<p>It wants to know what the general solution of the equation is in the circumstances given. </p>

<p>$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G\rho}{3} - \frac{k}{a^2}$$
$$ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G \rho_0}{a^3} - \frac{k}{a^2}$$</p>

<p>If the "second term dominates over the density term" (which it will when $a$ is large), then
$$ \dot{a} = -k$$
(but $k$ has a negative value)</p>

<p>Therefore $a(t) \propto t$.</p>

CC BY-SA 3.0
2017-07-21T14:29:41.047Z
2017-07-21T19:46:41.500Z
2017-07-21T19:46:41.500Z
ProfRob
<p>In Friedman equation:</p>

<blockquote>
  <p>the case k &lt; 0, with a Universe containing only matter (pressure = 0)
  so that $\rho = \rho_0/ a^3$ . What is the solution a(t) in a
  situation where the final term of the Friedmann equation dominates
  over the density term? How does the density of matter vary with time?</p>
</blockquote>

<p>The question from an introduction to modern cosmology from Andrew Liddle.</p>

<p>Friedmann Equation:</p>

<p>$$(\dot {a}/a)^2 = 8\pi G \rho / 3 - k/a^2$$ 
My problem is I don't even understand the question? what does the first question mean? the answer of the book is $a(t) \alpha t$. I am not even sure that it is the only answer. Can anyone say what the question wants and is the answer of the book, the only answer?</p>

<cosmology>
Solving Friedmann equation for specific conditions
titansarus
# Solving Friedmann equation for specific conditions

(Asked by: username_0 on 2017-07-21)

In Friedman equation:

> the case k < 0, with a Universe containing only matter (pressure = 0)
> so that $\rho = \rho\_0/ a^3$ . What is the solution a(t) in a
> situation where the final term of the Friedmann equation dominates
> over the density term? How does the density of matter vary with time?

The question from an introduction to modern cosmology from Andrew Liddle.

Friedmann Equation:

$$(\dot {a}/a)^2 = 8\pi G \rho / 3 - k/a^2$$
My problem is I don't even understand the question? what does the first question mean? the answer of the book is $a(t) \alpha t$. I am not even sure that it is the only answer. Can anyone say what the question wants and is the answer of the book, the only answer?

**Tags:** cosmology

---

## 1 answers

### Answer by username_1 on 2017-07-21. Score: 4 | Accepted answer

It wants to know what the general solution of the equation is in the circumstances given.

$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G\rho}{3} - \frac{k}{a^2}$$
$$ \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G \rho\_0}{a^3} - \frac{k}{a^2}$$

If the "second term dominates over the density term" (which it will when $a$ is large), then
$$ \dot{a} = -k$$
(but $k$ has a negative value)

Therefore $a(t) \propto t$.


---

