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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,

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

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

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

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.

$$theta = frac{D}{l}$$

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 .

We know that,

$$dtheta = frac{D}{d_A}$$

$$therefore D^2 = a(t)^2(r^2 dtheta^2) quad because dr^2 = 0; : dphi ^2 approx 0$$

$$Rightarrow D = a(t)rdtheta$$

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

$$dtheta = frac{D}{r_ca(t)}$$

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.

$$therefore dtheta = frac{D}{r_ca(t_1)}$$

Comparing this with the previous result, we get −

$$d_wedge = a(t_1)r_c$$

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

Therefore,

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

dA is the Angular Diameter Distance for the object.

Angular Diameter

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.

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