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
The Observed Angular Size
- in an expanding universe, the observed angular size α, will be different from the real (or actual) angular size at the time of observation, α~
- they are equal at the time of emission of the photons received during observations
Arc Length in terms of Observed Angular Size
l = a(te) sin(r(te)/a(te)) θob, κ=+1
r(te) θob, κ=0
a(te) sinh(r(te)/a(te)) θob, κ=-1
The Standard Bar Method
Friedmann’s Spatially Flat Universe With Dust
θob = l / r(te) -we can write θob in terms of z -this equation can then be used to check if the flat Friedmann's model with dust fits our universe => α = lHo/c 1/z, z<<1 lHo/2c * z, z>>1
Angular Size Distance
Definition
rang = l / θob
- this would be a real distance in a Eulidean space
- in our cosmological models, it reasonably approximates the spacetime distance only for very close sources, that is for z«1
Standard Bar Method
Second Derivation
-using the equation for r(te) in terms of z
θob(z) = l/r(te)
-where both θob and z are observable parameters
What can we use as a standard bar?
- the role of a ‘standard bar’ can be played by galaxies or clusters of galaxies but the fluctuations of the CMB have been the most useful so far
- they are located at huge distances corresponding to z~10^3
- and their predicted angular scale is very sensitive to the parameters of our cosmological models
CMB Data
- the best fit to CMB observations is given by models with Ωo~1
- which hints our universe may be flat
- since estimates based on the mass of visible matter give a much smaller critical parameter Ωo,vis=0.02h^(-2), this tells us that in addition to visible matter and radiation there must be some invisible matter components in the universe which account for most of its mass
The Standard Candle Method
Source Luminosity
-consider a source of electromagnetic radiation
-denote dEe as the amount of energy emitted by the source in time dte as measured in the source frame
-introduce the source luminosity:
L = dEe/dte
-this is not a directly observable parameter
The Standard Candle Method
Source Brightness
-we can measure directly the source birghtness or energy flux density, S
S = dEr~/dtrdA
-where dA is the surface element at the observers location and normal to the direction to the source
-dEr~ is the amount of energy emitted by the source which crosses surface element dA in time dtr
Standard Candle Method
What is the connection between S at the time of observation and L at the time of emission of the observed radiation?
-consider a sphere centred over the position of the light source with radius r equal to the distance to the observer
-denote dEr, the energy flowing across the sphere in time dtr
-when the source emission is isotropic, S is constant over the sphere and hence we can write:
dEr = A dEr~/dA = S A dtr
-in a transparent non-expanding universe, the energy dEr=dEe would cross the sphere during the time dtr=dte hence:
L=SA
Standard Candle Method
Euclidian vs Non-Euclidian Geometry
L=SA
-in a universe with Euclidian geometry, we would also have A=4πr²
=>
r = √[l/4πS]
-this shows how to deduce the distance to a source with known luminosity L based on the observed brightness S in a Euclidian universe
-in a universe with non-Euclidian geometry the distance given above is not the same as the actual distance between the observer and the radiation source
-we describe r as the luminosity distance
The Standard Candle Method in Modern Cosmology
Outline
- in an expanding universe with non-Euclidian geometry, three new features emerge:
i) energy redshift
ii) photons emitted during time interval dte at the source are received during a time dtr≠dte
iii) the area A is not given by the Euclidian formula
The Standard Candle Method in Modern Cosmology
i) energy redshift
-the energy of photons decreases as they travel across the universe
-a photon of wavelength λ has an energy that is inversely proportional to its wavelength:
Er = Ee [1+z]^(-1)
-where Er is the energy of the photon as measured by the observer and Ee is the energy of the photon as measured by the source
-due to the cosmological redshift, when the photons emitted during the time dte cross the sphere of radius r they will not be carrying energy dEe but only:
dEr = [1+z]^(-1) dEe
The Standard Candle Method in Modern Cosmology
ii) photons emitted during time interval dte at the source are received during a time dtr≠dte
-we have shown that dtr = dte [a(to)/a(te)] = dte (1+z) -if we combine these results: L = dEe/dte = (1+z)²dEr/dtr = (1+z)²SA OR S = 1/A[1+z]² * L
The Standard Candle Method in Modern Cosmology
iii) the area A is not given by the Euclidian formula
-we can get A(z) from r(z) for each FRW cosmological model
-for each model,
S = 1/A[1+z]² * L
predicts S(z) for sources with a given luminosity L
-this opens the possibility to test these models observationally
