How does the modulation index of a phase-modulated emission vary with RF carrier frequency?
Why Modulation index for angle modulation is the peak carrier phase deviation in radians, which equals peak frequency deviation divided by the modulating frequency. Both of those quantities are set by the modulating audio (its amplitude and its frequency), so the RF carrier frequency never enters the equation. In a transmitter that uses frequency multiplier stages, the deviation is multiplied by the same factor as the carrier, so the ratio, and hence the index, stays the same.
Watch out The idea that the index drops as carrier frequency rises comes from thinking a given deviation is a smaller percentage of a higher carrier, but modulation index compares deviation to the modulating frequency, not to the carrier.
Index = deviation / modulating frequency. The carrier frequency is not in the formula, so it cannot matter.