11

TheThe Darkness Darkness of the Universe:of the Universe:

Acceleration and DecelerationAcceleration and Deceleration

Eric Linder Lawrence Berkeley National Laboratory

22

Discovery! AccelerationDiscovery! Acceleration

33

acceler

ating

deceler

atingacc

elerating

decelerating

cf. Tonry et al. (2003)

Cosmic ConcordanceCosmic Concordance

• Supernovae alone

Accelerating expansion

> 0

• CMB (plus LSS)

Flat universe

> 0

• Any two of SN, CMB, LSS

Dark energy ~75%

44

Acceleration and Particle PhysicsAcceleration and Particle Physics

Key element is whether (aH)-1= å-1 is increasing or decreasing. I.e. is there acceleration: >0.

Also, å~aH~H/T~T/Mp for “classical” radiation, but during inflation this redshifts away and quantum particle creation enters.

a..

Com

ovin

g sc

ale

å-1

Time

horizon scale

Inflation

The conformal horizon scale (aH)-1 tells us when a comoving scale (e.g. perturbation mode) leaves or enters the horizon.

55

Acceleration = CurvatureAcceleration = Curvature

The Principle of Equivalence teaches that

Acceleration = Gravity = Curvature

Acceleration over time will get v=gh, so z = v = gh (gravitational redshift).

But, tt0 parallel lines not parallel (curvature)!

t0

t´Height

Time

66

Equations of MotionEquations of Motion

Expansion rate of the universe a(t)

ds2 = dt2+a2(t)[dr2/(1-kr2)+r2d2]

Friedmann equations

(å/a)2 = H2 = (8/3Mp2) [ m + ]

/a = -(4/3Mp2) [ m + +3p ]

Einstein-Hilbert action

S = d4x-g [ R/2 + L+ Lm ]

a..

77

Spacetime CurvatureSpacetime 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 (q=0).

Superacceleration (phantom models) is not (a) > 0, but (a/a) > 0, i.e. R > 1.

....

.. ... .

..

88

Today’s InflationToday’s Inflation

To learn about the physics behind dark energy we need to map the expansion history.

Subscripts label acceleration:

R = (1-q)/2

q = -a /å2

R =1/4 EdS R =1/2 acc R =1 superacc

a..

99

Equations of MotionEquations of Motion

Expansion rate of the universe a(t)

ds2 = dt2+a2(t)[dr2/(1-kr2)+r2d2]

Friedmann equations

(å/a)2 = H2 = (8/3Mp2) [ m + ]

/a = -(4/3Mp2) [ m + +3p ]

Einstein-Hilbert action

S = d4x-g [ R/2 + L+ Lm ]

a..

1010

Scalar Field TheoryScalar 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)

Energy density = (1/2) 2 + V() + (1/2)()2

Pressure p = (1/2) 2 - V() - (1/6)()2

.

.

1111

Scalar Field Equation of StateScalar Field Equation of State

Continuity equation follows KG equation

[(1/2) 2] + 6H [(1/2) 2 ] = -V

- V + 3H (+p) = -V

d/dln a = -3(+p) = -3 (1+w)

+ 3H = -dV()/d¨ ˙

Equation of state ratio

w = p/

Klein-Gordon equation (Lagrange equation of motion)

. . ..

...

1212

Equation of StateEquation of State

Reconstruction from EOS:

(a) = c exp{ 3 dln a [1+w(z)] }

(a) = dln a H-1 sqrt{ (a) [1+w(z)] }

V(a) = (1/2) (a) [1-w(z)]

K(a) = (1/2) 2 = (1/2) (a) [1+w(z)] .

1313

Equation of StateEquation of State

Limits of (canonical) Equations of State:

w = (K-V) / (K+V)

Potential energy dominates (slow roll)

V >> K w = -1

Kinetic energy dominates (fast roll)

K >> V w = +1

Oscillation about potential minimum (or coherent field, e.g. axion)

V = K w = 0

1414

Equation of StateEquation of State

Examples of (canonical) Equations of State:

d/dln a = -3(+p) = -3 (1+w)

= (Energy per particle)(Number of particles) / Volume = E N a-3

Constant w implies ~ a-3(1+w)

Matter: E~m~a0, N~a0 w=0

Radiation: E~1/~a-1, N~a0 w=1/3

Curvature energy: E~1/R2~a-2, N~a0 w=-1/3

Cosmological constant: E~V, N ~a0 w=-1Anisotropic shear: w=+1 Cosmic String network: w=-1/3 ; Domain walls: w=-2/3

1515

Expansion HistoryExpansion History

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.

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)

gravitational extensions or high energy physics

1616

Expansion HistoryExpansion History

For modifications H2, define an effective scalar field with

V = (3MP2/8) H2 + (MP

2H02/16) [ d H2/d ln a]

K = - (MP2H0

2/16) [ d H2/d ln a]

Example: H2 = A(m)n

w = -1+n

Example: H2 = (8/3) [g(m) - m]

w= -1 + (g-1)/[ g/m - 1 ]

1717

Weighing Dark EnergyWeighing Dark Energy

SN Target

1818

Exploring Dark EnergyExploring Dark Energy

1919

Dark Energy ModelsDark Energy Models

Scalar fields can roll:

1) fast -- “kination” [Tracking models]

2) slow -- acceleration [Quintessence]

3) steadily -- acceleration deceleration [Linear potential]

4) oscillate -- potential minimum, pseudoscalar, PNGB [V~ n]

2020

Power law potentialPower law potential

“Normal” potentials don’t work:

V() ~ n

have minima (n even), and field just oscillates, leading to EOS

w = (n-2)/(n+2)

n 0 2 4 ∞w -1 0 1/3 1

2121

OscillationsOscillations

Oscillating field

w = (n-2)/(n+2)

Take osc. time << H-1 and constant over osc.

2 = dt 2 / dt = d / d /

= 2 d [1-V/Vmax]1/2 / [1-V/Vmax]-1/2

If V = Vmax( /max)n then

w = -1 + 2 01dx (1-xn)1/2 / 0

1dx (1-xn)-1/2

= -1 +2n/(n+2)

. . .

K=0 Vmax=

.

Turner 1983

2222

Linear PotentialLinear Potential

Linear potential [Linde 1986]

V()=V0+

leads to collapsing universe, can constrain tc

a

t

curves of

2323

Tracking fieldsTracking fields

Criterion = VV/(V)2 > 1, d ln (-1)/dt <<H.

However, generally only achieves w0 > -0.7.

Successful model requires fast-slow roll.

Can start from wide variety of initial conditions, then join attractor trajectory of tracking behavior.

2424

QuintessenceQuintessence

Interesting models have dark energy:

1) dynamically important,

2) accelerating,

3) not

~ [(1+w)] ~ (1+w) HMp

Damped so H ~ V, and timescale is H-1.

Therefore ~ Mp.

Unless 1+w << 1, then << Mp and very hard to reconstruct potential.

.

.

..

2525

Dark Energy ModelsDark Energy Models

Inverse power law V() ~ -n

“SUGRA” V() ~ -n exp(2)

Running exponential V() ~ exp[- ()]

PNGB or “axion” V() ~ 1+cos(/f)

Albrecht-Skordis V() ~ [1+c1 +c22] exp(-)

“Tachyon” V() ~ [cosh()-1]n

Stochastic V() ~ [1+sin(/f)] exp(-)

...

2626

Tying HEP to CosmologyTying HEP to Cosmology

Accurate to 3% in EOS back to z=1.7 (vs. 27% for w1).

Accurate to 0.2% in distance back to zlss=1100!

Klein-Gordon equation + 3H = -dV()/d¨ ˙

w(a) = w0+wa(1-a)

2727

Scalar Field DynamicsScalar Field Dynamics

The cosmological constant has w=-1=constant. 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.

.

2828

Growth HistoryGrowth History

While dark energy itself does not cluster much, it affects the growth of matter structure.

Fractional density contrast = m/m evolves as

+ 2H = 4Gm

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 n=+2/3, i.e.

~ a

.. .

2929

Growth HistoryGrowth History

Growth rate of density fluctuations g(a) = (m/m)/a

€

g + [5 + 12

d ln H 2

d ln a ] ′ g a−1 + [3+ 12

d ln H 2

d ln a − 32 G Ωm (a)] ga−2 = S(a)

€

g + [3 + 2ℜ] ′ g a−1 + [1+ 2ℜ − 32 G Ωm (a)] ga−2 = S(a)

€

g + [ 72 − 3

2 w(a)Ωφ (a)] ′ g a−1 + 32 [1− w(a)]GΩφ (a) ga−2 = S(a)

€

g + [4 − q] ′ g a−1 + [2 − q − 32 G Ωm (a)] ga−2 = S(a)

3030

Gravitational PotentialGravitational Potential

Poisson equation

2(a)=4Ga2 m= 4Gm(0) g(a)

In matter dominated (hence decelerating) universe, m/m ~ a so g=const and =const.

Photons don’t interact with structure growth: blueshift falling into well matched by redshift climbing out.

Integrated Sachs-Wolfe (ISW) effect = 0.

3131

Inflation, Structure, and Dark EnergyInflation, 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 n=1 (Harrison-Zel’dovich).

CMB power spectrum

On large scales (low l), Sachs-Wolfe dominates and power l(l+1)Cl is flat.

3232

Deceleration and AccelerationDeceleration 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