A NEW MULTI-DIMENSIONAL GENERAL RELATIVISTIC NEUTRINO HYDRODYNAMICS CODE FOR CORE-COLLAPSE SUPERNOVAE. IV. THE NEUTRINO SIGNAL

The hydrodynamics module CoCoNuT is a general relativistic finite-volume solver using higher-order piecewise-parabolic-method reconstruction (Colella & Woodward 1984) and the approximate HLLC Riemann solver (Mignone & Bodo 2005) modified to avoid odd-even decoupling and the carbuncle phenomenon at strong shocks (Quirk 1994). The metric equations are solved in the extended conformal flatness approximation (xCFC) of Cordero-Carrión et al. (2009).

The neutrino transport module Vertex integrates the energy-dependent zeroth and first moment equations for neutrino and antineutrinos of all flavors using a variable Eddington factor technique (Rampp & Janka 2002). We resort to the "ray-by-ray-plus" approximation (Buras et al. 2006b) to make the multi-D transport problem tractable. In the ray-by-ray-plus approach, the radiation field is assumed to be axially symmetric around the radial unit vector, which effectively decouple the neutrino transport problem along different rays through the origin. However, the lateral advection of neutrinos with the fluid is included in an operator-split fashion, as is the lateral neutrino pressure gradient in the optically thick regime. It should be pointed out that the ray-by-ray-plus approximation does not imply that neutrinos propagate only in the radial direction, it merely implies that the energy flux vector is radial and that lateral gradients (except for advection terms) in the moment equations are neglected. The ray-by-ray-plus approach allows us to predict angular variations in the neutrino radiation field at least in rough qualitative agreement with full multi-angle transport (Ott et al. 2008; Brandt et al. 2011). Full multi-angle transport smears out angular variations in the radiation field at larger radii, especially outside the neutrinosphere. In order to obtain better quantitative estimates for the angular variations at large distances from the proto-neutron star, we therefore reprocess our simulation data to obtain observable neutrino fluxes using the method introduced by Müller et al. (2012c) as described in Section 4.1.

The moment equations solved along each ray fully include relativistic effects (Müller et al. 2010) and energy redistribution in "inelastic" or "non-conservative" scattering reactions. An up-to-date set of neutrino interactions rates, briefly summarized in Table 1, has been used for the simulations presented in this paper. All models were computed with the "full set" of Müller et al. (2012b).

Table 1. Neutrino Physics Input

Process	Reference
$\nu A \rightleftharpoons \nu A$	Horowitz (1997; ion–ion correlations)
	Langanke et al. (2008; inelastic contribution)
$\nu \ e^\pm \rightleftharpoons \nu \ e^\pm$	Mezzacappa & Bruenn (1993)
$\nu \ N \rightleftharpoons \nu \ N$	Burrows & Sawyer (1998)a
$\nu _e\ n \rightleftharpoons e^-\ p$	Burrows & Sawyer (1998)a
$\bar{\nu }_e \ p \rightleftharpoons e^+ \ n$	Burrows & Sawyer (1998)a
$\nu _e \ A^{\prime } \rightleftharpoons e^- \ A$	Langanke et al. (2003)
$\nu \bar{\nu } \rightleftharpoons e^- \ e^+$	Bruenn (1985), Pons et al. (1998)
$\nu \bar{\nu } \ NN \rightleftharpoons NN$	Hannestad & Raffelt (1998)
$\nu _{\mu ,\tau }\bar{\nu }_{\mu ,\tau } \rightleftharpoons \nu _{e}\bar{\nu }_{e}$	Buras et al. (2003)
$\stackrel{(-)}{\nu }_{\mu ,\tau }\stackrel{(-)}{\nu }_e \rightleftharpoons \stackrel{(-)}{\nu }_{\mu ,\tau }\stackrel{(-)}{\nu }_e$	Buras et al. (2003)

Note. ^aNote that these reaction rates account for nucleon thermal motions, phase-space blocking, energy transfer to the nucleon associated with recoil ("non-conservative" or "non-isoenergetic" scattering), and nucleon correlations at high densities. Moreover, we include the quenching of the axial-vector coupling at high densities (Carter & Prakash 2002), correction to the effective nucleon mass (Reddy et al. 1999), and weak magnetism effects (Horowitz 2002). However, we ignore nucleon potential effects (Martínez-Pinedo et al. 2012; Roberts et al. 2012a), which are of minor importance during the accretion phase (Martínez-Pinedo et al. 2012).

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We have simulated the collapse and post-bounce evolution of non-rotating progenitors with 8.1 M_☉, 9.6 M_☉, 11.2 M_☉, 15 M_☉, 25 M_☉, and 27 M_☉. The 8.1 M_☉ (u8.1) and 9.6 M_☉ (z9.6) stars (A. Heger 2012, private communication) are progenitors close to the lower mass limit for iron core formation at metallicities of Z = 10⁻⁴ and Z = 0, respectively. These stars explode very quickly with little help from convection and the SASI. Three progenitors (s11.2, s25, s27) have been taken from Woosley et al. (2002) and serve as examples for a relatively fast but rather weak convectively dominated explosion (s11.2), a fast SASI-dominated explosion (s27), and a SASI-dominated model (s25), which fails to explode until more than 600 ms after bounce and shows little promise for a successful explosion at later times. Model s15s7b2 from Woosley & Weaver (1995) illustrates the case of a late explosion. The dynamics of these models has already been discussed in much detail elsewhere (Müller et al. 2012a, 2012b, 2013; Janka et al. 2012).

The core-collapse simulations have been conducted using the equation of state of Lattimer & Swesty (1991) with a bulk incompressibility modulus of K = 220 MeV (LS220) for models z9.6, s25, and s27, and with K = 180 MeV (LS180) for u8.1, s11.2, and s15s7b2. For the neutron star masses encountered in these two cases, LS220 and LS180 yield very similar results because of very similar proto-neutron star radii as discussed in Müller et al. (2013), and the incompatibility of LS180 with the observed maximum neutron star mass of ≈2 M_☉ (Demorest et al. 2010) is therefore of no immediate concern. We expect only a weak dependence of the model dynamics and the neutrino signal on the choice of bulk incompressibility (K = 220 MeV versus K = 180 MeV) for these two EoS's.

Models s11.2 and s15s7sb2 have been simulated with a reduced angular resolution of 28 (64 zones) instead of 14 (128 zones). The setup of the six simulations is summarized in Table 2, including references for more details on the model evolution and dynamics.

Table 2. Model Setup

	Metallicity	Angular	Explosion	Time of		Progenitor	Simulation
Progenitor	Z/Z☉	Resolution	Obtained	Explosiona	EoS	Reference	Reference
u8.1	10−4	14	yes	175 ms	LS180	A. Heger (private communication)	Müller et al. (2012a)
z9.6	0	14	yes	125 ms	LS220	A. Heger (private communication)	Müller et al. (2013)
s11.2	1	28	yes	213 ms	LS180	Woosley et al. (2002)	Müller et al. (2012b)
s15s7b2	1	28	yes	569 ms	LS180	Woosley & Weaver (1995)	Müller et al. (2012b)
s25	1	14	no	⋅⋅⋅	LS220	Woosley et al. (2002)	Müller et al. (2013)
s27	1	14	yes	209 ms	LS220	Woosley et al. (2002)	Müller et al. (2012a)

Note. ^aDefined as the point in time when the average shock radius 〈r_sh〉 reaches 400 km.

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The first prominent feature in Figure 1 is the well-known electron neutrino or neutronization burst that occurs when the post-shock matter becomes optically thin shortly after bounce as the newly formed shock propagates outward ("'shock breakout"). As the post-shock region is far away from neutrino-less beta-equilibrium, electron captures on protons quickly produce a large number of electron neutrinos at this stage.

The neutronization burst signal in ν_e is rather homogeneous across the different progenitors (in agreement with Mayle et al. 1987; Liebendörfer et al. 2003; Kachelrieß et al. 2005) with a maximum luminosity of (3.3...4.4) × 10⁵³ erg s⁻¹, an interval bracketed by models s15s7b2 and s25 on the lower and upper end. There is even less spread in the peak mean energy of electron neutrinos $\langle E_{\nu _e}\rangle$ reached during the burst, which ranges from 12.9 MeV (s15s7b2) to 14.2 MeV (z9.6).

As the ν_e burst subsides, the flux of $\bar{\nu }_e$ and ν_μ/τ starts to rise. While the emission of $\bar{\nu }_e$ is initially suppressed due to a relatively high electron fraction around the neutrinosphere and in the cooling region above it (which implies strong electron degeneracy), leading to a delayed rise of $\bar{\nu }_e$'s compared to the ν_μ/τ's, the $\bar{\nu }_e$'s eventually reach a flux similar to the ν_e's. The excess of electron (anti)neutrino emission compared to the heavy flavor neutrinos is due the contribution from accretion and is hence progenitor-dependent; for the low-mass progenitors u8.1 and z9.6, it is never very pronounced, whereas the electron neutrino and antineutrino luminosities can be larger almost by a factor of two for more massive progenitors.

Interestingly, the different progenitors show a different luminosity hierarchy of electron neutrinos and antineutrinos during the early accretion phase with an early crossing around ∼60 ms after bounce in the case of models s11.2 and s15s7b2 where the $\bar{\nu }_e$ luminosity slightly exceeds the ν_e luminosity (a phenomenon observed even more strongly in the 12 M_☉ model simulated by Bruenn et al. 2013). This is a subtle effect that cannot be connected to the surface properties of the proto-neutron star. During this early phase, neutrinos are emitted from a very extended region, and for electron neutrinos, emission still dominates over absorption all the way to the shock due to the deleptonization of the infalling matter. By contrast, a considerable fraction of the electron antineutrinos is absorbed in the region where net heating develops later on. The different balance between emission and absorption in the post-shock region results in a considerable re-adjustment for the electron flavor luminosities from the proto-neutron star surface and deeper layers of the cooling region, as we show in Figure 4 for model s27, which does not show the early crossing of ν_e and $\bar{\nu }_e$ luminosities. Moreover, proto-neutron star convection affects electron neutrino and antineutrino luminosities differently at early stages, generally tilting the balance in favor of electron neutrinos by transporting lepton number from deeper layers into the neutrinosphere region (Buras et al. 2006b). The hierarchy of electron neutrino and antineutrino luminosities during the early phase thus seems almost accidental, and is certainly very sensitive to small variations in the models (such as the extent of the region affected by proto-neutron star convection and the exact temporal decay behavior of the mass accretion rate in the early post-bounce phase).

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During the later accretion phase, the electron neutrino and antineutrino luminosities are largely regulated by the mass accretion rate $\dot{M}$ (Fischer et al. 2009). For high $\dot{M}$, the total electron flavor luminosity is of the order of the accretion luminosity $L_{\nu _e}+L_{\bar{\nu }_e} \sim G M \dot{M}/ r_{\rm PNS}$ (M and r_PNS being the mass and radius of the proto-neutron star, respectively), though it may deviate by a few tens of percent.

As the accretion rate drops, the relative contribution of the electron neutrino and antineutrino flux from diffusion out of the deeper PNS core becomes more appreciable. Rapid changes in $\dot{M}$ due to the infall of composition interfaces in the progenitor result in pronounced steps in the luminosity of ν_e and $\bar{\nu }_e$ (see Figure 1). The heavy flavor neutrinos show a steady decline and are not sensitive to sudden variations in $\dot{M}$.

Although the neutrino luminosities during the accretion phase are related to the proto-neutron star parameters, quantitative deductions from the luminosities appear difficult. For the heavy flavor neutrinos, one can assume a simple gray body ansatz in terms of the neutron star surface temperature T_ν and radius $r_{\rm PNS}^2$,

$$\begin{equation} L_{\nu _{\mu /\tau }}=4 \pi \epsilon \sigma _{\rm fermi} r_{\rm PNS}^2 T_{\nu }^4. \end{equation} \tag{ 4 }$$

Here, σ_fermi = 4.50 × 10³⁵ erg MeV⁻⁴ s⁻¹ cm⁻² is the radiation constant for neutrinos as left- or right-handed fermions of zero degeneracy, T_ν is an effective surface temperature approximately given by $\langle E_{\bar{\nu }_e} \rangle / 3.15$, and the grayness factor accounts for deviations from black body emission. Figure 5 shows a considerable spread in across the progenitors as well as strong non-monotonic time variations. We obtain values in the range = 0.4...0.85, which is compatible with the findings of Hüdepohl et al. (2009) for an electron-capture supernova during the first second.

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Similarly, the accretion rate $\dot{M}$ can only be inferred very tentatively from the luminosities. The accretion luminosity L_acc is already difficult to separate from the diffusive (gray body) contribution to begin with. As long as it is sufficiently high, the excess of the electron (anti)neutrino luminosity over the (anti)neutrino luminosity of a single heavy flavor may be taken as a proxy for L_acc, which may then be related to the accretion rate $\dot{M}$ by introducing an efficiency parameter η,

$$\begin{equation} L_{\rm acc}\approx L_{\nu _e}+L_{\bar{\nu }_e}-2 L_{\nu _{\mu /\tau }} = \eta \frac{G M \dot{M}}{ r_{\rm PNS}}. \end{equation} \tag{ 5 }$$

The bottom panel of Figure 5 shows that for the less massive progenitors (u8.1, z9.6, s11.2) the subtraction of a diffusive component works up to shock revival at best, where η ≈ 1. Afterward, η increases steeply as the accretion rate $\dot{M}$ plummets. For models with an extended accretion phase continuing beyond the onset of the explosion (s15s7b2, s25, s27), we find η ≈ 0.5...1 most of the time even after shock revival.

It is noteworthy that even the (directionally averaged) luminosity shows marked temporal fluctuations in models with strong activity of the SASI and/or convection, such as s15s7b2, s25, and s27. These fluctuations are a consequence of oscillations in the angle-averaged shock radius, resulting in a periodic increase and decrease of the rate of mass accretion $\dot{M}_{\rm cool}$ into the cooling region as shown in Figure 6. However, in the observable signal for a fixed viewing angle these variations in the angle-integrated luminosity are dwarfed by spatio-temporal variations in the neutrino emission, which are discussed at length in Section 4.

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The directionally averaged neutrino mean energies show an almost monotonic rise. The infall of composition interfaces, however, can lead to short phases of stagnation or even drops in the mean energies, but the mean energies are much less affected by changes in the accretion rate than the luminosities.

In all but the two least massive progenitors (u8.1 and z9.6), we eventually observe a crossing of the mean energies of electron antineutrinos and heavy flavor neutrinos, and hence a violation of the canonical hierarchy of mean energies $\langle E_{\nu _{\mu /\tau }}\rangle > \langle E_{\bar{\nu }_{e}}\rangle > \langle E_{\nu _e}\rangle$. This feature has been found by many modern 1D and 2D neutrino-hydrodynamics simulations (Liebendörfer et al. 2004; Marek et al. 2009; Marek & Janka 2009) that include some or all of the relevant thermal equilibration processes for heavy flavor neutrinos (neutrino-electron scattering, nucleon bremsstrahlung, neutrino pair conversion, energy-exchanges in neutrino-nucleon scattering reactions).

These processes are one important cause for the crossing as they push the effective heavy flavor temperature $T_{\nu _{\mu /\tau },{\rm eff}} \approx \langle E_{\nu _{\mu /\tau }}\rangle / 3.15$ down close to the matter temperature in the roughly isothermal atmosphere of the proto-neutron star (where most of the electron neutrinos and antineutrinos are produced prior to the explosion), a process discussed at length in Raffelt (2001); Keil et al. (2003). However, this does not yet account for the crossing of the mean energies, which hinges on two other factors that are illustrated in Figure 7. Figure 7 shows the evolution of the mean flux temperature T_flux and the local spectral temperature T_{ν, loc} of the different neutrino species as they propagate out of the proto-neutron star atmosphere compared to the matter temperature at a time well after the crossing for model s25. The flux temperature T_flux for species i is defined in terms of the 1st angular moment H of the energy-dependent neutrino intensity in the comoving frame as

$$\begin{equation} T_{\rm flux} = \frac{1}{3.15} \frac{\int _0^\infty H \, {d}E_\nu }{\int _0^\infty E_\nu ^{-1} H \, {d}E_\nu }, \end{equation} \tag{ 6 }$$

i.e., the temperature T_flux is obtained from ratio of the neutrino energy and number flux assuming that the spectrum is a Fermi–Dirac spectrum with vanishing chemical potential. The local neutrino temperature T_{ν, loc} is defined similarly in terms of the 0th moment J as

$$\begin{equation} T_{\nu ,{\rm loc}} = \frac{1}{3.15} \frac{\int _0^\infty J \, {d}E_\nu }{\int _0^\infty E_\nu ^{-1} J \, {d}E_\nu }. \end{equation} \tag{ 7 }$$

T_flux reflects the thermodynamic conditions in the decoupling region, and with a negative temperature gradient in the outer neutron star atmosphere one could naturally expect the standard hierarchy, since the heavy flavor neutrinos decouple at smaller radii than electron antineutrinos.

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However, Figure 7 demonstrates a loophole in this argument. As already pointed out by Liebendörfer et al. (2004), a temperature inversion can occur in the cooling region, where ν_e and $\bar{\nu }_e$ are produced by charged-current processes. The flux temperature of electron antineutrinos can therefore be higher than that of the heavy flavor neutrinos. The asymptotic value of $T_{\bar{\nu }_e,{\rm flux}}$ roughly corresponds to the local temperature maximum at ∼22 km, whereas the flux temperature of ν_μ/τ's reaches its asymptotic value around the local temperature minimum at ∼20 km, which is about 0.5 MeV lower. The temperature inversion can easily be accounted for; it is a result of the competition between neutrino cooling and adiabatic compression in the accretion layer. Assuming spherical symmetry, stationarity, and Newtonian gravity, and neglecting self-gravity we can write the energy equation for the fluid in terms of the radial velocity v_r, the specific enthalpy h, the gravitational potential Φ, and the source term $\dot{q}$ due to neutrino heating/cooling as

$$\begin{equation} \frac{{d}(h +\Phi)}{{d}t}= v_r \frac{\partial (h +\Phi)}{\partial r}= \frac{\dot{q}}{\rho }. \end{equation} \tag{ 8 }$$

With baryon-dominated gas in the cooling region, the specific enthalpy is approximately 5/2k_bT/m_n (m_n being the nucleon mass) so that the gradient of h immediately reflects the temperature gradient, and the competition between cooling and adiabatic compression becomes obvious (note that both $\dot{q}$ and v_r are negative in the cooling region):

$$\begin{equation} \frac{5 k_b}{2m_n}\frac{\partial T}{\partial r}\sim \frac{\dot{q}}{v_r}-\frac{\partial \Phi }{\partial r}. \end{equation} \tag{ 9 }$$

As the density gradient in the atmosphere steepens, the accreted matter has to radiate away its gravitational binding energy while traversing an increasingly narrower cooling region. The balance is then tilted in favor of the term $\frac{\dot{q}}{v_r}$ in Equation (9), and a positive temperature gradient in the outer proto-neutron star atmosphere arises.

Moreover, electron (anti)neutrinos of different energies decouple at different radii because the absorption and scattering cross sections scale with the square of the neutrino energy, $E_\nu ^2$. The emerging energy-dependent luminosity dL/dE_ν is essentially determined by the equilibrium intensity $\mathcal {I}_{\rm eq}$ at an optical depth of ∼2/3 and by the effective emitting surface,

$$\begin{equation} \frac{{d}L}{{d}E_\nu } \propto r(\tau =2/3)^2 \times \mathcal {I}_{\rm eq}(\tau =3/2,E_\nu). \end{equation} \tag{ 10 }$$

Because of the relatively strong variation of $\mathcal {I}_{\rm eq}$ with temperature $\mathcal {I}_{\rm eq} \propto E_\nu ^3 [1+\exp (E_\nu /k_b T- \eta _{\rm eq})]$, the emission of neutrinos with energies near the spectral maximum in the region around the local temperature maximum is enhanced compared to that of low-energy neutrinos that decouple further inside and thus at lower temperatures. As a result, the mean energy of the emerging non-thermal spectrum can be even higher than one would assume based on the temperature in the decoupling region (see Figure 7, lower panel).

Neutrinos in the high-energy tail, on the other hand, are emitted from regions where there is again an appreciable negative temperature gradient, and the spectrum therefore declines faster than a thermal spectrum for neutrinos well above the average energy. If the temperature inversion in the accretion layer is strong, this "pinching" (Janka & Hillebrandt 1989a, 1989b; Raffelt 2001; Keil et al. 2003; Tamborra et al. 2012) is no longer effective enough to keep the electron antineutrino mean energy below that of the heavy flavor neutrinos during the late accretion phase. The root mean square energy, $\langle E_\nu ^2\rangle ^{1/2}$, however, is more affected by the high-energy tail and always shows the canonical hierarchy $\langle E_{\nu _e}^2\rangle ^{1/2} < \langle E_{\bar{\nu }_e}^2\rangle ^{1/2}<\langle E_{\nu _{\mu /\tau }}^2\rangle ^{1/2}$ (see also Marek & Janka 2009).

We should also mention that the precise instant of the crossing is probably affected by weak magnetism corrections (Horowitz 2002), which are different for ν_μ/τ and $\bar{\nu }_{\mu /\tau }$. Weak magnetism lowers the opacity for antineutrinos and leads to somewhat harder spectra (by about 1 MeV; Liebendörfer et al. 2003; Bruenn et al. 2013), and this effect would also occur in $\bar{\nu }_\mu$ and $\bar{\nu }_\tau$. The Vertex transport code does not distinguish heavy flavor neutrinos and antineutrinos in its current implementation, so that we obtain mean energies for $\bar{\nu }_{\mu /\tau }$ that are slightly too low. However, simulations that treat heavy flavor neutrinos and antineutrinos separately still show the crossing despite slightly higher heavy flavor antineutrino mean energies, albeit at slightly different times for ν_μ/τ and $\bar{\nu }_{\mu /\tau }$.

The gradual rise of the neutrino mean energies is clearly a consequence of the contraction of the proto-neutron star and its rising surface temperature. Surprisingly, little attention has been paid to the dominant parameters regulating the neutrinospheric temperature during the accretion phase so far. Figure 8 shows that for each model, the electron antineutrino mean energy scales remarkably well with the proto-neutron star mass M (defined by the baryonic mass contained in regions with densities exceeding 10¹¹ g cm⁻³),

$$\begin{equation} \langle E_{\bar{\nu }_e}(t) \rangle \propto M(t). \end{equation} \tag{ 11 }$$

The transient stagnation or decrease of the mean energy associated with the infall of composition interfaces causes slight deviations from the relation. There is also a spread of ≲ 20% between different progenitor models such that more massive progenitors (s25 and s27) tend to have a somewhat lower ratio $\langle E_{\bar{\nu }_e}(t) \rangle / M(t)$. To first order, Equation (11) still accounts reasonably well for the large variation of the neutrino mean energies (11 MeV to 20 MeV for $\bar{\nu }_e$) through the post-bounce phase for different progenitors.

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How can this scaling be accounted for, and how can it be reconciled with the fact that the nuclear equation of state also affects the neutrino mean energies considerably (Sumiyoshi et al. 2005; Marek et al. 2009; Marek & Janka 2009; O'Connor & Ott 2013)? In order to answer this question, we construct a simple model for the isothermal atmosphere of the proto-neutron star whose properties determine the emerging electron neutrino and antineutrino spectrum.

We first note that the transition from a steep negative temperature gradient in the roughly isentropic accretion mantle of the proto-neutron star to an almost flat temperature gradient in the cooling region (Figure 7) roughly coincides with the neutrinosphere for $\bar{\nu }_e$ (simply because efficient cooling can only occur at low optical depth τ). At the neutrinosphere radius R_ν, we have

$$\begin{equation} \tau =\int _{R_{\rm \nu }}^\infty \kappa _{\rm eff} \,{d}r \approx 1, \end{equation} \tag{ 12 }$$

where κ_eff is the effective energy-averaged opacity for electron antineutrinos. Since the dominant absorption and scattering opacity scales with the density ρ and the neutrino energy E_ν as $\rho E_\nu ^2$, this implies

$$\begin{equation} C \int _{R_{\rm \nu }}^\infty \rho T_\nu ^2 \,{d}r \approx 1, \end{equation} \tag{ 13 }$$

where T_ν is the temperature at the average neutrinosphere (and the effective temperature of the emitted electron antineutrinos), and C is a normalization constant. The density ρ(r) in this integral is in turn determined by the density ρ_ν at the base of the roughly isothermal, baryon-dominated atmosphere (cooling region and neutrinospheric region), the temperature T_ν, and the local gravitational acceleration $GM/R_\nu ^2$ at R_ν (see Mihalas & Weibel Mihalas 1984, Chapter 2),

$$\begin{equation} \rho \approx \rho _\nu \exp \left(-\frac{G M \left(r- R_\nu \right) m_n}{R_\nu ^2 k T_\nu } \right). \end{equation} \tag{ 14 }$$

Using these relations,³ we can determine that the effective optical depth τ_ν for electron antineutrinos at the base of the atmosphere scales as

$$\begin{equation} \tau _\nu \propto \frac{\rho _\nu R_\nu ^2 T_\nu ^3}{G M}. \end{equation} \tag{ 15 }$$

ρ_ν can be eliminated from this relation if we bear in mind that for r < R_ν, the stratification in the thick accretion mantle is roughly adiabatic because it is well-mixed by proto-neutron star convection. For densities sufficiently below nuclear saturation density and down to ρ_ν, where the nearly isentropic mantle smoothly joins the isothermal atmosphere, the plasma can be described as a mixture of relativistic and non-relativistic ideal gas components, and the adiabaticity constraint therefore leads to a power-law for the temperature,

$$\begin{equation} T = K(s) \rho ^{\gamma -1}, \end{equation} \tag{ 16 }$$

where γ ≈ 3/2 is the effective adiabatic index and K(s) is a constant depending solely on the entropy s. The specific entropy in the accretion mantle (and hence K(s)) does not vary considerably across progenitors and changes very slowly during the accretion phase for the following reasons: The entropy profile during the early post-bounce phase is determined by the shock propagation through the iron core, which is rather similar for different progenitors. During the subsequent post-bounce evolution, the matter piled up on the proto-neutron star surface then cools down to the bulk entropy of the accretion matter as it settles, and essentially just extends the accretion mantle with constant entropy. The rather high mass of the mantle and a thermal relaxation (cooling) time-scale on the order of seconds also help to keep the entropy relatively constant.

After eliminating ρ_ν, the condition τ_ν ≈ 1 at the neutrinosphere translates into a relation between the neutrinosphere temperature and the surface gravity at the base of the atmosphere:

$$\begin{equation} T_\nu ^5 \propto \frac{G M}{R_\nu ^2}. \end{equation} \tag{ 17 }$$

The neutrinosphere radius $R_\nu ^2$ is essentially determined by the proto-neutron star mass only through the mass–radius relationship for neutron stars with a cold inner core (s ≈ 1 k_b/nucleon) of ≈0.43...0.5 M_☉ (depending on the EoS) and an adiabatically stratified mantle with s ≈ 5k_b/nucleon. R_ν is a relatively steep function of the proto-neutron star mass, in contrast to cold neutron stars, where R ≈ const. over a wide range of masses for nucleonic EoSs (see, e.g., Figure 1 in Müller et al. 2012b for an example with a mantle of relatively low entropy). Indeed, the steepness of the mass–radius relation for cold neutron stars partly helps to understand why proto-neutron stars with a hot mantle contract strongly with increasing mass, as we shall see in the following.

In the core of a proto-neutron star above nuclear saturation density, repulsive nucleon-nucleon interactions dominate the pressure, and thermal effects are subdominant. Down to and even somewhat below nuclear saturation density, the density profile of the proto-neutron star therefore roughly follows a TOV solution for a cold neutron star. Somewhat below nuclear saturation density, the thermal pressure of the baryons becomes the dominant contribution, and the density profile transitions into the TOV solution for an adiabatic mantle, with a much more shallow decline of the density compared to cold neutron stars at comparable densities.

Since the density profile of cold neutron stars is very steep below saturation density, the "fitting radius" between the core and the mantle is always close to the radius of a cold neutron star with a baryonic mass identical to that of the core. The steepness of the mass–radius relation for cold neutron stars (Lattimer & Prakash 2007; Steiner et al. 2010) implies that the radius of the inner core remains roughly constant as the proto-neutron star accretes. This is illustrated in Figure 9, which shows the evolution of isosurfaces corresponding to nuclear saturation density and lower fiducial densities in the neutron star for model s25.

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The outer radius of the adiabatic mantle on top of the core, i.e., R_ν, not only depends on the radius of the inner core, but also on the gravitational acceleration in the mantle: The pressure at the base of the mantle, P_core, remains fairly constant as the proto-neutron star accretes; it is roughly given by the pressure of cold neutron star matter at the fitting density. As the proto-neutron star grows in mass and the surface gravity increases, only a more and more tenuous mantle can be supported by P_core, and therefore the mantle contracts. More quantitatively, we can argue that the extent of the mantle is regulated by the pressure scale-height ℓ, which is inversely proportional to the mass of the supranuclear core M_core, $\ell \propto P_{\rm core} /(M_{\rm core}^{-1})$. Since more and more of the proto-neutron star mass is concentrated in the core as its total baryonic mass increases, and since the self-gravity of the mantle also plays a role, R_ν actually declines even faster with M than M⁻¹. The rapid contraction of isosurfaces for densities of ∼10^11...12 g cm⁻³, is also shown in Figure 9.

Of course, the precise dependence of R_ν on M can only be determined by the solution of the TOV equation either for a pre-defined stratification or by using the dynamically evolving stratification in supernova simulations in conjunction with a solution of the stationary neutrino transport equations (which is both implicitly done as we evolve the models). Empirically, it turns out that well below the maximum neutron star mass, the solutions of the TOV equation (which is implicitly reproduced in our time-dependent simulations) for such a stratification can be approximated by a power law with index

$$\begin{equation} \frac{{d}\ln R_\nu }{{d}\ln M} \sim -2, \end{equation} \tag{ 18 }$$

as shown by Figure 9. This value for the power-law index suggests precisely the mass–temperature relation we find in our simulations,

$$\begin{equation} T_\nu \propto M. \end{equation} \tag{ 19 }$$

The actual TOV solution is, of course, not a power law. Nonetheless, deviations from this power-law behavior do not change the mass–temperature relation appreciably for two reasons: First, different power laws still give relatively similar neutron star radii simply because the proto-neutron star mass range is limited (see Figure 9). Furthermore, the high power of the temperature in Equation (17) implies that even a relative deviation of 30% between different power laws is reduced to a deviation of 11% in T_ν.

Naturally, this mass–temperature relation is modified by slight progenitor variations, by the varying strength of the accretion effect described in Section 3.3, and by gravitational redshift. Progenitor variations in the mass–temperature relation can be traced to slightly different entropies in the proto-neutron star convection zone. Higher entropies in more massive progenitors (s25, s27) lead to somewhat lower neutrino mean energies.

The equation of state is another important factor that regulates the neutrino mean energies through the compactness of the proto-neutron star (Sumiyoshi et al. 2005; Marek et al. 2009; Marek & Janka 2009; O'Connor & Ott 2013). This is not inconsistent with the mass–temperature relation (19). Different equations of state will lead to different mass–radius relations for the core and mantle, and hence affect the surface gravity at the neutrinosphere. Stiffer equations of state generally result in a smaller core mass, a more massive and extended mantle, and hence somewhat smaller neutrinosphere temperatures. However, this affects only the proportionality constant in the mass–temperature relation (19). Since relativistic effects in the mantle are only of moderate magnitude and equation-of-state variations mainly change the radius of the core dominated by nucleon interactions, similar power laws are expected for different equations of state as long as the core radius varies little with mass as suggested by neutron star radius measurements (Lattimer & Prakash 2007; Steiner et al. 2010). Even a different slope of the mass–radius relationship for proto-neutron stars with a hot accretion mantle does not strongly affect the relation expressed by Equation (19) because the neutrino temperature enters as $T_\nu ^5$ in the underlying relation (17). 1D simulations with the EoS of Shen et al. (1998) indeed show a similar scaling relation for the electron antineutrino mean energy (L. Hüdepohl et al., in preparation).

Different from artificial 1D explosion models, we find no clear fingerprint of shock revival in the angle-averaged neutrino luminosities and mean energies. Specifically, there is no abrupt drop in the electron neutrino and antineutrino luminosity associated with the onset of the explosion. Models where the shock moves out very rapidly and accretion is quenched rather abruptly (like u8.1 or z9.6) also have a rather low accretion luminosity to begin with so that a strong decline of the luminosity cannot be expected. For other progenitors (s11.2 and s27), the onset of the explosion is associated with a strong decline of the luminosity, but this is just a reflection of the infall of the shell interface that triggers shock revival. Moreover, the decline of the electron neutrino and antineutrino luminosity can also be rather gradual because much of the shocked material is still channeled down to the proto-neutron star through accretion downflows in these cases. In cases where the explosion is not associated with the infall of a shell interface (model s15s7b2), or where the accretion rate remains high (e.g., because of a strongly unipolar explosion geometry), there may even be no trace at all of shock revival in the total neutrino luminosity.

The angle-averaged mean energies of electron neutrinos and antineutrinos may show only weak fingerprint of shock revival: For models u8.1, z9.6, and s11.2, the mean energies stagnate or even decline slowly over extended periods (>100 ms), implying that there is almost no further accretion onto the proto-neutron star. However, there is still a considerable rise in the mean energies in models s15s7b2 and s27 due to the high mass accretion rates after shock revival.

In models s11.2 and s27, the explosion phase is characterized by stronger non-monotonic variations in the total luminosity and the angle-averaged mean energies than during the accretion phase. Such variations on intermediate time-scales are also associated with anisotropic neutrino emission and are discussed at length in Section 4.3

Since the ray-by-ray-plus transport does not account for lateral neutrino flux components outside the neutrinosphere, the information on each "ray" for angle (θ, φ) in 3D or latitude θ in 2D can only propagate radially. In reality, an observer outside the neutrinosphere would receive flux not just from one angular zone (i.e., ray), but from the whole radiating surface facing him. The observable neutrino luminosity along a specific direction (θ, φ) therefore differs from the asymptotic value of the "ray luminosity" 4πα²ϕ⁴F_eul(r, θ, φ)r² for (r → ∞) in the integrand of Equation (2). For this reason, we reconstruct the observable signal from the ray-dependent neutrino fluxes following the method of Müller et al. (2012c), which assumes a neutrinospheric emission law $\mathcal {I}(\mathbf {n}) \propto 1+3/2 \cos \gamma (\mathbf {n})$ for the neutrino intensity as a function of the angle γ between the radial direction and the direction vector n of the emitted neutrinos. The observable luminosity L_o(n) at infinity along the direction n is then given by the integral of $\mathcal {I}(\mathbf {n})$ over the visible emitting surface (surface elements dA),

$$\begin{equation} L_o (\mathbf {n}) = 2 \int _{\rm vis.surf.} F_{\rm eul} (\mathbf {r},\theta) \left(1+\frac{3}{2} \cos \gamma (\mathbf {n}) \right) \cos \gamma (\mathbf {n}) \,{d}A. \end{equation} \tag{ 20 }$$

Observable expectation values $\langle E_o^i (\mathbf {n}) \rangle$ of powers of the neutrino energy E can be obtained in a similar manner from the first angular moment of the neutrino radiation intensity:

$$\begin{eqnarray} &&\langle E_o^i (\mathbf {n}) \rangle \nonumber \\ &&\quad= \frac{ \int _{\rm vis.surf.} \int _0^\infty E^{i-1} H (E_\nu ,\mathbf {r},\theta) \left(1+\frac{3}{2} \cos \gamma \right) \cos \gamma \, {d}E \, {d}A}{ \int _{\rm vis.surf.} \int _0^\infty E^{-1} H (E_\nu ,\mathbf {r},\theta) \left(1+\frac{3}{2} \cos \gamma \right) \cos \gamma \, {d}E \, {d}A}. \nonumber \\ \end{eqnarray} \tag{ 21 }$$

In order to study the observability of temporal variations in the neutrino emission, we use L_o(n) and $\langle E_o^i (\mathbf {n}) \rangle$ to estimate the expected signal in IceCube (Abbasi et al. 2011; Salathe et al. 2012), which is best suited for detecting fast time variations among operating detectors thanks to its excellent temporal resolution and to the prospective high event rate (see Lund et al. 2010, 2012; Tamborra et al. 2013). Since a simplified detector model is fully sufficient for the purpose of demonstration, we estimate the excess rate $\mathfrak {R}$ per time bin (Δt = 1.6384 ms) over the background due to supernova neutrinos following Halzen & Raffelt (2009):

$$\begin{equation} \mathfrak {R}=186 \, {\rm bin}^{-1} \frac{\tilde{L}_o}{10^{52} \, {\rm erg} \, {\rm s}^{-1}} \frac{ \langle \tilde{E}_o^3 \ \rangle /\langle \tilde{E}_o\rangle }{225 \, {\rm MeV}^2} \left(\frac{10 \, {\rm kpc}}{d} \right)^2. \end{equation} \tag{ 22 }$$

Here, $\tilde{L}_o$, $\tilde{E}_o^3$, and $\tilde{E}_o$ are the electron antineutrino luminosity and the third and first energy moment of the distribution function as measured on Earth, and d is the distance to the supernova. The background rate $\mathfrak {R}_0$ is taken to be

$$\begin{equation} \mathfrak {R}_0=2200 \, {\rm bin}^{-1}. \end{equation} \tag{ 23 }$$

The luminosity $\tilde{L}_o$ and the energy moments $\langle \tilde{E}_o^i \rangle$ of electron antineutrinos emitted from the supernova may be modified by Mikheyev–Smirnov–Wolfenstein (MSW) resonances in the outer shells of the progenitor and by non-linear collective neutrino flavor conversion (see Duan & Kneller 2009; Duan et al. 2010 for a review). These effects partly depend on unknown neutrino parameters (mass hierarchy); and non-linear flavor conversion is not yet completely understood, in particular in the absence of axial symmetry in the radiation field (Mirizzi 2013; Raffelt et al. 2013). As a full exploration of the possible scenarios is beyond the scope of this paper, we estimate the IceCube signal under the optimistic assumption of a normal mass hierarchy and neglect non-linear flavor conversion (which is probably suppressed until the later neutrino-driven wind phase, see Chakraborty et al. 2011; Sarikas et al. 2012a, 2012b). Furthermore, we neglect any possible reconversion of neutrinos due to the Earth effect. Under these assumptions, the observable electron antineutrino luminosity and the energy moments relevant for the IceCube signal can be expressed in terms of the unmodified quantities L_o, 〈E_o〉 and $\langle E_o^3 \rangle$ (see Kachelrieß et al. 2005)

$$\begin{equation} \tilde{L}_o = \cos ^2 \theta _{12} L_{o,\bar{\nu }_e} + \sin ^2 \theta _{12} L_{o,\bar{\nu }_x}, \end{equation} \tag{ 24 }$$

$$\begin{equation} \tilde{E}_o = \cos ^2 \theta _{12} \langle E_{o,\bar{\nu }_e} \rangle + \sin ^2 \theta _{12} \langle E_{o,\bar{\nu }_x} \rangle, \end{equation} \tag{ 25 }$$

$$\begin{equation} \tilde{E}_o^3 = \cos ^2 \theta _{12} \langle E_{o,\bar{\nu }_e}^3 \rangle + \sin ^2 \theta _{12} \langle E_{o,\bar{\nu }_x}^3 \rangle, \end{equation} \tag{ 26 }$$

where sin ²θ₁₂ = 0.311 (Beringer et al. 2012). While we cannot hope to discuss the possible impact of flavor conversion on our findings in its entirety, this simplified approach at least reflects the fact that detectors will never measure the unoscillated $\bar{\nu }_e$ signal computed in our simulations. Using the unoscillated neutrino flux would result in a systematic overestimation of the signal-to-noise ratio of features in the time-frequency domain. Our simplified flavor conversion is intended to roughly represent the most optimistic case (maximum amplitude of fluctuations in the observed electron antineutrino flux) that can possibly be encountered.

The actual detector signal will also be subject to statistical fluctuations. We therefore compute simulated IceCube signals assuming a Poisson distribution with an expectation value of $\mathfrak {R}(t)$ in each bin. Such simulated signals for the four progenitor models s11.2, s15s7b2, s25, and s27 for observers located in the north and south polar directions and in the equatorial plane are shown in Figure 10. The assumed distance is d = 10 kpc, except for model s11.2 (d = 5 kpc). As the low-mass progenitors u8.1 and z9.6 do not exhibit pronounced spatio-temporal variations in the neutrino emission, we do not discuss these models any further in this section.

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Different from earlier work on the subject (Marek & Janka 2009; Lund et al. 2010; Brandt et al. 2011; Lund et al. 2012; Tamborra et al. 2013), we not only study the frequency spectrum for the entire signal or distinct phases of the evolution, but instead analyze the full time-frequency structure of the expected IceCube signals by means of a wavelet analysis. The wavelet transform χ(t, p) of the rate $\mathfrak {R}$ depends both on time t and on the period p, and is given in terms of the mother wavelet ψ by

$$\begin{equation} \chi (t,p)= \frac{1}{\sqrt{|p|}} \int _{-\infty }^{\infty } (\mathfrak {R}(t)+\mathfrak {R}_0) \ \psi ^\star \left(\frac{t^{\prime }-t}{p} \right) {d}t^{\prime }. \end{equation} \tag{ 27 }$$

For evaluating Equation (27), we use the discrete Poisson realization of the binned IceCube data (including the background), and we employ the Morlet wavelet as mother wavelet with a scaling parameter⁴ σ = 6:

$$\begin{eqnarray} \nonumber \psi (x) &=& \left(1+e^{-\sigma ^2}-2 e^{-3\sigma ^2/4}\right)^{-1/2} \pi ^{-1/4} e^{-x^2/2} \\ &&\times \left(e^{i \sigma x} - e^{-\sigma ^2/2} \right). \end{eqnarray} \tag{ 28 }$$

In order to assess the significance of features in the wavelet spectrogram, we consider the squared signal-to-noise ratio (S/N)² between the absolute square of the wavelet transform and the expectation value 〈|χ_noise|²〉 due to the background,

$$\begin{equation} ({\rm S/N})^2= \frac{|\chi |^2(t,p)}{\langle |\chi _{\rm noise}|^2 \rangle }. \end{equation} \tag{ 29 }$$

The computation of 〈|χ_noise|²〉 for Poissonian noise in different time bins is described in the Appendix. For the discrete wavelet transform, 〈|χ_noise|²〉 depends weakly on the period p as long as p ≫ Δt and p ≪ T (where T is length of the time series). To avoid spuriously high signal-to-noise ratios for p ≈ Δt, we always use the noise level at p ≈ 50 ms.

Wavelet spectrograms showing (S/N)² for some observer directions and different distances are shown in Figures 11, 12 (s25), 13 (s15s7b2), 14 (s11.2), and 15 (s27).

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The simulated real-time signals in Figure 10 show clearly discernible temporal variations on top of the secular evolution that we already discussed for the total angle-averaged luminosity. The wavelet spectrograms reveal that these variations are indeed caused by spatio-temporal modulations in the emission and not by white noise due the detector background. It is convenient to analyze the distinct fingerprints of the pre-explosion phase and the explosion phase separately.

The most remarkable feature of neutrino emission in the pre-explosion phase is seen in strongly SASI-dominated models like s25 and s27 (bottom panels of Figure 10), which exhibit a strong quasi-periodic modulation of the signal for observers located along the direction of the SASI sloshing motions (Marek & Janka 2009; Ott et al. 2008; Brandt et al. 2011; Lund et al. 2010, 2012; Tamborra et al. 2013). In a model like s15s7b2 (top right panel of Figure 10), in which the SASI is probably present but where strong convective overturn also develops, such modulations are still visible but far less pronounced. The modulations are particularly strong for a model like s25 that either fails to explode or would eventually explode at very late times.

In such a case, SASI activity may persist over several hundreds of milliseconds with little activity of parasitic instabilities on top of the SASI, and thus with a well-defined periodicity of the oscillations. This gives rise to a distinct pattern in the wavelet spectrograms (Figures 11 and 12) that shows a continuous decrease of the SASI period as the shock retracts. This decrease may be interrupted by sudden drops in the accretion rate associated with the infall of a composition shell interface in the progenitor. The changing conditions when such an interface falls through the shock cause a transient interruption of the quasi-periodic shock oscillations (Figure 11). In the case of model s25, the drop in the accretion rate is clearly reflected in the time-dependent neutrino signal (bottom left panel of Figure 10). If SASI activity resumes afterward, it may do so with a somewhat longer oscillation period than before, which is reflected as a small step in the frequency band in the bottom right panel of Figure 11.

The SASI-induced modulations of the neutrino emission are most pronounced for observers located along the direction of SASI sloshing, but even for observers viewing the supernova from an orthogonal direction, the time-frequency structure of the neutrino signal still shows fingerprints of SASI activity (top right panel of Figure 11), in particular if we assume a distance smaller than 10 kpc, as shown by the spectrograms in Figure 12. However, the frequency appears to be much broader in this case, reaching to significantly lower periods. The bottom panel of Figure 12 (which shows the spectrogram for an observer in the equatorial plane at 2 kpc) suggests that there may be two emission bands and that the dominant modulation frequency is roughly twice as large as for a polar observer. This is just a reflection of the fact that an excursion of the shock in either direction will affect the neutrino emission in the equatorial plane in the same manner. For a real detection, the relative orientation of the observer to a putative sloshing mode of the SASI would be unknown, but the presence or absence of a secondary emission peak at lower frequencies (longer periods) would allow a discrimination between the "real" SASI frequency band and the artificial "overtone" that occurs for an observer not located along the direction of the sloshing mode. For a pure spiral mode, this overtone would probably be absent, and the viewing angle would only affect the amplitude of the modulations.

The IceCube signal from a Galactic supernova could thus not only provide unambiguous evidence about SASI activity in the supernova core, but might even give some hints about the involved modes (sloshing versus spiral modes) and could allow us to directly follow the temporal evolution of the SASI frequency. This would also provide qualitative information about the shock radius and the proto-neutron star radius, which set the oscillation period of the SASI (Foglizzo et al. 2007; Scheck et al. 2008). Quite remarkably, it is possible to formulate a rather simple quantitative relation between the time-dependent period of the IceCube signal modulations and these two radii. If the SASI is due to an advective-acoustic cycle, its period is given by the sum of the advective and acoustic time-scales for perturbations traveling between the (angle-averaged) shock radius r_sh and the radius of maximum deceleration r_∇ (Foglizzo et al. 2007; Scheck et al. 2008):

$$\begin{equation} T_{\rm SASI} = \tau _{\rm adv}+\tau _{\rm ac} = \int _{r_\nabla }^{r_{\rm sh}} \frac{{d}r}{|v_r|}+ \int _{r_\nabla }^{r_{\rm sh}} \frac{{d}r}{c_s-|v_r|}. \end{equation} \tag{ 30 }$$

Here, v_r and c_s are the (average) radial velocity and the local sound speed. The velocity profile between r_sh and r_∇ is roughly linear,

$$\begin{equation} v_r \sim -\beta ^{-1} \sqrt{\frac{G M}{r_{\rm sh}}} \left(\frac{r}{r_{\rm sh}}\right), \end{equation} \tag{ 31 }$$

where β is the ratio between the post-shock and pre-shock density. Since the flow becomes very subsonic close to r_∇, the advection time-scale τ_adv will typically be the dominant term that decides about the scaling of T_SASI with the parameters of the accretion flow (M, r_sh, r_∇). Plugging in Equation (31) into Equation (30) and neglecting the acoustic time-scale results in

$$\begin{equation} T_{\rm SASI} \propto r_{\rm sh}^{3/2} M^{-1/2} \ln \left(\frac{r_{\rm sh}}{r_\nabla } \right). \end{equation} \tag{ 32 }$$

The radius of maximum deceleration r_∇ is somewhat difficult to infer from simulations, and its dependence on other proto-neutron star parameters (mass, core radius R_core, neutrinosphere radius, gain radius, surface temperature) is rather complicated since the location of the coupling region for vorticity perturbations and acoustic perturbations also depends on the density gradient in the cooling region (Scheck et al. 2008). However, we can formulate an empirical scaling law in terms of the proto-neutron star radius r_PNS (defined by a fiducial density of 10¹¹ g cm⁻³ as in Müller et al. 2012b; Bruenn et al. 2013; Suwa et al. 2013) for the SASI oscillation period:

$$\begin{equation} T_{\rm SASI} = 19 \, {\rm ms} \left(\frac{r_{\rm sh}}{100 \, {\rm km}}\right)^{3/2} \ln \left(\frac{r_{\rm sh}}{r_{\rm PNS}} \right). \end{equation} \tag{ 33 }$$

Here, we have ignored the relatively weak dependence on M in Equation (32), which gives a somewhat better fit to the observed SASI periods when using r_PNS instead of r_∇ to estimate the advection time-scale. The prediction for T_SASI is indicated in Figures 11, 13, 14 and 15 by a red curve. In Figure 12, T_SASI/2 is also shown.

While Figures 11 and 12 demonstrate that Equation (33) describes models with continuous SASI activity very well, one should also note that the models s15s7b2 (Figure 13) and s27 (Figure 15) show some activity around the expected frequency during the phases where the SASI is active (prior to the onset of the explosion at ∼125 ms in s27) and during the phase of strong shock retraction around 300...350 ms in s15s7b2). Even the supposedly convective model s11.2 (Figure 14) shows some (faint) broadband activity around the SASI frequency prior to the onset of the explosion (at p ≈ 35 ms at a post-bounce time of ≈100 ms). Among these exploding models with somewhat less extended and less regular SASI activity than model s25, model s27 is particularly noteworthy as it shows clear signs of an increasing shock oscillation period—and hence an increasing shock radius—from 100 ms onward. This is one of the characteristic features of exploding models, which we discuss in the following section.

However, different from a model like s25, the time-frequency structure of the neutrino signal of these models is a less robust indicator for SASI as the physical mechanism underlying the emission modulation. The broadband nature, the intermittent character of the signal modulations, and the temporal variability of the dominant frequency point to a significant and perhaps dominant role of convection in models like s15s7b2 and s11.2.

As discussed in Section 3.5, the directionally averaged neutrino fluxes cannot serve as robust indicators for the onset of the explosion. The relatively abrupt decline of the luminosity seen in artificial 1D explosion models is often absent in multi-D simulations. A step-like decline in the electron neutrino and antineutrino luminosities rather points to the infall of a composition interface.

Fortunately, the spatio-temporal variations of the neutrino-emission provide several clues for diagnosing the transition from the accretion phase to the explosion phase. However, these fingerprints are not immediately obvious from a superficial visual inspection of the simulated IceCube detection rates in Figure 10. Fluctuations in the observable neutrino signal are present both prior to and after shock revival, and the amplitudes in both phases are not dissimilar. The occurrence of sustained global emission anisotropies (e.g., in model s15s7b2, top right panel of Figure 10) is a qualitative difference to the pre-explosion phase. Such global anisotropies can result from one-sided accretion in very asymmetric explosions (see our discussion of model s15s7b2 in Müller et al. 2012b), but they can obviously not be observed directly.

Nevertheless, the transition to the explosion betrays itself by quantitative and qualitative changes in the fluctuating neutrino signal: Due to the expansion of the shock, the typical frequency of the fluctuations decreases (Figures 13 and 15), and the fluctuations lack the well-defined periodicity familiar from SASI-dominated models in the pre-explosion phase. In the wavelet spectrograms of the exploding models s11.2, s15s7b2 and s27 (intermittent) broadband activity for periods longer than ≳ 20 ms therefore dominates over short-period fluctuations. In our models, a typical period of fluctuations around ∼20...25 ms seems to provide a very rough dividing line between non-exploding and exploding models. Figure 16 illustrates that for model s15s7b2 the activity peak at p = 20 ms coincides very well with the phase of shock revival. A neutrino signal dominated by fluctuations in the range of 40...50 Hz is therefore probably a good indicator for shock revival.

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It is worth noting that even during the phase of shock revival, the spectrum of the emission fluctuations still shows a preferred frequency at least in some models, albeit that the peak is rather broad. Model s15s7b2, for example, shows strong signal fluctuations around 500 ms for an observer along the north polar axis. The typical period of these fluctuations is in rough agreement with the advection time-scale between the minimum shock radius and the proto-neutron star (Figure 13 and also Figure 15 for the corresponding feature in model s27). This could suggest that some advective-acoustic feedback mechanism (which may or may not be identified with the SASI) is still active at this stage and provides a preferred time-scale for variations in the accretion flow and the neutrino emission.

Early fallback of some of the shocked material onto the proto-neutron star also gives rise to characteristic signatures in the neutrino emission after the onset of the explosion, for which model s11.2 is a prime example. As already discussed in Müller et al. (2012b, 2013), the energy of the hot, neutrino-heated material is relatively low in this case so that the high-entropy bubbles fail to push out all the material swept up by the shock after the onset of the explosion. Much of the shocked material is therefore channeled onto the proto-neutron star through long-lived downflows, but on occasion, some of the swept-up material also penetrates the expanding high-entropy bubbles to form a new downflow as illustrated in Figure 17. Such a new downflow not only excites oscillations of the proto-neutron star surface that give rise to burst-like gravitational wave emission (Müller et al. 2013), but also carries fresh material into the cooling region.

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This results in a sudden jump in the neutrino emission, which would be observable by IceCube for a Galactic supernova at a distance of 5 kpc as can be seen in the top left panel of Figure 10 (with two prominent bursts and ∼780 ms and ∼820 ms). Although the mini-bursts in the neutrino-emission are more strongly pronounced for an observer in the south polar direction (i.e., above the hemisphere where the downflow develops), they are visible from any direction as the newly injected material is quickly redistributed across the whole cooling region. As the expected count rate rises by several hundred, these events are clearly distinguishable from statistical fluctuations in the signal, which are of the order of $\sqrt{\mathfrak {R}} \approx 63$ during the relevant phase. If the jump in the detection rate is statistically significant, such mini-bursts also leave a distinct trace in the wavelet spectrogram in the form of extended vertical stripes. These patterns are most clearly visible in the case of model s11.2 (Figure 14). Model s27 also shows such mini-bursts, albeit a little less sharp (Figure 15), e.g., at ∼550 ms and ∼650 ms. The detection of such signatures would not only indicate that an explosion has been initiated prior to the burst, but would also indicate that the explosion is (still) reasonably weak at this stage such that outflow and accretion downflows persist simultaneously for an extended period of time.