Cosmology

• Red Shifts

Cosmology

A very short note on confusing terminology...

The rest of this page is under revision.

Redshifts

We now have to consider three types of redshifts:

1. Doppler Redshift: This is due to the movement of an object through space. The Doppler shift will result in shorter wavelengths if the object is moving towards us (blue shift), and longer wavelengths if the object is moving away from us (redshift): $${\lambda }_{\mathrm{observed}}={\lambda }_{\mathrm{source}}\frac{(c-v_{\mathrm{source}})}{(c-v_{\mathrm{observer}})}$$

   or $$\Delta \lambda ={\lambda }_{\mathrm{source}}\frac{(v_{\mathrm{source}}-v_{\mathrm{observer}})}{(c-v_{\mathrm{observer}})}$$

2. Gravitational Redshift: This redshift occurs because photons must expend energy to climb out of a gravity well; those that reach the top and escape have less energy than when they started. Since wavelength depends on energy (E = ℏν = ℏλ/c). $$\underset{r\to \infty }{\lim }z(r)=\frac{1}{{(1-\frac{r_s}{R_e})}^{1/2}}-1$$ Here, r_s is the Schwarzschild radius of the mass, R_e is the radius at which the photon is emitted (R_e > r_s). The redshift is, as usual, z = Δλ/λ₀ = v/c.

   Consider what happens if the photon is emitted where R_e = r_s (at the Schwarzschild radius). In that case, we have (1 - 1)^½ = 0 in the denominator: the expression blows up and the amount of shift is infinitely large (no light escapes). If R_e >> r_s, the fraction goes to 0, and we have (1/1) -1 = 0: there is no redshift when the photon is emitted sufficiently far from the singularity. So gravitational redshift occurs only when R_e is within a range of r_s. For cosmological purposes, we can ignore gravitational redshift unless we are making observations within a reasonable distance of a gravitational mass sufficient to warp space significantly.

3. Cosmological Redshift: The cosmological redshift is due to the expansion of space itself. Again we have λ/λ₀ = 1 + z. If z = 3 for some observed object, then λ/λ₀ = 4; that is, the universe is four times the size it was when the photon was emitted by the object, due to expansion of space. Once we know the expansion factor and the recessional velocity, we can determine the distance to the object in light years, which gives us the length of time the photon has been traveling and the lookback age of the object.

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