Post on 20-Feb-2022
PHYS 2961 Lecture 05 2
What is Cosmology
● What is the content of the universe:● Today?● Long ago?● In the far future?
● How did the universe begin?
● How did the universe evolve?● What does that imply?
● What is the ultimate fate of the universe?
● Cosmology is the study of the universe as a whole
● It asks the “biggest” questions in nature
● It turns out, we can answer all of these questions to a large extent
● Physics principles drive theoretical models● Measurements give us the answers
PHYS 2961 Lecture 05 3
Redshift
Doppler effect:● Apparent change in wavelength due to relative motion
Recall the doppler redshift of light from special relativity:
But recall, with c = 1
Defining the redshift as
● Both Redshift and Blueshift are allowed● Depends on whether relative motion is moving together or apart● In cosmology (unlike astronomy), we only have to deal with redshift
PHYS 2961 Lecture 05 4
Hubble's Law:The Expanding Universe
● In 1929, Hubble measured the redshift of spectral lines in distant galaxies
● Discovered correlation between redshift and brightness
● Since brightness is related to distance, he derived a relation between velocity and distance
● H0 came from fit to data
Main Result:● Everything is moving apart
Play the movie in reverse:● Everything began at the same point● Big Bang!
PHYS 2961 Lecture 05 5
Validity of Hubble's Law
● Valid for small redshift
● Small redshift = close to Earth● At these small distances, the velocity – distance relationship is the
dominant effect
● Beyond this, other effects become important● Gravitational effects● Time dependence of H0
● But the correlation remains valid● The farther we look, the faster objects are moving away from us
Ignoring all other effects, we can use Hubble's law to estimate the age of the universe
● This is surprisingly a very good estimate● Hubble's Law must be a good starting point● Other cosmological effects must be perturbations
PHYS 2961 Lecture 05 6
Parameterizing the Expansion
● If objects are moving apart, the distance between two objects, D, changes with time
● We can parameterize this distance with two terms, one of which has the time dependence
● r is the comoving distance● Same value at all points in time● Independent of expansion
● R(t) is called the scale parameter● Accounts for all of the time dependence● Parameterizes the expansion of the universe● Using the definition of redshift, we can write the
scale parameter in terms of redshift
t = 0 is present
Note many texts introduce the parameter
Hubble's law gives
Which can be written
We will use R in this course
PHYS 2961 Lecture 05 7
The Cosmological Standard Model
● Developed by Friedmann, Lemaitre, Robertson, and Walker (FLRW)
● Consider some cosmologically large length scale● At this scale, the universe is isotropic and
homogeneous● Average over galaxy clusters and intergalactic space
● Using this for a matter distribution (FLRW metric) in the Einstein Field Equations from General Relativity
● Friedmann equation
● k describes the curvature of spacetime● There are 3 allowed values for k: +1, 0, -1
PHYS 2961 Lecture 05 8
Meaning of the Friedmann Equation
● Left hand term describes kinetic energy● v2 term, like ½ mv2
● First term on right describes classical gravitation● Total mass-energy density ρ
● Right-most term describes spacetime curvature● e.g. light bending in gravitational field● Changes dynamics of motion, straight lines → geodesics
PHYS 2961 Lecture 05 9
Example of Friedmann Equation
Consider a gaussian sphere with total mass M and a test mass m
becomes
Kinetic – Potential = constant
k=-1, negative curvatureKinetic > PotentialUniverse expands forever
k=+1, positive curvatureKinetic < PotentialUniverse collapses on itself
k=0, flat curvatureKinetic = PotentialExpansion asymptotically approaches zeroSee HW02 problems 3 and 4
PHYS 2961 Lecture 05 10
Curvature
Curvature can be understood by the analogy of lines on different 2D surfacesConsider a trianglePositive curvature makes the edges bulge outNegative curvature bends them inFlat curvature leaves it unchanged (Euclidean geometry)
PHYS 2961 Lecture 05 11
Open, Closed, Flat Universe
● The shape of spacetime can describe the fate of a universe dominated by matter and curvature
● Three simple cases for the fate● Open● Closed● Flat
● Measurements of our universe fine k=0● We live in a flat universe● However, we recently discovered that
the expansion of the universe is accelerating
● Dark Energy (2011 Nobel Prize)● This breaks this simple picture, and
changes the fate of the universe
PHYS 2961 Lecture 05 12
Conservation of Energy
In Cosmology, the first law of thermodynamics can be written
For an energy density ρ
→
PHYS 2961 Lecture 05 13
Equation of State
The pressure and energy density are related by an equation of state
● This is a general relationship● Different systems of energy have different equations of state
● Matter, photons, vacuum, etc
● This gives a relationship of the energy density to the scale factor● We can use this equation of state to describe each contribution to
the energy density of the universe in terms of R
PHYS 2961 Lecture 05 14
Sources of Energy Density
Most of cosmology can be described by four soucres of energy● Matter
● Massive fermions● Radiation
● Massless or ultra-relativistic particles● Photons, neutrinos
● Vacuum● Energy contained in the vacuum
● Curvature● An effective energy density that describes the curvature of spacetime● Here we will only treat the first three● Sufficient for the important results discussed in this course
Total Density Matter Radiation Vacuum
PHYS 2961 Lecture 05 15
Radiation
For radiation, there is a simple way to understand the energy density
Recall Einstein's relation
The energy of each photon scales with R-1
The photon density is the number of photons per unit volumeScales with R-3
The energy density scales with R-4
For photons, the equation of state is
So w = 1/3Gives the same result
PHYS 2961 Lecture 05 16
Radiation Dominated Universe
If the dominant contribution to the energy density is radiation, the Friedmann equation can be written
This gives the time evolution of the expansion of a radiation dominated universe
● So many high energy photons that we can ignore everything else!
● Only photon energy density is important
PHYS 2961 Lecture 05 17
What Happens as Radiation Universe Expands
Consequence of energy density
● As the universe expands, the density drops● One factor from volume change, R-3● One factor from “stretching out photons”
● Wavelength gets longer, energy drops● Gravitational redshift● Photons redshift as the universe expands
PHYS 2961 Lecture 05 18
Matter
For matter
For non relativistic matter, v << 1A Taylor's series expansion about small v gives w = 2/3
This is expectedEnergy density is the mass per unit volumeAs the universe expands, the volume grows with R3
But total mass is fixed
Note that the density still decreasesBut more slowly than for radiation
PHYS 2961 Lecture 05 19
Matter Dominated Universe
In the case where the dominant energy in the universe is matterThe Friedmann equation is
The expansion grows more quickly for a matter dominated universe than for a radiation dominated universeDue to the fact that as it expands, the energy density drops more slowlySince it's the energy density that drives the expansion, this is why the expansion is different
PHYS 2961 Lecture 05 20
Vacuum Energy
Crazy observation in quantum physicsThe vacuum contains energyExample: Casimir EffectTwo plates with only vacuum between them are attractedConsider the harmonic oscillator potentialEnergy levels:
For the vacuum, n = 0But E is not zero!This introduces the concept of vacuum energyUsing this, one can compute the casimir forceThis has been experimentally measured!
PHYS 2961 Lecture 05 21
Vacuum Energy Density
There's a lot of vacuum in the universeSo this tiny effect adds up to a huge contribution
Vacuum energy density should be constantNo matter how the universe expands, the vacuum is unchanged
There is always the same amount of vacuum in a given volume
From this, we conclude that
For this to happen, we need an equation of state with w = -1
This corresponds to a negative pressure
The vacuum exerts a negative pressure, driving expansion of spaceAs space expands, it creates more vacuumPositive feedback
PHYS 2961 Lecture 05 22
Vacuum Dominated Universe
● Consider a universe dominated by vacuum● This is the simplest case to solve
● The Friedmann equation is
● Here we see the positive feedback: Exponential growth● A vacuum energy dominated universe expands
exponentially forever
PHYS 2961 Lecture 05 23
Modeling the Real Universe
In general, the time evolution of the universe must be computed numericallyNo closed form solution for the case of multiple components simultaneouslyHowever, we can draw some conclusions based on these simple examples:
History of the universe:● Early universe
● Radiation dominated● Many, many photons, high energy density● This density drops the fastest (due to redshift)● Eventually too little radiation energy density
● Middle universe● Matter dominated● 47 kyr – 10 Gyr● After photons have cooled off, matter is the next dominant energy● This density also drops, but slower than radiation● Eventually too little matter energy density
● Late universe● Vacuum energy dominated● Dark Energy!● After all other energy density is too low, exponential growth● Ultimate fate of the universe
PHYS 2961 Lecture 05 24
Crude Description of Cosmological Evolution
● Treating each component as the “dominant” energy
● Derive time evolution for each epoch
● The real solution requires modeling transitions
● What happens when matter and vacuum energy are roughly equal (like today)
● We can only solve the Friedmann equation numerically for this