# I Does the number of causal patches increase bc of expansion

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1. Apr 28, 2017

### durant35

I'm having a hard time understanding some fundamental properties of our expanding universe.

It seems reasonable that globally the size of the universe increases bc of the expansion. However the size of each causal patch/observable universe stays the same. Following this logic it would seem that the number of causal patches/causally connected regions increases with time - but that seems very debatable since causal patches refer to fixed points relative to which matter content moves away from. So what is true? Does the number of patches increase or stays constant in spite of the increase of the volume of space?

Thank you.

2. Apr 28, 2017

### Bandersnatch

A few questions to your question:
1. What do you mean by causal patches, exactly? Future lightcones? Past lightcones? Something else? I'm asking because...
2. ...I don't understand in what way can a 'causal patch/observable universe stay the same'. Can you explain what you mean here?
3. Can you read lighcone graphs such as these?

3. Apr 28, 2017

### durant35

1) As you noticed, I am a bit lost in the terminology so let me try to explain. By a causal patch I mean a region inside an event horizon defined by the accelerated expansion, stuff which passes it can never contact us again.

Unfortunately, I don't know if that's a synonym for the observable universe, but I hope you now have better details.

2) I was assuming that the region I mentioned stays constant in size due to the expansion of space. I don't know if that's correct.

3) I don't know how to read the graphs that you attached.

:/

4. Apr 28, 2017

### Bandersnatch

Alright. The extent of the cosmic event horizon is not constant in time, although it does asymptotically approach a constant size. It is also not the same as the observable universe.
But I think I do get you question now. Give me until tomorrow to write a tutorial on reading those graphs, as I believe they'll make everything clear in a much more satisfactory fashion than just words could ever do.

5. May 2, 2017

### Bandersnatch

Sorry for the delay, I was preoccupied with family matters.
This post has turned into a venerable wall of text, so if you want just a short answer after all, there's a tl;dr; down the bottom.

I don't know what concepts you're familiar with, so I'll describe everything from the ground up. Skip those parts you'll find obvious.

The graphs we'll be discussing are from the following paper on cosmological misconceptions by Tamara Davis & Richard Lineweaver:
https://arxiv.org/abs/astro-ph/0310808
The animated versions are taken from this site:
http://yukterez.net/
Finally, you can use Jorrie's calculator to plot similar graphs yourself (just not mirrored):
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7-2017-02-08/LightCone_Ho7.html
The one picture with sideways light cones is from Ned Wright's cosmology tutorial website, I believe.

These space-time graphs represent standard LCDM cosmology and those from the paper use somewhat dated parameters (little has changed since then, though).

With that out of the way, let's start with what a light cone graph is.

Light cone graphs track null worldlines through space-time. Null (light-like) worldline is the path a beam of light would take between emission and reception events.
A light cone in a non-expanding space with two spatial dimensions suppressed would look like this:

(Usually there's also a future light cone - mirrored upside-down around the here and now, and representing events that will be in contact with the observer in the future (rather than the other way around). But the graphs below don't include these, so I left it out.)

To describe an event you need to name where and when it happened. All events on the vertical line with 0 proper distance are 'here' events - they have happened, are happening, or will happen where the observer is. All events on the horizontal 'now' line are 'now' events - they are happening right now everywhere. Their intersection is here and now.

The observer is currently seeing/being in causal contact with all events which happened at the light cone line. All events within the light cone were observed in the past.

This lets us define the observable universe as the size of the base of the light cone - these are the farthest and oldest events that we can currently see.

As time advances, the apex of the cone moves up, and its base covers more space.

In the case of non-expanding space this means that even though at any given time there is an infinite number of observers not in causal contact with each other, all they need to do is wait enough time to observe each other's events, as the following graph illustrates:

Each observer starts in their own causal patch, which then starts encompassing other observers. With infinite time to spare, each observer's observable universe can cover infinite spatial extent, and galaxies can be observed forever, throughout their histories.

But what happens to the observable universe in expanding space?

If we try and transplant light cones into an expanding space, we get a distorted teardrop-like shape:

(here, future light cones are drawn instead of past ones - but it doesn't matter)
Each line represents a separate observer (e.g. a galaxy) moving with the Hubble flow. Notice that at each point the path of light is locally parallel to light cones of observers it passes by. The less prominent the expansion (i.e. closer to any given observer), the less distorted the light cone is.

This shape can be seen on the first graph of the three from the paper, shown below. This one uses the most immediately-intuitive coordinates.
- Proper distance is the distance we think of in everyday life - what you'd get if you were to freeze the expansion and take a really long ruler to measure how far things are.
- Scalefactor $a$ represents how many times smaller/larger the universe was/will be than now. By size of the universe we mean any large distance whatsoever that we may want to measure and see how it changes. Scalefactor is there just to help us out - it's the time on the other side that matters.

Other information shown here includes:
- Hubble sphere - this is where recession velocity equals c
- Particle horizon - the distance to which emitters of the oldest observable light have receded to in the time it took it to reach the observer
- Event horizon - appearing due to presence of the cosmological constant; events outside the horizon will never be in causal contact with the observer. This graph in particular shows that the event horizon grows with time.
- The dotted lines with numerical labels (1, 3, 10 and 1000) represent selected points in space moving with the Hubble flow. It's best to think of these as other observers, or simply as some galaxies, whose flaring out paths represent moving away due to expansion. The numbers indicate redshift.

For an event to currently be in causal contact with (i.e. be seen by) the observer, it must lie on the light cone.
All events that happened inside the light cone were in causal contact with the observer in the past.
All events outside the light cone and within the event horizon will be in causal contact some time in the future - the closer it is to the event horizon, the later it will be observed.

The presence of the event horizon indicates that no amount of waiting can make the observer at the 0 line see all the events. For each galaxy (dotted line) there will come a time when it crosses the event horizon and no light emitted from that point onward will ever make it to the observer.

This will be better visible if we change the spatial coordinate system to comoving distances - it means that we stretch (or contract) any proper distance as many times as the universe has contracted/expanded by. For present time this doesn't change the distance value, as the scalefactor is defined as equal to 1 at present. As a result of this coordinate change, the bottom of the earlier light cone is splayed out, allowing us to see what's going on in more detail.
Notice that all galaxies (dotted lines) are now straight and parallel lines. This is because we've exactly factored out the expansion.

Once again we can see that as time advances, galaxies are leaving the event horizon, preventing any future contact with the observer.

The evolution of the light cone can be seen in the following animations:
http://yukterez.net/lcdm/lcdm-flrw-animation.gif

All these graphs have the annoying disadvantage of not continuing to infinite future (so we can't see what happens if we wait a really long time).

We can fix that by rescaling the time axis, so that it is progressively more compressed the higher it goes. The previously discussed graph will become distorted like so:

(The meaning and conversion of conformal time to proper time is not straightforward, so rather than fail to explain, I've included some marks to help eyeball the meaning)
Proper scaling recovers the shape of light cones from when we discussed static spaces.
This convenient feature makes it very easy to find answers to question such as what and when is observable? Just draw straight lines parallel to existing light cones.

It is also easy to judge what happens to causal patches:

So this figure above is the actual answer to the question asked.
As can be seen, each observer's observable universe always increases, but is limited in maximum size to the extent of the event horizon.
Two sufficiently nearby observers become causally connected and then separate.
Even though they separate, both observers will still keep seeing each other's signals, but in increasingly outdated state. I.e. if they ever enter each other's causal patch, they always remain in causal contact with some past version of each other. The key difference from the static case is that each faraway galaxy can be seen only for a small fraction of its history (that limited history is stretched in time until infinite future).
In the far future, all comoving observers end up within their own isolated patch, with only ghosts of other galaxies to look at.

I hope this clarifies more than it obfuscates

Last edited by a moderator: May 14, 2017