The open universe and the age of any matter universe

The open universe swaps sines and cosines for their hyperbolic cousins. Together with the flat and closed cases, it gives one smooth curve for the age of every matter-dominated universe.

Follows MIT 8.286 lecture 10MIT 8.286 Lecture 10: Introduction to Non-Euclidean SpacesOn MIT OpenCourseWare: the lecture and its readings

You need this first

The age of a closed universe from H and Ω8.286 · Lecture 9

Full reading path (14 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)
The age of a closed universe from H and Ω (8.286)

On this page · 4 sections

The open universe
Three kinds of universe on one graph
The age of any matter-dominated universe
Summary

Lecture 9 ended with the closed universe. Lecture 10 begins with a short review of the closed universe and then finishes the job: the open universe, and then the age of any matter-dominated universe, before turning to the geometry of curved space on the next page.

The open universe#

For an open universe $k < 0$ , so the closed-universe variable $a/\sqrt k$ would be imaginary. Guth avoids that by defining a positive constant:

Open universe: a positive constant

\kappa \equiv -k > 0, \qquad \tilde a \equiv \frac{a}{\sqrt\kappa}

In words

kappa is defined as minus k greater than 0, a tilde is defined as a over kappa

Source: 8.286 Lecture 10 slides, slide 5

The rest of the algebra parallels the closed case. Where the closed universe had sines and cosines, the open one gets hyperbolic sines and cosines:

Evolution of an open universe

c\,t = \alpha\,(\sinh\theta - \theta), \qquad \frac{a}{\sqrt\kappa} = \alpha\,(\cosh\theta - 1)

In words

c t equals alpha ( sinh theta minus theta ), a over kappa equals alpha ( cosh theta minus 1)

Source: 8.286 Lecture 10 slides, slide 5

The order of the terms is fixed by positivity: $\sinh\theta > \theta$ and $\cosh\theta > 1$ for $\theta > 0$ , just as $\theta > \sin\theta$ and $1 > \cos\theta$ in the closed case. (The slide and the lecture at first say "hypergeometric"; Guth corrects himself to the hyperbolic functions.)

Because $\cosh\theta$ grows without limit, an open universe expands forever. For large $\theta$ , $\sinh\theta \approx \cosh\theta$ , so $a$ grows in proportion to $t$ : the expansion settles into a constant rate.

Three kinds of universe on one graph#

Dividing $a/\sqrt{|k|}$ and $ct$ by $\alpha$ removes every parameter, so all closed universes collapse onto one curve, and all open ones onto another:

Loading graph…

Generated from the parametric solutions. The closed universe reaches a maximum of 2 and recollapses at ct/α = 2π; the open one grows forever, eventually at a constant rate. Near the Big Bang all three coincide.

The flat curve is a special case: $k = 0$ , so its normalisation on these axes is a matter of convention, chosen so that it looks like the limit of the other two. What matters is the shape near the start: all three curves merge at early times. Every matter-dominated universe starts out looking flat, with $\Omega$ very close to 1. Guth points out that this is the seed of the flatness problem, which he returns to later in the course; it becomes one of the motivations for inflation.

How $\Omega$ changes afterwards depends on the type. In a closed universe it rises from 1 toward infinity at maximum expansion (the density falls, but the critical density falls faster). In an open universe it falls from 1 toward zero.

The age of any matter-dominated universe#

Combining the three cases gives the age in terms of the two quantities astronomers measure, $H$ and $\Omega$ :

Age of a matter-dominated universe

|H|\,t = \begin{cases} \dfrac{\Omega}{2(1-\Omega)^{3/2}}\left[\dfrac{2\sqrt{1-\Omega}}{\Omega} - \sinh^{-1}\!\dfrac{2\sqrt{1-\Omega}}{\Omega}\right] & \Omega < 1 \\[2ex] \tfrac23 & \Omega = 1 \\[1ex] \dfrac{\Omega}{2(\Omega-1)^{3/2}}\left[\sin^{-1}\!\left(\pm\dfrac{2\sqrt{\Omega-1}}{\Omega}\right) \mp \dfrac{2\sqrt{\Omega-1}}{\Omega}\right] & \Omega > 1 \end{cases}

In words

|H| t equals omega over 2(1- omega ) to the power 3/2 [ 2 the square root of 1- omega over omega minus inverse sinh 2 the square root of 1- omega over omega ] when omega less than 1 ; or 2 over 3 when omega equals 1 ; or omega over 2( omega minus 1) to the power 3/2 [ inverse 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 ] when omega greater than 1

Source: 8.286 Lecture 10 slides, slide 6

The three formulas look completely different, yet they join into one smooth curve. Guth's explanation: with complex numbers, the open and closed formulas are the same function — imaginary arguments turn trigonometric functions into hyperbolic ones.

Loading graph…

Age in units of 1/H, computed from the three formulas (expanding phase). Open universes (left) are older, closed ones (right) younger, for the same H.

Worked example

An open universe of matter alone

Suppose the universe contained only matter, with $\Omega = 0.3$ (the measured matter density) and $H = 67.3$ km/s/Mpc. How old would it be?

The open formula

With $\Omega = 0.3$ : $2\sqrt{1-\Omega}/\Omega = 2\sqrt{0.7}/0.3 = 5.578$ and $\sinh^{-1}(5.578) = 2.420$ .

Put it together

$Ht = \dfrac{0.3}{2(0.7)^{3/2}}\,(5.578 - 2.420) = 0.256 \times 3.158 = 0.809$ .

In years

$1/H = 14.5$ billion years, so $t = 0.809 \times 14.5 = 11.8$ billion years — older than the flat case (9.7), but measurements show the universe is very nearly flat, so this is not the answer either. The real universe needs dark energy.

Summary#

For an open universe, $\kappa = -k > 0$ and $ct = \alpha(\sinh\theta - \theta)$ , $a/\sqrt\kappa = \alpha(\cosh\theta - 1)$ : it expands forever, eventually at a constant rate.
Scaled by $\alpha$ , all closed universes follow one curve and all open ones another; all matter universes look flat at early times — the seed of the flatness problem.
The age $Ht$ is one smooth function of $\Omega$ : above $2/3$ for open universes, below for closed ones.

Symbols on this page

Show the 19 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”).
$\kappa$ kappa: κ = −k, positive for an open universe. Say it: “kappa”.
$\Omega$ 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”.
$\approx$ approximately equal: Is approximately equal to. Say it: “is approximately”.
$\equiv$ identical to: Is defined as (or is identically equal to). Say it: “is defined as”.
$\to$ 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”.
$\times$ times: Times (multiplication). Between two vectors: the cross product. Say it: “times”.
$\sin$ sine: Sine of an angle. Say it: “sine”.
$\cos$ cosine: Cosine of an angle. Say it: “cos”.
$\sinh$ hyperbolic sine: Hyperbolic sine, (eˣ − e⁻ˣ)/2. Say it: “sinh” (sounds like “shine or sinch”).
$\cosh$ hyperbolic cosine: Hyperbolic cosine, (eˣ + e⁻ˣ)/2. Say it: “cosh” (sounds like “kosh”).
$\tilde a$ 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”.
$c$ 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”.

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

Where this leads

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Check your understanding

How does the open-universe solution differ from the closed one?
Why does Guth introduce $\kappa = -k$ for the open universe?
For the same Hubble rate, which matter-dominated universe is oldest?
At very early times, the scale factors of open, flat and closed universes…

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