Lecture 05 Cosmology - Rensselaer Polytechnic Institute

Lecture 05

Cosmology

Part I

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


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


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!


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


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


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


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


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


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)


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


Conservation of Energy

In Cosmology, the first law of thermodynamics can be written

For an energy density ρ

→


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


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


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


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


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


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


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


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!


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


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


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


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

Lecture 05 Cosmology - Rensselaer Polytechnic Institute

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