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I have read this question:

Effect of expansion of space on CMB

where Ted Bunn says:

For definiteness, let's consider a wave packet of electromagnetic radiation with some fairly well-defined wavelength. At some early time, it has a wavelength $\lambda_1$ and energy $U_1$. (I'm not calling it $E$ because I want to reserve that for the electric field.) After the Universe has expanded for a while, it has a longer wavelength $\lambda_2$ and a smaller energy $U_2$. (Fine print: wavelengths and energies are measured by a comoving observer -- that is, one who's at rest in the natural coordinates to use.) In fact, the ratios are both just the factor by which the Universe has expanded: $$ {\lambda_2\over\lambda_1}={U_1\over U_2}={a_2\over a_1}\equiv 1+z, $$ where $a$ is the "scale factor" of the Universe. $1+z$ is the standard notation for this ratio, where $z$ is the redshift.Just to be clear: by "amplitude" you mean the amplitude of a classical electromagnetic wave -- that is, the peak value of the electric field -- right? In that case, the answer is that the amplitude goes down.

But the CMB does have a certain redshift.

The cosmic microwave background has a redshift of z = 1089, corresponding to an age of approximately 379,000 years after the Big Bang and a comoving distance of more than 46 billion light years.

https://en.wikipedia.org/wiki/Redshift

Is this a contradiction? This does not explain whether the CMB's photons' wavelengths themselves are getting stretched as the universe expands.

Question:

  1. Do the CMB's photons wavelength get stretched as they travel through expanding space or is the CMB redshift constant?
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You need to remember Hubble's discovery - the farther away an object is, the more the light it emitted has been redshifted between it and you. Because light moves at finite speed, this also means that the time between when the light was emitted and observed also correlates with redshift. So, when someone says something "has a redshift" or is "at a redshift", they're saying the wavelength has been stretched by a factor of one plus the redshift since the light was first produced, and you can use the correlations to figure out how far away it is now, and how long the light took to get to you, among other things.

See Ned Wright's Cosmology Tutorial for more, including diagrams.

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I don't see the contradiction. The wavelength of light stretches with time in the way that you wrote. The wavenumber $k=2\pi/\lambda$ decreases accordingly.

What we mean when we say that the CMB has a redshift of 1089 is that it was emitted (i.e. released, when electrons and protons bound into atoms) at a time when the universe was 1090 times smaller than today. Since being emitted, its wavelength has therefore stretched, along with the universe, by a factor of 1090, to become the wavelength we see today, about 1 mm.

The CMB was emitted in all directions, and the CMB radiation we see at any given time is, from our perspective, all coming from the surface whose light took the same time to reach us.

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