Cosmology - Luminosity Distance

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As discussed in the previous chapter, the angular diameter distance to a source at red shift z is given by −

$$d_\wedge (z_{gal}) = \frac{c}{1+z_{gal}}\int_{0}^{z_{gal}} \frac{1}{H(z)}dz$$

$$d_\wedge(z_{gal}) = \frac{r_c}{1+z_{gal}}$$

where $r_c$ is comoving distance.

The Luminosity Distance depends on cosmology and it is defined as the distance at which the observed flux f is from an object.

If the intrinsic luminosity $d_L$ of a distant object is known, we can calculate its luminosity by measuring the flux $f$ which is determined by −

$$d_L(z) = \sqrt{\frac{L}{4\pi f}}$$

The Photon Energy gets red shifted.

$$\frac{\lambda_{obs}}{\lambda_{emi}} = \frac{a_0}{a_e}$$

where $\lambda_{obs}, \lambda_{emi}$ are observed and emitted wave lengths and $a_0, a_e$ are corresponding scale factors.

$$\frac{\Delta t_{obs}}{\Delta t_{emi}} = \frac{a_0}{a_e}$$

where $\Delta_t{obs}$ is observed as the photon time interval, while $\Delta_t{emi}$ is the time interval at which they are emitted.

$$L_{emi} = \frac{nhv_{emi}}{\Delta t_{emi}}$$

$$L_{obs} = \frac{nhv_{obs}}{\Delta t_{obs}}$$

$\Delta t_{obs}$ will take more time than $\Delta t_{emi}$ because the detector should receive all the photons.

$$L_{obs} = L_{emi}\left ( \frac{a_0}{a_e} \right )^2$$

$$L_{obs} < L_{emi}$$

$$f_{obs} = \frac{L_{obs}}{4\pi d_L^2}$$

For a non-expanding universe, luminosity distance is same as the comoving distance.

$$d_L = r_c$$

$$\Rightarrow f_{obs} = \frac{L_{obs}}{4\pi r_c^2}$$

$$f_{obs} = \frac{L_{emi}}{4 \pi r_c^2}\left ( \frac{a_e}{a_0} \right )^2$$

$$\Rightarrow d_L = r_c\left ( \frac{a_0}{a_e} \right )$$

We are finding luminosity distance $d_L$ for calculating luminosity of emitting object $L_{emi}$ −

  • Interpretation − If we know the red shift z of any galaxy, we can find out $d_A$ and from that we can calculate $r_c$. This is used to find out $d_L$.

  • If $d_L ! = r_c(a_0/a_e)$, then we can’t find Lemi from $f_{obs}$.

The relation between Luminosity Distance $d_L$ and Angular Diameter Distance $d_A.$

We know that −

$$d_A(z_{gal}) = \frac{d_L}{1+z_{gal}}\left ( \frac{a_0}{a_e} \right )$$

$$d_L = (1 + z_{gal})d_A(z_{gal})\left ( \frac{a_0}{a_e} \right )$$

Scale factor when photons are emitted is given by −

$$a_e = \frac{1}{(1+z_{gal})}$$

Scale factor for the present universe is −

$$a_0 = 1$$

$$d_L = (1 + z_{gal})^2d_\wedge(z_{gal})$$

Which one to choose either $d_L$ or $d_A$?

  • For a galaxy of known size and red shift for calculating how big it is, then $d_A$ is used.

  • If there is a galaxy of a given apparent magnitude, then to find out how big it is, $d_L$ is used.

Example − If it is given that two galaxies of equal red shift (z = 1) and in the plane of sky they are separated by 2.3 arc sec then what is the maximum physical separation between those two?

For this, use $d_A$ as follows −

$$d_A(z_{gal}) = \frac{c}{1+z_{gal}}\int_{0}^{z_{gal}} \frac{1}{H(z)}dz$$

where z = 1 substitutes H(z) based on the cosmological parameters of the galaxies.

Points to Remember

  • Luminosity distance depends on cosmology.

  • If the intrinsic luminosity $d_L$ of a distant object is known, we can calculate its luminosity by measuring the flux f.

  • For a non-expanding universe, luminosity distance is same as the comoving distance.

  • Luminosity distance is always greater than the Angular Diameter Distance.

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