COSMOLOGY – PHYS 30392
www.universetoday.com/51894/
GEOMETRY OF THE UNIVERSE Part II
http://www.jb.man.ac.uk/~gp/
[email protected]
Giampaolo Pisano - Jodrell Bank Centre for Astrophysics
The University of Manchester - February 2013
OBSERVING THE UNIVERSE Part II
One Page Summary
Cosmic Microwave Background Radiation
CMB as a perfect black body
CMB dipole, foregrounds and anisotropies
CMB theory in brief
Homogeneity and Isotropy
Definitions
Relation between them
Application of the Cosmological principle
The Expansion in the Universe
Galaxy redshift and Hubble’s law
Cosmological principle violation?
Homogeneous and isotropic expansion: visual and mathematical descriptions
Age of the Universe and Horizon distance
Dark sky paradox in numbers
Particles in the Universe
Barions, leptons and radiation
Barionic matter and Dark matter (2 definitions)
GEOMETRY OF THE UNIVERSE Part I
One Page Summary
The Equivalence Principle
Gravity (vs other forces) dominating at large scales
Newton’s laws
The Weak Equivalence Principle
Accelerated lift and lift in gravitational fields
Lift in uniform motion and lift in free fall
The Strong Equivalence Principle
General Relativity Concepts
Photons in gravitational fields
Newton’s vs Einsten’s views
Tidal forces
Possible distinction between inertial and free fall frames
Gravitational Redshift
Clocks in gravitational fields
Curved spaces
Flat space, negative and positive curvatures
Curved 3D spaces, metrics and properties
Variable curvature and tidal forces
This week: Example classes
Topics from the First 6 lectures
GEOMETRY OF THE UNIVERSE
The Equivalence Principle
General Relativity Concepts
Curved Spaces
→ The Robertson-Walker Metric
Proper Distance
Expansion and redshift
Expansion and CMB temperature
References: Ryden, Introduction to Cosmology – Par. 3.3
Serjeant, Observational Cosmology – Par. 1.3
Flat 4D Spaces: Minkowsky Space-Time
- Special Relativity tells us that space and time form a 4D continuum
- Distance between two events in (t, r,θ ,φ) and (t+dt, r+dr, θ+dθ , φ+dφ):
ds 2 = − c 2 dt 2 + dr 2 + r 2 dΩ 2 - Minkowski Metric
Euclidean metric
- A photon follows a special 4D geodesic, a null geodesic, such that:
ds 2 = 0
This metric is valid only if space-time is not curved
Without gravitational effects, Minkowski space-time is flat and static
Curved 4D Spaces: Robertson - Walker Metric 1/3
- A metric for an homogeneous, isotropic and expanding Universe is:
ds 2 = −c 2 dt 2 + a(t ) 2 [dr 2 + Sκ (r ) 2 dΩ 2 ] - Robertson-Walker Metric
- Where:
Uniform curved space metric
a(t) - Scale Factor
a(t0 )=1 (Today)
- Describes how distances expand/contract with time
t - Cosmic time
- Time measured by an observer who sees the Universe expanding
uniformly around him, i.e. by a Fundamental Observer
( r, θ, φ ) - Comoving coordinates
- Point coordinates that remain constant in time during an homogeneous
and isotropic expansion
Curved 4D Spaces: Robertson - Walker Metric 2/3
Note 1
- Homogeneous and isotropic expanding Universe implies:
All informations about its geometry contained within: a(t), κ, R0
- Inflating balloon 2D analogy for expanding Universe:
There is not a centre where the explosion came from
The radial coordinate is ‘time-like’, at the centre it was the
beginning of time, not a particular location
Explosion occured everywhere, at every point of space
Note 2
Scientific
American
Curved 4D Spaces: Robertson - Walker Metric 3/3
Note 3
- RW metric is homogeneous and isotropic:
Any point can be the origin: r = 0
Note 4
- RW metric has preferred inertial reference frames in which the
expansion of the Universe looks isotropic:
Cosmic rest frame : equivalent to the CMB reference frame
Note 5
- Why our bodies do not expand ?
At ‘human’ scales, things held together by e.m. forces
- Why planetary systems, galaxies or galaxy clusters do not expand ?
At those scales they are held together by gravity
RW metric holds only at very large scales (>100 Mpc)
GEOMETRY OF THE UNIVERSE
The Equivalence Principle
General Relativity Concepts
Curved Spaces
The Robertson-Walker Metric
→ Proper Distance
Expansion and redshift
Expansion and CMB temperature
References: Ryden, Introduction to Cosmology – Par. 3.4
Comoving Coordinates
- Consider galaxies during the expansion of the Universe
- The comoving coordinate system is
carried along with the expansion
- The galaxies remain at fixed coordinates
during the expansion
- Their comoving distance is constant
Time
The expansion is entirely taken care
of by the scale factor a(t)
Proper Distance
- We want to calculate the physical distance of a very far galaxy
- Suppose to be at the origin observing the galaxy
at comoving coordinates (r, θ, φ )
- The RW metric at a fixed time t is :
( r,θ,φ )
x
(0,0,0)
dt=0 → ds 2 = a (t ) 2 [dr 2 + Sκ (r )dΩ 2 ]
- Along the photon null geodesic the angles (θ, φ ) are constant:
→ ds = a (t )dr
- Integrating over the radial comoving coordinate:
r
d p (t ) = a(t ) ∫ dr
0
d p (t ) = a(t )r - Proper Distance
The distance measured by a ‘tape measure’ at a given cosmic time
Proper Distance: Hubble’s Law
- Proper Distance : length of the spatial geodesic between two points
when the scale factor is fixed at a value a(t)
- The proper distance rate of change is:
- At the current time t=t0 :
a&
&
&
d p = ar = d P
a
(r& = 0)
v P (t0 ) = H 0 d P (t0 ) - Hubble’s Law H 0 =  
a
a&
  t =t0
Linear relation between galaxy recession speed and proper distance
Note
- During the expansion, the radius of curvature of the Universe
increases at the same rate:
R(t ) = a (t ) R0
Proper Distance: Hubble Distance
- The velocity-distance relation implies a critical distance:
c
= 4300 ± 400 Mpc - Hubble Distance
v P = H 0 d P → d H (t0 ) =
H0
- Points with greater values will have: v P = d&P > c Superluminal speed !
Notes
- Galaxies further than 4300 Mpc move away at speeds greater than c !
- In Special Relativity : objects relative motion at v < c within static space
- In General Relativity : no objection having relative motions at v > c due
to the expansion of space
GEOMETRY OF THE UNIVERSE
The Equivalence Principle
General Relativity Concepts
Curved Spaces
The Robertson-Walker Metric
Proper Distance
→ Expansion and redshift
Expansion and CMB temperature
References: Liddle, Introduction to Modern Cosmology – Par. 5.2
Expansion and Redshift 1/3
- What is the relation between expansion redshift and expansion scale factor ?
- Let’s assume light emitted and detected by two very close objects:
γem
γobs
A
B
dr
- According to Hubble’s law :
dv = Hdr =
- For the Doppler effect we have:
dλ
λem
=
a&
dr
a
dv
λobs − λem
=
c
λem
- The time between emission and detection: dt =
dr
c
Expansion and Redshift 2/3
- Combining the equations:
- Integrating:
dλ
λem
=
dv a& dr a&
da
=
= dt =
c a c
a
a
ln λ = ln a + const
λ∝a
- Instantaneous wavelength
measured at any given time
λobs a(tobs )
=
λem a(tem )
As space expands wavelengths increase proportionally
Example
λobs = 2λem → a (tem ) =
a(tobs )
2
When the light was emitted the Universe was half the present size
Expansion and Redshift 3/3
Notes
- Result completely general Valid also for distant objects
(Rigorous treatment both in Liddle and Ryden books)
- We can imagine the wavelengths being ‘stretched’ by expansion:
The ‘amount of stretch’ tells us how much the Universe expanded
since the emission of light
- Using the redshift definition:
z=
λobs − λem
λem
→
λobs
= 1+ z
λem
1+ z =
a(tobs )
a (tem )
Redshift – Scale factor relation
GEOMETRY OF THE UNIVERSE
The Equivalence Principle
General Relativity Concepts
Curved Spaces
The Robertson-Walker Metric
Proper Distance
Expansion and redshift
→ Expansion and CMB temperature
References: Ryden, Introduction to Cosmology – Par. 2.5
Black Body Radiation 1/4: Definition
Black Body radiation : radiation which is in thermal equilibrium with matter
- Black-body radiation can be obtained keeping radiation inside an
enclosure at T until equilibrium is achieved:
T
T
Iν = Bν (T )
Bν (T )
- Small hole to measure the radiation without disturbing equilibrium
Properties
- Independent enclosure properties/shape
- Dependent only on the temperature T
- Homogeneous and isotropic
ε (ν , T ) - Photons energy density
Black Body Radiation 2/4: Planck’s Law and frequency limits
- The energy density of photons in the frequency range (ν,ν +dν) :
ε (ν , T )
- Planck’s law
Wien
Rayleigh
Jeans
∝ ν2
∝ e − hν
kT
8πh ν 3
ε (ν , T )dν = 3 hν kT dν
c e
−1
h: Planck constant
k: Boltzmann constant
ν
Low ν
High ν
hν << kT → e
hν kT
hν >> kT → e
hν kT
≈ 1 + hν kT
−1 ≈ e
hν kT
ε
R− J
8πν 2
(ν , T ) = 3 kT - Rayleigh-Jeans
c
limit
8πhν 3 − hν
e
ε (ν , T ) =
c3
W
kT
- Wien limit
hν max = 2.82 kT - ε(ν) as function of frequency
Max
λmaxT = 0.0029 [m ⋅ K ] - ε(λ) as function of wavelength
- Wien’s
displacement law
Black Body Radiation 3/4: Properties
- Integrating over all the frequencies:
- Total energy density
ε γ = αT 4
where: α =
- BB spectra at different T
( from Stefan-Boltzmann law )
π 2 k4
15 h 3c 3
= 7.56 ×10 −16 Jm −3K − 4
- In addition:
nγ = β T 3 - Number density of photons
2.404 k 3
7
− 3 −3
where: β =
=
2
.
03
×
10
m
K
2
3 3
π hc
Eγmean ≈ 2.70kT - Mean photon energy
Close to the spectrum ν peak
Monotonicity property:
Every curve at T lies entirely
above all the others at lower T
Black Body Radiation 4/4: Examples
http://solarscience.msfc.nasa.gov/
The Sun
- Approximate BB with T ~ 5800K :
−7
Eγmean ≈ 1.3 eV → λmax = 5 ×10 m
Visible light
Room T
mean
- Assume T ~ 310 K : Eγ
−6
≈ 0.07 eV → λmax = 9 ×10 m
Infrared light
Expansion and CMB Temperature 1/4
- Initially the Universe was hot, dense and opaque
Black Body
- Then when the temperature dropped to ~ 3000K:
Universe became transparent Emission CMB
- Today the CMB temperature is ~ 2.73K :
A factor 1100 lower
What is the relation between CMB temperature and
the expansion scale factor ?
Expansion and CMB Temperature 2/4
- Let’s consider a region of volume V expanding with the Universe: V ∝ a (t ) 3
- BB radiation in V as a gas of photons with energy density and pressure:
ε γ = αT
4
Pγ =
εγ
3
- Applying the First Law of Thermodynamics :
dQ: amount heat flowing in/out gas
dE: change of internal energy
dV: change volume box
P: pressure
dQ = dE + PdV
- The Universe is homogeneous same T everywhere, no heat flow:
dQ = 0
Expansion and CMB Temperature 3/4
- First law applied to an homogeneous expanding Universe:
dE
dV
= − P(t )
dt
dt
- For the CMB photons:
d
αT 4 dV
4
αT V = −
→
dt
3 dt
E = ε γ V = αT 4V
αT 4
P = Pγ =
3
(
α 4T 3
dT
4 4 dV
V = − αT
α 4T
dt
3
dt
3
- Since: V ∝ a (t ) 3
)
dT
dV
1
dV
V + αT 4
= − αT 4
dt
dt
3
dt
1 dT
1 dV
3a 2 da
→
=−
=− 3
T dt
3V dt
3a dt
d
d
→
(ln T ) = − (ln a)
dt
dt
1
T (t ) ∝
a (t )
CMB temperature – Scale factor relation
Expansion and CMB Temperature 4/4
- Comparing emission and observation times, we can rewrite it as:
TCMB (tobs ) a (tem )
=
TCMB (tem ) a (tobs )
CMB temperature - Scale factor relation
CMB temperature dropped by a factor 1100 because the Universe
expanded by the same factor
- Note: When emitted the CMB was a Cosmic Near-Infrared Background
a (tobs )
- Reminding the redshift - scale factor relation: 1 + z =
a(tem )
TCMB (tobs )
1
=
CMB temperature - Redshift relation
TCMB (tem ) 1 + z
The CMB was emitted at a redshift z ~1100
Next Topic: DYNAMICS OF THE UNIVERSE
The Friedmann Equation
The Fluid Equation
The Acceleration Equation
The Equation of State
The Cosmological Constant