X-ray background/peculiar velocities and the Big Bang

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X-ray background and peculiar velocities provide insights into the Big Bang model, primarily through the understanding of cosmic expansion and structure formation. The microwave background radiation, originating from about 300,000 years after the Big Bang, cooled to its current temperature of slightly under 3 K as the universe expanded. Galaxies exhibit peculiar velocities that are proportional to their distance, supporting the theory of an expanding universe that began nearly 14 billion years ago. While the diffuse X-ray background can be resolved into individual sources, its exact role in distinguishing cosmological models remains unclear. In contrast, statistical analysis of peculiar velocities can effectively test different cosmological models, particularly those involving dark matter and structure formation.
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I heard that the X-ray background and peculiar velocities provide evidence that leads scientists to believe in the Big Bang model. What exactly do they have to do with it?
 
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The backgound is not x-ray, but microwave. According to theory about 300,000 yrs. after big bang, the universe was cool enough for atmos to form. The universe was filled with radiation at a high temperature. As the universe expanded, the radiation cooled down to its present value (slightly under 3 deg. K). There doesn't seem to be any other explanation.

The velocities that you refer to are those of galaxies which are going away from us with speed proportional to distance - characteristic of an expanding universe, which started a little less than 14 bil. yrs ago.
 
There is actually a x-ray background that is diffuse at large scales, but that can be resolved in individual sources when observing with high angular resolution. The background is assumed to be the result of individual sources at high redshifts. There are however lots of unanswered questions regarding the x-ray background such as its relation to structure formation. I am not aware how this background can be used to discriminate between cosmological models.

On the other hand, statistical analysis of peculiar velocities can be a test of cosmological models or at least classes of models. This is because the model-dependent spectrum of perturbations determines the conditions for non-linear gravitational collapse and how structure is formed. Thus for example a cold dark matter model with a hierarchical bottom-up structure formation history predicts high peculiar velocities due to collisions and tidal forces.
 
https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
Why was the Hubble constant assumed to be decreasing and slowing down (decelerating) the expansion rate of the Universe, while at the same time Dark Energy is presumably accelerating the expansion? And to thicken the plot. recent news from NASA indicates that the Hubble constant is now increasing. Can you clarify this enigma? Also., if the Hubble constant eventually decreases, why is there a lower limit to its value?

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