Acceleration proportional to what?

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acceleration proportional to what??

Hello,

We know that according to Hubble's law V=HD that velocity of distant galaxies are proportional to distance from earth but what about acceleration?? Is acceleration of space with distant galaxies proportional to distance from earth and one another or different model works for it??
 

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Acceleration of space leads to deviations from Hubble's law.
The acceleration is a function of time. In the early universe, it was negative (so expansion slowed), now it is positive (so expansion is accelerating). Instead of the Hubble "constant" (which is not constant on cosmological timescales), it is better to interpret the evolution of the universe in terms of the scale factor. Without acceleration, the scale factor would increase with a constant derivative.

If you know the scale factor at the point where some light was emitted and the scale factor now (defined to be 1), you can calculate the redshift.
Example graph for the scale factor with various scenarios (red=accelerating) from NASA.
You can also calculate how bright those objects should appear to us, if you can estimate their true brightness. By comparing those observations with the model, you can measure the accelerated expansion. And this has been done with many measurements (from the supernova cosmology project)
 
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But what about Quintessence model of dark energy?? If we measure acceleration of galaxies and if it is different for different galaxies which are currently at different distance from earth than quintessence model is correct because it predict different dark energy density over place and over time otherwise cosmological constant model is true.
 
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Even the cosmological constant model leads to a time-dependence of the acceleration, as the decreasing matter/radiation density leads to a acceleration of the acceleration (3rd derivative of the scale factor).
If the time-dependence deviates from this prediction, then a time-dependent cosmological constant might be an explanation. But that will need much more precise observations.
 

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