Equation to give the lookback time as a function of redshift

[happyparticle]

Homework Statement

Inverting $z = H_0(t_0 - t_e) + (1 + \frac{q_0}{2}H_0^2(t_0 - t_e)^2)$ to give the lookback time as a function of redshift $t_0 - t_e = H_0^{-1}[z - (1 + \frac{q_0}{2})z^2]$

Relevant Equations

$H_0$ is the Hubble's constant
$q_0$ is the deceleration parameter
$z$ is the redshift

Hi,
I'm currently reading the introduction to cosmology second edition by Barbara Ryden and at the page 105, the author says we get $t_0 - t_e = H_0^{-1}[z - (1 + \frac{q_0}{2})z^2]$ by inverting $z = H_0(t_0 - t_e) + (1 + \frac{q_0}{2}H_0^2(t_0 - t_e)^2)$.

However, I can't figure out how she got this result.
Any help will be appreciate.

Thank you

 

[TSny]

Homework Statement: Inverting $z = H_0(t_0 - t_e) + (1 + \frac{q_0}{2}H_0^2(t_0 - t_e)^2)$ to give the lookback time as a function of redshift $t_0 - t_e = H_0^{-1}[z - (1 + \frac{q_0}{2})z^2]$
Relevant Equations: $H_0$ is the Hubble's constant
$q_0$ is the deceleration parameter
$z$ is the redshift

Hi,
I'm currently reading the introduction to cosmology second edition by Barbara Ryden and at the page 105, the author says we get $t_0 - t_e = H_0^{-1}[z - (1 + \frac{q_0}{2})z^2]$ by inverting $z = H_0(t_0 - t_e) + (1 + \frac{q_0}{2}H_0^2(t_0 - t_e)^2)$.

However, I can't figure out how she got this result.
Any help will be appreciate.

Thank you

I believe you have some typographical errors. The formula for $z$ should read $$z \approx H_0(t_0-t_e) + \left(\frac{1+q_0}{2} \right)H_0^2(t_0-t_e)^2.$$
This is an approximate expression that assumes $H_0(t_0-t_e)$ is small. So, the equation expresses $z$ to second order in $H_0(t_0-t_e)$.

For convenience, let $x = H_0(t_0-t_e)$ and $b = \large \frac{1+q_0}{2}$. So, we may write the relation as $$z \approx x+ bx^2$$ where $x$ is a small first-order term. When inverting this, you only need to get an approximate expression for $x$ in terms of $z$ that is accurate to second order in $z$.

Can you see a way to do that?

 

[happyparticle]

I can't see it. I tried a taylor's series but I don't get the same result.

I made a mistake. I think it works with a Taylor's series around z=0.

Thank you! I would not have seen it without your help.