The Friedmann equation (8.286)

The expanding-sphere page ended with a conserved quantity $E$ . In Lecture 7 Guth rewrites it in the form every cosmologist uses: the Friedmann equation. It relates how fast the universe expands to what it contains and to a single constant, $k$ , which will turn out to describe the shape of space.

From E to the Friedmann equation#

Start from the conserved quantity and the fact that matter density falls as $\rho = \rho_i/a^3$ :

The conserved quantity

E = \tfrac12\,\dot a^2 - \frac{4\pi}{3}\,\frac{G\rho_i}{a}

In words

E equals one half a dot squared minus 4 pi over 3 G rho sub i over a

Source: 8.286 Lecture 7 slides, slide 3

Multiply by $2/a^2$ and use $\rho_i/a^3 = \rho$ :

Rearranged

\left(\frac{\dot a}{a}\right)^2 - \frac{8\pi}{3}\,G\rho = \frac{2E}{a^2}

In words

( a dot over a ) squared minus 8 pi over 3 G rho equals 2E over a squared

Source: 8.286 Lecture 7 transcript

Nobody in cosmology actually talks about $E$ — that was Guth's own stepping stone. The conventional constant is

Curvature constant

k \equiv -\frac{2E}{c^2}

In words

k is defined as minus 2E over c squared

Source: 8.286 Lecture 7 transcript

and with it we arrive at the boxed result:

The Friedmann equation

\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi}{3}\,G\rho - \frac{k c^2}{a^2}

In words

( a dot over a ) squared equals 8 pi over 3 G rho minus k c squared over a squared

Source: 8.286 Lecture 7 transcript

Since $\dot a/a = H$ , the left side is the square of the Hubble rate. The equation says: the expansion rate is set by the density, corrected by the constant $k$ .

The Russian physicist Alexander Friedmann derived this equation from general relativity in 1922, in a paper on universes with positive $k$ , followed in 1924 by one on negative $k$ . (In the lecture Guth recalls the dates as 1922 and 1923; the second paper actually appeared in 1924.) Guth calls the pair below together "the Friedmann equations":

The two Friedmann equations (pressureless matter)

\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi}{3}G\rho - \frac{kc^2}{a^2}, \qquad \ddot a = -\frac{4\pi}{3}G\rho\,a

In words

( a dot over a ) squared equals 8 pi over 3 G rho minus kc squared over a squared , a double dot equals minus 4 pi over 3 G rho a

Source: 8.286 Lecture 7 transcript

Why Newton gives the exact answer#

It may seem surprising that a Newtonian calculation reproduces general relativity exactly. Guth's argument is homogeneity. Look at a small box, say one metre across. Inside it, the expansion velocities are tiny and gravity is weak, so general relativity must agree with Newton's theory there. But the universe is homogeneous: if you understand one small box, you understand every box, and so the whole universe. (This works for pressureless matter; radiation and dark energy need the relativistic additions mentioned above.)

Notches, and what k's number means#

So far $r$ and $r_i$ were in metres. Guth now makes the comoving coordinate's unit explicit: say a particle's coordinate $r_i$ is 5 notches if it was 5 metres from the origin at the initial time $t_i$ . Then

Physical radius with notches

r\,[\text{m}] = a(t)\,[\text{m/notch}] \times r_i\,[\text{notch}], \qquad a(t_i) = 1\ \text{m/notch}

In words

r [ m ] equals a(t) [ m/notch ] times r sub i [ notch ], a(t sub i ) equals 1 m/notch

Source: 8.286 Lecture 7 transcript

The initial time $t_i$ was arbitrary, so its only trace is the size of the notch. Checking the units of $k$ :

Units of k

[k] = \left[\frac{\dot a^2}{c^2}\right] = \frac{\text{s}^2}{\text{m}^2}\cdot\frac{\text{m}^2}{\text{notch}^2\,\text{s}^2} = \frac{1}{\text{notch}^2}

In words

[k] equals [ a dot squared over c squared ] equals s squared over m squared times m squared over notch squared s squared equals 1 over notch squared

Source: 8.286 Lecture 7 transcript

So the number $k$ depends on how big a notch is. Two conventions are common:

Make $a = 1$ today. Then a notch is a metre today (the convention of Barbara Ryden's textbook, for example).
Choose the notch so that $k = +1$ or $-1$ . The sign of $k$ cannot be changed by rescaling; if $k = 0$ , the notch is left arbitrary.

Either way, notches and the bare number $k$ never appear in any physical prediction. Physical results depend only on the sign of $k$ and on notch-free combinations such as $kc^2/a^2$ , which later turns out to measure the curvature of space. Guth thinks the $k = \pm1$ convention is the more common one in textbooks.

Worked example

Checking the rearrangement

Show that $\frac{2}{a^2}E$ gives the left side of the rearranged equation, and confirm that the Friedmann equation has consistent units.

Multiply out

$\dfrac{2}{a^2}\left(\tfrac12\dot a^2 - \tfrac{4\pi}{3}\dfrac{G\rho_i}{a}\right) = \dfrac{\dot a^2}{a^2} - \dfrac{8\pi}{3}\dfrac{G\rho_i}{a^3} = \left(\dfrac{\dot a}{a}\right)^2 - \dfrac{8\pi}{3}G\rho$ .

Units of each term

$(\dot a/a)^2$ has units of 1/s² (notches cancel). $G\rho$ : $\text{m}^3\,\text{kg}^{-1}\text{s}^{-2} \times \text{kg}\,\text{m}^{-3} = \text{s}^{-2}$ . And $kc^2/a^2$ : $\text{notch}^{-2} \times \text{m}^2\text{s}^{-2} \times \text{notch}^2\,\text{m}^{-2} = \text{s}^{-2}$ . All three agree.

Summary#

Multiplying $E$ by $2/a^2$ and defining $k = -2E/c^2$ gives the Friedmann equation, $(\dot a/a)^2 = \tfrac{8\pi}{3}G\rho - kc^2/a^2$ .
Friedmann derived it from general relativity in 1922; for pressureless matter, the Newtonian derivation is exact because a homogeneous universe is understood from any small region.
With comoving coordinates in notches, $a$ has units m/notch and $k$ has units 1/notch²; the bare number $k$ is not physical, but its sign and the combination $kc^2/a^2$ are.
With pressure, the first Friedmann equation survives; the $\ddot a$ equation changes.

Symbols on this page

Show the 15 symbols used here

$\pi$ 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”.
$\propto$ proportional to: Is proportional to: doubling one doubles the other. Say it: “is proportional to”.
$\equiv$ identical to: Is defined as (or is identically equal to). Say it: “is defined as”.
$\pm$ plus or minus: Plus or minus: both signs, or an uncertainty (67.3 ± 1.2). Say it: “plus or minus”.
$\times$ times: Times (multiplication). Between two vectors: the cross product. Say it: “times”.
$\cdot$ dot: Times (multiplication). Between two vectors: the dot product. Say it: “times”.
$\dot a$ dot: Rate of change with time: ȧ = da/dt. Say it: “dot”.
$\ddot a$ double dot: Second rate of change with time: ä = d²a/dt². Say it: “double dot”.
$a$ 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”.
$\rho_{i}$ initial density: The density when a = 1, so that ρ = ρᵢ/a³ for matter. Say it: “rho i”.

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

Check your understanding

How is the curvature constant

k

defined from the conserved quantity

E

Why can a purely Newtonian calculation give exactly the general-relativity result here?

In Guth's convention, what are the units of

k

Which of the two Friedmann equations still holds when the matter has pressure?

0 of 4 answered

The Friedmann equation

You need this first

From E to the Friedmann equation#

Why Newton gives the exact answer#

Notches, and what k's number means#

Checking the rearrangement

Multiply out

Units of each term

Summary#

Symbols on this page

Where this leads

Pages that build on this

Check your understanding

Flashcards