They do not need to be related. If the cloud is oscillating in some way, ie it may be pulsing at the fundamental, then parts of the cloud will have a higher mass (for that sub-volume) than others ... so it is possible that some parts of the cloud will exceed the Jeans mass (for that sub-volume) even if the entire cloud does not. However, the cloud will have a Jeans mass even if it is not pulsing.
During radiation domination the Jeans length is of order the horizon size but still longer than the wavelength of the fundamental tone, so that all subhorizon modes oscillate as sound waves. See https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/#refsDiscman said:
We see everything from the present horizon all the way down to smaller scales well-within the sound-horizon at decoupling. However, only those modes with wavelengths on order of and smaller than the horizon at decoupling are oscillating (see Figure 4 in the article I referenced earlier).Discman said:
Can you reference a particular figure where you see this? Often in the literature, the Hubble sphere is taken to correspond to the event horizon, even though this identification is only true in de Sitter space.
I agree with Wayne and I don't believe I've contradicted him: one has fluctuations immediately after inflation, but only those within the horizon have had time to complete oscillations as sound waves. There are of course some modes within the horizon at the end of inflation, but those that are much larger are effectively frozen -- this is a simple causality argument. The horizon is not a hard boundary, beyond which all modes are completely frozen; indeed, the fundamental mode has a wavelength twice the horizon size and it has completed a half oscillation. The horizon can, however, be used as a quick way of separating the effectively frozen modes from the acoustic waves, and the temperature spectrum illustrates this fact (broad central peak corresponds to fundamental mode, with the correlation dying off at larger scales).Discman said:
The fundamental mode acoustic wave is a type of sound wave that is characterized by its lowest frequency and longest wavelength. It is the simplest form of acoustic wave and is often referred to as the first harmonic.
The fundamental mode acoustic wave is generated by an oscillating source, such as a vibrating object or an electronic device. The oscillations of the source cause the surrounding medium, such as air or water, to vibrate and propagate the sound wave.
The fundamental mode acoustic wave has a frequency that is determined by the source and a wavelength that is determined by the properties of the medium. It travels at a constant speed and can be described by its amplitude, frequency, and wavelength.
The fundamental mode acoustic wave has many practical applications, such as in communication systems, medical imaging, and non-destructive testing. It is also used in musical instruments, such as string and wind instruments, to produce different pitches and tones.
The fundamental mode acoustic wave differs from other types of acoustic waves, such as higher harmonics or complex waves, in its frequency and wavelength. It is the simplest form of acoustic wave and serves as the basis for more complex sound waves.