Thinking in a cooling universe

Leans on The cost of forgetting, Black holes glow.

Will thinking keep getting cheaper?

Space is not quite empty. It is full of the afterglow of the Big Bang, light that has been travelling since the universe was young and hot. In 2540 that afterglow is cold: a few degrees above absolute zero. But the universe keeps stretching, and stretching cools it. Light that is pulled out to a longer wavelength is light with less energy in it, and a whole sky of such light is a colder sky.

That matters to a mind because of what you read on The cost of forgetting: forgetting a bit has to leave a little heat behind, and the heat is smaller the colder the place you dump it. Deep space is the coldest place that can take heat in bulk. So as the sky cools, the same joule pays for more forgetting.

A hundred billion years after 2540 the sky has fallen below a hundredth of a kelvin, and erasing a bit costs 336 times less than it does in the Vigil's time. This reduction motivates the hypothesis that a patient civilization would wait (Why they might be asleep).

Vesper, the Vigil's black hole, emits Hawking radiation at a temperature of 4.3 billionths of a kelvin (Black holes glow). Its temperature changes negligibly on these timescales. About 351 billion years after 2540, the cooling sky drops below that temperature, making Vesper warmer than its surroundings.

The Vigil's star runs out long before that. Ember, a small orange dwarf, has about 30 billion years of main-sequence time left at First Light in the representative track; Ember's actual age is unmeasured. The track selects a point that best matches the canonical radius and luminosity, then follows the star to turnoff. The Vigil's founding argument is that the hole holds far more usable energy than the star. A star radiates only a small fraction of its total rest-mass energy over its life; accretion onto Vesper at its spin can radiate 0.10 of the incoming gas's rest-mass energy. As a gross energy comparison, feeding the same 0.70 Suns of material to the hole would release 217 times the energy Ember would radiate over that representative remaining main-sequence interval. This comparison does not include the energy or engineering needed to lift the material from the star and deliver it to the hole challenge. After that, Vesper is the warmest engine left. Whether the Vigil ever does it is a choice the story leaves open fiction.

Guess first If space keeps stretching forever, does the sky keep getting colder forever?

No. The stretching itself puts a floor under the temperature. It is tiny, about 2 × 10⁻³⁰ kelvin, but it is not zero, and the sky never gets colder than that.

In 1979 the physicist Freeman Dyson argued that unlimited cooling could allow a mind to perform unlimited computation on a finite energy supply. It would lower its temperature, slow its computation and spend progressively longer periods hibernating while shedding heat. The discovery of accelerating expansion in the late 1990s challenged the cosmology behind that argument. If dark energy is a cosmological constant, the cosmological horizon emits radiation at a fixed temperature frontier. This sets a positive minimum erasure cost, so a finite energy supply permits only a finite number of erasures.

Under this cosmological model, the energy cost of erasure falls for a long time before reaching a fixed minimum. The energy a mind can ever reach is also limited, placing a finite upper bound on the irreversible computation it can perform.

What does a bit cost in each era?

The cosmic afterglow cools in step with the stretching of space. If space is 336 times wider than it was in 2540 (which is how far it will have stretched a hundred billion years later), the sky is 336 times colder:

sky temperature = (sky temperature in 2540) ÷ (how many times wider space has become)

Space has been stretching faster and faster for several billion years, and in the long run it grows like money at a fixed interest rate: by a factor of e (about 2.718) every 17.4 billion years. So in the far future the sky gets colder by that same factor in each interval, and the cooling is quick.

Put the forgetting price, kT ln 2, on top of that, and you get the table. All times are counted from the year 2540.

When	Sky temperature	Cost to forget one bit	Cheaper than in 2540 by
2540	2.73 K	2.6 × 10⁻²³ J	the baseline
100 billion years after 2540	0.0081 K	7.8 × 10⁻²⁶ J	336 times
351 billion years after 2540 (sky meets Vesper's glow)	4.3 × 10⁻⁹ K	4.1 × 10⁻³² J	6.4 × 10⁸ times
1 trillion years after 2540	2.7 × 10⁻²⁵ K	2.6 × 10⁻⁴⁸ J	1.0 × 10²⁵ times
About 1,204 billion years after 2540 (the afterglow fades to the floor)	2.2 × 10⁻³⁰ K	2.1 × 10⁻⁵³ J	1.2 × 10³⁰ times

The last row marks when the diluted afterglow falls to the temperature of cosmological horizon radiation. If dark energy remains constant, the sky approaches 2 × 10⁻³⁰ kelvin and the minimum erasure cost stops falling.

Two comparisons make the floor concrete.

A joule at the floor forgets 4.7 × 10⁵² bits. A joule spent in a room at 300 K forgets 3.5 × 10²⁰ bits. The floor is 1.2 × 10³⁰ times cheaper than the sky of 2540.
Vesper stays warm. Its glow is 1.9 × 10²¹ times hotter than the floor, for as long as it lasts. Left alone, a non-spinning hole of its mass lasts about 3.6 × 10⁷⁰ years. So after the crossover the hole is a hot source and no longer a cold sink, contrary to the idea on The black hole is not your freezer. The temperature here uses Vesper's spin; the lifetime does not.

The sky's temperature as space expands, on log axes. It falls past Vesper's own temperature and then flattens at the floor dark energy sets. Illustrative: the dashed segment is not computed, and the curve is computed only from the first marked point on.

As an energy comparison, suppose every bit represented by Vesper's horizon entropy were erased. The horizon, at Vesper's spin, represents 2 × 10⁷⁹ bits (Black holes glow). At the temperature floor, erasing that much information costs 4.0 × 10²⁶ joules, equivalent to 1.0 seconds of the Sun's output. Even this small positive cost gives a finite number of erasures when the energy supply is finite.

Open everything

Why does the sky cool, and why does Dyson's argument fail?Scaling arguments, a few integrals and the de Sitter temperature

The afterglow cools as one over the width of space

The afterglow is a gas of photons. Space stretches by a factor $a$ (the scale factor, equal to 1 in 2540). Two things happen to the gas:

The number of photons per cubic metre falls as $a^{-3}$ , because the same photons fill $a^3$ times the volume.
Each photon's wavelength stretches by $a$ , so its energy falls as $a^{-1}$ .

Energy per cubic metre therefore falls as $a^{-4}$ . A black-body gas has energy density $u = (4\sigma/c)\,T^4$ , so

$T^4 \propto a^{-4} \quad\Rightarrow\quad T(a) = \frac{T_0}{a}$

with $T_0$ the temperature in 2540, 2.7255 K. The same follows from the spectrum: every frequency is divided by $a$ together, which turns a Planck curve into another Planck curve at temperature $T_0/a$ .

Combine it with Landauer's $kT\ln 2$ per erased bit and the price of forgetting into the sky is

$E_{\text{bit}}(a) = \frac{kT_0 \ln 2}{a}$

How fast space stretches

For a flat universe with matter and a cosmological constant, the scale factor obeys

$a(t) \propto \sinh^{2/3}\!\left(\tfrac{3}{2} H_\Lambda t\right), \qquad H_\Lambda = H_0\sqrt{\Omega_\Lambda}$

where $t$ runs from the Big Bang. The numbers used on this page take $H_0$ = 67.7 km/s/Mpc, $\Omega_\Lambda$ = 0.69, $\Omega_m$ = 0.31, and ignore radiation, which does not matter this late. For small $t$ the sinh is its argument and $a \propto t^{2/3}$ , the ordinary matter-filled universe. For large $t$ , $\sinh x \approx e^{x}/2$ and

$a \propto e^{H_\Lambda t}$

The time $1/H_\Lambda$ is 17.4 billion years: the sky is colder by a factor of e in each such interval.

So $T \propto e^{-H_\Lambda t}$ in the far future. From 2540 to the arrival of the floor, the base-ten logarithm of the growth in $a$ is 30.1.

Dyson's bookkeeping

Dyson's biological scaling hypothesis says that a copy of a creature rebuilt to work at a lower temperature, in a correspondingly colder world, lives the same life with every vital process slowed by the same factor. So a mind's clock is set by its own temperature $\theta$ , which it is free to choose, so long as it stays warmer than the sky it dumps heat into. Its subjective time, counted in moments of thought, is

$u = f\int \theta\,dt$

with $f$ a constant. Each moment processes a fixed amount of information, and erasing it costs $k\theta\ln 2$ per bit, so the power the mind spends is

$P \propto \text{rate} \times \text{cost} \propto \theta^2$

One factor of $\theta$ is the slower clock, the other the cheaper erasure.

Let the mind cool as a power of time, $\theta \propto t^{-\alpha}$ . Running continuously, $u \propto \int^\infty t^{-\alpha}\,dt$ is endless for $\alpha \le 1$ while the energy $\int^\infty t^{-2\alpha}\,dt$ is finite for $\alpha > \tfrac12$ . Waste heat adds another constraint. The most a radiator of fixed size can shed into a cold sky falls as $\theta^3$ in Dyson's estimate, faster than the $\theta^2$ the mind produces. Below some temperature a continuously running mind cannot shed heat fast enough.

His way out is hibernation. The mind is awake for a fraction $g$ of the time and asleep the rest, while its radiator keeps working, and it shortens its waking hours in step with its temperature, $g \propto \theta$ . Then

$u \propto \int^{\infty} g\,\theta\,dt \propto \int^{\infty} t^{-2\alpha}\,dt, \qquad E \propto \int^{\infty} g\,\theta^2\,dt \propto \int^{\infty} t^{-3\alpha}\,dt$

Subjective time is endless when $\alpha \le \tfrac12$ , and the energy is finite when $\alpha > \tfrac13$ . Dyson took $\alpha = 3/8$ , which satisfies both conditions: the subjective-time integral diverges while the energy integral converges.

The only thing asked of the universe is that the sky keep getting colder than the mind, for ever. In the open, ever-slowing universe Dyson assumed, the afterglow falls as $1/t$ , much faster than $t^{-3/8}$ , so that holds.

Why accelerating expansion breaks it

With a cosmological constant the afterglow first falls fast, $T \propto e^{-H_\Lambda t}$ , but the sky as a whole never falls below $T_{dS}$ . A mind must stay warmer than its sky to get rid of heat, so $\theta$ can follow Dyson's schedule only until it reaches the floor, and after that every erasure costs at least $kT_{dS}\ln 2$ . Sleeping does not help: it spreads the thoughts out but does not make any of them cheaper. A finite budget $E$ therefore buys at most

$N_{\max} = \frac{E}{kT_{dS}\ln 2}$

erasures: 4.7 × 10⁵² per joule. The budget is finite because accelerated expansion places resources beyond a distance of about $c/H_\Lambda$ , roughly 17.4 billion light years, permanently out of reach.

Why Vesper's glow stays put and the sky does not

The Hawking temperature depends only on the black hole's mass and spin: $T = \hbar c^3/(8\pi G M k)$ without spin, and Vesper's spin of 0.70 lowers it to 0.83 of that (Black holes glow). Vesper's mass does not change on these timescales, and its spin is taken as fixed. The sky, by contrast, is $T_0/a$ . They meet when $a = T_0/T_{\text{Vesper}}$ : space 6.4 × 10⁸ times wider than in 2540. The growth law above puts that 351 billion years after 2540. After that, the black hole is warmer than its surroundings and radiates away, slowly, into a sky that keeps cooling to the floor.

Where the floor comes from

Gibbons and Hawking showed in 1977 that a universe with a cosmological constant has a horizon around every observer, at distance $r_H = c/H_\Lambda$ , and that it is hot in the same sense that a black hole's horizon is hot. For Schwarzschild the temperature is $T = \hbar\kappa/(2\pi k c)$ , with $\kappa$ the surface gravity at the horizon, $\kappa = c^4/(4GM)$ . That gives the Hawking temperature $\hbar c^3/(8\pi G M k)$ . For de Sitter space, the static patch has $f(r) = 1 - H_\Lambda^2 r^2/c^2$ , and the same recipe, $\kappa = \tfrac{c^2}{2}|f'(r_H)| = cH_\Lambda$ , gives

$T_{dS} = \frac{\hbar H_\Lambda}{2\pi k}$

which is 2.22 × 10⁻³⁰ K for the values above.

The interpretation is that a detector at rest in such a universe clicks as if it sat in a bath at $T_{dS}$ . This is the floor, and nothing a mind does can lower it. The two glows add, so there is no sharp moment when one takes over; the date in the table is when the diluted afterglow has fallen to $T_{dS}$ . After that the cosmic horizon's own radiation is what remains.

Is this settled, and can anything get round the floor?The papers, the loopholes and what is still argued

The papers

Dyson's "Time without end" (1979) is where the question begins: he asked whether life, taken as a physical process, could last forever in an open universe, and argued that it could, by cooling, slowing and hibernating. Adams and Laughlin (1997) then laid out the whole long timeline of an ageing universe, from the last star formation through to the evaporation of black holes. In their picture, ordinary star formation runs out of gas, and the smallest ordinary stars burn out, at about 1 × 10¹⁴ years after the Big Bang, and the black holes outlast everything else.

Dyson's paper assumed that the expansion keeps slowing, or at least does not accelerate. The discovery in the late 1990s that it accelerates changed the picture. In 2000 Lawrence Krauss and Glenn Starkman worked through what a cosmological constant does. They found that a universe can expand for ever and still give any civilization only a finite amount of conscious life, however long it lasts. They went further: setting aside exotic strong-gravity effects, they argued that the information any civilization can ever gather is finite even without a cosmological constant, so if minds are physical computers, none of them is eternal.

Gibbons and Hawking (1977) supply the temperature. Their paper treats the cosmological horizon the way Hawking had treated a black hole's, and finds particle creation and a thermodynamics for it.

Loopholes and their costs

Don't erase. A mind that never forgets pays no Landauer bill at all, whatever the sky. But it must keep every record, and memory is finite too. This trade is Computing without forgetting.
Dump the heat somewhere else. The Landauer bill depends on where the heat goes, and the sky is not the only possible place. Whether a patient civilization really gains the full factor of cooling by waiting is argued on Why they might be asleep.
A warmer neighbourhood. The table counts only the afterglow. Adams and Laughlin show that several trillion years on, other light (dying stars, dark matter annihilating, black holes glowing) outshines it. Local sources like these can only make a mind's surroundings warmer than the table, never colder, so the floor stays a floor.
Exotic gravity. Krauss and Starkman set aside strong gravitational effects that nobody has a tested theory of. If such effects exist, the finite bound could be dodged. frontier

What is still argued

The finite bound rests on dark energy being a true cosmological constant, the same value for all time. In 2025 the DESI collaboration measured the expansion history from millions of galaxies and quasars. On their own the data fit a constant dark energy well. Combined with the cosmic microwave background, they prefer a dark energy that changes with time at 3.1 sigma, and adding supernova surveys gives 2.8 to 4.2 sigma, depending on which survey. That is short of the five sigma physicists ask before calling something a discovery. In the favoured fit, dark energy has been weakening for the last few billion years. If it fades away for good, the floor goes with it and Dyson's question opens again; if it grows, the horizon closes in and the bound tightens. As of 2026 the ultimate fate of the universe is open. frontier

Applying this reasoning to minds requires further assumptions: a mind needs a physical computer, the computer can approach the sky's temperature, and its clock can scale with temperature as Dyson proposed. How well a working mind could satisfy these conditions remains uncertain.