Transcript Lecture 21 Redshifts - Models of the Universe

Lecture 21
Cosmological Models
ASTR 340
Fall 2006
Dennis Papadopoulos
Spectral Lines - Doppler
1 z 
em
obs
obs  em
z
em
Doppler Examples
Doppler Examples
Expansion Redshifts
= Rnow/Rthen
z=2 three times, z=10,
eleven times
Expansion Redshifts
Expansion - Example
Current Record Redshift
Hubbleology
• Hubble length DH=c/H ,
• Hubble sphere: Volume enclosed in Hubble
sphere estimates the volume of the Universe
that can be in our light-cone; it is the limit of the
observable Universe. Everything that could have
affected us
• Every point has its own Hubble sphere
• Look-back time: Time required for light to travel
from emission to observation
Gravitational Redshift
Interpretation of Hubble law in
terms of relativity
• New way to look at redshifts observed by Hubble
• Redshift is not due to velocity of galaxies
– Galaxies are (approximately) stationary in space…
– Galaxies get further apart because the space between
them is physically expanding!
– The expansion of space, as R(t) in the metric equation, also
affects the wavelength of light… as space expands, the
wavelength expands and so there is a redshift.
• So, cosmological redshift is due to cosmological
expansion of wavelength of light, not the regular
Doppler shift from local motions.
Relation between z and R(t)
• Using our relativistic interpretation of
cosmic redshifts, we write
Rpresent 
obs  
em
Remitted 
• Redshift of a galaxy is defined by

z
obs  em
em
• So, we have…
Rpresent
Rpresent  Remitted R
z
1 

Remitted
Remitted
R
Hubble Law for “nearby” (z<0.1) objects
• Thus
cR
(R/t)
cz 
 ct 
 dlighttravel  H
R
R
where Hubble’s constant is defined by

H
1 R 1 dR

R t R dt
• But also, for comoving coordinates of two
galaxies differing by space-time interval
d=R(t)Dcomoving , have
v= Dcomoving  R/t=(d/R)(R/t)
• Hence v= d H for two galaxies with fixed
comoving separation
Peculiar velocities
• Of course, galaxies are not precisely at fixed
comoving locations in space
• They have local random motions, called
“peculiar velocities”
– e.g. motions of galaxies in “local group”
• This is the reason that observational Hubble
law is not exact straight line but has scatter
• Since random velocities do not overall
increase with comoving separation, but
cosmological redshift does, it is necessary to
measure fairly distant galaxies to determine
the Hubble constant accurately
Distance determinations
further away
• In modern times, Cepheids in the Virgo galaxy
cluster have been measured with Hubble Space
Telescope (16 Mpc away…)
Virgo cluster
Tully-Fisher relation
• Tully-Fisher relationship (spiral galaxies)
– Correlation between
• width of particular emission line of hydrogen,
• Intrinsic luminosity of galaxy
– So, you can measure distance by…
• Measuring width of line in spectrum
• Using TF relationship to work out intrinsic luminosity of galaxy
• Compare with observed brightness to determine distance
– Works out to about 200Mpc (then hydrogen line becomes too
hard to measure)
Hubble time
• Once the Hubble parameter has been determined accurately, it
gives very useful information about age and size of the
expanding Universe…
• Recall Hubble parameter is ratio of rate of change of size of
Universe to size of Universe:
1 R 1 dR
H
R t

R dt
• If Universe were expanding at a constant rate, we would have
R/t=constant and R(t) =t(R/t) ; then would have
H=
(R/t)/R=1/t
• ie tH=1/H would be age of Universe since Big Bang

R(t)
t
Modeling the Universe
Chapter 11
BASIC COSMOLOGICAL
ASSUMPTIONS
• Germany 1915:
–
–
–
–
Einstein just completed theory of GR
Explains anomalous orbit of Mercury perfectly
Schwarzschild is working on black holes etc.
Einstein turns his attention to modeling the
universe as a whole…
• How to proceed… it’s a horribly complex
problem
How to make progress…
• Proceed by ignoring details…
– Imagine that all matter in universe is “smoothed” out
– i.e., ignore details like stars and galaxies, but deal
with a smooth distribution of matter
• Then make the following assumptions
– Universe is homogeneous – every place in the
universe has the same conditions as every other
place, on average.
– Universe is isotropic – there is no preferred direction
in the universe, on average.
• There is clearly large-scale structure
– Filaments, clumps
– Voids and bubbles
• But, homogeneous on very large-scales.
• So, we have the…
• The Generalized Copernican Principle… there
are no special points in space within the
Universe. The Universe has no center!
• These ideas are collectively called the
Cosmological Principles.
Key Assumptions
Riddles of Conventional Thinking
Stability
GR vs. Newtonian
Newtonian Universe
Expanding Sphere
Fates of Expanding Universe
Spherical Universe
Friedman Universes
Einstein’s Greatest Blunder
THE DYNAMICS OF THE
UNIVERSE – EINSTEIN’S MODEL
• Einstein’s equations of GR
8G
G 4 T
c
“G” describes the spacetime curvature (including its
dependence with time) of
Universe… here’s where we
plug in the RW geometries.
“T” describes the matter
content of the Universe.
Here’s where we tell the
equations that the Universe is
homogeneous and isotropic.
• Einstein plugged the three homogeneous/isotropic cases
of the FRW metric formula into his equations of GR to
see what would happen…
• Einstein found…
– That, for a static universe (R(t)=constant), only the spherical
case worked as a solution to his equations
– If the sphere started off static, it would rapidly start collapsing
(since gravity attracts)
– The only way to prevent collapse was for the universe to start off
expanding… there would then be a phase of expansion
followed by a phase of collapse
• So… Einstein could have used this to predict that the
universe must be either expanding or contracting!
• … but this was before Hubble discovered expanding
universe (more soon!)– everybody thought that
universe was static (neither expanding nor
contracting).
• So instead, Einstein modified his GR equations!
– Essentially added a repulsive component of gravity
– New term called “Cosmological Constant”
– Could make his spherical universe remain static
– BUT, it was unstable… a fine balance of opposing
forces. Slightest push could make it expand
violently or collapse horribly.
• Soon after, Hubble discovered that the
universe was expanding!
• Einstein called the Cosmological Constant
“Greatest Blunder of My Life!
• ….but very recent work suggests that he
may have been right (more later!)
Sum up Newtonian Universe