The Hubble parameter and the age of the universe

The Hubble parameter and the age
of the universe
Djapo Haris, 10.02.2005
RHI seminar WS 2004/2005
1
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
• Observations
I 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
• Summary
2
Robertson-Walker metric
• Cosmological Principle
• large scale homogeneity and isotropy
• space-time metric
ds2 = dt2 − a2 (t)
dr2
1 − kr 2
+ r 2 dθ 2 + sin2 θdφ
2
• k can only take discret values +1,-1 and 0
3
Expansion parameter
• expansion parameter a(t) for low redshift can be
approximated by a series
a(t) = a0
1
1
2
2
1 − H0 (t0 − t) − q0 H0 (t0 − t) − j0 H03 (t0 − t)3 + · · ·
2
3!
• t0 − t lookback time
• H(t) =
1 da(t)
a(t) dt
Hubble parameter
h
i−2
2
1 d a(t)
1 da(t)
• q(t) = − a(t)
deceleration parameter
dt2
a(t) dt
h
i−3
3
1 d a(t)
1 da(t)
• j(t) = a(t)
jerk parameter
dt3
a(t) dt
4
The Redshift
• The redshifts connection to expansion parameter
λ0 − λ e
a(t0 )
z=
=
−1
λe
a(te )
5
The Redshift
• The redshifts connection to expansion parameter
λ0 − λ e
a(t0 )
z=
=
−1
λe
a(te )
• Hubble’s (luminosity distance) law
z
1
dL (z) =
1 + [1 − q0 ] z
H0
2
1
k
2
−
1 − q0 − 3q0 + j0 + 2 2 z 2 + O(z 3 )
6
H 0 a0
5
The Redshift
• The redshifts connection to expansion parameter
λ0 − λ e
a(t0 )
z=
=
−1
λe
a(te )
• Hubble’s (luminosity distance) law
z
1
dL (z) =
1 + [1 − q0 ] z
H0
2
1
k
2
−
1 − q0 − 3q0 + j0 + 2 2 z 2 + O(z 3 )
6
H 0 a0
• completely model independent, only
Robertson-Walker metric assumed
5
Friedman-Lamaître equations
• cosmological equations of motion are derived from
Einstein’s equations
Rµν
1
− gµν R = 8πGTµν + Λgµν
2
6
Friedman-Lamaître equations
• cosmological equations of motion are derived from
Einstein’s equations
Rµν
1
− gµν R = 8πGTµν + Λgµν
2
• assuming that the matter content of the universe is a
perfect fluid
Tµν = −pgµν + (p + ρ)uµ uν
6
Friedman-Lamaître equations
• Friedmann-Lemaître equations
2
ȧ(t)
8πGρ
k
Λ
=
− 2 +
a(t)
3
a (t)
3
Λ 4πG
ä(t)
=
−
(ρ + 3p)
a(t)
3
3
µν
= 0 leads to
• Energy conservation via T;µ
ȧ(t)
(ρ + p)
ρ̇ = −3
a(t)
7
Standard model solutions
• expansion history of universe
• assume domination of one componet (radiation,
matter, cosmological constant)
• each componet distinguished by equation of state
parameter w = p/ρ
• integration of equation ρ̇ = −3(1 + w)ρȧ/a gives
ρ ∝ a−3(1+w)
8
Standard model solutions
• for w 6= −1, and neglecting the curvature and
cosmological terms, we have
a(t) ∝ t
2
3(1+w)
9
Standard model solutions
• for w 6= −1, and neglecting the curvature and
cosmological terms, we have
a(t) ∝ t
2
3(1+w)
• radiation dominated universe
1
1
−4
2
w = 3 , ρ ∝ a ; a ∝ t ; H = 12 t
9
Standard model solutions
• for w 6= −1, and neglecting the curvature and
cosmological terms, we have
a(t) ∝ t
2
3(1+w)
• radiation dominated universe
1
1
−4
2
w = 3 , ρ ∝ a ; a ∝ t ; H = 12 t
• matter dominated universe
2
−3
w = 0, ρ ∝ a ; a ∝ t 3 ; H = 23 t
9
Standard model solutions
• future universe expansion dominated by vacuum
energy
• equation of state w = −1
√Λ
• simple solution a(t) ∝ e 3 t
• w can depend on
time in this case
10
Standard model solutions
• future universe expansion dominated by vacuum
energy
• equation of state w = −1
√Λ
• simple solution a(t) ∝ e 3 t
• w can depend on
time in this case
10
Cosmology parameters (definitions)
• scaled Hubble parameter h, is defined via
H = 100 h−1 km s−1 M pc−1
11
Cosmology parameters (definitions)
• scaled Hubble parameter h, is defined via
H = 100 h−1 km s−1 M pc−1
• critical density, such that k = 0 when Λ = 0
ρc =
3H 2 (t)
8πG
= 1.88 × 10−26 h2 kg m−3
11
Cosmology parameters (definitions)
• scaled Hubble parameter h, is defined via
H = 100 h−1 km s−1 M pc−1
• critical density, such that k = 0 when Λ = 0
ρc =
3H 2 (t)
8πG
= 1.88 × 10−26 h2 kg m−3
• density parameter for pressureless matter Ωm =
ρ
ρc
11
Cosmology parameters (definitions)
• scaled Hubble parameter h, is defined via
H = 100 h−1 km s−1 M pc−1
• critical density, such that k = 0 when Λ = 0
ρc =
3H 2 (t)
8πG
= 1.88 × 10−26 h2 kg m−3
• density parameter for pressureless matter Ωm =
• density parametter of the vacuum Ωλ =
ρ
ρc
Λ
3H 2 (t)
11
Cosmology parameters (definitions)
• scaled Hubble parameter h, is defined via
H = 100 h−1 km s−1 M pc−1
• critical density, such that k = 0 when Λ = 0
ρc =
3H 2 (t)
8πG
= 1.88 × 10−26 h2 kg m−3
• density parameter for pressureless matter Ωm =
• density parametter of the vacuum Ωλ =
ρ
ρc
Λ
3H 2 (t)
• cosmological density parameter Ωtot = Ωm + Ωλ
11
Cosmology parameters (definitions)
• scaled Hubble parameter h, is defined via
H = 100 h−1 km s−1 M pc−1
• critical density, such that k = 0 when Λ = 0
ρc =
3H 2 (t)
8πG
= 1.88 × 10−26 h2 kg m−3
• density parameter for pressureless matter Ωm =
• density parametter of the vacuum Ωλ =
ρ
ρc
Λ
3H 2 (t)
• cosmological density parameter Ωtot = Ωm + Ωλ
• present day parameters
Λ
ΩM = 8πG
Ω
=
,
2 ρ0 ,
Λ
3H
3H 2
0
0
ΩK = − a2kH 2
0
0
11
Age of the universe
• three competing terms drive the expansion: a matter
term, cosmological term, and a curvature term
• Friedman-Lemaître equations implies
1 = Ω M + ΩΛ + ΩK
• ΩK measures how much the geometry differs from
that of flat spacetime
• deceleration paremeter is q0 = 12 ΩM − ΩΛ
• if matter density is too large, the universe will
recollaps before Λ-driven term becomes significant


0
0 ≤ ΩM ≤ 1
n h
io3
ΩΛ ≥
 4ΩM cos 1 cos−1 ( 1−ΩM + 4π
ΩM > 1
3
ΩM
3
12
Age of the universe
3
N
o
Bi
g
Ba
ng
• if ΩΛ is negative,
recollapse is inevitable
• Ωtot determines only
the geometry of the
universe
• for Ωtot = 1 q0 = 0 implies ΩM = 13
.5
-0
q 0=
0
q 0=
ting
lera
e
c
Ac
ng
rati
e
l
e
Dec
5
0.
q 0=
.7
% 95.
4% 68
.3%
1
99
ΩΛ
• Ωtot value has a
meaning for the
expansion only if
ΩΛ = 0
2
Expands to Infinity
0
Cl
Recollapses
os
Op
ed
en
Ω
to
t
-1
0.0
0.5
1.0
ΩM
^
ΩΛ=0
1.5
=1
2.0
2.5
13
Age of the Universe
• lookback time from redshift
t0 − t 1 =
=
H0−1
Z
H0−1
Z
z1
0
dz
(1 + z)H(z)
z1
0
dz
(1 + z) [(1 + z)2 (1 + ΩM z) − z(2 + z)ΩΛ ]
1/2
• the age of the universe is then
t0 =
H0−1
Z
∞
0
dz
(1 + z) [(1 + z)2 (1 + ΩM z) − z(2 + z)ΩΛ ]
1/2
14
Flat adiabatic ΛCDM
• inflation results in Gaussian, adiabatic, nearly scale
invariant primordial fluctuations
15
Flat adiabatic ΛCDM
• inflation results in Gaussian, adiabatic, nearly scale
invariant primordial fluctuations
• growth of fluctuations depends on properties of
matter and dark matter
15
Flat adiabatic ΛCDM
• inflation results in Gaussian, adiabatic, nearly scale
invariant primordial fluctuations
• growth of fluctuations depends on properties of
matter and dark matter
• for CDM structure form
bottom up, from small to
progressively larger
15
Flat adiabatic ΛCDM
• inflation results in Gaussian, adiabatic, nearly scale
invariant primordial fluctuations
• growth of fluctuations depends on properties of
matter and dark matter
• for CDM structure form
bottom up, from small to
progressively larger
• for HDM structure form
top down, from larger to
progressively smaller
15
Flat adiabatic ΛCDM
• inflation results in Gaussian, adiabatic, nearly scale
invariant primordial fluctuations
• growth of fluctuations depends on properties of
matter and dark matter
• for CDM structure form
bottom up, from small to
progressively larger
• for HDM structure form
top down, from larger to
progressively 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
1.0
1.0
0.5
0.5
∆(m-M) (mag)
∆(m-M) (mag)
• various models shown
0.0
-0.5
0.0
-0.5
Ground Discovered
HST Discovered
-1.0
)
dq/dz=0 (j0=0
eration, q0=-,
el
Constant Acc
0.5
Constant Deceleration
-1.0
0.0
Coasting, q(z)=0
Acceleration+Deceleration, q0=-, dq/dz=++
Acceleration+Jerk, q0=-, j0=++
0.5
1.0
z
, q0=+, dq/dz=0 (j =0)
0
1.5
2.0
1.0
, (+Ω M=
1.0)
+Ω M= Evolution ~ z
(
t
s
u
ay d
z gr
high-
0.5
0.0
-0.5
Ground Discovered
HST Discovered
)
q(z)=q0+z(dq/dz)
∆(m-M) (mag)
∆(m-M) (mag)
1.0
-1.0
0.0
-0.5
0.0
Empty (Ω=0)
ΩM=0.27, ΩΛ=0.73
"replenishing" gray Dust
0.5
ΩM =1.0
, ΩΛ =0
1.0
z
.0
1.5
2.0
17
CMB power spectrum
• the acoustic peaks arise from adiabatic compresion
of the photon-baryon fluid as it falls into preexisting
wells in gravitational potential
18
CMB power spectrum
• the peak characteristics are interpretes in terms of
flat adiabatic ΛCDM cosmological model
19
CMB power spectrum
• the peak characteristics are interpretes in terms of
flat adiabatic ΛCDM cosmological model
• the wells are enhanced by dark matter which cluster
19
CMB power spectrum
• the peak characteristics are interpretes in terms of
flat adiabatic ΛCDM cosmological model
• the wells are enhanced by dark matter which cluster
• first peak corespond to the scale of the mode that
has compressed once
19
CMB power spectrum
• the peak characteristics are interpretes in terms of
flat adiabatic ΛCDM cosmological model
• the wells are enhanced by dark matter which cluster
• first peak corespond to the scale of the mode that
has compressed once
• the second peak arises from a refraction phase of an
acoustic wave
19
CMB power spectrum
• the peak characteristics are interpretes in terms of
flat adiabatic ΛCDM cosmological model
• the wells are enhanced by dark matter which cluster
• first peak corespond to the scale of the mode that
has compressed once
• the second peak arises from a refraction phase of an
acoustic wave
• amplitude of the second peak decreases as ωb
increases
19
CMB power spectrum
• the peak characteristics are interpretes in terms of
flat adiabatic ΛCDM cosmological model
• the wells are enhanced by dark matter which cluster
• first peak corespond to the scale of the mode that
has compressed once
• the second peak arises from a refraction phase of an
acoustic wave
• amplitude of the second peak decreases as ωb
increases
• for l > 40 TT spectrum predicts the TE spectrum
19
CMB power spectrum
• the peak characteristics are interpretes in terms of
flat adiabatic ΛCDM cosmological model
• the wells are enhanced by dark matter which cluster
• first peak corespond to the scale of the mode that
has compressed once
• the second peak arises from a refraction phase of an
acoustic 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 reionized
electron 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 state
for dark matter
• implication for the nature of
the 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 cosmological
parameters
• combination with the result obtained from CMB
observations
Joint Constraints on Cosmological Parameters including CMB data
Constant w flat
WMAP+Main
w
ΩK
+LRG
−0.92 ± 0.30 −0.80 ± 0.18
···
···
Ωm h2 0.145 ± 0.014 0.135 ± 0.008
w = −1 curved
w = −1 flat
WMAP+Main
+LRG
WMAP+Main
+LRG
···
···
···
···
···
···
−0.045 ± 0.032 −0.010 ± 0.009
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 cosmic
parameters
• several other projects measure the same parameters,
such as Lyα, SDSS etc.
Various cosmological parameters from different sources
2000
2dFGRS
HST
WMAP
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
H0
h
km
s M pc
i
24
Future projects
• WMAP has presented only first-year results, there
should be four, and then there is PLANCK
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
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 this
cannot 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 ΛCDM
model best
• best estimate of the value for the Hubble’s constant is
H0 = 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 parameter
was 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/θ̇ = a0 r1
L 1/2
I the luminosity distance dL = 4πl
= a20 r1 /a(t1 )
• connection dL = (1 + z)dM = (1 + z)2 dA
27