Cosmology

This piece is especially for my younger self, whose curiosity about moons, stars, and space-time mechanics never faded.

An interesting question I came across : Assuming that the Universe continues to expand forever, without slowing down, how would we see the apparent size of distant galaxies evolve in the long run?

The surprising long‑term fate of distant galaxies in an eternally expanding universe

At first glance, it feels intuitive to say: “If the Universe keeps expanding forever, distant galaxies should just keep shrinking in the sky until they become invisible points.” That’s a very natural guess, but in a dark‑energy–dominated universe (like ours appears to be), the correct answer is more subtle and much more interesting: in the far future, the apparent size of a very distant galaxy approaches a finite, non‑zero value instead of shrinking to zero.

In other words, the galaxy’s image “freezes” at a certain angular size, very much like an object falling toward a black hole horizon appears to slow down and freeze from the point of view of a distant observer.

Why this happens: intuition first

To understand the result without getting lost in equations, it helps to separate two ideas:

1. The galaxy’s physical distance is growing.
2. Because of cosmic expansion, the proper distance between us and a distant galaxy increases over time.
3. Naively, you’d expect “farther away” to mean “smaller on the sky.”
4. But the light we see now was emitted long ago.
5. When we look at a distant galaxy, we’re not seeing it “now”; we’re seeing it as it was when it emitted the light we’re currently receiving.
6. In a universe dominated by dark energy, beyond a certain distance there is a cosmic event horizon: light emitted after a certain time can never reach us.
7. That means that as time goes on, the new light we receive from very distant galaxies was emitted earlier and earlier relative to our “now” in cosmic time, and never later than a certain limiting emission time.

The key effect is this: as the universe accelerates, there comes a point where we stop seeing “newer” images of a distant galaxy. We keep receiving photons, but they were all emitted within a finite window of the galaxy’s history. The apparent size of the galaxy then tends to the angular size corresponding to that limiting emission time. So the image doesn’t shrink to zero; it asymptotically approaches a fixed, non‑zero angular size.

What the math is doing in the de Sitter example

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The reasoning you quoted uses a very simple model of an accelerating universe: de Sitter spacetime in 1+1 dimensions (one time coordinate t and one space coordinate x). In that model, dominated purely by dark energy, the metric can be written as
ds2=dt2−e2tdx2ds2=dt2−e2tdx2.​

• Here, t is a time coordinate (scaled so that the Hubble parameter is effectively set to 1), and x is a comoving spatial coordinate.
• A galaxy sitting at fixed comoving coordinate x = D keeps that coordinate forever; the expansion is encoded in the factor e2te2t in front of dx2dx2.
• Light follows paths with ds2=0ds2=0, so for a light ray you get ∣dx∣=e−tdt∣dx∣=e−tdt.​
• Integrating that equation between the emission time t0t0 and the reception time t1t1 for a galaxy at comoving distance D gives a relation of the form
  D=e−t0−e−t1D=e−t0−e−t1.
  Solving for t0t0 yields:
  t0=−log⁡(e−t1+D)t0=−log(e−t1+D).

This tells you: “The light we receive at observer time t1t1 was emitted by the galaxy at time t0t0.”

Now the crucial step is to look at what happens as t1→∞t1→∞:

• As t1t1 gets very large, e−t1→0e−t1→0.
• So the expression simplifies to t0→−log⁡(D)t0→−log(D), which is a constant.

This means:

• As we go to later and later observer times, the emission time of the photons we receive from that galaxy approaches a fixed finite value t0∗=−log⁡(D)t0∗=−log(D).
• We never see light emitted after that limiting time, because beyond it, photons can no longer “catch up” with us due to the accelerated expansion.
• Therefore, the galaxy’s image on our sky corresponds to a bounded, finite epoch of its past. Our view of it stops updating after a while and effectively “freezes.”​
• Since the galaxy had a non‑zero physical size at that limiting emission time and the corresponding angular diameter distance does not diverge to infinity, the apparent angular size tends to a non‑zero constant. The galaxy never shrinks to an infinitesimal dot.

Analogy with a black hole horizon

This is closely analogous to the classic black hole picture:

• For a distant observer, an object falling toward the event horizon seems to slow down, redden, and freeze just above the horizon.
• The reason is also geometric: the light emitted later and closer to the horizon has an increasingly hard time escaping to the observer. In the limit, light emitted “at” the horizon never reaches them.
• In an accelerating universe with a cosmological horizon, ultra‑distant galaxies play a similar role:
• Their later light emissions cannot reach us once the expansion has accelerated enough.
• We keep receiving photons that were emitted earlier, up to a limiting emission time, and then our picture of that galaxy stops evolving