*3.2. The Effect of the Variable Electron Injection Rate*

Although theoretically the low-energy photon spectral index *α* can reach −2/3 caused by a decaying magnetic field, since this is a gradual process, *α* would be softer than −2/3 for *E* . *Ep*, which can be seen from Figure 3. In fact, *α* ∼ −1 can be fitted easily, but fitting *α* slightly smaller than −2/3 is difficult. We here consider a variable electron injection rate, which could induce *α* ∼ −2/3. The variable electron injection rate may be suggested by that the actual GRB shell is not homogeneous and presents a density profile, for example, a Gaussian density profile, inducing a rising electron injection rate. Nonetheless, we do not know its growing method clearly. Ref. [22] discussed this effect in their "toy box model", and suggested, because of a rising electron injection rate, *α* goes from −0.82 to −1.03, which is dependent on the growing power-law index *q* (where the injection rate ∝ *t* <sup>0</sup>*<sup>q</sup>* with *q* = 1, 2 or 3). Here we adopt similar expressions of the rising electron injection rate,

$$dN\_{\varepsilon,2} = 8\pi R^2 n\_1'(\gamma\_{21}\mathfrak{E}\_{21}/\gamma\mathfrak{E})\gamma^2 cdt \times \left(\frac{t}{t\_0}\right)^q \tag{18}$$

and

$$dN\_{\varepsilon,3} = 8\pi R^2 n\_4' (\gamma\_{34}\beta\_{34}/\gamma\beta) \gamma^2 cdt \times \left(\frac{t}{t\_0}\right)^q,\tag{19}$$

where the factor ( *t t*0 ) is to maintain the same electron injection number in the interval *t*<sup>0</sup> as that for the constant injection rate *q* = 0.

We show the electron distribution for a rising electron injection rate in the top panel of Figure 4. The rising electron injection would increase the electrons injected later, which would cool in a weaker magnetic field and pile up at .*γ* 0 *<sup>e</sup>*,*m*. This can result in a harder electron distribution and a relative spectrum. The slopes of the spectra are presented in the bottom panel of Figure 4. We can see that the slopes of the spectra tend to reach −4/3 more easily than in the constant electron injection case. In addition, a larger *q* would generate a harder low-energy spectral index.

**Figure 4.** The top panel shows the electron distributions in evolutional magnetic fields and different electron injection rising indices. We adopt the electron injection rising index, *q* = 0 (solid line), *q* = 1 (dotted line) and *q* = 2 (dashed line). The bottom panel shows the corresponding synchrotron spectral slopes for these electron distributions. The same parameters as in Figure 1 are taken for numerical calculations.
