What is the tradeoff in decreasing the frequency bandwidth when reducing noise of a resistor?

A resistor's Johnson (Inherent) noise can be modelled as an RMS voltage in series with the resistor: $$e_{rms}\triangleq\sqrt{4kRT\Delta f}$$ where k is Boltzmann's constant, R is resistance, T is absolute temperature in Kelvin, and delta-f is the frequency bandwidth of the circuit.

Mathematically speaking, decreasing the frequency bandwidth decreases e_rms and effectively reduces noise. However I question what the tradeoff of this operation could be. In other words, there has to be some disadvantage or issue if we decrease the frequency bandwidth.

• What do you mean? You can't just reduce the bandwidth on a real resistor. You can reduce the bandwidth of the measurement, but that's not reducing the inherent noise. If you are talking about reducing measured noise by reducing the bandwidth a circuit operates at, then the obvious disadvantage is reduced bandwidth. May 6, 2022 at 14:58
• can't we use some bandpass filter? @DKNguyen May 6, 2022 at 15:00
• Then you are asking about the disadvantage of reducing noised measured by reducing circuit bandwidth. See second half of my previous response. May 6, 2022 at 15:01