BRIGHT BROADBAND AFTERGLOWS OF GRAVITATIONAL WAVE BURSTS FROM MERGERS OF BINARY NEUTRON STARS

The next generation of gravitational wave (GW) detectors, such as the Advanced LIGO (Abbott et al. 2009), Advanced VIRGO (Acernese et al. 2008), and KAGRA (Kuroda et al. 2010) interferometers, are expected to detect GW signals from mergers of two compact objects. These gravitational wave bursts (GWBs) have a well-defined "chirp" signal, which can be unambiguously identified. Once detected, the GW signals would open a brand new channel for us to study the universe, especially the physics in the strong field regime. Due to the faint nature of GWs, an associated electromagnetic (EM) emission signal coinciding with a GWB in both trigger time and direction would increase the signal-to-noise ratio of the GW signal, and therefore would be essential for its identification.

One of the top candidates of GWBs is the merger of two neutron stars (i.e., NS–NS mergers; Taylor & Weisberg 1982; Kramer et al. 2006). The EM signals associated with such an event include a short gamma-ray burst (SGRB; Eichler et al. 1989; Rosswog et al. 2013; Gehrels et al. 2005; Barthelmy et al. 2005; Berger 2011), an optical "macronova" (Li & Paczyński 1998; Kulkarni 2005; Metzger et al. 2010), and a long-lasting radio afterglow (Nakar & Piran 2011; Metzger & Berger 2012; Piran et al. 2013). Numerical simulations show that binary neutron star mergers could eject a fraction of the materials, forming a mildly anisotropic outflow with a typical velocity of about 0.1–0.3c (where c is the speed of light) and a typical mass of about 10⁻⁴–10⁻² M_☉ (e.g., Rezzolla et al. 2011; Rosswog et al. 2013; Hotokezaka et al. 2013). The radioactivity of this ejecta powers the macronova and the interaction between the ejecta and the ambient medium is the source of radio afterglow. Usually, the merger product is assumed to be a black hole or a temporal hyper-massive neutron star which survives 10–100 ms before collapsing into the black hole (e.g., Rosswog et al. 2003, 2013; Aloy et al. 2005; Shibata et al. 2005; Rezzolla et al. 2011). Nonetheless, recent observations of Galactic neutron stars and NS–NS binaries suggest that the maximum neutron star mass can be high, which is close to the total mass of the NS–NS systems (Dai et al. 2006; Zhang 2013 and references therein). Indeed, for the measured parameters of six known Galactic neutron star binaries and a range of equations of state, the majority of mergers of the known binaries will form a massive millisecond pulsar and survive for an extended period of time (Morrison et al. 2004). When the equation of state of nuclear matter is stiff (see arguments in Dai et al. 2006; Zhang 2013 and references therein), a stable massive neutron star would form after the merger. This newborn massive neutron star would be differentially rotating. The dynamo mechanism may operate and generate an ultrastrong magnetic field (Duncan & Thompson 1992; Kluźniak & Ruderman 1998; Dai & Lu 1998b), so that the product is very likely a millisecond magnetar. Evidence of a magnetar following some SGRBs has been collected in the Swift data (Rowlinson et al. 2010; Rowlinson et al. 2013), and magnetic activities of such a post-merger massive neutron star have been suggested to interpret several X-ray flares and plateau phase in SGRBs (Dai et al. 2006; Gao & Fan 2006; Fan & Xu 2006).

Since both the GW signal and the millisecond magnetar wind are nearly isotropic, a bright EM signal can be associated with an NS–NS merger GWB regardless of whether there is an SGRB–GWB association (Zhang 2013). Even if there is an association, most GWBs would not be associated with the SGRB since SGRBs are collimated. Zhang (2013) proposed that the near-isotropic magnetar wind of a post-merger millisecond magnetar would undergo magnetic dissipation (Zhang & Yan 2011) and power a bright X-ray afterglow emission. Here we suggest that after partially dissipating the magnetic energy, a significant fraction ξ of the magnetar spin energy would be used to push the ejecta, which drives a strong forward shock into the ambient medium. The continuous injection of the Poynting flux into the blast wave modifies the blast wave dynamics and leads to rich radiation signatures (Dai & Lu 1998a; Zhang & Mészáros 2001; Dai 2004). Figure 1 presents a physical picture for several EM emission components appearing after the merger. Here we study the dynamics of such an interaction in detail and calculate broadband afterglow emission from this forward shock.

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The post-merger hyper-massive neutron star may be near the breakup limit, so that the total spin energy $E_{\rm {rot}}=(1/2)I \Omega _{0}^{2} \simeq 2\times 10^{52} I_{45} P_{0,-3}^{-2} \ {\rm erg}$ (with I₄₅ ∼ 1.5 for a massive neutron star) may be universal. Here P₀ ∼ 1 ms is the initial spin period of the proto-magnetar. Throughout the paper, the convention Q = 10ⁿQ_n is used for cgs units, except for the ejecta mass M_ej, which is in units of solar mass M_☉. Given nearly the same total energy, the spin-down luminosity and the characteristic spin-down timescale critically depend on the polar-cap dipole magnetic field strength B_p (Zhang & Mészáros 2001), i.e., L_sd = L_{sd, 0}/(1 + t/T_sd)², where $L_{\rm sd,0} \simeq 10^{49} \ {\rm erg\ s^{-1}} B^{2}_{p,15}R_{6}^{6}P_{0,-3}^{-4}$, and the spin-down timescale $T_{\rm {sd}} \simeq 2 \times 10^3\; {\rm s}\; I_{45} B_{p,15}^{-2} P_{0,-3}^2 R_6^{-6} \simeq E_{\rm {rot}}/L_{\rm sd,0}$, where R = 10⁶R₆ cm is the stellar radius.⁸

After the internal dissipation of the magnetar wind that powers the early X-ray afterglow (Zhang 2013), the remaining spin energy would be added to the blast wave. The dynamics of the blast wave depends on the magnetization parameter σ of the magnetar wind after the internal dissipation. Since for the confined wind magnetic dissipation occurs upon interaction between the wind and the ejecta, in this paper, we assume that the wind is still magnetized (moderately high σ), so that there is no strong reverse shock into the magnetar wind (Zhang & Kobayashi 2005; Mimica et al. 2009).⁹ As a result, the remaining spin energy is continuously injected into the blast wave with a luminosity L₀ = ξL_{sd, 0}, where ξ < 1 denotes the fraction of the spin-down luminosity that is added to the blast wave. The evolution of the blast wave can be described by a system with continuous energy injection (Dai & Lu 1998a; Zhang & Mészáros 2001).

The newly formed massive magnetar is initially hot. A Poynting-flux-dominated outflow is launched ∼10 s later, when the neutrino-driven wind is clean enough (Metzger et al. 2011). At this time, the front of the ejecta traveled a distance of ∼6 × 10¹⁰ cm (for v ∼ 0.2c), with a width Δ ∼ 10⁷ cm. The ultrarelativistic magnetar wind takes ∼2 s to catch up the ejecta and drives a forward shock into the ejecta. Balancing the magnetic pressure and the ram pressure of shocked fluid in the ejecta, one can estimate the shocked fluid speed as $v_s \sim 10^{-4} c L_{0,47}^{1/2} \Delta _7^{1/2} M_{\rm ej,-3}^{-1/2}$, which is in the same order of forward shock speed. Thus, the forward shock would cross the ejecta in around $t_\Delta \sim \Delta / v_s \sim 3 {\rm \,s} \,L_{0,47}^{-1/2} \Delta _7^{1/2} M_{\rm ej,-3}^{1/2}$. Note that when calculating magnetic pressure, we have assumed a toroidal magnetic field configuration in the Poynting flux, but adopting a different magnetic configuration would not significantly affect the estimate of t_Δ.

After the forward shock crosses the ejecta, the forward shock ploughs into the ambient medium. The dynamics of the blast wave during this stage is defined by energy conservation:¹⁰

$$\begin{eqnarray} &&L_{\rm {0}}t=(\gamma -1)M_{\rm ej}c^2+(\gamma ^{2}-1)M_{\rm sw}c^2, \end{eqnarray} \tag{ 1 }$$

where M_sw = (4π/3)R³nm_p is the swept mass from the interstellar medium. Initially, (γ − 1)M_ejc² ≫ (γ² − 1)M_swc², so the kinetic energy of the ejecta would increase linearly with time until t = min(T_sd, T_dec), where the deceleration timescale T_dec is defined by the condition (γ − 1)M_ejc² = (γ² − 1)M_swc². By setting T_dec ∼ T_sd, we can derive a critical ejecta mass

$$\begin{eqnarray} M_{\rm ej,c,1}&\sim& 10^{-3}\ M_{\odot }n^{1/8}I_{45}^{5/4}L_{0,47}^{-3/8} P_{0,-3}^{-5/2}\xi ^{5/4}\nonumber\\ &\sim& 10^{-3}\ M_{\odot }n^{1/8}I_{45}^{5/4}B^{-3/4}_{p,14}R_{6}^{-9/4}P_{0,-3}^{-1} \xi ^{7/8}, \end{eqnarray} \tag{ 2 }$$

which separates regimes with different blast wave dynamics. For a millisecond massive magnetar, the parameters I₄₅, R₆, and P_{0, −3} are all essentially fixed values. The dependence on n is very weak (1/8 power), so the key parameters that determine the blast wave parameters are the ejecta mass M_ej and the magnetar injection luminosity L₀ (or the magnetic field strength B_p). If M_ej < M_{ej, c, 1} (or T_dec < T_sd), then the ejecta can be accelerated linearly until the deceleration radius, after which the blast wave decelerates, but still with continuous energy injection until T_sd. Conversely, in the opposite regime (M_ej > M_{ej, c, 1} or T_sd < T_dec), the blast wave is only accelerated to T_sd, after which it coasts before decelerating at T_dec. In the intermediate regime of M_ej ∼ M_{ej, c, 1} (or T_dec ∼ T_sd), the blast wave decays after being linearly accelerated.

There is another critical ejecta mass which defines whether the blast wave can reach a relativistic speed. This is defined by E_rotξ = 2(γ − 1)M_ejc². Defining a relativistic ejecta as γ − 1 > 1, this second critical ejecta mass is

$$\begin{equation} M_{\rm ej,c,2} \sim 6\times 10^{-3}\ M_\odot I_{45} P_{0,-3}^{-2} \xi. \end{equation} \tag{ 3 }$$

An ejecta heavier than this would not be accelerated to a relativistic speed.

Below, we discuss four dynamical regimes.

Case I: M_ej < M_{ej, c, 1} or T_sd > T_dec. This requires both a small L₀ (or low B_p) and a small M_ej. We take an example with L₀ ∼ 10⁴⁷ erg s⁻¹ (B_p ∼ 10¹⁴ G) and M_ej ∼ 10⁻⁴ M_☉. To describe the dynamics in such a case, besides the spin-down timescale T_sd, we need three more characteristic timescales and the Lorentz factor value at the deceleration time

$$\begin{eqnarray} &&T_{\rm {dec}}\sim 4.4\times 10^{4}\, {\rm s}\, L_{0,47}^{-7/10}M_{\rm ej,-4}^{4/5}n^{-1/10}\nonumber \\ &&T_{\rm {N1}}\sim 3.6\times 10^{3}\, {\rm s}\, L_{0,47}^{-1}M_{\rm ej,-4}^{}\nonumber \\ &&T_{\rm {N2}}\sim 4.5\times 10^{7}\, {\rm s}\, L_{0,47}^{1/3}n^{-1/3}T_{\rm sd,5}^{1/3}\nonumber \\ &&\gamma _{\rm {dec}}\sim 12.2L_{0,47}^{3/10}\, M_{\rm ej,-4}^{-1/5}n^{-1/10}+1, \end{eqnarray} \tag{ 4 }$$

where T_N1andT_N2 are the two timescales when the blast wave passes the non-relativistic-to-relativistic transition line γ − 1 = 1 during the acceleration and deceleration phases. With these parameters, one can characterize the dynamical evolution of the blast wave (Figure 2(a)), as shown in Table 1. Based on the dynamics, we can quantify the temporal evolution of synchrotron radiation characteristic frequencies ν_a, ν_m, and ν_c, and the peak flux, F_{ν, max}. The evolutions of the characteristic frequencies are presented in Figure 2(b) and collected in Table 2.

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Table 1. Expression of the Lorentz Factor and Radius as a Function of Model Parameters in Different Temporal Regimes for All Dynamical Cases

	γ	R
Case I
t < TN1	$0.28L_{0,47}M_{\rm ej,-4}^{-1}t_{3}+1$	$3.2\times 10^{13}L_{0,47}^{1/2}M_{\rm ej,-4}^{-1/2}t_{3}^{3/2}$
TN1 < t < Tdec	$2.8L_{0,47}M_{\rm ej,-4}^{-1}t_{4}+1$	$4.6\times 10^{15}L_{0,47}^{2}M_{\rm ej,-4}^{-2}t_{4}^{3}$
Tdec < t < Tsd	$9.9L_{0,47}^{1/8}n^{-1/8}t_{5}^{-1/4}+1$	$5.9\times 10^{17}L_{0,47}^{1/4}n^{-1/4}t_{5}^{1/2}$
Tsd < t < TN2	$4.2L_{0,47}^{1/8}T_{\rm sd,5}^{1/8}n^{-1/8}t_{6}^{-3/8}+1$	$1.1\times 10^{18}L_{0,47}^{1/4}T_{\rm sd,5}^{1/4}n^{-1/4}t_{6}^{1/4}$
t > TN2	$0.4L_{0,47}^{2/5}T_{\rm sd,5}^{2/5}n^{-2/5}t_{8}^{-6/5}+1$	$3.7\times 10^{18}L_{0,47}^{1/5}T_{\rm sd,5}^{1/5}n^{-1/5}t_{8}^{2/5}$
Case II
t < TN1	$0.08\xi M_{\rm ej,-4}^{-1}T_{\rm sd,3}^{-1}t_{}+1$	$1.7\times 10^{10}\xi ^{1/2} M_{\rm ej,-4}^{-1/2}T_{\rm sd,3}^{-1/2}t_{}^{3/2}$
TN1 < t < Tsd	$83.3\xi M_{\rm ej,-4}^{-1}T_{\rm sd,3}^{-1}t_{3}+1$	$4.2\times 10^{17}\xi ^{2} M_{\rm ej,-4}^{-2}T_{\rm sd,3}^{-2}t_{3}^{3}$
Tsd < t < TN2	$14.8\xi M_{\rm ej,-4}^{-1}T_{\rm sd,3}^{3/8}t_{5}^{-3/8}+1$	$1.3\times 10^{18}\xi ^{2} M_{\rm ej,-4}^{-2}T_{\rm sd,3}^{3/4}t_{5}^{1/4}$
t > TN2	$1.4\xi ^{16/5} M_{\rm ej,-4}^{-16/5}T_{\rm sd,3}^{6/5}t_{8}^{-6/5}+1$	$7.1\times 10^{18}\xi ^{8/5} M_{\rm ej,-4}^{-8/5}T_{\rm sd,3}^{3/5}t_{8}^{2/5}$
Case III
t < TN1	$0.02\xi M_{\rm ej,-3}^{-1}T_{\rm sd,3}^{-1}t_{}+1$	$7.8\times 10^{9}\xi ^{1/2} M_{\rm ej,-3}^{-1/2}T_{\rm sd,3}^{-1/2}t_{}^{3/2}$
TN1 < t < Tsd	$16.7\xi M_{\rm ej,-3}^{-1}T_{\rm sd,3}^{-1}t_{3}+1$	$1.7\times 10^{16}\xi ^{2} M_{\rm ej,-3}^{-2}T_{\rm sd,3}^{-2}t_{3}^{3}$
Tsd < t < Tdec	$16.7\xi M_{\rm ej,-3}^{-1}+1$	$1.7\times 10^{17}\xi ^{2} M_{\rm ej,-3}^{-2}t_{4}^{}$
Tdec < t < TN2	$3.5\xi ^{1/8}n^{-1/8}t_{6}^{-3/8}+1$	$7.2\times 10^{17}\xi ^{1/4} n^{-1/4}t_{6}^{1/4}$
t > TN2	$0.2\xi ^{2/5}n^{-2/5}t_{8}^{-6/5}+1$	$2.8\times 10^{18}\xi ^{1/5} n^{-1/5}t_{8}^{2/5}$

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Table 2. Temporal Scaling Indices of Various Parameters in Different Temporal Regimes for All Dynamical Cases

	γ − 1	R	νa	νm	νc	Fν, max
Case I: L0 ∼ 1047 erg s−1, Mej ∼ 10−4 M☉
t < Tma1	1	$\frac{3}{2}$	$\frac{5p+4}{2(p+4)}$	$\frac{5}{2}$	$-\frac{7}{2}$	5
Tma1 < t < TN1	1	$\frac{3}{2}$	$\frac{1}{10}$	$\frac{5}{2}$	$-\frac{7}{2}$	5
TN1 < t < Tdec	1	3	$\frac{11}{5}$	4	−6	11
Tdec < t < Tsd	$-\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{5}$	−1	−1	1
Tsd < t < Tma2	$-\frac{3}{8}$	$\frac{1}{4}$	0	$-\frac{3}{2}$	$-\frac{1}{2}$	0
Tma2 < t < TN2	$-\frac{3}{8}$	$\frac{1}{4}$	$-\frac{3p+2}{2(p+4)}$	$-\frac{3}{2}$	$-\frac{1}{2}$	0
t > TN2	$-\frac{6}{5}$	$\frac{2}{5}$	$\frac{2-3p}{p+4}$	−3	$-\frac{1}{5}$	$\frac{3}{5}$
Case II: L0 ∼ 1049 erg s−1, Mej ∼ 10−4 M☉
t < Tma1	1	$\frac{3}{2}$	$\frac{5p+4}{2(p+4)}$	$\frac{5}{2}$	$-\frac{7}{2}$	5
Tma1 < t < TN1	1	$\frac{3}{2}$	$\frac{1}{10}$	$\frac{5}{2}$	$-\frac{7}{2}$	5
TN1 < t < Tsd	1	3	$\frac{11}{5}$	4	−6	11
Tsd < t < Tma2	$-\frac{3}{8}$	$\frac{1}{4}$	0	$-\frac{3}{2}$	$-\frac{1}{2}$	0
Tma2 < t < TN2	$-\frac{3}{8}$	$\frac{1}{4}$	$-\frac{3p+2}{2(p+4)}$	$-\frac{3}{2}$	$-\frac{1}{2}$	0
t > TN2	$-\frac{6}{5}$	$\frac{2}{5}$	$\frac{2-3p}{p+4}$	−3	$-\frac{1}{5}$	$\frac{3}{5}$
Case III: L0 ∼ 1049 erg s−1, Mej ∼ 10−3 M☉
t < Tma1	1	$\frac{3}{2}$	$\frac{5p+4}{2(p+4)}$	$\frac{5}{2}$	$-\frac{7}{2}$	5
Tma1 < t < TN1	1	$\frac{3}{2}$	$\frac{1}{10}$	$\frac{5}{2}$	$-\frac{7}{2}$	5
TN1 < t < Tsd	1	3	$\frac{11}{5}$	4	−6	11
Tsd < t < Tdec	0	1	$\frac{3}{5}$	0	−2	3
Tdec < t < Tma2	$-\frac{3}{8}$	$\frac{1}{4}$	0	$-\frac{3}{2}$	$-\frac{1}{2}$	0
Tma2 < t < TN2	$-\frac{3}{8}$	$\frac{1}{4}$	$-\frac{3p+2}{2(p+4)}$	$-\frac{3}{2}$	$-\frac{1}{2}$	0
t > TN2	$-\frac{6}{5}$	$\frac{2}{5}$	$\frac{2-3p}{p+4}$	−3	$-\frac{1}{5}$	$\frac{3}{5}$

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Following the standard procedure in Sari et al. (1998), we derive the synchrotron radiation characteristic frequencies and the peak flux density at T_dec,

$$\begin{eqnarray} &&\nu _{\rm {a,dec}}\sim 5.0\times 10^{8}\, {\rm Hz}\, L_{0,47}^{3/50}M_{\rm ej,-4}^{4/25}n^{29/50}\epsilon _{e,-1}^{-1}\epsilon _{B,-2}^{1/5}\nonumber \\ &&\quad\quad\quad\times\left(\frac{p-1}{p-2}\right)(p+1)^{3/5}f(p)^{3/5}\nonumber\\ &&\nu _{\rm {m,dec}}\sim 1.3\times 10^{14}\, {\rm Hz}\, L_{0,47}^{6/5}M_{\rm ej,-4}^{-4/5}n^{1/10}\epsilon _{e,-1}^{2}\epsilon _{B,-2}^{1/2}\left(\frac{p-2}{p-1}\right)^{2}\nonumber \\ &&\nu _{\rm {c,dec}}\sim 9.6\times 10^{14}\, {\rm Hz}\, L_{0,47}^{1/5}M_{\rm ej,-4}^{-4/5}n^{-9/10}\epsilon _{B,-2}^{-3/2}\nonumber \\ &&F_{\rm {\nu,max,dec}}\sim 1.7\times 10^{5} \,\mu {\rm Jy}\, L_{0,47}^{3/10}M_{\rm ej,-4}^{4/5}n^{2/5}\epsilon _{B,-2}^{1/2}D_{27}^{-2}, \end{eqnarray} \tag{ 5 }$$

where f(p) = (Γ((3p + 22)/12)Γ((3p + 2)/12))/(Γ((3p + 19)/12)Γ((3p − 1)/12)). With the temporal evolution power-law indices of these parameters (Table 2), one can calculate the X-ray, optical, and radio afterglow light curves. Note that there are two more temporal segments listed in Table 2, since ν_a crosses ν_m twice at

$$\begin{eqnarray} T_{\rm {ma1}}&\sim& 1.4\times 10^{2}\, {\rm s}\, L_{0,47}^{-5/4}M_{\rm ej,-4}^{5/4}n^{1/8}\epsilon _{e,-1}^{-5/4}\epsilon _{B,-2}^{-1/8}\left(\frac{p-2}{p-1}\right)^{-5/4}\nonumber \\ &&\times(p+1)^{1/4}f(p)^{1/4},\nonumber\\ T_{\rm {ma2}}&\sim& 1.9\times 10^{8}\, {\rm s}\, L_{0,47}^{1/5}n^{-2/5}T_{\rm sd,5}^{1/5}\epsilon _{e,-1}^{2}\epsilon _{B,-2}^{1/5}\left(\frac{p-2}{p-1}\right)^{2}\nonumber\\ &&\times(p+1)^{-2/5}f(p)^{-2/5}, \end{eqnarray} \tag{ 6 }$$

respectively. We present the light curves in the X-ray (Figure 2(d)), optical, and radio (10 GHz) bands (Figure 2(c)). The distance is taken as 300 Mpc, the detection horizon of the Advanced LIGO.

Case II: M_ej ∼ M_{ej, c, 1} or T_sd ∼ T_dec. The dynamics and the expressions of the characteristic parameters become simpler:

$$\begin{eqnarray} T_{\rm {dec}}&\sim& T_{\rm sd}\nonumber \\ T_{\rm {N1}}&\sim& 12 {\rm \,s\,} \xi ^{-1}M_{\rm ej,-4}^{}T_{\rm sd,3}\nonumber \\ T_{\rm {N2}}&\sim& 1.3\times 10^{8} \,{\rm s}\, \xi ^{8/3}M_{\rm ej,-4}^{-8/3}T_{\rm sd,3}\nonumber \\ \gamma _{\rm {sd}}&\sim& 83.3\xi M_{\rm ej,-4}^{-1}+1. \end{eqnarray} \tag{ 7 }$$

The temporal indices of the evolutions of ν_a, ν_m, ν_c,  and F_{ν, max} are listed in Table 2, and the expressions of γ and R are shown in Table 1.

As examples, we consider L₀ ∼ 10⁴⁹ erg s⁻¹ (B_p ∼ 10¹⁵ G) versus M_ej ∼ 10⁻⁴M_☉, which satisfies T_sd ∼ T_dec.

Similar to Case I, we have

$$\begin{eqnarray} &&\nu _{\rm {a,sd}}\sim 2.2\times 10^{9}\, {\rm Hz}\, \xi ^{11/5}L_{0,49}^{-3/5}M_{\rm ej,-4}^{-8/5}n^{4/5}\epsilon _{e,-1}^{-1}\epsilon _{B,-2}^{1/5}\left(\frac{p-1}{p-2}\right)\nonumber \\ &&\quad\quad\quad\times(p+1)^{3/5}f(p)^{3/5}\nonumber \\ &&\nu _{\rm {m,sd}}\sim 2.7\times 10^{17}\, {\rm Hz}\, \xi ^{4}M_{\rm ej,-4}^{-4}n^{1/2}\epsilon _{e,-1}^{2}\epsilon _{B,-2}^{1/2}\left(\frac{p-2}{p-1}\right)^{2}\nonumber \\ &&\nu _{\rm {c,sd}}\sim 8.6\times 10^{14}\, {\rm Hz}\, \xi ^{-4}M_{\rm ej,-4}^{4}n^{-3/2}T_{\rm sd,3}^{-2}\epsilon _{B,-2}^{-3/2}\nonumber \\ &&F_{\rm {\nu,max,sd}}\sim 2.4\times 10^{8} \,\mu {\rm Jy}\, \xi ^{11}L_{0,49}^{-3}M_{\rm ej,-4}^{-8}n^{3/2}\epsilon _{B,-2}^{1/2}D_{27}^{-2}\nonumber \\ &&T_{\rm {ma1}}\sim 1.4\times 10^{-1} \,{\rm s}\, \xi ^{-1}L_{0,49}^{-1/4}M_{\rm ej,-4}^{5/4}n^{1/8}T_{\rm sd,3}\epsilon _{e,-1}^{-5/4}\epsilon _{B,-2}^{-1/8}\nonumber \\ &&\quad\quad\quad\times\left(\frac{p-2}{p-1}\right)^{-5/4}(p+1)^{1/4}f(p)^{1/4}\nonumber \\ &&T_{\rm {ma2}}\sim 2.5\times 10^{8} \,{\rm s}\, \xi ^{6/5}L_{0,49}^{2/5}M_{\rm ej,-4}^{-8/5}n^{-1/5}T_{\rm sd,3}^{}\epsilon _{e,-1}^{2}\epsilon _{B,-2}^{1/5}\nonumber \\ &&\quad\quad\quad\times\left(\frac{p-2}{p-1}\right)^{2}(p+1)^{-2/5}f(p)^{-2/5}. \end{eqnarray} \tag{ 8 }$$

The expressions of γ and R as well as the power-law indices for this case are also presented in Tables 1 and 2, respectively. The dynamics, typical frequency evolution, and the light curves are presented in Figure 3. We note that in this case (and Case III), the synchrotron radiation properties are very sensitive to M_ej and ξ.

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Case III: M_{ej, c, 1} < M_ej < M_{ej, c, 2} (T_sd < T_dec). As an example, we take B_p ∼ 10¹⁵ G and M_ej ∼ 10⁻³M_☉.

For this example, the dynamics and the expressions of the characteristic parameters become

$$\begin{eqnarray} T_{\rm {dec}}&\sim& 1.5\times 10^{4} \,{\rm s\,} \xi ^{-7/3}M_{\rm ej,-3}^{8/3}n^{-1/3}\nonumber \\ T_{\rm {N1}}&\sim& 59.9 \,{\rm s}\, \xi ^{-1}M_{\rm ej,-3}^{}T_{\rm sd,3}\nonumber \\ T_{\rm {N2}}&\sim& 2.7\times 10^{7} \,{\rm s}\, \xi ^{1/3}n^{-1/3}\nonumber \\ \gamma _{\rm {sd}}&\sim& 16.7\xi M_{\rm ej,-3}^{-1}+1 \end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray} &&\nu _{\rm {a,sd}}\sim 1.6\times 10^{8}\, {\rm Hz}\, \xi ^{8/5}M_{\rm ej,-3}^{-8/5}n^{4/5}T_{\rm sd,3}^{3/5}\epsilon _{e,-1}^{-1}\epsilon _{B,-2}^{1/5}\left(\frac{p-1}{p-2}\right)\nonumber \\ &&\quad\quad\quad\times(p+1)^{3/5}f(p)^{3/5}\nonumber \\ &&\nu _{\rm {m,sd}}\sim 4.5\times 10^{14}\, {\rm Hz}\, \xi ^{4}M_{\rm ej,-3}^{-4}n^{1/2}\epsilon _{e,-1}^{2}\epsilon _{B,-2}^{1/2}\left(\frac{p-2}{p-1}\right)^{2}\nonumber \\ &&\nu _{\rm {c,sd}}\sim 5.3\times 10^{17}\, {\rm Hz}\, \xi ^{-4}M_{\rm ej,-3}^{4}n^{-3/2}T_{\rm sd,3}^{-2}\epsilon _{B,-2}^{-3/2}\nonumber \\ &&F_{\rm {\nu,max,sd}}\sim 6.5\times 10^{2} \,\mu {\rm Jy}\, \xi ^{8}M_{\rm ej,-3}^{-8}n^{3/2}T_{\rm sd,3}^{3}\epsilon _{B,-2}^{1/2}D_{27}^{-2}\nonumber \\ &&T_{\rm {ma1}}\sim 1.0 {\rm \,s\,} \xi ^{-5/4}M_{\rm ej,-3}^{5/4}n^{1/8}T_{\rm sd,3}^{5/4}\epsilon _{e,-1}^{-5/4}\epsilon _{B,-2}^{-1/8}\left(\frac{p-2}{p-1}\right)^{-5/4}\nonumber \\ &&\quad\quad\quad\times(p+1)^{1/4}f(p)^{1/4}\nonumber \\ &&T_{\rm {ma2}}\sim 9.9\times 10^{7} {\rm \,s} \,\xi ^{1/5}n^{-2/5}\epsilon _{e,-1}^{2}\epsilon _{B,-2}^{1/5}\left(\frac{p-2}{p-1}\right)^{2}\nonumber \\ &&\quad\quad\quad\times(p+1)^{-2/5}f(p)^{-2/5}. \end{eqnarray} \tag{ 10 }$$

The power-law indices of various parameters for this case are also collected in Table 2, and the dynamics, frequency evolutions, and light curves are presented in Figure 4.

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Case IV: M_ej > M_{ej, c, 2}. In this case, the blast wave never reaches a relativistic speed. The dynamics is similar to Case III, with the coasting regime in the non-relativistic phase. The dynamics for a non-relativistic ejecta and its radio afterglow emission have been discussed in Nakar & Piran (2011). Our Case IV resembles what is discussed in Nakar & Piran (2011), but the afterglow flux is much enhanced because of the larger total energy involved.

For all the cases, bright broadband EM afterglow emission signals are predicted. The light curves typically show a sharp rise around T_sd, which coincides with the ending time of the X-ray afterglow signal discussed by Zhang (2013) due to internal dissipation of the magnetar wind. The X-ray afterglow luminosity predicted in our model is generally lower than that of the internal dissipation signal, but the optical and radio signals are much brighter. In some cases, the R-band magnitude can reach 11th at 300 Mpc, if M_ej is small enough (so that the blast wave has a high Lorentz factor) and the medium density is not too low. The duration of detectable optical emission ranges from 10³ s to a year timescale. The radio afterglow can reach the Jy level for an extended period of time, with peak reached in the year timescale. These signals can be readily picked up by all-sky optical monitors and radio surveys. The X-ray afterglow can also be picked up by large field-of-view imaging telescopes such as the ISS-Lobster.

Since these signals originate from interaction between the magnetar wind and the ejecta in the equatorial directions, they are not supposed to be accompanied with SGRBs and some internal-dissipation X-ray afterglows (Zhang 2013) in the free wind zone. Due to a larger solid angle, the event rate for this geometry (orange observer in Figure 1) should be higher than the other two geometries (red and yellow observers in Figure 1). However, the brightness of the afterglow critically depends on the unknown parameters such as M_ej, B_p (and hence L₀), and n. The event rate also crucially depends on the event rate of NS–NS mergers and the fraction of mergers that leave behind a massive magnetar rather than a black hole.

This afterglow signal is much stronger than the afterglow signal due to ejecta–medium interaction with a black hole as the post-merger product (Nakar & Piran 2011). The main reason is the much larger energy budget involved in the magnetar case. Since the relativistic phase can be achieved, both the X-ray and optical afterglows are detectable, which peak around the magnetar spin-down timescale (10³–10⁵ s). The radio peak is later, similar to the black hole case (Nakar & Piran 2011), but the radio afterglow flux is also much brighter (reaching Jy level) due to a much larger energy budget involved. The current event rate limit of >350 mJy radio transients in the minutes-to-days timescale at 1.4 GHz is <6 × 10⁻⁴ deg⁻² yr⁻¹ (Bower & Saul 2011), or <20 yr all sky. In view of the large uncertainties in the NS–NS merger rate and the fraction of millisecond magnetar as the post-merger product, our prediction is entirely consistent with this upper limit. Because of their brightness, these radio transients can be detected outside the Advanced LIGO horizon, which may account for some sub-mJy radio transients discovered by the Very Large Array (Bower et al. 2007).

Recently, Kyutoku et al. (2012) proposed another possible EM counterpart of GWB with a wide solid angle. They did not invoke a long-lasting millisecond magnetar as the merger product, but speculated that during the merger process a breakout shock from the merging neutron matter would accelerate a small fraction of surface material, which reaches a relativistic speed. Such an outflow would also emit broadband synchrotron emission by shocking the surrounding medium. Within that scenario, the predicted peak flux is lower and the duration is shorter than the EM signals predicted in Zhang (2013) and this work due to a much lower energy carried by the outflow.

Detecting the GWB-associated bright signals as discussed in this paper would unambiguously confirm the astrophysical origin of GWBs. Equally important, it would suggest that NS–NS mergers leave behind a hyper-massive neutron star, which gives an important constraint on the neutron star equation of state. With the GWB data, one can infer the information of the two neutron stars involved in the merger. Modeling afterglow emission can give useful constraints on the ejected mass M_ej and the properties of the post-merger compact objects. Therefore, a combination of GWB and afterglow information would shed light into the detailed merger physics, and in particular, provide a probe of massive millisecond magnetars and stiff equations of state for neutron matter.

We thank Yi-Zhong Fan and Jian-Yan Wei for stimulating discussions. We acknowledge the National Basic Research Program ("973" Program) of China under grant Nos. 2009CB824800 and 2013CB834900. This work is also supported by the National Natural Science Foundation of China (grant Nos. 11033002 and 10921063) and by NSF AST-0908362. X.F.W. acknowledges support by the One-Hundred-Talents Program and the Youth Innovation Promotion Association of Chinese Academy of Sciences.