The number of galaxies is always for the observable universe - about 92 billion light years in diameter if I remember correctly - not the entire universe. From the observed flatness of the observable universe we know that the entire universe is at least about 100 times larger in volume assuming it is isotropic.

What defines the horizon?

Is the horizon spherical?

It is spherical and it is the distance light could travel in the 13.7 billion years since the big bang, or more precisely since when the universe got cold enough to allow electrons and protons to form atoms which in turn made the universe transparent to light and which occurred about 400,000 years after the big bang. Note that the radius is about 46 billion light years and not only 13.7 billion light years because the universe continued to expand while the light was on its way to us. This also changed the frequency of the light and so the light emitted at the edge of the observable universe at the time of recombination is the cosmic microwave background we receive today.

> What defines the horizon?

The speed of light. Things beyond the horizon are things where the light hasn't had time to reach us yet.

> Is the horizon spherical?

Given that the speed of light is isotropic, yes.

>> What defines the horizon?

> The speed of light. Things beyond the horizon are things where the light hasn't had time to reach us yet.

No. Things within the horizon are things where the light hasn't had time to reach us yet.

Things beyond the horizon are things where no amount of time will ever enable the light to reach us. That's what makes it "unobservable".

That seems wrong to me but maybe I am just misunderstanding you. The light from the particle horizon is by definition the light arriving here now. Whether light emitted on either side of the particle horizon will reach us depends on whether the emission happens within the cosmological event horizon which only coincide with the particle horizon in a non-accelerating universe and changes over time in an [de]accelerating universe.

If we move away from an observer in our accelerating universe we first pass objects that can still emit light that can reach the observer, then we pass the cosmological event horizon and enter a region where objects were once able to emit light reaching the observer but which are now moving away to fast for they light emitted now to ever reach the observer and finally we pass the particle horizon and enter the unobservable universe where objects were never able to emit light able to reach us.

This may sound contradictory but one mus keep in mind that objects between the cosmological event horizon and the particle horizon emitted light in the past that is still on its way to us so that we can still observe those objects for some time but they can just no longer emit light that will reach us.

Also note that I avoided talking about the inside and outside of horizons because I am not sure what side is usually called the inside respectively the outside. From the point of view of an observer he is inside the horizon, on the other hand in analogy with a black hole where the horizon prevents light reaching an observer outside the horizon we would have to call the observer outside of the horizon. I think the former point of view is the common one but I am absolutely not sure about that.

EDIT: Replaced Hubble horizon with cosmological event horizon as pointed out by cygx.

Physically, the Hubble 'horizon' is mostly meaningless. What you have in mind is the cosmological event horizon.

Both can be defined in terms of certain velocities reaching c: The hubble sphere is located where the change in proper distance at constant cosmological time reaches the speed of light (aka recession velocities), the cosmological event horizon is located where relative velocities as evaluated by parallel transport along the light path does so.

Thanks, updated my comment accordingly.

I stand corrected.

If it is infinite then yes. We don't know that the universe is infinite.

The visible universe is not infinite, and it is both shrinking and growing at the same time.

How is it shrinking?

The space between various parts (if there is more than we can see) is growing faster than the speed of light, eventually it will get to the point where even nearby galaxies disappear.

Nearby galaxies will only disappear for practical purposes: Their light emission will continue to reach us, it just gets increasingly redshifted. In fact, if we were able to detect them, we would continue to discover more and more galaxies, but the total number will asymptotically approach a fixed value (ignoring the change in numbers due to galactic evolution).

No, I think flukus is right. The space itself between galaxies is expanding (and the expansion is accelerating), which means, so eventually even our closest neighbour Andromeda will disappear behind our observable horizon and be eternally inaccessible, in all regions of the electromagnetic spectrum. It's counter-intuitive, but a good way to visualize is by imagining miniature galaxies printed on an expanding balloon (with the expansion speeding up), and then imagining what happens to a photon leaving galaxy A and travelling "towards" galaxy B - you'll notice that if the balloon is expanding rapidly enough, the photon ends up moving away relative to both A and B.

Instead of an expanding balloon, just use a flat piece of paper as your model of space-time (after dimensional reduction):

Due to conformal flatness of the FLRW metric, it's possible to represent the Hubble flow as vertical lines, the spatial surfaces at constant cosmological time as horizontal lines and light rays as straight lines angled at 45°. The geodesics for massive particles not following the Hubble flow however would be more complicated, but are of no particular interest.

The big bang happens at the lower edge of the paper, the upper edge represents t=inf. The observable universe is bounded by two light rays that meet at infinity. In our toy model with 1 spatial dimension, if an observer follows the Hubble flow along a vertical line, the number of observable galaxies will be proportional to the width of the intersection of the cone formed by light rays and the horizontal line representing space at the time when galaxies first formed.

As ASCII, it looks like this:

    conformal time
          ^
          |           "us"
    t=inf +------------x------------+
          |           /|\           |
          |          / | \ <------------ cosmic event horizon
          |         /  |  \         | (bounds observable universe)
          |        /   |   \        |
    t=now +-------/----x----\-------+
          |      /    /|\    \      |
          |     /    / | \ <------------ light emitted by farthest
          |    /    /  |  \    \    |   visible galaxy as it formed
          |   /    /   |   \    \   | 
          +--x----x----|----x----x--+ <---- time of galaxy formation
          | /:    :    |    :    :\ |
          |/ :    :    |    :    : \|
      t=0 +--:----:----+----:----:--+---> comoving distance 
             :    :         :    :
             :    |<------->| <--- currently visible galaxies
             :                   :
             |<----------------->| <--- asymptotically visible galaxies

Galaxies that are visible right now will continue to be visible forever in theory, but may become undetectable in practice due to increasing redshift. They will also not be seen to cross any 'edge of the universe', but instead appear to become fainter, redder and frozen in time.

As time goes by, we will continue to observe new galaxies being formed, which increases the total number of theoretically visible galaxies. In practice, redshift may again prevent us from doing so in practice.

Sure, that makes sense. I hadn't thought about time dilation effects near the horizon. It's also kind of fascinating. Thanks for correcting me!

I thought the generally accepted position was that the universe is finite but expanding?

The observable universe is finite as according to our current models, light emitted beyond a certain distance will never reach us. The global size beyond the oberservational limit is harder to tackle. You're right that even though the observable universe is finite, it's volume is potentially increasing forever.