Mass That ClicksEpisode 6 ／ Once again, the c·t lens legitimately works

Is there a "lower wall" to mass ── the opposite direction from last time, closing in from the size of the universe

The Smallest Mass, IR Side:
The Floor Set by the Size of the Universe Last time, the interior of particles (UV) set the neutrino's lightness. This time, the reverse ── the size of the universe (IR).
The lower wall of mass is deeper than "cannot be measured": it was the physical boundary of frozen vs. oscillating. Here $R_h=ct$ legitimately works.

Tools you'll need: the floor from Episode 1, division, $R_h=ct$ and sampling from Cosmology That Clicks Hook equation: $m_{\min}\sim \hbar/(c^2 t)$

In Episode 5, the internal structure of particles (UV) set the neutrino's outlandish lightness. This time, from the completely opposite direction ── the size of the universe (IR) ── we close in on mass. The question is simple: is there a "lower wall" to mass? The answer is yes. And that wall is at once a "limit of measurability" and a physical boundary of whether a field freezes or oscillates. Here, our sister series' $R_h=ct$ enters, this time legitimately, right inside the equation.

01Is there a lower wall to mass? ── three entrances that all reach the same place

Estimate the lower bound on mass in three independent ways, and all of them land in the same place.

Compton wavelength = size of the universe: when the wavelength $\hbar/(mc)$ exceeds the cosmic horizon $R_H$, a particle's quantum spread is larger than the universe.
Energy–time uncertainty: the longest usable time is the age of the universe $t$. The smallest measurable energy is $\hbar/t$.
One oscillation: if a mass's oscillation period exceeds the age of the universe, it can never complete even one oscillation.

The universe's IR floor (here R_h=ct enters)

$$m_{\min}\sim\frac{\hbar}{c\,R_H}=\frac{\hbar}{c\,(c\,t)}=\frac{\hbar H_0}{c^2}\approx 1.4\times10^{-33}\ \text{eV}$$

In Episode 4, $c\cdot t$ stood in the background as "the arena where it works (scale invariance)." This time it legitimately appears right inside the equation, as $R_H=c\,t$ ── because the size of the universe $R_H$ is the distance light travels over the age of the universe $t$, that is, exactly $c\cdot t$.

02The hidden card ── this is deeper than "cannot be measured." It freezes.

Inside the expanding universe, a field of mass $m$ behaves like a pendulum with friction.

Equation of motion for a field in the expanding universe

$$\ddot\phi + \underbrace{3H\dot\phi}_{\text{Hubble friction}} + \Big(\frac{mc^2}{\hbar}\Big)^2\phi = 0$$

The middle term $3H\dot\phi$ is Hubble friction (the viscosity of cosmic expansion). Here the behavior splits into two.

The boundary of frozen vs. oscillating

$mc^2/\hbar > H$ (heavy) → it oscillates. An ordinary massive particle.
$mc^2/\hbar < H$ (light) → overdamped. Frozen, unable to move. Dark-energy-like.
The boundary: $m=\hbar H/c^2$ ── exactly the IR floor.

So this wall is not only a "limit of measurement," it is a line of physical phase transition. A field lighter than this can never complete a single oscillation before the universe changes through expansion; pinned down by Hubble friction, it freezes. Episode 1's "cannot oscillate even once" is rigorously backed up by an equation of motion with a viscous term.

Figure: the time evolution of a field in the expanding universe. Use the slider to change the mass $m$ (as a ratio, with the IR floor $\hbar H/c^2$ set to 1). Heavier than the floor and it oscillates; lighter and it freezes under Hubble friction.

Mass m ／ IR floor (ratio, with $\hbar H/c^2$ set to 1)2.50

Move m and it switches between frozen and oscillating.

oscillates (heavy = matter-like) freezes (light = dark-energy-like)

This is not speculation but a working tool ── during inflation, "light fields freeze and fluctuate"; axions (which begin oscillating and become dark matter the instant $H$ drops below $m$); quintessence (which keeps \(m

03The two meanings of R_h=ct

Put in $R_H=c\,t$, and the floor is $m_{\min}\sim\hbar/(c^2 t)$. This equation has two meanings.

(a) Measurement window (the low-frequency end of sampling) The longest observable time $t$ sets the lowest measurable frequency = the smallest mass. Mirroring how, in Cosmology That Clicks, Episode 5 "Sampling," there was a wall of $fa finite observation time \(t$ creates a wall of $f>1/t$ on the low-frequency (= low-mass) side. It is the cleanest implementation of "because the universe is finite, the bandwidth is finite."

(b) Dynamics (it actually freezes) At time $t$, a field lighter than $H\sim1/t$ is still frozen. As $t$ grows and $H$ drops, a field of a certain mass thaws and begins to oscillate (this is how axions start moving). It is not a matter of measurement; the field actually behaves that way.

Both are $\sim10^{-33}$ eV now, and both drop as $1/t$. The earlier the universe, the higher the floor of smallest mass was ── because $t$ was smaller. Following Episode 4's "arena where $c\cdot t$ works," this time $c\cdot t$ sets even the floor's change over time.

04What lives in the basement

A basement stretching another 30 orders of magnitude below the neutrino ($\sim0.05$ eV). Here are its residents, real and proposed.

~2×10⁻³ eVthe dark-energy scale ($(\rho_\Lambda)^{1/4}$) ── the star of next time

~10⁻²² eVfuzzy dark matter (proposed): its de Broglie wavelength is galaxy-sized. The lightest particle seriously proposed.

< 10⁻²⁷ eVupper bound on the photon mass (galactic magnetic fields) ── only up to here can it be distinguished from zero

~10⁻³²⁻³³ eVupper bound on the graviton mass (cosmology) ── if its Compton wavelength = the horizon, it exactly touches the floor

~1.4×10⁻³³ eVthe IR floor $\hbar H_0/c^2$ ── the bottom of the basement. This is the outer limit of the "smallest mass humanity can know."

What's striking is that even the photon cannot be distinguished from zero for masses lighter than $10^{-27}$ eV. The range $10^{-27}$–$10^{-33}$ eV is a "band unknowable even to the photon," and the IR floor is its outer edge. It is symbolic, too, that the cosmological upper bound on the graviton just touches the floor.

◇ ◇ ◇

05Reaching the same place from the opposite direction to Episode 5

Line up Episode 5 (UV) and this one (IR), and the same words appear twice, in opposite directions.

The smallest mass always points to a "largest"

Episode 5, UV side: $m_\nu\sim v^2/M$ ── smallest mass ↔ largest energy (grand unification $10^{14}$ GeV).
Episode 6, IR side: $m_{\min}\sim\hbar/(c^2 t)$ ── smallest mass ↔ largest length (the cosmic horizon $c\,t$).

One points to the ceiling of high energy (UV), the other to the edge of the universe (IR). The two "smallest masses" seem to come from utterly different physics ── yet next time, the two shake hands at a single point: meV. In fact meV is the geometric mean of the IR floor and the Planck mass, $\sqrt{m_{\rm IR}\cdot M_{\rm Pl}}\approx$ meV. Exactly halfway between UV and IR. That mystery is the star of the finale.

06Low masses don't disappear ── the drop in resolution is about the "high-energy end"

Here let's answer a common question ── "If $c\cdot t$ lowers the speed of light and the universe's 'resolution' drops, won't low-energy masses become harder and harder to see?" It's intuitive, but actually the reverse is true. The trap lies in tying low energy to "fine things."

Low energy ≠ fine

$$\text{low energy}=\text{low frequency}=\text{long wavelength}=\textbf{big and sluggish}$$

A mass's Compton wavelength is $\lambda=\hbar/(mc)$. The lighter it is, the longer and "bigger" the wavelength. What a coarse grid struggles with is short wavelength = high energy = heavy things; long wavelength = low energy = light things are, if anything, the side a coarse world is good at.

The physics gives the same conclusion. To measure a mass and distinguish it from zero, its rest energy $mc^2$ must oscillate at least once over the age of the universe $t$ ── so the smallest measurable rest energy is $\sim\hbar/t=\hbar H$. Since $t$ increases, this drops as $1/t$. As time passes, ever-lighter masses become visible. Section 03's "the floor drops as $1/t$" is exactly this ── expansion is an ally of low mass.

Why it feels like things "become invisible" ── the trap of units In the $c\cdot t$ gauge, substituting $c\propto1/t$ into $m_{\min}=\hbar/(c^2 t)$ makes it look like it increases as $\hbar t$, producing the illusion that "light masses are pushed out." But this is the trap of reading a dimensionful "mass" in a gauge where $c$ varies. What decides measurability is the dimensionless $mc^2 t/\hbar$ (frequency × time), which is gauge-independent and returns "the floor drops (lighter is easier to see)." Just as the watchword says ── $c\cdot t$ is a rephrasing of coordinates; read the physics in dimensionless terms.

What truly becomes harder to see over time is the opposite high-energy (heavy) end ── the universe cools, the surrounding energy drops, and heavy particles are no longer produced. The reality behind "resolution drops" is this retreat of the high-energy end. And on the light side, as we'll see in Bonus ②, the rest mass is invariant (it does not redshift), so low masses do not fade away and vanish.

An honest line

The IR floor is an operational (unmeasurable) and dynamical (frozen) floor, not "mass is quantized in steps of $10^{-33}$ eV," nor "there is one particle sitting there." It is a horizon of the knowable / the behaviorable. It is a different thing from the once-discussed "smallest bit of a finite computer = mass" ── that diagnosis is in Bonus ①.

The scale $\sim10^{-33}$ eV is robust, but the coefficient (the $2\pi$, the choice of horizon) is ambiguous by order unity. This floor is also standard physics and does not require $c\cdot t=\text{constant}$ ($R_H\sim ct$ holds to order of magnitude for any expansion). This lens's contribution is not "creating the floor" but giving the natural $1/t$ reading $\hbar/(c^2 t)$. We won't overclaim. And the once-stated "the gap gets heavier with $t$" is backwards; the real floor drops with $t$.

Practice problems (solvable with today's material)

State the condition for a field of mass $m$ to "freeze," in the language of the equation of motion.
See the answer
$mc^2/\hbar < H$ ($m<\hbar H/c^2$). Hubble friction $3H\dot\phi$ beats the restoring force, becoming overdamped; the field can never complete a single oscillation before the universe changes, and it freezes. It behaves like dark energy.
In one phrase, why does the IR floor drop as $1/t$?
See the answer
Because $m_{\min}\sim\hbar/(c\,R_H)$, and with the horizon $R_H=c\,t$, we get $m_{\min}\sim\hbar/(c^2 t)$. The longer the age of the universe $t$, the lower the floor ── the earlier the universe, the higher the floor was.
Does the IR floor mean that "mass is quantized in steps of $10^{-33}$ eV"?
See the answer
No. This is an operational and dynamical floor (a horizon) of "cannot be measured / freezes," not a quantization of mass, nor a particle actually sitting there. It only means that a mass lighter than $10^{-33}$ eV cannot be distinguished from zero by any observer in this universe.

Episode 6 summaryThe lower wall was set by the size of the universe

A lower bound on mass really exists ── $m_{\min}\sim\hbar H_0/c^2\sim10^{-33}$ eV. Its true nature is "the boundary of freezing vs. oscillating under Hubble friction," a two-in-one of a measurement window (the low-frequency end of sampling) and dynamics (the freeze line). With $R_h=ct$ it drops as $1/t$, so it was higher in the past. At the bottom of the basement, the graviton's and photon's mass upper bounds touch, and that is the outer limit of the "smallest mass humanity can know."

In Episode 5 (UV) and Episode 6 (IR), we saw "the smallest mass points to a largest" from both sides ── one to grand unification, the other to the cosmic horizon. And next time, these two meet at meV. meV is the geometric mean of the IR floor and the Planck scale. Toward that single point where smallest and largest shake hands ── at last, in the finale.

Next time ── the finale "The smallest and the largest shake hands at meV." Episode 5 (UV) and Episode 6 (IR) meet at the single point of meV. We derive by hand the UV–IR boundary that gravity imposes (from the black-hole limit, $\rho_\Lambda\lesssim M_{\rm Pl}^2/L^2$), put in $L=ct$, and get meV. The dark dimension, and the cosmological constant problem ── to the last door, left open.

This document is Episode 6 of the "Mass That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. The cosmological IR scale $\hbar H_0/c^2\approx1.4\times10^{-33}$ eV, the equation of motion for a scalar field in the expanding universe $\ddot\phi+3H\dot\phi+(mc^2/\hbar)^2\phi=0$ and the frozen/oscillating boundary from Hubble friction ($m\sim\hbar H/c^2$; the exact critical damping is $mc^2/\hbar=\tfrac32 H$), the axion misalignment mechanism, quintessence, free-streaming, and the upper bounds on the photon and graviton masses are all established physics / current research topics. The IR floor is an operational and dynamical lower bound, not a quantization of mass. The scale $\sim10^{-33}$ eV is robust, but the coefficient has an order-unity ambiguity, and this lower bound holds in standard cosmology without assuming $c\cdot t=\text{constant}$ ($R_H\sim ct$ holds generally to order of magnitude). The local speed of light is invariant. The figure is a schematic from numerical integration in units of $H=1$. ── To print, use your browser's "Print" and "Save as PDF" (in the print version, the slider and answers are static and hidden).

Print / save as PDF: Ctrl+P (⌘+P on Mac). On screen, the slider lets you watch a field lighter than the floor freeze up. "See the answer" opens each solution.