (cid:96) polarization
ν
Since the BAO feature is measured from the galaxy CMB data is added to this baseBAO dataset, the com-
two-point correlation function, to avoid double counting bination is referred to as basepolBAO, see Tab. II. The
of information, when considering the base and basepol comparisonbetweenbasePK andbaseBAO,aswellasbe-
datasetswedonotincludetheDR11CMASSBAOmea- tween basepolPK and basepolBAO, gives insight into the
surements, as the DR11 CMASS and DR12 CMASS vol- roleplayedbylarge-scalestructuredatasetsinconstrain-
umes overlap. However, if we drop the DR12 CMASS ingneutrinomasses. Inparticular,itallowsforanassess-
powerspectrumfromourdatasets,weareallowedtoadd ment of the relative importance of shape information in
DR11 CMASS BAO measurements without this leading theformofthepowerspectrumagainstgeometricalinfor-
to double-counting of information. Therefore, for com- mationintheformofBAOmeasurementswhenderiving
10
the neutrino mass bounds. For clarity, all the denomi- in tension with the CMB-only H value by 3σ. We refer
0
nations of the combinations of datasets we consider are to the corresponding prior as H073p02. 9
summarized in Tab. II.
All the BAO measurements used in this work are tab-
ulated in Tab. III. Note that we do not include BAO E. Optical depth to reionization
measurements from the DR7 main galaxy sample [141]
or from the cross-correlation of DR11 quasars with the The first generation of galaxies ended the dark ages of
Lyα forest absorption [142], and hence our results are the Universe. These galaxies emitted UV photons which
not directly comparable to other existing studies which gradually ionized the neutral hydrogen which had ren-
included these measurements. dered the Universe transparent following the epoch of
recombination, in a process known as reionization (see
e.g. Ref. [149] for a review). So far, it is not entirely
D. Hubble parameter measurements clear when cosmic reionization took place. Cosmological
measurements can constrain the optical depth to reion-
Direct measurements of H are very important when ization τ, which, assuming instantaneous reionization (a
0
consideringboundsonM . WithCMBdataalone,there very common useful approximation), can be related to
ν
exists a strong degeneracy between Mν and H0 (see e.g. the redshift of reionization zre.
[143]). WhenM isvaried,thedistancetolastscattering Early CMB measurements of τ from WMAP favored
ν
changes as well. Defining ωb ≡ Ωbh2, ωc ≡ Ωch2, ωm ≡ an early-reionization scenario (zre =10.6±1.1 in the in-
Ω h2, ω ≡ Ω h2, ω ≡ Ω h2, within a flat Universe, stantaneous reionization approximation [150]), requiring
m r r ν ν
this distance is given by: the presence of sources of reionization at z (cid:38) 10. This
result was in tension with observations of Ly-α emit-
(cid:90) zdec dz ters at z (cid:39) 7 (see e.g. [151–155]), that suggest that
χ=c (1,7)
(cid:113)
0 ω (1+z)4+ω (1+z)3+(cid:0)1− ωm(cid:1) reionization ended by z (cid:39) 6. However, the results deliv-
r m h2 ered by the Planck collaboration in the 2015 public data
release, using the large-scale (low-(cid:96)) polarization obser-
where ω = ω +ω +ω . The structure of the CMB
m c b ν vations of the Planck Low Frequency Instrument (LFI)
acoustic peaks leaves little freedom in varying ω and
c [91]incombinationwithPlancktemperatureandlensing
ω . Therefore, for what concerns the distance to the last
b data, indicate that τ =0.066±0.016 [39], corresponding
scattering, a change in M can be compensated essen-
ν to a significantly lower value for the redshift of instan-
tially only by a change in h or, in other words, by a
taneous reionization: z = 8.8+1.2 (see also [156] for an
changeinH0. ThissuggeststhatMν andH0arestrongly assessment of the role roef the c−le1a.1ning procedure on the
anti-correlated: the effect on the CMB of increasing M
ν lower estimate of τ, and [157] for an alternative indirect
can be easily compensated by a decrease in H , and vice
0
method for measuring large-scale polarization and hence
versa.
constrain τ using only small-scale and lensing polariza-
In light of the above discussion, we expect a prior on
tion maps), and thus reducing the need for high-redshift
the Hubble parameter to help pinning down the allowed
sources of reionization [158–162].
valuesofM fromCMBdata. Here,weconsidertwodif-
ν The optical depth to reionization is a crucial quan-
ferentpriorsontheHubbleparameter. Thefirstpriorwe
titywhenconsideringconstraintsonthesumofneutrino
considerisbasedonareanalysisofanoldermeasurement
masses, the reason being that there exist degeneracies
based on the Hubble Space Telescope, the original mea-
surement being H = (73.8±2.4) km s−1Mpc−1 [144]. between τ and Mν (see e.g. [17, 21, 28, 88, 163–165]). If
0 we consider CMB data only (focusing on the TT spec-
The original measurement showed a ∼ 2.4σ tension
trum), an increase in M , which results in a suppression
with the value of H derived from fitting CMB data ν
0 of structure, reduces the smearing of the damping tail.
[39, 70]. The reanalysis, conducted by Efstathiou in
This effect can be compensated by an increase in τ. Due
Ref. [145], used the revised geometric maser distance
to the well-known degeneracy between A and τ from
to NGC4258 of Ref. [146] as a distance anchor. This s
CMB temperature data (which is sensitive to the combi-
reanalysis obtains a more conservative value of H =
(70.6 ± 3.3) km s−1Mpc−1, which agrees with the0ex- nation Ase−2τ), the value of As should also be increased
accordingly. However,thevalueofA alsodeterminesthe
tracted H value from CMB-only within 1σ. We refer to s
0 overall amplitude of the matter power spectrum, which
this prior as H070p6.
is furthermore affected by the presence of massive neu-
The second prior we consider is based on the most re-
trinos, which reduce the small-scale clustering. If, in
cent HST 2.4% determination of the Hubble parameter
inRef.[147]. Thismeasurementbenefitsfrommorethan
twice the number of Cepheid variables used to calibrate
luminosity distances, with respect to the previous anal-
9 We do not include here the latest 3.8% determination of H0
ysis [144], as well as from improved determinations of by the H0LiCOW program. The measurement, based on grav-
distance anchors. The measured value of the Hubble pa- itational time delays of three multiply-imaged quasar systems,
rameter is H0 = (73.02±1.79) km s−1Mpc−1, which is yieldsH0=71.9+−23..40 kms−1Mpc−1 [148].
