ALAN HEAVENS I N S T I T U T E F O R A S T R O N O M Y U N I V E R S I T Y O F E D I N B U R G H

A F H @ R O E . A C . U K T H I R D T R R 3 3 W I N T E R S C H O O L

P A S S O D E L T O N A L E ( I T A L Y ) 6 - 1 1 D E C E M B E R 2009

Overview Lecture 2 Theory for Observers, Observations for

Theorists

Causality issue

  Big Bang theory has a number of puzzles, including the uniformity of the microwave background

  Temperature is uniform to 10-5, but opposite sides of sky (at z=1000) should not have been in causal contact (size of particle horizon ~2ct; t=300,000 years; DA=14 Mpc, angular size of horizon ~1 degree)

Inflation

  Inflation solves the problem of causality, by having a period of exponential accelerated expansion in the Early Universe.

  With exponential expansion, the comoving Hubble radius c (aH)-1 decreases with time.

  This is roughly the comoving distance a photon travels as the scale factor increases by e:

Hubble Radius

  Comoving description:

  With exponential expansion early on, photons can travel large comoving distances, solving the causality problem

  Needs >60 e-foldings   A decreasing Hubble radius

requires acceleration: This early inflation (t~10-35 s) has vastly higher energy density than current vacuum energy.

Inflation by scalar fields

  Scalar fields ϕ(x)   If quanta are massive, then

  with

  gives the Klein-Gordon equation:

  Neglecting spatial derivatives

Scalar fields in cosmology

  Scalar field characterised by an action

  with equations of motion determined by δS=0.   Technical point: S must be a scalar. d4x is not. Extra

term involving g=det(gµν) makes d4x √(-g) a scalar; thus requiring L to be a scalar.

  Euler-Lagrange equations:

  To get the Klein-Gordon equation, we need a Lagrangian given by: L =

12∂µφ ∂µφ− m2c4

22φ2

=Newton’s law

S =

dt(T − V )

Lagrangian and conserved current

  L does not depend explicitly on xµ. This symmetry leads to conserved quantity (Noether current)

  Generalise V so that   This gives   Now compare with a perfect fluid:

  Gives

Equation of state

  Equation of state of a homogeneous scalar field is

  So we get inflation if

Equations of motion: expanding coords

  The Euler-Lagrange equations give

  Approximations: homogeneous, and ‘slow-roll’:

  These give

  Friedmann’s equation gives

Inflation dynamics

  Slow-roll conditions:

  Inflation ends when ϕ is near bottom, and slow-roll conditions are not satisfied.

  Universe reheats to high temperature

≡ mpl

16π

V

V

2

1; η ≡m2

pl

8π

V

V; |η| 1

Ending inflation

  Time derivatives become more important as field approaches the minimum

  Rather than slow-rolling, the field oscillates, damped by the friction term

  Coupling with matter fields produces particles, reheating the Universe

3Hφ

Fluctuations from Inflation (heuristic)

 ϕ fluctuates because of quantum effects, with an r.m.s. (from QFT: sea Peacock)

  Parts of the Universe with different ϕ exit inflation at different times, differing by

  Hence there is a fractional variation in energy densities on the Hubble radius scale of

  While V ≈ constant, H ≈ constant, so horizon-scale perturbations are roughly constant. This gives scale-invariant fluctuations, δH

2 ≈ constant.   Power spectrum of density fluctuations: P (k) ∝ kn; n 1

Spectrum of fluctuations

  More carefully, variations in V give a tilt. Fluctuations of a certain wavenumber k cross the Hubble radius when k-1 = 1/(aH). H ≈ constant, so

  writing

  gives the tilt of the spectrum as

  Typically n<1.

d ln(δ2H

)d ln k

= ad ln(δ2

H)

da=

1H

d ln(δ2H

)dt

=φ

H

d ln(δ2H

)dφ

Number of e-foldings

  In order to account for the observed uniformity in the Universe, there need to be at least 60 e-foldings

  Flat potentials will generally inflate.   N depends on slow-roll parameters   Leads to a consistency relation

between N, n and tensor modes r in CMB

N =

Hdt = − 8π

m2pl

φfinal

φinit

dφV

V

Komatsu et al 2006

Perturbations

  Perturbed FRW metric, in Newtonian gauge (observers follow unperturbed paths)

  η is conformal time. Φ is potential perturbation. Ψ is curvature perturbation. These are scalar perturbations (vector [vorticity] and tensor [gravitational waves] modes also possible).

  For fluid-like behaviour (no anisotropic stress), Φ=Ψ   If v << c, only Φ matters.

ds2 = a2(η)(1 + 2Φ)dη2 − (1− 2Ψ)dxidxi

η ≡

dt

a(t)|Φ| 1, |Ψ| 1

Newtonian perturbation theory

  x= proper coordinate; r = comoving coordinate

  Continuity equation

  Change to comoving coordinates:

x = a(t)rx = ar + arx = ar + 2ar + ar

∂ρ

∂t

x

+ ρ∇x · x = 0

∇x =1a∇;

∂

∂t

x

=∂

∂t+ u ·∇; u ≡ r

Zero-pressure perturbations

  Let the overdensity δ be defined by

  Then continuity becomes

  Zero-order terms give   And linear terms (in δ or u) give   Equation of motion gives   Take divergence, add the Poisson equation

which gives   Finally

ρ(r, t) = ρ0(t) [1 + δ(r, t)]

ρ0 ∝ a−3

δ +∇ · u = 0 . ≡ ∂/∂t

∇2xΦ = 4πGρ

x = −∇xΦ ar + 2au + au = −∇a

(Φ + δΦ)

δ + 2H δ − 4πGρ0δ = 0∇2(Φ + δΦ) = 4πGρ0a

2(1 + δ)

∂[ρ0(1 + δ)]∂t

+ u ·∇[ρ0(1 + δ)] + ρ0(1 + δ)∇ · (r + au) = 0

(GR)

Solutions

  Flat, matter-dominated (Einstein-de Sitter) Universe: so H=2/3t, and

  Trial solution δ α tp gives p=2/3 or -1. Growing solution is then

  In general, in GR,   (with a very small dependence on Λ)

a ∝ t2/3 8πGρ0/3 = H2

δ +23t

δ − 23t2

δ = 0

δ ∝ t2/3

d ln δ

d ln a≡ f Ω0.56

m

Complications

  We cannot assume Ωm ≈ 1 and ignore other components. Growth is slower.

  In the early Universe, radiation is important, and growth rate is δ α a2 on super-Hubble radius scales. On smaller scales, growth is suppressed (Meszaros effect: perturbations are driven by Dark Matter, which is subdominant; expansion timescale is set by radiation, and it is short, because ρrad is large)

  Photon-electron coupling prevents sub-Hubble radius growth of baryons (sound speed ≈ c/√3)

- they oscillate.

Growth

Baryon and photon amplitude depends on phase at recombination, which depends on the wavenumber

Transfer Function

  T(k) defined so the linear power spectrum is

  Super-Hubble radius modes grow like a2 before equality.

  Modes which enter Hubble radius before equality don’t grow, so are suppressed by T=(aentry/aeq)2. Since k=aH on entry and H α 1/a2 (since a α t1/2), we find

P (k) ∝ T 2(k)Pinitial(k)

T (k) ∝ k−2 k →∞

Net effect

  Linear power spectrum

Neutrino streaming

  Neutrino oscillation experiments give

  So at least some neutrinos have non-zero mass. Since they interact only weakly/gravitationally, they can leave small perturbations and partially erase them.

  kT = 1.7 x 10-4 eV now

∆m212 = 7.9+1.0

−0.8 × 10−5eV2

|∆m223| = 2.2+1.1

−0.8 × 10−3eV2

Error on sum with Planck and Euclid weak lensing: 0.037eV (de Bernardis et al 2009)

Neutrino free-streaming length

  Consider flat Universe and radially-moving neutrino.

  Hence   Geodesic equation gives   Substitution gives

  When the neutrinos are relativistic and the Universe radiation-dominated, r α t1/2. At late times, when the neutrinos are non-relativistic dr/dt α t-4/3, so r converges. Free-streaming length is 112 (mν/eV)-1 Mpc (if masses equal)

ds2 = c2dτ2 = c2dt2 −R2(t)dr2

1 = t2 − R2

c2r2 . = d/dτ

R2r = Ac = constantdr

dt=

r

t=

Ac

R2

1 + R2r2

c2

=Ac

R2

1 + A2

R2

Neutrino effects

  Suppression on small scales

  Note:

  is the contribution to the variance of the matter from unit ln(k)

∆2(k) ≡ 12π2

k3P (k)

Present-day power spectrum

  Nonlinear effects add extra power on small scales

  Smith et al. (2003) fitting formula

Observations

  2dF, SDSS   Baryon oscillations seen   Turnover not yet

definitively seen   This is the galaxy power

spectrum   A bias is often assumed:

  Quadratic measures such as the power spectrum scale as

δn

n= b

δρ

ρ

Pgal(k) = b2P (k)

Bias?

  Galaxies are not necessarily where the mass is

On large scales, detailed statistical analysis shows galaxies and mass DO follow the same distribution (Verde et al 2002; Seljak et al 2005)

Peculiar velocities

  Deviations from Hubble flow depend on matter fluctuations. With an independent distance estimator, can be measured

  The variance in this, in patches of size R, is

  where W is the window function for the patch.   Since we measure Pgal (k), peculiar velocities usually

tell us about

u

u2 =

H20f

2

2π2

d3kP (k)W 2(kR)

β ≡ f(ΩM )b

Redshift Distortions

  More effective is to detect peculiar velocities by their effect on the redshift of galaxies

  At small distances,

  Radial distortions distort the correlation function, or power spectrum, so that it is not isotropic in ‘redshift space’.

  Measures β   Can get bias-independent measure:

cz = Hx + au

Guzzo et al 2005

f2(Ωm)P (k) = β2Pgal(k)

Taylor et al 2001