Cosmology - Chapter 6

Cosmology

Chapter 6: Cosmology Part 1: the large scale homogeneous

Universe

Some important things for Life about (homogeneous) cosmology

•  The Universe as we know it is old (13.8 Gyr), but had a finite beginning

•  The (observable) Universe is large, but finite in size

•  Baryonic material is a minority component of matter, but is not zero

•  The primordial composition was H and 4He

•  The Universe is expanding and not currently in thermal equilibrium, with a characteristic temperature that is very low (2.7K)

•  The Universe will continue to expand for a long time

•  The average density of atoms is low (of order 1 m-3)

Expansion

Expansion of the Universe Hubble (1929) redshift-distance relation

z obs em

em

=−λ λλ

z Hcd= 0

z

vcvc

z vc

z=+

−− ⇒ = <<

1

11 1for

v H d= 0

The “redshift” is the shift of spectral features in distant galaxies to longer wavelengths Relativistic Doppler effect (but see later)

Hubble’s Law

v ∝ d is a signature of an expanding “space”, but note:

•  This does not imply we are at the center

•  Scatter is due to “peculiar motions” of individual galaxies – typically 200 kms-1 relative to the smooth expansion flow

Hubble’s relation verified to z > 1 by e.g. Supernovae Type 1a

c.f. z < 0.003 in his original data

The extragalactic background light

3K Cosmic Microwave Background (CMB) dominates the radiation content of the Universe

CMB has Planck Black Body spectrum at T= 2.728 ± 0.002 K, exquisite precision. But, matter and radiation are not in thermal equilibrium today (there is no matter at 2.73 K!)

Also, CMB radiation is isotropic at the level of 10-3 (dipole term due to our own 600 kms-1 motion w.r.t. CMB frame, otherwise10-5).

CMB dipole

Photon to baryon ratio in the universe is high nγ/nB ~ 2×109

Out in space and back in

time

US Appearance of “inside of fog

bank”

The Big Bang

transparent

opaque

The last scattering surface of the CMB At very early times, matter and radiation were indeed in thermal equilibrium via creation/annihilation reactions of particle/anti-particle pairs.

Prior to 300,000 years after Big Bang, all matter was ionized plasma (T > 4500 K) and Universe was effectively opaque due to scattering of light off charged particles. Afterwards, matter was generally neutral (at least for a while) and the Universe was transparent, as it still is today. Since CMB has been around since the beginning, the visual impression is of being located at the center of a clearing in a fog bank. We “see” the “Last Scattering Surface” as a section through the Universe at immense distances as it was 3 x 105 yr after Big Bang.

Isotropy of the CMB is really a statement about the homogeneity of the Universe 300,000 yr after Big Bang.

p+

The Minkowski metric in Special Relativity is: 2222 dldtcds −=

Δ Δ Δ Δ Δs g x x g x xiji j

iji j

ji

2

0

3

0

3

= ≡==∑∑

ds c d R d S dk2 2 2 2 2 2= − +τ τ ω ω γ( )( ( ) )

!!"

#$$%

&+

−−= 22

22

2222

/1)( γττ dr

AkrdrRdcds

The most general metric will be of the form:

But, metrics for homogeneous and isotropic, but possibly expanding, Universes must have the form of the Robertson-Walker metric (often seen written in one of two forms)

Scale factor of expanding Universe

Effects of the curvature of a τ

= constant surface

Roberston-Walker metric for expanding isotropic homogeneous Universe

Comoving radial

coordinate ω

Conventional angular

coordinates θ,φ

time

A 2-d analogue of our 3-d Universe illustrates •  Isotropic expansion with v ∝ d •  Patterns and geometry/topology remain constant:

“comoving radius of curvature” cannot change •  There is no “center” (or “preferred galaxy”) and

the Universe itself is expanding, not “into anything”

Volume of Universe will be finite if (uniformly) positively curved, but will be formally infinite if (uniformly) negatively curved. Volume will be definitely “very large” if it is more or less flat in our observable region. Maps of galaxies in comoving

space at different τ

Our “observable region” (horizon = light travel distance in age of Universe) will likely be finite.

Some straightforward consequences of the RW metric

)()()1(2

1

ττ

RRz =+

RR

H

=

2RRRq

−=

constant=RT

The redshift of galaxies:

Hubble’s “constant” (= “parameter)” Normalised first derivative of R(τ)

The “deceleration parameter”, normalised second derivative of R(τ)

Behaviour of a black body radiation field

Radius of curvature of the Universe, linked to its volume if positively curved

radius = RA

Natural explanation of 2.7K microwave background: The Universe was in thermal equilibrium at very early times, in a hot dense state, but is no longer. But the black body T will simply decline as R-1 once thermal equilibrium is lost.

The Friedmann equation Solution of General Relativity field equations** for homogenous isotropic Universe gives the Friedmann equation linking the dynamics of the expansion R(τ), the ρ(R) “equation of state”, and the curvature c2/A2

!"

#$%

& Λ+−= 2

2

222

338 R

AkcRGR ρ

π

- scale factor from the Robertson Walker metric

- derivative w.r.t. cosmic time τ

-  density

-  curvature as in Robertson Walker metric: A = comoving radius of curvature (which is fixed for all time), and k = ± 1 to fix sign.

-  Einstein’s “cosmological constant” (equivalent to a ρ = constant term, a.k.a. “false vacuum”, i.e. a form of “Dark Energy”)

R!Rρ

kc2

A2

Λ

What are the possibilities for ρ(R), to solve the Friedmann equation for R(τ), since kc2/A2 is constant. ρM ∝ R-3 matter, either baryonic matter or non-radiative “dark matter”

ργ ∝ R-4 radiation, or other relativistic particles

ρΛ = constant “false vacuum” … and possibly others

Note: different R dependences mean that different components will dominate at different R. Also can imagine a given component changing from one R dependence to another, e.g. relativistic particles lose energy à non-relativistic.

Dynamical effects on R(τ)

Easy to solve Friedmann equation, e.g. if density is dominated by one component (and if curvature is negligible, then we get:

ρM ∝ R-3 R ∝ τ 2/3 Decelerated expansion (as you’d expect)

ργ ∝ R-4 R ∝ τ ½ Decelerated expansion (as you’d expect)

ρΛ = constant R ∝ eHτ (Exponentially) accelerated expansion under gravity!

What are the possibilities for ρ(R), to solve the Friedmann equation for R(τ), since kc2/A2 is constant.

We can parameterize the density of different components i in terms of Ωi, which is simply their density normalised to a “critical density” that is given by the expansion rate, H, and gravitational constant, G.

€

Ωi =ρiρc

ρc =3H 2

8πGΩ−1= Ωii∑⎛

⎝⎜

⎞

⎠⎟ −1=

kc2

A21!R2= k c

H⎛

⎝⎜

⎞

⎠⎟2 1RA( )2

Friedmann Equation then becomes:

•  c/H is of order the size of the horizon (see later), so |Ω-1| is the ratio of the horizon scale to the (physical) radius of curvature RA, i.e. the “importance” of the curvature.

•  If the expansion decelerates, R-dot decreases and |Ω-1| therefore

increases, i.e. Ω diverges away from unity, and curvature becomes more important, and vice versa. If the expansion accelerates then Ω converges towards unity.

The following diagrams look at three cases that were, 20 years ago, all more or less consistent with observations Ωm = 1: “Einstein-de Sitter” universe, dominated by

matter, preferred then theoretically. Ωm = 0.25: Open, low density universe, preferred then

observationally. Ωm = 0.25 + ΩΛ = 0.75: “concordance cosmology”

This is the one that is now strongly indicated by observations

Ωm = 1 Einstein-de Sitter Ωm = 0.25; ΩΛ = 0.75 “Concordance” Ωm = 0.25 Open Universe

All currently plausible models have R ~ 0 at some finite time in the past

i.e. BIG BANG “creation event”

Age since Big Bang is of order H0

-1 (almost exactly so for concordance model)

Note: late epoch acceleration of the current concordance model is barely noticeable


Comoving distance ω(z) converges at some finite value of ω for very high z – concept of “horizon”

The region of the Universe that we can “see” (physically interact with) is finite and has radius of order (c/H)

Horizon distance that light has travelled since redshift z (in units of c/H0)

ωH (t) =cR(t)0

t∫ dt


All models approach Ωm = 1 in the past (at very early times Ωrel =1).

This makes our treatment of early Universe simple….. e.g. universe was matter dominated with ρm= 3H2/8πG

Universe turns around and recollapses

Universe rapidly becomes “empty”

Three matter-dominated models with Ωm = 1.1, 1.0 and 0.9 today

Problem for Life:

Ωm > 1 Universe quickly turns around and recollapses

Ωm < 1 Universe cannot make structures like galaxies (it is “empty” and gravitationally instability fails – see later)

Development of Life needs a Universe which had Ω very very close to Ω = 1, to produce a long-live Universe able to form structures

log R/Rtoday

The divergence of Ωm from unity:

The existence of non-baryonic “dark matter”

•  First evidence: “missing mass” in systems on scales of galaxies and larger

Van Albada et al 1985 NGC 3198

Galaxy rotation curves v(r) indicate mass has density going as r-2, whereas distribution of visible material goes as r-3.

Dynamical measurements of galaxies in clusters, hydrostatic equilibrium of hot gas, gravitational lensing, all consistently indicate mass ~ 7× mass of visible baryons

•  Analysis of primordial nucleosynthesis in early Universe constrains photo-baryon ratio, and sets ΩB ~ 0.04, compared with total ΩM ~ 0.27

•  Existence of non-baryonic dark matter required to ensure survival of density fluctuations during phase when Universe was a fully ionized plasma.

•  Non-baryonic dark matter dominated Universe provides extremely good quantitative match to CMB fluctuations and structure in the Universe today.

Not surprising in retrospect? We primarily observe the Universe with electromagnetic radiation ...

There is little doubt that normal baryonic matter is a minority (~ 15%) of matter in the Universe.

Dark matter (effectively) interacts only through gravity (and possibly through Weak Force). No good for Life!

The challenge for Physics •  There are no candidate DM particles within the Standard Model of Particle

Physics. •  There are very many candidates outside the Standard Model, e.g.

•  axions (10-5 eV) •  sterile neutrinos (wide range of masses) •  super-symmetric WIMPs (100 GeV) •  primordial black holes (very massive)

Measuring the parameters that describe our Universe on the largest scales

H Expansion rate of the Universe (H ~ 70 kms-1Mpc-1) Need to measure distances to objects with known redshift •  “distance ladder” approach based on parallax of nearest stars •  Well-understood astrophysical systems (e.g. expanding SN

explosions)

q Deceleration/Acceleration of the expansion Need to measure deviations from Hubble’s Law at large distances •  SN1a, assuming they are standard

candles of fixed brightness

SN1a gave clear indication* (in 2000) for accelerating expansion now ΩΛ ~ 0.7


Ωm matter density of the Universe today (Ωm ~ 0.3) •  growth rate of structures •  measure “mass-to-luminosity” ratio in e.g. rich

clusters of galaxies and apply to the measured luminosity density of the Universe

Ωb matter density of baryonic matter Big Bang Nucleosynthesis: in early Universe (t < 200 sec), nuclear reactions p + n à 4He + trace of 2H, 3He, 7Li Observed ratios beautifully consistent

with matter density (today) of Ωb = 0.04

Aside: differences with stellar nucleosynthesis? •  plenty of free neutrons at t ~ 1 sec •  all over in ~ 100 seconds: τ(sec) ~ T-2(1010K)

Measuring the parameters that describe our Universe on the largest scales Ω  Total Ω = ΣΩi, also given by curvature

of space: So, we could measure the angular size of something of known physical size at great distance

Length scale of peak density fluctuations in universe on CMB last scattering surface is well understood theoretically. Boomerang circumpolar balloon experiment (1999)

Peak at 1 degree scale à Ω = 1 Curvature is very small within our observable volume


Important consistency check with constraints on the age of the Universe

τ  Current age of the Universe is constrained from ages of things in it using astrophysical models

•  Masses of (highest mass) surviving stars in globular clusters (12.5±1.5 Gyr)

•  Temperatures of (coolest) white dwarfs (8.5±1.5 Gyr in disk) •  Radioactive dating of radioactive isotopes, using assumed

production history (10-20 Gyr) •  Age of the Earth itself (4.6 Gyr)

All these different estimates involve very different physics, but are all consistent with age ~ 13.7 Gyr, with globular clusters forming about 1 Gyr after Big Bang, disk of our Galaxy after about 5 Gyr and so on.

Details of CMB fluctuations are a Rosetta Stone of almost all of the cosmological parameters (WMAP, Planck missions)

The “Concordance Cosmology” H = 71 ± 3 kms-1Mpc-1

Curvature = - 0.02 ± 0.02 ΩM = 0.27 ± 0.05 of which

ΩB = 0.044 ± 0.004 ΩΛ = 0.73 ± 0.03 n = 0.95 ± 0.02 σ8 = 0.74 ± 0.05


Basic cosmology - some first implications for Life (a)  The Universe is quite big:

•  1011 galaxies within observable region, but radius of curvature ≥ 100 × observable region, so total volume is almost certainly ≥ 106 larger.

and quite old (13.8 Gyr since Big Bang) •  time for multiple generations of stars and chemical enrichment of gas •  temperature of the radiation is very low (2.7 K) so matter and

radiation are far from equilibrium today. and had Ωm ~ 1 for an extended period, •  allowing galaxies and other structure to form

(b) The present-day average density of baryonic matter is low (or order 1 atom per m3) but is non-zero. We might have expected exact symmetry between particles and anti-particles in the early hot Universe, producing no residual of matter today.

These inter-related facts all follow from the fact that Ω was set to be very close to unity at early times. How was this done? INFLATION

How did the hot dense Big Bang universe come to have non-zero Baryon number? BARYOGENESIS

Inflation Why is the Universe large and long-lived?

The concept of inflation:

“Inflation” is a period of very rapid, accelerating expansion at the beginning of the Big Bang - i.e. t ~ 10-35 sec.

Proposed ca. 1980 to “explain” several shortcomings of standard Big Bang scenario, by invoking new Physics at very high energies (of order 1015 GeV)

What is the relevance of Inflation for Life in the Universe?

•  It naturally sets Ω very close to 1 at early times, producing a long-lived large Universe;

•  It naturally explains why different parts of the Universe are the same (same physics etc);

•  Next lecture: the small (quantum) density fluctuations present during inflation will grow into the large scale structure seen in the Universe today (e.g. galaxies). The “spectrum” of fluctuations predicted from inflation matches what we see.

Aside: “Dark Energy” today is behaving like inflation in accelerating the Universe again, but with an energy density that is 10-100 times lower(!) Are these two phenomena related ??

Inflation: Setting Ω = 1 at early times From the Friedmann equation, in any decelerating Universe, Ω progressively diverges from Ω ~ 1. But in an accelerating Universe, Ω progressively converges towards Ω ~ 1.

Ω =1+ kc2

A2"

#$

%

&'1R2=1+ k

c /H( )2

(AR)2

Idea of “inflation” was introduced in 1980. •  Postulate that at early times (Ε ~ 1016 GeV), ρ was dominated by a non-

zero energy-level of the vacuum, ρvac. Originally GUT symmetry-breaking field, but works better more general “inflaton field”.

•  This produces exponential expansion R ∝ eHτ, with H =8πG3

ρvac⎛

⎝⎜

⎞

⎠⎟1/2

~ constant

•  At the end of inflation, the energy in ρvac was thermalized into hot matter and radiation ρr, from then on a standard “decelerating” hot Big Bang universe.

•  Beyond this basic idea, there are a great variety of more speculative ideas about how it got into this state, how it got out of it, how ρvac may have evolved, and so on.

Idea of inflation pre-dated by 20 years the discovery of the “similar” Dark Energy, that is now emerging to (again?) dominate density of the Universe. Are they connected? But, note the energy scale is ×1027 different.

An attraction of Inflation: the “Horizon Paradox” In a decelerating Universe, ωH increases monotonically. For instance, the mass of matter within the horizon increases as τ (during matter-dominated era) and as τ3/2 (during radiation dominated era).

f(t)

ω

A

0

B

O

τ=0

A and B have not, at the epoch at which we observe them to be, had a chance to exchange information, if

the Universe has always been decelerating!

Inevitable consequence: all that we see today within our horizon was, at earlier times, in many disjoint regions that were apparently always outside of each other’s horizons

ττ

ωτ

τ

dRc

e ∫=2

1)(

Changing R(τ) at early times (and having accelerating universe) enables all these areas to have been in causal contact at early times.

Baryogenesis Why is there any matter in the Universe?

Baryogenesis and the matter-antimatter asymmetry

•  Matter and anti-matter are equally respectable in Physics.

•  Yet, the Universe today contains almost no (naturally occuring**) anti-matter at all. Dominance of matter over antimatter is established on at least galactic scales and is almost certainly “universal”.

•  At high energies, matter and anti-matter pairs were in equilibrium with creation and annihilation reactions: e.g

•  As energies fell during the expansion, pair-annihilation dominated over pair-creation. Therefore, present day photon-particle ratio (η ~ 10-9) reflects an initial small asymmetry in number of particles and anti-particles before the last annihilation occurred.

•  If this had not been present, all particles and all anti-particles would have anihilated and the Universe would have contained no particles (or anti-particles) at all ⇒ Life presumably impossible!

•  Inflation strongly predicts net baryon (and lepton) number precisely equal to zero after inflation.

Where did the non-zero baryon (and lepton) number of the Universe come from

nnp

p

≈ +1 η

€

e + e ↔ 2γ

Postulate X-boson with two decay paths that do not conserve B (OK in GUTs(?) but never so far observed)

X qq B L

X ql B L

r

r

!→! = − =

! →! = + =−

1 1

12 2

23

0

13

1

X qq B L

X ql B L

r

r

!→! − = + − =

! →! − = − − = −−

1 1

12 2

23

0

13

1

( ) ( )( )21

2121

)()1()1(

BBrrBrBrBrrB

−−=

−−−−+=εPostulate CP-violation so that branching ratios of X and anti-X are different (seen in Nature but not understood)

Imagine loss of equilibrium (freeze-out) so that back reactions do not destroy B gained in this way.

When? mX ~ 1014 GeV, implying 10-34 sec, i.e. very soon after inflation

Sakharov’s three conditions

Are there unknown connections?

Strange co-incidence: ΩB ~ ΩCDM And today at least: ~ ΩΛ

Is this coincidence telling us something? Would expect very different physics: WIMP etc. for Dark Matter, Inflaton field at 1016 GeV, GUTS and CPT violation.

Note: Dark Matter, Inflation and Baryogenesis all represent unknown (or at least untested, and possibly untestable) Physics, lying outside of the “Standard Model + GR”, for which the Universe itself may represent the only evidence.

Or is it simply anthropic? Arguably there would be no Life as we know it: •  if ΩΒ to low to allow cooling to form galaxies or if ΩCDM to low to prevent

Silk eradication of density fluctuations on galaxy scales (i.e. need to have 10-3 > ΩΒ/ ΩM < 10 ?)

•  if Λ dominated at z > 100 (i.e. ΩΛ > 106 ΩM) …. more next week

Cosmology - Chapter 6

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