# Can the age of the universe at the time of emitted light be calculated with Hubble's Law?

Let there be an object being observed, distance D away. D equals the speed of light times the time it takes the light to arrive (but space expands while the light is propagating, which is most likely the cause for the following discrepancy I think). T is the current age of the universe in years and t is the age at the time that the light was emitted.

$$v_r = H_0D$$ $$t = \frac{D}{c}$$ $$\frac{v_r}{c} = H_0\frac{D}{c} \approx z \implies t \approx \frac{z}{H_0}$$ This would also assume that the Hubble parameter is constant, possibly another cause for the discrepency (I believe) Between this and the age proposed by the FLRW equations;

$$a\propto t^{2/3} \implies t\propto a^{1.5} \implies t = T\dfrac{1}{(1+z)^{1.5}}$$

And since the current Hubble time is $\frac{1}{H_0}$, $$z= \dfrac{1}{(1+z)^{1.5}}$$ which is obviously only true for certain redshifts. Even if it was assumed that $a\propto t$ the two relations would not hold for all redshifts. Am I correct that this is because of distance and/or $H_0$ not being constant or is there another mistaken assumption?

• You have figured the time using two different models, your own simplified model and Friedman's model. Naturally you obtained two different results reflecting the differences between the models. Setting these different results equal to each other is meaningless. The equation for z that you derived in the end has no logical meaning and doesn't reflect any physical reality, but is a mere consequence of an improper equation. May 20, 2018 at 17:27
• That's what I figured, but what I'm wondering is what separates the two models. Is it that space expands as the light is moving towards us, meaning d isn't constant with the model I used? What I'm trying to ask is why is 'my' model wrong, if that makes sense. May 21, 2018 at 18:15
• You are using a formula for a fixed distance, but the distance is not fixed. Does this give a wrong result? The only way to tell is to do this the proper way. You must start your model with a spacetime metric, then solve the Lagrange equations for geodesics. The formula for the null (lightlike) geodesics will give you the result you are looking for. Without a metric you have no model, as you don't know the topology or geometry of spacetime or how exactly the universe expands. Check out the Milne model. It is fascinating, if you dig deeper than just a Wiki page. Walker of FLRW was his student. May 21, 2018 at 19:42
• Thanks, while waiting for answers I have looked around and found some evidence for that. I don't know anything past calc 3 so I'm not aware if that's enough but I'll try my best. May 23, 2018 at 3:08