Holographic Dark Energy
Preety Sidhu
5 May 2006
Black Holes and Entropy
• Black holes are
“maximal entropy
objects”
• Entropy of a black hole
proportional to surface
area of event horizon
• Max entropy for volume
of space goes as
bounding surface area,
not mass
The Holographic Principle
• All information about a physical system in some
region of space is encoded in its boundary surface,
not its volume
• Like all the information in a room encoded in its
walls
Information Entropy
• Information entropy (or
Shannon entropy) measure
of “randomness” or
“uncertainty” in a signal
• Thermodynamic entropy
like amount of Shannon
entropy “missing” between
classical macroscopic
variables and full
microscopic description of
system’s state
• Entropy ultimately
measured in bits or nats
• 1 bit = (ln 2) nats  0.69
nats
• 1 nat ~ 4 Planck areas
• Total bits related to
matter/energy degrees of
freedom
• Maximum info density, for
given volume, about
enclosed particle states
• Matter cannot be infinitely
subdivided
Holographic Cosmology
• Related to the (poorly
understood) principles of
quantum gravity
• Bekenstein max entropy for
weakly self-gravitating
physical system [4D flat
spacetime]:
S ≤ 2πER
• Taken to be max
holographic entropy for
universe
Sizes and Scales
• In quantum field theory
– UV cutoff: short wavelength, high energy bound
– IR cutoff: long wavelength, low energy bound
– Related by limits set by black hole formation
• UV limit ~ Planck length
• IR limit ~ “size of universe”
– Particle horizon: largest comoving distance from which
light could have reached observer today
– Event horizon: largest comoving distance from which
light will ever reach observer
Vacuum Fluctuations
• Uncertainty principle for quantum vacuum energy
fluctuations, with N degrees of freedom
• Holographic principle sets N within UV and IR
cutoffs
• One degree of freedom is a maximum entropy of
one Boltzmann unit k
• Corresponds to
Ω = π/4
(z) = -1+(1- π/4)z
• No adjustable parameters, consistent with recent
cosmological observations
Holographic Bounds
• Holographic entropy bound violated for
closed universe of particle horizon size
• Can solve for closed universe:
– Replace particle horizon with event horizon
– Add negative pressure component
– Or Hubble scale IR cutoff with non-minimal
coupling to scalar field