Cosmology & the Big Bang

AY16 Lecture 20, April 15, 2008

Mathematical Cosmology, con’t Determination of Cosmological Parameters Inflation & the Big Bang

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

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.

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

The Hubble Constant:

• H0 = *current* expansion rate

• = (velocity) / (distance)

• = (km/s) / (Megaparsecs)

• named after Edwin Hubble who

discovered the relation in 1929.

The story of the Hubble Constant (never called

that by Hubble!) is the “Cosmological Distance Ladder” or the “Extragalactic Distance Scale”

Basically, we need distances & velocities to galaxies and other things.

Velocities are easy --- pick a galaxy, any galaxy, get spectrum with moderate resolution, R ~ 1000 (i.e λ/R ~ 5Å)

N.B. R = Linear Reciprocal Dispersion, get line centroids to ~ 1/10 R ~ 0.5Å/5000Å ~ 1 part in 104 ~ 30 km/s

Spectral features in galaxies

Velocity Measurement

Radial Velocities (stars, galaxies) now usually measured by cross-correlation techniques pioneered by Simkin (1973), Schechter (1976) & Tonry & Davis (1979). Accuracy depends on Signal-to-Noise and resolution. Typically, for S/N > ~ 20, errors are ~ 10% of Δλ, where (remember)

R = λ/Δλ

Distances are Hard!

Hubble’s original estimates of galaxy distances were based on brightest stars which were based on Cepheid Variables

Distances to the LMC, SMC, NGC6822 & eventually M31 from Cepheids.

Find the brightest stars and assume they’re the same (independent of galaxy type, etc.)

CepheidsPretty Good Distance Indicators --- Standard Candles from

the Period-Luminosity (PL)

relation: L ≈ P3/2 PLC relationMV = -2.61 - 3.76 log P +2.60 (B-V)but ya gotta find them!

H0 circa 1929 ~ 600 km/s/Mpc Wrong!

1. Hubble’s galactic calibrators not classical Cepheids.

2. At large distances, brightest stars confused with star clusters.

3. Hubble’s magnitude scale was off.

•

P-L Relation, LMC

deVaucouleurs ‘76Cosmological Distance Ladder

Cosmological Distance Ladder

Find things that work as distance indicators (standard candles, standard yardsticks) to greater and greater distances.

Locally: Primary Indicators

Cepheids MB ~ -2 to -6

RR Lyrae Stars MB ~ 0

Novae MB ~ -6 to -9

Calibrate Cepheids via parallax, moving cluster = convergent point method, expansion parallax Baade-Wesselink, main sequence (HR diagram) fitting.

Secondary Distance Indicators Brightest Stars (XX??) Tully-Fisher (+ IRTF) Planetary Nebulae LF Globular Cluster LF

Supernovae of type Ia

Supernovae of type II (EPM)

Fundamental Plane (Dn-σ)

Faber-Jackson

Surface Brightness Fluctuations

Red Giant Branch Tip

Luminosity Classes (XXX)

HII Region Diameters (XXX)

HII Region Luminosities (???)

Lemaitre 1927

Hubble 1929

Oort 1932Baade 1952

Tully-Fisher•

Surface Brightness

Fluctuations

Tonry & Schneider

Baade-Wesselink --- EPM

EPM = Expanding Photospheres Method

Basically observe and expanding/contracting object at two (multiple) times. Get redshift and get SED. Then

L1 = 4πR12σT1

4 & L2 = 4πR22σT2

4

and R2 = R1 + v δt (or better ∫ vdt)

Fukugita, Hogan & Peebles 1993

•

HST H0 Key Project Team

WFPC2

footprint

Cepheid Light Curves N1326a

Matching P-L Relations

IC4182 (HST) MW (Ground)

(matter):0. Baryons from Nucleosynthesis

1. Sum up Starlight (count stars and/

or count galaxies)

2. Count and Weigh Galaxies

3. Use Global techniques:

Large Scale Structure

Large Scale Flows

Big Bang Nucleosynthesis

For

H0=70km/s/Mpc

(baryons)

~ 0.04

Ωmatter:Measure luminosity density = (sum of

all galaxies x their luminosity) per

unit volume (l/v) = L

Measure mean mass-to-light ratio for

galaxies (M/L)

Multiply: Mass density = (M /L) x (L)

How do we measure the Luminosity density?

Redshift Surveys + Φ(M)

Measure the Galaxy Luminosity Function

For a typical flux (magnitude = mL) limited

survey, we can see a galaxy of absolute magnitude M to a distance

r = 10 ( mL -M - 25)/5 Mpc

V(M) = 4/3 π r3 (Survey Solid Angle)

then

Φ(M) = dN(M)/dM = N(M,M±dM/2)/V(M)

Φ(M) or Φ(L) is the number density of galaxies of a given magnitude or luminosity in a sample.

Early forms:

N

M

Holmberg

Hubble

Zwicky

Abell Form (circa 1960) = two power laws

Now use the Schechter LF Form:

Φ(L) dL = φ* (L/L*)α exp(-L/L*) d(L/L*)

or

Φ(M)dM = 0.4 φ* log[dex 0.4(M*-M)]α+1

exp[-dex 0.4(M*-M)] dM

where φ* = normalization (# / Mpc3)

α = faint end slope

M*, L* = characteristic mag or luminosity

Schechter Form

The Schechter form for the LF is derived from Press-Schechter formalism for self-similar galaxy formation (more later).

Is integrable(!) solution:

L = φ* L* Γ(α+2) a Gamma function

Galaxy Luminosity Function:

Luminosity Density The Luminosity Density is then just the

integral of the luminosity function:

L = ∫ L Φ(L) dL

or

L = ∫ L(M) Φ(M) dM

(either way works)

∞

0

∞

0

Luminosity DensityTypical numbers:

B band log L = 26.65

R band log L = 26.90

K band log L = 27.20

In units of ergs s-1 Hz-1 Mpc-3 for H0=70,

in Solar Units LB = 1.2 x 108 L/ Mpc3

Galaxy Masses and M/Ls

Galaxies are weighed via a large number of techniques:

(a) Disk Dispersion (more later)(b) Rotation Curves(c) Velocity Dispersions (d) Binary Galaxies(e) Galaxy Groups

(f) Galaxy Clusters

Virial Theorem /Projected Mass

Hydrostatic Equilibrium

Gravitational Lensing

(g) Large Scale Flows

(e) Cosmic Virial Theorem

Galaxy Field Velocity Dispersion

In all cases, L = Σ LGal in the system.

(b) Rotation Curves

½ m1v(r)2 (sin i)2 GM(r)m1

M(r) = (sin i)2

With m1 = test particle mass, i = inclination,

r = radius, v(r) = rotation speed at r

r r2

=

v(r)2 r

G 2

(d) Binary Galaxies

Must Model Projection Effects!

M ~

i = inclination angle

φ = orbital velocity angle

1

cos3i cos2φ

Abell 2142Hot Gas in X-rays

Strong Gravitational Lensing

Galaxy Flows:

Observed galaxy “velocity” is composed of

several parts

VO = VHubble + Vpeculiar + Vgrav + LSR

and

VP/VH = (1/3) () 0.66

• Blue 1000 < V < 2000 km/sBlue 1000 < V < 2000 km/s

•

The Local Supercluster

VIRGO

The Local Supercluster

We have an infall measure for the LSC and from redshift surveys we have a pretty good measure of δρ/ρ:

VP ~ 250 km/s

VH = 1100 + 250 km/s = 1350

δρ/ρ ~ 2.5

Ω ~ 0.25

In terms of M/LB Ratios M/L populations ~ 1-5 M/L rotation/dispersion ~ 10 M/L galaxy satellites ~ 25 M/L binaries ~ 50 M/L galaxy groups ~ 100 M/L Clusters ~ 400 M/L CVT ~ 3-500 M/L Flows ~ 500

What’s This Saying?

(1) M/L maxes out ~ 450,

ΩG = ΩM = 0.25 ± 0.05

(2) M/L grows with scale?! Gravitating matter seems to be distributed

on a scale somewhat larger than galaxies. and there’s more of it than Baryons

Non-Baryonic Dark Matter exists

Cosmological Constant:

Cosmological Constant = Lambda

is measured by observing the

geometry of the Universe at large

redshift (distance)

Supernovae as standard candles

CMB Fluctuations vs Models

Essence Project, 2004

Levels of Certainty in Science

You bet:

A Dime = $0.1

Your Dog = $100

Your House = $100,000

Your Firstborn = $100,000,000 ….

each x 1000 (except in New York and Boston where everything is x 10!!!)

WMAP Microwave Sky

Best Fit

b=0.04

CDM=0.27

=0.71

T=1.02

+/- 0.02

Large scale geometry:CMB Fluctuations as measured by

WMAP indicate that ΩT is

very nearly unity (1.02 +/- 0.02) the Universe is FLAT

ΩΛ = ΩT - ΩM

Contents:= (density of the Universe)/

(closure density)

= 1.02 +/- 0.02

(total) = (baryons) +

(neutrinos) + (Cold dark matter)

+ (Dark Energy)

Contents:Omega (stars) =0.005 +/- 0.002

Omega(baryons) = 0.044 +/- 0.004

Omega(neutrinos) < 0.008

Omega(CDM) = 0.23 +/- 0.04

Omega(Dark Energy) = 0.73 +/- 0.04

Omega(Total) = 1

Contents of the Universe

0.71

0.24

0.005 0.045

Age of the Universe:

Ages of the Oldest things: stars,

galaxies, star clusters

Cosmological expansion age :

~ (1/H0) x geometric factors

Cosmological Age Calculation In FRW Cosmologies, the age of the

Universe is calculated from

τ0 = -H0-1 ∫

Where the terms are fairly self explanatory. We need to know H0, ΩM and ΩΛ

(1+z)[(1+z)2(ΩMz+1) – ΩΛz(z+2)]1/2

dz

∞

0

The empty model has 0 = H0-1

The SCDM Flat model has 0 = (2/3) H0-1

For the general case (with a CC), the full form is:

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

]

The Age of Flat Universes

H0/ΩΛ 0.0 0.6 0.7 0.8

55 11.9 15.1 17.1 18.5 65 10.0 12.7 14.5 16.2 70 9.4 11.9 13.6 15.1 75 8.7 11.1 12.6 14.0

Where Ωtotal = 1.00000, and the ΩΛ = 0 models

are the Standard CDM models in Gyr

Alternatives

ΩM = 0.3, ΩΛ = 0

gives τ0 = 0.79 H0-1 = 11.8 Gyr for H=65

(no Lambda)

ΩM = 0.25, ΩΛ = 0.6

gives τ0 = 0.97 H0-1 = 14.6 Gyr for H=65

(minimal Lambda)

JPH’s Favorite Guess Today: H0 = 70 +/- 5 km/s/Mpc

The Universe is going to expand forever

Its current age is around

14 Billion Years, and

There is a good chance its FLAT with a

Cosmological constant =

(Lambda) ~ 0.7