Lecture 2 - ICCUBicc.ub.edu/~liciaverde/TALKS/cosmo2.pdf · Dynamical Estimation of Galaxy Masses...
Transcript of Lecture 2 - ICCUBicc.ub.edu/~liciaverde/TALKS/cosmo2.pdf · Dynamical Estimation of Galaxy Masses...
Lecture 2
Friedmann Equations The content of the Universe (Cosmological parameters; intro) Distances
Licia Verde Introduction to Cosmology
http://icc.ub.edu/~liciaverde/cernlectures.html
• In GR space tells mass how to move, mass tells space how to curve.
• Suspicion: a(t) related to content of Universe? • Really need GR but we do Newton…
Friedmann equations (1)
integrate kinetic potential
Substitute and divide each side by
Symmetric under time reversal
Start with an expanding sphere: How does it evolve of U>0 ? How does it evolve of U<0 ?
Need GR to go further
Birkhoff theorem
Friedmann equations 1(continued)
U was related to spatial curvature
Total matter-energy density: ε/c2
Make the Universe flat Today (and at any time using (t) for 0)
Define: Density parameter, gives curvature
Mass-energy gravity
curvature
ρ
Friedmann equations 2 One eq.; 2 unknowns a(t) , ρ(t), need a relation between the two
Expanding universe, adiabatic
Consider a spherical chunk of Universe as in the sphere of the example before
Fluid equation
Friedman and fluid equations are ENERGY CONSERVATION
Friedmann Equations 3 Let’s combine the two to get a useful eqn
Multiply Friedmann by a2 derive wrt t Divide by
Use fluid eq.
Acceleration equation!
If P=0 and ε>0 … So if you start with an expanding Universe….
And what would it take to make it accelerate?
If the Universe started off expanding stuff should slow down the expansion!
.
Friedmann equations NOT independent!
If you want to solve for
need more info…
Interesting cases:
Non-relativistic matter
Radiation
Cosmological constant
Accelerating fluid This is weird…… but looks like we are stuck with it
Einstein and the cosmological constant Poisson equation
If static
If static Empty… humm….. or
Does not get diluted…
Vacuum energy…
Einstein….
I told you it was weird….
The universe composition Matter; radiation; curvature;etc…
One for each component
Matter… Radiation… (redshift is back) Cosmological constant…
Use Friedmann equation
You can solve it!
Exercise: If it was only one component Ansatz: If
The farther object you can see…. Horizon! (we’ll get back to this later)
Exercise: If it was only Λ
Grows exponentially and c/H<<dp!
Composition of the Universe
Radiation…
Content: Dark matter
There is more than meets the eye In the solar system sun + planets
Mass-to-light ratio Let’s consider galaxies
optical light Gas (H,21cm)
rotating
Dynamical Estimation of Galaxy Masses From Kepler’s and Newton’s laws, the rotation speed of an atom of gas (mass mgas) around a galaxy (mass M) is given by
(M + mgas) P2 = M P2 = a3
where P and a are the period and semi-major axis of the orbit, and the atom of gas is much less massive than the galaxy.
Putting this all together then yields
v = 2 π r / P
If the atom’s orbit is circular, then a is the radius of the circle (r), and, according to Kepler’s 2nd law, the atom’s speed is constant. So
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v = 2π M ⋅1r
or M ∝ v2 r
Dynamical Estimation of Galaxy Masses According to Newton’s laws, once outside a galaxy, the rotation velocity of gas should decrease with distance. But that’s not observed!
There must be a lot of mass in the outer regions of galaxies that we are not observing!
DARK MATTER
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v = 2π M ⋅1r
Dynamical Estimation of Galaxy Masses Virtually all spiral galaxies have these “flat” rotation curves. The outer regions of spirals must be dominated by dark matter.
"In a spiral galaxy, the ratio of dark-to-light matter is about a factor of ten. That's probably a good number for the ratio of
our ignorance-to-knowledge."
Mass-to-light ratio ~10
Vera Rubin
Mass measurements in galaxy clusters For a group of objects, there always must be balance between gravity and velocity. Too little velocity, and gravity takes over, making the cluster smaller. Too much velocity, and the objects escape the group’s gravity, causing the group to evaporate.
By measuring the Doppler shifts of galaxies in a cluster, you can measure the cluster’s gravity. And from gravity, you get mass.
Virial theorem
Coma cluster
Kinetic energy =-1/2 potential energy
whoa
Mass-to-light ratio ~100 to 300
The Mass of Clusters As Fritz Zwicky noticed in 1933, galaxies in clusters move much too fast for the amount of matter we see. The clusters must contain a lot of unseen matter providing extra gravity.
Once again – dark matter
& X-rays
Mass bends space-time
Global geometry local geometry
Gravitational lensing
A light ray can be deflected by gravity. The greater the gravity, the greater the bending. As a result, a collection of matter can act as a gravitational lens.
Gravitational Lenses In the case of a background point source, the result might be multiple images of the object. In the case of a larger source (i.e., a galaxy), the result can be arcs and arclets.
In this image, the blue areas show background galaxies, while the red represents foreground mass.
Gravitational Lenses The greater the mass, the greater the gravity, and the greater the gravitational lens effect. A massive object can cause a large deflection in the light path. (It also greatly amplifies the light.)
Example of a Gravitational Lens
Gravitational lensing
b
HA!
Gravitational Mass Measurements By estimating the distance to the lens and the object, one can measure the deflection angle of the light. This angle allows you to estimate mass.
When astronomers do this, they find a lot more matter than is visible in the galaxy or the cluster. Galaxies and clusters contain dark matter.
Abell 2218
Reconstructed dark matter distribution of cluster Abell 2218
MACHO project
Planets searches
So what is the dark matter?
• Nucleosynthesis • Theories of particle physics • Structure formation
Must be something unknown to Earth Weakly interacting!
For the particle physicists among you: is DM the only direct indication we have of physics beyond the standard model?
New-er evidence Bullet cluster
REAL DATA!
Computer-SIMULATION
DIY dark matter Underground detectors
Underground astronomers
DIY dark matter Underground detectors
Underground astronomers
Computer simulation of dark matter distribution
Radiation
Back to Friedmann
If you have a mix of components, chances are that at different times in the life of the Universe different components dominate
MATTER + CURVATURE ONLY
closed
flat
open
Density is destiny!!
A ZOO OF POSSIBILITIES…..
lookback time
Evolution….
This is a BIG puzzle
Distances, for completeness
Luminosity distance: In an expanding universe, distant galaxies are much dimmer than you would normally expect because the photons of light become stretched and spread out over a wide area.
Angular diameter distance: In an expanding universe, we see distant galaxies when they were much younger and much closer to us
Comoving Distance is the opposite of the Angular Diameter Distance - it tells us where galaxies are now rather than where they were when they emitted the light that we now see
The Light Travel Time Distance represents the time taken for the light from distant galaxies to reach us
For small distances all four distance scales converge and become the same
Distances in useful format
Comoving, line-of-sight distance
Comoving transverse distance
Distances, useful format
Angular diameter distance
Warning: not additive!
http://xxx.lanl.gov/pdf/astro-ph/9905116v4 http://www.icosmo.org/
Distances, useful format
Luminosity distance
Comoving Volume
Parameters that govern the global geometry of space-time
Parameters that govern the expansion rate
Parameters that characterize inhomogeneities
{The smooth Universe
Inhomogeneous Universe {
Derived parameters Parameterizing our ignorance { Beyond the minimal model: “extra” parameters
Different type of parameters
GR geometry, fate of Universe, content (but not much details)
Statistically speaking: clustering, galaxies etc…
test: which is which?
The standard cosmological model: parameters
The smooth Universe: content
Gives the global geometry/curvature
Key concepts today
• The content of the universe (amount and type of stuff) govern its geometry and expansion history
• Friedmann Equations (with these you go everywhere!)
• Content of the Universe and evidences for dark matter
• Distances & Useful formulae