Flashcards in 6. Distance and Redshift Relations in Cosmology Deck (30)

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1

##
FRW spacetime metric at present time

Metric

###
g = ds² = -c²dtxdt + g(3)

-where:

g(3) = dl²

= ao²{dχxdχ/[1-ϰχ²] + χ²(dθxdθ + sin²θdφxdφ)}

2

##
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]

3

##
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

4

## 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?'

5

##
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

6

##
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)

7

## 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)

8

##
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

9

##
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

10

##
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

11

## 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

12

## 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

13

## 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??

14

##
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

15

##
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

16

## 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

17

## 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

18

##
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

19

##
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

20

##
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

21

## 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

22

## 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

23

##
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

24

##
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

25

##
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

26

##
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

27

##
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

28

##
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

29

##
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

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