A short side-trip between main episodes. On not being fooled by a name.
Cosmology has a famous number called the "Hubble constant" \(H_0\). It represents how vigorously the universe is expanding, and it shows up often in the news. Because "constant" is in the name, you'd think it's an unchanging value — but in fact, the Hubble constant changes over time. The name is a small lie. Using Episode 1's equation, let's verify that with a single derivative. A short side-trip.
The Hubble constant \(H\) is how fast the size of the universe \(a\) is increasing "per unit of itself" — the rate of increase. In symbols, it's the rate of change of \(a\), namely \(\dot a\) (\(a\) differentiated with respect to time), divided by \(a\) itself.
The point is that it's a "rate," not a "speed." For example, if it grows 2% per year, \(H\) is 0.02/yr. The "present value" of this \(H\) is that famous \(H_0\). So — is this rate really constant?
In the universe of Episodes 1 and 4 (\(c\cdot t=\text{constant}\)), the universe grows in a straight line, proportional to time. That is, the size is \(a(t)=k\,t\) (\(k\) a constant). Let's put this into the definition.
The size a and its rate of change
$$a(t) = k\,t \qquad\Rightarrow\qquad \dot a = k\quad(\text{constant})$$Form the rate H
$$H = \frac{\dot a}{a} = \frac{k}{k\,t} = \frac{1}{t}$$The \(k\) cancels cleanly, giving \(H = 1/t\). The larger the age \(t\), the smaller \(H\) — inverse proportionality. When the universe is young, \(H\) is large (it swells vigorously); as it ages, it gentles. So the Hubble "constant" is no constant at all, but a quantity that decreases with time.
In the figure below, watch \(H=1/t\) actually decrease with time. Move the slider for "how old the universe is now" and you can read off the Hubble "constant" for that era.
The reason lies in how the name arose. The Hubble constant appeared as "the constant of proportionality by which, when you observe now, every galaxy recedes faster the farther it is." At "one instant" surveying the sky, distance and recession speed are cleanly proportional, and that proportionality constant is \(H_0\). It is a constant with respect to space (the same in every direction), but not a constant with respect to time — that's the confusing part.
The Hubble constant is a constant in the sense of "at that instant, the same value everywhere in space." It is not a constant in the sense of "unchanging as time passes." That's why the present value carries a deliberate "now" subscript, \(H_0\).
So, precisely, calling it the "Hubble parameter \(H(t)\)" is correct, and specialists distinguish it that way. "Constant" is a leftover of the historical name. Don't be fooled by the name; return to the definition (\(\dot a/a\)), and one derivative shows it decreasing as \(H=1/t\) — that was this little tale.
\(H=1/t\) exactly holds only in a straight-line-expansion universe (Episode 1's model). The real universe expands in a slightly more complex way, so \(H\) and \(1/t\) aren't exactly equal but close (\(H_0 t_0\) is about 1). Still, the conclusion "the Hubble constant changes with time" is the same in every standard model — even if the name is "constant," the substance moves.
Compute the Hubble constant \(H=\dot a/a\) in a straight-line-expansion universe and you get \(H=1/t\). Inversely proportional to the age of the universe, decreasing with time — no "constant" at all. "Constant" meant "the same everywhere in space at that instant," not "unchanging in time."
In physics, a name sometimes drifts from reality. When it does, what you can rely on is not the name but the defining equation. Return to \(H=\dot a/a\) and one derivative reveals the truth. Don't be fooled by the name; trust the definition — a short side-trip, but a habit that serves you forever.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you read off the value of H.