# Speed of light through the ISM and Wavelength

Articles written about fast radio bursts describe the signal's short-wavelength components arriving before its longer-wavelength components, suggesting energy-dependent time delays in the interstellar medium (ex. http://www.scientificamerican.com/article/a-brilliant-flash-then-nothing-new-fast-radio-bursts-mystify-astronomers/).

Why are lower-energy signals slowed slightly more than higher-energy signals when passing through clouds of electrons? Also, how do we know about this slowing effect if it appears to be so small (what objects or events have exhibited the time difference)?

My first-order logic guessed that higher energy radiation is slowed more, remembering the mnemonic "Blue bends best" from optics. I assume these intergalactic clouds behave quite differently from a solid prism.

When electromagnetic waves travel through a plasma, they excite currents in the free charged particles. In such cases it can be shown (using Maxwell's equations) that the waves propagate with a relationship between their frequency $$\omega$$ and "wavenumber" $$k = 2\pi/\lambda$$ given by $$\omega^2 = \omega_p^{2} + c^2k^2,$$ where $$\omega_p$$ is known as the "plasma" frequency and equals $$(n_e e^2/\epsilon_0 m_e)^{1/2}$$ for the electrons in the plasma (i.e. it depends on the electron number density $$n_e$$.).
Now, if you have a bunch of photons emitted as a pulse, the relevant velocity is the group velocity given by $$v_g=d\omega/dk$$. So $$v_g = \frac{c^2 k}{(\omega_p^{2} +c^2 k^2)^{1/2}} = c\left(1 - \frac{\omega_p^2}{\omega^2}\right)^{1/2}$$
This converges to $$c$$ when the frequencies are high (wavelengths are short), but is slower when frequencies are low (wavelengths are long).