Post on 27-Apr-2020
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)
[
dr2
1 − kr2+ r2
(
dθ2 + sin2 θdφ2)
]
• k can only take discret values +1,-1 and 0
3
Expansion parameter
• expansion parameter a(t) for low redshift can beapproximated by a series
a(t) = a0
[
1 − H0(t0 − t) − 1
2q0H
20 (t0 − t)2 − 1
3!j0H
30 (t0 − t)3 + · · ·
]
• t0 − t lookback time
• H(t) = 1a(t)
da(t)dt Hubble parameter
• q(t) = − 1a(t)
d2a(t)dt2
[
1a(t)
da(t)dt
]
−2deceleration parameter
• j(t) = 1a(t)
d3a(t)dt3
[
1a(t)
da(t)dt
]
−3jerk parameter
4
The Redshift
• The redshifts connection to expansion parameter
z =λ0 − λe
λe=
a(t0)
a(te)− 1
• Hubble’s (luminosity distance) law
dL(z) =z
H0
1 +1
2[1 − q0] z
− 1
6
[
1 − q0 − 3q20 + j0 +
k
H20a2
0
]
z2 + O(z3)
• completely model independent, onlyRobertson-Walker metric assumed
5
The Redshift
• The redshifts connection to expansion parameter
z =λ0 − λe
λe=
a(t0)
a(te)− 1
• Hubble’s (luminosity distance) law
dL(z) =z
H0
1 +1
2[1 − q0] z
− 1
6
[
1 − q0 − 3q20 + j0 +
k
H20a2
0
]
z2 + O(z3)
• completely model independent, onlyRobertson-Walker metric assumed
5
The Redshift
• The redshifts connection to expansion parameter
z =λ0 − λe
λe=
a(t0)
a(te)− 1
• Hubble’s (luminosity distance) law
dL(z) =z
H0
1 +1
2[1 − q0] z
− 1
6
[
1 − q0 − 3q20 + j0 +
k
H20a2
0
]
z2 + O(z3)
• completely model independent, onlyRobertson-Walker metric assumed
5
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
6
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
6
Friedman-Lamaître equations
• Friedmann-Lemaître equations
(
a(t)
a(t)
)2
=8πGρ
3− k
a2(t)+
Λ
3
a(t)
a(t)=
Λ
3− 4πG
3(ρ + 3p)
• Energy conservation via T µν;µ = 0 leads to
ρ = −3a(t)
a(t)(ρ + p)
7
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)
8
Standard model solutions
• 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
2t
• matter dominated universew = 0, ρ ∝ a−3; a ∝ t
23 ; H = 2
3t
9
Standard model solutions
• 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
2t
• matter dominated universew = 0, ρ ∝ a−3; a ∝ t
23 ; H = 2
3t
9
Standard model solutions
• 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
2t
• matter dominated universew = 0, ρ ∝ a−3; a ∝ t
23 ; H = 2
3t
9
Standard model solutions
• future universe expansion dominated by vacuumenergy
• equation of state w = −1
• simple solution a(t) ∝ e√
Λ3t
• w can depend ontime in this case
10
Standard model solutions
• future universe expansion dominated by vacuumenergy
• equation of state w = −1
• simple solution a(t) ∝ e√
Λ3t
• w can depend ontime in this case
10
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
a20H
20
11
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
a20H
20
11
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
a20H
20
11
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
a20H
20
11
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
a20H
20
11
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
a20H
20
11
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
4ΩM
cos[
13 cos−1(1−ΩM
ΩM+ 4π
3
]3ΩM > 1
12
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
ΩM
-1
0
1
2
3
ΩΛ
68.3
%
95.4
%
99.7
%
No Big
Bang
Ωtot =1
Expands to Infinity
Recollapses ΩΛ=0
Open
Closed
Accelerating
Decelerating
q0=0
q0=-0.5
q0=0.5
^
13
Age of the Universe
• lookback time from redshift
t0 − t1 = H−10
∫ z1
0
dz
(1 + z)H(z)
= H−10
∫ z1
0
dz
(1 + z) [(1 + z)2(1 + ΩMz) − z(2 + z)ΩΛ]1/2
• the age of the universe is then
t0 = H−10
∫
∞
0
dz
(1 + z) [(1 + z)2(1 + ΩMz) − z(2 + z)ΩΛ]1/2
14
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
15
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
15
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
15
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
15
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
15
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
16
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
-1.0
-0.5
0.0
0.5
1.0
∆(m
-M)
(mag
)
HST DiscoveredGround Discovered
0.0 0.5 1.0 1.5 2.0z
-1.0
-0.5
0.0
0.5
1.0
∆(m
-M)
(mag
)
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)
-1.0
-0.5
0.0
0.5
1.0
∆(m
-M)
(mag
)
HST DiscoveredGround Discovered
0.0 0.5 1.0 1.5 2.0z
-0.5
0.0
0.5∆(
m-M
) (m
ag)
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
17
CMB power spectrum
• the acoustic peaks arise from adiabatic compresionof the photon-baryon fluid as it falls into preexistingwells in gravitational potential
18
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
19
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
19
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
19
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
19
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
19
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
19
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
19
Matter content
• matter content is determined in several projects
• universe evolves along the line of Ωtot = 1, if it is flat
20
Equation of state
• constraints on the equation of statefor dark matter
• implication for the nature ofthe dark energy
21
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
22
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
23
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
H0
h
kms Mpc
i
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
24
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)
25
Future projects• WMAP has presented only first-year results, there
should be four, and then there is PLANCK• high redshift supernova
surveys will continue
• SNAP (SuperNovaAcceleration Probe)
25
Future projects• WMAP has presented only first-year results, there
should be four, and then there is PLANCK• high redshift supernova
surveys will continue• SNAP (SuperNova
Acceleration Probe)
25
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
26
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 =(
L4πl
)1/2= a2
0r1/a(t1)
• connection dL = (1 + z)dM = (1 + z)2dA
27