Luminosity Distance: Finding the Connection

[Apashanka]

If photons are emmitted from a galaxy A at t_e to galaxy B (receives at present time a=1,t₀) ,x being the comoving distance ,and current distance be d_L
L(t₀)=a(t_e)²L(t_e)
Flux(Φ₀)=a(t_e)²L(t_e)/4πd_L²
In comoving frame ∅=L(t_e)/4πx²
Is there any way to relate Φ and ∅ to find the luminousity distance d_L??

 

[kimbyd]

Wikipedia has a good page:
https://en.wikipedia.org/wiki/Distance_measures_(cosmology)

I don't think you need to do anything special in the above case. The relative scale factor between the emission and observation is sufficient (this gives you the redshift).

 

[Apashanka]

Wikipedia has a good page:
https://en.wikipedia.org/wiki/Distance_measures_(cosmology)

I don't think you need to do anything special in the above case. The relative scale factor between the emission and observation is sufficient (this gives you the redshift).

My question is how to find the luminousity distance from the above two equations by relating them.

 

[Apashanka]

If photons are emmitted from a galaxy A at t_e to galaxy B (receives at present time a=1,t₀) ,x being the comoving distance ,and current distance be d_L
L(t₀)=a(t_e)²L(t_e)
Flux(Φ₀)=a(t_e)²L(t_e)/4πd_L²
In comoving frame ∅=L(t_e)/4πx²
Is there any way to relate Φ and ∅ to find the luminousity distance d_L??

Or the other way if n_e be the no. density of photons at t_e then flux will be
Φ(t_e)=n_ecE(t_e),
Φ(t_o)=n_oca(t_e)E(t_e)
where n_o is the no. density of photons at the observed time.=n_ea(t_e)³/a(t₀)³
c is the speed of light.(constt. throughout the expansion)

Φ(t_e) is the same as in the comoving coordinate =∅ .
If a(t_o)=1(present time)
then Φ(t_o)=a(t_e)⁴n_ecE(t_e)
,Φ(t_o)=a(t_e)⁴∅Is this the way that they are related ??,if so then from the quoted part above d_L can be calculated which is d_L=x/a(t_e)??

 

[PeterDonis]

@Apashanka you have already been given a good answer and a reference by @kimbyd. You need to read the article linked to and then ask further questions if you are having trouble with something in that article, or you need to address questions specifically to what @kimbyd said in post #2 (pay particular attention to the last paragraph).

 

[Apashanka]

@Apashanka you have already been given a good answer and a reference by @kimbyd. You need to read the article linked to and then ask further questions if you are having trouble with something in that article, or you need to address questions specifically to what @kimbyd said in post #2 (pay particular attention to the last paragraph).

Actually what I mean to say is that is the formula relating Φ(t_o) and ∅ valid so that d_L comes out to be x/a(t_e) ??
As in the Wikipedia page @kimbyd has provided d_L vs d_A is given ,d_L alone is not given as function of z

 

[kimbyd]

Actually what I mean to say is that is the formula relating Φ(t_o) and ∅ valid so that d_L comes out to be x/a(t_e) ??
As in the Wikipedia page @kimbyd has provided d_L vs d_A is given ,d_L alone is not given as function of z

What's the scale factor at emission?
What's the scale factor at observation?

That's all you need. That and the definition of redshift: $z + 1 = 1/a$.

 

[Apashanka]

What's the scale factor at emission?
What's the scale factor at observation?

That's all you need. That and the definition of redshift: $z + 1 = 1/a$.

Yes.
Is relation between Φ(t_o) and ∅ correct.??
Is so then d_L will be x(1+z)

 

[kimbyd]

Yes.
Is relation between Φ(t_o) and ∅ correct.??
Is so then d_L will be x(1+z)

It's hard for me to say because I'm a little unsure of your notation. But I think you're getting closer at least.

 

[Apashanka]

It's hard for me to say because I'm a little unsure of your notation. But I think you're getting closer at least.

The arguments are this which I gave

Or the other way if n_e be the no. density of photons at t_e then flux will be
Φ(t_e)=n_ecE(t_e),
Φ(t_o)=n_oca(t_e)E(t_e)
where n_o is the no. density of photons at the observed time.=n_ea(t_e)³/a(t₀)³
c is the speed of light.(constt. throughout the expansion)

Φ(t_e) is the same as in the comoving coordinate =∅ .
If a(t_o)=1(present time)
then Φ(t_o)=a(t_e)⁴n_ecE(t_e)
,Φ(t_o)=a(t_e)⁴∅Is this the way that they are related ??,if so then from the quoted part above d_L can be calculated which is d_L=x/a(t_e)??

 

[kimbyd]

The arguments are this which I gave

I'm still not sure what you mean by ∅. Usually this character refers to the empty set.

Is this supposed to refer to the comoving distance from us to the galaxy where the signal is observed?

 

[Apashanka]

I'm still not sure what you mean by ∅. Usually this character refers to the empty set.

Is this supposed to refer to the comoving distance from us to the galaxy where the signal is observed?

No ∅ is the flux received by the observer in the co-moving frame.
If Φ(t_e) be the emmited flux at t_e by the emmiter ,observer in the comoving frame will receive the same , that's why Φ(t_e)=∅

I have mentioned this in post #10

 

[kimbyd]

No ∅ is the flux received by the observer in the co-moving frame.
If Φ(t_e) be the emmited flux at t_e by the emmiter ,observer in the comoving frame will receive the same , that's why Φ(t_e)=∅

I have mentioned this in post #10

Okay. I read your statement as this was the co-moving coordinate.

Generally the emitted flux and observed flux are wildly different, but I think it's okay if you're talking about a uniform photon gas. In which case yes, the flux falls off as $1/a^4$ as time increases.

But it sounds like you're not referring to a uniform photon gas here (as in the CMB), but rather one galaxy's emitted light. In which case you have to take into account how much space the light spreads out into. This is why when computing the luminosity distance, an integral over the expansion is required.

 

[Apashanka]

Okay. I read your statement as this was the co-moving coordinate.

Generally the emitted flux and observed flux are wildly different, but I think it's okay if you're talking about a uniform photon gas. In which case yes, the flux falls off as $1/a^4$ as time increases.

But it sounds like you're not referring to a uniform photon gas here (as in the CMB), but rather one galaxy's emitted light. In which case you have to take into account how much space the light spreads out into. This is why when computing the luminosity distance, an integral over the expansion is required.

Okay thanks
But sir will the relation between Φ(t_o) and ∅ remains same for uniform photon gas and for the case of light emmited from galaxy,??

 

[Apashanka]

In which case you have to take into account how much space the light spreads out into. This is why when computing the luminosity distance, an integral over the expansion is required.

Will you give some suggestion regarding this

 

[Apashanka]

One simple question ,the angular diameter distance d_A=xa, and luminosity distance is d_L=x/a(t_e),t_e is the emmision time.
For d_L=(1+z)²d_A ,whether a in the d_A is a(t_e)??
x is the comoving distance.
z being the redshift corresponding to emmision time.