On Expanding

On Expanding

“The line that fits you exactly at one moment is the one guaranteed to leave you everywhere else. That leaving is not a failure. It is how the bend gets drawn.”

The Reach

Do you remember the first line they ever taught you in school?

Mine began as a joke. Read the first thing anyone learns of lines, y = mx + b, as a confession: output equals sensitivity times input, plus the baseline you were already standing on. Slope as responsiveness, intercept as the Center you start from, the perceiver doing the facing. A toy.

And then the toy stands up and turns out to be a law.

For the slope, write the letter astronomers use. For the input, write distance. The line becomes Hubble’s law: the speed at which a galaxy recedes, proportional to how far away it already is. The humble reach learned at fourteen is the exact local description of an expanding cosmos. Galaxies are not hurled apart through a waiting emptiness; the distance between them lengthens while each one rests where it is. Nothing has to move. The ground holding everything grows.

Run the line far enough and you get recession faster than light: impossible for any traveler, untroubling for the growing itself, because it was never travel.¹ The line does not break out there. At any one moment it is straight all the way out, and the recession past light is the line telling the truth at a distance nobody could cross. What the line cannot do is last. Its slope belongs to its moment. It is what a single instant of Intent looks like when you let it run.

The Bend

The rate of the growing is not a constant. Draw the size of the universe against time (astronomers call it the scale factor), and at every instant there is one straight line that touches the curve there and then parts from it: the reach of that moment. Gather all of those touching lines, one for every instant across Time, and the curve they wrap is the true history of the growing.² No one tangent contains it. The history is recovered only from the accumulation of reaches that each touch once and let go.

This is the standing condition I keep circling back to. Thoughts are tangents. A life is the curve their failures cannot help but draw.

Every Jay is a tangent too: the felt moment a perceiver lives in meeting something, leaning toward coherence with the whole or away from it and weighing as much as it is meant. Each one touches once. A person is a great many of them, and the bend they wrap, over years, is a character.

The Inflection

There is a moment in this history, and it is datable. For its first several billion years the expansion slowed. Matter ran the universe then, and gravity did what gravity does. It gathered matter toward matter and held it: gas into stars, stars into galaxies, galaxies into clusters. The universe was gathering locally, and while it gathered, the bend was concave and the reach kept shortening.

Then, some five or six billion years ago, the matter had spread thin enough across a volume that never stopped growing that the energy of empty space, which does not thin, began to set the pace.³ The expansion began to accelerate. The bend switched sides.

Between the slowing and the opening there is a single point where the curvature passes through zero, where, for an instant, the curve goes straight and tells the exact truth about itself. An inflection sitting in cosmic history like a held breath. The deceleration inflected: it handed itself over smoothly to acceleration without any break at all. Nothing that had gathered came apart. The galaxies kept their stars. Only the pace of the room around them changed.

The one place the reach is briefly, perfectly honest is the place the bend pauses long enough to be told straight.

Unscreenable

Now the through-line. Electric charge can be screened. Gravity cannot; of the long-range forces, it is the one nothing anywhere refuses. The energy of empty space cannot be walled out either. It sits in every cubic meter there is, in your kitchen as much as between the galaxies.

But there is a limit to what it does, and the limit is gravity’s work. Wherever gravity has gathered something and holds it, the expansion does not pull it apart: a planet, a solar system, a galaxy, the little knot of galaxies we live in. They keep their size; only the gaps between them grow.⁴

So there are two motions in the cosmos. Gravity is coherence holding close. Expansion is reach. One keeps what has come together; the other makes room around it.

Read that through a life. A Trace, what a life sends rippling out through Time, has both motions in it. What a person holds close, they keep: the grooves worn by facing the same way again and again, until the facing is character. What they send out reaches, and once it has gone, nobody can wall it off. That cuts both ways. A cruel life reaches too; negative lives leave infamy, where kind ones leave renown. The reach does not care what it carries. Coherence alone was never the aim. A mob coheres. The aim is to cohere toward the good: to be happy, and to help the people near you be happy too.⁵

The Negative Pressure

Here is the jewel, and it is a contemplative one. What drives the acceleration lives in the emptiest vacuum itself, present in space that has been swept of every particle. Its pressure is negative, a tension equal and opposite to its energy, and in Einstein’s gravity, pressure itself has weight. Give gravity enough negative pressure and it stops gathering and starts spreading.⁶ The void is not straining outward like a balloon. It is so thoroughly itself, all the way through, that its own weight comes out as room.

The letter for it is Λ, and the letter has a stranger history than its job. Einstein wrote it into his field equations in 1917 to keep the universe from growing or shrinking, because a static cosmos was what everyone then assumed. Then the universe turned out to be expanding, and the term looked unneeded. In 1934 Lemaître read it another way, as the energy of the vacuum itself, with exactly the negative pressure that goes with it.⁷

My framework borrows the same letter for something else: the floor beneath the reading, what the gathering returns when nobody in the room is coordinated at all. It is small, and almost never nothing. It is no term added at the end. I call it grace, as what remains: no one earns it and no one can spend it down; nobody carries it into the room, and a crowd scattered every which way falls toward it rather than all the way to nothing. I will not tell you the two are one thing. The cosmic Λ is a fixed density of energy; the framework’s floor is a typical value that falls as a crowd grows. What they share is a refusal I find beautiful (neither will let the emptiness be nothing) and a letter, which I took on purpose.

The empty is not empty. Its fullness is the engine.

The contemplatives found their own version of this long before the telescopes. Dzogchen speaks of a ground that is open rather than absent, the space in which everything arises. The Tao Te Ching calls the Way empty, and says that, used, it is never used up.⁸ They got there by sitting still, quieting the mind until the noise thinned. The most thoroughly emptied place in existence turns out to be the one place that cannot stop giving.

From Gathering to Reach

The whole arc, then, runs from gathering to reach. While matter ran things, the universe spent itself holding close, building the galaxies and most of their stars, and slowing the expansion as it did. When the matter had thinned enough, the reach took over the pace, and what had gathered stayed gathered, carried apart into a wider room. The inflection is the moment the gathering made room for the reach.

A life can own the same inflection. There is a long season of gathering (people, habits, the grooves a facing wears in), and from inside, it can feel like slowing down. Then a point comes, smoothly and without announcement, where what you gathered no longer takes all your strength to hold, and the Trace begins to open out. Nothing you held close has to be let go for that to happen.

The Measure

Now the arithmetic, which I have been leaning on without writing it down.

Each perspective enters the equation carrying two things: how strongly it is held, its intensity, and the way it is turned, its facing, which is its Intent. Lay every perspective present over every other and you have the gathering. Divide by the total intensity in the room and you have it in proportion: how sharp the shared world is, a size that tops out at one, and which way it faces. Each of us lives that whole from our own angle to it. Turned the same way, you read as high as the room is sharp: near plus one in a room that agrees. Crosswise, near nothing. Dead against, the same amount below zero. The absolute direction never shows. Turn every facing together and the room turns with them, but no reading changes. A facing counts only by how it sits against the others, which is what I mean when I say phase is Intent, set down as plainly as I can.⁹

Now, the growing. And here the equation says two things at once, and I want both of them on the table.

The first is a cost. Your share of the whole thins as the room fills. The total you are divided by is everyone’s intensity added together, so every arrival makes your part of the average smaller; in a crowd of a million holding their facings about as hard as you, your say in which way the room turns is about a millionth.

But look at the equation’s last line (note 9), the one for what you live. It holds the room’s sharpness and your angle to the room, and nothing else. There is no separate term for how loud you are, since loudness enters only as weight, and none for how big the room has become. A million new perspectives arrive and your hand in shaping the whole shrinks. Your reading of it is still the sharpness of the room times how you are turned.

The second is that the ground gets quieter. A crowd with no coordination still hums along, typically, at a little under one over the square root of how many there are. But look at what that means as the crowd grows. The hum drops. The background thins. An aligned Trace stands out against an ever-stiller ground. Its own strength is unchanged; everything failing to agree has sunk lower, and the floor with it.

Why the ground gets quieter: things that agree add up; things that disagree wander.

Line a thousand arrows up the same way and you have traveled a thousand lengths. Scatter a thousand arrows and you have traveled about thirty: the square root, more or less, because a scattered walk turns back on itself as often as it goes on. So coherent things grow like the count, and incoherent things grow like its root, and the gap between those two rates widens forever.

Divide each by the thousand. The agreeing crowd still reads one. The scattered crowd comes to about three hundredths, typically, and less every time the crowd doubles.

It is tempting to say the cosmos turned for the same reason: that the universe began to accelerate because agreement grows like a count and disagreement like a root. It did not. The universe turned because matter thins as the space it fills grows, falling off with the cube of the stretch, while the energy of the vacuum stays the same in every cubic meter however many cubic meters there are. Sooner or later the one that does not thin must set the pace.¹⁰ That is physics, and it owes nothing to my arithmetic. What I keep is the image, and I mark it as mine: in both, something that does not dilute outlasts something that does, and the crossing was always coming. The only open question was when.

And a life can take the image for what it is worth. You do not have to grow louder than the noise. You have to stay in tune while the noise disperses, which in this arithmetic it does on its own.

The Legible Window

And here is the strangest thing the expansion gives, and it comes with a grief attached. Because the growing accelerates, the most distant galaxies are crossing a cosmological horizon and receding beyond any possible reach of light. Their images thin out of the sky. Run the clock far enough forward and every galaxy outside our own merged neighborhood will have slipped from view. Observers in that deep future will open their eyes onto a small, dark, local sky and find next to no evidence of the hundreds of billions of galaxies, no afterglow of the beginning, no readable record that the cosmos was ever vast or ever expanding at all.¹¹ The universe is erasing the testimony of its own history from view.

Which means legibility is not a permanent property of existence. It is a window, a finite epoch in Time when the traces of the whole can still be read, and we happen to be living inside it, late enough that the light has arrived and early enough that it has not yet fled. To read the cosmos is a privilege with an expiration. It gives a strange new footing to the old conviction that a life that holds together leaves legible Traces, whichever way it faces: legibility itself is local, and on the clock.

But follow the Trace one step past the grief.

What recedes past the horizon does not cease. The galaxies beyond the edge go on, holding their stars together and drifting farther off, and have only become unreadable from here.

Legibility is local; the Trace is not. The light that leaves our sky is released, going on writing in a script no instrument of ours will ever recover. The closing of the window is not the ending of the writing.

So that is what the first line they ever taught me turned out to be for. It is the reach, doing at the scale of everything what it has done since the joke: touching once, telling the truth for an instant, and letting go so the bend can carry on being drawn.

Supplemental note

Parallels and Lineage

Within the On ——ing series, this is the essay that turns to the cosmos. It is the companion of On Gravitating, which reads gravity as the one long-range force nothing can screen and as one-signed accumulation; the two essays agree on a single picture, in which gravity is coherence holding close and the cosmic expansion is reach, and bound systems take no part in the expansion. Each can be read on its own. The functional is taken as seated in On Explaining, where Λ is named the floor beneath the reading.

The cosmology drawn upon is standard: Hubble’s law, exact in proper distance at a given cosmic time; the discovery of the accelerating expansion in 1998–99; the transition from deceleration to acceleration, which follows from matter diluting while the vacuum’s energy density stays constant; dark energy’s negative pressure and the history of Λ from Einstein’s static universe to Lemaître’s vacuum energy; and the far-future horizon described by Krauss and Scherrer. Whether dark energy is exactly constant is an open question. The material is used for its structure, read under the Quantum Lens and never as a physics of mind, and nothing in the essay claims that cosmology follows from the framework’s arithmetic. The comparison between coherent sums growing like a count and scattered ones like its square root is the framework’s own image, set beside the cosmology and not offered as its cause.

The essay carries one ethical step that rides with the functional throughout the series: coherence carries whatever it is given. The reach of a Trace runs in both directions; the aim the series proposes is to cohere toward the good. The contemplative parallels (the Dzogchen ground and the Tao Te Ching’s inexhaustible emptiness) are named as kin in attitude only.

Notes

1. In proper distance, Hubble’s law v = H(t) · D is exact at any one cosmic time, out to any distance. Galaxies beyond the distance at which v reaches the speed of light recede faster than light without violating special relativity, because the recession is the growth of the distance between them, not motion through space. Hubble, “Relation between Distance and Radial Velocity”; Davis and Lineweaver, “Expanding Confusion.”

2. The scale factor a(t) gives the size of the universe relative to today. The tangent to its curve at each instant has slope ȧ, and the family of all those tangents envelops the curve. The Hubble parameter is that slope divided by the size, H = ȧ ⁄ a; the lines of Hubble’s law themselves all pass through the origin and are not the tangents. The letters here are cosmology’s own: a is the scale factor, not the reading aᵢ of note 9, and the Ωs of note 3 are density parameters, not the set of everyone present. Ryden, Introduction to Cosmology.

3. The acceleration was discovered in the light of distant supernovae. Riess et al., “Observational Evidence”; Perlmutter et al., “Measurements of Ω and Λ.” Measurements of the transition redshift range from about 0.4 to 0.7: 0.46 ± 0.13 from supernovae (Riess et al., “Type Ia Supernova Discoveries”) and 0.4 ± 0.1 from the ages of passively evolving galaxies (Moresco et al., “A 6% Measurement”). In the standard flat model, with matter about 0.315 of the present total, the transition falls at z = (2Ω_Λ ⁄ Ω_m)^1/3 − 1 ≈ 0.63, about six billion years ago. Planck Collaboration, “Planck 2018 Results. VI.” The account here begins after the brief early eras of radiation and inflation.

4. Whatever is bound, by gravity or by electromagnetic forces, stays out of the cosmological expansion: the effect on it is too small to matter and does not make it grow. Cooperstock, Faraoni, and Vollick, “Cosmological Expansion on Local Systems.” The same picture is read from gravity’s side in Johnson, On Gravitating.

5. The good here is judged as the series judges it, by the Golden Rule taken both ways: do for others what you would want done for you, and do not do to them what you would not want done to you, each weighed from where the other stands.

6. The acceleration equation is ä ⁄ a = −(4πG ⁄ 3)(ρ + 3p ⁄ c²). For the vacuum, p = −ρc², so ρ + 3p ⁄ c² = −2ρ, which is negative, and the expansion accelerates. Negative pressure is a tension; it is its contribution to gravity, not any outward push, that drives the acceleration. Ryden, Introduction to Cosmology. Whether dark energy is exactly a constant is open: DESI’s baryon-acoustic-oscillation measurements, combined with the cosmic microwave background and supernovae, hint that it may be weakening over time. The preference stands at about three to four standard deviations, depending on which supernova sample is used, short of the conventional five for a discovery. Adame et al., “DESI 2024 VI”; Abdul-Karim et al., “DESI DR2 Results. II.”

7. Einstein added the constant in 1917 as a lowercase λ, his letter and not the series’ λ for the reading, so that the universe could stand still. Einstein, “Kosmologische Betrachtungen”; O’Raifeartaigh et al., “Einstein’s 1917 Static Model.” Lemaître first read it as the energy density of the vacuum, writing that one “must associate a pressure p = −ρc² to the density of energy ρc² of vacuum.” Lemaître, “Evolution of the Expanding Universe.” In the framework, the floor is what a scattered room falls toward: for facings drawn independently, all equally weighted, |L| averages near 0.89 ⁄ √N, a typical value and not a bound, and two equal halves turned exactly against each other can read zero. Rayleigh, “Resultant of a Large Number of Vibrations”; Johnson, On Explaining. The two share a letter by analogy only.

8. Tao Te Ching, chap. 4; wording varies by translation. In Dzogchen the ground (gzhi) is described as primordially pure and open. Buddhist emptiness should not be read as this kind of fullness across the board: in Madhyamaka, all things are empty of inherent nature (svabhāva), and emptiness names that lack, not a substrate. Hayes, “Madhyamaka.” Readings of emptiness as a fullness belong to the later tathāgatagarbha and “other-emptiness” (gzhan stong) traditions.

9. The functional’s five lines, their letters seated as in On Explaining: Ψ(τ) = Σⱼ αⱼ(τ) · e^iφⱼ(τ); Ξ(τ) = Σⱼ αⱼ(τ); L(τ) = Ψ(τ) ⁄ Ξ(τ), with |L| ≤ 1; Θᵢ(τ) = arg L(τ) − φᵢ(τ); aᵢ(τ) = |L(τ)| · cos Θᵢ(τ). A perspective’s weight in the average is αᵢ ⁄ Ξ, which falls as others arrive; multiplying every α by one number leaves L as it was, and says nothing about newcomers. The reading aᵢ has no separate term for intensity or for the count of perspectives; both reach it only through |L| and arg L. Add N unit arrows at random facings and the sum grows as √N in root-mean-square size, with a mean near 0.89√N; line them up and it is N. Rayleigh, “Resultant of a Large Number of Vibrations.” The complex phase is a structural device taken from the form of wave interference; it makes no quantum claim and no prediction, and it measures nothing. Johnson, On Explaining.

10. Matter’s density falls as a⁻³ as the universe grows; the vacuum’s stays constant. The expansion accelerates once the vacuum’s density exceeds half that of matter. Ryden, Introduction to Cosmology.

11. Krauss and Scherrer, “Return of a Static Universe.” In their account, observers a hundred billion years from now and beyond will see only their own merged galaxy, and in time will find the cosmic microwave background redshifted beyond detection. The picture assumes that the vacuum’s energy stays constant; if dark energy is weakening (note 6), the far future is less certain.

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