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

€

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

€

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