Define modulation index. Why is it kept low? What is the role of a bandpass filter?

In order to transmit some message over long distance, amplitude modulation is used, where a initially low frequency message signal is superimposed on a high frequency carrier wave as a result a new wave is formed having high frequency whose amplitude is varying in accordance with message signal

As shown in figure

The equation of waveform of message signal is

m(t) = A_msinω_mt

where A_m is the amplitude of message signal , ω_m is the angular frequency of the message signal

The equation of waveform of Carrier wave is

c(t) = A_csinω_ct

where cis the amplitude of Carrier wave, ω_c is the angular frequency of the Carrier wave

the equation of waveform of amplitude modulated wave having high frequency of carrier wave and variation in amplitude in accordance with message signal is

c_m(t) = (A_c + A_msinω_mt)sin ω_ct

or can be written as

c_m(t) = A_c (1 + A_m/A_c sinω_mt)sin ω_ct

so this ratio of amplitude of original message signal to amplitude of carrier wave is called modulation index