how it all begins raul abramo - transregio 2009 from...

raul abramo - transregio 2009how it all begins

From inflation to the CMB to today’s universe

I - How it all begins

Raul AbramoPhysics Institute - University of São Paulo

[email protected]

mailto:[email protected]

mailto:[email protected]


Very brief cosmic history

energy

380.000 yrs

200 s

time

1 MeV

1 eV

redshift

109

103

0 15Gy

BBN

Decoupling(surf. last

scattering)


Some crucial observations to understand our Universe:

• Cosmic microwave background(COBE, WMAP, Boomerang, Dasi, QUAD, ... PLANCK...)

• Matter distribution over large scales – “Matter Spectrum”


radiationmatter:

z~104

today

• History of cosmic domination:

z≅1089: “decoupling” (recombination)D

ark

Ene

rgy


Some numbers I will often use...

• Age of the Universe today: T0 = (14 ± 0.5) Gy

• Density: ρ0 = (1.9 ± 0.15) h2 x 10-29 g cm-3

• Hubble parameter: H0 = 100 h Km s-1 Mpc-1

h = 0.72 ± 0.05

• Baryon fraction: Ωb h 2 = (ρb / ρtot) h 2 = 0.024 ± 0.003

• Radiation

(photons and massless neutrinos): Ωr = 2.5 x 10-5 h -2

1 pc = 3,26 l.yr. 1 Mpc = 3,1 x 1024 cm

... use to compute, e.g., matter-radiation equality:

• Matter (dark matter + baryons) Ωm = 0.2 - 0.3


• The physics of the CMB involves propagation and scattering of photons

• The CMB is also the most distant direct observation we have of the universe in its infancy, hence it is a key observable to test correlations and causality over the largest observable scales of the Universe

• So, let’s review some of the basic facts about the propagation of light and, therefore, of causality in a Friedmann-Robertson-Walker spacetime

• Light propagates over null geodesics. For a radial light ray:


t

d

• In an FRW spacetime, proper distances for light-speed signals can be finite even when the travel time extends arbitrarily into the past or into the future.

• For instance, let’s take a decelerating FRW:

This spacetime can be continued to the past only down to t=0 (when a=0). Then:

dHp is the maximum physical distance a light ray can cover if it was emmitted at some arbitrary time in the past. This means that the past light cone in this scenario is bounded, and cannot be extended beyond that limit.

a

t


• This maximal distance is called a horizon. Since in this case (p<1) the horizon refers to a truncation of the PLC, it is a past-like horizon, a.k.a. a particle horizon. This horizon is usually approximately equal to the curvature radius of the FRW, r ~ 1/R-1/2 ~ H-1 - i.e., the Hubble radius!

• The particle horizon separates observers which never had causal contact prior to the time t. Therefore, when there is a particle horizon, the Universe can be separated into regions which are (up to that time) causally disconnected

• Since the Universe has been, for most of its history, dominated by either radiation (p=1/2) or matter (p~2/3), if that were true down to t=0 then our particle horizon today would be:

Ex: compute the particle horizon at the time of decoupling (t~380.000 y, z~1100), assuming that p=1/2. A: ~200 Kpc.

???

How can the CMB be so

homogeneous over the whole sky

???


But consider, instead, what happens if the upper limit is take to be tf → ∞, and take the lower limit to be t.

This distance would then correspond to the maximal length that separates two objects such that they could exchange a light-speed signal emmitted at time t. If that maximal distance is not infinity, then there an event horizon:

• Now take an accelerating scale factor:

a

t

We still have some initial time t=0, however:

is an arbitrarily large distance as we take the lower limit ti → 0 , and hence there is no particle horizon in this case.


• The physical significance of a horizon is profound, as it clearly marks causality boundaries:

A particle horizon sets a limit to the past light-cone of observers at time t: pairs of observers separated by a distance larger than dpH at time t have never been in causal contact before t.

An event horizon sets a limit to the future light-cone of observers at time t: pairs of observers separated by a distance larger than deH at time t will never again be in causal contact after t.

comoving distances

t2deH(t2)

t1

comoving distances

t0

0

dpH(t1)

tdec

t1


• The physical meaning of an event horizon is that it marks the boundary beyond which observers lose the possibility of causal connection in the future.

• What happens when the accelerated expansion doesn’t last forever? Mathematically, there isn’t an event horizon anymore - but still we can define the notion of an effective horizon:

• Like the notion of thermal equilibrium, what really matters is the time interval during which there is no causal contact, compared to the typical times for other (causal) physical processes

v = c

v = c

v = c

v = c


Can do that with scalar fields!

Hubble “drag”Potential

φ

V(φ)

If the kinetic energy => “slow roll” :is << than the potential energy

With slow-roll, V(φ)

works like a time-varying

Λ

Eq. motion(Klein-Gordon)

small

Background, φ(t):


Explicit example: the “power-law” model characterized by the mass scales M and s:

Using these into Friedman’s and Klein-Gordon’s equations:

The solution (up to a transient) is:

with:

Where:

0 = ! + 3H! + V,!

3H2 = 8"G

!12!2 + V

"

a(t) =!

t

t0

"p

, H =p

t

!(t) = !0 logt

t0Just choose p (i.e., s) sufficiently large and inflation will ensue


How it all started: inflationary particle creation

- the ultimate “free lunch”!

QUANTUM MECHANICS:

" ΔE Δt > h/2π "

Vacuum is filled with virtual pairs of particles, which survive during

brief moments before beingannihilated back to vacuum

Guth 1980Starobinsky 1979

Linde 1982-85

Mukhanov & Chibisov 1979-1980

Hawking Guth & Pi

...


Virtual pairs quantum fluctuations

Right here, right now:


Virtual pairs in an expanding background (accelerated spacetime)

Horizon H-1

Inflation (accelerated expansion) converts virtuais pairs into

real ones

accelerated expansion separates the pairs

A contraction followed by expansion (bounce) would have a

similar effect


Quantum fluctuations of a scalar field + metric perturbations during inflation

• After diagonalizing and integrating by parts, quadratic action:

• Solutions to modes @ U.V.

(k2>>µ2) and I.R. (k2<<µ2):

Good reviews:Bassett, Tsujikawa & Wands astro-ph/0507632

L. Sriramkumar, arXiv:0904.4854

3H2 = 8!G V (")" + 3H" + V ! = 0


• Quantization:

• Vacuum: ak|0>=0 But:

=> Mixing of positive- and negative-energy modes

=> Amplification of zero-point energy by the “external field” (acc. expansion)

=> ~ Particle creation

•Inflation generates inhomogeneities in the “primeval soup”


λphys

t

H-1

End of inflation

UVIR

“UV”

Today

v!!k + [k2 + µ2(!)]vk = 0

• Curvature perturbations:

Constant on large scales (“IR”)

convenient f/ normalization!

Rk =vk

z= !k +

H

"#"k


• The primordial spectrum of curvature (-> density) perturbations:

!R(|"x ! "x!|) = "R("x)R("x!)#

!R!kR!!k!" # 2!2k"3!2

R(k) "(#k $ #k#)

=!

d3k

(2!)3/2

d3k!

(2!)3/2e"i!k·!xei!k!·!x!

!R!kR#!k!"

! !R(r) =!

dk

k!2R(k)

sin kr

kr, r = |"x" "x!|

It will also be useful to know the inverse of this relation (“Hankel transform”):

!2R(k) =

2!

k3

!dr r2 sin kr

kr"R(r)

• Many times we also define a dimensionful power spectrum:

PR(k) = 2!2k!3!2R(k)! |Rk|2


• Convergence arguments limit the form of the spectrum (Zel’dovich,

Harrison, ...), so this function must be nearly scale-invariant:

0 ! !!k < 2" , P (!) =12"

R!k = ei"!kAR(!k)

PGauss(AR) ! e! A2

R4!2k!3!2

R

• Rk arise from quantized harmonic oscillators their statistic is that of

harmonic oscillators in their ground state GAUSSIAN!

!2R(k) = !2

R(k0)!

k

k0

"nS!1+...

P (!)

!2!

PG(AR)

AR

Nearly scale-invariant spectrum of Gaussian perturbations (Zel’dovich, ‘70s)

Considering higher-order interactions

lead to deviations from gaussianity!


• The metric also has matter-independent fluctuations - gravity waves:

• Their relative power is also a key observable:

• Both density (“scalar”) and gravity wave (“tensor”) perturbations impact

directly the CMB, and it is useful to set a “pivot” scale to determine the spectra:

Amplitude nailed by WMAP

!2h(k) = 2

k3

2!2|hk|2

!2R(k) = !2

R(k0)!

k

k0

"nS!1+...

, k0 = 0.002Mpc!1

!2h(k) = !2

h(k0)!

k

k0

"nT +...

r =!2

h(k0)!2R(k0)

!2R(k0) ! 2.2" 10!9


• Density perturbations (“scalar”) and gravity waves (“tensor”) are both

generated during inflation (or some other early-Universe free restaurant).

• Although density perturbations have been “digested” by processes at

low energies, gravity waves come straight from the GUT era!

• However, scalar and tensor perturbations test very different physical

scales:

!V (!)/M2

pl

V !(!)/H

!

!V (!)/M2

pl

Mpl

! 1Mpl

V 3/2

V !

! 1M2

pl

"V

M2pl = G!1


10-35 s 10-33 s 3. 105 y 1.5 1010 y

quantum

fluctuation

classic

al

fluctuation

perturbation

in density &

temperature

Andromeda

We live here

(Adapted from Lineweaver 1997)

Summarizing: from quantum fluctuations to CMB, galaxies etc.


• And how does inflation subvert the causality bounds of usual radiation/matter cosmology?

t0

dpH(t1)

tdec

t1

Inflation: the ultimate past-light-cone democracy


Tomorrow: Cosmic Microwave Background Radiation

• Z ≅ 1200 ⇔ Tγ ≅ 10 eV

• Universe starts to become

neutral

• Photons start to “free

stream”

• While this happens,

density and temperature

fluctuations are imprinted on

the CMB photons

• We detect these photons

today - with their initial

energies redshifted by the

same factor 1/(1+z)

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