Download Cosmology: Einstein

Dark Energy Cosmology
Robert Caldwell
Dartmouth College
INPE Winter School
September 12-16, 2005
Physics & Astronomy
Dartmouth College
Hanover, New Hampshire
Dartmouth Cosmology, Gravitation, and Field Theory Group
Cosmology: the physics of the universe
goals
• understand the origin, evolution, and future of the universe
• determine the composition and distribution of matter and energy
• test gravity on the largest scales
• test the physics of the dark sector: dark matter and energy
• introduce framework and tools of relativistic cosmology
• introduce dark energy cosmology
Cosmology: current status
2005:
Numerous observations and experiments support the hypothesis
that our universe is filled by some sort of dark energy which is
responsible for the cosmic acceleration.
Low density
Little curvature
Accelerating
simple explanation?
Cosmology: current status
Freedman & Turner, Rev Mod Phys 75 (2003) 1433
Special Relativity
Postulate 1
The laws of nature and the results of experiments in a
given reference frame are independent of the
translational motion of the system.
Postulate 2
The speed of light is finite and independent of the
motion of the source.
SR spacetime
Lorentz invariance
t’
t
x
x’
Relativistic Physics
Action: free particle
Geodesic equation
Particle Kinematics
Special Relativity
Experiment: search for preferred-frame effects
eg CMB
laboratory
Mansouri & Sexl, 1977
SR:
Special Relativity
Experiments:
Michelson-Morley: orientation dependence
Kennedy-Thorndike: velocity dependence
Ives-Stillwell: contraction, dilation
Stanwix et al, PRL 95 (2005) 040404
Wolf et al, PRL 90 (2003) 060402
Saathoff et al, PRL 91 (2003) 190403
Precision tests of
Lorentz Invariance
Special Relativity
Experiments:
isotropy of the speed of light
Muller et al, PRL 91 (2003) 020401
Standard Model Extensions (SME)
overview: Bluhm hep-ph/0506054
Muller et al, PRL 91 (2003) 020401
(astrophysical)
Kostelecky et al, PRL 87 (2001) 251304
boost-invariance of neutron
Cane et al, PRL 93 (2004) 230801
Special Relativity
Theory: Lorentz Invariance violations
The idea of a smooth, continuous spacetime breaks down
near the Planck scale
in many theories of quantum gravitational phenomena. As a
practical consequence, such theories predict violations of
Lorentz Invariance in the form of the dispersion relation
overview: Mattingly gr-qc/0502097
Kip Thorne 1994 "Black holes and time
machines: Einstein's outrageous legacy"
• Forbidden decays now allowed
• Relativistic -factor has different meaning
• Possible CPT violation?
numerous astrophysics,
cosmology implications
Special Relativity
Theory: Lorentz Invariance violations
These effects may have greater implications within the full qft of
the standard model
p
k
due to LI violating terms
Implies that different fields would have values
of c that vary by as much as 10%
Collins et al, PRL 93 (2004) 191301
minimal mixing with standard model?
Myers & Pospelov, gr-qc/0402028
Curved-Spacetime Physics
1.
Spacetime, the set of all events, is a 4D manifold with a metric (M,g).
2.
The metric is measurable by rods and clocks.
3.
The metric of spacetime can be put in the Lorentz form momentarily at
any particular event by an appropriate choice of coordinates.
4.
Freely-falling particles, unaffected by other forces, move on timelike
geodesics of the spacetime.
5.
Any physical law that can be expressed in tensor notation in SR has
exactly the same form in a locally-inertial frame of a curved spacetime.
Curved-Spacetime Physics
Action: free particle
Geodesic equation
Gravitation & Inertia
Curved-Spacetime Physics
Newtonian limit: a slowly-moving particle in a weak, stationary grav. field
Curved-Spacetime Physics
Poisson equation
Energy density belongs to the
source stress-energy tensor
Gravitational potential belongs
to the spacetime metric
Laws of fundamental physics as
2nd order differential equations
Curvature
conventions
Wald 1984
examples
metric: -+++
Weak Fields
CODATA Rev Mod Phys 77 (2005) 1
gravitational potentials due to nearby sources
Earth
Sun
Galactic Center
Weak Fields
trace-reversed field
select Lorentz gauge
Cosmology: Einstein
“Cosmological Considerations on the General Theory of Relativity”
Einstein (1917) [see Cosmological Constants, eds. Bernstein & Feinberg]
Copernican Principle gives way to Cosmological Principle
Model the universe as spatially homogeneous, isotropic
Mach’s influence on relativity and inertia
No inertia relative to spacetime, but inertia of masses
relative to one another
Observation and experiment
Account for the small kinetic motions of stars and nebulae
First mathematical model of the universe in general relativity.
Cosmology: Einstein
This is a facsimile of Einstein’s lecture notes for a course he
taught on general relativity in 1919. The final topic of the
course was cosmology, which he had begun to investigate
only two years earlier. On these pages he describes his
methods in constructing the first mathematical model of
cosmology in general relativity. This universe contains nonrelativistic matter, stars and nebulae in agreement with the
contemporary observations, but is spatially-finite following his
failure to find boundary conditions satisfying Mach’s Principle.
The current approach to theoretical cosmology is very similar,
striking a balance between empirical and theoretical inputs.
Einstein’s lecture notes: 1919
Cosmology: Einstein
Homogeneous & Isotropic:
and stationary: A, B depend on r only.
Inertia and Gravitation
inertia
gravitation
require:
test particle momentum, from geodesic equation
Cosmology: Einstein
No Boundary:
spatial-surfaces are closed three-spheres
Matter Content:
Problem!
t-t equation can balance, but not i-i
Cosmology: Einstein
Imbalanced Gravitation:
How to ensure that stars and nebulae reach equilibrium?
~ Newton’s objection to an infinite universe
: a screening length
.
.
.
“…the newly introduced universal constant defines both the mean
density of distribution  which can remain in equilibrium and also the
radius … of the spherical space.”
Cosmology: Einstein
Einstein’s lecture notes: 1919
“Much later, when I was discussing cosmological problems with
Einstein, he remarked that the introduction of the cosmological term
was the biggest blunder of his life.” (George Gamow)
The cosmic expansion of the universe had not been discovered at
the time, so he proceeds to build a static universe by adding “” to
the field equations, a cosmological constant. This seemed to solve a
problem that troubled even Newton -- why did the entire universe
not collapse under its own gravitational attraction. In his view, the 
introduces a screening length which cuts off the influence of the
gravitational potential beyond a certain radius, thereby allowing the
motion of stars and nebulae to approach equilibrium. In fact, the
compact geometry of this model sets a maximum distance for the
reach of gravity, whereas the addition of  only increases largescale kinetic motions. The gravitational field equations momentarily
balance, like a tug-of-war, but this model universe is unstable to
expansion or collapse.
Willem de Sitter found inspiration in Einstein’s model, even though
he wrote that the introduction of  “detracts from the symmetry and
elegance of Einstein's original theory, one of whose chief attractions
was that it explained so much without introducing any new
hypothesis or empirical constant.” Shortly after the publication of
Einstein’s results, he proposed a model universe containing only .
This idealized solution features accelerated cosmic expansion, and
closely resembles the origin of the Big Bang in the inflationary era,
or our future in a Big Chill.
Cosmology: deSitter
“On Einstein’s Theory of Gravitation and Astronomical Consequences”
deSitter (1917) [see Cosmological Constants, eds. Bernstein & Feinberg]
Copernican Principle gives way to Cosmological Principle
Mach’s influence on relativity and inertia
Observation and experiment
Universe contains no matter: stars and nebulae as test particles
Confound Machians: relativity of inertia without “distant stars”
meet Einstein’s requirements
Idealized spacetime of the cosmological constant
Action Principle
Einstein-Hilbert
metric variation
g,  variation
Palatini identity
Action Principle
Einstein’s cosmological term seems inevitable!
“The genie () has been let out of the bottle”