Cosmology Part I
What is Cosmology Cosmology is the study of the universe as a whole It asks the biggest questions in nature 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? It turns out, we can answer all of these questions to a large extent Physics principles drive theoretical models Measurements give us the answers 2
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 3
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! 4
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 5
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 Hubble's law gives 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 Which can be written We will use R in this course 6
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 7
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 8
Example of Friedmann Equation Consider a gaussian sphere with total mass M and a test mass m k=-1, negative curvature Kinetic > Potential Universe expands forever becomes k=+1, positive curvature Kinetic < Potential Universe collapses on itself Kinetic Potential = constant k=0, flat curvature Kinetic = Potential Expansion asymptotically approaches zero See HW02 problems 3 and 4 9
Curvature Curvature can be understood by the analogy of lines on different 2D surfaces Consider a triangle Positive curvature makes the edges bulge out Negative curvature bends them in Flat curvature leaves it unchanged (Euclidean geometry) 10
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 11
Conservation of Energy In Cosmology, the first law of thermodynamics can be written For an energy density ρ 12
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 13
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 14
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 volume Scales with R-3 The energy density scales with R-4 For photons, the equation of state is So w = 1/3 Gives the same result 15
Radiation Dominated Universe So many high energy photons that we can ignore everything else! Only photon energy density is important 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 16
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 17
Matter For matter For non relativistic matter, v << 1 A Taylor's series expansion about small v gives w = 2/3 This is expected Energy density is the mass per unit volume As the universe expands, the volume grows with R3 But total mass is fixed Note that the density still decreases But more slowly than for radiation 18
Matter Dominated Universe In the case where the dominant energy in the universe is matter The Friedmann equation is The expansion grows more quickly for a matter dominated universe than for a radiation dominated universe Due to the fact that as it expands, the energy density drops more slowly Since it's the energy density that drives the expansion, this is why the expansion is different 19
Vacuum Energy Crazy observation in quantum physics The vacuum contains energy Example: Casimir Effect Two plates with only vacuum between them are attracted Consider the harmonic oscillator potential Energy levels: For the vacuum, n = 0 But E is not zero! This introduces the concept of vacuum energy Using this, one can compute the casimir force This has been experimentally measured! 20
Vacuum Energy Density There's a lot of vacuum in the universe So this tiny effect adds up to a huge contribution Vacuum energy density should be constant No 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 space As space expands, it creates more vacuum Positive feedback 21
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 22
Modeling the Real Universe In general, the time evolution of the universe must be computed numerically No closed form solution for the case of multiple components simultaneously However, 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 23
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 24