The age of a closed universe from H and Ω

Astronomers measure H and Ω, not the cycloid's α and θ. Translating gives the age of a closed universe — and shows that Ω climbs from 1 to infinity and back over its life.

Follows MIT 8.286 lecture 9MIT 8.286 Lecture 9: The Dynamics of Homogeneous Expansion, Part VOn MIT OpenCourseWare: the lecture and its readings

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Full reading path (13 pages)

Vectors (8.01)
Position and velocity in one dimension (8.01)
Acceleration in one dimension (8.01)
Motion in two dimensions and projectiles (8.01)
Newton's laws of motion (8.01)
The universal law of gravitation (8.01)
The homogeneous expanding universe (8.286)
Why a static universe cannot work (8.286)
The expanding sphere and the equation for a(t) (8.286)
The Friedmann equation (8.286)
Open, closed and flat universes (8.286)
The closed universe as a cycloid (8.286)
The age of the universe and the Big Bang singularity (8.286)

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From H and Ω to the cycloid
The age formula
Summary

The cycloid solution describes a closed universe with the parameter $\alpha$ and the development angle $\theta$ . Astronomers cannot measure either. What they can measure is the Hubble rate $H$ and the density relative to critical, $\Omega$ . The second half of Lecture 9 translates one description into the other.

From H and Ω to the cycloid#

Write the density as a fraction of the critical density, $\rho = \Omega\rho_c$ with $\rho_c = 3H^2/(8\pi G)$ , so $\tfrac{8\pi}{3}G\rho = H^2\Omega$ . Then the Friedmann equation in the rescaled variable $\tilde a = a/\sqrt k$ reads $H^2 = H^2\Omega - c^2/\tilde a^2$ , which gives the size and then $\alpha$ :

Size and α from H and Ω

\tilde a = \frac{c}{|H|\sqrt{\Omega - 1}}, \qquad \alpha = \frac{c\,\Omega}{2|H|\,(\Omega - 1)^{3/2}}

In words

a tilde equals c over |H| the square root of omega minus 1 , alpha equals c omega over 2|H| ( omega minus 1) to the power 3/2

Source: 8.286 Lecture 9 transcript

The absolute value $|H|$ is needed because $H$ is negative while the universe contracts. Putting these into $\tilde a = \alpha(1 - \cos\theta)$ , almost everything cancels:

Ω and the development angle

\cos\theta = \frac{2 - \Omega}{\Omega}, \qquad \Omega = \frac{2}{1 + \cos\theta}

In words

cos theta equals 2 minus omega over omega , omega equals 2 over 1 plus cos theta

Source: 8.286 Lecture 9 transcript

This has a striking consequence. As $t \to 0$ ( $\theta \to 0$ ), $\Omega \to 1$ : even a closed universe starts out arbitrarily close to the critical density. $\Omega$ then grows, reaching 2 at $\theta = \pi/2$ and becoming infinite at maximum size ( $\theta = \pi$ ). There the expansion stops, $H = 0$ , so the critical density $3H^2/(8\pi G)$ is zero while the actual density is not; for that instant the universe is momentarily static. During the collapse the story runs in reverse, and $\Omega$ returns to 1 at the Big Crunch.

The age formula#

Time follows from $ct = \alpha(\theta - \sin\theta)$ , with $\sin\theta = \pm 2\sqrt{\Omega - 1}/\Omega$ . Guth writes the result with an inverse sine:

Age of a closed universe

t = \frac{\Omega}{2|H|(\Omega - 1)^{3/2}}\left\{\arcsin\!\left(\pm\frac{2\sqrt{\Omega - 1}}{\Omega}\right) \mp \frac{2\sqrt{\Omega - 1}}{\Omega}\right\}

In words

t equals omega over 2|H|( omega minus 1) to the power 3/2 arc sine ( plus or minus 2 the square root of omega minus 1 over omega ) minus or plus 2 the square root of omega minus 1 over omega

Source: 8.286 Lecture 9 slides, slides 6–7

The signs inside and outside the braces are always opposite. Because the inverse sine has many branches, one must pick the right one for the phase of the universe's life. Guth's table on slide 7:

Quadrant of θ	Phase	Ω	Sign	θ range
1	expanding	1 → 2	upper	0 → π/2
2	expanding	2 → ∞	upper	π/2 → π
3	contracting	∞ → 2	lower	π → 3π/2
4	contracting	2 → 1	lower	3π/2 → 2π

If our universe were closed and matter-dominated, it would be in the first quadrant. (In quadrants 2–4, "arcsin" must be read as the branch matching the θ range, not the calculator's principal value. Equivalently, while the universe expands, $\theta = \arccos\big((2 - \Omega)/\Omega\big)$ picks the right angle automatically; the plot below uses that form.)

Loading graph…

Age in units of 1/H for a closed matter-only universe, computed from the cycloid solution. As Ω → 1 it approaches the flat value 2/3; denser universes are younger for the same H.

Worked example

A universe with twice the critical density

Suppose a matter-only universe had $\Omega = 2$ and $H = 67.3$ km/s/Mpc today. Find its age, its total lifetime and how much bigger it will get.

Development angle

$\cos\theta = (2 - 2)/2 = 0$ , so $\theta = \pi/2$ (first quadrant, expanding).

Age

$\dfrac{\Omega}{2(\Omega - 1)^{3/2}} = 1$ , so $Ht = \theta - \sin\theta = \pi/2 - 1 = 0.571$ . With $1/H = 14.5$ billion years, $t = 8.3$ billion years — younger than the flat universe's 9.7.

Lifetime

$t_\text{total} = 2\pi\alpha/c = \dfrac{2\pi\,\Omega}{2|H|(\Omega - 1)^{3/2}} = 2\pi/H = 91$ billion years from Big Bang to Big Crunch.

Maximum size

Now $\tilde a = \alpha(1 - \cos\tfrac{\pi}{2}) = \alpha$ ; at maximum, $2\alpha$ . The universe will only double in size — in agreement with $a_{\max} = \Omega/(\Omega - 1) = 2$ found earlier from the conserved energy.

Summary#

Using $\rho = \Omega\rho_c$ , the cycloid parameters become $\tilde a = c/(|H|\sqrt{\Omega - 1})$ and $\alpha = c\Omega/[2|H|(\Omega - 1)^{3/2}]$ .
$\Omega = 2/(1 + \cos\theta)$ : a closed universe starts at $\Omega = 1$ , reaches $\infty$ at maximum size, and returns to 1 at the Big Crunch.
The age involves an inverse sine whose branch depends on the phase; for $\Omega = 2$ , $t = (\pi/2 - 1)/H \approx 0.571/H$ .
Denser closed universes are younger than a flat one with the same $H$ ; as $\Omega \to 1$ the age approaches $\tfrac23 H^{-1}$ .

Symbols on this page

Show the 22 symbols used here

$\alpha$ alpha: A length constant of the parametric (cycloid) solution, as in ct = α(θ − sin θ). Say it: “alpha”.
$\theta$ theta: The parameter (development angle) of the parametric solution: it runs from 0 at the Big Bang. Say it: “theta” (sounds like “thay-ta or thee-ta”).
$\pi$ pi: The ratio of a circle's circumference to its diameter, π ≈ 3.14159. Say it: “pi”.
$\rho$ rho: Density: mass (or energy divided by c²) per unit volume. ρ_c is the critical density, ρ̄ the mean density. Say it: “rho”.
$\Omega$ capital omega: The density parameter Ω = ρ/ρ_c. Subscripts pick one ingredient: Ω_m matter, Ω_vac (or Ω_Λ) vacuum energy, Ω_r radiation, Ω_k curvature. Say it: “omega”.
$\infty$ infinity: Infinity: larger than any number. Say it: “infinity”.
$\approx$ approximately equal: Is approximately equal to. Say it: “is approximately”.
$\to$ arrow: Goes to (tends to), or becomes. Say it: “goes to”.
$\pm$ plus or minus: Plus or minus: both signs, or an uncertainty (67.3 ± 1.2). Say it: “plus or minus”.
$\mp$ minus or plus: Minus or plus: the opposite sign to a ± in the same formula. Say it: “minus or plus”.
$\max$ maximum: Maximum: the largest value. Say it: “max”.
$\sin$ sine: Sine of an angle. Say it: “sine”.
$\cos$ cosine: Cosine of an angle. Say it: “cos”.
$\arcsin$ arcsine: Inverse sine: the angle whose sine is the given number. Say it: “arc sine”.
$\arccos$ arccosine: Inverse cosine: the angle whose cosine is the given number. Say it: “arc cos”.
$\tilde a$ tilde: A modified version of a quantity (ã, k̃); its meaning is given where it is introduced. Say it: “tilde”.
$a$ a: The scale factor a(t): how much the universe has expanded. Physical distance = a × comoving distance (Guth's units: metres per notch). Say it: “a”.
$H$ H: The Hubble parameter H = ȧ/a: the expansion rate of the universe at a given time. Say it: “H”.
$G$ G: Newton's gravitational constant, G ≈ 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻². Say it: “G”.
$c$ c: The speed of light, c ≈ 2.998 × 10⁸ m/s. Say it: “c”.
$k$ k: The curvature constant: k > 0 closed, k = 0 flat, k < 0 open (Guth's units: 1/notch²). Say it: “k”.
$\rho_{c}$ critical density: The critical density, 3H²/(8πG): the density of a flat universe (about 9 × 10⁻²⁷ kg/m³ today). Say it: “rho c”.

Tap or click a symbol in a formula to see the same explanation. All symbols: symbols and notation.

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The open universe and the age of any matter universe8.286 · Lecture 10

Check your understanding

In a closed universe, how does $\Omega$ change from the Big Bang to the moment of maximum size?
A closed universe has $\Omega = 2$ today. How does its age compare with that of a flat universe with the same $H$ ?
Which value of $\Omega$ corresponds to development angle $\theta = \pi/2$ ?

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