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It is said that before the big bang there was no before, as space/time was created in the big bang. The big bang creates a rapid expansion of space/time... so how fast is rapid?

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    $\begingroup$ Welcome to Physics.SE! Please note that we expect a modicum of prior effort behind the questions posted here. Since the answer to that question can be found by literally typing "speed of expansion of space" into Google (albeit the first number it gives you is slightly off), this doesn't reflect as showing prior effort. $\endgroup$ – Jim Oct 1 '15 at 17:21
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Your question is very easy to answer because the answer is just the value of the Hubble constant (or its inverse the Hubble time). The trouble is that it's hard to explain to a non-physicist what the answer means. I'll have a go, but you may find it hard going.

When we are describing the expansion of the universe we can't simply talk about its size. That's because the size of the universe is unknown, and indeed it may be infinite in which case it doesn't have a size at all. We get round this by describing the expansion using the distance between objects in the universe.

Suppose there is a galaxy 10 billion light years away. If the universe is expanding this means the distance to the galaxy will be increasing. So if the universe expands by a factor of two the distance to the galaxy would be 20 billion light years. Conversely, if the universe were contracting then the distance to the galaxy would be decreasing. If the universe contracted to a half then the distance to the galaxy would be 5 million years.

Mathematically we describe the expansion using a scale factor, which we write as $a(t)$. We define the scale factor to be one right now, so if the scale factor increases to two it means everything is twice as far away and the universe has doubled in size. If the scale factor decreases to $0.5$ it means everything is twice as close and the universe has halved in size. We calculate the way the scale factor varies with time using general relativity, and the results look like this (taken from this question):

Scale factor

Given that describing the expansion of the universe is quite complicated, there isn't a simple answer to your question How fast does space/time travel?, but we could for example as how long it takes the universe to double in size. That seems like a reasonable criterion for the speed of the expansion. This doubing time is called the Hubble time, and we can calculate it from the scale factor. The variation of the Hubble time looks like this (from the same question as the diagram above):

Hubble time

The $y$ axis shows the ratio of the Hubble time to its current value of about $14.5$ billion years. So $2$ on the $y$ axis means the Hubble time is $29$ billion years and $0.5$ on the $y$ axis means a Hubble time of $7.25$ billion years.

The graph shows that just after the Big Bang the Hubble time was very small, so the universe was doubling in size every few seconds. Right now it's doubling in size every $14.5$ billion years, and in the future the doubling time settles down to a constant value of around $17$ to $18$ billion years.

I started out by measuring the Hubble parameter, which I'm sure you've heard of. The Hubble parameter is just the reciprocal of the Hubble time. So a big Hubble parameter means the universe is expanding rapidly and a small Hubble parameter means the universe is expanding slowly. The useful thing about the Hubble parameter is it tells us how fast distant galaxies are moving away from us. The velocity $v$ of some object at a distance $d$ is given by:

$$ v = H\, d $$

where $H$ is the Hubble parameter. This is known as Hubble's law. The Hubble parameter has been decreasing with time, so in the past objects $10$ billion light years away were moving faster than they are now.

So there are lots of ways to describe how fast the universe is expanding, but no simple way. However I hope this has made things a bit clearer without melting too many brain cells.

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How fast does space/time travel?

Fast. But note that there's no travelling per se, and that space expands. Have a google on that. Spacetime is a "space at all times" abstract thing. Here's a depiction of it, courtesy of NASA:

enter image description here

It looks like a champagne flute on its side, without a handle or base. The time axis goes from left to right, and the diameter increases. Let's say the real universe is spherical and getting bigger all the time. If you took a slice out of the middle every billion years and lined up all your slices left to right like slices of bread, the shape you'd end up with would be like this champagne flute. Every vertical line represents a billion years.

It is said that before the big bang there was no before, as space/time was created in the big bang. The big bang creates a rapid expansion of space/time... so how fast is rapid?

Fast. The big bang is the bright spot on the left where the champagne flute begins. The period after that is said to be when inflation occurred, when the universe expanded very very fast. However nobody knows this for sure, there's no actual evidence for inflation. There's is however good evidence that the universe is expanding. You can find articles like this about that:

"The most precise measurement ever made of the speed of the universe's expansion is in, thanks to NASA's Spitzer Space Telescope, and it's a doozy. Space itself is pulling apart at the seams, expanding at a rate of 74.3 plus or minus 2.1 kilometers (46.2 plus or minus 1.3 miles) per second per megaparsec (a megaparsec is roughly 3 million light-years). If those numbers are a little too much to contemplate, rest assured that's really, really fast. And it's getting faster all the time".

But that doesn't really tell you how fast. To get a feel for that, check out the observable universe on Wikipedia:

"The comoving distance from Earth to the edge of the observable universe is about 14 gigaparsecs (46 billion light years or 4.3×1026 meters) in any direction".

The radius of the observable universe is 46 billion light years. The actual universe could be bigger than that, but we have no evidence that it is, and we certainly have no evidence that the universe is infinite. So for now let's go with the 46 billion light years, and visualize the universe as an expanding sphere:

enter image description here CCASA image of the observable universe by Azcolvin429, see Wikipedia

The radius has expanded from some small size to at least 46 billion light years in 13.8 billion years. So how fast is it expanding? On average it's at least 3.333333 times the speed of light. How's that for fast? And that's just for starters, because it's getting faster. And see articles like this which claim the diameter of the universe is at least 156 billion light years.

"All the distance covered by the light in the early universe gets increased by the expansion of the universe," explains Neil Cornish, an astrophysicist at Montana State University. "Think of it like compound interest". Need a visual? Imagine the universe just a million years after it was born, Cornish suggests. A batch of light travels for a year, covering one light-year. "At that time, the universe was about 1,000 times smaller than it is today," he said. "Thus, that one light-year has now stretched to become 1,000 light-years". All the pieces add up to 78 billion-light-years. The light has not travelled that far, but "the starting point of a photon reaching us today after traveling for 13.7 billion years is now 78 billion light-years away," Cornish said. That would be the radius of the universe, and twice that - 156 billion light-years - is the diameter. That's based on a view going 90 percent of the way back in time, so it might be slightly larger. "It can be thought of as a spherical diameter is the usual sense," Cornish added comfortingly.

I like the compound interest myself. I don't like inflation, or the infinite universe, but that compound interest I like.

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