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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

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Dark energy cosmology l.jpg

Dark Energy Cosmology

Robert Caldwell

Dartmouth College

INPE Winter School

September 12-16, 2005

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Physics & Astronomy

Dartmouth College

Hanover, New Hampshire

Dartmouth Cosmology, Gravitation, and Field Theory Group

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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

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Cosmology: current status

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


simple explanation?

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Cosmology: current status

Freedman & Turner, Rev Mod Phys 75 (2003) 1433

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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





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Relativistic Physics

Action: free particle

Geodesic equation

Particle Kinematics

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Special Relativity

Experiment: search for preferred-frame effects

eg CMB


Mansouri & Sexl, 1977


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Special Relativity


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

Precision tests of Lorentz Invariance

Saathoff et al, PRL 91 (2003) 190403

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Special Relativity


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


Kostelecky et al, PRL 87 (2001) 251304

boost-invariance of neutron

Cane et al, PRL 93 (2004) 230801

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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

Kip Thorne 1994 "Black holes and time machines: Einstein's outrageous legacy"

overview: Mattingly gr-qc/0502097

  • Forbidden decays now allowed

  • Relativistic -factor has different meaning

  • Possible CPT violation?

numerous astrophysics, cosmology implications

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Special Relativity

Theory: Lorentz Invariance violations

These effects may have greater implications within the full qft of the standard model



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

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Curved-Spacetime Physics

Spacetime, the set of all events, is a 4D manifold with a metric (M,g).

The metric is measurable by rods and clocks.

The metric of spacetime can be put in the Lorentz form momentarily at any particular event by an appropriate choice of coordinates.

Freely-falling particles, unaffected by other forces, move on timelike geodesics of the spacetime.

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.

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Curved-Spacetime Physics

Action: free particle

Geodesic equation

Gravitation & Inertia

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Curved-Spacetime Physics

Newtonian limit: a slowly-moving particle in a weak, stationary grav. field

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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

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Wald 1984

metric: -+++


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Weak Fields

CODATA Rev Mod Phys 77 (2005) 1

gravitational potentials due to nearby sources



Galactic Center

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Weak Fields

trace-reversed field

select Lorentz gauge

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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.

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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 non-relativistic 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

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Cosmology: Einstein

Homogeneous & Isotropic:

and stationary: A, B depend on r only.

Inertia and Gravitation




test particle momentum, from geodesic equation

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Cosmology: Einstein

No Boundary:

spatial-surfaces are closed three-spheres

Matter Content:

t-t equation can balance, but not i-i


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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.”

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Cosmology: Einstein

“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 large-scale 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.

Einstein’s lecture notes: 1919

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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

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Action Principle


metric variation

g,  variation

Palatini identity

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Action Principle

Einstein’s cosmological term seems inevitable!

“The genie () has been let out of the bottle”