THE FIRST GAMMA-RAY BURSTS IN THE UNIVERSE

We calculate density profiles for outbursts in Pop III H ii regions with ZEUS-MP (Whalen & Norman 2006) in the same manner as in M12. We treat stellar winds and outbursts as time-dependent inflows at the inner boundary of a one-dimensional spherical grid:

$$\begin{equation} \rho \, = \, \frac{\dot{m}}{4 \pi {r_{\mathrm{ib}}}^2 v_{\mathrm{w}}}, \end{equation} \tag{ 2 }$$

where r_ib is the radius at the inner boundary and v_w is the wind velocity. Outbursts are generated by increasing $\dot{m}$ and lowering v_w. At the beginning of a run the grid is initialized with one of the three H ii region densities. The mesh has 32,000 uniform zones and extends from 3.084 × 10¹⁰ cm to 9.252 × 10¹⁴ cm (∼60 AU) for He merger simulations and from 10⁻⁵ pc to 0.2 pc for other mergers. We impose outflow conditions on the outer boundary, and the grid is domain decomposed into eight tiles, with 4000 zones per tile and 1 tile per processor.

Radiative cooling can flatten shells into cold, dense structures that strongly affect the evolution of the GRB jet. Although there are no metals or dust in our primordial ejections, the shell can still cool by X-ray emission and H and He lines in the shocked gas. Our ZEUS-MP models include collisional excitation and ionization cooling by H and He, recombinational cooling, H₂ cooling, and bremsstrahlung cooling, with our nonequilibrium H and He reaction network providing the species mass fractions (H, H⁺, He, He⁺, He^{2 +}, H⁻, H$^{+}_{2}$, H₂, and e⁻) needed to calculate these collisional cooling processes. The hydrodynamics in our models are always evolved on the lesser of the cooling and Courant times to capture the effect of cooling on the structure of the flow. We neglect the effect of ionizing radiation from the progenitor on winds and shells. This treatment is approximate, given that the star illuminates the flow and that its luminosity evolves over time. However, the energy deposited in the flow by ionizations is small in comparison to its bulk kinetic energy and is unlikely to alter its properties in the vicinity of the burst.

One important difference between our shell profiles and those of M12 is that a fast wind does not precede the ejection of the envelope and later detach from its outer surface to create a very low-density rarefaction zone ahead of the shell. The outburst instead plows up the much higher density H ii region, as shown in Figure 3. Note that there are significant differences in the structures of shells ejected just before the burst and ejections 2000 yr before the burst. Radiative cooling has inverted the density profile inside the shell at later times and created a much larger density jump at the interface between the shell and the termination shock due to the wind that piles up at its inner surface. These structural differences, together with the distance of the shell from the explosion, have important consequences for GRB light curves as we discuss below.

We show afterglow light curves for Pop III GRBs in uniform-density H ii regions at z = 20 in Figure 4. In all three plots, sensitivity limits are shown for the appropriate instruments, beginning at the minimum integration time for each one that would result in a detection. The peak flux occurs at later times for lower frequencies. Fluxes are highest in the NIR, reaching ∼10 mJy for a 10⁵³ erg burst in an n = 10 cm⁻³ H ii region. They are somewhat lower in the radio and X-ray, although the radio flux falls off only gradually after reaching its peak. In the NIR and X-ray, the flux scales roughly with the burst energy, but in the radio, the flux increases by a factor of ∼50 with each decade in energy. Also, the NIR and X-ray afterglows are brightest in the highest densities at early times, but after a few hours, this trend is reversed. There is much less variation in flux with ambient density in the radio afterglows.

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The peak flux at a given frequency is greatest in the NIR at ∼3 μm, and it decreases monotonically above and below this frequency. Pop III GRB afterglows in H ii regions will thus most easily be detected in current and proposed NIR instruments. The afterglow reaches a peak at or before ∼10⁻¹ days but falls off gradually enough that instruments such as GRIPS and WFIRST could detect a 10⁵¹ erg burst for up to about 2 days and a 10⁵³ erg burst for up to 10 days. The JANUS NIRT would detect a 10⁵³ erg burst for about 10 days, while a 10⁵¹ erg burst would be just at the threshold of detectability. The EXIST IRT would see NIR afterglows for 0.5–25 days at z ∼ 20, and the JWST could detect these events for 2–80 days, depending on energy and H ii region density.

In the radio portion of the spectrum, the VLA could just barely detect a 10⁵¹ erg burst in a 0.1 cm⁻³ H ii region, while the same burst would be visible to the SKA for about an hour. At the opposite extreme, a 10⁵³ erg GRB in a 10 cm⁻³ H ii region would be detectable by the VLA for nearly 80 days while SKA would see the same afterglow for ∼200 days. Because the peak flux is reached earlier at higher frequencies, it will be difficult for ALMA to detect even the brightest afterglow for more than about a day. LOFAR will not see these explosions because its sensitivity falls below their peak fluxes. At 80 MHz, the peak flux only reaches ∼10⁻² mJy, well below the LOFAR 5σ sensitivity of 80 mJy for an eight hour integration.

GRBs in H ii regions do not produce very bright X-ray afterglows. The X-ray instruments that are sensitive to the lowest frequency X-rays will be the most suitable for detecting these events. The MXT, HET, and WFI aboard SVOM, EXIST, and LOBSTER, respectively, plus the GBM on Fermi, with their high sensitivities and low minimum observable frequencies, will most easily detect Pop III GRBs. Only the bursts with the highest energies would be visible. A 10⁵¹ erg GRB would reach a maximum flux of only ∼10⁻⁴ mJy at 3 × 10⁹ GHz, well below the detection thresholds of every current and proposed X-ray mission except for the Fermi GBM. However, a 10⁵³ erg event could produce an afterglow with a flux of ∼10⁻¹ mJy at 10⁹ GHz immediately after the end of the prompt emission phase. This flux would then fall off slowly enough for SVOM, EXIST, and LOBSTER to detect the afterglow for nearly a day. We note that the FERMI LAT is not well suited to hunting for high-redshift GRB afterglows because of its high threshold frequency, 4.8 × 10¹² GHz, and modest sensitivity.

Light curves for GRBs in r⁻² winds are shown in Figure 5. As noted earlier, these environments are expected in many binary merger scenarios in which the hydrogen envelope of the star has been driven by stellar winds to radii that are beyond the reach of the jet before its afterglow dims below visibility. Like GRBs in H ii regions, afterglows from explosions in stellar winds are brightest in the NIR at about 3.0 μm, with peak fluxes falling off above and below this frequency. However, due to the higher densities at smaller radii, afterglow fluxes at frequencies above the radio are greater in winds than in H ii regions for a given burst energy. Fluxes in the radio are dimmer in winds than in H ii regions because they peak at later times, when the jet is at larger radii and lower densities than the H ii regions. All three fluxes scale roughly with the energy of the burst after a few hours, when the brightest afterglows are generally in the most diffuse envelopes.

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A 10⁵³ erg burst in a 10⁻⁴ M_☉ yr⁻¹ wind reaches a maximum NIR flux of ∼1 Jy 100 s after the burst, making it readily detectable for up to 10 days with EXIST. GRIPS and WFIRST could extend this window out to 10–20 days, and this event would be visible to JANUS NIRT for almost a day. JANUS, however, with its somewhat lower sensitivities, would not be able to see the lowest energy bursts in the NIR because their fluxes fall so quickly after the burst. EXIST would see these GRBs for up to half a day. The most sensitive of the instruments, the JWST NIRCam, could observe the least energetic GRBs for a day and the most energetic ones for 100 days, providing detailed followup to an initial detection in X-rays.

Besides being somewhat dimmer than in H ii regions, radio afterglows in winds reach peak fluxes at later times: 1–30 days instead of 0.1–3 days. The radio flux also peaks at later times in higher mass loss rates. A 10⁵³ erg burst in a 10⁻⁶ M_☉ yr⁻¹ stellar wind reaches a peak flux of ∼1 mJy at 5 GHz after 0.8 days. It would be possible to detect such an afterglow with the VLA from about 0.1–80 days after the burst, and SKA would extend this range out to 200 days. Neither ALMA nor LOFAR would be able to detect a Pop III GRB afterglow at z = 20 in a stellar wind. The least energetic GRBs in the strongest winds are marginally detectable by VLA for a day and would be visible to SKA for about 10 days.

X-ray afterglows for GRBs in stellar winds are much brighter than those H ii regions. A 10⁵³ erg GRB at z = 20 produces fluxes greater than 20 mJy at 10⁹ GHz for 2.5 hr. This emission would be easily detectable by SVOM, EXIST, LOBSTER, JANUS, Swift, and the Fermi GBM. The afterglow would be visible to the JANUS XRFM for nearly a half an hour and to Fermi for almost two hr. At the other extreme, a 10⁵¹ erg GRB in a 10⁻⁶ M_☉ yr⁻¹ wind would produce an afterglow that would only reach ∼5 × 10⁻³ mJy at 10⁹ GHz. This event would only be detectable for ∼200 s with the Fermi GBM, and would likely not be found by other instruments.

We show X-ray, NIR, and radio light curves for Pop III GRB jets crashing into massive shells ejected by helium merger events in Figure 6. In each case, the 5 M_☉ shell, which has been driven into H ii region densities of 0.1, 1, or 10 cm⁻³ by a stellar wind with $\dot{m} =$ 10⁻³ M_☉ yr⁻¹, has a radius of ∼2 AU. The GRB jet breaks out into the r⁻² wind before colliding with the shell. The high densities in the wind at these small radii decelerate the jet to mildly relativistic speeds in ∼10 minutes in the local frame. Because the time between the ejection of the shell and the burst is short, the wind has not had time to pile up and form a termination shock at the inner layer of the shell. The jet therefore transitions directly from the wind to the hydrogen shell.

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A mildly relativistic reverse shock forms at the contact discontinuity between the wind and the shell, which then steps back through the jet in the frame of the jet. A new forward shock develops with a lower Lorentz factor than that of the original forward shock but that is still, in general, mildly relativistic. The new forward shock advances into the shell and immediately encounters another density jump, where it produces another pair of forward and reverse shocks. This cycle continues, with several to dozens of shock pairs eventually being formed. By the time the first forward shock reaches the interior of the shell, where the density is nearly constant and no more shocks are produced, all forward shocks have become non-relativistic and all reverse shocks have retreated through the forward shocks that created them.

The jet remains in the shell for the entire 100 day simulation. When it finally emerges, there may be a slight re-collimation of the jet due to the low density of the surrounding medium. Any rebrightening of the light curve would be minimal, however, due to the low density of the medium. We find that the H ii region density beyond the shell has essentially no effect on the afterglow light curve because the structure of the shell has not yet been altered by the relic H ii region.

Light curves for Pop III GRBs in He merger shells are quite different from those in winds and H ii regions. Their structure in a given band varies strongly with the energy of the burst and also across the bands themselves. They also exhibit much more variation over time, with sharp drops that are sometimes preceded by flares. The peak flux is again in the NIR, but there is also a large initial X-ray flux. Radio emission is suppressed by synchrotron self-absorption in the high densities in the shell, so detecting He merger GRBs at any frequency below the IR will be nearly impossible with current or proposed instruments. The flux at all frequencies is essentially quenched a short time after the jet enters the shell.

The high densities in the shell (black plot in the left panel of Figure 3) produce a large NIR flux. A 10⁵¹ erg GRB reaches a peak flux of 10 mJy at 150 s at 3.0 μm, while a 10⁵³ erg GRB peaks at nearly 100 mJy. Although they are visible to all the NIR detectors in Table 1, they will probably only be detected by satellites whose onboard X-ray instruments are triggered by the event because their NIR fluxes are so short-lived. There would not be enough time to slew any ground-based instruments to capture the NIR afterglows of GRBs due to He mergers. This problem would be mitigated in cases in which there is more time between the ejection and the burst, since delays of up to several days would not alter the structure of the shell but would allow followup in the NIR from the ground. Note that fluxes from more energetic bursts are quenched sooner because the jet reaches the shell in less time. Higher burst energies imply larger ejecta masses for a given Lorentz factor, and they are not decelerated by the wind as much as jets with lower masses.

At 83 keV, the flux from a 10⁵³ erg burst reaches 10 mJy just after the end of the prompt emission phase. This GRB would be visible to SVOM, JANUS, LOBSTER, EXIST, and Fermi until the jet collides with the shell at t_obs ≲ 0.1 days. The flux then drops by many orders of magnitude, in some cases after producing a flare that lasts for a few seconds to a few minutes. GRB afterglows with strong, transient X-ray and IR fluxes that end within a few hours would be clear signatures of a Pop III GRB from a He merger, particularly if there are flares.

We plot afterglow light curves for Pop III GRBs in dense shells ejected in binary mergers in Figure 7. The hydrogen shell has a mass of 5 M_☉ and is driven to a radius of ∼0.2 pc by a 10⁻⁵ M_☉ yr⁻¹ wind in H ii region densities of 0.1, 1, and 10 cm⁻³. The jet initially breaks out into the wind and then reaches the termination shock at about 10 days, where the wind has piled up at the inner surface of the shell. By this time the jet has decelerated to Γ ∼ 10. A mildly relativistic reverse shock forms at the interface of the free-streaming and shocked winds and then propagates back into the jet. At the same time, a mildly relativistic forward shock advances into the shocked wind. As with the He merger shell, a series of forward and reverse shock pairs form at the interface between the free-streaming and shocked winds until the leading forward shock has fully advanced into the piled-up wind at the inner surface of the shell. The jet, now only mildly relativistic, reaches the shell itself at several hundred to 1000 days, at which time it becomes non-relativistic. A series of shock pairs are again created as the jet enters the shell as in the He merger case. The jet eventually breaks out of the hydrogen shell and into the H ii region, but only after several thousand days.

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As shown in Figure 7, the afterglow light curves are similar to those in stellar winds out to 20–50 days. Although the jet collides with the shocked wind after ∼5 days, the effect on the light curve is minimal. Only when the jet reaches the hydrogen shell at 20–50 days does the light curve deviate from that of a simple wind. In the X-ray band, the afterglow falls below the detection limits of current and proposed instruments before the jet collides with the shell. In the IR, the flux is only 10⁻⁵–10⁻⁴ mJy when the shell is reached, making the transition into the shell barely detectable with the JWST or perhaps WFIRST, and then only for the most energetic bursts. Likewise, at 5 GHz the afterglow flux is only ∼10⁻² mJy when the jet crashes into the shell. The VLA may barely be able to detect the sudden drop in flux when the jet collides with the shell, but the SKA will detect it. Note that in some cases the drop in flux will be preceded by a flare in the radio. Pop III GRBs due to binary mergers would look like events in simple winds, even when the shell is relatively close to the progenitor because the X-ray and NIR fluxes would fall below detectability before plummeting when the jet reaches the shell. However, they could still be distinguished from GRBs in winds by followup observations in the radio at later times, which could capture the abrupt drop in flux. Even the least energetic Pop III GRB from a binary merger will be visible in the NIR to all the instruments in Table 1 for at least a day.

Our models show that what primarily governs the light curves of Pop III GRBs due to He and binary mergers for a given energy are the mass loss rates after the ejection (and hence the density of the wind envelope) and the time between the ejection and the burst. The mass of the shell does not matter, as long as it can fully quench the jet.

At least half of the GRBs with observed afterglows exhibit some sort of flaring in their light curves (see, e.g., Roming et al. 2012). The leading contender for the origin of these flares is late-time central engine activity. The collision of the relativistic jet with strong density features has also been suggested as the cause of some flares, like those in our He and binary merger models (M12). However, some have questioned if flares really occur when a GRB jet collides with a dense shell (Gat et al. 2013). When the jet encounters the shell, a pair of forward and reverse shocks is formed, as noted earlier. The new forward shock continues into the shell while the reverse shock steps back through the old forward shock in the frame of the shock. Material swept up by the new forward shock does not mix with material that was swept up prior to the collision because of the contact discontinuity between the reverse shock and the new forward shock. The M12 afterglow model, which employs the swept mass approximation to model jet evolution, may therefore overestimate the mass that radiates at the post-collision Lorentz factor. Gat et al. (2013) argue that this could lead to overestimates of the flux at the interface between the shocked wind and the hydrogen shell, and hence a flare when none should exist.

The Gat et al. (2013) model accounts for the radial structure of the jet when modeling the passage of the reverse shock through the original forward shock during a collision with a density jump. However, their model does not resolve the density structure at the inner surface of a shell. It instead treats the inner surface as a single density jump of factor a = 10⁵. The radial structure of a jet that has collided with a single jump a is not the same as for a jet that has traversed a series of smaller jumps that sum up to a. Indeed, when we apply our model to the jets and density profiles in Gat et al. (2013), it does not produce any flares either.

Figure 8 shows the radial structure of a jet after it has crashed into the 5 M_☉ He merger shell in our study. The jet does not instantaneously transition into the shell, but instead it encounters a series of density jumps over ∼two hr. Consequently, the structure of the jet becomes highly complex, with the vast majority of the swept-up mass being confined to a narrow region at the leading edge of the jet. The potential smearing out of flares due to the radial extent of the jet is therefore far less than that predicted by the Gat et al. (2013) model. A relativistic blast wave colliding with this massive shell will thus produce a flare, even though the reverse shock and the curvature of the blast wave may reduce its amplitude.

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If the GRB jet encounters an abrupt change in the density of the surrounding medium, Equation (A1), which assumes self-similar expansion of the ejecta, no longer applies. When the jet collides with the jump, a contact discontinuity forms between the material that was swept up by the jet prior to the collision and the material that was swept up afterward. A reverse shock forms at the contact discontinuity and backsteps into the jet in the frame of the jet. A forward shock also forms just past the jump and propagates into the medium beyond it. The contact discontinuity also moves forward, but at a much lower velocity as the jet continues to advance into the new medium and plow it up.

Although several analytic treatments exist for the evolution of the jet and its component shocks at a density jump (Nakar & Granot 2007; Dai & Lu 2002; Gat et al. 2013), they all assume that the jet will be relativistic upon collision with the jump, which is not always true for our density profiles. We must therefore resort to the general Blandford & McKee (1976) and Taylor (1950) equations for the jump conditions in order to self-consistently evolve the jet evolution in any medium.

When the jet encounters a density jump, the medium surrounding the jump can be partitioned into four regions (Figure 9). Region I is the undisturbed material beyond the density jump; region II is the postshock material behind the forward shock; region III is the region of the jet through which the reverse shock has passed; and region IV is the unshocked jet material. The contact discontinuity forms between regions II and III. For a relativistic or mildly relativistic jet, the jump conditions follow from the conservation of energy (wγ²β), momentum (wγ²β² + p), and particle number (nγβ) flux densities across the jump in the frame of the shock, where w and p are the enthalpy and pressure of the gas, respectively, and γ and β are the Lorentz factor and velocity of the gas particles. The pressure of the gas is

$$\begin{equation} p = (\hat{\gamma } - 1)(e - \rho), \end{equation} \tag{ A8 }$$

where $\hat{\gamma }$ is the adiabatic index, e = γnm_pc² is the energy density of the gas and ρ = nm_pc² is the rest-frame gas density. The enthalpy w = e + p.

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For any two adjacent regions 1 and 2 in regions I–IV described above, the jump conditions for relativistic and mildly relativistic jets are (Blandford & McKee 1976)

$$\begin{equation} \frac{e_2}{n_2} = \gamma _2\frac{w_1}{n_1}, \end{equation} \tag{ A9 }$$

$$\begin{equation} \frac{n_2}{n_1} = \frac{\hat{\gamma _2}\gamma _2 + 1}{\hat{\gamma _2} - 1}, \end{equation} \tag{ A10 }$$

and

$$\begin{equation} \Gamma ^2 = \frac{(\gamma _2 + 1)[\hat{\gamma _2}(\gamma _2 - 1) + 1]^2}{\hat{\gamma _2}(2 - \hat{\gamma _2})(\gamma _2 - 1) + 2}, \end{equation} \tag{ A11 }$$

where Γ is the Lorentz factor of the new forward shock. There are three known quantities: the Lorentz factor of the jet when it crashes into the density jump and the densities n_I and n_IV of regions I and IV, respectively. From these three quantities and the requirement that the pressure and energy density must be equal in regions II and III (due to the presence of a contact discontinuity between them), the Lorentz factors of the forward shock, the reverse shock, and the contact discontinuity can be determined from the three jump conditions.

Given these Lorentz factors, an infinitesimal time after the leading edge of the jet collides with the jump, the subsequent motion of the forward shock is determined from energy conservation with Equation (4) of Nakar & Granot (2007). The evolution of the reverse shock is set by the fact that the energy density remains constant across the contact discontinuity. The Lorentz factor that satisfies this requirement is found numerically. Meanwhile, the initial shock, which has no knowledge of the contact discontinuity, continues to expand adiabatically.

Previous authors have assumed the jet to be relativistic when it encounters the jump. This in general is true for simple density profiles where one or perhaps a few density jumps are present. In our much more complicated and realistic profiles, however, there can be dozens of density jumps. If the jump scale factor a = n/n₀ (where n₀ and n are the densities of the medium before and after the jump, respectively) is > 1, the Lorentz factor of the new forward shock will usually be lower than that of the initial shock. If the new forward shock encounters a second density jump, then another forward shock will be produced with an even lower Lorentz factor. The forward shock can eventually become non-relativistic when it encounters a subsequent jump, and the Blandford–McKee conditions will no longer apply. The flow will instead obey the Sedov–von Neumann–Taylor jump conditions for a Newtonian fluid (Taylor 1950):

$$\begin{equation} \frac{\rho _1}{\rho _0} = \frac{\hat{\gamma } - 1 + (\hat{\gamma } + 1)y_1}{ \hat{\gamma } + 1 + (\hat{\gamma } - 1)y_1}, \end{equation} \tag{ A12 }$$

$$\begin{equation} \frac{U^2}{a^2} = \frac{1}{2\hat{\gamma }}[\hat{\gamma } - 1 + (\hat{\gamma } + 1)y_1], \end{equation} \tag{ A13 }$$

and

$$\begin{equation} \frac{u_1}{U} = \frac{2(y_1 - 1)}{\hat{\gamma } - 1 + (\hat{\gamma } + 1)y_1}, \end{equation} \tag{ A14 }$$

where ρ₀ and ρ₁ are the densities before and after the jump, respectively; $\hat{\gamma } = 5/3$ is the adiabatic index y₁ = p₁/p₀; p₀ and p₁ are the pressures behind and ahead of the jump; a is the sound speed; U is the velocity of the forward shock; and u₁ is the velocity of the fluid behind the forward shock. The velocities of the forward and reverse shocks and the contact discontinuity can be found by solving these jump conditions numerically if the velocity of the initial shock and the pressures of regions I and IV are known. Our ZEUS-MP models provide the temperature of region IV at each grid point, and the temperature of region I can be calculated by assuming equipartition of thermal and kinetic energies within the jet. The pressures in these regions are then easily obtained from the temperatures by the ideal gas law.

Our afterglow model in M12 essentially ignores the presence of density jumps and instead assumes that the evolution of the jet is dependent only on the total mass it sweeps up. The mass swept up by the jet is assumed to instantly mix with previously swept-up material, with a prompt change in the Lorentz factor along the entire jet. In reality, any collision with external structures produces a reverse shock that steps backward through the jet. Until the reverse shock reaches a location in the jet, that location is not aware that any interaction has taken place. Consequently, any sharp feature in the light curve that might be produced by a density jump will be somewhat smoothed out by a time equal to the jet-crossing time of the reverse shock.

If we assume that a constant fraction _B of the total fireball energy is stored in magnetic fields, then the equipartition magnetic field strength at the shock boundary is (i.e., Huang et al. 2000):

$$\begin{equation} \frac{B^{\prime 2}}{8\pi } = \epsilon _B \frac{\hat{\gamma }\Gamma + 1}{\hat{\gamma } - 1} (\Gamma - 1)n(r)m_pc^2, \end{equation} \tag{ A15 }$$

where _B is the fraction of the burst energy stored in magnetic fields, n(r) is the number density of the medium at radius r, and m_p is the proton mass.

The electrons that are injected into the shock are assumed to have a velocity distribution N(γ)∝γ^−p with a minimum Lorentz factor γ_m. The minimum Lorentz factor of the injected electrons is (Huang et al. 2000)

$$\begin{equation} \gamma _m = \epsilon _e(\Gamma - 1)\frac{m_p(p-2)}{m_e (p-1)} + 1, \end{equation} \tag{ A16 }$$

where m_e is the electron mass. Electrons with a Lorentz factor γ_e emit synchrotron radiation at a characteristic frequency (Rybicki & Lightman 1979)

$$\begin{equation} \nu (\gamma _e) = \frac{\gamma }{1 - \beta _e} \frac{3q_eB}{4\pi m_ec}, \end{equation} \tag{ A17 }$$

where $\beta _e = \sqrt{{{1 - \gamma _e^{-2}}}}$, and q_e is the electron charge. The injection break frequency ν_m corresponds to the peak emission frequency of the injected electrons, and can be found by substituting γ_e = γ_m into Equation (A17).

Relativistic electrons in the shock cool radiatively through IC scattering and synchrotron emission on a comoving frame timescale (Panaitescu & Kumar 2000)

$$\begin{equation} t^{\prime }_{\rm rad}(\gamma) = \frac{t^{\prime }_{\rm syn}}{Y+1}, \end{equation} \tag{ A18 }$$

where Y is the Compton parameter. The synchrotron cooling timescale $t^{\prime }_{\rm syn}$ of an electron is equal to the ratio of its energy E to the synchrotron power P it radiates:

$$\begin{equation} t^{\prime }_{\rm syn} = \frac{E}{P} = \frac{1}{\gamma _e\beta _e^2} \frac{6\pi m_e c}{\sigma _T B^{\prime 2}}, \end{equation} \tag{ A19 }$$

where σ_T is the Thomson cross section for electron scattering. Equations (A18) and (A19) can be solved to find the Lorentz factor for electrons that cool on a timescale equal to the observer frame age of the remnant, γ_c:

$$\begin{equation} \gamma _c = \frac{6\pi m_ec^2}{B^{\prime 2}\sigma _e(Y + 1)t^{\prime }}. \end{equation} \tag{ A20 }$$

The frequency of the cooling break, ν_c, is found by substituting γ_e = γ_c in Equation (A17).

A.3.1. Fast-cooling Electrons

Electrons in the jet can also cool by adiabatic expansion of the gas. When cooling times for electrons with Lorentz factor γ_m are less than the age of the jet (ν_c < ν_m, where ν_c is the frequency of the cooling break), the electrons in the jet lose a significant portion of their energy through emission of radiation and are said to be radiative, or fast-cooling. On the other hand, if the cooling time is greater than the age of the jet (ν_c > ν_m) the electrons do not lose much energy to radiation and are said to be adiabatic, or slow-cooling.

To calculate Y, we only account for one upscattering of the synchrotron photons. If the electrons injected into the shock are fast-cooling and the frequency of the absorption break ν_a < min(ν_m, ν_c), then Y is approximately (Panaitescu & Meszaros 2000)

$$\begin{equation} Y_r = \gamma _{\rm m}\gamma _{\rm c}\tau _{\rm e}, \end{equation} \tag{ A21 }$$

where a constant of order unity is ignored and τ_e, the optical depth to electron scattering, is

$$\begin{equation} \tau ^{\prime }_e = \frac{\sigma _em(r)}{4\pi m_{\rm p}r^{\prime 2}}. \end{equation} \tag{ A22 }$$

The medium becomes optically thick to synchrotron self-absorption at the absorption break frequency, ν_a. When both the injection break and the cooling break lie in the optically thick regime, Y becomes

$$\begin{equation} Y_r = Y_* = \tau ^{\prime }_e\left(C_2^{2-p}\gamma _c^7\gamma _m^{7(p-1)}\right)^{1/(p+5)}, \end{equation} \tag{ A23 }$$

where $C_2 \equiv 5q_e\tau ^{\prime }_e/\sigma _eB^{\prime }$ (Panaitescu & Meszaros 2000).

A.3.2. Slow-cooling Electrons

If the electrons are slow-cooling, Y becomes

$$\begin{equation} Y_a = \tau ^{\prime }_{\rm e}\gamma _{\rm i}^{p-1}\gamma _{\rm c}^{3-p}, \end{equation} \tag{ A24 }$$

as long as ν_a < min(ν_m, ν_c) (Panaitescu & Meszaros 2000) and 1 < p < 3 (here, we have again ignored a constant of order unity). If the injection and cooling breaks lie in the region of the spectrum that is optically thick to synchrotron self-absorption, then Y is the same as in the corresponding fast-cooling case and

$$\begin{equation} Y_a = Y_*. \end{equation} \tag{ A25 }$$

At lower frequencies, the medium in which the jet propagates becomes optically thick to synchrotron self-absorption. F_ν then becomes ∝ν² at some absorption break frequency ν_a where the optical depth to self-absorption is τ_ab = 1. The frequency of the absorption break depends on the cooling regime of the electrons (fast or slow) and on the order and values of the injection and cooling breaks. In the fast-cooling regime (Panaitescu & Meszaros 2000),

$$\begin{equation} \nu ^{\prime }_{{\rm a}, {\rm fast}\hbox{-}{\rm cooling}} = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}C_2^{3/10}\gamma _c^{-1/2}, & \gamma _a < \gamma _c < \gamma _m \\ (C_2\gamma _c)^{1/6}, & \gamma _c < \gamma _a < \gamma _m \\ \left(C_2\gamma _c\gamma _m^{p-1}\right)^{1/(p+5)}, & \gamma _c < \gamma _m < \gamma _a, \\ \end{array}\right. \end{equation} \tag{ A26 }$$

and in the slow-cooling regime

$$\begin{equation} \nu ^{\prime }_{{\rm a}, {\rm slow}\hbox{-}{\rm cooling}} = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}C_2^{3/10}\gamma _m^{-1/2}, & \gamma _a < \gamma _m < \gamma _c \\ \left(C_2\gamma _m^{p-1}\right)^{1/(p-4)}, & \gamma _m < \gamma _a < \gamma _c \\ \left(C_2\gamma _m^{p-1}\gamma _c\right)^{1/(p+5)}, & \gamma _m < \gamma _c < \gamma _a. \\ \end{array}\right. \end{equation} \tag{ A27 }$$

In order to produce light curves, we must first find the time dependence of Γ(r), n(r), and m(r). Equation (A1) can be solved numerically for Γ(r), and Equation (A4) can then be used to relate the observer time t_obs to the jet position r, allowing us to rewrite the equations defining the three break frequencies in terms of t_obs, Γ(t_obs), n(t_obs), and M(t_obs). Given the three break frequencies and the peak flux density, analytical light curves can then be calculated from the radio to the γ-ray regions of the spectrum. If ν_a < min(ν_m, ν_c), then the peak flux density $F_{ \nu,\oplus }^{\rm max}$ occurs at the injection break if ν_m < ν_c and at the cooling break if ν_m > ν_c:

$$\begin{equation} F_{\nu,\oplus }^{\rm max} = \frac{\sqrt{3}\phi _{\rm p}}{4\pi D^2}\frac{q_e^3\beta _m^2}{m_{\rm e}c^2}\frac{\Gamma B^{\prime }m(r)}{m_{\rm p}}, \end{equation} \tag{ A28 }$$

where ϕ_p is a factor calculated by Wijers & Galama (1999) that depends on the value of p, and D = (1  +  z)^−1/2D_l, where D_l is the luminosity distance to the source (Panaitescu & Kumar 2000). The flux at any frequency ν (ignoring relativistic beaming and the spherical nature of the emitting region) has been derived by Sari et al. (1998) and Panaitescu & Kumar (2000) as described below.

A.5.1. Fast-cooling Electrons

When the electrons are in the fast-cooling regime, the peak flux occurs at the cooling break as long as ν_a < ν_c:

$$\begin{equation} F_{\nu, \oplus } = F_{\nu,\oplus }^{\rm max} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}(\nu /\nu _a)^2(\nu _a/\nu _c)^{1/3}, & \nu < \nu _a \\ (\nu /\nu _c)^{1/3}, & \nu _a < \nu < \nu _c \\ (\nu /\nu _c)^{-1/2}, & \nu _c < \nu < \nu _m \\ (\nu /\nu _m)^{-p/2}(\nu _m/\nu _c)^{-1/2}, & \nu _m < \nu. \end{array}\right. \end{equation} \tag{ A29 }$$

If the medium is optically thick to synchrotron self-absorption at the cooling break frequency, the maximum flux moves to the absorption break frequency. Between the absorption and cooling breaks, F_ν∝ν^5/2 but becomes ∝ν² below the cooling break:

$$\begin{equation} F_{\nu, \oplus } = F_{\nu,\oplus }^{\rm max} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}(\nu /\nu _c)^2(\nu _c/\nu _a)^{5/2}, & \nu < \nu _c \\ (\nu /\nu _a)^{5/2}, & \nu _c < \nu < \nu _a \\ (\nu /\nu _a)^{-1/2}, & \nu _a < \nu < \nu _m \\ (\nu /\nu _m)^{-p/2}(\nu _m/\nu _a)^{-1/2}, & \nu _m < \nu. \end{array}\right. \end{equation} \tag{ A30 }$$

In canonical afterglow models that assume a uniform-density environment, the cooling and injection breaks move to lower frequencies over time. Eventually, both can lie below the absorption break; but they are far too late in the evolution of the burst to be relevant to anything but the radio afterglow and long after the time at which the electrons in the jet have transitioned to the slow-cooling regime. In the more realistic density profiles in our models, the extremely high density that the jet can encounter as it passes through a thick shell causes it to abruptly transition from highly relativistic to Newtonian expansion. The decrease in Γ leads to a sharp drop in the injection break frequency, while the increased density of the medium leads to a higher magnetic field amplitude, which in turn causes a drop in the cooling break frequency. The result is that the frequency of the absorption break can be several orders of magnitude higher than the cooling and injection break frequencies as the jet passes through the shell. Multiple transitions between the fast and slow electron cooling regimes can also occur. In a thick shell, when ν_a > ν_m and the electrons are in the fast-cooling regime,

$$\begin{equation} F_{\nu, \oplus } = F_{\nu,\oplus }^{\rm max} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}(\nu /\nu _c)^2(\nu _c/\nu _a)^{5/2}, & \nu < \nu _c \\ (\nu /\nu _a)^{5/2}, & \nu _c < \nu < \nu _a \\ (\nu /\nu _a)^{-p/2}, & \nu _m < \nu. \end{array}\right. \end{equation} \tag{ A31 }$$

A.5.2. Slow-cooling Electrons

Our models yield the same flux as the canonical wind models until the jet collides with a shocked wind that has piled up behind a thick shell. If it does not reach the shocked wind in the first few hours, the electrons in the leading edge of the jet transition to the slow-cooling regime, with ν_a ≪ ν_m and

$$\begin{equation} F_{\nu, \oplus } = F_{\nu,\oplus }^{\rm max} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}(\nu /\nu _a)^2(\nu _a/\nu _m)^{1/3}, & \nu < \nu _a \\ (\nu /\nu _m)^{1/3}, & \nu _a < \nu < \nu _m \\ (\nu /\nu _m)^{-(p-1)/2}, & \nu _m < \nu < \nu _c \\ (\nu /\nu _c)^{-p/2}(\nu _c/\nu _m)^{-(p-1)/2}, & \nu _c < \nu. \end{array}\right. \end{equation} \tag{ A32 }$$

If ν_m < ν_a < ν_c,

$$\begin{equation} F_{\nu, \oplus } = F_{\nu,\oplus }^{\rm max} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}(\nu /\nu _m)^2(\nu _m/\nu _a)^{5/2}, & \nu < \nu _a \\ (\nu /\nu _a)^{5/2}, & \nu _a < \nu < \nu _m \\ (\nu /\nu _a)^{-(p-1)/2}, & \nu _m < \nu < \nu _c \\ (\nu /\nu _c)^{-p/2}(\nu _c/\nu _a)^{-(p-1)/2}, & \nu _c < \nu. \end{array}\right. \end{equation} \tag{ A33 }$$

As noted above, as the jet passes through a dense shell, it can pass through multiple transitions between fast and slow electron cooling. When the electrons are in the slow-cooling regime and ν_a > ν_c,

$$\begin{equation} F_{\nu, \oplus } = F_{\nu,\oplus }^{\rm max} \left\lbrace \begin{array}{@{}l@{\quad }l@{}}(\nu /\nu _m)^2(\nu _m/\nu _a)^{5/2}, & \nu < \nu _m \\ (\nu /\nu _a)^{5/2}, & \nu _m < \nu < \nu _a \\ (\nu /\nu _a)^{-p/2}, & \nu _a < \nu. \end{array}\right. \end{equation} \tag{ A34 }$$

Low-frequency radiation can be boosted to higher energies in the presence of relativistic electrons through the process of IC scattering. The contribution to the light curve at any frequency ν can easily be determined by using the expressions for the synchrotron light curve at the IC frequency

$$\begin{equation} \nu _{\rm IC} = \frac{\nu }{\Gamma ^2(1+\beta ^2)}. \end{equation} \tag{ A35 }$$

The IC flux density at frequency ν is then

$$\begin{equation} F_{\nu {\rm , IC}} = \tau _e F_{\nu _{\rm IC} {\rm , syn}}, \end{equation} \tag{ A36 }$$

where $F_{\nu _{\rm IC} {\rm , syn}}$ is the synchrotron flux density at frequency ν_IC.

The spherical nature of a GRB afterglow can substantially alter the observed light curve (Fenimore et al. 1996). The burst ejecta is initially ultra-relativistic, with 100 ≲ Γ ≲ 1000. Radiation that is emitted by ejecta moving directly toward the observer (i.e., along the line connecting the observer and the progenitor) will be more highly beamed than radiation emitted by material that is not moving directly toward the observer. There will also be a delay in the arrival time of photons that are emitted from regions of the jet not traveling directly toward the observer, as these regions are further away from the observer. The beaming of radiation from material moving toward the observer leads to two effects. First, a higher flux is observed than would be expected if beaming was ignored. Second, the edge of the jet, which does not move directly toward the observer and is therefore not as highly beamed as the center of the jet, is not visible to the observer until Γ ∼ 1/θ_j. The delay in arrival time of light emitted by material not moving directly toward the observer tends to smooth out sharp features in the light curve of the jet. The method described above and in M12 neglects the spherical nature of the emitting region but can be slightly modified to account for it. The total luminosity $L^{\prime }_{\nu ^{\prime }}$ radiated by the electrons at comoving frame frequency ν' must first be found. As shown by Wijers & Galama (1999), the power per unit frequency transforms as $P_\nu = \Gamma P ^{\prime }_{\nu ^{\prime }}$, implying that

$$\begin{equation} L^{\prime }_{\nu ^{\prime }} = 4\pi \frac{D^2 F_{\nu,\oplus }}{\Gamma }, \end{equation} \tag{ A37 }$$

where D is the luminosity distance to the source and F_ν is the flux density detected at a distance D from the source by an observer in the isotropic frame if the spherical nature of the emitting region is neglected. The observed flux density is then

$$\begin{equation} F_{\nu } = \frac{1}{4\pi D^2} \int\!\!\!\int \limits _{\Omega _j}{\frac{L^{\prime }_{\nu ^{\prime }}[r(\theta)]\mathcal {D} ^3}{\Omega _j}d\cos {\theta }\ d\phi }, \end{equation} \tag{ A38 }$$

where $\mathcal {D} = 1/{\Gamma (1-\beta \cos {\theta })}$ is the Doppler factor and Ω_j = 2π(1 − cos θ_j) is the solid angle subtended by the jet (Moderski et al. 2000). Equation (A38) must be numerically integrated over the equal arrival time surface defined by

$$\begin{equation} t = \int {\frac{(1 - \beta \cos {\theta })}{c\beta }dr} = {\rm constant}. \end{equation} \tag{ A39 }$$

Let (r(t), θ) refer to a point on the leading edge of the jet, with 0 ⩽ θ ⩽ θ_j and θ = 0 referring to the center of the jet and the direction to the observer. Radiation emitted by material at larger θ will be detected by the observer at progressively later times. For a given observer time t, the jet radius r at which radiation being detected from jet angle θ was emitted can be found by numerically integrating Equation (A39) over r with θ held constant until the desired value of t is reached. Equation (A38) is then used to find the total observed flux density by integrating over θ from 0 to θ_j. Note that, due to the large cosmological redshifts that we study, another correction must be made to transform the isotropic frame flux density at a given frequency to the reference frame of an Earthbound observer.