Flavor ratios of astrophysical neutrinos interacting with stochastic gravitational waves having arbitrary spectra
aa r X i v : . [ a s t r o - ph . H E ] S e p Flavor ratios of astrophysical neutrinos interactingwith stochastic gravitational waves having arbitraryspectrum
Maxim Dvornikov ∗ Pushkov Institute of Terrestrial Magnetism, Ionosphereand Radiowave Propagation (IZMIRAN),108840 Troitsk, Moscow, Russia
Abstract
We study the evolution and oscillations of fixed massive neutrinos interacting withstochastic gravitational waves (GWs). The energy spectrum of these GWs is Gaussianand the correlator of the amplitudes is arbitrary. We derive the equation for the densitymatrix for flavor neutrinos in this case. In the two flavors approximation, this equationcan be solved analytically. We find the numerical solution for the density matrix in thegeneral case of three neutrino flavors. We consider merging binary black holes as sourcesof stochastic GWs with realistic spectra. Both normal and inverted mass orderings areanalyzed. We discuss the relaxation of the neutrino fluxes in stochastic GWs emittedmainly by supermassive black holes. In this situation, we obtain the range of energies andthe propagation lengths for which the relaxation process is the most efficient. We discussthe application of our results for the observation of fluxes of astrophysical neutrinos.
Flavor transformations of a neutrino beam, called neutrino flavor oscillations, recently con-firmed experimentally (see, e.g. Ref. [1]), are the direct indication to the nonzero massesof these particles and mixing between different neutrino states. Various external fields areknown to contribute, or even enhance, neutrino oscillations [2]. The gravitational interaction,in spite of its weakness, is supposed to modify the neutrino oscillations process.Neutrino oscillations in gravitational fields were first considered in Ref. [3] and, subse-quently, in multiple papers. Many of them are reviewed in Ref. [4]. A gravitational field issupposed to modify the phase of a massive neutrino wave function [5]. This phase obeys theHamilton-Jacobi equation, which should be written down in curved spacetime.It is important to study neutrino oscillations in gravitational fields considering not onlystatic fields, but also time dependent gravitational backgrounds like a gravitational wave(GW). It is inspired by the recent GWs detection reported in Ref. [6]. Now multiple sourcesof GWs, mainly as coalescing binary compact objects, are catagolized in Ref. [7]. Sources ofGWs are expected to emit significant neutrino fluxes [8]. Thus, emitted GWs can influence thepropagation and oscillations of astrophysical neutrinos. There are active searches of neutrinosemitted by merging binaries, which are the sources of GWs [9]. ∗ [email protected]
1e studied neutrino flavor and spin (i.e. transitions between active left and sterile rightstates) oscillations under the influence of gravitational fields in Refs. [10–14]. Neutrino spinoscillations in GWs were considered in Ref. [13]. The evolution of the spin of a fermion inGW was discussed in Ref. [15]. The action of GWs on the propagation and flavor oscillationsof astrophysical neutrinos was studied in Refs. [14, 16].In the present work, we continue our research in Ref. [14]. We study the evolution ofa beam of fixed massive neutrinos interacting with stochastic GWs with arbitrary energyspectrum. In Sec. 2, we derive the equation for the density matrix for flavor neutrinos andsolve it analytically in the two flavors approximation. The numerical solution of this equationfor the general situation of the three neutrino flavors is obtained in Sec. 3, where we studyastrophysical applications. We consider coalescing binary BHs as sources of stochastic GWs.We obtain the maximal energy and the minimal propagation length, at which the fluxes offlavor neutrinos reach their asymptotic values. We discuss our results and consider theirimplication for the observation of astrophysical neutrinos in Ref. 4.
We study the system of three flavor neutrinos p ν e , ν µ , ν τ q with nonzero mixing. One candiagonalize the mass matrix of such neutrinos using the neutrino mass eigenstates ψ a , a “ , ,
3, which have the definite masses m a . Flavor and mass eigenstates are related by thematrix transformation, ν “ U ψ, (1)where U is the mixing matrix.In the most general situation of three neutrino flavors ν “ p ν e , ν µ , ν τ q , the mixing matrixin Eq. (1) can be parameterized in the following form [17]: U “ ¨˝ c s ´ s c ˛‚ ¨ ¨˝ c s e ´ i δ CP ´ s e i δ CP c ˛‚ ¨ ¨˝ c s ´ s c
00 0 1 ˛‚ , (2)where c ab “ cos θ ab , s ab “ sin θ ab , θ ab are the corresponding vacuum mixing angles, and δ CP is the CP violating phase. The values of these parameters can be found in Ref. [18]. Themixing matrix takes more simple form in the frequently used two flavors approximation, U “ ˆ cos θ sin θ ´ sin θ cos θ ˙ , (3)where θ is the only mixing angle.We suppose that these flavor neutrinos move in curved spacetime with background gravi-tational field in the form of a plane gravitational wave with circular polarization propagatingalong the z -axis. The interval in this case has the form [19],d s “ g µν d x µ d x ν “ d t ´ p ´ h cos φ q d x ´ p ` h cos φ q d y ` x d yh sin φ ´ d z , (4)where h is the dimensionless amplitude of the wave, φ “ p ωt ´ kz q is the phase of the wave, ω is frequency of the wave, and k is the wave vector. In Eq. (4), we use the Cartesian coordinates x µ “ p t, x, y, z q . 2he dynamics of flavor oscillations ν α Ø ν β is described by the effective Schrodingerequation, i ν “ H f ν, H f “ U H m U : , H m “ H p vac q m ` H p g q m , (5)where H f and H m are the effective Hamiltonians in the flavor and mass bases, H p vac q m “ E diag p m , m , m q is the part of the Hamiltonian responsible for vacuum oscillations, and E is the mean neutrino energy.Using the results of Ref. [20], we have derived in Ref. [14] the contribution to H m fromthe neutrino interaction with GW. We assume that ωL | β a ´ β b | ! , a, b “ , . . . , , (6)where L is the neutrino propagation distance, β a “ p { E a is the velocity of the mass eigenstate, p is the mean neutrino momentum, and E a “ a m a ` p is the energy of the mass eigenstate.In this case, H p g q m have only diagonal components which have the form, ´ H p g q m ¯ aa “ ´ p h E a sin ϑ cos 2 ϕ « hA m a E , (7)where ϑ and ϕ are the spherical angles fixing the neutrino momentum with respect to theGW wave vector and A p ϑ, ϕ q “ sin ϑ cos 2 ϕ . In Eq. (7), we keep only the linear term in h and take that p " m a . Using Eqs. (5) and (7), we get that H f “ H ` H , H “ U H p vac q m U : , H “ ξH , (8)where ξ “ hA . Note that H in Eq. (8) is the constant matrix.Now we suppose that neutrinos interact with stochastic GWs, i.e. the angles ϑ and ϕ , aswell as h are random functions of time. In this case, it is convenient to study the evolution ofthe density matrix ρ rather than the wave function ν . In this approach, the diagonal elementsof ρ are the probabilities to detect a certain flavor in a neutrino beam. Using the results ofRef. [21], we get the equation for ρ I p t q “ exp p i H t q ρ p t q exp p´ i H t q in the form,i ρ I “ r H I , ρ I s , (9)where H I “ exp p i H t q H exp p´ i H t q “ H .The formal solution of Eq. (9) can be represented in the form of a series, which shouldbe averaged over a certain time interval. We suppose that the amplitudes of GW formthe Gaussian stochastic process. Thus, only even terms in this series survive since all oddcorrelators, like x h p t q h p t q h p t qy etc, are vanishing. Eventually one has x ρ I y p t q “ ρ ´ @ A D r H , r H , ρ ss ż t d t ż t d t f p| t ´ t |q` @ A D r H , r H , r H , r H , ρ ss ż t d t ż t d t ż t d t ż t d t ˆ r f p| t ´ t |q f p| t ´ t |q ` f p| t ´ t |q f p| t ´ t |q ` f p| t ´ t |q f p| t ´ t |qs ´ . . . , (10)where ρ “ ρ I p q “ ρ p q is the initial density matrix, f p| t ´ t |q “ x h p t q h p t qy is thecorrelator of the GW amplitude, and @ A D “ π ż π d ϑ ż π d ϕA p ϑ, ϕ q “ . (11)3s the mean value of the angle factor squared.To derive Eq.(10) we assume that both ϑ and ϕ have the δ -correlated Gaussian distribu-tions, which is a reasonable assumption since stochastic GWs intersect a neutrino trajectoryrandomly. However, unlike Ref. [14] we do not assume that the amplitude of GW has thesame distribution, i.e. f p| t ´ t |q ‰„ δ p t ´ t q is the arbitrary function.After lengthy but straightforward calculations we transform Eq. (10) to the form, x ρ I y p t q “ ρ ´ a @ A D r H , r H , ρ ss ` ` a @ A D˘ r H , r H , r H , r H , ρ ss ´ . . . , (12)where a p t q “ ż t d t ż t d t f p| t ´ t |q “ ż t d t p t ´ t q f p| t |q . (13)One can check that Eq. (12) is the formal solution of the following equation:dd t x ρ I y p t q “ ´ g p t q @ A D r H , r H , x ρ I y p t qss , (14)where g p t q “ ż t d t f p| t ´ t |q , (15)which is the generalization of the results of Ref. [14] for the arbitrary correlator f p| t ´ t |q of the amplitudes of GWs. If we choose f p| t ´ t |q “ τ @ h D δ p t ´ t q , where τ is thephenomenological correlation time, and use Eqs. (11) and (15), then Eq. (14) reproducesEq. (2.17) in Ref. [14].Let us study the two flavors approximation. In this case, H “ ∆ m E p σ n q , n “ p sin 2 θ, , ´ cos 2 θ q , where ∆ m “ m ´ m . We can sum analytically the series in Eq. (12). Indeed, x ρ I y p t q “ ρ ´ r ρ ´ p σ n q ρ p σ n qs λ ˆ ´ λ ` λ ´ . . . ˙ “ ” ρ ´ ` e ´ λ ¯ ` p σ n q ρ p σ n q ´ ´ e ´ λ ¯ı , (16)where λ “ a @ A D ` ∆ m ˘ { E . If we choose the δ -correlated Gaussian distribution, then a “ τ t @ h D and Eq. (16) reproduces the corresponding result of Ref. [14].The correlation function x h p t q h p qy “ f p| t |q can be expressed in terms of the spectraldensity S p f q as [22] x h p t q h p qy “ ż d f cos p πf t q S p f q , (17)where f is the frequency measured in Hz. Instead of the spectral density in Eq. (17) it isconvenient to consider the energy density of stochastic GWs Ω p f q per logarithmic frequencyinterval with respect to the closure density of the universe ρ c “ H πG “ . ˆ ´ GeV ¨ cm ´ [23], S p f q “ Gρ c πf Ω p f q , (18)4here H is the Hubble constant and G “ . ˆ ´ GeV ´ is the Newton constant. Thefunction g p t q in Eq. (15) takes the form, g p t q “ Gρ c π ż d ff sin p πf t q Ω p f q . (19)It should be used in the differential Eq. (14). In this section, we study flavor transformations of a neutrino beam under the influence ofrealistic stochastic GW background. This problem was studied previously in Ref. [14], wherecoalescing black holes (BHs) were considered. However, the correlator of GW amplitudes wasassumed in Ref. [14] to be of the form, x h p t q h p t qy „ δ p t ´ t q . Spectra of realistic stochasticGW backgrounds were mentioned in Ref. [26] to be approximated by power laws, Ω p f q „ f α ,with the frequency f being in a confined region. Thus, the approximation made in Ref. [14]is quite rough.We should specify the initial condition for Eq. (14), ρ I p q “ ρ p q . We study the evolutionof astrophysical neutrinos created in decays of charged pions [24]. In this case, one hasthat the fluxes of flavor neutrinos at a source are p F ν e : F ν µ : F ν τ q S “ p q . Thus ρ I p q “ diag p { , { , q .First, we study the GW background from coalescing supermassive BHs (SMBHs). In thiscase, we can approximate Ω p f q by [25]Ω p f q “ Ω , if f min ă f ă f max , , otherwise, (20)where Ω „ ´ , f min „ ´ Hz, and f max „ ´ Hz. Note that this value of Ω does notviolate the constraint established in Ref. [26]. Using Eq. (20), we get g p t q in Eq. (19) in theexplicit form, g p t q “ ´ Gρ c Ω π " sin p πf max t q ´ p πf max t q f ´ sin p πf min t q ´ p πf min t q f ` πt „ cos p πf max t q f ´ cos p πf min t q f ` p πt q r Ci p πf max t q ´ Ci p πf min t qs * , (21)where Ci p x q “ γ ` ln x ` ∫ x t ´ t d t is the cosine integral and γ « , @ A D “ ,Ω , and f min , max given above. The neutrino energy is in the range E “ p ˜ q MeV andthe propagation distance is L “ H p g q m „ E ´ in Eq. (7). Indeed, one gets that A ρ p I q kk E p t q practically coincides with A ρ p I q kk E p q “ diag p { , { , q at E “
10 GeV.Comparing the present results with the findings of Ref. [14], where the same problemwas studied, we establish the more significant effect of GWs on the relaxation of the density5 t ′ ρ ρ ρ (a) t ′ ρ ρ ρ (b) Figure 1: The diagonal elements of x ρ I y based on the numerical solution of Eq. (14) versus t “ t { L . Here we adopt the normal mass ordering with ∆ m “ . ˆ ´ eV , ∆ m “ . ˆ ´ eV , θ “ . θ “ . θ “ .
15, and δ CP “ .
77. The neutrino beampropagation distance is L “ E “ MeV; and (b) E “
10 GeV. t ′ ρ ρ ρ (a) t ′ ρ ρ ρ (b) Figure 2: The same as in Fig. 1, but for the inverted mass ordering with ∆ m “ . ˆ ´ eV , ∆ m “ ´ . ˆ ´ eV , θ “ . θ “ . θ “ .
15, and δ CP “ . E “ MeV; and (b) E “
10 GeV. 6 t ′ F ν e F ν µ F ν τ (a) t ′ F ν e F ν µ F ν τ (b) Figure 3: Fluxes of flavor neutrinos based on the numerical solution of Eq. (14) with g p t q inEq. (21) for E “ MeV and L “ E, MeV x F ν e , ‘ y @ F ν µ , ‘ D x F ν τ , ‘ y L and greater E . This discrepancy can be explained by the underestimation of the relaxationtime τ in Ref. [14]. In the present work, we take into account the spectrum of stochastic GWsexactly. Thus there is no need to approximate the correlator by a δ -function.The elements of x ρ I y are not the measurable quantities. The fluxes of flavor neutri-nos F ν λ are proportional to the diagonal elements of the total density matrix x ρ y p t q “ exp p´ i H t q x ρ y I p t q exp p i H t q . In Fig. 3, we show F ν λ for L “ E “ MeV.We have demonstrated in Figs. 1(b) and 2(b) that GWs does not contribute to the evolutionof the density matrix at E “
10 GeV. Thus we do show the fluxes for E “
10 GeV.It is difficult to distinguish the fluxes for different flavors in Fig. 3 by sight because ofthe rapid vacuum oscillations. However, if one averages the signal over the length of theseoscillations, one gets the fluxes in a detector x F ν λ , ‘ y which turn out to be different for differentflavors. This additional averaging is equivalent to the situation when we study not only theaction of stochastic GWs on the neutrino beam, but also consider randomly distributedneutrino sources.The values of x F ν λ , ‘ y for different E are given in Tables 1 and 2. Here we take that L “ E “ MeV, and high, E “
10 GeV, energies. We remind that we established above that, at E “
10 GeV, stochastic GWspractically do not contribute to the relaxation of neutrino fluxes. Therefore, the differenceof the fluxes in the first and the second rows in Tables 1 and 2 is owing to the neutrinointeraction with stochastic GWs. One can see that this difference be up to 2 %. It is maximalfor the inverted mass ordering. 7 , MeV x F ν e , ‘ y @ F ν µ , ‘ D x F ν τ , ‘ y ω “ ω max “ πf max , E “ MeV and L “ α “ { p f q “ $&% Ω min ´ ff min ¯ { , if f min ă f ă f max , , otherwise, (22)where f min “ ´ Hz, f max “ Hz, and Ω min “ ´ [25]. Unfortunately, it is not possibleto express the function g p t q in Eq. (19) in the explicit form for Ω p f q in Eq. (22).We have solved Eq. (14) for this case. The value of x ρ I y p t q turn out to be unchangedfor f min , max and Ω min given above for both normal and inverted mass orderings. Thus theevolution of x ρ I y p t q is similar to that shown in Figs. 1(b) and 2(b). We have checked E downto 1 MeV and L up to 1 Gpc.Thus, merging BHs with stellar masses as sources stochastic GWs do not lead to a signifi-cant relaxation of fluxes of astrophysical neutrinos. Such sources of GWs are more abundantthan SMBHs studied above. Nevertheless the typical frequencies of the spectrum of GWsemitted are much higher that in case of SMBHs. It is reason why this kind of GWs does notcontribute to the relaxation of neutrino fluxes. In the present work, we have studied the evolution of three mixed flavor neutrinos accountingfor their interaction with stochastic GWs with arbitrary spectrum. In Sec. 2, we have derivedEqs. (14) and (15) for the density matrix for flavor neutrinos. This equation generalizes theresult of Ref. [14], where the δ -correlator of the GW amplitudes was assumed.In realistic situations, the spectral density of stochastic GWs is a certain function ofthe frequency in a confined frequency range. It leads to the correlator of amplitudes notnecessarily proportional to the δ -function. Thus, the approximation made in Ref. [14] is quiterough.The estimate of the correlation time τ , or the typical frequency of the spectrum ˜ f „ τ ´ ,in Ref. [14] is the main source of the inexactitude of the description of the relaxation ofneutrino fluxes. Indeed, g „ ˜ f ´ in Eq. (19) and ˜ f is in quite broad range. Therefore, slightlychanging ˜ f , we get a significant variation of the relaxation length of the neutrino fluxes.In the present work, we avoided this uncertainty. The lower part of the spectrum Ω p f q turns out to give the main contribution to the relaxation of the fluxes. It can explain muchfaster relaxation of the fluxes compared to Ref. [14].We have considered the application of our results for the evolution of fluxes of astro-physical neutrinos in Sec. (3). We have studied merging binary BHs as sources of stochastic8ravitational waves. We have considered two cases: SMBHs and BHs with stellar masses. Inthe case of SMBHs the relaxation distance is L „ MeV. Thus, the effect of the relaxation of neutrino fluxescan be important for supernova (SN) neutrinos in our galaxy.The fluxes of SN neutrinos were reported in Ref. [27] to be modified by the mixing betweenactive and hypothetical sterile neutrinos. We have demonstrated that the interaction of activeneutrinos with stochastic GWs background with realistic characteristics can also modify theobserved SN neutrino fluxes at the few percent level. Perhaps, the predicted effect can beobserved by the existing [28] or future [29, 30] neutrino telescopes.Merging BHs with stellar masses are more abundant than coalescing SMBHs. Howeverthe lower frequency of their spectra are much higher than that of SMBHs; cf. Eqs. (20)and (22). Thus the effect of relaxation of neutrino fluxes is smaller in this case. This fact wasconfirmed by numerical simulations.We have revealed that the neutrino interaction with stochastic GWs results in the re-laxation of neutrino fluxes. However, it is not the only random factor affecting the observedfluxes of astrophysical neutrinos. In a realistic situation, a neutrino telescope detects particlesemitted by multiple randomly distributed sources.In Sec. 3, we have accounted for this factor by considering neutrinos with high energies E “
10 GeV, for which the action of GWs on neutrino oscillations is negligible. Comparingthe cases E “ MeV and E “
10 GeV, one could extract the contribution of stochasticGWs to the mean neutrino fluxes, which is up to 2 %; cf. Tables 1 and 2. Perhaps, one coulddetect stochastic GWs through a precise measurement of astrophysical neutrino fluxes ratherthan using a direct technique described in Ref. [31].Accounting for the interaction with stochastic GWs and averaging over the positions ofrandomly distributed neutrino sources, we have obtained that the fluxes at a detector are notequal: p F ν e : F ν e : F ν e q ‘ ‰ p q . This problem of the flavor content of astrophysicalneutrinos was studied in Ref. [32]. The fluxes of ultra high energy neutrinos at a detectorwere found in Ref. [32] to depend on the channel of production of these particles.In the present work, we have demonstrated that the interaction of astrophysical neutrinoswith stochastic GWs can result in the deviation of the flavor ratio at a detector from thevalue p q . The predicted fluxes are not excluded by the recent observation of ultrahigh energy astrophysical neutrinos reported in Ref. [33]. Plans to improve sensitivity in thedetermination of the flavor ratio of astrophysical neutrinos are outlined in Ref. [34].In summary, we have studied the evolution of fluxes of astrophysical neutrinos interactingwith stochastic GWs having an arbitrary energy spectrum emitted by randomly distributedrealistic binary BHs. The consideration of the nontrivial spectrum allowed us to significantlyreduce the relaxation distance traveled by a neutrino beam for different flavors to reachthe asymptotic values. We could also increase the neutrino energy, for which the relaxationbecomes sizable. Now the effect can be potentially observed even for SN neutrinos propagatingwithin our Galaxy. References [1] The Borexino Collaboration, M. Agostini et al., Improved measurement of B so-lar neutrinos with 1 . ¨ y of Borexino exposure, Phys. Rev. D , 062001 (2020)[arXiv:1709.00756]. 92] M. Fukugita and T. Yanagida, Physics of Neutrinos and Applications to Astrophysics (Berlin, Springer, 2003).[3] D. V. Ahluwalia and C. Burgard, Gravitationally induced quantum mechanical phasesand neutrino oscillations in astrophysical environments, Gen. Rel. Grav. , 1161–1170(1996) [gr-qc/9603008].[4] G. G. Luciano and L. Petruzziello, Testing gravity with neutrinos: From classical toquantum regime, to be published in Int. J. Mod. Phys. D [arXiv:2007.08664].[5] N. Fornengo, C. Giunti, C. W. Kim, and J. Song, Gravitational effects on the neutrinooscillation, Phys. Rev. D (1997) 1895–1902 [hep-ph/9611231].[6] LIGO Scientific Collaboration and Virgo Collaboration, B. P. Abbott et al., Observationof gravitational waves from a binary black hole merger, Phys. Rev. Lett. , 061102(2016) [arXiv:1602.03837].[7] LIGO Scientific Collaboration and Virgo Collaboration, B. P. Abbott et al., GWTC-1:A gravitational-wave transient catalog of compact binary mergers observed by LIGOand Virgo during the first and second observing runs, Phys. Rev. X , 031040 (2019)[arXiv:1811.12907].[8] P. M´esz´eros, D. B. Fox, C. Hanna, and K. Murase, Multi-messenger astrophysics, NatureRev. Phys. , 585–599 (2019) [arXiv:1906.10212].[9] IceCube Collaboration, M. G. Aartsen et al., IceCube search for neutrinos coincident withcompact binary mergers from LIGO-Virgo’s first gravitational-wave transient catalog,Astrophys. J. Lett. , L10 (2020) [arXiv:2004.02910].[10] M. Dvornikov, Neutrino spin oscillations in gravitational fields, Int. J. Mod. Phys. D ,1017–1034 (2006) [hep-ph/0601095].[11] M. Dvornikov, Neutrino spin oscillations in matter under the influence of gravitationaland electromagnetic fields, J. Cosmol. Astropart. Phys. 06 (2013) 015 [arXiv:1306.2659].[12] M. Dvornikov, Spin oscillations of neutrinos scattered off a rotating black hole, Eur.Phys. J. C , 474 (2020) [arXiv:2006.01636].[13] M. Dvornikov, Neutrino spin oscillations in external fields in curved space-time, Phys.Rev. D , 116021 (2019) [arXiv:1902.11285].[14] M. Dvornikov, Neutrino flavor oscillations in stochastic gravitational waves, Phys. Rev.D , 096014 (2019) [arXiv:1906.06167].[15] Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, General treatment of quantumand classical spinning particles in external fields, Phys. Rev. D , 105005 (2017)[arXiv:1708.05601].[16] G. Koutsoumbas and D. Metaxas, Neutrino oscillations in gravitational and cosmologicalbackgrounds [arXiv:1909.02735].[17] C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics (Oxford,Oxford University Press, 2007), pp. 111–116.1018] P. F. de Salas, D. V. Forero, S. Gariazzo, P. Mart´ınez-Mirav´e, O. Mena, C. A. Ternes,M. T´ortola, and J. W. F. Valle, 2020 Global reassessment of the neutrino oscillationpicture [arXiv:2006.11237].[19] P. Hoyng,
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