Photons from neutrinos: the gamma ray echo of a supernova neutrino burst

When a star undergoes core collapse, a vast amount of energy is released in a ~10 s long burst of neutrinos of all species. Inverse beta decay in the star's hydrogen envelope causes an electromagnetic cascade which ultimately results in a flare of gamma rays - an"echo"of the neutrino burst - at the characteristic energy of 0.511 MeV. We study the phenomenology and detectability of this flare. Its luminosity curve is characterized by a fast, seconds-long, rise and an equally fast decline, with a minute- or hour-long plateau in between. For a near-Earth star (distance D<1 kpc) the echo will be observable at near future gamma ray telescopes with an effective area of 10^3 cm^2 or larger. Its observation will inform us on the envelope size and composition. In conjunction with the direct detection of the neutrino burst, it will also give information on the neutrino emission away from the line of sight and will enable tests of neutrino propagation effects between the stellar surface and Earth.


INTRODUCTION
A core collapse supernova is the most powerful neutrino emitter known so far.The ∼ 10 s-long burst of thermal neutrinos emitted from the outskirts of the collapsed core is the main cooling mechanism, and is a powerful diagnostic tool of the physics that takes place in the very dense and hot region deep inside the star.
Interestingly, one of the best supernova neutrino detectors is the most abundant element in the universe, Hydrogen.Indeed, the process of inverse β decay (IBD), νe + p → e + + n, has a relatively large, well known, cross section (Vogel & Beacom 1999;Strumia & Vissani 2003) and, depending on the type of detector, it can Cecilia.Lunardini@asu.edumkm7190@psu.eduprovide information on the energies and arrival times of the individual neutrinos detected.This simple, reliable method has found application in water and liquid scintillator detectors (Scholberg 2012), and it was used in the first and only detection of supernova neutrinos, the burst from SN1987A (Hirata et al. 1987;Bionta et al. 1987;Alekseev et al. 1988).Its evolution has been driven by the need of having larger detector masses; e.g., about O(100) kt mass of water is needed for high statistics detection of supernovae beyond our galaxy.
The concept of Hydrogen as detector leads to an idea: why not use the vast mass of Hydrogen in or near the star itself as detector?This question was first studied several decades ago, when it was observed in Bisnovatyi- Kogan et al. (1975); Ryazhskaya (1999) that inverse beta decay in the hydrogen envelope of a collapsing star leads to a transient signal of positron annihilation (e + + e − → γ + γ) signatures, mainly in the form of 0.511 MeV gamma rays (Lu & Qian 2007) 1 .Due to the geometry of the system (Figure 1), these gamma rays arrive at Earth as an echo, spread over a characteristic time ∆t ∼ R/c (with R being the star's radius), relative to the neutrino burst.In those early studies, the predicted luminosity of this echo was considered too low for observation, and therefore this phenomenon was largely ignored since.
In this letter, we present a modern study of the gamma ray echo of a supernova neutrino burst.There are two main elements of novelty.The first is a prediction of the gamma ray light curve, and its dependence on the main parameters.The second is the discussion of the potential of upcoming gamma-ray surveys to observe the echo from nearby core-collapse supernovae, and extract important information from it.With improved, nextgeneration, gamma ray telescopes like COSI (Tomsick 2021) (already funded), AMEGO (Caputo et al. 2019;Kierans 2020), and AMEGO-X (Caputo et al. 2022), detecting this 511 keV signal will soon be a realistic possibility.
The paper is organized as follows.We discuss the formalism of the gamma ray echo, along with its timedependent flux in section 2. The detectability of the echo, and relevant backgrounds are discussed in section 3. We summarize and discuss future prospects in section 4.

FORMALISM
To fix the ideas, we focus on the case which is most favorable for detection, where the gamma rays originate near the surface of the star, and propagate without absorption to Earth (attenuation will be discussed below).Let's begin by estimating the total photon flux at Earth.We assume a spherically symmetric star, and model the flux of νe reaching its surface (after flavor conversion, see, e.g.Duan & Kneller 2009 for a review) as having total energy E ν,tot = 5 10 52 ergs.
The commonly used "alpha spectrum" is assumed for its energy distribution (Keil et al. 2003;Tamborra et al. 2012), with the first two momenta being ⟨E ν ⟩ = 15 MeV and ⟨E2 ν ⟩ = 293.2MeV 2 (corresponding to the shape parameter α = 2.3, where (1 + α) −1 = ⟨E 2 ν ⟩/⟨E ν ⟩ 2 − 1).For simplicity, we use a time-independent spectrum, so the total number of νe emitted is, simply, N ν = E ν,tot /⟨E ν ⟩ ≃ 2.1 10 57 .Two cross sections are relevant here: one is the spectrum-averaged IBD cross section, ⟨σ IBD ⟩ = 2.05 10 −41 cm 2 , the other is for the Compton scattering of gamma rays, σ C = 3 10 −25 cm 2 (Rybicki & Lightman 1986), which is the main channel of photon absorption at the energies of interest (Lu & Qian 2007).As an approximation, we consider that the emerging flux of gamma rays is entirely due to the νe that interact in the outermost layer of the star, a very thin shell of width equal to the gamma ray Compton optical depth (l C ∼ O(10 9 ) cm, l C ≪ R, see Lu & Qian 2007).One can then express the number of positrons produced in this layer, and the corresponding number of gamma rays from positron propagation that leave the star as (Ryazhskaya 1999): where Y p ∼ 1 and Y e ∼ 1 are the proton and electron fractions in the stellar matter; η γ ∼ 2 is the effective number of 0.511 MeV gamma rays produced per positron (see below), and the factor 1/2 accounts for the fact that half of the gamma rays propagates inwards and is absorbed.By symmetry arguments, it follows immediately that the time-integrated photon flux at Earth is 2 We estimate the uncertainty on N + and N γ to be about ∼ 50% in either direction due to the uncertainty in the neutrino spectrum parameters.

Time-dependent gamma ray flux
To compute the expected time-dependent gamma ray flux and its energy spectrum, it is necessary to consider a specific stellar environment and model the positron propagation in detail.Here we follow the extensive discussion in Lu & Qian (2007), where a precise estimate for η γ is found.There, the values Y p = 0.7 and Y e = 0.85 are used.It is shown that positron annihilation is the main channel of gamma ray production, dominating over the secondary channel -the emission of 2.22 MeV gamma rays from neutron capture -by roughly two orders of magnitude in flux.For a hydrogen envelope in thermodynamic equilibrium at temperature T ∼ 10 4 K and density ρ ∼ 10 −8 g cm −3 , it was found that positron thermalization is -in the vast majority of cases -fast, occurring over a typical time scale of ∼ 10 −2 s, with the excitation of free electrons being the dominant energy loss mechanism.Direct positron annihilation with free electrons is the main absorption process, and the probability that it occurs before thermalization is estimated to be P f a ≃ 0.1 (see eq. ( 53) in Lu & Qian (2007)).Therefore, roughly a fraction P ∼ 1 − P f a ≃ 0.9 of all positrons annihilate after thermalization.A more detailed calculation, which includes several other energy loss and absorption processes, and the formation of positronium states, leads to P ≃ 0.87 (see fig. 3 and related text in Lu & Qian (2007)).The annihilation of thermalized positrons results in a gamma ray spectrum that is centered at E γ = 0.511 MeV, with width (full width at half maximum) ∆E γ ≃ 2 keV.Accounting for the fact that each annihilation produces two photons, we therefore estimate η γ = 2P = 1.74 which will be used here as reference value.
Let us now describe the expected gamma ray lightcurve at Earth, for a star at distance D, and a given neutrino number luminosity L ν (t) = dN ν /dt.The gamma ray flux at Earth, Φ γ , is obtained by integrating over the visible surface of the star, and by considering that photons reach the detector with a time delay that increases with their angular distance, θ, from the line of sight (see Figure 1).We find the expression: which is consistent with Equation (1), and where c is the speed of light, and B is a box function normalized to 1: B(y) = c/R for 0 ≤ y ≤ R/c, and B(y) = 0 elsewhere.The second line of Equation (3) emphasizes that the echo is described by a convolution operation (see, e.g., the formalism in Dwek et al. 2021).Here t = 0 is set to be the start of the neutrino burst as observed at Earth.

Results
Using the expression in Equation 3, the gamma ray flux can be computed for specific scenarios of neutrino emission.For illustration, we model the neutrino luminosity as a truncated exponential: where the limit t 0 → +∞ (no truncation) well approximates the case of a neutron-star forming collapse, where the proto-neutron star cools smoothly by neutrino emission over several tens of seconds.The case t 0 ≲ 1 s could describe a collapse with direct black-hole formation (failed supernova, see, e.g.O'Connor & Ott 2011; Pejcha & Thompson 2015;Ertl et al. 2016), for which the neutrino emission is truncated sharply when the neutrinosphere falls within the gravitational radius.Here we take t 0 = 1 s (for black hole formation) and τ = 3 s.The description in Equation ( 4) captures the main features of the luminosity curve over a multi-second timescale, which is sufficient for the present scope.We note that fast fluctuations of L ν (t) (over a time scale 0.1 s or less) such as those expected in the first second or so of the neutrino burst (Foglizzo 2001;Blondin et al. 2003;Foglizzo 2002) would in any case be smoothed out by the integration in Equation ( 3), and therefore have a negligible effect on the gamma ray lightcurve.
The result for the case of a neutron-star forming collapse is sufficiently simple, and is given by: The results are shown in Figure 2 and 3 (solid curves).
The expression in Equation ( 5) describes the "phases" of the star as seen by an observer at Earth (see illustration in Figure 2): first, there is an increase in flux (0 ≤ t ≤ R/c), when the surface of the star becomes bright in a circle around the line of sight, and the circle expands.After a time comparable with the neutrino emission timescale, t ∼ 3τ , the luminosity of the echo has reached a plateau.This behavior describes the phase where the gamma-ray emitting region of the star -as seen at Earth -is made of an expanding annulus where the intensity of emission is at its maximum, whereas the region near the line of sight emits less intensely due to the decline on L ν .The plateau lasts until t = R/c, when the entire visible face of the star has become bright in gamma rays; at later times the star still appears completely illuminated, but the gamma ray flux declines over a timescale ∼ τ because all the points on its surface are receiving a neutrino flux that is past its peak luminosity.
As shown in Equations ( 3) and ( 5), and Figure 3, the echo becomes fainter and longer for larger envelope radii; for a reference radius R = 10 13.5 cm and distance D = 1 kpc to the star, we estimate a duration of R/c ≃ 10 3 s (approximately 17 minutes) and maximum flux Φ γ ∼ 10 −6 cm −2 s −1 .
The case of a failed supernova -for which the analytical result is complicated, and will be omitted for simplicity -is described in Figure 3 (dotted lines).Qualitatively, the behavior is similar to the previous case, with the difference that the transition between phases is sharper, reflecting the sudden drop of L ν .In this case, an observer at Earth would see a sharp boundary between a fully illuminated annulus and a completely dark circle centered at the line of sight.Note that, by construction, the total energy emitted in neutrinos is the same for the two types of collapses.Therefore, due to the shorter time scale of the emission, for a failed supernova the rise phase of the echo is more luminous and could be more easily observed.

Attenuation
Let us briefly discuss the attenuation of the emitted gamma rays due to propagation in the circumstellar medium (CSM) and interstellar medium (ISM).The attenuation factor is η abs = exp − ds κ ν ρ , where κ ν , and ρ are the monochromatic opacity and density of the medium respectively, and the integral is performed along the line of sight.For simplicity, we approximate the CSM using the radially averaged density and temperature profiles from Georgy et al. (2013), where typical values are in the range: −29 ≲ log(ρ/g/cm 3 ) ≲ −22.5, 2.5 ≲ log(T /K) ≲ 8.5.For these, the 0.511 MeV photon interactions are dominated by Compton scattering (in the Klein-Nishina regime): κ ν ≈ 0.071 cm 2 /g.Absorption by atoms and ions is negligible, being most effective at lower energy (see Figure 4, where the corresponding spectral lines are shown).We find that attenuation is practically negligible; for example η abs − 1 ≃ 1.5 10 −6 .Predicted gamma ray lightcurves for a neutron-star-forming collapse (NSFC, solid lines) and a direct black-holeforming collapse (BHFC, dotted lines), at distance D = 1 kpc and different stellar radii (labels on curves).Dashed: νe flux for the two cases (see right vertical axis for scale).For all the results, we used Eν,tot = 5 10 52 ergs for the total energy emitted in νe.The galactic background is shown as well.For both signal (one case only, for illustration) and background, the shadings represent the uncertainties discussed in the text.for a CSM of 15 pc width, and η abs − 1 ≃ 2.2 10 −4 for propagation in the ISM over a distance of 1 kpc.

DETECTABILITY
In this section, we focus on the detectability of the gamma ray echo and discuss the relevant backgrounds.The echo is detectable in principle if it produces at least one photon signal in a detector at Earth: N s = AΦ γ,tot ≳ 1, where A is the effective area of the telescope, which we assume to be pointing in the direction of the star (incidence angle θ i = 0).Using Equation (2), we obtain the detectability condition (for fixed neutrino flux parameters): Current or upcoming instrument typically have A ≲ 10 2 cm 2 (see, e.g, COSI Tomsick 2021 and 511-CAM Shirazi et al. 2023), and would therefore only be able to observe an echo from a very nearby collapse with an exceptionally luminous neutrino emission.In particular, the NASA funded detector COSI has A > 20 cm 2 as design sensitivity at 511 keV, along with a 25% field of view of the sky and an angular resolution of < 4.1 • .With a factor of 2 improvement on its design specification (A = 40 cm 2 ), COSI would be able to see the echo from stars at D ∼ 0.2 kpc, like supernova candidates Betelgeuse and ϵ Pegasi.For the largest telescopes of The detectability of the echo depending on the distance, D, and the stellar radius, R, for a telescope with A = 103 cm 2 area.The horizontal (solid) lines correspond to a fixed number of signal events Ns (numbers on curves).The region outside the shaded areas is where the signal can be distinguished from the background (see text), assuming a (fixed) galactic background flux Φ gal = 3 10 −6 cm −2 s −1 (darker shading) and Φ gal = 3 10 −5 cm −2 s −1 (lighter shading) respectively.The markers represent the nearby stars for which the radii are known, and correspond to different intervals of realistic (directionspecific) signal-to-background ratio; see legend and Table 1.

Backgrounds
Realistically, detection requires the signal to be distinguishable above the relevant backgrounds, mainly the diffuse galactic background at E γ = 0.511 MeV, which peaks in the direction of the Galactic Center (see, e.g., Diehl 2013;Roland 2016;Diehl et al. 2021;Frontera et al. 2021).We do not consider possible contamination of an echo signal from other nearby sources like gamma ray bursts, other supernovae, supernova remnants, low mass X-ray binaries, and others.Positron annihilation from competing processes in the stellar envelope is unlikely to contribute, as the temperature of the envelope is well below the threshold for thermally producing electron-positron pairs, and positrons from (post-collapse) radionuclides near the core of the star would be located too deep to affect the echo over its short timescale.
Let us estimate the diffuse galactic background at E γ = 0.511 MeV.We take the value dΦ gal /dΩ ≃ 3 10 −4 cm −2 s −1 sr −1 as a reference for this flux away from the galactic center (see results in Skinner et al. 2015, for galactic latitude and longitude b = 0 • and |l| = 60 • ).Taking the published angular resolution of AMEGO, δθ ≃ 3 • 3 (corresponding to a solid angle δΩ = π(δθ) 2 ≃ 10 −2 sr), we obtain a flux Φ gal ≃ 3 10 −6 cm −2 s −1 , which is comparable to the maximum value predicted for the echo for R ≳ 10 13 cm and D = 1 kpc (see Figure 3).This indicates that realistically, a distance below the kpc scale is needed for a robust identification of the echo.A rough criterion for the signal to be detectable above the background is that the number of signal photon counts exceeds the one of the background over the duration of the echo: N s /N B ≳ 1, where N B ≃ Φ gal AR/c4 .Numerically, we get: If N s ≳ 10, one can use the (less stringent) requirement that the signal exceeds a 3σ Gaussian fluctuation of the background:

Prospects for detection amongst nearby supernova candidates
Since a detection is possible only for near-Earth collapses, we have examined the known hydrogen-rich supernova candidate stars within a radius of 1 kpc; they are listed in Table 1 (taken from Mukhopadhyay et al. 2020 with some updates, see table caption).For each, we report the estimated distance and radius, the value of Φ gal in the direction of the star and our result for N s /N B .The direction-specific Φ gal is evaluated from Skinner et al. (2015) (see Figure 2 there).In the absence of a a detailed three dimensional model of the galactic background, the angular separation between the galactic center and the star has been calculated only along the galactic longitude (l), assuming latitude b = 0.For stars with angular separation exceeding 90 • , we take the fixed value Φ gal = 3.0 10 −7 cm −2 s −1 , which is an overestimate, and therefore leads to conservative conclusions.
The conditions for detectability, Equations ( 6) and (7) (and its extension to Gaussian statistics, see above), are illustrated in Figure 5, for two fixed values of the galactic background flux, Φ gal = 3 10 −6 cm −2 s −1 and Φ gal = 3 10 −5 cm −2 s −1 .The candidate stars in Table 1 are shown as well, color coded according to the realistic (direction-specific) signal-to-background ratio given in the Table .From the figure, it appears that the detection of the echo is possible -with a telescope having A ≳ 10 3 cm -for several candidates.Many of them are in a fortunate location, where the galactic background is low.Examples are Betelgeuse and Rigel, for which the angular separation from the galactic center is ∼ 157 • and ∼ 123 • respectively, and therefore the background is overestimated.Rigel is also favored by is relatively compact size, R ≃ 5 10 12 cm, which implies a higher peak of the echo flux (see Figure 3).In contrast, Antares -which is similar in distance and radius to Betelgeuse -is disfavored by its proximity to the Galactic Center, which causes a higher background (Φ gal ≃ 2.3 10 −5 cm −2 s −1 ) and lower signal-to-background ratio.

DISCUSSION AND FUTURE PROSPECTS
The observation of the gamma ray echo could reveal information that would otherwise be inaccessible.In particular, the echo provides information on the νe flux passing through the stellar surface facing the observer.Therefore, the comparison with the detected neutrino burst would test the intensity of the neutrino emission away from the line of sight.It would also allow to search for exotic effects that might affect the neutrinos between the star and Earth, like neutrino decay, conversion into sterile states, scattering on Dark Matter, etc.Our predicted gamma ray flux might serve as a reference for searches of gamma-ray-producing effects beyond the Standard Model, like axion-photon conversion (see, e.g.Chattopadhyay et al. 2023 for a related idea).The echo could also provide independent estimates of the progenitor star's features, mainly the radius and envelope composition.Its initial, rising phase could also contribute to early alerts of the collapse, preceding the explosion (or collapse into black hole) of the star.
This work can be extended in many ways, to obtain more realistic predictions.For example, post-main sequence massive stars undergo radial pulsations with periods of 100 − 1000 days (Goldberg et al. 2020), which would change the envelope structure, and therefore influence the lightcurve of the gamma ray echo.One could also use numerical results for the stellar envelope structure and composition and for the neutrino luminosity, that might include deviations from spherical symmetry.Different, more realistic forms of the time profile of the neutrino luminosity could be examined (e.g., a power-law form, see Suwa et al. 2021).Including secondary branches of positron propagation, and the contribution of the layers of the star deeper than the gamma ray optical depth may result in more optimistic predictions for the gamma ray flux.The idea of a gamma ray echo could be extended to other set-ups, for example involving supernova progenitors with Carbon-or Oxygenrich envelopes -where gamma rays can be produced by neutrino-nucleus scattering -, or envelope-stripped stars where the echo might be due to a detached hydrogen shell (for example, the shell between two companion stars, see, e.g., Pejcha et al. 2022).The latter, however, might give very faint echoes due to their more extended structure (larger R).
In conclusion, we have presented a modern rendering of the idea of supernova neutrinos producing a gamma ray echo.This phenomenon is conceptually interesting, because the neutrinos take the unusual role of being the source of an electromagnetic signal, and is also attractive as a realistic target of observation for future large gamma ray telescopes with sub-MeV capability.It adds another facet to the very rich landscape of multimessenger astronomy.Table 1.Nearby red and blue supergiants and their estimated radii, distances and positions; adapted from Mukhopadhyay et al. (2020).Also given are the angular separation from the galactic center (along l, assuming b = 0), the associated 511 keV galactic background, and the predicted signal-to-background ratio.We only list stars to which our scenario applies, namely, stars that have a hydrogen envelope and for which the radius R is known.

Figure 1 .
Figure 1.Geometry of a gamma ray echo.

Figure 2 .
Figure2.Example of neutrino (dashed, vertical axis on the right) and gamma ray (solid, vertical axis on the left) lightcurves.See legend for the stellar radius and distance to Earth.Included are time snapshots of the star, showing the part that shines in gamma rays as seen by an observer at Earth (regions in white in the black disks).
Figure3.Predicted gamma ray lightcurves for a neutron-star-forming collapse (NSFC, solid lines) and a direct black-holeforming collapse (BHFC, dotted lines), at distance D = 1 kpc and different stellar radii (labels on curves).Dashed: νe flux for the two cases (see right vertical axis for scale).For all the results, we used Eν,tot = 5 10 52 ergs for the total energy emitted in νe.The galactic background is shown as well.For both signal (one case only, for illustration) and background, the shadings represent the uncertainties discussed in the text.

Figure 4 .
Figure 4.A schematic graph illustrating the opacities of the ISM corresponding to the various values of photon energy.The regime of interest for this work (0.511 MeV) is shown with an arrow.