Mathematical details will be left to ASTR 328....

When observing objects at high redshift, simple Euclidian geometry no longer works. We have to take into account the shape and expansion history of the universe, which various in different cosmologies, i.e., as a function of H0, OmegaM, and OmegaL. This means we have to know the cosmology to derive physical properties, or conversely, if we know physical properties we can infer the cosmological parameters.

So how do we measure luminosity, angular size, and surface brightness at cosmological distances? DON'T SIMPLY USE HUBBLE'S LAW!

Luminosity distance

Normally, flux = Luminosity/(4piD

First, define a coordinate distance that depends on the scale factor R and the comoving distance r. The energy flux you get from an object at that position is:

- reduced because of normal inverse square law of light: f ~ L/4pi(Rr)2
- reduced again becase the photons are being redshift and therefore reduced in energy: f ~ 1/(1+z)
- reduced yet again because photons arrive more slowly due to cosmological time dilation: f ~ 1/(1+z)

For OmegaL=0 universes, dL is analytic in form, but for universes with arbitrary OmegaM and OmegaL, it is not. Need to do numerical integration.

But in all cosmologies, for z<<1 this can be done using simple Taylor expansions to get

r(z) ~ (cz/H

so

d

where q

Then we just use dL in place of the usual d in the distance modulus equation (from Carroll and Ostlie):

Angular size distance

Normally, the angular size of an object is given by theta=size/distance (using the small angle approximation). Under cosmology, we write

theta = size/(Rr) and then note that R = 1/(1+z), so

theta = size(1+z)/r

If we define the angular size distance as d

theta = size / d

like usual. Defined this way, we have

d

for any cosmology. From Carroll and Ostlie:

Surface brightness

Notice that things get fainter and fainter and high redshift, but their angular size behaves differently -- so surface bightness will change with redshift!

Since observed surface brightness is flux/(angular size)

SB ~ f/theta

for any cosmology.

So surface brightness drops as redshift increases. At z=2, SB is down by 3

The above discussion about apparent brightnesses and the luminosity distance assumes you are detecting all the flux at all wavelengths -- i.e, measuring the bolometric flux. We never do this. We measure brightness over a range of wavelengths, ie through a filter. In terms of wavelengths, redshift both shifts and stretches light we see from distant objects. How does this complicate things?

- Redshift changes the emitted wavelength we observe: lambda
_{obs}= lambda_{em}(1+z) - Redshift changes the range of emitted wavelengths we observe. deltalambda
_{obs}= deltalambda_{em}(1+z)

at z=0 | at z=2 | |

central wavelength (Angstroms) | 5500 | 1833 |

bandpass (Angstroms) | 1000 | 333 |

So when we observe high redshift objects, we are seeing them at intrinsically different (bluer) wavelengths. If you know the spectrum of an object you can correct for this, using a so-called "K-correction". If you don't know what the spectrum is, you have to guess or assume a K-correction.

Or, you can observe in redder (infrared) filters as you look at objects at higher reshift.

Put cosmology and bandshifting together and what do you get? Consider the luminosity profile of a spiral galaxy:

Finally, how would galaxy evolution change things -- what did today's spiral galaxies look like when they were young? What about giant elliptical galaxies like M87?