Cosmology & the Big Bang

AY16 Lecture 19, April 10, 2008

Introduction to Cosmology

Basic Principles

Fundamental Observations

The FRW Metric

“My husband adheres to the Big Bang theory of creation.”

COSMOLOGYThe BIG Picture!

The cosmological model dominates much of extragalactic astronomy, in fact much of astronomy & astrophysics, even physics.

What is Cosmology? “The study of the large scale structure & evolution of the Universe”

What is Cosmogony? “The study of the origin of observable structures”

Cosmology is perhaps the oldest real science.

Its tied to our “World View.”

Changing Worldviews Age Universe----------------------------------------------------------------------------------

10,000 years BC --- The Valley you lived in

1,000 years BC --- Your Kingdom

300 years BC --- The Mediterranean

(for Egypto-Greco-Romans, at least!)

100 years AD --- The Earth + Celestial Sphere

400 years ago --- The Solar System

100 years ago --- The Milky Way

75 years ago --- The “Modern” Universe

(2 Billion Light -Years in *radius*)

Today --- An Infinite Universe

(the visible part has a Radius of ~15 Gly)

A Brief History of Extragalactic Astronomy:

~ 1750 Early Rumblings of “Island Universes”

from I. Kant, T. Wright, P. Laplace.

This seems to have been forgotten soon after.

1800’s Catalogs of Things but no understanding.

de la Caille, Messier, Herschel3, Dreyer

William, Caroline & John

~ 1875 The Discovery of the Galaxy --- Kapteyn’s Universe

1910 Removal of the Solar-Centric view

1900-1920 Shapley and the Great Debate

1907 Bohlin --- M31 Parallax

1918 Van Maanen --- M31 Parallax

1885 S Andromeda = SN1885a

large reflectors + photographic plates

1920 The Shapley-Curtis debate

Shapley + Globular Clusters + Cepheids

1924 Hubble & the Hooker --- NGC 6822 Cepheids, eventually M31 Cepheids

1910-1930 Theory! Einstein, Friedmann,

deSitter, Lemaitre, Tolman, Robertson …

1922 Opik’s M31 Mass-to-Light ratio

L = 4πr2 GMm/r = ½ mv2

M = ½ v2r/G,

so M/L = ½

and Opik estimated D of M31 at 450 kpc.

v2 1 1G 4πD

1929 Hubble (+Slipher) Velocity-Distance Law

1930’s Hubble’s Classification Scheme for Galaxies (Tuning Fork Diagram)

N.B. Absolutely necessary but wrong interpretation, set galaxy evolution back 20 years!

Hubble’s Galaxy Counts (Humason)

1937 Zwicky & Smith DARK MATTER

1940’s Galactic Dust, Stellar Populations, Hubble Diagram

1948 Gamow & the Hot Big Bang

1950’s deVaucouleurs’ Local “Supergalaxy”

Rubin: Flows

Dicke: CM

HMS Velocities + Hubble Diagram

Baade & Sandage: H0 revisions

Minkowski: Radio galaxies

1960’s The Hubble Constant Debate

Tinsley: Stellar Evolution Galaxy Evolution

Greenstein & Schmidt: Quasars

Arp: Peculiar Galaxies

Spinrad &Taylor : Population Synthesis

Page: Galaxy Masses

1970’s Stability & Halos

Starbursts

H0!!! q0!!!

First Feeble Redshift Surveys

CMB Dipole

Galaxy Clusters & X-Rays

Gravitational Lenses

Galaxy Formation

1980’s Large-Scale Structure

Large Scale Flows & Cold Dark Matter

Galaxy Counts

H0!!!!

IRAS & Dusty Starbursts

1990’s COBE: 2.723 K + fluctuations

Biased galaxy Formation

Unified AGN Models

Λ!!!!!

Concordance Cosmology

HST and galaxy evolution

2000+ The Cosmic Web

Reionization

First Light

COSMOLOGY is a modern subject:

Today based on Principles & Observables

The basic framework for our current

view of the Universe rests on ideas and discoveries (mostly) from the early 20th century.

Basics:

Einstein’s General Relativity

The Copernican Principle

Fundamental Principles:

• Cosmological Principle: (a.k.a. the Copernican principle). There is no preferred place in space --- the Universe should look the same from anywhere

The Universe is HOMOGENEOUS and ISOTROPIC.

we believe this is true to zeroth order

(i.e. on large scales, yes, on small scales, no)

A variant of the CP is

• The Perfect Cosmological Principle:

The Universe is also the same in time. The STEADY STATE Model (XXX it’s demonstrably wrong)

• The Anthropic Cosmological Principle:

We see the Universe in a preferrred state (time etc.) --- when Humans can exist.

the ACP is almost the opposite of the PCP.

it leads to the Goldilocks Universe:

Not too hot, Not too cold

Not too dense, Not too empty

Not too young, Not too old….

• Relativistic Cosmological Principle: The Laws of Physics are the same everywhere and everywhen.

(!!!) absolutely necessary, often assumed and forgotten. (!!!)

Fundamental Observations:• The Sky is Dark at Night (Olber’s P.)

this implies there must be some limit to the observable Universe.

• The Universe is generally Expanding

It’s not static.

galaxies appear to be moving away from us --- and each other.

Olber’s Paradox

Hubble’s Discovery of Expansion

• The Universe is Homogeneous on large scales --- there exists an almost isotropic microwave background (the CMB) of T~3K

a.k.a. relic radiation

• The Universe is not Empty. It has stuff in it, stuff consistent with a hot origin (the Universe has a temperature), i.e. contents consistent with nuclear physics operating in an initially hot, dense medium

COBE Fluctuations

t/t < 10-5, i.e. much smoother than a baby’s bottom!

Observational Cosmology consists of taking these bases to build a more

detailed picture of the structure and evolution of the Universe.

Sometimes to (1) Feed Theorists

(2) Kill Theories

(3) Explore

generally support Gamow’s hot big bang model

COSMOLOGICAL FRAMEWORK:

The Friedmann-Robertson-Walker

Metric

+

The Cosmic Microwave Background

= THE HOT BIG BANG

The Big Bang

WRONG!

WRONG!!!

WRONG ??

THE TRUTH BEHIND THE BIG BANG THEORY

T

How my wife describes my job! RIGHT!

Mathematical CosmologyThe simplest questions are Geometric.

How is Space measured?

Standard 3-Space Metric:

ds2 = dx2 + dy2 + dz2

= dr2 + r2dθ2 + r2sin2

θ d2

In Cartesian or Spherical coordinates in

Euclidean Space.

Now make our space Non-Static, but

“homogeneous” & “isotropic”

ds2 = R

2(t)(dx

2 + dy2 + dz

2)

And then allow transformation to a more general geometry (i.e. allow non-Euclidean geometry) but keep isotropic and homogeneous:

ds2 = (1+1/4kr2)-2 (dx2+dy2+dz2)R2(t)

where r2 = x2 + y2 + z2, and k is a

measure of space curvature.

Note the Special Relativistic Minkowski Metric

ds2 = c2dt2 – (dx2 +dy2 + dz2)

So, if we take our general metric and add the 4th

(time) dimension, we have:

ds2 = c2dt2 – R2(t)(dx2 +dy2 + dz2)/(1+kr2/4)

or in spherical coordinates and simplifying,

ds2 = c2dt2 – R2(t)[dr

2/(1-kr

2) +

rdsind

which is the (Friedmann)-Robertson-Walker Metric, a.k.a. FRW

• The FRW metric is the most general,

non-static, homogeneous and isotropic

metric. It was derived ~1930 by Robertson and Walker.

R(t), the Scale Factor, is an unspecified function of time (which is usually assumed to be continuous)

and k = 1, 0, or -1 = the Curvature Constant

For k = -1 or 0, space is infiniteinfinite

K = +1

Spherical

c < r

K = -1

Hyperbolic

c > r

K = 0

Flat

c = r

Consider Expansion in an Isotropic Universe

l 0l = l 0

a(t-t 0)

a(t) = Universal Expansion factor

Expansion is self-similar and produces a

change in the frequency of received radiation. If t0 = now, here, observed & t1 =

the time at which light is emitted from a distant object in the scaled universe:

t0

te = t1

te+ΔteR0(t) l

R(t)lt0+Δt0

a(t)δt

1/Δte = υ1 frequency of emitted radiation

= =

so υ1, λ1 = lab or rest frequency/wavelength and

υ0, λ0 = observed frequency/wavelength

and if we define z ≡ redshift =

Then 1 + z = =

υ0 a(t1) λ1

υ1 a(t0) λ0

λ0 – λ1

λ0R(t0) a(t0)

R(t1) a(t1)

For small z we can interpret this “redshift” in terms of a Doppler shift, cz = v, or Doppler velocity. For small Δt, if r0 = 0 (set origin to

us, the observer), we have:

r1 = c (t0 - t1)/R(t0)

d = r1 R(t0) = c(t0 -t1) the distance

and thus

cz = c(t0 – t1) 1 dR(t1)R(t0) dt

so v = cz = H0d

where H0 = ≈

is the definition of Hubble’s Constant.

Note that this is true for small z only. This formula for distance is NOT subject to special relativity. The convention is to quote apparent velocities as v = cz.

dR(t0) 1 dR(t1) 1

dt R(t0) dt R(t1)

The apparent (radial) velocity of any object is made

up of three parts

v = vH + vP + vG

vH = the cosmological stretching of the metric,

a.k.a. the Hubble Flow

vP = the component of the “peculiar” velocity w.r.t.

the Hubble Flow that is an actual space velocity. vP is Doppler, so computations of dynamical properties like cluster velocity dispersions that result from vP do require the (1+v2/c2) –1/2 correction.

vG = the gravitational redshift, usually tiny

Expansion Age & Hubble Distance

The Hubble Constant has units of inverse time; its actually also a measure of the expansion age of the Universe:

τH = H0-1 = 9.78x109 h-1 years = 3.09x1017 h-1 s

where H0 = 100 h km/s/Mpc

And the Hubble Distance is

DH = c/H0 = 3000 h-1 Mpc = 9.26x1025 h-1 m

What about the scale factor R(t)?

R(t) is specified by Physics we can use Newtonian Physics (the

Newtonian approximation) but now General Relativity holds.

Start with Einstein’s (tensor) Field Equations

Gg and

GRgR

Where

is the Stress Energy tensor

R is the Ricci tensor

g is the metric tensor

G is the Einstein tensor

and R is the scalar curvature

RgR = g

is the Einstein Equation

The vector/scalar terms of the Tensor Field

Equations give the linear form Einstein’s Equations:

(dR/dt)2/R

2 + kc

2/R

2 = 8Gc2+c

2/3

energy density CC

2(d2R/dt

2)/R + (dR/dt)

2/R + kc

2/R

2 =

-8GPc3+c2

pressure term

And Friedmann’s Equations:

(dR/dt)2

= 2GM/R + c2R

2/3 – kc

2

kc2 = Ro

2[(8G/3)o – Ho]2

if = 0 (no Cosmological Constant)

or

(dR/dt)2/R2 - 8Go /3 =c

2/3 – kc

2/R2

which is known as Friedmann’s Equation

Note that if we assume Λ = 0, we have

(d2R/dt2)/R = (ρ + 3P)

and in a matter dominated Universe, ρ >> P

So we can define a critical density by combining the cosmological equations:

ρC = =

4πG3

3 R2.

8πG R2

3H02

8πG

And we define the ratio of the density to the

critical density as the parameter

Ω ≡ ρ/ρC

For a matter dominated, Λ=0 cosmology,

Ω > 1 = closed Ω = 1 = flat, just bound Ω < 1 = openThere are many possible forms of R(t), especially

when Λ and P are reintroduced. Its our job to find the right one!

Λ = 0

Some of possible forms are:

Big Bang Models:

Einstein-deSitter k=0 flat, open & infinite

expands

Friedmann-Lemaitre k=-1 hyperbolic “

“ “ k=+1 spherical, closed

finite, collapses

Leimaitre Λ ≠0 k=+1 spherical, closed

finite, expands

Non-Big Bang Models

Eddington-Lemaitre Λ≠0 k=+1 spherical, closed, finite, static then expands

Steady State k=0 flat, open,

infinite, stationary

deSitter k=0 empty, no singularity, open, infinite

k =

≡ Radius of Curvature of the Universe

H02 (Ω0 – 1) + 1/3 Λ0

c2

R(t)

t

F-L,0

E-L

F-L,C

L

SS,dS

EdSA Child’s Garden

of Cosmological Models

Harrison’s Classes

Geometric:

Closed

Open

Kinematic:

Bang

Whimper

Static

Oscillating

Cosmology is now the search for three numbers:

• The Expansion Rate = Hubble’s Constant

= H0

• The Mean Matter Density = Ωmatter

• The Cosmological Constant = ΩΛ

Taken together, these three numbers describe the geometry of space-time and its evolution. They also give you the Age of the Universe.

The best routes to the first two are in the Nearby Universe:

H0 is determined by measuring distances and redshifts to galaxies. It changes with time in real FRW models so by definition it must be measured locally.

(matter) is determined locally by (1)

a census, (2) topography, or (3) gravity versus the velocity field (how things move in the presence of lumps).

The cosmological constant is determined by

measuring geometry on large scales --- e.g. by the supernovae Hubble Diagram (distance versus redshift)

Or

by a difference technique, e.g. CMB shows us that the Universe is Flat, but ΩM is only 0.3,

so ……

Lookback Time

For a Friedmann-Lemaitre Big-Bang Model, the lookback time as a function of redshift is

τL = H0-1 ( ) for q0=0; Λ=0

= 2/3 H0-1 [1 – (1 + z)-3/2] for q0=1/2, Λ=0

z1+z

Cosmological Age Calculation For models w/o a Cosmological Constant, for

q0 = 0 0 = H0-1

q0 = ½ 0 = (2/3)H0-1

q0 > ½ 0 = H0-1

(1/(1-2q0) + ..)

where q0 = /2 (if = 0)

For the general case (with a CC), the full

form is:

0 = -H0-1 ∫

0

(1+z)[(1+z)2(mz+1) –

(z(z+2))] -1/2

dz

and a good approximation is

0 = (2/3) H0-1 sinn-1

[(|1-a|/a)1/2

]

/ |[1-a]|1/2

∞

Where

a = matter -0.3*total + 0.3

and

sinn-1 = sinh-1 if a </= 1

= sin-1 if a > 1

(from Carroll, Press and Turner, 1992)

Also, for a flat model with L,

0 = (2/3)H0-1

-1/2

ln[(1+1/2

)/(1-)1/2

]

Classical Cosmological TestsUse observations to test/measure the

cosmological model:

Observables: Apparent magnitudes

Redshifts

Angular Diameters

Number Counts

Ages of Things (rocks, stars, star clusters)

(1) Magnitude vs Redshift (The Hubble Diagram)

(assume or find a standard candle)

log

cz

V magnitude (corrected)

q00-1

1/2

1

(2) Counts vs Magnitude (Evolution?)

(3) Angular Diameter vs Redshift (Yardstick)

(4) Age Consistency (rocks vs expansion age)

(5) Density versus Redshift (galaxy evolution

again)