Download Cosmological Constant

Chapter 23
Cosmology
(the study of the big and the small)
Fundamental Particles Primer (the small)
Hadrons – interact via strong
force and have internal structure
(made of quarks)
Baryons – massive (ex.
proton and neutron)
composite fermions
Mesons – less massive
composite bosons
Quark combination for:
Proton = uud
Neutron = udd
Leptons – do not interact by
strong force but obey Pauli
exclusion (ex. electron and
neutrino)
Photons – massless with energy E = hc/λ
Quark properties
The Cosmological Principle (the big) : the universe is
homogeneous and isotropic on sufficiently large scales
The universe looks
pretty much like this
everywhere – “walls”
and “voids” are present
but no larger structures
are seen….
It follows that the
Universe has no “edge”
or center.
But is the Universe the same at all times?
Recall universal expansion:
recession velocity = Ho x distance
(Hubble Law)
The cosmological principle does not imply that the Universe
is constant at all times.
This was once thought to be the case - Steady State Universe
Universal expansion points to a beginning of the Universe
and implies that the Universe is changing over time…
The Hot Big Bang model – Universe has expanded from an
initial hot and dense state to its current cooler and lowerdensity state
First Cosmological Observation:
The night sky is dark! (Olbers’ Paradox)
Every line of sight would eventually hit a star in an infinite and
eternal Universe and the sky would be always bright
 Number of stars increases by r2 for each shell
 brightness decreases by r2 for each shell
 Brightness per shell is a constant!
The fact that the sky is not
uniformly bright indicates that
either
• the Universe has a finite size
and/or
• the Universe has a finite age
Second Cosmological Observation:
Hubble Law
Indicates homogeneous, isotropic expansion where
r(t) = a(t) ro
ro = r(to) is the separation between two points at the current time to
a(t) is a dimensionless scale factor
Distance between two points increases with velocity
Which takes form of the Hubble law v(t) = H(t) r(t) where
H(t) is the Hubble parameter and Ho = H(to) is the Hubble Constant
Thus, the primary resolution to Olbers’ paradox is that the Universe has a finite
age  stars beyond the horizon distance ro ~ c/Ho are invisible to us because
their light hasn’t had time to reach us.
Third Cosmological Observation:
Cosmic Microwave Background
•At the time of the Big Bang, the
Universe was very HOT – emitting
gamma-rays – the hot dense universe
radiates as a blackbody.
•This radiation has been redshifted by
the expansion of the Universe so that
the peak of the radiation is now at
radio/microwave wavelengths.
•Current temperature is To = 2.727 K
Third Cosmological Observation:
Cosmic Microwave Background
The Universe was
opaque to light at the
time of the Big Bang
(T>104 K) due to high
particle density,
ionization, and
scattering
When cooled to ~3000 K during expansion, recombination of atoms allowed
photons to pass without as much scattering or absorbing.
At current temp (To = 2.7 K), CMB is cooled by a factor of 1100.
T ~ 1/a(t)  temperature is inversely proportional to the scale factor
λ ~ a(t)  wavelength is directly proportional to the scale factor
Discovery of the CMB
• First observed (inadvertently) in 1965 by
Arno Penzias and Robert Wilson at the Bell
Telephone Laboratories in Murray Hill, NJ
• Detected excess noise in a radio receiver
peak emission at BB temp = 3 degrees
• In parallel, researchers at Princeton were
preparing an experiment to find the CMB.
• When they heard about the Bell Labs result
they immediately realized that the CMB had
been found
•The result was a pair of papers in the Physical Review: one by Penzias and Wilson
detailing the observations, and one by Dicke, Peebles, Roll, and Wilkinson giving the
cosmological interpretation.
•Penzias and Wilson shared the 1978 Nobel prize in physics
•Almost immediately after its detection, the Steady State theory was dead
These three cosmological observations
underpin the concept of the Hot Big Bang
model – Universe has expanded from an initial
hot and dense state to its current cooler and
lower-density state
…but how exactly does the Universe expand?
Newtonian Gravity in Cosmology
What can we learn about the evolution of an expanding Universe by applying
Newtonian gravity?
Assume Isotropy  Universe is spherically symmetric from any point – spherical
volume evolves under its own influence and homogeneity  (r) = constant
For a test mass moving on the surface
of a sphere with mass M at position r
(d2r/dt2) = -GM/r(t)2
Multiply each side by dr/dt and integrate
over time to get
½ (dr/dt)2 = GM/r(t) + k
(23.31)
(23.27)
Divide each side by (roa)2/2 to get
(23.28)
Where k is an integration constant.
Now let the sphere have any old mass
and radius
M = (4π/3) ρ(t) r(t)3 & use r(t) = a(t) ro
H(t)2 =
Friedmann Equation
(23.32)
½ (dr/dt)2 = GM/r(t) + k
(23.28)
The future of a self-gravitating sphere depends on the sign of k
k>0
Right hand side of equation is always positive
Left hand side is always greater than zero
Expansion continues forever!
Open or Unbound Universe
k<0
Right hand side is zero when r reaches
some maximum rmax = GM/(k)
After this, the Universe starts to collapse
Closed or Bound Universe
k=0
The Universe expands at an ever decreasing rate (dr/dt)  0 as t  infinity
Borderline Universe or Marginally Bound
H(t)2 =
(23.32)
For a given value of H(t) there is a critical mass density for which k=0,
where gravity is just sufficient to halt the expansion
ρc = 3H(t)2/(8πG)
At the present time:
Ho = 70 km/s/Mpc, then
ρc,o ≈ 1.4 x 1011 Msun/Mpc3
(about one H atom in 200 L volume of space)
The fate of the
Universe
depends on its
density…
High density = enough
matter to gravitationally
halt expansion and cause
gravitational collapse
Low density = not enough
gravitational attraction to
stop expansion…it goes on
forever
Except in the case of a positive Cosmological Constant –
introduced by Einstein in context of General Relativity
Cosmology and General Relativity
Einstein’s principle of equivalence led to the understanding that
space-time is curved in the presence of a gravitational field. The
mass/energy density of the Universe determines the geometry of
space-time over large scales in the Universe.
Describe curvature of space (in 2-d) in terms of angles/areas of a
β
triangle.
If the angles of a triangle are α, β, γ then
α
γ
2
α + β + γ = 180° + (κA/rc )
where κ is the curvature constant, A is the areas of the triangle and rc is
the radius of curvature
κ = +1
Bound/Closed Universe
Sphere  finite area, no edge/boundary
κ = -1
Open Universe
Hyperboloid  infinite area, no edge/boundary
κ=0
Flat/Marginally Bound/Critical Universe
Plane  infinite area, no edge/boundary
How is space curved?
180
Deviations of α + β + γ from 180 degrees are tiny unless the area of the
triangle is comparable to r2c,o. Some simple observations show that if the
Universe is curved, the radius of curvature must be close to the Hubble
distance, c/Ho ~ 4300 Mpc. To see why, consider looking at a galaxy with
diameter D a distance d away.
If κ = 0, α = D/d (small angle formula)
If κ > 0, α > D/d and mass-energy content acts like a magnifying lens
when d = π rc,o, galaxy size would fill the sky – not observed!
galaxy would also be seen at d+Co, d+2Co, etc. where Co = 2πrc,o
If κ < 0, α < D/d and Universe acts like demagnifying lens
Objects at d much greater than rc will be exponentially tiny
Since galaxies are resolved in angular size to distances comparable to Hubble
distance, and the above effects are not seen, we conclude radius of curvature is
comparable to or larger than Hubble distance. Universe consistent with flat.
Metrics of Space-time
Compute distances between 2 events or objects in 4-d space-time
Distances in 3-d space are given by the metric (omitting parentheses)
Including time and switching to spherical coordinates gives Minkowski Metric
used in SR
For an expanding, flat Universe we use the Robertson-Walker Metric
Where a is the scale factor of the Universe and r, θ, φ are the comoving
coordinates of a point in space. If the expansion of the universe is
homogeneous and isotropic, comoving coordinates are constant with time.
The photons from distant galaxies follow null geodesics (where dl=0).
Proper distance: length of geodesic
between two points when the time and
scale factor are fixed (assume flat
Universe with θ and φ constant)
Since proper distance is impossible to measure, we measure photons from the
galaxy emitted at time te < to, remembering that photons follow null geodesics.
Relating this to something we can observe…
Redshift – which tells us scale factor at te
Typo in book! Should be λo/λe = a(to)/a(te)
In GR, functional form of a(t), curvature constant κ and rc,o are
determined by the Field Equations. These equations link curvature to
energy density and pressure at each point in space-time through the
relativistic Friedmann Equation
Newtonian
Main differences with Newtonian form:
mass density ρ  energy density u
(photons have energy and contribute as well as massive particles)
kκ
(positively/negatively curved space rather than open/closed Universe)
Addition of Λ/3 term on right – Cosmological Constant (with units 1/time2)
This term adds a constant (with time) energy density to the Universe 
History of the Cosmological Constant
Newton understood the nature of gravity on a Universe that contains
matter. He postulated that either
1) The Universe is infinite in volume and mass (keeping it from gravitational collapse).
2) The Universe is expanding fast enough to overcome the attractive force of gravity.
3) The Universe is collapsing (either #2 or #3 imply a beginning of the Universe).
Since Newton believed the universe was eternal and unchanging, he believed that the
universe was therefore infinite.
Solving Einstein’s field equations resulted in the constant of integration Λ. This energy
density could balance the mass energy density and allow for a static Universe that
contains matter – seemed a great finding to support the idea of a Steady-State Universe!
However, once Hubble’s observations of an expanding Universe were
discovered in 1920, Einstein realized a non-zero Λ was unnecessary (#2 above)
and called it his “greatest blunder”.
In fact, both Hubble and Einstein were correct – the Universe is expanding and
changing which does point to a beginning (BB), but this does not exclude the
possibility of a non-zero cosmological constant and, in fact, current estimates
predict that it is non-zero.
Rewrite Friedmann
equation in terms of
energy density
components
ur = radiation density (from relativistic particles like photons)
um = matter density (non-relativistic like protons, electrons, WIMPS)
uΛ = lambda density (vacuum density from cosmological constant)
Flatness of Universe implies ur + um + uΛ = uc (critical density)
Components of Universe often given in terms of a dimensionless
Density Parameter Ω(t) = u(t)/uc(t)
where Ω<1 is a negatively curved Universe and Ω>1 is positively curved.
Thus, knowing how the Universe expands with time requires knowing how
much energy density is in radiation, matter and Λ today and how radiation
and matter density evolve with time.