Lecture 21Cosmological Models

ASTR 340Fall 2006

Dennis Papadopoulos

Spectral Lines - Doppler

obs em

em

z

1 em

obs

z

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

• Redshift of a galaxy is defined by

• So, we have…

em

emobsz

obs RpresentRemitted

em

zRpresentRemitted

1Rpresent Remitted

Remitted

RR

Hubble Law for “nearby” (z<0.1) objects

• Thus

where Hubble’s constant is defined by

• 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

H 1R

Rt

1RdRdt

cz cRR

ct(R /t)R

dlight travel H

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:

• 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

H 1R

Rt

1RdRdt

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

TG 4

8cG

“G” describes the space-time 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