The Hubble parameter and the age of the...

The Hubble parameter and the ageof the universe

Djapo Haris, 10.02.2005

RHI seminar WS 2004/2005

Overview• Theoretical Model of Cosmology

I Robertson-Walker metric

I The Redshift

I Friedmann-Lemaître equations

I Standard model solutions

I Cosmology parameters (definitions)

I Flat adiabatic ΛCDM

• ObservationsI Hubble’s law (observations)

I CMB power spectrum

I Matter content

I Equation of state

I Matter power spectrum

I Cosmology parameters (observations)

I Future projects

• Summary2

Robertson-Walker metric

• Cosmological Principle

• large scale homogeneity and isotropy

• space-time metric

ds2 = dt2 − a2(t)

1 − kr2+ r2

dθ2 + sin2 θdφ2)

• k can only take discret values +1,-1 and 0

Expansion parameter

• expansion parameter a(t) for low redshift can beapproximated by a series

a(t) = a0

1 − H0(t0 − t) − 1

20 (t0 − t)2 − 1

30 (t0 − t)3 + · · ·

• t0 − t lookback time

• H(t) = 1a(t)

da(t)dt Hubble parameter

• q(t) = − 1a(t)

d2a(t)dt2

da(t)dt

−2deceleration parameter

• j(t) = 1a(t)

d3a(t)dt3

da(t)dt

−3jerk parameter

The Redshift

• The redshifts connection to expansion parameter

z =λ0 − λe

a(te)− 1

• Hubble’s (luminosity distance) law

dL(z) =z

2[1 − q0] z

1 − q0 − 3q20 + j0 +

z2 + O(z3)

• completely model independent, onlyRobertson-Walker metric assumed

The Redshift

z =λ0 − λe

a(te)− 1

dL(z) =z

2[1 − q0] z

1 − q0 − 3q20 + j0 +

z2 + O(z3)

The Redshift

z =λ0 − λe

a(te)− 1

dL(z) =z

2[1 − q0] z

1 − q0 − 3q20 + j0 +

z2 + O(z3)

Friedman-Lamaître equations

• cosmological equations of motion are derived fromEinstein’s equations

Rµν − 1

2gµνR = 8πGTµν + Λgµν

• assuming that the matter content of the universe is aperfect fluid

Tµν = −pgµν + (p + ρ)uµuν

• cosmological equations of motion are derived fromEinstein’s equations

Rµν − 1

2gµνR = 8πGTµν + Λgµν

• assuming that the matter content of the universe is aperfect fluid

Tµν = −pgµν + (p + ρ)uµuν

• Friedmann-Lemaître equations

=8πGρ

3− k

a2(t)+

3− 4πG

3(ρ + 3p)

• Energy conservation via T µν;µ = 0 leads to

ρ = −3a(t)

a(t)(ρ + p)

Standard model solutions

• expansion history of universe

• assume domination of one componet (radiation,matter, cosmological constant)

• each componet distinguished by equation of stateparameter w = p/ρ

• integration of equation ρ = −3(1 + w)ρa/a gives

ρ ∝ a−3(1+w)

• for w 6= −1, and neglecting the curvature andcosmological terms, we have

a(t) ∝ t2

3(1+w)

• radiation dominated universew = 1

3 , ρ ∝ a−4; a ∝ t12 ; H = 1

• matter dominated universew = 0, ρ ∝ a−3; a ∝ t

23 ; H = 2

a(t) ∝ t2

3(1+w)

3 , ρ ∝ a−4; a ∝ t12 ; H = 1

23 ; H = 2

a(t) ∝ t2

3(1+w)

3 , ρ ∝ a−4; a ∝ t12 ; H = 1

23 ; H = 2

• future universe expansion dominated by vacuumenergy

• equation of state w = −1

• simple solution a(t) ∝ e√

• w can depend ontime in this case

• future universe expansion dominated by vacuumenergy

• equation of state w = −1

• simple solution a(t) ∝ e√

• w can depend ontime in this case

Cosmology parameters (definitions)

• scaled Hubble parameter h, is defined viaH = 100 h−1 km s−1 Mpc−1

• critical density, such that k = 0 when Λ = 0

ρc = 3H2(t)8πG = 1.88 × 10−26 h2 kg m−3

• density parameter for pressureless matter Ωm = ρρc

• density parametter of the vacuum Ωλ = Λ3H2(t)

• cosmological density parameter Ωtot = Ωm + Ωλ

• present day parametersΩM = 8πG

3H20ρ0, ΩΛ = Λ

3H20, ΩK = − k

ρc = 3H2(t)8πG = 1.88 × 10−26 h2 kg m−3

3H20ρ0, ΩΛ = Λ

3H20, ΩK = − k

ρc = 3H2(t)8πG = 1.88 × 10−26 h2 kg m−3

3H20ρ0, ΩΛ = Λ

3H20, ΩK = − k

ρc = 3H2(t)8πG = 1.88 × 10−26 h2 kg m−3

3H20ρ0, ΩΛ = Λ

3H20, ΩK = − k

ρc = 3H2(t)8πG = 1.88 × 10−26 h2 kg m−3

3H20ρ0, ΩΛ = Λ

3H20, ΩK = − k

ρc = 3H2(t)8πG = 1.88 × 10−26 h2 kg m−3

3H20ρ0, ΩΛ = Λ

3H20, ΩK = − k

Age of the universe

• three competing terms drive the expansion: a matterterm, cosmological term, and a curvature term

• Friedman-Lemaître equations implies1 = ΩM + ΩΛ + ΩK

• ΩK measures how much the geometry differs fromthat of flat spacetime

• deceleration paremeter is q0 = 12ΩM − ΩΛ

• if matter density is too large, the universe willrecollaps before Λ-driven term becomes significant

ΩΛ ≥

0 0 ≤ ΩM ≤ 1

13 cos−1(1−ΩM

ΩM+ 4π

]3ΩM > 1

Age of the universe

• if ΩΛ is negative,recollapse is inevitable

• Ωtot determines onlythe geometry of theuniverse

• Ωtot value has ameaning for theexpansion only ifΩΛ = 0

• for Ωtot = 1 q0 = 0 im-plies ΩM = 1

30.0 0.5 1.0 1.5 2.0 2.5

No Big

Ωtot =1

Expands to Infinity

Recollapses ΩΛ=0

Closed

Accelerating

Decelerating

q0=-0.5

q0=0.5

Age of the Universe

• lookback time from redshift

t0 − t1 = H−10

∫ z1

(1 + z)H(z)

= H−10

∫ z1

(1 + z) [(1 + z)2(1 + ΩMz) − z(2 + z)ΩΛ]1/2

• the age of the universe is then

t0 = H−10

(1 + z) [(1 + z)2(1 + ΩMz) − z(2 + z)ΩΛ]1/2

Flat adiabatic ΛCDM

• inflation results in Gaussian, adiabatic, nearly scaleinvariant primordial fluctuations

• growth of fluctuations depends on properties ofmatter and dark matter

• for CDM structure formbottom up, from small toprogressively larger

• for HDM structure formtop down, from larger toprogressively smaller

Hubble’s law (observations)

• Hubble’s law with distances, goes only up to z < 0.1

• mB apparent magnitude in the blue filter

• average Hubble diagram, ∆z < 0.01

Hubble’s law (observations)

• HSTkey project

• ∆(m−M) magnitude residual from empty cosmology

• SN1999ff, lucky observation of a supernova at z = 1.7

• various models shown

HST DiscoveredGround Discovered

0.0 0.5 1.0 1.5 2.0z

q(z)=q0+z(dq/dz)

Coasting, q(z)=0

Constant Acceleration, q0=-, dq/dz=0 (j0=0)

Acceleration+Deceleration, q0=-, dq/dz=++Acceleration+Jerk, q0=-, j0=++

Constant Deceleration, q0=+, dq/dz=0 (j0=0)

HST DiscoveredGround Discovered

0.0 0.5 1.0 1.5 2.0z

0.5∆(

ΩM=1.0, ΩΛ=0.0

high-z gray dust (+ΩM=1.0)Evolution ~ z, (+ΩM=1.0)

Empty (Ω=0)ΩM=0.27, ΩΛ=0.73"replenishing" gray Dust

CMB power spectrum

• the acoustic peaks arise from adiabatic compresionof the photon-baryon fluid as it falls into preexistingwells in gravitational potential

CMB power spectrum

• the peak characteristics are interpretes in terms offlat adiabatic ΛCDM cosmological model

• the wells are enhanced by dark matter which cluster

• first peak corespond to the scale of the mode thathas compressed once

• the second peak arises from a refraction phase of anacoustic wave

• amplitude of the second peak decreases as ωb

increases

• for l > 40 TT spectrum predicts the TE spectrum

• for l < 20 TE spectrum is produced by reionizedelectron scattering the CMB quadrupole

CMB power spectrum

increases

CMB power spectrum

increases

CMB power spectrum

increases

CMB power spectrum

increases

CMB power spectrum

increases

CMB power spectrum

increases

Matter content

• matter content is determined in several projects

• universe evolves along the line of Ωtot = 1, if it is flat

Equation of state

• constraints on the equation of statefor dark matter

• implication for the nature ofthe dark energy

Matter power spectrum

• survey of very large volumes required

• peak arises as a result of the sound waves in matter

• primarely sensitive to Ωm and h

Cosmology parameters (observations)

• "state of the art" constants on cosmologicalparameters

• combination with the result obtained from CMBobservations

Joint Constraints on Cosmological Parameters including CMB data

Constant w flat w = −1 curved w = −1 flat

WMAP+Main +LRG WMAP+Main +LRG WMAP+Main +LRG

w −0.92 ± 0.30 −0.80 ± 0.18 · · · · · · · · · · · ·

ΩK · · · · · · −0.045 ± 0.032 −0.010 ± 0.009 · · · · · ·

Ωmh2 0.145 ± 0.014 0.135 ± 0.008 0.134 ± 0.012 0.136 ± 0.008 0.146 ± 0.009 0.142 ± 0.005

Ωm 0.329 ± 0.074 0.326 ± 0.037 0.431 ± 0.096 0.306 ± 0.027 0.305 ± 0.042 0.298 ± 0.025

h 0.679 ± 0.100 0.648 ± 0.045 0.569 ± 0.082 0.669 ± 0.028 0.696 ± 0.033 0.692 ± 0.021

n 0.984 ± 0.033 0.983 ± 0.035 0.964 ± 0.032 0.973 ± 0.030 0.980 ± 0.031 0.963 ± 0.022

Cosmology parameters (observations)

• recent improvement in determination of cosmicparameters

• several other projects measure the same parameters,such as Lyα, SDSS etc.

Various cosmological parameters from different sources

2000 2dFGRS HST WMAP

kms Mpc

65 ± 8 76.6 ± 3.2 71 ± 8 71 ± 4

t0 [Gyr] 9 − 17 · · · · · · 13.7 ± 0.02

Ωb 0.045 ± 0.0057 0.042 ± 0.002 · · · 0.044 ± 0.004

Ωm 0.4 ± 0.2 0.231 ± 0.021 0.29+0.05−0.03 0.27 ± 0.04

ΩΛ 0.71 ± 0.14 · · · 0.71 0.73 ± 0.004

Ωtot 1.11 ± 0.07 · · · · · · 1.02 ± 0.02

w · · · < −0.52 −1.02+0.13−0.19 < −0.78

Future projects• WMAP has presented only first-year results, there

should be four, and then there is PLANCK

• high redshift supernovasurveys will continue

• SNAP (SuperNovaAcceleration Probe)

should be four, and then there is PLANCK• high redshift supernova

surveys will continue

• SNAP (SuperNovaAcceleration Probe)

should be four, and then there is PLANCK• high redshift supernova

surveys will continue• SNAP (SuperNova

Acceleration Probe)

Summary

• universe is described by RW metric

• universe is, most likely, geometrically flat, but thiscannot be experimentally proven

• universe expansion is not linear

• universe is accelerating propelled by dark energy

• acceleration commenced at 0.5 < z < 1

• current observations of the universe the ΛCDMmodel best

• best estimate of the value for the Hubble’s constant isH0 = 71 ± 4

• the age of the universe is ' 13.7 Gyr

Appendix A

• measure radial coordinate r1

• looking back to time t1 when expansion parameterwas a(t1)

• r1,t1 and a(t1) cannot be directly measured

• directly measurable quantities are:

I the angular diameter distance; dA = D/θ = a(t1)r1

I the proper motion distance; dM = u/θ = a0r1

I the luminosity distance dL =(

)1/2= a2

0r1/a(t1)

• connection dL = (1 + z)dM = (1 + z)2dA

The Hubble parameter and the age of the...

Documents

Transcript of The Hubble parameter and the age of the...