Download Lecture 20, PPT version

Outline - April 8, 2010
• Cosmological Principle - what is it, when is it valid?
• Space and Time as physical quantities
• Meaningless Cosmological Questions
• More implications of Hubble’s Law
• Using Hubble’s Law to estimate the age of the universe
• Big Bang predictions - motivation for the remaining chapters in
the book
The Cosmological Principle
On a large enough scale, the universe is
both isotropic and homogeneous
ISOTROPY: There is no preferred direction in space. (All
directions are alike.)
HOMOGENEITY: One randomly-chosen large volume of
the universe will have the same physical properties (and
identical physical laws) as another randomly-chosen
large volume of the universe. (All places are alike.)
2-Dimensional Examples of Isotropy and Homogeneity
• Surface of plain white“cue” ball used for playing pool (billiards)
• Infinite forest of identical trees
Warning: at some level all analogies fail…
Isotropic Forest
Anisotropic Forest (trails = “preferred direction”)
Four 2-Dimensional Universes
• Which are isotropic on a large scale (same in all directions)?
• Which are anisotropic on a large scale (have preferred
directions)?
• Which are homogeneous on a large scale (all places are
alike)?
• Which are inhomogeneous on a large scale?
• Which satisfy the Cosmological Principle?
Obviously isotropic and homogeneous
Satisfies the Cosmological Principle
Homogeneous on a large scale (the pattern repeats
over and over), but is obviously anisotropic
Does not satisfy Cosmological Principle
Homogeneous on a large scale, but is
actually anisotropic (look at the diagonals!)
Does not satisfy the Cosmological Principle
On a large scale, this is both homogeneous and isotropic
Satisfies the Cosmological Principle
This is actually a computer simulation of how galaxies may have formed in
the universe after the Big Bang. It matches the real, observed universe
remarkably well.
Yellow = lots of galaxies present, black = very few galaxies present.
How Large is “Large Enough”?
Answer: 1.5 billion light years (or more!!)
You need to cut out a cube of the universe that’s 1.5 billion ly X 1.5
billion ly x 1.5 billion ly to guarantee the same number of galaxies and
same number of galaxy clusters that you would find in a cube of the
same size, but very far away from the original cube.
Cosmological Principle, II
Is the cosmological principle nothing more than a convenient
assumption without which we would be unable to make
progress in studying the entire universe?
NO!! (it’s a testable hypothesis)
1. Maps of the locations of galaxies in the universe (same average
number of galaxies, etc. as long as length scale is at least 1.5
billion light years)
2. Cosmic Microwave Background Radiation (CMBR) will turn out to
be a strong proof of isotropy of universe
Unsettling Consequence of Cosmological Principle
There can be NO CENTER to the universe!
Where is the “center” of the SURFACE of an unmarked sphere?
Working Definition for “The Universe”
The universe is that which contains and subsumes1 all of the laws
of nature and everything subject to those laws.
(John Hawley & Katharine Holcomb, “Foundations of Modern Cosmology”)
1subsume:
to classify within a larger category or under a general principle
The universe contains all that is physical,
including space and time.
Space & Time as Physical Quantities
Recall from General Relativity:
Time is affected by gravity (clock on a high-gravity planet runs
slower than clock high up in the atmosphere)
Space is affected by gravity (“warped” by massive objects, causes
light rays to bend around on curved paths, causes planets to orbit
about sun on “natural” curves)
Anything that is affected by the physical laws
must, itself, be physical!
Meaningless Question
What is the color of the sound of a trumpet?
Meaningless Cosmology Questions
• What happened before the Big Bang?
• What exists outside the universe?
These questions assume attributes (“before”, “outside”) which posit
space and time as being distinct from the universe. This is not
acceptable in the context of “physical cosmology”.
Time did not exist before the universe
and space does not exist outside it!
Hubble’s Law Revisited
v = H0 d
This is a text
box
H0 is called “Hubble’s constant”
H0 = 71 km/s/Mpc or H0 = 22 km/s/Mly (these are really weird units)
Hubble’s Law Revisited
Hubble’s law does not say that “the universe is expanding into space”.
What Hubble’s Law does say is that space itself is expanding.
Implications of Hubble’s Law / expansion of space:
1. The universe was smaller and denser in the past
2. The universe was hotter in the past
3. The universe had a specific beginning
How old should the universe be?
Run the “Hubble Movie” backwards:
• some time in the past all galaxies would have been on top of each other!
• if universe has been expanding at constant rate for all time, then all galaxies
would have been on top of each other at time equal to 1/H0
Distance between any two galaxy clusters at the present day:
distance = speed x time (the standard formula)
speed = H0 x distance (Hubble’s Law, specifically)
Substitute Hubble’s Law into distance formula:
distance = H0 x distance x time
Cancel factors of distance and rearrange:
time = 1/H0
“Hubble Age” = 1/H0
The time on the previous slide is the time it would have taken the
galaxy clusters to reach their current separation, having started
from being on top of each other, and is an estimate of the age of
the universe known as the Hubble Age.
Current best measurements of H0 give a value of 71 km/s/Mpc, which
translates into a Hubble Age of about 14 billion years, which agrees
reasonably well with the estimated ages of the oldest stars in the
universe.
The Cosmological Redshift
A subtle point:
• the cosmological redshift is not a simple Doppler shift
• Doppler shifts result from sources of waves moving through space;
cosmological redshift happens because distances between sources of
waves (i.e., galaxies emitting light) increase due to space itself
getting bigger
• as photons travel through the universe they are STRETCHED as the
fabric of spacetime expands
Stretched Photon (Balloon Analogy)
This has a profound consequence for the relative importance of
mass and light for the total “energy budget” of the universe.
Density
When we talk about “density” what we usually mean is the “mass density”:
mass density = (amount of mass) / volume
We can equally define an “energy density” which is just
energy density = (amount of energy) / volume
Note: the “energy” could be the energy of light (E = hc /)
or the energy of matter (E = mc2)
Affect of Expansion on Energy Density of Mass
Consider a 1-meter cube of space that contains 1 kg of hydrogen gas.
The mass-energy density of the hydrogen gas is
D = mc2 / volume = (1 kg)x(3x108 m/s)2 / (1 m)3 = 9x1016 J/m3
Now let the cube expand so that its sides are 2 meters long, and don’t
let any of the gas escape from the cube. The mass-energy density of the
hydrogen gas in the bigger cube is
D = mc2 / volume = (1 kg)x(3x108 m/s)2 / (2 m)3 = 1.125x1016 J/m3
So, the mass-energy density has decreased by a factor of 8 = 23 (i.e.,
three factors of length).
What about the energy density of light?
What happens to the energy of a photon when you increase the
wavelength?
The photon loses energy!
Consequence: as the universe expands, the density of “radiant energy” (i.e., light)
decreases faster than the density of “mass energy”.
For light, the energy density decreases by FOUR factors of length, not three!!
There are fewer photons per unit volume, but the energy of each individual
photon has also decreased from its original value.
Matter Dominated vs. Radiation Dominated Universe
At the present day, there is roughly 20,000 times more energy density in the form
of mass/matter. We live at a time when the universe is “Matter Dominated”.
In the far distant past there would have been a time when there was more energy
density in the form of light/radiation. This would have been a time when the
universe was “Radiation Dominated”.
The Big Bang Concept
The universe began in an extremely hot,
extremely dense state
A fantastic notion, but also a truly testable hypothesis.
Big Bang Predictions
• The universe had a specific beginning
• The sky should be dark at night
• Initially, the universe was extremely hot, dense, and opaque
• The universe has evolved/changed over time
• The universe is expanding
• Cosmic objects (such as stars) should have a chemical composition
that is roughly 75% hydrogen and 25% helium
What evidence exists?
Darkness at Night (a.k.a Olbers’ Paradox)
Consider a universe that is:
• infinite in extent
• infinite in age
• filled uniformly with identical stars (or galaxies) that have been
shining for all time
Result: the sky should ALWAYS be bright
Suppose the stars extend
to infinite distances
Consider stars at
distance, d. What will
the combined
brightness of all stars
at distance, d, be?
All lines of sight eventually
intersect a star
At a given distance, the brightness
of each individual star decreases
as (1/distance)2 BUT the number of
stars at that distance increases as
(distance)2
Total amount of light coming from a
given distance is the number of
stars at that distance times the
brightness of each star at that
distance, so amount of light you
see coming from any particular
distance is a CONSTANT value
because the factors of distance
cancel each other!!
The sky should NEVER be dark (but we all know it’s dark at night, so at
least one of our starting assumptions was WRONG)!