If we know the universe is expanding in whatever direction we look, can't we reasonably estimate where the 'center' of the Universe is?
Is the rate of expansions in all directions the same?
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The speed that a galaxy moves away from you is larger the further away the galaxy already is. This is most consistent with an expanding universe.
I find the following mental picture helpful: Imagine a balloon, with lots of little dots painted on it. If you inflate the balloon, then every point is moving away from every other point. Where would the "center" be? Pretty much everywhere.
Wikipedia kindly presents four models you can use to visualize the expansion of the universe. All of these are meant to explain why we think the universe is expanding, yet we think there is no center. Everything† moves away from everything else, with expanding velocity proportional to the distance.
- In the "ant on a rubber rope model" one imagines an ant (idealized as pointlike) crawling at a constant speed on a perfectly elastic rope which is constantly stretching. If we stretch the rope in accordance with the ΛCDM scale factor and think of the ant's speed as the speed of light, then this analogy is numerically accurate—the ant's position over time will match the path of the red line on the embedding diagram above.
- In the "rubber sheet model" one replaces the rope with a flat two-dimensional rubber sheet which expands uniformly in all directions. The addition of a second spatial dimension raises the possibility of showing local perturbations of the spatial geometry by local curvature in the sheet.
- In the "balloon model" the flat sheet is replaced by a spherical balloon which is inflated from an initial size of zero (representing the big bang). A balloon has positive Gaussian curvature while observations suggest that the real universe is spatially flat, but this inconsistency can be eliminated by making the balloon very large so that it is locally flat to within the limits of observation. This analogy is potentially confusing since it wrongly suggests that the big bang took place at the center of the balloon. In fact points off the surface of the balloon have no meaning, even if they were occupied by the balloon at an earlier time.
- In the "raisin bread model" one imagines a loaf of raisin bread expanding in the oven. The loaf (space) expands as a whole, but the raisins (gravitationally bound objects) do not expand; they merely grow farther away from each other.
† Bound objects—in particular galaxies—are held together by binding forces like gravitation and therefore do not expand internally.
I think there are two different questions in your question. Or maybe it is better to say that before asking your question, another question should be considered first.
Your question: Can we estimate where the center of the universe is?
You also ask about the rate of expansion but I think we have to address the question above first.
And maybe your question implies this question. But I think it is easier to separate the two and first ask:
Is there a center of the universe?
According to the first line of this Wikipedia entry: https://en.m.wikipedia.org/wiki/History_of_the_center_of_the_Universe
"The center of the Universe is a concept that lacks a coherent definition in modern astronomy; according to standard cosmological theories on the shape of the universe, it has no center."
So the current theories are that it has no center. And the article is a good summary of the history of theories that the earth is the center, the sun is the center, our galaxy the milky way is the center, etc.
So, the current "theories" are that there is no center. And since they all are still currently theories, without complete scientific proof, the answer really is: "We dont know."
And it is widely accepted that we cant even see the entire universe because it is believed that past a certain point (in all directions) the universe is expanding outward (from earths reference point) faster than the speed of light, therefore our view of the universe is limited by the speed of light. So, maybe we can't see a center that possibly exists beyond our observable universe. But the current theories are that there is no center.
Now, the rate of expansion in all directions has been observed to be the same in all directions which might seem to support the theory that the earth is the center of the universe. But it is believed that you would see the same rate of expansion around you from any point within the universe, which leads to the theory that space itself is expanding, not the actual distances between galaxies.
I like how it is described here.
The big bang was an,
"explosion of space, not an explosion in space...
..., our view of the universe is limited by the speed of light and the finite time since the Big Bang. The observable part is very large, but it is probably very small compared to the whole universe, which may even be infinite. We have no way of knowing what the shape of the universe is beyond the observable horizon, and no way of knowing whether the cosmological principle has any validity on the largest distance scales possible.
If the Big Bang were an ordinary explosion in an already existing space, we would be able to look out and see the expanding edge of the explosion with empty space beyond. Instead, we see back towards the Big Bang itself and detect a faint background glow from the hot primordial gases of the early universe. This "cosmic microwave background radiation" is uniform in all directions. This tells us that it is not matter that is expanding outwards from a point, but rather it is space itself that expands evenly.
It is important to stress that other observations support the view that there is no centre to the universe, at least insofar as observations can reach. The fact that the universe is expanding uniformly would not rule out the possibility that there is some denser, hotter place that might be called the centre, but careful studies of the distribution and motion of galaxies confirm that it is homogeneous on the largest scales we can see, with no sign of a special point to call the centre...
If the entire universe turned out to have a centre on some scale beyond the observable universe, such a centre might turn out to be just one of many "centres" on much larger scales, just as the centre of our galaxy did before.
In other words, although the standard Big Bang models describe an expanding universe with no centre, and this is consistent with all observations, there is still a possibility that these models are not accurate on scales larger than we can observe. We still have no real answer to the question "Where is the centre of the universe?"."
We can't locate the centre of the universe because there isn't one. Every part of the universe was part of the big bang and there are numerous duplicates on Physics SE that address this. e.g.
What is our location relative to the Big Bang?
Is the rate of expansions in all directions the same?
You are essentially asking whether the universe is isotropic? Most cosmological models assume that on large scales ($>100$ Mpc), that this is true.
Observationally, it is - to a very high degree of precision - once you remove the motion of the Galaxy with respect to the cosmic microwave background rest frame. But there are intriguing recent, and highly contested results, that suggest the rate of expansion, and particularly the acceleration of the rate of expansion, do show some anisotropies (e.g. Feindt et al. 2013; Colin et al. 2019) that may just be due to a large part of our local part of the universe being part of some bulk flow (or something else).