Determination of neutrino mass hierarchy by 21 cm line and CMB B-mode polarization observations

0.51 MB·English

by  Yoshihiko Oyama

Determination of neutrino mass hierarchy by 21 cm line and CMB B-mode
polarization observations
Yoshihiko Oyama,1 Akie Shimizu,1 and Kazunori Kohri1,2
1The Graduate University for Advanced Studies (SOKENDAI), 1-1 Oho, Tsukuba 305-0801, Japan
2Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba 305-0801, Japan
(Dated: January 10, 2013)
We focus on the ongoing and future observations for both the 21 cm line and the CMB B-mode
polarizationproducedbyaCMBlensing,andstudytheirsensitivitiestotheeffectivenumberofneu-
trinospecies,thetotalneutrinomass,andtheneutrinomasshierarchy. Wefindthatcombiningthe
CMBobservationswith futuresquarekilometerarraysoptimized for21cmlinesuchasOmniscope
3
can determine the neutrino mass hierarchy at 2σ. We also show that a more feasible combination
1 ofPlanck + Polarbear and SKAcan strongly improveerrorsof theboundson thetotal neutrino
0 mass and theeffectivenumberof neutrinospecies to be∆Σmν ∼0.12 eV and ∆Nν ∼0.38 at 2σ.
2

n
a I. INTRODUCTION tions. ForforecastsbyfutureCMBobservations,seealso
J Refs. [32, 33].

9
Since the discoveries of neutrino masses by Super- In addition, by observing power spectrum of cosmo-

Kamiokande throughneutrino oscillationexperiments in logical 21 cm radiation fluctuation, we will be able to
]
O 1998, the standard model of particle physics has been obtain independent useful information for the neutrino
C forced to change to theoretically include the neutrino masses [34–37]. That is because the 21 cm radiation
masses. is emitted (1) long after the CMB epoch (at a redshift
.
h So far only mass-squared differences of the neutri- z 103 ) and (2) before an onset of the LSS forma-
p nos have been measured by neutrino oscillation exper- tio≪n. The former condition (1) gives us information on
-
o iments, which are reported to be ∆m2 m2 m2 = smaller neutrino mass ( . 0.1 eV). The latter condition
r 7.59+0.19 10−5eV2 [1] and ∆m2 21 ≡m22 −m21 = (2)meanswecantreatonlyalinearregimeofthematter
st 2.43−+00..1231 ×10−3eV2 [2]. However,3a2bs≡olute3va−lues2and perturbation,whichcanbeanalyticallycalculatedunlike
a −0.13 × the LSS case.
theirhierarchicalstructure(normalorinverted)havenot
[
been obtained yet although information for them is in- In actual analyses, it should be essentialthat we com-

3 dispensable to build such new particle physics models. binedataofthe21cmwiththatoftheCMBobservations
v
In particle physics, some new ideas and new future because they complementary constrain cosmological pa-
3
experiments based on those ideas have been proposed rameter spaces each other. Leaving aside minded neu-
2
2 to observe the absolute values and/or the hierarchy of trino parameters, for example, the former is quite sen-
5 neutrinomasses,e.g.,throughtritiumbetadecayinKA- sitive to the dark energy density, but the latter is rela-
. TRINexperiment[3],neutrinolessdouble-betadecay[4], tively insensitive to it. On the other hand, the former
5
0 atmospheric neutrinos in the proposed iron calorimeter has only a mild sensitivity to the normalization of the
2 at INO [5, 6] and the upgrade of the IceCube detector matterperturbation,butthe latterhasanobvioussensi-
1 (PINGU) [7], and long-baseline oscillation experiments, tivitytoitbydefinition. Inpioneeringworksby[36],the
: e.g.,NOνA[8],J-PARCtoKorea(T2KK)[9,10]andOki authors tried to constrain the neutrino mass hierarchy
v
i island (T2KO) [11], and CERN to Super-Kamiokande by combining Planck satellite with future 21 cm obser-
X with high energy (5 GeV) neutrino beam [12]. vations in case of relatively degenerate neutrino masses
r On the other hand, such nonzero neutrino masses af- Σmν 0.3 eV.
a ∼
fectcosmologysignificantlybecauserelativisticneutrinos Here, however,we additionally include analyses of the
prohibittheperturbationfromevolving,duetofollowing CMB B-mode polarization produced by a CMB lensing.
tworeasons. Firstofall,ingeneralrelativity,thedensity This gives us more detailed information on the matter
perturbation of a relativistic particle can hardly evolve powerspectrumat laterepochs, whichmeans it has bet-
at all before it becomes nonrelativistic. Second a rela- ter sensitivities for smaller neutrino masses down to .
tivistic neutrino erases its own density perturbation up 0.1 eV. That is essential to distinguish the normal hi-
toahorizonscalethroughitsfreestreamingateverycos- erarchy from the inverted one. In particular we adopt
mictime. Bymeasuringspectraofdensityperturbations ongoing and future CMB observations such as Polar-
by using observations of cosmic microwave background bear andCMBPol,whichhavemuchbettersensitivities
(CMB) anisotropies and large-scale structure (LSS), we to the B-mode. For ongoing and future 21 cm observa-
couldconstrainthetotalneutrinomassΣm [13–29]and tions,weadoptMWA,SKAandOmniscopeexperiments.
ν
the effective number of neutrino species N [22, 23, 25– We forecast possible allowed parameter regions for both
ν
31]. So far the robust upper bound on Σ m has been neutrinomassesandeffective numberofneutrinospecies
ν
obtainedtobe Σm <0.62eV(95%C.L.)(see Ref.[26] when we use the above-mentioned ongoing and future
ν
and references therein) by these cosmological observa- observations of the 21 cm and the CMB. In particu-
2
lar we propose a nice combination of neutrino masses, Here P (k,z) is defined by δT (k)δT∗(k′)
r =(m m )/Σm to make the mass hierarchy bring (2π)3δ3(2k1 k′)P (k), where δT h Tb T¯bis thie de≡-
ν 3 1 ν 21 b b b
− − ≡ −
to light as is explained in the text. viationfromaspatiallyaveragedbrightnesstemperature
T¯,P isthematterpowerspectrum,andµ=kˆ nˆ isthe
b δδ
·
cosine of the angle between the wave number k and the
II. 21 CM RADIATION line of sight. In principle, T¯b can be calculated although
it depends on the unknown ionization and thermal his-
tory. Therefore we treat T¯ as a free parameter to be
Here we briefly reviewbasic methods to use the 21cm b
measured.
line observations as a cosmological probe. For further
The power spectrum P (k,z) and the comoving wave
details, we refer readers to Refs. [34, 38]. 21
number k are not directly measured by the observations
of 21 cm radiation [45, 46]. Instead, here we define u as
theFourierdualofΘ θ eˆ +θ eˆ +∆feˆ ,whereθ and
i i j j k i
A. Power spectrum of 21 cm radiation ≡
θ determine an angular location on the sky plane and
j
∆f shows the frequency difference from the central red-
The21cmlineoftheneutralhydrogenatomisemitted shift of a z bin. The vector u and its function P (u,z)
21
by hyperfine splitting of the 1S ground state due to an aredirectlymeasuredbythe observations. Relationships
interactionof magnetic moments of protonand electron. betweenu u eˆ +u eˆ +u eˆ andkarerepresentedby
i i j j || k
≡
Spin temperature T of neutral hydrogen gas is defined u u eˆ +u eˆ =d (z)k =2πL/λ,andu =y(z)k .
S ⊥ i i j j A ⊥ || ||
≡
through a ratio between number densities of hydrogen Here ” ” denotes the vector component perpendicular
⊥
atom in the 1S triplet and 1S singlet levels, n1/n0 to the line of sight. ” ” denotes the component in the
≡ k
(g1/g0)exp(−T⋆/TS). where T⋆ ≡ hc/kBλ21 = 0.068 K line of sight. dA(z) is the comoving angular diameter
with λ21( 21 cm) being the wave length of the 21 cm distance to a given redshift. y(z) = λ21(1+z)2/H(z)
≃
line at a rest frame, and g1/g0 = 3 is the ratio of spin means the conversion factor between comoving distance
degeneracy factors of the two levels. A difference be- intervals and frequency intervals ∆f. L is the baseline
tweenthe observed21cmline brightnesstemperature at vector of an interferometer. λ = λ (1 + z) denotes
21
redshift z and the CMB temperature TCMB is given by the observed wave length of the redshifted 21 cm line.
In u space, the power spectrum P (u,z) is defined by
21
T (x) 27x (1+δ ) Ωbh2 0.15 1+z 1/2 hδTb(u)δTb∗(u′)i≡(2π)3δ3(u−u′)P21(u). Then, the re-
b ≈ HI b 0.023 Ω h2 10 lation between P21(u,z) and P21(k,z) is given by
(cid:18) (cid:19)(cid:18) m (cid:19)
TS TCMB H(z)/(1+z) 1
− mK, (1) P (u,z)= P (k,z). (3)
×(cid:18) TS (cid:19)(cid:18) dvk/drk (cid:19) 21 d2A(z)y(z) 21
where xHI is the neutral fraction of hydrogen, δb is the We perform our analyses in terms of P21(u,z) since this
hydrogendensityfluctuation,anddvk/drk isthegradient quantity is directly measurable without any cosmolog-
of the proper velocity along the line of sight due to both ical assumptions. For methods of foreground removals,
the Hubble expansion and the peculiar velocity. seealsorecentdiscussionsaboutindependentcomponent
In general, Tb is sensitive to details of intergalactic analysis(ICA)algorithm,FastICA[47]whichwillbede-
medium(IGM).However,withafewreasonableassump- velopedintermsoftheongoingLOFARobservation[48].
tions we can omit this dependence [39–41]. At an epoch
of reionization (EOR) long after star formation begins,
X-raybackgroundproducedbyearlystellarremnantshas
B. Effects of neutrino masses on power spectrum
heatedtheIGM.ThereforeagaskinetictemperatureT
K
couldbemuchhigherthanthe CMBtemperatureT .
CMB
The massive neutrinos affect the growthof the matter
Furthermore the star formation produces a large back-
densityperturbationmainlyduetofollowingtwophysical
ground of Lyα photons sufficient to couple T to T via
S K
mechanisms.[49]. Firstofall,amassiveneutrinoν (even
theWouthuysen-Fieldeffect[42,43]. Inthisscenario,we i
withitslightmassm .0.3eV)becomesnonrelativistic
are justified in taking TCMB TK TS at z . 10, so νi
≪ ∼ at T m and has contributed to the energy density
that Tb does not depend on TS. ∼ νi
of cold dark matter (CDM), which changes the matter-
In addition, we adopt following assumptions for the
radiation equality epoch and has changed an expansion
EOR in the same manner as [36, 44]. If the IGM is fully
rateoftheuniversesincethattime. Whenwefixthetotal
neutral, fluctuations of the 21 cm radiation arise only
mass of neutrinos Σm (.0.3 eV), only the latter effect
fromdensity fluctuations. In this limit, we canwrite the ν
is effective. Second, the matter density perturbations on
power spectrum of the 21 cm line brightness fluctuation
smallscalescanbesuppressedduetotheneutrinos’free-
P (k) as [36]
21 streaming. As long as neutrinos are relativistic, they
travel at speed of light, and their free-streaming scales
P (k,z)=T¯2(z) 1+µ2 2P (k,z). (2)
21 b δδ are approximately equal to the Hubble horizon. Then
(cid:0) (cid:1)
3
the free-streaming effect erases their own perturbations for observations of the 21 cm signal. In case of Omnis-
within such scales. cope, we take all of antennae distributed with a filled
Compared with the standard ΛCDM models where nucleus according to [45].
threemasslessactiveneutrinosareassumed,wewillcon- To forecast 1 σ errors of cosmological parameters, we
sider two more freedoms. First one is an introduction use the Fisher matrix formalism [56]. For the observa-
of the effective number of neutrino species N , which tions of the 21 cm signal, the Fisher matrix for cosmo-
ν
counts generations of relativistic neutrinos before the logical parameters p is expressed as [57]
i
matter-radiationequality epochand shouldnotbe equal
to three. Second one is the neutrino mass hierarchy. It F21cm = 1 ∂P21(u) ∂P21(u) , (5)
is clear that a change of N affects the epoch of matter- ij [δP (u)]2 ∂p ∂p
ν pixels 21 (cid:18) i (cid:19)(cid:18) j (cid:19)
radiationequality. Ontheotherhand,theneutrinomass X
hierarchy affects both the free-streaming scales and the where we sum only over half the Fourier space. The
expansion rate as was mentioned above [50]. In terms of Fisher matrix determines the errors of the parameter p
i
the observations of the 21 cm signal, the minimum cut- to be
off of the wave number is given by k = 2π/(yB)
min
6 × 10−2hMpc−1 (see Sec IIC) while the wave num∼- ∆pi (F−1)ii. (6)
ber corresponding to the neutrino free-streaming scale ≥
is k . 10−2hMpc−1. Therefore the main feature of q
free The error of the power spectrum measurement δP (u)
21
the modificationofthe matterdensityfluctuationdue to
in a pixel at u consists of a sum of the sample variance
the change of the mass hierarchy comes from the mod-
and the thermal detector noise. It is expressed as
ification of the cosmic expansion when we fix the total
matter density at the present time. P (u)+P (u )
21 N ⊥
δP (u)= , (7)
21 N1/2
c
C. Forecasting methods and interferometers where N = 2πk ∆k ∆k V(z)/(2π)3 is the number of
c ⊥ ⊥ k
independent modes in an annulus summing over the az-
Here we summarize future observations of the 21 cm imuthal angle, V(z) = d (z)2y(z)B FOV is the survey
A
signals emitted at the EOR. We also provide a brief re- volume,Bisthebandwidth,andFOV×( λ2/A )denotes
e
≈
view ofthe Fishermatrixformalismfor the 21cmobser- the field of view of the interferometer. For each exper-
vations. We consider MWA [51], SKA [52] and Omnis- iment, we take account of the presence of foregrounds
cope [53] for future observations. The summary of the and adopt a cutoff at 2π/(yB) k [57]. We also take
k
≤
detailed specifications is listed in Table I. Each interfer- a maximum value of k to be k = 3hMpc−1 beyond
max
ometer has its own different noise power spectrum, which nonlinear effects become important and exclude
all information for k <k.
max
λ2(z)Tsys(z) 2 1 For each experiment, we assume a specific redshift
P (u ,z)= , (4)
N ⊥ A (z) t n(u ) range as follows [44]. We consider Ly-α forests in ab-
(cid:18) e (cid:19) 0 ⊥
sorptionspectra ofquasarsandassume that reionization
which affects sensitivities to the 21 cm signals. Here occurred sharply at z = 7.5. For an upper limit on the
T 280[(1 + z)/7.4]2.3K is the system tempera- accessible redshift range, we take it to be z . 10 be-
sys
≃
ture [54], t is the total observation time, and A is the cause of increasing foregrounds and uncertainty in the
0 e
effective collecting area of each antenna tile. The effect spin temperature at higher redshifts. For the above rea-
of the configuration of the antennae is encoded in the sons,weassumethattheobservedredshiftrangeofEOR
number density of baseline n(u ). In order to calculate is7.8 10.2. OnlyforMWA,weassumeasingleredshift
⊥
−
n(u ), we have to assume a realization of antenna den- slice centered at z =8.
⊥
sity profiles for each interferometer. For MWA, we take When we calculate the Fisher matrix, we choose
500 antennae distributed with a filled nucleus of radius the following basic set of cosmological parameters:
20 m surrounded by the remainder of the antennae dis- the energy density of matter Ω h2, baryon Ω h2,
m b
tributed with an r−2 antenna density profile out to 750 dark energy Ω , the scalar spectral index n , the
Λ s
m [55]. For SKA, we distribute 20% of a total of 5000 scalar fluctuation amplitude A (the pivot scale is
s
antennae within a 1 km radius and take the antennae taken to be k = 0.002 Mpc−1), the reionization
pivot
distributedwithanucleussurroundedbyanr−2 antenna optical depth τ, Helium fraction Y , and the to-
He
density profile in the same way as those of MWA. These tal neutrino mass Σm = m + m + m . Fidu-
ν 1 2 3
antennae are surrounded by a further 30% of the total cial values of these parameters (except for Σm )
ν
antennae in a uniform density annulus of outer radius 6 are adopted to be (Ω h2,Ω h2,Ω ,n ,A ,τ,Y ) =
m b Λ s s He
km[36]. The remainderofthe antennaeisdistributed at (0.147,0.023,0.7,0.95,24 10−10,0.1,0.24). We set a
×
largerdistancessparselyto be usefulforpowerspectrum range of the fiducial value of Σm to be Σm = 0.05
ν ν
−
measurements. Finally, we consider Omniscope that is 0.3 eV. Besides these parameters, the brightness tem-
afuturesquare-kilometercollectingareaarrayoptimized perature of 21 cm radiation T¯(z) can be taken as a free
b
4
Experiment Nant Ae([mz=2]8) L[mmi]n L[kmmax] [FdOegV2] z Experiment [GνHz] [µ∆KT−T′] [µ∆KP−P′] θFW[′]HM fsky
MWA 500 14 4 1.5 π162 7.8-8.2 Planck [32] 70 137 195 14 0.65
SKA 5000 120 10 5 π5.62 7.8-10.2 100 64.6 104 9.5 0.65
Omniscope 106 1 1 1 2.1×104 7.8-10.2 143 42.6 80.9 7.1 0.65
Polarbear [64] 150 - 8 3.5 0.017
TABLE I: Specifications for each interferometers. Lmin CMBPol (EPIC-2m) [65] 70 2.96 4.19 11 0.65
(Lmax) is the minimum (maximum) baseline. For MWA, we 100 2.29 3.24 8 0.65
assume a single redshift slice centered at z = 8. For SKA
150 2.21 3.13 5 0.65
andOmniscope,theobservedredshiftrangeisz=7.8−10.3,
andwedividetherangeintofiveredshiftsliceswiththickness
∆z ≈ 0.5. For each experiment, bandwidth is B = 8 MHz, TABLE II: Experimental specifications of Planck, Polar-
bear and CMBPol assumed in this study. Here ν is the
and weassumed observations for 8000 h on two places in the
spkoyr.tioWnaelatsosuλm2efotrhMatWthAeaenffdecStKivAe.cFololercOtimngniasrceoapeA,eboisthprAoe- o1b′×se1r′vpaitxioenl,∆frePqPuiesntchye,p∆oTlaTriizsatthioentesmenpseitriavtiutyrepseern1s′i×tiv1′itpyixpeelr,
and FOV are fixed. θFWHM is the angular resolution defined as the full width at
half-maximum, and f is the observed fraction of the sky.
sky
We use ℓmax = 2000 for Polarbear, and ℓmax = 2500 for
Planck and CMBPol.
parameter. In this study, we adopt the fiducial values
of T¯(z) to be T¯(8),T¯(8.5),T¯(9),T¯(9.5),T¯(10) =
b b b b b b
(26,26,27,27,28)in units of mK.
(cid:0) (cid:1) III. CMB
Additionally, we separately study following two cases:
(A) Effective number of neutrino species A. CMB and neutrino
We add one more parameter of the effective number CMB power spectra are sensitive to neutrino masses.
of neutrino species Nν to the fiducial set of the param- Therearethreeeffectsthatprovidedetectablesignalsfor
eters. The fiducial value of this parameter is set to be the neutrino masses: (1) the transition from relativistic
Nν =3.04. In this analysis, we assumed three species of neutrinotononrelativisticone,(2)smoothingofthemat-
massive neutrinos + an extra relativistic component. terperturbationbyitsfree-streaminginsmallscales,and
(3)variationoflensedCMBpowerspectra. FutureCMB
experiments are expected to set stringent constraints on
(B) Neutrino mass hierarchy
the sum of neutrino masses [49, 62]. In particular, the
lasteffectisuniqueintheCMBB-modepolarizationpro-
In a cosmological context, many different parameter- ducedbyaCMBlensing. Hereweproposetocombinethe
izations of the mass hierarchy have been proposed [58– CMB experiments with the 21 cm line observations. As
61]. Weadoptr (m m )/Σm [61]asanadditional wewillseeinSectionIV,thecombinedapproachresolves
ν 3 1 ν
≡ −
parameter to nicely discriminate the true neutrino-mass degeneracyamongsomekeycosmologicalparametersand
hierarchypatternfromtheotherbetweenthenormaland ismorepowerfulthanindividualCMBmeasurements. In
inverted hierarchies. The normal and inverted mass hi- addition, it is notable that we are able to detect the ef-
erarchies mean m < m m and m m < m , fectivenumberofneutrinospecies[63]anddeterminethe
1 2 3 3 1 2
≪ ≪
respectively. We add r to the fiducial set of the pa- neutrino mass hierarchy.
ν
rameters. r becomes positive (negative) for the nor-
ν
mal (inverted) hierarchy. It should be a remarkably nice
point that the difference between r ’s of these two hier-
ν
archies becomes larger as the total mass Σm becomes
ν B. Sensitivity and Analysis of the CMB
smaller. Therefore rν is particularly useful for distin- experiments
guishing the mass hierarchy. In Fig 2 we plot behaviors
of r as a function of Σm . Note that there is a lowest
ν ν In this study, we choose Planck [32], Polarbear [64]
value of Σm which depends on a type of the hierarchies
ν
andCMBPol[65]asexamplesofCMB experiments. Ex-
by the neutrino oscillationexperiments, i.e., 0.1 eV for
∼ perimental specifications we assumed are summarized in
the inverted hierarchy and 0.05 eV for the normal hi-
∼ Table II.
erarchy. Therefore, if we could obtain a clear constraint
like 0.05 eV Σm 0.10 eV, the hierarchy should be In our analysis for the CMB, we also take the same
ν
obviously no≤rmal wi≪thout any ambiguities. As will be fiducial model (Ω h2,Ωh2,Ω ,n ,A ,τ,Y ) as that of
m Λ s s He
shown later, however, we can discriminate the hierarchy the 21 cm line experiments (see previous section). We
even when 0.10 eV .Σm . evaluate errors of cosmological parameters by using
ν
5
Fisher matrix, which is given by [56] Σmν [eV] Nν
Fiducial value 0.05 3.04
(2ℓ+1)
FCijMB = 2 fsky Planck + Polarbear 0.146 0.282
Xl + MWA 0.114 0.240
∂C ∂C
Trace C−1 ℓC−1 ℓ . (8) + SKA 0.0592 0.189
× (cid:20) ℓ ∂pi ℓ ∂pj (cid:21) + Omniscope 0.0226 0.0753
CMBPol 0.0538 0.0929
Here C is a covariance matrix constructed by using
ℓ
CMBpowerspectraCX(X=TT,EE,TE),deflectionan- + SKA 0.0276 0.0827
ℓ
glespectrumCdd,crosscorrelationbetweentemperature + Omniscope 0.0131 0.0438
ℓ
and deflection angle CTd, and noise power spectra NX
ℓ ℓ
and Ndd, where Cdd is calculated by a lensing poten- TABLE III: 1-σ experimental uncertainties of Σmν and Nν,
tial [68ℓ] and is relaℓted with CBB 1. We compute Ndd definedby ∆pi=p(F−1)ii.
ℓ ℓ
by using a public code FUTURCMB [69] which adopts
the quadratic estimator [68]. In this algorithm, Ndd is
ℓ
reconstructedbyNY (Y=TT,EE,BB). The covariance
ℓ (1) 2 ℓ<25, 2000<ℓ 2500
matrix in the Fisher matrix is expressed as ≤ ≤
NY,Planck+PB =NY,Planck (13)
CTT+NTT CTE CTd ℓ ℓ
ℓ ℓ ℓ ℓ
Cℓ = CℓTE CℓEE+NℓEE 0 . (9) (2) 25 ℓ 2000
CTd 0 Cdd+Ndd ≤ ≤
 ℓ ℓ ℓ 
  NY,Planck+PB =[1/NY,Planck+1/NY,PB]−1 (14)
where NY is expressed by using both a beam size ℓ ℓ ℓ
ℓ
σbeam(ν) = θFWHM(ν)/√8ln2 and an instrumental sen- Since Polarbear observes only CMB polarizations,
sitivity ∆Y(ν) to be the temperature noise powerspectrumNTT,Planck+PB is
ℓ
−1 equal to NTT,Planck.
1 ℓ
NY = , (10) In order to combine the CMB experiments with the
ℓ " ν NℓY(ν)# 21cmline experiments,wecalculatethe combinedfisher
X
matrix to be
where
F21cm+CMB FCMB+F21cm. (15)
NY(ν)=∆2(ν)exp ℓ(ℓ+1)σ2 (ν) . (11) ≃
ℓ Y beam
Herewedidnotuseinformationforapossiblecorrelation
For∆ (ν)and∆ (ν),wec(cid:2)ommonlyuse∆ (cid:3)(ν)listed
EE BB PP between fluctuations of the 21 cm and the CMB.
in Table II. Ndd is calculated by NTT, NEE, and NBB.
ℓ ℓ ℓ ℓ
In case of Planck and Polarbear, we combine both
the experiments,andassumethat the 1.7%regionofthe
IV. RESULTS
whole sky is observed by both the experiments, and the
remaining 63.3%(= 65% 1.7%) region is observed by
− Inthissection,wenumericallyevaluatehowwecande-
Planckonly. ThereforeweevaluateatotalFishermatrix
FCMB by summing the two Fisher matrices, termine(A)theeffectivenumberofneutrinospecies,and
(B)theneutrinomasshierarchy,bycombiningthe21cm
FCMB =FPlanck(f =0.633) line observations (MWA, SKA, or Omniscope) with the
sky
CMBexperiments (Planck+ Polarbear,orCMBPol).
+FPlanck+PB(f =0.017), (12)
sky To obtainFisher matrices we use CAMB [66,67] for cal-
culationsofCMBanisotropiesC andmatterpowerspec-
whereFPlanck isthe Fishermatrixofthe regionobserved l
tra P .
by Planck only and FPlanck+PB is that by both Planck δδ
and Polarbear.
In addition, we calculate noise power spectra
NY,Planck+PB of the CMB polarization (Y= EE or BB) A. Constraints on Nν
ℓ
in FPlanck+PB with the following operation.
In Fig.1, we plot contours of 90%confidence levels
(C.L.) forecasts in Σm -N plane. The fiducial values
ν ν
of neutrino parameters, N and Σm , are taken to be
ν ν
N =3.04 and Σm =0.1 eV (upper two panels), which
1 ByperformingapubliccodeHALOFIT[66,67],wehavechecked ν ν
corresponds to the lowest value of the inverted hierar-
thatmodificationsbyincludingnonlineareffectsforevolutionsof
chy model, or Σm =0.05 eV (lower two panels), which
thematterpowerspectrumaremuchsmallerthantypicalerrors ν
inouranalyses andnegligibleforparameter fittings. corresponds to the lowest value of the normal hierarchy
6
FIG. 1: Contours of 90% C.L. forecasts in Σmν-Nν plane, by adopting Planck + Polarbear + each 21 cm experiment (left
twopanels),orCMBPol+each21cmexperiment(righttwopanels). Fiducialvaluesofneutrinoparameters,Nν andΣmν,are
taken to be Nν =3.04 and Σmν =0.1 eV (for upper two panels) or Σmν =0.05 eV (for lower two panels). The dashed line
means the constraint obtained by only a CMB observation such as Planck + Polarbear alone (left two panels), or CMBPol
alone (right two panels). The severer constrains are obtained by combining theCMB with a 21 cm observation such as MWA
(outer solid, only for left panels), SKA (middlesolid), and Omniscope (innersolid), respectively.
model. Adding the 21 cm experiments to the CMB ex- more severely.
periment,weseethatthereisasubstantialimprovement The case of Σm = 0.1 eV to be fiducial (upper two
ν
for the sensitivities to Σm and N . That is because panels) corresponds to the lowest value for the inverted
ν ν
severalparameterdegeneraciesarebrokenbythosecom- hierarchywhenweuseoscillationdata. Thenitisnotable
binations, e.g., in particular T and As were completely that CMBPol + SKA can detect the nonzero neutrino
b
degenerate only in 21 cm line measurements. Therefore mass. Of course, Planck + Polarbear + Omniscope
it is essential to add the CMB to the 21 cm experiment and CMBPol + Omniscope can obviously do the same
to be vital for breaking those parameter degeneracies. job.
If each CMB experiment is combined with SKA or On the other hand, the case of Σmν = 0.05 eV to
Omniscope, the corresponding constraint can be signifi- be fiducial (lower two panels), which corresponds to the
cantly improved. We showed numerical values of those lowestvalueforthenormalhierarchy,onlyPlanck+Po-
errors in Table III in case that the fiducial values are larbear + Omniscope or CMBPol + Omniscope can
taken to be N = 3.04 and Σm = 0.05 eV. On the detect the nonzero neutrino mass.
ν ν
otherhand,comparingthosevalueswiththecurrentbest
bounds for Σm + N model, which give Σm < 0.89
ν ν ν
eV and N = 4.47+1.82 obtained by CMB (WMAP) + B. Constraints on neutrino mass hierarchy
ν −1.74
HST(Hubble Space Telescope) + BAO [28], we find that
the ongoing and future 21 cm line + the CMB obser- Next we discuss if we will be able to determine the
vation will be able to constrain those parameters much neutrinomasshierarchiesbyusingthoseongoingandfu-
7
FIG. 2: Forecasts of 2σ errors on rν = (m3−m1)/Σmν constrained by both the 21 cm and the CMB observations in case
of the inverted hierarchy to be fiducial (left), and the normal hierarchy to be fiducial (right). The constrains are obtained
by combining Omniscope with Planck + Polarbear (thick dashed lines), and Omniscope with CMBPol (thick solid lines),
respectively. Allowed parameters on rν by neutrino oscillation experiments are plotted as two bands for the inverted and the
normal hierarchies, respectively (the name of each hierarchy is written in the close vicinity of the line). The solid line inside
theband is the fiducial valueof rν as a function of Σmν.
ture 21 cm and CMB observations. In Fig. 2 we plot Fig.3,weplotcontoursof90%C.L.inΣm r planein
ν ν
−
2σ errors of the parameter r (m m )/Σm con- order to show errors of Σm along the x-axis for typical
ν 3 1 ν ν
≡ −
strained by both the 21 cm and the CMB observations fiducial values. As is clearly shown in Fig. 2, actually
in caseofthe invertedhierarchyto be fiducial (left), and those combinations of the observations will be able to
the normal hierarchy to be fiducial (right). It is notable determine the neutrino mass hierarchy to be inverted or
that the difference between r ’s of these two hierarchies normalforΣm .0.13eVorΣm .0.1eVat90%C.L.,
ν ν ν
becomes larger as the total mass Σm becomes smaller. respectively. Althoughthedeterminationispossibleonly
ν
Therefore, r is quite useful to distinguish a true mass ataroundΣm . (0.1)eV,thoseresultsshouldberea-
ν ν
O
hierarchy from the other. Allowed parameters on r by sonable. That is because a precise discrimination of the
ν
neutrinooscillationexperimentsareplottedastwobands mass hierarchyitself may have no meaning if the masses
for the invertedand the normalhierarchies,respectively. are highly degenerate, i.e., if Σm 0.1 0.3 eV.
ν
≫ −
The thin solid lines inside the bands are the experimen- OnceaclearsignatureΣm 0.1eVweredetermined
ν
≪
tal mean values by oscillations, one of which is taken byobservationsorexperiments,itshouldbeobviousthat
to be a corresponding fiducial value of r as a function the hierarchy must be normal without any ambiguities.
ν
of Σm in eachanalysis. The constrainsare obtained by On the other hand if the hierarchy were inverted, we
ν
combiningOmniscopewithPlanck+Polarbear(thick could not determine it only by using Σm . However, it
ν
dashed lines) and Omniscope with CMBPol (thick solid isremarkablethatourmethodisquiteusefulbecausewe
lines), respectively. candiscriminate the hierarchyfromthe other evenif the
For 0.3 eV . Σmν the mass eigenvalues m1, m2, and fiducial values were Σmν & 0.1 eV for both the normal
m are almost degenerate. Therefore the difference be- and inverted cases. This is clearly shown in Fig 3. In
3
tween two hierarchies has little influence on the matter case that a fiducial value of Σmν is taken to be the low-
power spectrum. Therefore the constraints on r are est values in neutrino oscillation experiments, the upper
ν
significantly weak compared with the difference between left(right)figureindicatesthatevenCMBPol+SKAcan
them, and then we cannot distinguish the true hierarchy discriminate the inverted (normal) mass hierarchy from
from the other. the normal (inverted) one.
Onthe otherhandhowever,the differenceincreasesas
Σm decreasesdowntom 0.1eV. Byusingthisprop-
ν ν
erty,theCMB(Planck+Po∼larbearorCMBPol)+the V. CONCLUSIONS
21 cm (Omniscope) observations can constrain the neu-
trinomasshierarchyseverely. TypicallyerrorsofΣm at We have studied how we can constrain effective num-
ν
around Σm = 0.1 eV are given by ∆Σm = 0.0087 eV ber of neutrino species N , total neutrino masses Σm ,
ν ν ν ν
for Planck + Polarbear, and ∆Σm = 0.0069 eV for and neutrino mass hierarchy by using the 21 cm obser-
ν
CMBPol at 1σ, respectively. Therefore, the error of the vations(MWA,SKA,andOmniscope)andthe CMBob-
x-axis is negligible compared with that of y-axis. In servations (Planck, Polarbear, and CMBPol). It is
8
FIG. 3: Contours of 90% C.L. forecasts in Σmν-rν plane, by adopting CMBPol + each 21 cm experiment. Fiducial value
of Σmν and the mass hierarchy (diagonal cross) are taken to be: Σmν = 0.1 eV and the inverted (for left upper panel),
Σmν =0.12 eVandtheinverted(forleft lower panel),Σmν =0.06 eVandthenormal(forrightupperpanel),Σmν =0.1eV
and thenormal (for right lower panel). The short dashed lines mean theconstraints obtained by only a CMBPol observation,
and the long dashed line means the one by a SKA + CMBPol observation for 24000 h on four places in the sky. The outer
(inner) solid line means combining the CMBPol with SKA (Omniscope). Allowed parameters on rν by neutrino oscillation
experimentsare plotted in thesame way of Fig. 2.
essential to combine the 21 cm with the CMB B-mode archy from the other by constraining r . As was clearly
ν
polarization produced by a CMB lensing to break var- showninFig.2,thecombinationsoftheCMB(Planck+
ious degeneracies in cosmological parameters when we Polarbear or CMBPol)+ the 21 cm (Omniscope) will
perform multiple-parameter fittings. be able to determine the hierarchyto be invertedornor-
mal for Σm . 0.13 eV or . 0.1 eV at 2σ, respectively.
About the constraints on Σm –N plane, for a fidu- ν
ν ν
Furthermore, if the fiducial value of Σm is taken to be
cial value Σm = 0.1 eV which corresponds to the low- ν
ν
the lowest value in the neutrino oscillation experiments,
est value in the inverted hierarchy, we have found that
CMBPol + SKA, Planck + Polarbear + Omniscope evenCMBPol+SKA candetermine the masshierarchy.
and CMBPol + Omniscope can detect the nonzero neu-
trino mass. For a fiducial value Σm = 0.05 eV, which
ν In this study we have taken the simplified model of
corresponds to the lowest value in the normal hierarchy,
reionization. In case of more likely detailed modeling of
Planck+Polarbear + OmniscopeorCMBPol+ Om-
reionization [45], it was pointed out that the constraints
niscope can detect the nonzero neutrino mass.
on cosmological parameters may moderately change at
As for the determination of the neutrino mass hier- 10 50 %. Fortunately, this effect is comparatively
∼ −
archy, we have proposed a new parameter r = (m small and should not be fatal to constrain the neutrino
ν 3
−
m )/Σm and studied how to discriminate a true hier- mass hierarchy in the current analyses.
1 ν
9
VI. ACKNOWLEDGMENT sions. ThisworkissupportedinpartbyGrant-in-Aidfor
Scientific research from the Ministry of Education, Sci-
WethankM.Hazumi,K.Ioka,H.Kodama,R.Nagata, ence, Sports, and Culture (MEXT), Japan, No. 245516
T.Namikawa,M.TakadaandA.Taruyaforusefuldiscus- (A.S.), Nos. 21111006,22244030,and 23540327(K.K.).
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