The%20Darkness%20%20of%20the%20Universe:

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The Darkness of the Universe Acceleration
and Deceleration
Eric Linder Lawrence Berkeley National Laboratory
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Discovery! Acceleration
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Cosmic Concordance

• Supernovae alone ? Accelerating
  expansion
• ? gt 0
• CMB (plus LSS)
• ? Flat universe
• ? ? gt 0
• Any two of SN, CMB, LSS
• ? Dark energy 75

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Acceleration and Particle Physics
The conformal horizon scale (aH)-1 tells us when
a comoving scale (e.g. perturbation mode) leaves
or enters the horizon.
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Acceleration Curvature
The Principle of Equivalence teaches that
Acceleration Gravity Curvature
Acceleration ? over time will get vgh, so z
v gh (gravitational redshift). But, t??t0 ?
parallel lines not parallel (curvature)!
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Equations of Motion
Expansion rate of the universe a(t) ds2
?dt2a2(t)dr2/(1-kr2)r2d?2 Friedmann
equations (å/a)2 H2 (8?/3Mp2) ?m ??
/a -(4?/3Mp2) ?m ?? 3p?
Einstein-Hilbert action S ?d4x?-g R/2?
L? Lm
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Spacetime Curvature
Ricci scalar curvature R R?? 6 a/a
(å/a)2 6 ( a/a H2) Define reduced
scalar curvature R R/(12H2) (1/2) 1 aa/
å2 (1/2)(1-q) Note that division
between acceleration and deceleration occurs for
R 1/2 (q0). Superacceleration (phantom models)
is not (a) gt 0, but (a/a) gt 0, i.e. R gt 1.
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Todays Inflation
To learn about the physics behind dark energy we
need to map the expansion history.
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Equations of Motion
Expansion rate of the universe a(t) ds2
?dt2a2(t)dr2/(1-kr2)r2d?2 Friedmann
equations (å/a)2 H2 (8?/3Mp2) ?m ??
/a -(4?/3Mp2) ?m ?? 3p?
Einstein-Hilbert action S ?d4x?-g R/2?
L? Lm
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Scalar Field Theory
Scalar field Lagrangian - canonical, minimally
coupled L? (1/2)(???)2 - V(?) Noether
prescription ? Energy-momentum tensor T??(2/?-g)
?(?-g L )/?g?? Perfect fluid form (from RW
metric)
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Energy density ?? (1/2) ? 2 V(?)
(1/2)(??)2 Pressure p? (1/2) ? 2 - V(?) -
(1/6)(??)2
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Scalar Field Equation of State
Equation of state ratio w p/? Klein-Gordon
equation (Lagrange equation of motion)
Continuity equation follows KG equation (1/2)?
2 6H (1/2)? 2 -V ? - V 3H (?p)
-V d?/dln a -3(?p) -3? (1w)
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Equation of State
Reconstruction from EOS ?(a) ?? ?c exp 3
?dln a 1w(z) ?(a) ?dln a H-1 sqrt ?(a)
1w(z) V(a) (1/2) ?(a) 1-w(z) K(a)
(1/2)? 2 (1/2) ?(a) 1w(z)
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Equation of State
Limits of (canonical) Equations of State w
(K-V) / (KV) Potential energy dominates (slow
roll) V gtgt K ? w -1 Kinetic energy dominates
(fast roll) K gtgt V ? w 1 Oscillation about
potential minimum (or coherent field, e.g.
axion) ?V? ?K? ? w 0
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Equation of State
Examples of (canonical) Equations of State
d?/dln a -3(?p) -3? (1w) ? (Energy per
particle)(Number of particles) / Volume E N
a-3 Constant w implies ? a-3(1w) Matter
Ema0, Na0 ? w0 Radiation E1/?a-1, Na0 ?
w1/3 Curvature energy E1/R2a-2, Na0 ?
w-1/3 Cosmological constant EV, N a0 ?
w-1 Anisotropic shear w1 Cosmic String
network w-1/3 Domain walls w-2/3
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Expansion History
Observations that map out expansion history a(t),
or w(a), tell us about the fundamental physics of
dark energy. Alterations to Friedmann framework
? w(a)
Suppose we admit our ignorance H2 (8?/3) ?m
?H2(a) Effective equation of state w(a) -1 -
(1/3) dln (?H2) / dln a Modifications of the
expansion history are equivalent to time
variation w(a). Period.
gravitational extensions or high energy physics
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Expansion History
For modifications ?H2, define an effective scalar
field with V (3MP2/8?) ?H2 (MP2H02/16?) d
?H2/d ln a K - (MP2H02/16?) d ?H2/d ln a
Example ?H2 A(?m)n w -1n Example ?H2
(8?/3) g(?m) - ?m w -1 (g?-1)/ g/?m - 1
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Weighing Dark Energy
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Exploring Dark Energy
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Dark Energy Models

• Scalar fields can roll
• fast -- kination Tracking models
• slow -- acceleration Quintessence
• steadily -- acceleration deceleration Linear
  potential
• oscillate -- potential minimum, pseudoscalar,
  PNGB V ?n

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Power law potential
Normal potentials dont work V(?) ?n have
minima (n even), and field just oscillates,
leading to EOS w (n-2)/(n2) n 0 2 4 8 w -
1 0 1/3 1
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Oscillations
Oscillating field w (n-2)/(n2) Take osc. time
ltlt H-1 and ? constant over osc. ??2? ?dt ?2
/ ?dt ?d? ? / ?d? /? 2? ?d? 1-V/Vmax1/2
/ 1-V/Vmax-1/2 If V Vmax(? /?max)n then ?w?
-1 2 ?01dx (1-xn)1/2 / ?01dx (1-xn)-1/2
-1 2n/(n2)
Turner 1983
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Linear Potential
Linear potential Linde 1986 V(?)V0?? leads to
collapsing universe, can constrain tc
curves of ?
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Tracking fields
Can start from wide variety of initial
conditions, then join attractor trajectory of
tracking behavior.
Criterion ? VV?/(V?)2 gt 1, d ln (?-1)/dt ltltH.
However, generally only achieves w0 gt -0.7.
Successful model requires fast-slow roll.
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Quintessence

• Interesting models have dark energy
• dynamically important,
• accelerating,
• not ?
• ?(1w)? ?(1w) HMp
• Damped so H V?, and timescale is H-1.
• Therefore ?? Mp.
• Unless 1w ltlt 1, then ?? ltlt Mp and very hard to
  reconstruct potential.

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Dark Energy Models
Inverse power law V(?) ?-n SUGRA V(?) ?-n
exp(??2) Running exponential V(?) exp-
?(?)? PNGB or axion V(?) 1cos(?/f)
Albrecht-Skordis V(?) 1c1? c2?2
exp(-??) Tachyon V(?) cosh(??)-1n Stochastic
V(?) 1sin(?/f) exp(-??) ...
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Tying HEP to Cosmology
w(a) w0wa(1-a)
Accurate to 3 in EOS back to z1.7 (vs. 27 for
w1). Accurate to 0.2 in distance back to
zlss1100!
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Scalar Field Dynamics
The cosmological constant has w-1constant.
Essentially no other model does. Dynamics in the
form of w/H w? dw/dln a can be detected by
cosmological observations. Dynamics also
implies spatial inhomogeneities. Scale is given
by effective mass meff ?V? This is of order H
10-33 eV, so clustering difficult on subhorizon
scales. Vaguely detectable through full sky
CMB-LSS crosscorrelation.
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Growth History

• While dark energy itself does not cluster much,
  it affects the growth of matter structure.
• Fractional density contrast ? ??m/?m evolves as
• ? 2H ? 4?G?m ?
• Sourced by gravitational instability of density
  contrast, suppressed by Hubble drag.
• Matter domination case
• ? a-3 t-2, H (2/3t). Try ? tn.
• Characteristic equation n(n-1)(4/3)n-(3/2)(4/9)0
  . Growing mode n2/3, i.e. ? a

..
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Growth History
Growth rate of density fluctuations g(a)
(??m/?m)/a
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Gravitational Potential
Poisson equation ?2?(a)4?Ga2 ??m 4?G?m(0)
g(a) In matter dominated (hence decelerating)
universe, ??m/?m a so gconst and ?const.
Photons dont interact with structure growth
blueshift falling into well matched by redshift
climbing out. Integrated Sachs-Wolfe (ISW)
effect 0.
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Inflation, Structure, and Dark Energy

• Matter power spectrum
• Pk ? (??m/?m)2 ? kn
• Scale free (primordially, but then distorted
  since comoving wavelengths entering horizon in
  radiation epoch evolve differently - imprint
  zeq).
• Potential power spectrum
• ?2 ?L L4 ? (??m/?m)2 ?L L4 k3Pk L1-n
• Scale invariant for n1 (Harrison-Zeldovich).
• CMB power spectrum
• On large scales (low l), Sachs-Wolfe dominates
  and power l(l1)Cl is flat.

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Deceleration and Acceleration
CMB power spectrum measures n-1 and
inflation. Nonzero ISW measures breakdown of
matter domination at early times (radiation) and
late times (dark energy). Large scales (low l)
not precisely measurable due to cosmic variance.
So look for better way to probe decay of
gravitational potentials. Next The
Darkness of the Universe 3 Mapping Expansion
and Growth