Neutrino masses and ordering via multimessenger astronomy

We define the theoretical framework and deduce the conditions under which multi-messenger astronomy can provide useful information about neutrino masses and their ordering. The framework uses time differences between the arrival of neutrinos and the other light messenger, i.e. the graviton, emitted in astrophysical catastrophes. We also provide a preliminary feasibility study elucidating the experimental reach and challenges for planned neutrino detectors such as Hyper-Kamiokande as well as future several megaton detectors. This study shows that future experiments can be useful in testing independently the cosmological bounds on absolute neutrino masses. Concretely the success of such measurements depends crucially on the available rate of astrophysical events and further requires development of high resolution timing besides the advocated need of megaton size detectors.


I. INTRODUCTION
The fascinating discovery by the LIGO Collaboration [1] of ripples in the fabric of space-time-the gravitational waves (GWs), first anticipated by Einstein a century agoshows us a completely new way of exploring the Universe. GWs carry detailed information about astrophysical catastrophes and can provide a clear reference time for multimessenger astronomy. In the next decade we therefore expect great advances from the experimental particle physics searches, on Earth and in space.
It is therefore timely to ask whether it is possible to use these new extraordinary experimental achievements as new tools to help settle some of the open issues in particle physics.
We know that the Standard Model (SM) cannot be the ultimate theory of Nature since the neutrino sector and dark matter are not yet properly accounted for. In fact, the nature of the three light active neutrinos ν i (i ¼ 1, 2, 3) with definite mass m i is unknown. To date, neutrinos can still be Dirac fermions if particle interactions conserve some additive lepton number, e.g., the total lepton charge L ¼ L e þ L μ þ L τ . However, if the total lepton charge is violated, they can have a Majorana nature [2,3]. The only feasible experiment, so far, that can unveil the nature of massive neutrinos is neutrinoless double beta ðββÞ 0ν decay (see, e.g., Ref. [4] for a review).
Another pressing question to answer is how light the neutrinos are. 1 Experimental evidence of neutrino oscillations, and thus the existence of at least three neutrino states, forces us to include them in the SM and to give them small mass differences [8]. However, oscillation experiments are not sensitive to their masses. The fact that their masses are tiny, when compared to other SM particles, comes from cosmology, where an upper bound on the sum of the active neutrinos P i m i < 0.23 eV can be established [9]. More recently, more stringent limits have been obtained through the Lyman alpha forest power spectrum, P i m i < 0.12 eV [10]. These constraints will be further tested independently by other experiments such as beta decay and neutrinoless double beta decay. Future large scale structure surveys like the approved EUCLID [11] will allow us to constrain P i m i down to 0.01 eV when combined with Planck data.
The enormous disparity between the neutrino masses and those of the charged leptons and quarks suggests that the neutrino masses might be related to the existence of a new fundamental mass scale in particle physics, associated with the existence of new physics beyond that predicted by the SM. The so-called seesaw mechanism [12] gives an appealing explanation of neutrino mass generation, at the same time explaining the smallness of their masses and of their possible Majorana nature, through the existence of heavier fermionic SM singlets. It can also serve as a stepping stone for the explanation of the observed baryon asymmetry in the Universe through leptogenesis [13].
The detection of GW150914 [1] has already ignited the experimental neutrino community (see, e.g., the null search results of ANTARES and ICE-CUBE [14]), and the next-generation kilometer-scale laser-interferometric GW detectors such as aLIGO [15], aVIRGO [16], and KAGRA [17] will have a strong impact on multimessenger astronomy.
The goal of this work is to investigate whether experiments, making use of GW detection in combination with the associated neutrino (and photon) counterparts, can * langaeble@cp3.sdu.dk † meroni@cp3.sdu.dk ‡ sannino@cp3.dias.sdu.dk 1 For a discussion on neutrino speed measurements on Earth, see, e.g., Refs. [5][6][7]. make a dent in understanding the ordering of neutrino masses.
It has been established in the past literature [18][19][20][21][22][23][24][25][26], and more recently in Ref. [27], that valuable information on the neutrino masses can be obtained by investigating the time delay between the observation of neutrinos and gravitational waves emitted in astrophysical events such as supernovae. In the meantime, neutrino physics has entered the precision era with the determination of the reactor mixing angle θ 13 [28,29]. Furthermore, different experiments, ranging from cosmological surveys to particle experiments, are constraining the absolute neutrino mass to be less than 0.1 eV. 2 It is therefore timely to think about this subject in a new light.
Despite the progress in neutrino physics, current experiments cannot yet decide on the neutrino mass ordering and their absolute mass scale.
In the following we will briefly review the current status of neutrino ordering and mixing. Then we explore the conditions under which multimessenger astronomy can reveal or constrain the neutrino mass ordering and absolute mass. We conclude with a preliminary feasibility investigation.

II. NEUTRINO ORDERINGS: CURRENT STATUS
Current available neutrino oscillation data [30] (see Table I) are compatible with two types of neutrino mass spectra. These depend on the sign of Δm 2 3l (l ¼ 1, 2) and are summarized below: (i) Spectrum with normal ordering (NO): (ii) Spectrum with inverted ordering (IO): It should be kept in mind that Δm 2 31 ðNOÞ ¼ jΔm 2 32 ðIOÞj, where the notation is self-explanatory. Depending on the value of the lightest neutrino mass, m min , the neutrino mass spectrum can be as follows: We denote solar and atmospheric square mass differences, respectively, as Δm 2 21 and Δm 2 3l . The current cosmological bounds strongly disfavor the degenerate regime. However, these results should be further tested independently, for example, by beta decay and neutrinoless double beta decay experiments.

III. MULTIMESSENGER ASTRONOMY
The detection of GWs is a crucial test of general relativity and, as already discussed in the literature, it is also important to deduce other relevant physical properties. This new information can be derived when comparing, for example, their propagation velocity with those of photons and neutrinos coming from the same astrophysical source.

A. Setup
Let us start by considering a potential observation of an astrophysical catastrophe. Using the same notation of [27], we denote with T g ≡ L=v g , T ν i ≡ L=v ν i and T γ ≡ L=v γ , respectively, the time of propagation of a GW, a given neutrino mass eigenstate, and photons with group velocities v g , v ν i , and v γ . Following Fig. 1 a GW is emitted at the time t E g from a source at distance L and detected on Earth at t g . Similarly, we have emission and detection times for photons and neutrinos. For instance, astrophysical catastrophes like the merging of a neutron star binary or the core bounce of a core-collapsed supernova (SN) are believed to follow this pattern. The difference of the arrival times between the GWs and neutrinos, τ obs ≡ t ν − t g , or the GW and a photon, τ γ obs ≡ t γ − t g , are both observables, which can be positive or negative for an early or late arrival of a GW. Typically the emission times of the three signals (GW, γ and ν) do not coincide. 3 For instance, in the supernova explosion SN1987A [32], the neutrinos arrived approximately 2-3 hours before the associated photons.
Let us assume now that a neutrino is emitted at where we assume that the different species of neutrinos are produced with a common energy value E. If a given neutrino is produced by a source at a distance L, the time-of-flight delay Δt i with respect to a massless particle, emitted by the same source at the same time, is Here we do not take into account cosmic expansion since we consider sources at low redshift, z ≪ 0.1. This causes an error of less than 5%. From the expression in (2), we observe that larger distances and small neutrino energies are needed in order to maximize the experimental sensitivity. For distances around 50 kpc (SN1987A) and an energy of 10 MeV, a neutrino with a mass of 0.07 eV (the upper current absolute mass scale inferred from the Planck Collaboration [9]) would arrive ∼10 −4 s later than a massless particle. Similar to (2) we express the time delay between the arrival of two neutrino mass eigenstates as with Δm 2 ij ¼ m 2 i − m 2 j and to leading order in m 2 c 4 =E 2 . We note that in this limit the time intervals do not depend on the absolute neutrino mass scale but solely on the square mass differences which are determined experimentally (see Table I).

B. Disentangling neutrino mass ordering
We are now equipped with the needed information to address our overarching quests. From (3) we observe that if the detector uncertainty is 10 −3 s, we are able to disentangle the atmospheric (solar) squared mass differences with a signal coming from a distance larger than 0.8 (26) Mpc, assuming neutrinos have an energy of about 10 MeV. These distances decrease if we lower the neutrino energy.
This means that for neutrinos with an average energy of 10 MeV, the delay time of the heaviest neutrino mass eigenstate with respect to the lightest is larger than 10 −3 s independently of the absolute neutrino mass scale and hierarchy, for distances larger than ∼0.8 Mpc. Therefore, assuming an accuracy of 10 −3 s, the relevant sources are those at distances larger than 0.8 Mpc. With better time accuracy the distance decreases linearly.
We show in Fig. 2 the time delay (for each mass eigenstate) Δt i considering NO and IO (left and right panels, respectively) as a function of the lightest neutrino mass, setting the neutrino energy to 10 MeV and the distance of the source to 1 Mpc. The physically relevant arrival time differences between neutrino mass eigenstates Δt ν i ν j can be readily determined from Fig. 2. In this figure we also show the future sensitivity on the absolute neutrino mass of the β decay experiment KATRIN [33], which is expected to be around 0.2 eV, and the constraints given by the Planck Collaboration on the sum of the light active FIG. 1. GW, neutrino and photon propagation in time. 3 In alternative theories of gravity the three particles under study-photons, gravitons and neutrinos-can couple to different effective metrics. In this case the Shapiro delay is not the same for the three signals [31]. In this work, however, we assume the same coupling to the metric for all the signals. P i m i ≤ 0.23 eV 95% C.L. In Table II we produce relevant benchmark neutrino time lapses considering two different source distances for different values of the lightest neutrino mass for 10 MeV neutrinos. Table III shows the substantial gain in time lapse for the distances of 1 Mpc but with a neutrino mass energy of 5 MeV, which is still within experimental reach [34]. For illustrative purposes we give similar data for 10 Mpc in parentheses.
From Fig. 2 we observe that for a given distance and energy, the NO and IO spectra differ by having different time delay patterns. We note that for IO the delay between the two heaviest mass eigenstates is equivalent to the time lapse between the first two lighter mass eigenstates for NO. If we consider a conservative time accuracy of 10 −4 s for the next generation of detectors, 4 the time lapse differences between NO and IO will not be distinguishable.
However, in addition to the time information, the ratio between the amplitudes of the different neutrinos reaching the detector can also be measured. Since the distances considered here are very large, neutrinos will reach the detector incoherently such that the time integrated arrival probability is where α and β are flavor eigenstates. In fact, this expression holds true whenever the time arrival differences among the three mass eigenstates are smaller than the detector time resolution. However, when Δt ν i ν j is larger than the detector resolution, each mass eigenstate ν i can be detected independently and will interact with the detector with probability 5  This accuracy is conservative compared with an estimate based on the uncertainty on the vertex reconstruction, which is about 3 m for Hyper-Kamiokande [34]. In order to obtain a global time, when comparing with other experiments, a higher uncertainty is expected. 5 We work in the regime of incoherence. Defining σ xP (σ xD ) as the spatial width of the production (detection) neutrino wave packet, we work under the assumption that jðv j − v k ÞL=cj ≫ maxðσ xP ; σ xD Þ, with v i and v j the two group velocities of the two wave packets of neutrino mass eigenstates ν i and ν j .
For simplicity, here we do not consider matter effects which could, in principle, take place in the propagation through the Earth itself. In Fig. 3 we illustrate a possible pattern of neutrino detection following (5), using the time differences reported in Table III. Depending on the source, its distance and the experimental time sensitivity, the figure shows that, at least in principle, one can observe interesting time patterns reflecting the neutrino ordering and mixing.
So far we have discussed the basic setup and argued that neutrino detectors on Earth can help disentangle the neutrino ordering, when observing distant astrophysical catastrophes. We now discuss information that can be obtained when comparing time differences with respect to the other light messengers.

C. Absolute neutrino masses from time differences
The attractive idea to use a multisignal approach was put forward in [27], where the authors translate a potential SN signal of GWs and neutrinos into limits on the speed of GWs and on the absolute neutrino mass scale. We define which implies where i denotes the ith neutrino mass eigenstate. The deviation from the speed of light for GWs and neutrinos reads with From the definition (6) it follows that where, as already defined earlier, T 0 ¼ L=c. If in (10) we consider an uncertainty in the time of emission of neutrinos, τ ν int , in order to detect the GW and the neutrino signal, we must have and using (10) to the first order in δ ν and δ g , one finds Using the inequality above and assuming τ ν int ∼ 10 ms (typical time for a SN burst) and an energy equal to the energy threshold of HK, E ν ¼ 7 MeV, we show in Fig. 4 the δ g dependence on the lightest neutrino mass for a reference distance of L ¼ 1 Mpc (grey region). In principle, detectors with a lower energy resolutions, such as JUNO (E th ν ¼ 1.806 MeV), could test lower values of m min and could probe neutrino mass up to ∼0.02 eV for distances around 1 Mpc, which are at least an order of magnitude lower than present cosmological limits and the perspective upper limit from KATRIN (see orange region in Fig. 4).
Last, we notice that limits on v g can also be obtained from high energetic events or from the requirement of Lorentz invariance. In fact, if the GW velocity is subluminal, then cosmic rays lose their energy via gravitational Cherenkov radiation and cannot reach the Earth. The fact that ultra-high-energy cosmic rays are observed on Earth limits the GW propagation speed to be assuming that the cosmic rays have galactic origin (extragalactic) [35]. Further independent constraints on Lorentz violation can therefore be set when observing photon and gravitational waves. Reference [36] attempted this by combining the event GW150914 in GWs with the observation made by the Fermi Gamma-Ray Burst Monitor [37] of a transient photon source in apparent coincidence [36], v g − c < 10 −17 c: There are serious concerns about the true correlation between the two events. Nevertheless, if one recasts the limits in Eqs. (13) and (14), one obtains Independent bounds on δ g are important since they allow for a more precise interpretation of Fig. 4 in terms of m min . So far we have discussed the time difference measurement between a neutrino and a GW. Similarly, one can imagine a time difference to emerge if, rather than a GW, one were to detect a photon. If all messengers were simultaneously detected and assuming a unique source by using (2), within experimental resolution, the following consistency condition must hold:

IV. PRELIMINARY FEASIBILITY STUDY AND CONCLUSION
So far we have been concerned with the theoretical setup, and since the framework presented here relies on distant sources, we now perform a preliminary study of the actual experimental feasibility. In the following, we will not discuss the distribution or the expected number of various kinds of astrophysical events, but instead focus on the number of detected neutrinos assuming a specific source at a given distance. From the analysis above it is clear that three parameters are vital to increase the time lapse between mass eigenstates: the distance from the source L, the energy of the emitted neutrino, E ν , and the absolute neutrino mass m min . Conversely, the larger the distance is, the smaller the rate is. As a consequence, if the neutrino counterparts of events like GW150914 are emitted by the source, it would be hard, if not impossible, to detect them on Earth.
As a benchmark investigation we concentrate on the next generation of neutrino detection experiments such as the 1 Mton Hyper-Kamiokande (HK) in Japan [34], which has already sparked interest in the astrophysical community. Astrophysical catastrophes like the merging of a neutron star-black hole (NS-BH) binary or the core bounce of a core-collapsed supernova are expected to produce a total neutrino output carrying an overall energy of approximately 10 53 erg. For such an event one expects an integrated time flux per squared meter of about 3 × 10 11 ðd=MpcÞ −2 m −2 on Earth. Despite the fact that a large number of neutrinos will reach Earth because of their low cross section, only a tiny fraction will be detected. Previous studies [38] indicate that HK can detect 1-2 neutrino events per year from supernovae in the range up to 10 Mpc.
However, our theoretical analysis only made use of the neutrinos emitted during the initial burst from the source, which can be determined by integrating the following neutrino detection rate over the relevant time interval: where n p is the number of protons in the target; E ν;e are, respectively, the (anti)neutrino and the (electron) positron energy of the event; F ðE ν ; tÞ is the flux per unit time, area and energy; and ϵ is the detector efficiency. Finally σ 0 ðE e ; E ν Þ ¼ dσ=dE e is the differential cross section of the process under study. We will assume the efficiency of the detector to be 100% for energies larger than the energy threshold of the detector, E ν > E th ν . Our estimates assume a typical energy in neutrinos emitted from astrophysical sources within the initial burst to be of the order of ∼10 51 erg, as well as a mean neutrino energy hEν e i ∼ 12 MeV. From a SN at a distance d, HK (0.74 Mton, E th ν ¼ 7 MeV, E th e ¼ 4.5 MeV) would expect the following number of detected neutrinos (indicated by λ ES ) via neutrino-electron elastic scattering (ES) processes: FIG. 4. Plot of δ g as a function of the lightest neutrino mass considering τ ν int ∼ 10 ms using Eq. (12). We consider the HK and JUNO energy thresholds, E ν ¼ 7 MeV (grey region) and where the initial burst is primarily ν e from the neutronization process. Similarly, from a NS-BH merger, where the burst consists mostly ofν e , we get, via inverse beta decay (IBD) [39], For such low rates it is useful to estimate the actual detection probability as a function of the distance from the source. To assess this, we use the Poisson probability to detect n events as P n ¼ λ n e −λ =n!, where λ is the expected number of events, given in Eq. (18) or Eq. (19). In Fig. 5 we show, as an illustrative example, the detection probability for IBD resulting from requiring at least one, two and 10 events per burst, indicated, respectively, with blue, red and black curves. In our estimates we use the energy range 7-30 MeV. The plot shows the HK detection probability for ν e for a NS-BH merger (solid line), as well as the one for a hypothetical 5 Mton detector (dashed line) (see, e.g., Ref. [40]). We observe that even for ∼1 Mpc and a 7-30 MeV energy range, one can still observe ∼1 event.
These estimates show that it is possible to reach phenomenologically interesting neutrino mass differences from sources at ∼1 Mpc provided one can combine more than one Mton experiment. In order to compute the expected annual rate of detected neutrino events, one has to combine the above analysis with the annual rate of relevant astrophysical events. The annual rate of SNs is expected to be 1=3 yr −1 within 4 Mpc [38], while the rate for NS-BH mergers is more uncertain, with an expected rate of 10 2 -10 3 yr −1 within 1 Gpc. In the future this rate will be constrained by LIGO [41].
We stress that we have used conservative estimates, for example, in the total energy emitted with the neutrino burst. Another parameter relevant for our analysis is the time resolution in neutrino detection which, in the future, can be expected to fall below one millisecond. If this is the case it would allow sources as close as 100 kpc to become relevant for our analysis. In this case the neutrino flux increases by 2 orders of magnitude.
To conclude, we derived the theoretical and phenomenological conditions under which multimessenger astronomy can disentangle or further constrain the neutrino mass ordering. We have also argued that it can provide salient information on the absolute neutrino masses. We added a preliminary feasibility study to substantiate and further motivate our theoretical analysis. We have seen that future experiments can also be useful in independently testing the cosmological bounds on neutrino absolute masses. However, this requires high resolution timing and a significant increase in the combined fiducial volume compared to the current Cherenkov water detectors.
Conversely one can use future results on neutrino properties to provide detailed information about astrophysical sources simultaneously emitting GWs, photons and neutrinos, and possibly lower uncertainties in the emitted multimessenger signal from the source.

ACKNOWLEDGMENTS
The CP3-Origins center is partially funded by the Danish National Research Foundation, Grant No. DNRF90.
Note added.-While our work was under review, related papers on the propagation time of ultrarelativistic particles appeared in the literature [42,43], which provide relevant details for a high precision application of the presented framework.
FIG. 5. Detection probability of neutrinos versus distance from the source to Hyper-Kamiokande (solid lines) and to a hypothetical future 5 Mton experiment (dotted lines) [40] using a 7-30 MeV energy range. Blue, red and black curves represent the detection probability resulting from requiring observation of at least one, two and ten events per burst, respectively.