Multi-Dimensional Solution of Fast Neutrino Conversions in Binary Neutron Star Merger Remnants
PPrepared for submission to JCAP
Multi-Dimensional Solution of FastNeutrino Conversions in BinaryNeutron Star Merger Remnants
Ian Padilla-Gay, Shashank Shalgar, and Irene Tamborra
Niels Bohr International Academy and DARK, Niels Bohr Institute, University of Copen-hagen, Blegdamsvej 17, 2100, Copenhagen, DenmarkE-mail: [email protected], [email protected], [email protected]
Abstract.
Fast pairwise conversions of neutrinos are predicted to be ubiquitous in neutronstar merger remnants with potentially major implications on the nucleosynthesis of the ele-ments heavier than iron. We present the first sophisticated numerical solution of the neutrinoflavor evolution above the remnant disk within a (2+1+1) dimensional setup: two spatialcoordinates, one angular variable, and time. We look for a steady-state flavor configurationabove the remnant disk. Albeit the linear stability analysis predicts flavor instabilities atany location above the remnant disk, our simulations in the non-linear regime show that fastpairwise conversions lead to minimal neutrino mixing ( < a r X i v : . [ a s t r o - ph . H E ] N ov ontents The coalescence of a neutron star (NS) with another NS or a black hole (BH) leads to thebirth of a compact binary merger. Gravitational waves (GW) from a binary neutron starmerger have been detected by the LIGO and Virgo Collaborations, the GW170817 event,together with the multi-wavelength electromagnetic counterpart [1–3]. The multi-messengerdetection of GW170817 has confirmed theoretical predictions according to which compactbinary mergers are the precursors of short gamma-ray bursts (sGRBs), one of the mainfactories where the elements heavier than iron are synthesized—through the rapid neutron-capture process ( r -process)—and power kilonovae (electromagnetic transients bright in theoptical and infrared wavebands) [4–9].The GW170817 observation has shed light on the poorly explored physics of NS mergers.However, a robust theoretical understanding of the physics of these objects is still lackingand three-dimensional general-relativistic magnetohydrodynamical simulations with detailedneutrino transport are not yet available. In particular, the role of neutrinos is especiallyunclear despite the fact that a copious amount of neutrinos is produced in the coalescence.Neutrinos should affect the cooling of the merger remnant, as well as the overall ejectacomposition, and contribute to power sGRBs [5, 10–20].A crucial ingredient possibly affecting the neutrino reaction rates and energy depositionis the neutrino flavor conversion physics, currently neglected in most of the literature onthe subject. Besides the ordinary interactions of neutrinos with matter [21, 22], in compactbinary mergers, the neutrino density is so high that ν – ν interactions cannot be neglected,– 1 –imilarly to the case of core-collapse supernovae [23–25]. A characteristic feature of compactbinary mergers is the excess of ¯ ν e over ν e due to the overall protonization of the mergerremnant [11, 26, 27]. As a consequence, a matter-neutrino resonance can occur as the matterpotential cancels the ν – ν potential [28–34].In addition to the matter-neutrino resonance, ν – ν interactions can be responsible forthe development of fast pairwise conversions [35–37]. The latter can be triggered by theoccurrence of electron lepton number (ELN) crossings in the neutrino angular distributionsand could lead to flavor conversions on a time scale G F | n ν e − n ¯ ν e | − , where G F is the Fermiconstant and n ν e ( n ¯ ν e ) is the local number density of ν e (¯ ν e ). Reference [38] pointed outthat fast pairwise conversions could be ubiquitous above the remnant disk because of theaccretion torus geometry and the natural protonization of the remnant leading to an excessof ¯ ν e over ν e .Whether flavor equipartition is achieved as a consequence of fast pairwise conversionsis a subject of intense debate, also in the context of core-collapse supernovae [39–47]. If fastpairwise conversions lead to flavor equilibration in compact binary mergers, the nucleosyn-thesis of the heavy elements in the neutrino-driven wind can be drastically affected, and thefraction of lanthanides boosted with major implications for the kilonova observations [48].The possible consequences of fast pairwise conversions on the physics of compact mergersjustify a modeling of the flavor conversion physics that goes beyond the predictions of thelinear stability analysis [37, 49, 50].Building on Ref. [41], we present the first sophisticated modeling of fast pairwise con-versions in the non-linear regime above the disk of merger remnants. We rely on a (2+1+1)dimensional setup: we track the neutrino flavor evolution in two spatial coordinates, oneangular variable, and time. We solve the equations of motion of (anti)neutrinos in the ab-sence of collisions and aim to investigate the flavor evolution of the neutrino-dense gas abovethe disk of the remnant, searching for a steady-state configuration of flavor. Our goal is toidentify the location and extent of regions with significant flavor conversion.This work is organized as follows. In Section 2, we introduce the 2D box configurationthat we adopt to model the neutrino emission and propagation above the merger remnantdisk. The neutrino equations of motion and the semi-analytical tools to explore the eventualoccurrence of flavor instabilities in the context of fast pairwise conversions are introduced inSec. 3. In Sec. 4, we present our findings on fast pairwise conversions above the massive NSremnant disk in the non-linear regime; we also explore how the steady-state flavor configu-ration is affected by variations of the input model parameters. In Sec. 5, we investigate theflavor oscillation physics above a BH remnant disk. Finally, our conclusions are presentedin Sec. 6. The routine adopted to take into account the effects of neutrino advection in thepresence of flavor conversions is outlined in Appendix A. Appendix B instead provides detailson the convergence of our results for the adopted spatial resolution. Given the numerical challenges involved in the modeling of the flavor conversion physicswithin a realistic astrophysical framework, we here focus on a simpler toy model inspiredby the ones adopted in Refs. [38, 41]. We model the neutrino emission above the remnantdisk in a 2D box with width L x and height L y , with L x = L y ≡ L = 80 km, as sketched inFig. 1. Although this is a small patch of the overall region above the merger remnant, it islarge enough to explore the development and evolution of fast pairwise conversions. First,– 2 –e model a NS-disk remnant; our findings are extended to the case of a BH-disk remnant inSec. 5. At the bottom edge of the grid ( y = 0) in Fig. 1, we locate a thin neutrino source, S ν ,of length R = L/ ν e neutrinosphere. Similarly, we considera source of ¯ ν e , S ¯ ν , of length ¯ R = 75% R . The neutrino and antineutrino emission surfaces arecentered on ( x = L/ , y = 0). The neutrino source S ν is such that x ∈ [ L/ − R, L/ R ]and similarly for S ¯ ν with the replacement R → ¯ R .Our choice of the S ν size with respect to the one of S ¯ ν is guided by hydrodynamicalsimulations of a massive NS-disk [11]. Although it is well known that the decoupling surfacesof ν e and ¯ ν e are spatially well separated, see e.g. [27, 48], we assume that the neutrinospheresof ν e and ¯ ν e are coincident and the decoupling occurs suddenly for the sake of simplicity. Aswe will discuss later, this has an impact on the formation of ELN crossings, but it does notaffect the overall flavor conversion picture above the remnant disk.We also assume that non-electron flavors are generated through flavor conversions only.In the case of NS-disk remnants, a small amount of non-electron (anti)neutrinos is naturallyproduced in the NS-disk remnant (see, e.g., Refs. [11, 51]); in this case, our extreme assump-tion enhances the likelihood of having flavor conversions and, as we will discuss in Sec. 4, itdoes not affect our overall conclusions. Our ansatz closely mimics the BH-disk remnant caseinstead (see Sec. 5 and, e.g., Ref. [52]).As for the boundary conditions in our 2D box, we assume that, except for the edge con-taining the (anti)neutrino sources, the other edges of the 2D box act as sinks for (anti)neutrinos.Since the (anti)neutrinos continuously flow from the sources into the sinks, the total numberdensity of neutrinos and antineutrinos is conserved.We work in a two-flavor approximation, ( ν e , ν x ), and denote with ν x a mixture of thenon-electron flavors. In order to describe the neutrino and antineutrino fields, we rely on2 × x, y ) point in the 2D box: ρ ( (cid:126)x, θ, t ) = (cid:18) ρ ee ρ ex ρ ∗ ex ρ xx (cid:19) and ¯ ρ ( (cid:126)x, θ, t ) = (cid:18) ¯ ρ ee ¯ ρ ex ¯ ρ ∗ ex ¯ ρ xx (cid:19) . (2.1)The diagonal terms of the density matrix encode the flavor content information and areproportional to the (anti)neutrino number densities in ( x, y ); the off-diagonal terms are con-nected to the probability of flavor transitions, as we will discuss in the next section. Assuch, we normalize the density matrices in the following way: tr( ρ ) = 1 and tr( ¯ ρ ) = a . Theparameter a takes into account the asymmetry between neutrinos and antineutrinos, and wetake a = 2 . (cid:104) E ν e (cid:105) = (cid:104) E ¯ ν e (cid:105) (cid:39)
20 MeV to mimic typical average energies in the proximity of the(anti)neutrino decoupling region; as shown in Ref. [43], the assumption of a monoenergeticdistribution reproduces the flavor outcome that one would obtain when an energy distribu-tion centered on (cid:104) E ν e , ¯ ν e (cid:105) is considered. As a consequence, the neutrino distribution, for eachpoint in the ( x, y ) box and at the time t , is defined by the emission angle θ (see Fig. 1).In order to model the physics of fast pairwise conversions, we need to take into accountthe (anti)neutrino angular distributions. We assume that the emission surfaces of neutrinosand antineutrinos are perfect black-bodies and (anti)neutrinos are uniformly emitted in theforward direction across the source, i.e., θ ∈ ( − π/ , π/
2) with θ measured with respect tothe y direction (see Fig. 1). In order to guarantee that the emitting surfaces are Lambertian– 3 – igure 1 : Schematic representation of our merger remnant setup in a 2D box of size L x = L y = L . The neutrino source ( S ν , in red) and the antineutrino one ( S ¯ ν , in blue) have widths2 R and 2 ¯ R and are centered on ( x, y ) = ( L/ , ν e and ¯ ν e in the forward direction only i.e. θ ∈ ( − π/ , π/ θ : ρ ee ( θ ) = cos θ × (cid:40) x ,ν ∈ S ν exp (cid:16) ( x − L/ ∓ R ) σ (cid:17) otherwise , (2.2)¯ ρ ee ( θ ) = a cos θ × (cid:40) x , ¯ ν ∈ S ¯ ν exp (cid:16) ( x − L/ ∓ ¯ R ) σ (cid:17) otherwise , (2.3)where σ , ¯ σ smooth the edges of S ν and S ¯ ν and are set to 20% R and 20% ¯ R , respectively.By projecting the neutrino and antineutrino angular distributions from the sources onany ( x, y ) point in the 2D box, we obtain the contour plots in Fig. 2 for the resultant angle-integrated density matrices of ν e and ¯ ν e , (cid:82) dθρ ee ( (cid:126)x, θ ) and (cid:82) dθ ¯ ρ ee ( (cid:126)x, θ ), in the absence offlavor conversions (see also Sec. 3) for the NS-disk remnant configuration. One can see thatthe neutrino density gradually decreases as one moves from S ν and S ¯ ν towards the edges ofthe box.In order to explore the variation of the (anti)neutrino angular distributions across the2D box, Fig. 3 displays the angular distributions of ν e and ¯ ν e in the points A, B, and Chighlighted in Fig. 2. The width of the ELN crossings varies as one moves away from S ν and S ¯ ν , with implications on the flavor conversion physics, as we discuss in Sec. 3.– 4 –
20 40 60 80 x [km]020406080 y [ k m ] R ρ ee dθ × − x [km]020406080 y [ k m ] R ¯ ρ ee dθ × − NS-disk remnant: no oscillations
Figure 2 : Contour plots of the angle-integrated density matrices, ρ ( (cid:126)x, θ, t ) (on the left)and ¯ ρ ( (cid:126)x, θ, t ) (on the right), in the absence of flavor conversions for the NS-disk remnantconfiguration. This configuration can be obtained by solving Eqs. 3.1 and 3.2 for H ( θ ) =¯ H ( θ ) = 0, i.e. the time evolution of ρ ( (cid:126)x, θ, t ) and ¯ ρ ( (cid:126)x, θ, t ) is completely determined by theadvective operator (cid:126)v · (cid:126) ∇ (see Sec. 3 for more details). The quantities (cid:82) ρ ee dθ and (cid:82) ¯ ρ ee dθ are normalized to the maximum total particle number in the box [ (cid:82) ( ρ ee + ¯ ρ ee + 2 ρ xx ) dθ ].The coordinates of the points A, B, and C marked on the plane are: ( x, y ) (cid:39) (56 ,
1) km,(67 ,
1) km, and (72 ,
1) km, respectively. The (anti)neutrino density gradually decreases asone moves away from S ν and S ¯ ν . In this section, we introduce the equations of motion adopted to track the flavor evolutionabove the NS-disk remnant in our 2D setup. In order to gauge the role of neutrino flavor con-versions above the NS-disk remnant, we discuss the variation of the ν – ν interaction strengthacross the 2D box and introduce the instability parameter to characterize the depth of theELN crossings. We also adopt the linear stability analysis to compute the growth rate of theflavor instabilities in the regions with the largest instability parameter. The (anti)neutrino field is described through the density matrix approach introduced inSec. 2. Neglecting collisions, the flavor evolution of neutrinos and antineutrinos is describedby the following set of equations of motion (EoM): i (cid:18) ∂∂t + (cid:126)v · (cid:126) ∇ (cid:19) ρ ( (cid:126)x, θ, t ) = [ H ( θ ) , ρ ( (cid:126)x, θ, t )] , (3.1) i (cid:18) ∂∂t + (cid:126)v · (cid:126) ∇ (cid:19) ¯ ρ ( (cid:126)x, θ, t ) = [ ¯ H ( θ ) , ¯ ρ ( (cid:126)x, θ, t )] , (3.2)where the advective term, (cid:126)v · (cid:126) ∇ , is proportional to the velocity of (anti)neutrinos, whichwe assume to be equal to the speed of light, and is tangential to the neutrino trajectory.– 5 – . . . . . ρ ee ( θ ) , ¯ ρ ee ( θ ) Location A ρ ee ¯ ρ ee . . . . . ρ ee ( θ ) , ¯ ρ ee ( θ ) Location B − π/ − π/ π/ π/ θ . . . . . ρ ee ( θ ) , ¯ ρ ee ( θ ) Location C
Figure 3 : Angular distributions of ρ ee (red) and ¯ ρ ee (blue) in A, B and C (see Fig. 2). Thepresence of two disjoint grey areas imply the existence of ELN angular crossings. The angle-dependent density matrix elements are normalized to the maximum total particle number in( x, y ) A , B , C : ( ρ ee + ¯ ρ ee + 2 ρ xx ). As one moves away from S ν and S ¯ ν , the width of the ELNcrossings varies with implications on the flavor conversion physics.The contour plots of the angle-integrated density matrices of ν e and ¯ ν e in Fig. 2 can beobtained by solving Eqs. 3.1 and 3.2 when the right hand side of both EoMs is vanishing,i.e., (anti)neutrinos do not change their flavor.The neutrino Hamiltonian is H ( θ ) = ω (cid:18) − cos 2 θ V sin 2 θ V sin 2 θ V cos 2 θ V (cid:19) + H νν ( (cid:126)x, θ ) , (3.3)with the first term depending on the vacuum frequency ω = 0 . − , where ω = ∆ m / (cid:104) E ν e , ¯ ν e (cid:105) ,∆ m is the atmospheric squared mass difference and (cid:104) E ν e , ¯ ν e (cid:105) the average mean energy of ν e ’sand ¯ ν e ’s introduced in Sec. 2. The vacuum mixing angle is θ V = 10 − ; note that we assumea very small mixing to effectively ignore the matter potential [53]. The second term of theHamiltonian is the ν – ν interaction term: H νν ( (cid:126)x, θ ) = µ ( | (cid:126)x | ) (cid:90) dθ (cid:48) (cid:2) ρ ( (cid:126)x, θ (cid:48) , t ) − ¯ ρ ( (cid:126)x, θ (cid:48) , t ) (cid:3) (cid:2) − cos( θ − θ (cid:48) ) (cid:3) . (3.4)The potential, µ ( | (cid:126)x | ), parametrizes the strength of neutrino-neutrino interactions for eachpoint ( x, y ) in the box and its functional form is defined in Sec 3.2. The Hamiltonian ofantineutrinos, ¯ H ( θ ), is identical to H ( θ ) except for the following replacement: ω → − ω [54].– 6 –he integration over dθ (cid:48) is a consequence of our 2D setup. In a 3D box, the integration over dθ (cid:48) would be replaced by an integration over the solid angle. We have checked, however, thatthe integration over d cos θ that would arise in an azimuthally symmetric 3D system virtuallygives the same results as our 2D setup (see also Sec. 3.2). The ν – ν interaction potential varies across our 2D box, by taking into account the dilutionof the (anti)neutrino gas as we move away from the sources S ν and S ¯ ν . We parametrize it as µ ( | (cid:126)x | ) = µ η ( | (cid:126)x | ) , (3.5)where η ( | (cid:126)x | ) is a scaling function, and µ = 10 km − is the ν – ν interaction strength at theneutrinosphere [38].Since the modeling of flavor evolution in 3D is computationally challenging at present,we mimic the 3D setup by solving the EoM in 2D while taking into account the dilution ofthe neutrino gas in 3D. For an observer located at ( x, y ), the distance d above the source, S ν, ¯ ν , can be computed as d = dy/ cos θ , where dy is the vertical displacement from the sourceto ( x, y ), see Fig. 1. For observers that are not located above the source, the dilution of theflux is determined by the distance d . With this convention, the scaling function η is definedas η = (cid:32) − (cid:112) ( R/d ) + 1 (cid:33) (cid:34) arccos (cid:32) (cid:112) ( R/d ) + 1 (cid:33) − (cid:115) − R/d ) + 1 (cid:35) − . (3.6)To better understand the role of η , let us look at one limiting case for an observer along theaxis of symmetry. When dy (cid:29) R , η ∝ ( R/dy ), hence H νν ∝ ( R/dy ) for a 3D bulb model,as expected [54]. Figure 4 shows µ ( | (cid:126)x | ) in our 2D box (see Eq. 3.5). At the (anti)neutrinoemission surfaces, µ assumes the maximum value ( µ ) and drops as a function of the distancefrom the source. A favorable condition for the development of fast pairwise conversions is the presence of ELNcrossings between the angular distributions of ν e and ¯ ν e [37]. To this purpose, the “instabilityparameter” has been introduced in Ref. [41] to gauge the growth rate of flavor instabilities;the latter being dependent on the depth of the ELN crossings [55, 56]: ζ = ρ tot I I ( I + I ) , (3.7)where ρ tot is the total particle number defined as (cid:82) [ ρ ee + ¯ ρ ee + 2 ρ xx ] dθ and the factors I , are defined as I = (cid:90) π/ − π/ Θ [ ρ ee ( θ ) − ¯ ρ ee ( θ )] dθ and I = (cid:90) π/ − π/ Θ [¯ ρ ee ( θ ) − ρ ee ( θ )] dθ ; (3.8)the Heaviside function, Θ, is vanishing for ρ ee ( θ ) − ¯ ρ ee ( θ ) < ζ vanishes when the ELN crossing is zero. Theinstability parameter is a useful predictor of the growth rate of the off-diagonal components– 7 –
20 40 60 80 x [km]020406080 y [ k m ] . . . . . . . . µ [ k m − ] Figure 4 : Contour plot of the neutrino self-interaction strength, µ ( | (cid:126)x | ), in the 2D box. Theneutrino-neutrino potential is maximum in the proximity of the (anti)neutrino source and itgradually decreases as the distance from the (anti)neutrino sources increases.of the density matrices and, therefore, of the flavor instabilities (see Sec. 3.4 of Ref. [41] formore details).The left panel of Fig. 5 shows a contour plot of the instability parameter in the absenceof flavor conversions across our 2D box. One can see that ζ is large in the proximity of theedges of the neutrino emitting surfaces ( x (cid:39) ,
65 km and y ∈ [0 ,
15] km) and it graduallydecreases as we move away from the sources, since the (anti)neutrino gas dilutes and the ELNcrossings become less prominent. As a consequence, and by taking into account that µ ( | (cid:126)x | )decreases as we move away from S ν and S ¯ ν (see Eq. 3.5), we should expect fast pairwiseconversions to possibly occur where the ζ parameter is larger. Also, it is worth noticing that ζ is approximately zero in the central region of the emitting sources ( x ∈ [20 ,
60] km and y ∈ [0 ,
15] km), this is mostly a consequence of the fact that we assume the neutrinospheres of ν e and ¯ ν e to be coincident with each other, despite differing in width. Similarly to Ref. [38],we expect to find a suppression of the flavor instabilities in the proximity of the emittingsurfaces around the polar region ( ζ is very small in our case) and a growth of the instabilitiesat larger distances from the source ( ζ becomes larger).Note that flavor conversions affect the (anti)neutrino angular distributions. Hence,the instability parameter shown in Fig. 5 can be dynamically modified by fast pairwiseconversions. However, the plot provides with insights on the regions where flavor conversionsmay have larger effects, as we will see in Sec. 4. In order to explore the growth of the off-diagonal term in the density matrices, and thereforethe development of fast pairwise conversions, we first rely on the linear stability analysisto analytically predict the growth rate of the flavor instabilities [49, 50]. Note that, given– 8 –
20 40 60 80 x [km]020406080 y [ k m ] . . . . . . ⇣ p a r a m e t e r
60 65 70 75 80 x [km]05101520 y [ k m ] . . . | I m ( ⌦ ) | / µ [ ] Figure 5 : Left:
Contour plot of the instability parameter ζ (see Eq. 3.7) in the 2D box in theabsence of flavor conversions for the ν e and ¯ ν e distributions in Fig. 2. The ELN crossings aresignificant just above the edges of the emitting sources. Right:
Contour plot of | Im( ω ) | /µ for the homogeneous mode for the benchmark NS-disk model in one of the regions where theinstability parameter is the largest (see dashed lines in the left panel). A maximum growthrate | Im( ω ) | = 0 . µ (cid:39) ω = 0 in this section; suchassumption is justified since ω (cid:54) = 0 would mainly affect the non-linear regime of fast pairwiseconversions [43].We linearize the EoM and track the evolution of the off-diagonal term through thefollowing ansatz ρ ex ( θ ) = Q ( θ ) e − i Ω t and ¯ ρ ex ( θ ) = ¯ Q ( θ ) e − i Ω t , (3.9)where Ω = γ + iκ represents the collective oscillation frequency for neutrinos and antineu-trinos. If Im(Ω) (cid:54) = 0, then the flavor instability grows exponentially with rate | Im(Ω) | ,leading to fast pairwise conversions [37]. Note that we look for temporal instabilities for thehomogeneous mode ( (cid:126)k = 0), as these are the ones possibly leading to fast pairwise conver-sions in extended regions [38]; by adopting a similar disk setup, Ref. [38] found that spatialinstabilities occur in much smaller spatial regions than the temporal instabilities.The off-diagonal component of Eq. 3.1 is i ∂∂t ρ ex ( θ ) = H ee ( θ ) ρ ex ( θ ) + H ex ( θ ) ρ xx ( θ ) − ρ ee ( θ ) H ex ( θ ) − ρ ex ( θ ) H xx ( θ )= H ee ( θ ) ρ ex ( θ ) − ρ ee ( θ ) H ex ( θ ) , (3.10)where we have assumed ρ xx ( t = 0 s) = ¯ ρ xx ( t = 0 s) = 0. By substituting Eq. 3.9 in theequation above and solving for Q ( θ ), we obtain Q ( θ ) = ρ ee ( θ ) (cid:82) dθ (cid:48) [ Q ( θ (cid:48) ) − ¯ Q ( θ (cid:48) )][1 − cos ( θ − θ (cid:48) )] − Ω µ + (cid:82) dθ (cid:48) [ ρ ee ( θ (cid:48) ) − ¯ ρ ee ( θ (cid:48) )][1 − cos ( θ − θ (cid:48) )] . (3.11)– 9 – similar procedure follows for ¯ Q θ (see Eqs. 3.2 and 3.9). Then, combining the expressionsfor Q ( θ ) and ¯ Q ( θ ), we have Q ( θ ) − ¯ Q ( θ ) = (cid:90) dθ (cid:48) (cid:16) ρ ee ( θ ) − ¯ ρ ee ( θ ) − Ω µ + A ( θ ) (cid:17) [ Q ( θ (cid:48) ) − ¯ Q ( θ (cid:48) )][1 − cos ( θ − θ (cid:48) )] , (3.12)where A ( θ ) = (cid:82) dθ (cid:48) [ ρ ee ( θ (cid:48) ) − ¯ ρ ee ( θ (cid:48) )][1 − cos ( θ − θ (cid:48) )]. From the equation above, it must betrue that Q ( θ ) − ¯ Q ( θ ) = (cid:34) ρ ee ( θ ) − ¯ ρ ee ( θ ) − Ω µ + A ( θ ) (cid:35) ( a − b cos θ − c sin θ ) , (3.13)where a, b, c are unknown coefficients. By substiting Eq. 3.13 in Eq. 3.12, we obtain a systemof equations for the coefficients a , b , and c . Since the variable θ (cid:48) is a dummy variable, wereplace it by θ : abc = I [1] −I [cos θ ] −I [sin θ ] I [cos θ ] −I [cos θ ] −I [cos θ sin θ ] I [sin θ ] −I [cos θ sin θ ] −I [sin θ ] abc = M abc , (3.14)where the functional I [ f ] is I [ f ] = (cid:90) dθ (cid:34) ρ ee ( θ ) − ¯ ρ ee ( θ ) − Ω µ + A ( θ ) (cid:35) f ( θ ) . (3.15)The system of equations has a not trivial solution ifdet(M −
1) = 0 . (3.16)The latter equation is polynomial in the frequency Ω. To search for instabilities, we need tolook for the solutions with Im(Ω) = κ (cid:54) = 0. We then use the SciPy module [57] in Python tofind the roots numerically.The right panel of Fig. 5 shows the growth rate, | Im(Ω) | /µ , for a region of our 2D boxwhere the instability parameter is the largest (see the highlighted region in the left panel ofFig. 5). In the region of the 2D box corresponding to the edges of S ν , | Im(Ω) | /µ (cid:39) . .
06; if we compare our findings to the ones reported in the top panel of Fig. 3 of Ref. [38],we obtain a roughly comparable growth rate of the flavor instability. We should highlightthat we assume the ν e and ¯ ν e neutrinospheres to be exactly coincident with each other(although having different widths) while a two-disk model was considered in Ref. [38]; thisquantitatively affects the depth of the ELN crossings in the polar region above the remnantin the proximity of the source. We also note that we model differently the edges of the(anti)neutrino sources and the (anti)neutrino angular distributions with respect to Ref. [38]and this causes differences in the shape of the unstable regions above the NS-disk remnant. Most of the existing work in the context of neutrino flavor conversions above the remnantdisk focuses on exploring the phenomenology of slow collective oscillations and the matter-neutrino resonance [28–34]. The only existing literature on fast pairwise conversions in merger– 10 –emnants relies on the linear stability analysis to explore whether favorable conditions forfast conversions exist above the remnant disk [38, 48], as also discussed in Sec. 3.4. In thissection, we present the results of the numerical evolution in the non-linear regime of fastpairwise conversions above the NS-disk remnant and discuss the implications for the mergerphysics. We then generalize our findings by exploring the parameter space of the possible ν e –¯ ν e asymmetries expected above the NS-disk remnant and the relative ratio between thesize of the ν e and ¯ ν e sources. We solve the EoM introduced in Sec. 3 for the box setup described in Sec. 2 by following theprocedure outlined in Sec. 3.2 of Ref. [41]. In the numerical runs, we adopt N x = N y = 50number of bins for the x − y grid and N θ = 300 angular bins to ensure numerical convergence.In order to quantify the amount of flavor mixing, we introduce the angle integratedsurvival probabilities P ( ν e → ν e ) = (cid:82) dθ [ ρ ee ( (cid:126)x, t ) − ρ xx ( (cid:126)x, t = 0 s)] (cid:82) dθ [ ρ ee ( (cid:126)x, t = 0 s) − ρ xx ( (cid:126)x, t = 0 s)] , (4.1)and similarly for P (¯ ν e → ¯ ν e ) with the replacement ρ → ¯ ρ . Figure 6 shows the survivalprobabilities of ν e and ¯ ν e as functions of time for the three selected ( x, y ) locations (A, B,and C) in the 2D box, see Fig. 2. One can easily see that fast pairwise conversions takesome time to develop, but then they reach a “steady-state” configuration and the survivalprobability stabilizes, despite smaller scale oscillations, without changing dramatically.In the presence of flavor conversions, for each ( x, y ) point in the 2D box, flavor conver-sions develop on a time scale shorter than the advective time scale [41]. To take into accountthe (anti)neutrino drifting through the 2D box, for each ( x, y ) location in the 2D box, wetranslate the time-averaged neutrino and antineutrino density matrices from each spatial binto the neighboring bins after a time ∆ t (cid:39) O (10 − ) s, i.e. after the flavor conversion prob-ability in ( x, y ) has reached a steady-state configuration; we keep all the parameters withineach spatial bin unchanged, except for following the flavor conversions for smaller time in-tervals. This procedure is implemented in an automated fashion as described in Appendix Aand it naturally allows to recover the flavor configuration shown in Fig. 2 in the absence offlavor conversions. We stress that our procedure automatically allows to take into accountthe dynamical evolution of the angular distributions as a function of time, due to neutrinosstreaming from the neighboring bins.As seen in Fig. 6 the (anti)neutrino occupation numbers oscillate around an averagevalue after the system has reached the non-linear regime. In an astrophysical system, at agiven point in space, only the time-averaged occupation numbers are the relevant quantitiesas long as the size of the region over which neutrinos and antineutrinos are emitted is largerthan the length scale over which neutrinos and antineutrinos oscillate. The aforementionedcondition should always be satisfied above the remnant disk because of the short time-scalesover which fast flavor conversions occur.It is worth noticing that, while Fig. 2 represents the resultant angular distributions of ν e and ¯ ν e in the absence of flavor conversions across the 2D box, by streaming the oscillated(anti)neutrinos to their neighboring bins, we also modify the angular distributions dynami-cally. In Ref. [41], it was shown the neutrino advection smears the ELN crossings hinderingthe development of fast pairwise conversions; such an effect would eventually become efficienton time scales longer than ∆ t , i.e. after the steady-state configuration has been reached in– 11 – . . . . P ( ν e → ν e ) Location CLocation BLocation A0 1 2 3 4 5 6 7Time [10 − s]0 . . . . P ( ¯ ν e → ¯ ν e ) Figure 6 : Temporal evolution of the survival probabilities of ν e (top) and ¯ ν e (bottom)for the three selected locations A, B and C shown in Fig. 2 (see Eq. 4.1). After a certaintime, ∆ t (cid:39) O (10 − ) s, the survival probabilities have reached a steady-state configuration.Minimal flavor mixing is achieved for all three locations.our 2D box. Moreover, the ELN crossings in our system are assumed to be self-sustained intime because of the disk geometry and its protonization.For a selection of points close to S ν and S ¯ ν , we have also tested that the growth rateobtained from the linear stability analysis in Sec. 3.4 perfectly matches the linear regime ofthe numerical solution. In particular, for the point B in Fig. 2, we find | Im(Ω) | /µ = 0 . | Im(Ω) | /µ = 0 .
049 by relying on the stability analysis.As discussed in Ref. [43], the dependence of the linear growth rate on ω is overall negligible. Figure 6 shows the temporal evolution of the ν e and ¯ ν e survival probabilities as discussed inSec. 4.1. Even though flavor unstable solutions are predicted to exist almost at any locationabove the disk of the remnant and the linear stability analysis suggests a large growth rate,as shown in Fig. 5, our results show that fast pairwise conversions lead to a few percentvariation in the flavor transition probability. At most, an average value of P ( ν e → ν e ) (cid:39) . P (¯ ν e → ¯ ν e ) (cid:39) .
90, due to the lepton number conservation.To better explore the development of fast pairwise conversions as a function of theemission angle, the top panel of Fig. 7 shows ρ ee and ¯ ρ ee (respectively proportional to the ν e and ¯ ν e number densities) as functions of the emission angle θ for the observer locatedin B in Fig. 2. The angular distributions are displayed at t = 0 s ( i , dashed lines) andat t = 7 . × − s ( f , solid lines) when the flavor conversions have reached a steady-state configuration (see Fig. 6 and Appendix A). Initially, the ELN crossings are large for– 12 – . . . . ρ ee ( θ ) , ¯ ρ ee ( θ ) ν e ¯ ν e . . . . . | ρ f ee ( θ ) − ρ i ee ( θ ) | / ρ i ee ( θ ) ν e ¯ ν e − π/ − π/ π/ π/ θ . . . . ρ xx ( θ ) , ¯ ρ xx ( θ ) ν x ¯ ν x ρ, ¯ ρ diagonal terms at location B Figure 7 : Top:
Density matrix elements ρ ee ( θ ) (in red) and ¯ ρ ee ( θ ) (in blue) as functionsof the emission angle θ at t = 0 s (dashed curves, i ) and at t = 7 . × − s (solid curves, f ) at the selected location B of Fig. 2. The density matrix elements are normalized to ρ tot . Middle:
Relative difference between the final state ( f ) and initial state ( i ) of ρ ee ( θ ) (red)and ¯ ρ ee ( θ ) (blue), respectively. Bottom:
Same as the top panel, but for the density matrixelements ρ xx ( θ ) (black) and ¯ ρ xx ( θ ) (green). Fast pairwise conversions only affect the tail ofthe angular distributions of ν e and ¯ ν e inducing minimal changes. θ ∈ [ π/ , π/ ν e and ¯ ν e . As t increases, the angulardistributions of ν e and ¯ ν e most prominently evolve around the angular bins in the proximityof the ELN crossing, as highlighted in the middle panel of Fig. 7, until the density matricesreach a stationary value. In the bottom panel of Fig. 7, one can see that newly formed ν x and¯ ν x angular distributions peak in a very narrow θ interval where the ELN crossings occur. As aresult, ν x (¯ ν x ) will predominantly propagate outwards and away from the remnant symmetryaxis, thus having a marginal impact on the polar region of the system.Figure 8 summarizes our findings across the 2D box by displaying contours of the angle-integrated density matrix elements for neutrinos (on the left) and antineutrinos (on the– 13 – y [ k m ] R ρ ee dθ . . . . . . . × − y [ k m ] R ρ xx dθ . . . . . . × − x [km]01530456075 y [ k m ] R | ρ ex | dθ × − R ¯ ρ ee dθ × − R ¯ ρ xx dθ . . . . . . × − x [km] R | ¯ ρ ex | dθ × − NS-disk remnant: with oscillations
Figure 8 : Contour plots of the angle-integrated elements of the density matrices ρ ( (cid:126)x, θ, t )(one the left) and ¯ ρ ( (cid:126)x, θ, t ) (on the right) for the NS-disk remnant configuration normalizedto the maximum total particle number within the box i.e. (cid:82) dθ ( ρ ee + ¯ ρ ee + 2 ρ xx ), after∆ t = 10 − s, after the flavor distribution has reached a steady-state configuration. Threeselected locations (A, B, and C) are highlighted (see Fig. 2). The red box defines the regionscanned with higher spatial resolution, see Appendix B for details. As also shown in Fig. 5,fast pairwise conversions are more prominent near the edges of the neutrino sources. A smallamount of ν x ’s and ¯ ν x ’s is produced through fast pairwise conversions within the narrowopening angles at the edges of the neutrino surfaces. The ELN crossings are almost vanishingalong the axis of symmetry leading to practically no flavor conversions in the polar regionabove the NS-disk remnant. – 14 –ight) for the NS-disk remnant configuration when the steady-state configuration for flavorconversions is reached. The top panels are almost identical to the ones in Fig. 2 becauseof the overall small amount of flavor conversions despite the large instability parameter andgrowth rate (see Fig. 5). From the middle and the bottom panels, we can clearly see thatflavor conversions occur in the region at the edges of the emitting surfaces where ζ is larger,but they have a negligible role in the polar region above the remnant disk where the neutrino-driven wind nucleosynthesis could be affected [48].For completeness, Appendix B includes results of a high resolution run performed inthe red box in Fig. 8. The overall amount of flavor conversions is comparable in the low andhigh resolution simulations; for this reason, we have chosen to rely on simulations with lowerresolution in order to explore a larger region above the remnant disk.By comparing Figs. 5 and Fig. 8, we conclude that the high linear growth rate of fastpairwise conversions does not imply an overall large flavor conversion in the non-linear regime.However, we should stress that ours is the first numerical study of fast pairwise conversionsabove the merger remnant in the non-linear regime; as such, for the sake of simplicity, we haveneglected the collision term in EoM. The collision term may potentially play a significantrole, also because it generates a backward flux of (anti)neutrinos [42, 58], which is neglectedin our setup. A better refined modeling of the neutrino conversion physics may affect theflavor outcome with implications for the r -process nucleosynthesis [38, 48]. In order to gauge how our findings are modified for different configurations of the NS-diskremnant, in this section we vary the ν e –¯ ν e asymmetry parameter, a (see Eq. 2.3) within therange allowed from hydrodynamical simulations [32], as well as the relative ratio between thesizes of S ¯ ν and S ν ( ¯ R/R ).Figure 9 shows a contour plot of the maximum of the instability parameter ζ (Eq. 3.7,computed in the absence of flavor conversions) computed across the 2D box for each ( ¯ R/R, a ).The markers in Fig. 9 highlight three disk configurations that we have evolved numerically;the NS-disk configuration introduced in Sec. 2 is correspondent to ( ¯
R/R, a ) = (0 . , .
4) (bluecircle in Fig. 9). The conversion probability in the steady-state regime is P ( ν e → P ν x ) (cid:39) . R/R [green diamond, P ( ν e → P ν x ) (cid:39) .
04] and even more for the red triangle,where the instability parameter is maximal [ P ( ν e → P ν x ) (cid:39) . R/R ratio not realizable in astrophysical environments. Ourfindings suggest that flavor equilibration due to fast pairwise conversions is never achieved inour setup, despite the large growth rate predicted by the linear stability analysis. In addition,the regions in the 2D box that are most unstable are located in the proximity of the edges ofthe neutrino source, and no flavor conversions occur in the polar region above the NS-disk.
We now extend our exploration of the phenomenology of fast pairwise conversions to theBH-torus configuration. In this case, the neutrino (antineutrino) source, S BH ν ( S BH¯ ν ), isidentical to the NS-disk remnant case except for an inner edge located at R BH = 1 / R [27] incorrespondence of the innermost stable circular orbit, i.e. the sources do not emit particles for x ∈ [ L/ − R BH , L/ R BH ]; all other model parameters are identical to the ones introducedin Sec. 2. We observe that, in the case of the BH remnant, the neutrino and antineutrinos– 15 –verage energies are slightly higher than in the case of the massive NS remnant, see e.g. [48,52]. However, since minimal variations in ω do not affect the final flavor configuration [43],we keep ω unchanged for simplicity.Figure 10 shows the resultant angle-integrated density matrices for neutrinos and an-tineutrinos. By comparing Figs. 8 and 10, we can see that differences appear in the proximityof the inner source edges and just above the polar region, but the flavor distributions arecomparable at larger distances from the source. Also, in this case, the most unstable regionsappear in the proximity of the source external edges and minimal flavor conversions take placein the polar region, although more pronounced than for the NS-disk remnant configuration(see the bottom panels of Fig. 10 and Fig. 8).Our findings suggest that the flavor equipartition assumption adopted in Ref. [48] toexplore the implications on the nucleosynthesis of the heavy elements is difficult to achievefor our BH-disk configuration, despite the large growth rate predicted by the linear stabilityanalysis (see Sec. 3.4). Similarly to the NS-disk configuration, the most unstable regionsare located in the proximity of the edges of the neutrino sources. However, in this case,minimal conversions occur in the polar region above the BH-disk where the neutrino windmay dominate the r -process outcome; these findings are in rough agreement with the unstableregions reported in the top panel of Fig. 7 of Ref. [48] where a growth rate of the same order R/R ν e –¯ ν e a s y mm e tr y p a r a m e t e r , a . . . . . . . ρ tot xx /ρ tot ee ’ . × − ρ tot xx /ρ tot ee ’ . × − ρ tot xx /ρ tot ee ’ . × − . . . . . ζ m a x Figure 9 : Contour plot of the maximal value of the instability parameter (Eq. 3.7) in theparameter space defined by the relative ratio between the ¯ ν e and ν e source sizes ( ¯ R/R ) and the ν e –¯ ν e asymmetry parameter ( a ). For each point in the parameter space, corresponding to aNS-disk configuration, the maximum value of ζ is computed in the absence of oscillations. Inorder to gauge the overall amount of flavor conversion, the three colored diamonds representthree NS-disk configurations for which we have tracked the flavor evolution numerically. Thetransition probability is reported in the legend for each of the three selected configurations.A slightly larger transition probability is obtained for smaller ¯ R/R ratios.– 16 – y [ k m ] R ρ ee dθ . . . . . . . × − y [ k m ] R ρ xx dθ . . . . . . × − x [km]01530456075 y [ k m ] R | ρ ex | dθ × − R ¯ ρ ee dθ × − R ¯ ρ xx dθ . . . . . . × − x [km] R | ¯ ρ ex | dθ × − BH-disk remnant: with oscillations
Figure 10 : Same as Fig. 8 but for the BH-disk remnant configuration. Despite the differencesin the source geometry, the final flavor configuration is comparable to the NS-disk remnantconfiguration. This suggests that the specific details of the source geometry do not lead tofinal state flavor configurations that are dramatically different.of the one plotted in the right panel of Fig. 5 was obtained.It is worth noticing that Ref. [48], by studying the evolution of the BH-torus as afunction of time, reported an excess of ν e with respect to ¯ ν e in the polar region at late times(e.g., after 20 ms) as a result of the dynamical evolution of the merger remnant. This effect– 17 –s not taken into account in our simplified BH-disk, since we focus on a smaller time interval[∆ t (cid:39) O (10 − ) s] and we do not take into account modifications of the neutrino emissionproperties due to the dynamical evolution of the remnant. Neutron star merger remnants are dense in neutrinos, and the occurrence of electron leptonnumber (ELN) crossings between the angular distributions of ν e and ¯ ν e seem to be ubiquitousas a natural consequence of the disk protonization and the source geometry. If fast pairwiseconversions of neutrinos should occur, leading to flavor equipartition, this has been shownto lead to major consequences for the synthesis of the elements heavier than iron and therelated kilonova observations [38, 48]. However, the existing literature on the subject focuseson predicting the existence of eventual flavor unstable regions by relying on the linear stabilityanalysis.In the light of the possible major implications for the source physics, for the first time,we solve the flavor evolution above the disk remnant in a (2+1+1) dimensional setup: twospatial coordinates, one angular variable, and time. This is the first computation of fastpairwise conversions above the merger disk in the non-linear regime.For simplicity, we adopt a two-dimensional model with two coincident ν e and ¯ ν e neu-trinospheres, and a different size for the two sources. We look for the final steady-stateconfiguration in the presence of fast pairwise conversions by neglecting the collisional term inthe equations of motion and by mimicking a configuration where a massive neutron star sitsat the center of the remnant disk (NS-disk configuration) and a configuration with a blackhole remnant (BH-disk configuration). In addition, we scan the parameter space of the pos-sible disk model parameters predicted by hydrodynamical simulations to test the robustnessof our findings.We find that the most unstable regions favoring the occurrence of fast pairwise con-versions are located in the proximity of the edges of the neutrino emitting surfaces. Only aminimal flavor change occurs in the polar region above the merger remnant in the BH-diskconfiguration, but flavor conversions are almost absent in the surroundings of the polar regionin the NS-disk configuration. Fast pairwise flavor conversions are triggered early on and asteady-state configuration for the flavor ratio (modulo small high frequency modulations) isreached within O (10 − ) s.Even though flavor unstable solutions are predicted to exist almost at any location abovethe disk of the remnant with a large growth rate, as already shown in the literature, our resultspoint towards minimal flavor changes ( < r -process nucleosynthesis. However, our findings should betaken with caution given the approximations intrinsic to our modeling. An interplay betweenfast and slow ν – ν interactions in the context of the matter-neutrino resonance [28–34] mayoccur, and a full solution of the flavor evolution in 3D may change the flavor outcome yetagain.This work constitutes a major step forward in the exploration of fast pairwise conversionsin the context of compact merger remnants from a quantitative perspective. Our findingssuggest that a complete modeling of the neutrino flavor conversion physics should be takeninto account in order to make robust predictions for the electromagnetic emission expectedby the merger remnant and its aftermath. – 18 – cknowledgments We thank Meng-Ru Wu for useful comments on the manuscript. This project was supportedby the Villum Foundation (Project No. 13164), the Danmarks Frie Forskningsfonds (ProjectNo. 8049-00038B), the Knud Højgaard Foundation, and the Deutsche Forschungsgemein-schaft through Sonderforschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro- andParticle Physics” (NDM).
A The evolution algorithm
In order to explore the flavor configuration achieved in our 2D box after a certain time ∆ t ,we take into account neutrino advection in the EoM and aim to look for a “steady-state”flavor configuration, i.e. for a configuration where the survival probability of (anti)neutrinoshas reached a constant value as a function of time except for small oscillations around thatvalue. In this Appendix, we describe the algorithm adopted to transport the (anti)neutrinogas through the advective operator ( (cid:126)v · (cid:126) ∇ ) in the EoM.As sketched in Fig. 11, we evolve in time the flavor content in the different S i regionsin the box, individually and sequentially. We start from the one closest to the (anti)neutrinosources, S ν and S ¯ ν , through S N y at the opposite edge of the box.In our algorithm, the time-averaged density matrices are transported from S i to S i +1 ,if a steady-state configuration of flavor conversions has been reached, e.g. when the averagevalues of | ρ ex | , | ¯ ρ ex | do not change more than a few percent ( (cid:39) S N y is reached. Figure 11 : Schematic representation of the algorithm implemented to determine the fi-nal steady-state configuration reached in our 2D system The neutrino and antineutrinodecoupling regions, S ν and S ¯ ν , are plotted in red and blue, respectively. The regions S i ( i = 1 , , ..., N y ) that reach a steady-state flavor configuration as a function of time areshown in green. – 19 – − s] − − − − − − l og ( | R ρ e x ( θ ) d θ | ) Location CLocation BLocation A − s] − − − − − l og ( | R ρ e x ( θ ) d θ | ) Location B
Figure 12 : Top:
Temporal evolution of | (cid:82) ρ ex dθ | (solid) and | (cid:82) ¯ ρ ex dθ | (dashed) matrixelements at the locations A, B and C, see Figs. 2 and 6. The exponential growth of theoff-diagonal terms, and therefore, of the flavor instabilities, develops within a few ns. At alater stage, the system becomes highly non-linear and reaches an approximate steady-state. Bottom : Temporal evolution of | (cid:82) ρ ex dθ | (location B), but tracking its temporal evolutionfor almost an order of magnitude longer. The dashed lines highlight the small variation ofthe transition probability, which allows to compute a steady-state flavor configuration.The sequential and pixel-by-pixel time evolution of the box is well motivated by physicalarguments, namely, by the fact that ν – ν interactions occur locally. A neutrino located at( x, y ) can only affect its nearest-neighbouring background neutrinos at ( x ± δx, y ± δy ), where δx, δy are infinitesimal displacements (the length of δx, δy being set by the pixel length). Inaddition, the fact that we only stream neutrinos from the sources towards the opposite edgesof the simulation box, and do not propagate them backwards, guarantees that a steady-stateconfiguration is always achieved throughout the box.– 20 – Spatial resolution
In this appendix, we discuss the convergence of our results, especially regarding the spatialresolution adopted in this work. In order to do this, we solve the EoMs in a smaller box of8 × (high resolution run), corresponding to the red box in Fig. 8, while maintainingthe same number of grid points and all other input quantities unchanged. We follow theflavor evolution for 5 × − s by including neutrino advection at each time step. The redbox has been located in one of the most unstable regions above the emitting surfaces, hencewe expect that our test on the convergence of our results will provide an estimation of thelargest possible error in the prediction of the flavor conversion probability.Figure 13 shows a contour plot of the angle integrated density matrix elements, ρ ee ( (cid:126)x, θ, t )and ¯ ρ ee ( (cid:126)x, θ, t ), at 5 × − s for the region highlighted by the red box in Fig. 8. The toppanels have been obtained by using higher spatial resolution in the 8 × box, while thebottom panels represent the red box in Fig. 8. The overall amount of flavor conversion iscomparable in the low and high resolution cases. However, due to the better spatial resolu-tion, small scale structures develop across a small patch in the high-resolution run (see greenpatches in the top left panel of Fig. 13) in correspondence of the unstable regions found inthe right panel of Fig. 5. It is important to notice that the occurrence of a relatively largerconversion rate in a smaller spatial region in Fig. 13 does not lead to a spread of the flavorinstability to nearby spatial bins. The overall flavor conversion rate averaged over a largearea is thus unaffected by the presence of small scale structures.Figure 14 shows the time evolution of the angle-integrated ρ ex for point B in the highresolution run (in red) and in the low resolution run (in green), and for point D in the highresolution run (in orange, see Fig. 14). Location D has been chosen as representative of themost unstable region in the top panels of Fig. 13. One can see that the error in predictingthe amount of flavor conversions in our low resolution runs is less than 1% across the regioninspected in Fig. 13 and up to 10% for the small stripe showing the largest flavor conversions.Our findings are not surprising as the angular distributions for nearby bins are very similarto each other.Importantly, any spatial correlation between nearby spatial cells is averaged out by theadvective term in the non-linear regime, as discussed in Ref. [41]. Hence, the small localizedregion with slightly larger flavor conversions surrounding point D, in the high resolution runin Fig. 13, does not affect our overall conclusions. We stress that the red box in Fig. 8corresponds to the region with the largest amount of flavor conversions, hence our spatialresolution allows to obtain even more accurate results for any remaining location above theremnant disk. Given the negligible difference in the overall flavor outcome between the tworuns with different spatial resolution, we choose to adopt the coarser grid throughout thepaper since it allow us to explore a larger region above the remnant disk and better gaugethe role of neutrinos in compact binary mergers.Our results thus show the limitation of intuitive conclusions that can be drawn byrelying on the linear stability analysis which imply a strong correlation between variousspatial points. The collective nature of the flavor evolution is less manifest in the non-linearregime; this allows to perform numerical simulations over a coarser simulation grid than onemay anticipate. – 21 – x [km]02468 y [ k m ] R ⇢ ee d✓ . . . . . . . ⇥
60 62 64 66 68 x [km] R ¯ ⇢ ee d✓ ⇥ NS-disk remnant: high resolution
60 62 64 66 68 x [km]02468 y [ k m ] R ⇢ ee d✓ . . . . . . . ⇥
60 62 64 66 68 x [km] R ¯ ⇢ ee d✓ ⇥ NS-disk remnant: low resolution
Figure 13 : Analogous to Fig. 8, contour plots of the angle-integrated elements of the densitymatrices, ρ ee ( (cid:126)x, θ, t ) (on the left) and ¯ ρ ee ( (cid:126)x, θ, t ) (on the right) for the NS-disk configuration.The simulation domain is defined through a 8 × spatial grid for the top panels (highresolution run) and a 80 ×
80 km grid for the bottom panels (low resolution run, adoptedthroughout the paper); in both cases, the plotted region corresponds to the red box in Fig. 8.Two selected locations (B and D) are used to inspect the temporal evolution of the survivalprobability, see Fig. 14. The presence of small spatial structure does not affect the overallflavor evolution in the neighboring regions. References [1]
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