Finite size source effects and the correlation of neutrino transition probabilities through supernova turbulence
aa r X i v : . [ h e p - ph ] F e b Finite size source effects and the correlation of neutrino transition probabilitiesthrough supernova turbulence
James P. Kneller ∗ and Alex W. Mauney † Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA (Dated: October 8, 2018)The transition probabilities describing the evolution of a neutrino with a given energy along someray through a turbulent supernova are random variates unique to each ray. If the source of theneutrinos were a point then all neutrinos of a given energy and emitted at the same time whichwere detected in some far off location would have seen the same turbulent profile therefore theirtransition probabilities would be exactly correlated and would not form a representative sample ofthe underlying parent transition probability distributions. But if the source has a finite size thenthe profiles seen by neutrinos emitted from different points at the source will have seen differentturbulence and the correlation of the transition probabilities will be reduced. In this paper westudy the correlation of the neutrino transition probabilities through turbulent supernova profilesas a function of the separation δx between the emission points using an isotropic and an anisotropicpower spectrum for the random field used to model the turbulence. We find that if we use anisotropic power spectrum for the random field, the correlation of the high (H) density resonancemixing channel transition probability is significant, greater than 0.5, for emission separations of δx = 10 km, typical of proto neutron star radii, only when the turbulence amplitude is less than C ⋆ ∼ δx = 10 km. In contrast, there is significant correlation in the low (L) density resonantand non-resonant channels even for turbulence amplitudes as high as 50%. Switching to anisotropicspectra requires the introduction of an ‘isotropy’ parameter k I whose inverse defines the scale belowwhich the field is isotropic. We find the correlation of all transition probabilities, especially the Hresonance channel, strongly depends upon the choice of k I relative to the long wavelength radialcutoff k ⋆ . The spectral features in the H resonance mixing channel of the next Galactic supernovaneutrino burst may be strongly obscured by large amplitude turbulence when it enters the signaldue to the finite size of the source while the presence of features in the L and non resonant mixingchannels may persist, the exact amount depending upon the degree of anisotropy of the turbulence. PACS numbers: 47.27.-i,14.60.Pq,97.60.Bw
I. INTRODUCTION
The neutrino signal from the next core-collapse super-nova in our Galaxy will give us an unprecedented op-portunity to peer into the heart of an exploding starand confront our current paradigm of how these starsexplode with observations. But decoding the messagewill be no easy task because the neutrino signal will haveexperienced so many flavor-changing events on the tripfrom proto-neutron star to our detectors that scramblethe information, see for example Kneller, McLaughlin &Brockman [1] and Lund & Kneller [2]. The first flavorchanging effect the signal experiences is due to neutrinoself interactions / collective effects in the region up to ∼ ∗ Electronic address: [email protected] † Electronic address: [email protected] modelling [12, 36–39]. Finally, there is the possibilityof Earth matter effects leaving an imprint in the signalthough a recent study expects this effect to be minimal[40].What makes decoding the signal even more of a chal-lenge is that the neutrinos we receive at a given instantand with a given energy will not have experienced thesame flavor evolutionary history. The neutrinos arrivingat a detector will have been emitted from different loca-tions at the source and both the neutrino collective andthe MSW+turbulence effects will vary from trajectory totrajectory. Starting with Duan et al. [5], the self inter-action effects in calculations where the neutrino emissionover the source is assumed to be spherically symmetrichave been seen to be ‘angle dependent’ in the sense thata neutrino following a pure radial trajectory differs fromone emitted at an angle relative to the normal. Presum-ably allowing for aspherical source emission would onlymake the trajectory dependence even stronger. Similarlythe MSW plus turbulence effects are also trajectory de-pendent. If we temporarily cast aside the turbulence andfocus on the gross structure of the explosion i.e. the low-est angular multipole moments, an aspherical passage ofthe shock through the star, by itself, leads to a line-of-sight dependence. But, one must recall that we will notobserve the neutrinos from a supernova at widely differ-ent lines of sight, all our detectors are here on Earth. Thesize of the source is of order the proto-neutron star ra-dius, i.e. ∼
10 km while the shock effects show up in thesignal when the shock has propagated out to r ∼ km.As long as the curvature of the shock is over a lengthscalegreater than the source size the neutrinos which appear inour detectors will all have seen essentially the same pro-file. When one re-inserts the turbulence into the profile,one realizes this approximation may no longer be validbecause turbulence extends to much smaller lengthscaleseven when the shock is far out in the stellar mantle. Thedensity profile along two, essentially parallel lines of sightto a distant detector separated by ∼
10 km will no longerbe negligibly dissimilar and one must consider how thedissimilarity of the profiles propagates to the neutrinos.Any correlation will lead to a potential new feature ofthe neutrino signal.It has been shown that the transition probabilities fora single neutrino - the set of probabilities that relatesthe initial state to the state after passing through thesupernova - is not unique when turbulence is insertedinto a profile: it will depend upon the exact turbulencepattern seen by the neutrino as it travelled through thesupernova [38, 39, 41]. Those transition probabilities aredrawn from distributions whose properties will dependupon the stage of the explosion, the character of the tur-bulence, and the neutrino energy and mixing parameters.If the coherence of two neutrinos emitted at the sametime and with the same energy but from different loca-tions is small then the final states are uncorrelated andone would expect that the flux at a detector would justbe the mean of whatever distribution describes the tran-sition probabilities multiplied by the initial spectra. Butif the coherence is high then all the neutrinos will havethe same set of transition probabilities which one mightexpect to ‘scintillate’ together as the turbulence evolves.Of course, this ignores the issue of energy resolution andtemporal binning of the signal that becomes necessarybecause of the limited statistics.The purpose of this paper is to consider the issue of fi-nite source size and the correlation of the neutrino transi-tion probabilities along parallel trajectories through tur-bulent supernova profiles. Our calculations expand uponthe work of Kneller & Volpe [39] and Kneller & Mauney[41] upon which we rely heavily for the techniques usedto calculate the turbulence effects and as context for ourresults. We begin by describing the calculations we un-dertook paying particular attention to the constructionof the random fields used to model the turbulence. Thebasic approach to determining the effects of turbulenceare then demonstrated, followed by the computation ofthe transition probability correlation as a function of theseparation between the emission points. We finish bysummarizing our findings and discuss the implicationsfor the Galactic neutrino burst signal.
II. DESCRIPTION OF THE CALCULATIONS
The neutrino transition probabilities are the set ofprobabilities of measuring some neutrino state ν i given aninitial neutrino state ν j i.e P ( ν j → ν i ) = P ij . We shalldenote antineutrino transition probabilities by ¯ P ij . Ifthe S -matrix relating the initial and final wavefunctionsis known then these probabilities are just the square am-plitudes of the elements of S . The S -matrix is calculatedfrom the Schrodinger equation ı dSdr = H S (1)where H is the Hamiltonian. The Hamiltonian is the sumof the vacuum contribution H and the MSW potential V which describes the effect of matter. The vacuum Hamil-tonian is diagonal in what is known as the ‘mass’ basisand in this basis H is defined by two mass squared dif-ferences δm ij = m i − m j and the neutrino energy E .The mass basis is related to the flavor basis by the Maki-Nakagawa-Sakata-Pontecorvo [42, 43] unitary matrix U .The most common parametrization of U is in terms ofthree mixing angles, θ , θ and θ , a CP phase andtwo Majoranna phases.The MSW potential V is diagonal in the flavor basisbecause matter interacts with neutrinos based on theirflavor. The neutral current interaction leads to a contri-bution to V which is common to all flavors. This maybe omitted because it leads only to a global phase whichis unobservable. The charged current potential only af-fects the electron flavor neutrino and antineutrinos andis given by √ G F n e ( r ) where G F is the Fermi constantand n e ( r ) the electron density.In matter the two contributions to H means neitherthe mass nor the flavor states diagonalize the matrix. Butthere is a basis known as the matter basis which does di-agonalize H i.e. for a given value of the electron densitythere is a matrix ˜ U such that ˜ U † H ˜ U = K where K is thediagonal matrix of eigenvalues. When the MSW poten-tial vanishes the matter basis becomes the mass basis upto arbitrary phases. The matter basis is the most usefulfor studying the evolution of neutrinos through matterbecause it removes the trivial adiabatic MSW transitionand it will be the basis we use to report our results inthis paper. We refer the reader to Kneller & McLaughlin[44] and Galais, Kneller & Volpe [14] for a more detaileddescription of the matter basis.We now turn our attention to the turbulent densityprofiles through which we shall send our neutrinos. Asusual, we shall model the turbulence by multiplying a tur-bulence free density profile by a Gaussian random field.Since the spatial extent of the neutrino emission, of or-der 10 km, is much smaller than the radial location of theturbulence, of order r ∼ − km, we shall ignoreany curvature of the density profile features and use aplane-parallel model for the supernova. The z axis of ourCartesian co-ordinate system is aligned with the radialdirection of the profile. The profile we adopt is from a × × r [cm] × -23 × -22 V e [ e r g ] FIG. 1: The turbulence free MSW potential as a functionof distance through a supernova taken from a hydrodynam-ical simulation. The vertical lines indicate the positions ofthe reverse and froward shock in the profile. The horizontaldashed-dotted line is the two-flavor resonance density for a25 MeV neutrino with mixing angle sin θ = 0 . δm = 3 × − eV one-dimensional hydrodynamical simulation of a super-nova taken from Kneller, McLaughlin & Brockman [1].This profile is shown in figure (1) and is the same one usedin Kneller & Mauney [41]. The figure shows the presenceof two shocks: the forward shock at r s and the reverseshock at r r . In multi-dimensional simulations of super-nova both these shock fronts are aspherical and fluid flowthrough the distorted shocks leads to strong turbulencein the region between them. Our selection of this profilealso determines the neutrino energy we shall use since wewish the neutrinos to have an H resonance density thatdoes not intersect the shocks. Therefore we pick 25 MeVfor the neutrino energy and the reader may observe thatthe two-flavor resonance density for a 25 MeV, shown inthe figure, does not intersect the shocks as required.The turbulence is inserted by multiplying the profile inthe region between the reverse and forward shocks by afactor 1 + F ( r ) where F ( r ) is a three-dimensional Gaus-sian random field with zero mean. The random field isrepresented by a Fourier series, that is F ( r ) = C ⋆ tanh (cid:18) r − r r λ (cid:19) tanh (cid:18) r s − rλ (cid:19) × N k X n =1 p V n { A n cos ( k n · r ) + B n sin ( k n · r ) } . (2)In this equation the parameter C ⋆ sets the amplitudeof the fluctuations while the two tanh functions are in-cluded to suppress fluctuations close to the shocks andprevent discontinuities. The parameter λ is a dampingscale which we set to λ = 100 km. The random part of F appears in the the second half of equation (2) because theset of co-efficients { A } and { B } are independent standardGaussian random variates with zero mean. The k n area set of wavenumbers and, finally, the paramaters V n arek-space volume co-efficients. The method of fixing the N k k ’s, V ’s, A ’s and B ’s for a realization of F is ‘variant C’ of the Randomization Method described in Kramer,Kurbanmuradov, & Sabelfeld [45] which we have general-ized to three dimensions. The Randomization Method ingeneral partitions the space of wavenumbers into N k re-gions and from each we select a random wavevector usingthe power-spectrum, E ( k ), as a probability distribution.The volume paramaters V n are the integrals of the powerspectrum over each partition if the power spectrum isnormalized to unity. Variant C of the RandomizationMethod divides the k-space so that the number of par-titions per decade is uniform over N d decades startingfrom a cutoff scale k ⋆ . Throughtout this paper we shalluse a wavenumber cutoff k ⋆ set to twice the distance be-tween the shocks i.e. k ⋆ = π/ ( r s − r r ). The logarithmicdistribution of the modes increases the efficiency of thealgorithm in the sense that we can use a ‘small’ valueof N k and also the agreement between the exact statis-tical behavior of the field and that of an ensemble ofrealizations is uniform over some range of lengthscalesi.e. it is scale invariant. This feature is important forour study because the oscillation wavelength of the neu-trinos is constantly changing as the density evolves. Theminimum lengthscale we need to cover has been shownby Friedland & Gruzinov [38] and Kneller, McLaughlin& Patton [46] to be the reduced oscillation wavelengthsfor the neutrinos and antineutrinos i.e. λ ij = 1 / | δk ij | and ¯ λ ij = 1 / | δ ¯ k ij | - where δk ij and δ ¯ k ij are the differ-ences between the eigenvalues i and j of the neutrinosand antineutrinos respectively. Kneller & Mauney [41]showed the wavelengths in the turbulence region were oforder 1 km or greater which is approximately four ordersof magnitude smaller than the shock separation. Thismeans we need to pick N d ≥ A. The power spectrum
The final component of our calculations we have yetto discuss is the power spectrum E ( k ). In this paperwe shall consider two power spectra and our first choice,due to its simplicity, is a normalized three-dimensional,isotropic inverse power-law spectrum given by E ( k ) = ( α − πk ⋆ (cid:18) k ⋆ | k | (cid:19) α +2 Θ( | k | − k ⋆ ) . (3)for | k | ≥ k ⋆ where | k | is the magnitude of the wavevector k . Throughout this paper we shall adopt the Kolmogorovspectrum where α = 5 /
3. The one dimensional powerspectrum for the k z component of the wavevector is E ( k z ) = ( α − αk ⋆ (cid:18) k ⋆ | k z | (cid:19) α Θ( | k z | − k ⋆ )+ ( α − αk ⋆ Θ( k ⋆ − | k z | ) (4)which differs from the one dimensional power spectrumused by Kneller & Mauney [41] because for | k z | ≥ k ⋆ the power is suppressed by the factor 1 /α and the onedimensional spectrum is non-zero for | k z | ≤ k ⋆ . The two-point correlation function B ( δ r ) for this choice of a power spectrum depends only the magnitude of the separation, δr , and may be calculated analytically to be B ( δr ) = ı ( α − π k ⋆ δr ) α − n exp (cid:16) ıπα (cid:17) Γ( − α, ıπ k ⋆ δr ) − exp (cid:16) ıπα (cid:17) Γ( − α, − ıπ k ⋆ δr ) o (5)where Γ( n, x ) is the incomplete Gamma function. Thereis one last quantity to determine: the number of N k ofelements in the sets of random wavenumbers, coefficientsand volumes. To find this quantity we compare the sta-tistical properties of an ensemble of random field realiza-tions with the exact expressions as a function of the ratio N k /N d for a given N d . The statistical property we com-pute is the second order structure function G ( δ r ) whichis given by G ( δ r ) = h F ( r + δ r ) − F ( r ) i (6)where δ r is the separation between two points. The func-tion G ( δ r ) is related to the two-point correlation func-tion B ( δ r ) via G ( δ r ) / − B ( δ r ). For the isotropicpower spectrum both G and B are only functions ofthe magnitude of δ r and the correlation function is givenabove. In figure (2) we show the ratio R ( δr ) of the numer-ically calculated structure function for the isotropic ran-dom field to the exact solution as a function of the scale k ⋆ δr when we use either N k = 50 wavenumbers spreadover N d = 5 decades or N k = 90 wavenumbers over N d = 9 decades. The numerical calculation is the aver- × -13 × -11 × -9 × -7 × -5 × -3 × -1 × k * δ r R ( δ r) FIG. 2: The ratio of the structure function G ( δr ) as a func-tion of k ⋆ δr for two randomly orientated points in a 3-D ho-mogeneous and isotropic Gaussian random field to the exactstructure function. The two curves in the figure correspondto { N k , N d } = { , } (blue solid) and { N k , N d } = { , } .At every k ⋆ δr we generated 30 ,
000 realization of the field andthe error bar on each point is the standard deviation of themean F ( r + δ r ) − F ( r ).. age of 30 ,
000 realizations of the turbulence and the errorbar on each point is the error on the sample mean. Thefigure indicates that the method we use to generate ran-dom field realizations reproduces the analytic results forthe structure function very well and with high efficiencybecause good agreement between the statistics of the en-semble and the exact result requires just N k /N d = 10. Infact we find even N k /N d ratios of just N k /N d ∼ − N k /N d = 10.But isotropic and homogeneous three-dimensional tur-bulence is perhaps not a realistic scenario for supernovabecause the gravitational potential and the general fluidflow are in the radial direction. Only on sufficiently smallscales should the turbulence become isotropic. This divi-sion into large and small lengthscales indicates we shouldpartion the power-spectrum so that for | k z | ≥ k I thespectrum is isotropic, where k I is the isotropy scale, be-tween k ⋆ ≤ | k z | ≤ k I the spectrum is anisotropic andthen below the cutoff scale, | k z | ≤ k ⋆ , the power spec-trum should be set to zero since there should be no modeson scales larger than 1 /k ⋆ . For | k z | ≥ k I where the spec-trum is isotropic we use a power spectrum resemblingequation (3) E ( k ) = α ( α − πk ⋆ (cid:18) k ⋆ | k | (cid:19) α +2 Θ( | k | − k I ) . (7)Note the additional factor of α in the numerator. For k ⋆ ≤ | k z | ≤ k I we write the spectrum as the product E ( k x , k y , k z ) = E ( k x , k y ) × E ( k z ). The spectrum E ( k z ) ischosen to be a continuation of the inverse power-law givenabove while the spectrum in the xy directions, E ( k x , k y ),is the spectrum of the isotropic/homogeneous region inthese directions fixed at | k z | = k I . The spectrum for k ⋆ ≤ | k z | ≤ k I is thus E ( k x , k y , k z ) = α ( α − πk ⋆ (cid:18) k ⋆ | k z | (cid:19) α (cid:18) k I k x + k y + k I (cid:19) α/ Θ( k I − | k | ) Θ( | k | − k ⋆ ) . (8)The reader may verify the power spectrum definedby equations (7) and (8) is normalized. Thisanisotropic three-dimensional power spectrum yields aone-dimensional spectrum along the z direction given by E ( k z ) = ( α − k ⋆ (cid:18) k ⋆ | k z | (cid:19) α Θ( | k z | − k ⋆ ) (9)for | k z | ≥ k ⋆ which is exactly the same as the one-dimensional spectrum used in Kneller & Mauney [41].There is no analytic formula for the two-point structurefunction for randomly orientated separations using this power spectrum but if we consider the two-point struc-ture function of the random field for points orientatedalong the z direction then we can compute that in thisdirection B ( δz ) = ( α − π k ⋆ δz ) α − n exp (cid:16) ıπα (cid:17) Γ(1 − α, ıπ k ⋆ δz ) + exp (cid:16) ıπα (cid:17) Γ(1 − α, − ıπ k ⋆ δz ) o . (10)Compared to the isotropic spectrum above, thisanisotropic spectrum differs in important ways. First,even if we set k I = k ⋆ we observe that the lack of powerin the region | k z | ≤ k ⋆ means we have to compensateby increasing the structure / decreasing the correlationby the factor α . This increase is the reason for the ap-pearance of the extra factor α in equation (7). Next,as we increase the ratio f I = k I /k ⋆ , we push more andmore of the structure of the field in the xy direction toever smaller scales reducing even further the correlationof the field at some fixed non-radial separation δx com-pared to the isotropic case. This extra power at smallscales can be seen in figure (3) which is a plot of the ra-tio of the one-dimensional two-point structure functionin the x direction relative to the structure function alongthe z direction at the same separation scale for three val-ues of f I . As promised, when f I = 1 there is an equalamount of structure in the field along both radial and × -7 × -5 × -3 × -1 k * δ x G ( δ x ) / G ( δ z = δ x ) FIG. 3: The ratio of the structure function G ( δx ) as a func-tion of k ⋆ δx for two points aligned along the x direction tothe structure function G ( δz ) of two points aligned along the z direction at δx = δz . The three curves in the figure corre-spond to k I = k ⋆ (solid), k I = 10 k ⋆ (dashed) and k I = 100 k ⋆ (dot-dashed). At every k ⋆ δx we generated 30 ,
000 realizationof the field and the error bar on each point is the standarddeviation of F ( x + δx ) − F ( x ). The structure function G ( δz )was computed using the correlation function given in (10)and the relationship G ( δz ) / − B ( δz ). The inputs tothe random field generator were N k = 90, N d = 9. non-radial directions but as f I increases we push moreand more of the structure of the field in the xy directionto smaller scales.The anisotropic power spectrum we have constructedmeans the turbulence along different parallel rays is lesscorrelated than the turbulence along two rays at thesame separation when the power spectrum is isotropic.If that’s the case then the transition probabilities for theneutrinos travelling along those two rays should also beless correlated and below we quantify the decrease. III. RESULTS
Now that we have the random fields to model theturbulence we are all set to generate turbulent profilesand send neutrinos and antineutrinos through them. Toachieve higher efficiency we follow six neutrinos and sixantineutrinos simultaneously through every realization ofthe turbulence with one neutrino and one antineutrinoemitted at x ∈ { , , , , , } cm. Each timewe generate a new realization we end up with a differentset of transition probabilities so by repeating the calcula-tion many times - in our case a minimum of one thousandtimes but often much larger - we can create an ensembleof transition probabilities of size N from each emissionpoint. Once we have our ensemble we can then go aheadand compute means h P ij ( x ) i , variances V ij ( x ), and, ofcourse, correlations ρ ij ( δx ) = h P ij ( x ) P ij ( x + δx ) i − h P ij ( x ) i h P ij ( x + δx ) i p V ij ( x ) V ij ( x + δx ) (11)The correlation of the antineutrino transition probabil-ities will be denoted as ¯ ρ ij . In the large N limit theerror on the correlation is expected to be σ ρ = (1 − ρ ) / √ N −
1. Combining the results from the six emis-sion points we can form fifteen separations δx so fifteencorrelations but two points must be remembered: first,groups of them will cluster e.g. we will have a value forthe correlation at δx = 10 km but also two more at δx = 9 km and δx = 9 . P f( P ) FIG. 4: The frequency distribution of the transitionprobability P for each of the neutrino emission points x . From bottom to top the emission points are x =0 , , , , , cm. The turbulence amplitudeis set to C ⋆ = 30%, we used N k = 50, N d = 5 for the 3-D tur-bulence field generator, and the neutrino mixing parametersused are those given in the text with sin θ = 0 . - we should expect the results to be similar within eachcluster - and also they give us an indication if the error inthe results are comparable to the expected, large-N error σ ρ given above.We also need to specify the neutrino mixing parame-ters we have used. The hierarchy will be set to normaland we shall comment on how our results translate to theinverted hierarchy. As discussed, the neutrino energy willbe fixed at E = 25 MeV, typical of supernova neutrinoenergies and we shall set the neutrino mixing parame-ters to be δm = 8 × − eV , δm = 3 × − eV ,sin θ = 0 .
83, and sin θ = 1. The recent measure-ments of the last mixing angle θ by T2K [47], DoubleChooz [48], RENO [49] and Daya Bay [50] are all in theregion of θ ≈ ◦ . We shall adopt this value for the ma-jority of this paper but this result is sufficiently new thatwe shall show on occasion results with multiple values of θ in order to put this result in context.Finally, the turbulence amplitude C ⋆ will be allowedto vary but we shall focus upon larger values. With themeasurement of a large value of θ the turbulence effectsare negligible for amplitudes of order C ⋆ ∼
1% [41].
40 50 60 N k 〈 P 〉 , 〈 P 〉 , 〈 P 〉 FIG. 5: The mean of the transition probability P (circles), P (squares) and P (triangles) of the neutrinos emitted at x = 0 as a function of the parameter N k keeping the ratio N k /N d fixed at N k /N d = 10. The error bars are not theerror on the mean but rather the standard deviation of thesamples. The turbulence amplitude is set to C ⋆ = 30% andsin θ = 0 . A. The point source statistics
Before we show our results for the correlation of thetransition probabilities as a function of the emission sep-aration, we consider first the statistical properties of theensembles for each emission point. In addition to beinginteresting in their own right and useful as a reference,these calculations allow us to test that our 3D randomfield generator is working properly because the ensem-bles for each point of emission should be consistent andindependent of x .In figure (4) we show the frequency distribution of P for the six emission locations x using the mixing para-maters given above, C ⋆ = 30%, N k = 50, N d = 5 andsin (2 θ ) = 0 . N = 3265 for each emission point. In eachpanel of the figure the reader will observe that the transi-tion probability is almost uniformly distributed - there isa slight decrease in the frequency of higher values of P - but, more importantly, there is no observed trend with x . A closer inspection of figure (4) also hints at somecorrelation: the bottom few panels of the figure are verysimilar. We have reproduced this calculation for otherchoices of the N k and N d paramaters. The results areshown in figure (5) where we plot the mean values andstandard deviations of P , P and P for ensembles ofneutrinos emitted at x = 0 as a function of the param-eter N k keeping the ratio N k /N d fixed at N k /N d = 10.There is no discernable trend with N k and we march onconfident that setting N k = 50 and N d = 5 does not biasour results.We now allow the values of C ⋆ and θ to float and con-sider both the isotropic and anisotropic power spectrum.The evolution of the transition probability means as afunction of C ⋆ for the two power spectra and two choices C * 〈 P 〉 × -4 × -3 × -2 × -1 〈 P 〉 × -4 × -3 × -2 × -1 〈 P 〉 C * × -9 × -6 × -3 〈 P 〉 × -5 × -3 × -1 〈 P 〉 × -4 × -3 × -2 × -1 〈 P 〉 FIG. 6: Left figure, the mean of the transition probabilities P - top panel - P - center panel - and P - bottom panel - as afunction of C ⋆ for neutrinos emitted from a single point. The right figure is the mean of the distributions for the antineutrinotransition probabilities ¯ P - top panel - ¯ P - center panel - and ¯ P - bottom panel as a function of C ⋆ for antineutrinosemitted from a single point. In all panels the curves correspond to either sin θ = 4 × − (squares) or sin θ = 0 . of θ are shown in figure (6). There are many inter-esting trends discussed in detail in Kneller & Mauney[41]. Large amplitude turbulence works its way throughto affect every mixing channel, not just the H resonancechannel P , as promised so that by C ⋆ = 0 . h P i ∼ h P i ∼ h P i ∼ h ¯ P i ∼ h ¯ P i ∼ h ¯ P i ∼ θ = 9 ◦ . The only neu-trino mixing channel with reasonable sensitivity to θ is the H resonance channel P and even then the dis-parity in h P i at C ⋆ ∼ . C ⋆ ∼ .
3. Incontrast the antineutrinos are very sensitive to θ evenat large turbulence amplitudes: the expectation value for P varies by a factor of ∼ θ is changed fromsin θ = 4 × − to sin θ = 0 .
1, ¯ P and ¯ P onthe other hand change by ∼ − P , the isotropic power spectrum givesvalues of h P ij i which are smaller than the anisotropicspectrum. The neutrinos are more sensitive to the tur-bulence when the power spectrum is anisotropic becausethe neutrinos are sensitive to the amplitude of the turbu- lence modes of order the neutrino oscillation wavelength[38, 46] which is typically in the range of ∼
10 km in theH resonance region. The anisotropic spectrum removedall power for the fluctuations in the radial direction at thelong wavelengths above 1 /k ⋆ - which is of order 10 kmin our calculation - and to compensate we needed to in-crease the power on the smaller wavelengths which meansand effective increase of their amplitude. In fact we al-ready know the exact amount the amplitude is effectivelyincreased because we pointed out the 1 /α factor thatappears in the one dimensional power spectrum in theisotropic case compared to the one-dimensional spectrumderived from the anisotropic turbulence. Thankfully, ourexpectations are confirmed by figure (6) because the in-crease of all the mixing channels except P is on the ex-pected scale of α . The isotropy scale paramater k I , whichsets the scale in the radial direction below which the tur-bulence is isotropic, does not play a role for these pointsource statistics. The one-dimensional power spectrumalong the radial direction is independent of the isotropyscale k I which can be seen when comparing equations(4) and (9). So if the one-dimensional power spectrum isindependent of k I then the effect of switching the powerspectrum from isotropic to anisotropic is solely due to theremoval of radial long-wavelength fluctuations. The tran-sition probability P behaves slightly differently but is ρ ρ ρ ρ × × × δ x [cm] × × × δ x [cm] × × × δ x [cm] ρ FIG. 7: The correlation of the transition probabilities through isotropic turbulence of various turbulence amplitudes as afunction of the distance between emission points δx . From top row to bottom the correlations are for P , P , P , ¯ P and¯ P . The turbulence amplitudes are C ⋆ = 10% (left column), C ⋆ = 30% (center column) and C ⋆ = 50% (right column). Thevalues of θ are sin θ = 4 × − (squares joined by a solid line), sin θ = 10 − (triangles joined by a dot-dashed line),sin θ = 4 × − (diamonds joined by a double dot-dash line) and sin θ = 0 . entirely consistent with the understanding of the effectsin the other channels. At smaller amplitudes and thesmaller value of θ there is no effect of the power spec-trum switch upon h P i because the depolarization limithas been reached. At the larger mixing angle depolariza-tion has not achieved and switching the power spectrumleads to the effects as seen in P and P . The two-flavordepolarization limit is reached for the sin θ = 0 . C ⋆ ∼ h P i = 1 /
3. Whatever the mixing angle used,we see that the mean value h P i as a function of C ⋆ using the anisotropic spectrum begins the transition atsmaller C ⋆ than the same calculation using the isotropicspectrum because of the increased amplitude of the smallscale fluctuations in the former case. B. The correlation through isotropic turbulence
We now turn to the correlation of the transition proba-bilities as a function of the distance between the emissionpoints and consider first the case of the isotropic powerspectrum. Our result for the correlation of the transitionprobabilities, except ¯ P , as a function of the separation δx at various values of θ and turbulence amplitudes C ⋆ is shown in figure (7). ¯ P is excluded is because it isdifficult to calculate its correlation reliably. What onenotices immediately about the results are that ρ , ρ ,¯ ρ and ¯ ρ all show little sensitivity to either θ or C ⋆ - which is in contrast to figure (6). The reason for thelack of sensitivity of these correlations to θ and C ⋆ isexplained by the exponential distributions these transi-tion probabilities possess. Both the turbulence ampli-tude and the mixing angle simply ‘rescale’ the ensembleof transition probabilities and, as equation (11) shows,this rescaling cannot alter the correlation. One also seesthat the correlation of all these transition probabilities ishigh, & .
5, for all separations δx .
100 km.In contrast the correlation of P is sensitive to both θ and C ⋆ . When C ⋆ is of order C ⋆ ∼
10% the sen-sitivity to θ arises because the distributions of P atthe different mixing angle choices are very different: forsin θ = 4 × − the distribution is uniform, forsin θ = 0 . P . As C ⋆ increases the sensitivity disappears becausethe distributions at each value of θ become similar:this is the same behavior seen in figure (6). Finally,for C ⋆ = 10% the currently preferred value of θ givesgreater correlation at a given seperation than smaller val-ues of θ . The correlation ρ is high for δx .
10 km,a scale of order the proto-neutron star diameter, at for C ⋆ = 10% and decreases rapidly as C ⋆ increases. For C ⋆ & . P of two neutrinosemitted from points on the proto-neutron star separatedby a distance greater than δx & × × × × × δ x [cm] ρ ρ ρ FIG. 8: The correlation of the transition probabilities P - top panel - P - center panel - and P - bottom panel -through anisotropic turbulence as a function of the separationbetween neutrino emission points. The turbulence amplitudeis set at C ⋆ = 30% and sin θ = 0 .
1. In each panel the cor-relation of the transition probabilities through the isotropicturbulence is shown as the solid line. The other curves in eachpanel correspond to different values of the ratio f I = k I /k ⋆ : f I = 1 are squares joined by long dashed lines, f I = 10 aretriangles joined by dash-dot lines, and f I = 100 are diamondsjoined by dot double-dash lines. The error bars on each datapoint are estimated using the large N limit prediction. C. The correlation through anisotropic turbulence
The change to the mean point source transition proba-bilities when switching to an anisotropic power spectrumis both understandable and measurable but, overall, theeffects are small and of the order of factors of α i.e. theamplitude by which the small scale fluctuations in theanisotropic spectrum increased in amplitude comparedto the isotropic spectrum. That insensitivity no longerholds when we examine the correlations of the transi-tion probabilities because these quantities are functionsof the isotropy scale paramater k I . The correlations ofthe transition probabilities as a function of the separa-tion between the emission points is strongly sensitive tothe amount of turbulence power in the perpendicular di-rections and increasing k I relative to the fixed scale k ⋆ shifts the power from long wavelength, small k x and k y ,0 ρ ρ ρ ρ × × × δ x [cm] ρ × × × δ x [cm] × × × δ x [cm] FIG. 9: The correlation of the transition probabilities P , P , P , ¯ P and ¯ P as a function of the separation betweenemission points δx . The mixing angle θ was set at sin θ = 0 .
1. The left column of panels is for a turbulence amplitude of C ⋆ = 10%, the central column for C ⋆ = 30%, and the rightmost column is C ⋆ = 50%. In each panel the squares joined by thesolid lines are for f I = 1, the triangles joined by dash-dot lines are f I = 10, and the diamonds joined by dash-double dot linesare f I = 100. f I = k I /k ⋆ are shownin figure (8) for the case C ⋆ = 0 . θ = 0 . f I = 1 the difference between the isotropic andanisotropic power spectra are minimal but at larger ra-tios of the two scales the correlation at some given sep-aration δx drops noticeably in all three channels thoughthe reduction in the correlation of P and P is notas severe as that for the transition probability P . Itis still the case that the correlation of P and P attypical proto-neutron star radii of δx ∼
10 km is largerthan 0.5 if f I .
10 for this particular mixing angle choiceand turbulence amplitude. Pushing even more power tosmaller scales would lead to minimal correlation of thesetwo transition probabilities. In the case of P , the cor-relation at δx ∼
10 km is already small for this mix-ing angle and turbulence amplitude even in the isotropicand f I = 1 cases so pushing more power of the fluctua-tions in perpendicular directions to smaller wavelengthscompletely removes the correlation of P over the proto-neutron star radial scale.If we now vary the turbulence amplitude we generatefigure (9) which is, again, for a mixing angle of sin θ =0 .
1. Examining the results we quickly observe the samegeneral trends with changes in f I as seen in figure (8):increasing f I reduces the correlation with ρ affected toa greater degree than ρ , ρ , ¯ ρ and ¯ ρ . Likewise,the trends seen in figure (7) for changes in C ⋆ are alsoreproduced. IV. SUMMARY AND CONCLUSIONS
Supernova turbulence and its effects upon both theflavor composition of neutrinos that pass through it andtheir correlations depend upon many numerous param-eters one needs to introduce to describe the turbulence.All affect the result and here we try to succinctly sum-marize our results. For a neutrino energy of 25 MeV andusing a supernova density profile taken from a simulation 4 . P as a function of the emission separation δx drops considerably as C ⋆ increases for both the casesof isotropic and anisotropic turbulence. If the turbulenceamplitude is of order C ⋆ ∼ . P for neutrinos emitted from op-posite sides of the proto-neutron star, i.e. separated by ∼
10 km, is marginal for the isotropic spectrum and forthe anisotropic only when f I .
10. For C ⋆ & . ∼
10 km isjust too different to permit any correlation of this tran-sition probability in the supernova neutrino burst signal.If we switch to an inverted hierarchy then it will be thetransition probability ¯ P which behaves this way.In contrast, the correlation of the transition probabil-ities P , P , ¯ P and ¯ P in a normal hierarchy as afunction of emission separation is largely independentof C ⋆ and θ . The correlation decreases as the ratio f I = k I /k ⋆ increases but remains significant for sepa-rations of order the proto-neutron star radius even for f I ∼ P , P , ¯ P and ¯ P . These mixing channels, particularly P and¯ P , are the most promising for observing flavor scin-tillation assuming the energy resolution of our neutrinodetectors does not wash out the effect and the temporalcorrelation remains high. Acknowledgments
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