because radiation can go into earth from the secondary side and thats what common mode noise is.
I am not so sure. But I am willing to learn.
My understanding is as follows:
Common mode noise is associated with a common mode voltage. This becomes important whenever you are using a differential source (for example an op-amp or a transformer). An ideal op-amp or a transformer uses only differential voltage as input and the individual voltages are ignored.
For a real device, the actual potentials are important and that is not because of noise. Internal structure requires that the common mode voltage must be kept within some reasonable limits. For a practical transformer, the core saturation can put a limit or for an air core transformer, the dielectric breakdown may be considered.
For a transformer the output is also differential. This is not the noise we are talking about.
Consider this gedanken: you are using a battery as the input to a DC-DC isolated supply. How does the earth come in the picture?
In my understanding, earth is acting just like a capacitor- it can source or sink some reasonable amount of charge.
Noise is basically a voltage source and is related to impedance. Unfortunately common mode potential can be noisy.
What happens if you remove the bypass Y capacitor (as far as the noise is considered)?