70% Dark Energy
Dark Energy
Sean Carroll, Caltech
SSI 2009
25% Dark
5% Ordinary
Evidence for Dark Energy
Vacuum Energy and the Cosmological Constant
Dynamical Dark Energy and Quintessence
Was Einstein Right?
1. Evidence for Dark Energy
Dec. 1997: Something was in the air!
- age of the universe
- absence of power on small scales
- measurements of matter density
Theorists had a favorite model: a flat universe,
full of matter (ordinary + cold dark), with primordial
scale-free perturbations.
That model couldn’t be right! Something had to give -“flat,” “cold,” “scale-free,” or perhaps even “matter.”
The Friedmann equation with matter and radiation:
Multiply by a2 to get:
If a is increasing, each term
on the right is decreasing;
we therefore predict the
universe should. be
decelerating (a decreasing).
> Big Bang <
o of the universe, using type Ia supernovae as
standardizable candles.
SN 1994d
Result: supernovae are
dimmer than expected.
The universe is not
decelerating at all,
it’s accelerating.
Can’t be explained by
matter and radiation.
[Riess et al.; Perlmutter et al.; Knop et al.]
What could make the universe accelerate? From the
Friedmann equation, we need something that
doesn’t dilute away as the universe expands.
Call it dark energy.
> Big Bang <
If the dark energy density evolves as
then a DE-dominated universe obeys
which implies acceleration for
But people usually use the “equation-of-state parameter”
so that acceleration happens for
Fun non-Euclidean fact: “constant expansion rate” = “acceleration.”
The expansion rate is described by
the Hubble constant, H, relating the
distance of a galaxy to its velocity.
Einstein tells us that the Hubble
constant (squared) is proportional
to the energy density .
If  is constant (vacuum energy),
H will be constant. But the distance
d to some particular galaxy will be
increasing, so from v = Hd its
apparent velocity will go up:
it will accelerate away from us.
How can we check this idea?
Density parameter,  :
Then, if we know  we can instantly infer the geometry
of space:
Matter (ordinary + dark) only accounts for  ≈ 0.3, implying
negative curvature. Triangles should add up to < 180o.
CMB temperature anisotropies provide a standard ruler.
They were produced about 400,000
years after the Big Bang,and should
be most prominent at a physical
size of 400,000 light years across.
Tot = [peak(deg)]-1/2.
Observation: peak = 1o.
The universe is flat:
Tot = 1 .
[Miller et al.; de Bernardis et al; WMAP]
M = 0.3,
L = 0.7 .
2. Vacuum Energy (the Cosmological Constant)
What we know about dark energy:
smoothly distributed through space
varies slowly (if at all) with time
 ≈ constant (w ≈ -1)
Dark energy could be exactly
constant through space and
time: vacuum energy (i.e.
the cosmological constant L).
(artist's impression
of vacuum energy)
Energy of empty space.
People sometimes pretend there is a difference
between a cosmological constant,
and a vacuum energy,
There’s not; just set
Problem One:
Why is the vacuum
energy so small?
We know that virtual particles
couple to photons (e.g. Lamb
shift); why not to gravity?
Naively: vac = ∞, or at least vac = EPl/LPl3 = 10120 vac(obs).
Problem Two:
Why is the vacuum
energy important now?
We seem to be living in a
special time. Copernicus
would not be pleased.
Could we just be lucky?
The Gravitational Physics Data Book:
Newton's constant:
G = (6.67 ± 0.01) x 10-8 cm3 g-1 sec-2
Cosmological constant:
L = (1.2 ± 0.2) x 10-55 cm-2
If we set h = c = 1, we can write
G = EPlanck-2 and vac = Evac4 , and
EPlanck = 1027 eV ,
Different by 1030.
Evac = 10-3 eV .
Supersymmetry can squelch the vacuum energy; unfortunately,
in the real world it must be broken at ESUSY ~ 1012 eV.
Typically we would then expect
which is off by 1015. But if instead we were able to predict
it would agree with experiment. (All we need is a theory
that predicts this relation!)
1027 eV
1012 eV
10-3 eV
Is environmental selection at work?
String theory has extra
dimensions, with a vast
“landscape” of ways to hide
them. Perhaps 10500 or more.
The “constants of nature”
we observe depend on the
shape and size of the
compact manifold.
Everything changes from
one compactification to
the next, including the
value of the vacuum energy.
[Bousso & Polchinski; Kachru et al.]
Maybe each compactification actually exists somewhere.
Regions outside our observable universe, where the laws
of physics and constants of nature appear to be different.
In that case, vacuum
energy would be like
the weather; not a
fundamental parameter,
but something that
depends on where you
are in the universe.
Therefore (so the reasoning goes), it's hardly surprising
that we find such a tiny value of the vacuum energy –
regions where it is large are simply inhospitable.
3. Dynamical Dark Energy (Quintessence)
Dark energy doesn’t vary quickly, but maybe slowly.
[Wetterich; Peebles & Ratra;
Caldwell, Dave & Steinhardt; etc.]
This is an observationally interesting possibility.
Might be relevant to the cosmological constant problem
or the coincidence scandal -- somehow.
A problem: mass.
An excitation of the quintessence field is
a quintessence particle:
In quantum field theory, we don’t see the “bare”
particle; we see the collective effect of the sum
over fluctuating (virtual) quantum fields.
The effect of these virtual particles is to drive
the mass up! Unless there is a symmetry or other
physics that cuts it off.
Every particle we have observed has a symmetry
keeping its mass low. (The Higgs is a mystery.)
A field with a large mass rolls
quickly down its potential.
Quintessence requires
That’s very small. A new fine-tuning.
A related problem: interactions.
If A couples to B, and B to C, A should couple to C.
It’s hard to keep a new field completely isolated;
it should couple to Standard Model particles.
torsion-balance experiment
Coupling to a low-mass (long-range)
field implies a fifth force of nature,
which can be searched for in
laboratory experiments.
[Adelberger et al.]
Also: gradual evolution
of physical constants as the
field evolves.
Limit: couplings must be
suppressed by ~ 105 MP.
[Webb et al.]
Both fine-tunings -- mass and interactions -- can be
addressed in one fell swoop, by imagining a
slightly broken symmetry
[Frieman et al; Carroll]
Then the quintessence
is a pseudo-NambuGoldstone boson,
with a cosine potential
and naturally small
mass and interactions.
But one interaction is allowed -- a parity-violating
term of the form
, coupling quintessence to
the electromagnetic fields.
This interaction produces cosmological birefringence:
polarization vectors rotate as they travel through
the evolving scalar field.
WMAP 5-year data:
Radio galaxies also provide
interesting constraints.
1. A cosmological constant fits the data, at the
expense of a dramatic fine-tuning.
2. Dynamical models introduce new fine-tunings,
in the form of the small mass and couplings of
the new scalar field.
3. Dynamical models have not yet shed any light on
the cosmological constant problem or the
coincidence scandal.
4. Modified Gravity
Simplest possibility: replace
[Carroll, Duvvuri,
Trodden & Turner 2003]
The vacuum in this theory is not flat
space, but an accelerating universe!
But: the modified action brings a
new tachyonic scalar degree of
freedom to life. A scalar-tensor theory of gravity.
Scalar-Tensor Gravity
Introduce a scalar field (x) that determines the
I strength of gravity. Einstein's equation
is replaced by
variable “Newton's constant”
extra energy-momentum from 
The new field (x) is an extra degree of freedom;
an independently-propagating scalar particle.
The new scalar doesn’t
interact directly with
matter, because we say
so. But it does influence
the metric.
A natural value for the
Brans-Dicke parameter
 would be
where  = 1 is GR.
[Chiba 2003]
Experiments imply
 > 40,000 .
Loophole: the Chameleon Effect.
Curvature contributes to
the effective potential
for . With the right
bare potential, the field
can be pinned (with
large mass) in dense
regions, e.g. the galaxy.
Deviations from GR can be
hidden on sub-galactic scales.
[Khoury & Weltman;
Hu & Sawicki]
Dvali, Gabadadze, & Porrati (DGP) gravity: an infinite
extra dimension, with gravity stronger in the bulk;
5-d kicks in at large distances.
4-d gravity
5-d GR
5-d gravity suppressed by rc ~ H0-1
~ Hgravity
suppressed by rc ~ H0-1
r* = (rS rc2)1/3
rS = 2GM
4-d GR
[Dvali, Gabadadze & Porrati 2000]
Self-acceleration in DGP cosmology
The DGP version of the Friedmann equation is (naturally):
This exhibits self-acceleration: for  = 0, there is a
de Sitter solution with H = 1/rc = constant. However:
The acceleration is somewhat mild; think weff ~ -0.7.
Inconsistent with present data at about 5.
Fluctuations of the brane have negative energies
(ghosts). Hard to fix this problem.
1. We would expect GR to be modified on short
scales, not on long scales, but it could happen.
2. f(R) gravity can fit the data, but only through
various fine-tunings (over and above the
cosmological constant and coincidence problems)
and the chameleon mechanism.
3. DGP gravity doesn’t really fit the data , and has
issues with negative-energy ghosts.
Bottom line:
Dark energy is probably a cosmological constant.
Gravity is probably described by GR on large scales.

Sean Carroll