Appearance
12.8 — The Shape and Fate of Everything
Two questions that were once the same question and are no longer.
"Is the universe finite or infinite?" and "will it expand forever or recollapse?" were, in the textbooks of the 1980s, answered by one number: the density. Dark energy broke the connection.
The geometry of space
Three possibilities, and Chapter 6.7 built the mathematics.
Positive curvature (k > 0). Like the surface of a sphere. Triangle angles sum to more than 180°, circles have circumference less than 2\pi r, parallel lines converge. Finite volume, no boundary — you could travel in a straight line and return to your starting point.
Zero curvature (k = 0). Euclidean. Triangles sum to 180° exactly.
Negative curvature (k < 0). Saddle-shaped. Triangles sum to less than 180°, circles have circumference greater than 2\pi r, parallel lines diverge.
The measurement
Chapter 12.6 explained the method. The sound horizon at recombination is a known physical length — a standard ruler — and the angle it subtends depends on the geometry of the space the light crossed.
In a closed universe, light paths converge and the spot appears larger. In an open universe they diverge and it appears smaller.
The first acoustic peak sits at multipole \ell = 220, which corresponds to about 1° — exactly the flat prediction.
\boxed{\Omega_k = 0.001 \pm 0.002}
Flat to within 0.4 %.
And that is not a proof of exact flatness. A curvature radius of, say, 100 times the observable universe would be indistinguishable from flat. The measurement sets a lower bound on the curvature radius, not a value.
What flatness would mean if exact: infinite spatial extent, in the simplest topology.
Finite or infinite
This is not decided by curvature.
A flat universe is infinite only if its topology is trivial. A flat space can be finite if it wraps around — like the classic video game screen where leaving the right edge brings you back on the left. In three dimensions this is a 3-torus, and it is perfectly flat and finite.
How would you tell? If the universe were small enough to wrap within the observable region, light could have circled it and you would see the same structures in different directions.
The search: look for matching circles in the CMB. If the universe wraps, the sphere of last scattering intersects itself, and pairs of circles on opposite parts of the sky would show identical temperature patterns.
None found. The constraint is that the universe is larger than about 1.2 times the observable diameter — at least 100 billion light years across.
\boxed{\text{The universe is at least as big as what we can see, and whether it is infinite is unknown.}}
And there is a hint pointing the other way. The CMB shows less power on the largest angular scales than the standard model predicts, which could indicate a finite size — or could be a statistical fluke, since there is only one sky and only a few independent large-scale modes to measure. Cosmic variance means this can never be resolved by better instruments.
The observable universe versus the universe
Two things that are constantly conflated.
The observable universe is the region from which light has had time to reach us: 46.5 billion light years in radius (Chapter 12.5), 93 billion across, containing roughly 2\times10^{12} galaxies.
The universe is everything, and its size is unknown.
One inflation-based estimate. If inflation lasted long enough to solve the flatness problem with the observed precision, the whole universe is at least 10^{23} times the observable radius. Some models give far more.
And a useful correction to intuition: every observer everywhere has their own observable universe, centred on themselves. There is nothing special about the centre of ours.
Horizons
Three distinct horizons, and confusing them causes most of the misunderstanding about cosmology.
The particle horizon — the boundary of what we can see. 46.5 billion light years, and growing.
The Hubble sphere — where the recession speed equals c. 14 billion light years.
Objects beyond the Hubble sphere are visible. This surprises people. A photon emitted from beyond it initially moves away from us in proper distance, but the Hubble sphere itself expands, eventually overtakes the photon, and the photon then makes progress inwards. Most of the galaxies we observe were beyond the Hubble sphere when they emitted their light.
The event horizon — the boundary beyond which light emitted today will never reach us.
d_{\text{event}} = c\int_{t_0}^{\infty}\frac{dt}{a(t)}
In a decelerating universe this integral diverges — everything eventually becomes visible.
With dark energy, a grows exponentially and the integral converges:
d_{\text{event}} \approx 16\ \text{billion light years}
\boxed{\text{Anything currently beyond } z \approx 1.6 \text{ is emitting light that will never reach us.}}
And the situation is getting worse.
Galaxies are crossing the event horizon continuously. Once a galaxy's recession exceeds the rate at which light can close the gap, it is gone forever — its image redshifts, dims and freezes, exactly as an object falling into a black hole does (Chapter 12.3).
The observable universe, measured in comoving coordinates, is shrinking.
Cosmology in the far future
A genuinely unsettling consequence.
In about 150 billion years, everything outside the Local Group — which is gravitationally bound and will not expand — will have crossed the event horizon.
An astronomer then would see: a single merged elliptical galaxy (the remnant of the Milky Way, Andromeda and their companions), surrounded by absolute darkness.
No other galaxies. No redshift–distance relation. No cosmic microwave background — by then redshifted to wavelengths longer than the horizon, and undetectable in principle.
\boxed{\text{They would conclude that the universe consists of one static island in an infinite void.}}
They would have no way to discover the expansion, the Big Bang, or dark energy. The evidence would be gone.
Krauss and Scherrer made this point in 2007, and it raises a question that cannot be dismissed: are we currently in a similarly privileged epoch, with some crucial evidence already beyond reach?
We know of at least one case. Inflation erased whatever came before it. And the universe before recombination is opaque to light — only neutrinos and gravitational waves can carry information from earlier, and neither has been detected from that epoch.
The fate
The expansion history is determined by \Omega_m, \Omega_\Lambda and w.
With current measurements — \Omega_m = 0.315, \Omega_\Lambda = 0.685, w = -1 — the universe expands forever, accelerating.
The eras
| Era | Time | What happens |
|---|---|---|
| Stelliferous | 10^{6}–10^{14} y | Stars form and shine |
| Degenerate | 10^{14}–10^{40} y | Only remnants remain |
| Black hole | 10^{40}–10^{100} y | Black holes evaporate |
| Dark | >10^{100} y | Nothing left |
We are 13.8 billion years in — about 10^{10} — so essentially at the very beginning.
The stelliferous era ends
Star formation is already declining, having peaked about 10 billion years ago at roughly 30 times the current rate. The gas is being used up and locked into remnants.
In about 100 billion years, star formation ceases in most galaxies.
The last stars to die will be red dwarfs. Chapter 12.1 gave their lifetimes: a 0.1M_\odot star lives about 10^{13} years.
\boxed{\text{The last star goes out at about } 10^{14}\ \text{years} - \text{ten thousand times the current age of the universe.}}
And red dwarfs are fully convective, so they burn essentially all their hydrogen rather than just the core's 10 %. They do not become red giants; they slowly brighten, turn blue, and then fade.
The degenerate era
Nothing but white dwarfs, neutron stars, black holes, brown dwarfs and cold planets.
White dwarfs cool. With no energy source they radiate their residual heat, and after 10^{15} years they become black dwarfs at a few kelvin. None exists yet — the universe is not old enough for even the oldest white dwarf to have cooled.
Galaxies evaporate. Stellar encounters gradually redistribute energy, and by 10^{19} to 10^{20} years most remnants have been ejected while the rest have spiralled into the central black hole.
And if protons decay (Chapter 8.7) with a lifetime above 10^{34} years, all ordinary matter eventually disintegrates — around 10^{40} years.
If they do not decay, matter survives longer, and quantum tunnelling still does the job eventually: all matter tunnels into iron over about 10^{1500} years, and then into black holes over about 10^{10^{76}} years.
The black hole era
Chapter 12.3 gave the evaporation time:
t \approx 2\times10^{67}\left(\frac{M}{M_\odot}\right)^3\ \text{years}
A stellar black hole: 10^{67} years.
A supermassive one at 10^{9}M_\odot: 10^{94} years.
The largest that will ever form, from galaxy cluster mergers: about 10^{100} years.
Each ends with a burst of gamma rays as the final mass evaporates in a fraction of a second.
The dark era
After 10^{100} years: photons, neutrinos, electrons and positrons, at ever-decreasing density.
Heat death — thermodynamic equilibrium, maximum entropy, no free energy anywhere, no process that can extract work (Chapter 3.5).
And the expansion continues, diluting everything towards nothing.
Dyson's question, from 1979: can any form of information processing continue indefinitely? In an expanding universe with dark energy, the answer appears to be no — the event horizon has a finite temperature, giving an irreducible thermal background, and any computation eventually loses to it.
Alternative fates
All depend on dark energy behaving differently from a cosmological constant.
The Big Rip
If w < -1 — "phantom" dark energy — its density increases with expansion.
The rip time:
t_{\text{rip}} \approx \frac{2}{3|1+w|H_0\sqrt{1-\Omega_m}}
For w = -1.1:
t_{\text{rip}} = \frac{2}{3(0.1)(2.27\times10^{-18})\sqrt{0.685}} = \frac{2}{5.63\times10^{-19}} = 3.55\times10^{18}\ \text{s}
= 113\ \text{billion years}
The sequence in the final months, working backwards from the end:
| Time before rip | What is torn apart |
|---|---|
| 60 My | Galaxy clusters |
| 40 My | The Milky Way |
| 3 months | The solar system |
| 30 minutes | The Earth |
| 10^{-19} s | Atoms |
Because the dark energy density grows without bound, it eventually overwhelms every binding force in turn — gravity, then chemistry, then the nuclear forces.
Current data: w = -1.03 \pm 0.03. Consistent with -1, and phantom values are not excluded. DESI's hint of evolving dark energy (Chapter 12.7) includes phantom-like behaviour in the past.
The Big Crunch
Requires \Omega_m > 1 or dark energy that reverses sign.
Ruled out by current data, though a quintessence field whose potential turns negative could do it.
Vacuum decay
Chapter 8.4 raised this. The measured Higgs and top masses put the electroweak vacuum in a metastable region — stable against small perturbations, able to tunnel to a lower state.
If it happens anywhere, a bubble of true vacuum expands at essentially the speed of light, and inside it the particle masses and couplings are different. Atoms would not exist.
Calculated lifetime: above 10^{100} years, and the calculation is exquisitely sensitive to the top quark mass at the 0.1 % level.
No warning is possible, since the bubble wall travels at c.
Not a practical concern. Worth knowing because it is a real prediction of measured parameters rather than speculation.
The far-future timeline
| Time (years) | Event |
|---|---|
| 10^{9} | Sun's brightening ends complex life on Earth |
| 4.5\times10^{9} | Milky Way–Andromeda merger |
| 5\times10^{9} | Sun becomes a red giant |
| 10^{11} | Star formation ends |
| 1.5\times10^{11} | Local Group is alone in an empty sky |
| 10^{14} | Last star dies |
| 10^{15} | Planets ejected from orbits |
| 10^{19} | Galaxies evaporate |
| 10^{34}–10^{40} | Protons decay, if they do |
| 10^{67} | Stellar black holes evaporate |
| 10^{100} | Largest black holes evaporate |
| 10^{10^{56}} | Random fluctuations may briefly produce structure |
| 10^{10^{10^{56}}} | Poincaré recurrence, if the universe is finite |
The last two entries are speculative and depend on whether the universe's state space is finite. They are included because they appear in the literature and it is worth knowing what they mean and how seriously to take them.
Boltzmann brains
A genuine problem in cosmology, not a curiosity.
In an eternal universe at thermal equilibrium, random fluctuations eventually produce anything, including a self-aware observer complete with false memories — assembled by chance from the vacuum.
The trouble is the counting. Producing a fluctuation is exponentially suppressed in its entropy, so small fluctuations vastly outnumber large ones. A single brain with memories is enormously more likely than an entire universe with a 13.8-billion-year history containing that brain.
\boxed{\text{If Boltzmann brains form, they should outnumber ordinary observers by an unimaginable factor.}}
And then you should expect to be one — which means your memories are false and your observations are random, which undermines the reasoning that got you there.
This is treated as a reductio. A cosmological model that predicts Boltzmann brains dominate is considered to have failed a consistency test.
Proposed escapes: the universe has a finite lifetime; vacuum decay occurs before enough time passes; de Sitter space does not actually produce such fluctuations; or the required physics is not there.
It is not resolved, and it is used as a constraint on cosmological models.
What is honestly known
Established:
- The universe is flat to within 0.4 %.
- It is at least 100 billion light years across.
- It has been expanding for 13.8 billion years.
- The expansion is currently accelerating.
- The observable region contains about 2\times10^{12} galaxies.
Well supported:
- The expansion continues forever with a cosmological constant.
- Star formation ends in about 10^{11} years.
Extrapolation:
- Everything beyond about 10^{14} years, which assumes the physics holds unchanged over timescales 10^{4} times longer than any that has elapsed.
Unknown:
- Whether the universe is finite.
- Whether dark energy is constant.
- Whether protons decay.
- What happens after all the black holes evaporate.
Where this shows up in your life
Nowhere, on any timescale that matters. The Sun has 5 billion years and Earth's habitability perhaps 1 billion.
What it does is put things in scale. The universe is 13.8 billion years old and its stelliferous era lasts 10^{14} years. We are in the first 0.01 % of the era in which stars shine, which is itself a vanishing fraction of the whole.
And there is a genuine point about epistemology. The far-future astronomer sees no evidence of the Big Bang, and would be wrong about cosmology through no fault of their own. That is a reason for humility about what we cannot currently see, and a specific reminder that absence of evidence is sometimes a consequence of when you happened to look.
What the next chapter fixes
Everything in this Part has been established physics or careful extrapolation. The next chapter deliberately crosses into what fiction proposes, and checks each idea against the equations rather than dismissing it. Chapter 12.9 works through wormholes, warp drives, time travel and the multiverse — what the mathematics actually permits, what it forbids, what would be needed to build any of it, and where the line between "difficult" and "impossible" genuinely falls.