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

We focus on the ongoing and future observations for both the 21 cm line and the CMB B-mode polarization produced by a CMB lensing, and study their sensitivities to the effective number of neutrino species, the total neutrino mass, and the neutrino mass hierarchy. We find that combining the CMB observations with future square kilometer arrays optimized for 21 cm line such as Omniscope can determine the neutrino mass hierarchy at 2 sigma. We also show that a more feasible combination of Planck + POLARBEAR and SKA can strongly improve errors of the bounds on the total neutrino mass and the effective number of neutrino species to be Delta Sigma m_nu ~ 0.12 eV and Delta N_nu ~ 0.38 at 2 sigma.


Introduction
Since the discoveries of neutrino masses by Super-Kamiokande through neutrino oscillation experiments in 1998, the standard model of particle physics has been forced to change to theoretically include the neutrino masses.
On the other hand, such nonzero neutrino masses affect cosmology significantly because relativistic neutrinos prohibit the perturbation from evolving, due to following two reasons. First of all, * Corresponding author.
In addition, by observing power spectrum of cosmological 21 cm radiation fluctuation, we will be able to obtain independent useful information for the neutrino masses [34][35][36][37]. That is because the 21 cm radiation is emitted (1) long after the CMB epoch (at a redshift z 10 3 ) and (2) before an onset of the LSS formation. The former condition (1) gives us information on smaller neutrino mass ( 0.1 eV). The latter condition (2) means we can treat only a linear regime of the matter perturbation, which can be analytically calculated unlike the LSS case.
In actual analyses, it should be essential that we combine data of the 21 cm with that of the CMB observations because they complementary constrain cosmological parameter spaces each other. Leaving aside minded neutrino parameters, for example, the former is quite sensitive to the dark energy density, but the latter is relatively insensitive to it. On the other hand, the former has only a mild sensitivity to the normalization of the matter perturbation, but the latter has an obvious sensitivity to it by definition. In pioneering works by [36], the authors tried to constrain the Here, however, we additionally include analyses of the CMB B-mode polarization produced by a CMB lensing. This gives us more detailed information on the matter power spectrum at later epochs, which means it has better sensitivities for smaller neutrino masses down to 0.1 eV. That is essential to distinguish the normal hierarchy from the inverted one. In particular we adopt ongoing and future CMB observations such as Polarbear and CMBPol, which have much better sensitivities to the B-mode. For ongoing and future 21 cm observations, we adopt MWA, SKA and Omniscope experiments. We forecast possible allowed parameter regions for both neutrino masses and effective number of neutrino species when we use the above-mentioned ongoing and future observations of the 21 cm and the CMB. In particular we propose a nice combination of neutrino masses, r ν = (m 3 − m 1 )/Σm ν to make the mass hierarchy bring to light as is explained in the text.

21 cm radiation
Here we briefly review basic methods to use the 21 cm line observations as a cosmological probe. For further details, we refer readers to Refs. [34,38].

Power spectrum of 21 cm radiation
The 21 cm line of the neutral hydrogen atom is emitted by hyperfine splitting of the 1S ground state due to an interaction of magnetic moments of proton and electron. Spin temperature T S of neutral hydrogen gas is defined through a ratio between number densities of hydrogen atom in the 1S triplet and 1S singlet levels, n 1 /n 0 ≡ (g 1 /g 0 ) exp(−T /T S ). where T ≡ hc/k B λ 21 = 0.068 K with λ 21 ( 21 cm) being the wave length of the 21 cm line at a rest frame, and g 1 /g 0 = 3 is the ratio of spin degeneracy factors of the two levels. A difference between the observed 21 cm line brightness temperature at redshift z and the CMB temperature T CMB is given by where x HI is the neutral fraction of hydrogen, δ b is the hydrogen density fluctuation, and dv /dr is the gradient of the proper velocity along the line of sight due to both the Hubble expansion and the peculiar velocity. In general, T b is sensitive to details of intergalactic medium (IGM). However, with a few reasonable assumptions we can omit this dependence [39][40][41]. At an epoch of reionization (EOR) long after star formation begins, X-ray background produced by early stellar remnants has heated the IGM. Therefore a gas kinetic temperature T K could be much higher than the CMB temperature T CMB . Furthermore the star formation produces a large background of Lyα photons sufficient to couple T S to T K via the Wouthuysen-Field effect [42,43]. In this scenario, we are justified in taking T CMB T K ∼ T S at z 10, so that T b does not depend on T S .
In addition, we adopt following assumptions for the EOR in the same manner as [36,44]. If the IGM is fully neutral, fluctuations of the 21 cm radiation arise only from density fluctuations. In this limit, we can write the power spectrum of the 21 cm line brightness fluctuation P 21 (k) as [36] (2) is the deviation from a spatially averaged brightness temperatureT b , P δδ is the matter power spectrum, and μ =k ·n is the cosine of the angle between the wave number k and the line of sight. In principle,T b can be calculated although it depends on the unknown ionization and thermal history. Therefore we treatT b as a free parameter to be measured.
The power spectrum P 21 (k, z) and the comoving wave number k are not directly measured by the observations of 21 cm radiation [45,46]. Instead, here we define u as the Fourier dual of Θ ≡ θ iêi + θ jê j + fê k , where θ i and θ j determine an angular location on the sky plane and f shows the frequency difference from the central redshift of a z bin. The vector u and its function P 21 (u, z) are directly measured by the observations. Relationships between u ≡ u iêi + u jê j + u ê k and k are represented by u ⊥ ≡ u iêi + u jê j = d A (z)k ⊥ = 2π L/λ, and u = y(z)k . Here "⊥" denotes the vector component perpendicular to the line of sight. " " denotes the component in the line of sight. d A (z) is the comoving angular diameter distance to a given redshift.
means the conversion factor between comoving distance intervals and frequency intervals f . L is the baseline vector of an interferometer. λ = λ 21 (1 + z) denotes the observed wave length of the redshifted 21 cm line. In u space, the power spectrum . Then, the relation between P 21 (u, z) and P 21 (k, z) is given by We perform our analyses in terms of P 21 (u, z) since this quantity is directly measurable without any cosmological assumptions. For methods of foreground removals, see also recent discussions about independent component analysis (ICA) algorithm, FastICA [47] which will be developed in terms of the ongoing LOFAR observation [48].

Effects of neutrino masses on power spectrum
The massive neutrinos affect the growth of the matter density perturbation mainly due to following two physical mechanisms. [49]. First of all, a massive neutrino ν i (even with its light mass m ν i 0.3 eV) becomes nonrelativistic at T ∼ m ν i and has contributed to the energy density of cold dark matter (CDM), which changes the matter-radiation equality epoch and has changed an expansion rate of the universe since that time.
When we fix the total mass of neutrinos Σm ν ( 0.3 eV), only the latter effect is effective. Second, the matter density perturbations on small scales can be suppressed due to the neutrinos' free-streaming. As long as neutrinos are relativistic, they travel at speed of light, and their free-streaming scales are approximately equal to the Hubble horizon. Then the free-streaming effect erases their own perturbations within such scales.
Compared with the standard ΛCDM models where three massless active neutrinos are assumed, we will consider two more freedoms. First one is an introduction of the effective number of neutrino species N ν , which counts generations of relativistic neutrinos before the matter-radiation equality epoch and should not be equal to three. Second one is the neutrino mass hierarchy. It is clear that a change of N ν affects the epoch of matterradiation equality. On the other hand, the neutrino mass hierarchy affects both the free-streaming scales and the expansion rate as was mentioned above [50]. In terms of the observations of the 21 cm signal, the minimum cutoff of the wave number is given wave number corresponding to the neutrino free-streaming scale Table 1 Specifications for each interferometers. L min (L max ) is the minimum (maximum) baseline. For MWA, we assume a single redshift slice centered at z = 8. For SKA and Omniscope, the observed redshift range is z = 7.8-10.3, and we divide the range into five redshift slices with thickness z ≈ 0.5. For each experiment, bandwidth is B = 8 MHz, and we assumed observations for 8000 h on two places in the sky. We assume that the effective collecting area A e is proportional to λ 2  is k free 10 −2 h Mpc −1 . Therefore the main feature of the modification of the matter density fluctuation due to the change of the mass hierarchy comes from the modification of the cosmic expansion when we fix the total matter density at the present time.

Forecasting methods and interferometers
Here we summarize future observations of the 21 cm signals emitted at the EOR. We also provide a brief review of the Fisher matrix formalism for the 21 cm observations. We consider MWA [51], SKA [52] and Omniscope [53] for future observations. The summary of the detailed specifications is listed in Table 1. Each interferometer has its own different noise power spectrum, which affects sensitivities to the 21 cm signals. Here T sys 3 K is the system temperature [54], t 0 is the total observation time, and A e is the effective collecting area of each antenna tile. The effect of the configuration of the antennae is encoded in the number density of baseline n(u ⊥ ). In order to calculate n(u ⊥ ), we have to assume a realization of antenna density profiles for each interferometer. For MWA, we take 500 antennae distributed with a filled nucleus of radius 20 m surrounded by the remainder of the antennae distributed with an r −2 antenna density profile out to 750 m [55]. For SKA, we distribute 20% of a total of 5000 antennae within a 1 km radius and take the antennae distributed with a nucleus surrounded by an r −2 antenna density profile in the same way as those of MWA. These antennae are surrounded by a further 30% of the total antennae in a uniform density annulus of outer radius 6 km [36]. The remainder of the antennae is distributed at larger distances sparsely to be useful for power spectrum measurements. Finally, we consider Omniscope that is a future square-kilometer collecting area array optimized for observations of the 21 cm signal. In case of Omniscope, we take all of antennae distributed with a filled nucleus according to [45].
To forecast 1 σ errors of cosmological parameters, we use the Fisher matrix formalism [56]. For the observations of the 21 cm signal, the Fisher matrix for cosmological parameters p i is expressed as [57] F 21 cm where we sum only over half the Fourier space. The Fisher matrix determines the errors of the parameter p i to be The error of the power spectrum measurement δ P 21 (u) in a pixel at u consists of a sum of the sample variance and the thermal detector noise. It is expressed as where N c = 2πk ⊥ k ⊥ k V (z)/(2π ) 3 is the number of independent modes in an annulus summing over the azimuthal angle, is the bandwidth, and FOV (≈ λ 2 /A e ) denotes the field of view of the interferometer. For each experiment, we take account of the presence of foregrounds and adopt a cutoff at 2π /(yB) k [57]. We also take a maximum value of k to be k max = 3h Mpc −1 beyond which nonlinear effects become important and exclude all information for For each experiment, we assume a specific redshift range as follows [44]. We consider Ly-α forests in absorption spectra of quasars and assume that reionization occurred sharply at z = 7.5.
For an upper limit on the accessible redshift range, we take it to be z 10 because of increasing foregrounds and uncertainty in the spin temperature at higher redshifts. For the above reasons, we assume that the observed redshift range of EOR is 7.8-10.2. Only for MWA, we assume a single redshift slice centered at z = 8. Additionally, we separately study following two cases: (A) Effective number of neutrino species.
We add one more parameter of the effective number of neutrino species N ν to the fiducial set of the parameters. The fiducial value of this parameter is set to be N ν = 3.04. In this analysis, we assumed three species of massive neutrinos + an extra relativistic component.

(B) Neutrino mass hierarchy.
In a cosmological context, many different parameterizations of the mass hierarchy have been proposed [58][59][60][61]. We adopt r ν ≡ (m 3 − m 1 )/Σm ν [61] as an additional parameter to nicely discriminate the true neutrino-mass hierarchy pattern from the other between the normal and inverted hierarchies. The normal and inverted mass hierarchies mean m 1 < m 2 m 3 and m 3 m 1 < m 2 , respectively. We add r ν to the fiducial set of the parameters. r ν becomes positive (negative) for the normal (inverted) hierarchy. It should be a remarkably nice point that the difference between r ν 's of these two hierarchies becomes larger as the total mass Σm ν becomes smaller. Therefore r ν is particularly useful for distinguishing the mass hierarchy. In Fig. 2 we plot behaviors of r ν as a function of Σm ν . Note that there is a lowest value of Σm ν which depends on a type of the hierarchies by the neutrino oscillation experiments, i.e., ∼ 0.1 eV for the inverted hierarchy and ∼ 0.05 eV for the normal hierarchy. Therefore, if we could obtain a clear constraint like 0.05 eV Σm ν 0.10 eV, the hierarchy Table 2 Experimental specifications of Planck, Polarbear and CMBPol assumed in this study.
Here ν is the observation frequency, TT is the temperature sensitivity per 1 × 1 pixel, PP is the polarization sensitivity per 1 × 1 pixel, θ FWHM is the angular resolution defined as the full width at half-maximum, and f sky is the observed fraction of the sky. We use max = 2000 for Polarbear, and max = 2500 for Planck and should be obviously normal without any ambiguities. As will be shown later, however, we can discriminate the hierarchy even when 0.10 eV Σm ν .

CMB and neutrino
CMB power spectra are sensitive to neutrino masses. There are three effects that provide detectable signals for the neutrino masses: (1) the transition from relativistic neutrino to nonrelativistic one, (2) smoothing of the matter perturbation by its freestreaming in small scales, and (3) variation of lensed CMB power spectra. Future CMB experiments are expected to set stringent constraints on the sum of neutrino masses [49,62]. In particular, the last effect is unique in the CMB B-mode polarization produced by a CMB lensing. Here we propose to combine the CMB experiments with the 21 cm line observations. As we will see in Section 4, the combined approach resolves degeneracy among some key cosmological parameters and is more powerful than individual CMB measurements. In addition, it is notable that we are able to detect the effective number of neutrino species [63] and determine the neutrino mass hierarchy.

Sensitivity and analysis of the CMB experiments
In this study, we choose Planck [32], Polarbear [64] and CMBPol [65] as examples of CMB experiments. Experimental specifications we assumed are summarized in Table 2.
In our analysis for the CMB, we also take the same fiducial model (Ω m h 2 , Ωh 2 , Ω Λ , n s , A s , τ , Y He ) as that of the 21 cm line experiments (see previous section). We evaluate errors of cosmological parameters by using Fisher matrix, which is given by [56] Here C is a covariance matrix constructed by using CMB power spectra C X (X = TT, EE, TE), deflection angle spectrum C dd , cross correlation between temperature and deflection angle C Td , and noise power spectra N X and N dd , where C dd is calculated by a lensing potential [68] and is related with C BB . 1 We compute 1 By performing a public code HALOFIT [66,67], we have checked that modifications by including nonlinear effects for evolutions of the matter power spectrum are much smaller than typical errors in our analyses and negligible for parameter fittings.
N dd by using a public code FUTURCMB [69] which adopts the quadratic estimator [68]. In this algorithm, N dd is reconstructed by N Y (Y = TT, EE, BB). The covariance matrix in the Fisher matrix is expressed as where N Y is expressed by using both a beam size σ beam (ν) = θ FWHM (ν)/ √ 8 ln 2 and an instrumental sensitivity Y (ν) to be where For EE (ν) and BB (ν), we commonly use PP (ν) listed in Table 2.
N dd is calculated by N TT , N EE , and N BB . In case of Planck and Polarbear, we combine both the experiments, and assume that the 1.7% region of the whole sky is observed by both the experiments, and the remaining 63.3% (= 65%-1.7%) region is observed by Planck only. Therefore we evaluate a total Fisher matrix F CMB by summing the two Fisher matrices, where F Planck is the Fisher matrix of the region observed by Planck only and F Planck+PB is that by both Planck and Polarbear.
In addition, we calculate noise power spectra N Y,Planck+PB of the CMB polarization (Y = EE or BB) in F Planck+PB with the following operation.
In order to combine the CMB experiments with the 21 cm line experiments, we calculate the combined fisher matrix to be Here we did not use information for a possible correlation between fluctuations of the 21 cm and the CMB.

Results
In this section, we numerically evaluate how we can deter-

Constraints on N ν
In Fig. 1, we plot contours of 90% confidence levels (C.L.) forecasts in Σm ν -N ν plane. The fiducial values of neutrino parameters, N ν and Σm ν , are taken to be N ν = 3.04 and Σm ν = 0.1 eV (upper two panels), which corresponds to the lowest value of the inverted hierarchy model, or Σm ν = 0.05 eV (lower two panels), which corresponds to the lowest value of the normal hierarchy model. Adding the 21 cm experiments to the CMB experiment, we see that there is a substantial improvement for the sensitivities to Σm ν and N ν . That is because several parameter degeneracies are broken by those combinations, e.g., in particular T b and As were completely degenerate only in 21 cm line measurements. Therefore it is essential to add the CMB to the 21 cm experiment to be vital for breaking those parameter degeneracies. If each CMB experiment is combined with SKA or Omniscope, the corresponding constraint can be significantly improved. We showed numerical values of those errors in Table 3 in case that the fiducial values are taken to be N ν = 3.04 and Σm ν = 0.05 eV.
On the other hand, comparing those values with the current best bounds for Σm ν + N ν model, which give Σm ν < 0.89 eV and N ν = 4.47 +1.82 −1.74 obtained by CMB (WMAP) + HST (Hubble Space Telescope) + BAO [28], we find that the ongoing and future 21 cm line + the CMB observation will be able to constrain those parameters much more severely.
The case of Σm ν = 0.1 eV to be fiducial (upper two panels) corresponds to the lowest value for the inverted hierarchy when we use oscillation data. Then it is notable that CMBPol + SKA can detect the nonzero neutrino mass. Of course, Planck + Polarbear + Omniscope and CMBPol + Omniscope can obviously do the same job.
On the other hand, the case of Σm ν = 0.05 eV to be fiducial (lower two panels), which corresponds to the lowest value for the normal hierarchy, only Planck + Polarbear + Omniscope or CMBPol + Omniscope can detect the nonzero neutrino mass.

Constraints on neutrino mass hierarchy
Next we discuss if we will be able to determine the neutrino mass hierarchies by using those ongoing and future 21 cm and CMB observations. In Fig. 2 we plot 2σ errors of the parameter r ν ≡ (m 3 − m 1 )/Σm ν 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). It is notable that the difference between r ν 's of these two hierarchies becomes larger as the total mass Σm ν becomes smaller.
Therefore, r ν is quite useful to distinguish a true mass hierarchy from the other. Allowed parameters on r ν by neutrino oscillation experiments are plotted as two bands for the inverted and the normal hierarchies, respectively. The thin solid lines inside the bands are the experimental mean values by oscillations, one of which is taken to be a corresponding fiducial value of r ν as a function of Σm ν in each analysis. The constrains are obtained by combining Omniscope with Planck + Polarbear (thick  Allowed parameters on r ν by neutrino oscillation experiments are plotted in the same way of Fig. 2. dashed lines) and Omniscope with CMBPol (thick solid lines), respectively.
For 0.3 eV Σm ν the mass eigenvalues m 1 , m 2 , and m 3 are almost degenerate. Therefore the difference between two hierarchies has little influence on the matter power spectrum. Therefore the constraints on r ν are significantly weak compared with the difference between them, and then we cannot distinguish the true hierarchy from the other.
On the other hand however, the difference increases as Σm ν decreases down to m ν ∼ 0.1 eV. By using this property, the CMB (Planck + Polarbear or CMBPol) + the 21 cm (Omniscope) observations can constrain the neutrino mass hierarchy severely. Typically errors of Σm ν at around Σm ν = 0.1 eV are given by Σm ν = 0.0087 eV for Planck + Polarbear, and Σm ν = 0.0069 eV for CMBPol at 1σ , respectively. Therefore, the error of the x-axis is negligible compared with that of y-axis. In Fig. 3, we plot contours of 90% C.L. in Σm ν -r ν plane in order to show errors of Σm ν along the x-axis for typical fiducial values. As is clearly shown in Fig. 2, actually those combinations of the observations will be able to determine the neutrino mass hierarchy to be in- Once a clear signature Σm ν 0.1 eV were determined by observations or experiments, it should be obvious that the hierarchy must be normal without any ambiguities. On the other hand if the hierarchy were inverted, we could not determine it only by using Σm ν . However, it is remarkable that our method is quite useful because we can discriminate the hierarchy from the other even if the fiducial values were Σm ν 0.1 eV for both the normal and inverted cases. This is clearly shown in Fig. 3. In case that a fiducial value of Σm ν is taken to be the lowest values in neutrino oscillation experiments, the upper left (right) figure indicates that even CMBPol + SKA can discriminate the inverted (normal) mass hierarchy from the normal (inverted) one.

Conclusions
We have studied how we can constrain effective number of neutrino species N ν , total neutrino masses Σm ν , and neutrino mass hierarchy by using the 21 cm observations (MWA, SKA, and Omniscope) and the CMB observations (Planck, Polarbear, and CMBPol). It is essential to combine the 21 cm with the CMB Bmode polarization produced by a CMB lensing to break various degeneracies in cosmological parameters when we perform multipleparameter fittings.
About the constraints on Σm ν -N ν plane, for a fiducial value Σm ν = 0.1 eV which corresponds to the lowest value in the inverted hierarchy, we have found that CMBPol + SKA, Planck + Polarbear + Omniscope and CMBPol + Omniscope can detect the nonzero neutrino mass. For a fiducial value Σm ν = 0.05 eV, which corresponds to the lowest value in the normal hierarchy, Planck + Polarbear + Omniscope or CMBPol + Omniscope can detect the nonzero neutrino mass.
As for the determination of the neutrino mass hierarchy, we have proposed a new parameter r ν = (m 3 − m 1 )/Σm ν and studied how to discriminate a true hierarchy from the other by constraining r ν . As was clearly shown in Fig. 2, the combinations of the CMB (Planck + Polarbear or CMBPol) + the 21 cm (Omniscope) will be able to determine the hierarchy to be inverted or normal for Σm ν 0.13 eV or 0.1 eV at 2σ , respectively. Furthermore, if the fiducial value of Σm ν is taken to be the lowest value in the neutrino oscillation experiments, even CMBPol + SKA can determine the mass hierarchy.
In this study we have taken the simplified model of reionization. In case of more likely detailed modeling of reionization [45], it was pointed out that the constraints on cosmological parameters may moderately change at ∼ 10-50%. Fortunately, this effect is comparatively small and should not be fatal to constrain the neutrino mass hierarchy in the current analyses.