6. Distance and Redshift Relations in Cosmology Flashcards
FRW spacetime metric at present time
Metric
g = ds² = -c²dtxdt + g(3)
-where:
g(3) = dl²
= ao²{dχxdχ/[1-ϰχ²] + χ²(dθxdθ + sin²θdφxdφ)}
FRW spacetime metric at present time
dl
-along the radial direction we have dθ=dφ=0 so:
g(3) = dl² = ao² dχxdχ/[1-ϰχ²]
=>
dl = ao dχ/√[1-ϰχ²]
FRW spacetime metric at present time
distance r to source (as measured by observer)
-if χo=0 and χe are the coordinates of the observer and a source of light respectively, then the distance r to the source is given by: r(χo,χe) = ao ∫ dχ/√[1-ϰχ²] -where the integral is from χo to χe -for χo=0, this gives: r=ao*arcsin(χe) if ϰ=1 r=ao*χe if ϰ=0 r=ao*arcsinh(χe) if ϰ=-1
χ vs. Z
-in astronomical observations we cannot measure χe so a more practical question is ‘what is the distance from us to a remote galaxy with redshift z?’
Distance in Terms of Redshift
Derivation
-to measure distance with redshift, we need to know not only the current geometry of the universe but also the history of its expansion prior to the time of observation
-this implies computing χe(z)
-along the world-lines of photons we have:
c²dt² = a²(t) dχ²/[1-ϰχ²]
-where t spans from time of emission te to time of observation to
=>
c dt = - a(t) dχ/√[1-ϰχ²]
-with a -ve since dχ<0 if dt>0
-integrate both sides and sub into equation for r
Distance in Terms of Redshift
Equation
r(z) = c ao ∫ dt/a(t)
- where the integral is between te and to
- so to calculate distances in terms of redshift we need to know a(t)
a(t) for Friedmann’s flat universe with dust
-we have ϰ=0 (flat) and P=0 (dust)
=>
a(t) = ao [t/to]^(2/3)
r(z) for Friedmann’s flat universe with dust
Derivation
-sub expression for a(t) into the equation for distance in terms of redshift:
r(z) = c ao ∫ dt/a(t)
-then use the generalised Hubble Law to swap the time terms for terms in z
-note that other models of the universe will give different solutions for r(z) but the method is the same
r(z) for Friedmann’s flat universe with dust
z«1
-for z«1, r(z) reduces to the original Hubble law:
r(z) ≈ c/Ho z
r(z) for Friedmann’s flat universe with dust
z -> +∞
-in general, r(z) grows slower with z
r(z) -> 2c/Ho as z -> +∞
-this tells us that in order to be seen, a light source mus be located at a distance r
How far does a photon produced at the birth of the universe travel?
-the generalised Hubble law:
te = to [1+z]^(-3/2)
-shows that as te->0, z->+∞
-thus z->+∞ corresponds to an emission produced at the time of the big bang
-since the speed of light is finite, a photon can travel only a finite distance since the Big Bang and hence there must be a limit on how far we can see
How far can a photon travel during the lifetime of the universe?
-using the FRW metric with the origin at the point of emission, along the photon’s world-line, we can write:
c dt = +a(t) dχ/√[1-ϰχ²]
-with + since now dχ>0 for dt>0
-integrate from t=0 to arbitrary time t
- we know that the distance from the origin at this time is:
rh(t) = a(t) ∫dχ/√[1-ϰχ²]
-so
rh(t) = c a(t) ∫ du/a(u)
-where the integral is from 0 to t
-sub in a(t) for a particular model to get the cosmological horizon for that model
Causality Paradox of Friedmann’s Cosmology
-the existence of the cosmological horizon poses the causality paradox of Friedmann’s cosmology, how can the universe be uniform if it consists of causally disconnected parts??
Standard Bar Method
Outline
-consider a bar of length L a distance r from observer where r»L
-suppose this bar is perpendicular to the line of sight of the observer, the angular size α, with α«1, of the bar is the angle between the geodesics connecting the observer with the end points of the bar
-in Euclidian geometry we have:
l = rα, r=l/α
-where l is the arc length
Standard Bar Method
Arc Length, l
-start with the FRW spacetime metric
-choose a coordinate system such that its origin is at the observer and the arc is aligned with a θ coordinate line, we then have dr=0 and dφ=0
-then we have, for small angular size Δθ
l = a sin(r/a) Δθ, κ=+1
rΔθ, κ=0
a sinh(r/a) Δθ, κ=-1
