How do we know that the universe expands?

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Redshift is a key indicator of the universe's expansion, but it is not the sole reason for this belief. Modern cosmology relies on multiple lines of evidence, including the cosmic microwave background's temperature and anisotropy, galaxy evolution observations, and elemental abundances consistent with Big Bang Nucleosynthesis. Additional supporting data comes from the Lyman alpha forest, gravitational lensing time delays, and Baryon Acoustic Oscillations observed in galaxy surveys. Historical models like the steady state theory have been largely dismissed due to the accumulation of overwhelming evidence supporting the current cosmological model. The understanding of the universe's expansion continues to evolve as new evidence emerges.
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Is redshift the only reason? And why cannot redshift indicate whatever else than speed?
 
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Redshift isn't a reason why the universe expands; the fact that light is redshifted simply tells us that the universe is expanding.

What else do you suppose it can indicate?
 
Maybe a bad formulation. I try again: Is redshift the only reason that we believe that the universe is expanding?
 
In the 1930's yes this would be true. But we have many pieces of evidence today. These include:
* The temperature and anisotropy angular power spectrum of the cosmic microwave background
* The observation of galaxy evolution with redshift
* The observation of elemental abundances that agrees with Big Bang Nucleosynthesis
* The Lyman alpha forest (in Quasar spectra) evolution with redshift and the Gunn-Peterson test
* Time delays in gravitational lens systems (which give us a measure of the Hubble constant as a funciton of redshift)
* Observation of the Baryon Acoustic Oscillation peak seen in galaxy redshift surveys (evidence of the hot origin of the early universe)
* The observation of the evolution of structure in the Universe with redshift
* Agreement between the amplitude of structure in the CMB and the Universe today (When evolved forward with simple theory)

There are many more but that should keep you busy. It is not on a whim that the ideas of modern cosmology have been developed, they have been developed precisely because of the overwhelming evidence. Alternative ideas were reasonable when there was less evidence, for instance Fred Hoyle's steady state model, however new evidence ruled these out leaving the current model as the main contender at present, but that could all change with new evidence.
 
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