Cosmology - Angular Diameter Distance

In this chapter, we will understand what the Angular Diameter Distance is and how it helps in Cosmology.

For the present universe −

$Omega_{m,0} : = : 0.3$

$Omega_{wedge,0} : = : 0.69$

$Omega_{rad,0} : = : 0.01$

$Omega_{k,0} : = : 0$

We’ve studied two types of distances till now −

Proper distance (lp) − The distance that photons travel from the source to us, i.e., The Instantaneous distance.

Comoving distance (lc) − Distance between objects in a space which doesn’t expand, i.e., distance in a comoving frame of reference.

Distance as a Function of Redshift

Consider a galaxy which radiates a photon at time t₁ which is detected by the observer at t₀. We can write the proper distance to the galaxy as −

$$l_p = int_{t_1}^{t_0} cdt$$

Let the galaxy’s redshift be z,

$$Rightarrow frac{mathrm{d} z}{mathrm{d} t} = -frac{1}{a^2}frac{mathrm{d} a}{mathrm{d} t}$$

$$Rightarrow frac{mathrm{d} z}{mathrm{d} t} = -frac{frac{mathrm{d} a}{mathrm{d} t}}{a}frac{1}{a}$$

$$ herefore frac{mathrm{d} z}{mathrm{d} t} = -frac{H(z)}{a}$$

Now, comoving distance of the galaxy at any time t will be −

$$l_c = frac{l_p}{a(t)}$$

$$l_c = int_{t_1}^{t_0} frac{cdt}{a(t)}$$

In terms of z,

$$l_c = int_{t_0}^{t_1} frac{cdz}{H(z)}$$

There are two ways to find distances, which are as follows −

Flux-Luminosity Relationship

$$F = frac{L}{4pi d^2}$$

where d is the distance at the source.

The Angular Diameter Distance of a Source

If we know a source’s size, its angular width will tell us its distance from the observer.

$$ heta = frac{D}{l}$$

where l is the angular diameter distance of the source.

θ is the angular size of the source.

D is the size of the source.

Consider a galaxy of size D and angular size dθ.

We know that,

$$d heta = frac{D}{d_A}$$

$$ herefore D^2 = a(t)^2(r^2 d heta^2) quad ecause dr^2 = 0; : dphi ^2 approx 0$$

$$Rightarrow D = a(t)rd heta$$

Changing r to r_c, the comoving distance of the galaxy, we have −

$$d heta = frac{D}{r_ca(t)}$$

Here, if we choose t = t₀, we end up measuring the present distance to the galaxy. But D is measured at the time of emission of the photon. Therefore, by using t = t₀, we get a larger distance to the galaxy and hence an underestimation of its size. Therefore, we should use the time t₁.

$$ herefore d heta = frac{D}{r_ca(t_1)}$$

Comparing this with the previous result, we get −

$$d_wedge = a(t_1)r_c$$

$$r_c = l_c = frac{d_wedge}{a(t_1)} = d_wedge(1+z_1) quad ecause 1+z_1 = frac{1}{a(t_1)}$$

Therefore,

$$d_wedge = frac{c}{1+z_1} int_{0}^{z_1} frac{dz}{H(z)}$$

d_A is the Angular Diameter Distance for the object.

Points to Remember

If we know a source’s size, its angular width will tell us its distance from the observer.

Proper distance is the distance that photons travel from the source to us.

Comoving distance is the distance between objects in a space which doesn’t expand.