# How can bubble universes collide in eternal inflation?

In eternal inflation, how can bubble universes collide if the space between them is exponentially stretching and moving them further and further away?

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## 2 Answers

Light travels through space always at the speed of light locally, but remember that on top of that space itself is stretching as well. This makes the "God's eye view" velocity (which is not anything you can measure directly, but only compute) greater than the speed of light. By "God's eye view" I mean a velocity which is defined in terms of some particularly nice coordinate systems but that have no true physical significance: faster than light travel and signalling are still forbidden.

Here is a "conformal diagram" showing the situation of a bubble collision:

This is a spacetime diagram where the exponential expansion has been "factored out". This means that the units of time and distance get larger as you go up the page. Time runs from bottom to top and of course I'm only showing one of the spatial dimensions. Light travels on 45 degree diagonals on such a diagram. The fact that light goes at 45 degree diagonals means that local measurements always give the same speed of light $c$. But because the units of time and distance are changing from place to place the total distance / time calculation from, say, a blue dot to the red dot doesn't have to give $c$. Also note that for our purposes the horizontal line is just to guide the eye.

I've highlighted two bubble nucleation events with blue dots. You can see that the bubbles expand at the speed of light and eventually collide at the red dot. What happens next depends on the details of the model. I've shown a simple case where the "A" bubble dominates over the "B" bubble (which might be the case if the vacuum energy of A is smaller than B for example). A and B are both infinitely big bubble universes (I've really squashed space down to fit it all on the diagram!) but they see very different things. No observer in B is ever aware of the existence of A: no light signals from A can reach into the interior of B. On the other hand A can see a bubble collision in the past.

I can mathematically show that light can travel faster than $c$ in the "God's eye view" frame, though I won't attempt to compute a bubble collision for you ;). This will only help if you know a little general relativity. The metric for a spatially flat exponentially expanding universe is

$$\mathrm{d}s^{2}=-c^2\mathrm{d}t^{2}+a^{2}\left(t\right)\left(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}\right),$$

where $a(t) = \exp(Ht)$ is the scale factor. Light travels on a null geodesic. We can take a light ray in the $+x$ direction for simplicity, and we get

$$c^2 \dot{t}^2 = a^2(t) \dot{x}^2,$$

where the derivative is with respect to an arbitrary affine parameter. Rearranging and integrating we get

$$x(t)=\int\frac{c\mathrm{d}t}{a\left(t\right)}=c\frac{1-\mathrm{e}^{-Ht}}{H}.$$

This is the comoving coordinate of the light ray as a function of time. The proper distance from $x=0$ is just the scale factor times this:

$$s\left(t\right)=c\frac{\mathrm{e}^{Ht}-1}{H},$$

and the "God's eye view" speed is

$$\frac{\mathrm{d}s}{\mathrm{d}t}=c\mathrm{e}^{Ht}$$

which can become arbitrarily faster than $c$. Despite this note that the light ray cannot reach everything because as $t\to\infty$, $x\to c/H$ which is finite. This is called the cosmological horizon. There are observers who will never get your message, no matter how long you wait, even though your signal can travel an arbitrarily large distance over an arbitrarily long time. General relativity is weird like that. But also notice that in the short time limit (so thinking about local measurements) the speed goes to $c$, so general relativity is not so weird after all.

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This is a less mathematical version of Michael's answer.

It isn't the case that two growing bubble universes are automatically causally disconnected. Even in an inflating universe any spacetime point has a past light cone of non-zero size consisting of all the spacetime points causally linked to it. A bubble universe nucleating anywhere within the past light cone can collide with our point.

Even if the two bubble universes nucleate at exactly the same time they may still collide. As Michael describes, in the inflating universe the causal horizon is at a distance of $c/H$. Provided the two universes nucleate at a distance of less than $c/H$ their relative velocity will be less than $c$ and they can collide.

Incidentally, if you're interested the original paper by Garriga, Guth and Vilenkin calculating the bubble universe collision frequency can be found here.

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