Edmund Bertschinger

MIT Department of Physics andKavli Institute for Astrophysics and

Space Research

General Relativity and Applications3. Cosmology

2

Focus on Two Cosmological Questions

How do we measure the geometry of space?

What might dark energy be, and how might we find out?

3

Symmetry and the Geometry of Space

Translational and rotational symmetry allow space to have 1 of 3 possible local geometries (Robertson and Walker, 1930s)

Euclidean (angles in triangle) =

Hyperbolic (Gauss & Lobachevski, ~1830) (angles in triangle) <

Spherical (angles in triangle) >

4

Measuring the curvature of space

Hohenhagen

Brocken

Inselberg

1821 result: (angles in triangle) = within measurement uncertainties

Gauss surveyed three mountains in the Alps.

5

Modern test of Euclid’s 5th postulate

s = r()

Angular distance r() depends on spatial curvatureE.g. unit 2-sphere, s = sin(colatitude)*longitude

Euclidean: K = 0Hyperbolic: K < 0Spherical: K > 0K = (1)H0

2 = ( 1)(1.3x1026 m)-2

Requirements:1. Standard meterstick s2. Angular measurement 3. Radial measurement

6

Acoustic Measurement using Cosmic Microwave Background Radiation

The cosmic microwave background (CMB) radiation is the thermal (blackbody) radiation emitted by the hot dense gas formed in the Big Bang.

Burst of sound waves emitted at t = 0 reaches distance cstd at time td creating a visible perturbation of the CMB

S=cstd

7NASA/WMAP Science Team

The Cosmic Microwave Background RadiationFrom COBE (1994) to WMAP (2003)

All-sky maps of blackbody temperatureScale is 0.2 mK about 2.725K

8

Acoustic lengthscale from random pattern of fluctuations

Decompose initial fluctuations into random set of localized sources

Each source produces circular wavefronts expanding at sound speed

Correlation function C(r)=<I(x)I(x+r)> detects size of the wavefronts.

rcorr rcorr

Bashinsky & Bertschinger 2001,2002

9

Intrinsic+Gravitational Integrated Sachs-Wolfe

280 Mpc or 1.2° on the sky

(Exact calculations of LCDM model using LINGER with full diffusion)

Images of the CMB perturbation caused by a point potential excitation

10

Angular Correlations of CMB Anisotropy

dts

cs

Each point on the photosphere contains information from within the acoustic radius s = 147 Mpc (for the standard model)

)(cos)12(4

1)(

2cos21

21

lnnl

Pl

ClnT

Tn

T

TC

Two-point correlation function:

zrecomb

=1080 re

com

b =

14 G

pc

11

CMBR Angular Power Spectrum

Top: Temperature fluctuations vs. angular scale

Bottom: Cross-correlation of temperature and linear polarizationvs. angular scale

From Bennett et al. 2003, WMAP

12

Results forC(), usingfull numericalintegration(except formagentacurve =semi-analytic)

Bashinsky & Bertschinger 2001

Eisenstein et al. 2004

13

Result: Curvature Measurement

s = 147 Mpc (acoustic radius, from known trecomb)

0.598 degrees (WMAP measurement)

r() = s/ = 14100 Mpc (angular distance to photosphere)

= 14000 Mpc (photosphere distance, from trecomb)

r() = to within 1% (within uncertainties) Space is indistinguishable from Euclidean!

|K| < (1.3x1026 m)-2

Improvement on Gauss’s 1821 measurement by 1034

Hu & White 1996Weinberg 2001Spergel et al. 2003

14

Focus on Two Cosmological Questions

How do we measure the geometry of space?

What might dark energy be, and how might we find out?

15

Dark Matter and Dark Energy

Dark Matter: Invisible stuff that gravity draws into galaxies. Crucial element of galaxy formation.

Dark Energy: Invisible stuff that gravity does not draw into galaxies. Instead, this substance accelerates the recent expansion of the Universe.

To date the sole evidence for these substances comes from astrophysics. Despite their similar names, they probably are completely unrelated.

16

What might dark energy be?

A cosmological constant? Proposed by Einstein in 1917; his “greatest blunder” Energy density of empty space!

A new form of mass-energy? “Quintessence” – particles whose de Broglie wavelength is

billions of light years!

A modification of General Relativity? String theory or other physics might change gravity across

the observable universe

Maybe GR is correct but our understanding of it is wrong.

17

How might we find out?

Geometric methods “Expansion history of the Universe” measured using

distance and redshift of tracer objects. Dark energy has repulsive gravity causing the

expansion to accelerate. Supernovae, “baryon acoustic oscillations”

Perturbation methods Measure how primordial density perturbations grow

with time. Dark energy’s repulsive gravity slows down growth of

perturbations. Galaxy clusters, weak gravitational lensing

18

Dark Energy Drives Galaxies Apart

Kinetic Energy + Gravitational Energy = Constant

For sphere of radius R(t), v(t) = dR/dt,R(t) m

M(t)

Dark Energy dilutes little or not at all as it expands!

or

constant v increases as R and t increase,acceleration!

19

Mechanics of Dark Energy

Einstein (1917) static universe

20

Dark Energy in General Relativity Uniform expansion, one scale factor

Spatial curvature constant K K=0 K<0 K>0

Friedmann Equation (1922)

21

Energetics of dark energy

Energy density , pressure p

Adiabatic expansion: d(a3)+pd(a3)=0

“Equation of state” w = p/

Dark energy has w < -1/3 so a2 increases

22

A brief history of dark energy

1917 Einstein:1917 de Sitter: , “static” universe

1922-27 Lemaitre: drives expansion

1929 Hubble: The universe expands1967 Zel’dovich:1981 Guth: primordial dark energy (inflation)1995 Krauss & Turner: “The Cosmological

Constant is Back”1998 Competing groups announce cosmic

acceleration (Perlmutter et al., Riess et al.)

23

Geometric measurement of DE

Measure using “standard candles” (e.g. exploding white dwarfs), rulers (e.g. baryon acoustic length, Eisenstein et al. 2005), or gravitational radiation sirens (Holz & Hughes 2005)

Measure K from CMB acoustic length, combine with today from galaxies; then DE = – clustered

Both methods assume Friedmann is valid.

24

Seismology: Using waves to probe structure and composition of Earth

P waves: longitudinal, P waves: longitudinal, acoustic acoustic (Primary, (Primary, Pressure)Pressure)

S waves: transverse,S waves: transverse, shearshear (Secondary, (Secondary, Shear) Shear)

Seismic waves revealed the liquid core in 1906 (Oldham).

J. Louie, U. Nevada Reno

25

Ultra-low frequency seismology

Big Bang

Matter era

Neutral era

Gravitational waves(transverse)

Today

Acoustic waves(longitudinal)

(re)combination

Primordial Perturbations Probe Structure and Composition of the

Universe

WMAP,2001

Gravitational amplification

Light-cone diagram of spacetime

26

Probing DE with density perturbations

Evolution of galaxy clustering, e.g.abundance of galaxy clusters(found by X-ray emission or radio

shadows – Sunyaev-Zel’dovich effect)

“Weak” gravitational lensing

27

Abell 2218, Hubble Space Telescope

28

Gravitational potential perturbations

Scalar (density) perturbation (x,t)

29

Fundamental assumptions of theoretical cosmology

Spacetime has effectively 4 dimensions and on scales of Gpc is approximately homogeneous, isotropic, and uniformly expanding.

Light travels along null geodesics.

More general than GR

Some theory of gravity holds (e.g. General Relativity)

30

Long-wavelength perturbations

(x,t)

K<0 K>0

Absent ultra-long range forces, long-wavelength perturbations

must evolve like isolated universes

Hubble or Jeans Length

31

For example, in GR …

* If no entropy, shear stress, or other long-range perturbations.

Einstein equations yield a wave equation whose exact solution for long wavelengths is

Lyth & Wands 2003Bertschinger 2006, in preparation

32

Perturbations, or homogenous evolution?

This solution exactly matches the unperturbed evolution a Robertson-Walker universe, provided that the Friedmann equation holds!

Long-wavelength perturbations =Robertson-Walker + Friedmann

33

Consequences

Perturbation evolution measures the same thing as standard candles/rulers, unless:

Pressure or stress forces act

Other long-range forces act

34

Prospects for Deciphering Dark Energy

Case Geometric methods Dynamic methods

yes nothing new

Peculiar T yes nothing newNew force maybe, if wavelength-dependent

Misunderstanding consistency checkof GR

35

We don’t know what the dark energy is! But we want to find out!

Why is the energy density of the vacuum so small, when virtual particles should make it huge? If not Huge, why not Zero?

Why is dark energy comparably abundant to matter today, when it was negligible 10 billion years ago? Why now?

What is the ultimate fate of our Universe?