 Solution

In the cold underdense model, $1-v_g/c\simeq n/(2n_c)$ and ${\gamma }_{\varphi }^2\simeq n_c/n=500$. A relativistic electron slips through half an accelerating sinusoidal wavelength over $L_d\simeq {\gamma }_{\varphi }^2{\lambda }_p\simeq 8.35$ mm. For $E_z=E_0\sin \varphi$ and injection at the start of that half-cycle, integration over $0<\varphi <\pi$ givesInjection at the field crest and acceleration only to its next zero gives half this value, about 0.51 GeV; this explains a common factor-two convention in dephasing estimates. Multiplying the peak field by the entire dephasing length is instead an upper bound, not the sinusoid's actual integral. These are ideal wave-amplitude estimates: $a_0<1$ need not excite $E_0$, and finite pump depletion, pulse evolution and injection physics may reduce the gain. Without a specified nonlinear wake shape, the data warrant an order-GeV cold-wave upper scale, not a unique precision maximum for an actual experiment.