Standard Cosmological Models

[For background, read chapters 10 & 11 from Hawley & Holcomb]

Recap

We have discussed:

• The discovery of a universe beyond our galaxy

• The distance ladder & scale sizes

• Hubbles law, distance velocity (v=Hd)

• That the universe is expanding

• Germany 1915:– Einstein just completed theory of GR– Explains anomalous orbit of Mercury perfectly– Schwarzschild is working on black holes etc.– Einstein turns his attention to modeling the

universe as a whole…

Consider now how the first models for the universe developed…

How to make progress…

• Take the following assumptions– Universe is homogeneous – Universe is isotropic

– We have already learned the latter comprise the Cosmological Principle

homogeneous - same average properties everywhere

isotropic- looks the same in all directions

Cosmological Principles

Recap: Copernican Cosmological Principle On a large (billion ly) scale, the

universe is both homogeneous & isotropic (in 3-D space)

& Perfect Cosmological Principle On a large scale, the universe is

both homogeneous & isotropic (in space

AND time)

Observational evidence for homogeneity

Matter is distributed uniformly on large scales.

Las CampanasRedshift surveygot z for > 20,000 galaxies

get distances from redshifts - can see that homogeneity was true at all epochs

and isotropy

Thus the Cosmological Principle is thus supported by observations (averaging over large scales)

So assume Cosmological Principle is OK if we ignore details like stars & galaxies and deal with matter distribution averaged over large scales

POSSIBLE GEOMETRIES FOR THE UNIVERSE

• The Cosmological Principles constrain GR to give us the possible geometries for the space-time that describes Universe on large scales

• So, we need to find curved 4-d space-times which are both homogeneous & isotropic to match the observed universe…

• We need a solution of General Relativity which describes the universe

DYNAMICS OF THE UNIVERSE – EINSTEIN’S MODEL

• Back to Einstein’s equations of GR– For now, ignore cosmological constant

geometry=mass/energy

TG4

8

c

Gπ=

“G” describes the curvature (including its dependence with time) of Universe… here’s where we plug in the geometry

“T” describes the matter content of the Universe. Here’s where we tell the equations that the Universe is homogeneous and isotropic.

POSSIBLE GEOMETRIES FOR THE UNIVERSE

• Assuming that gravity is the only long-range force

• Einsteins spherical model - simple solution of GR but this model showed the universe would quickly collapse in on itself due to gravity. Thus, he postulated a long-range repulsive force, , to make a static universe spherical solution to GR

• GR geom. of spacetime = mass - energy could go on rhs as some new property of matter/energy, as it needs to be a repulsive force, can be thought of as a negative energy term

The Cosmological Constant

• Einstein used then to counter gravity and stop his spherical model from rapid collapse

• This however proved to be unstable - model could be set up to start as static but did not remain static with time

• When the universe was discovered to be expanding he withdrew this model as his biggest blunder!

• If not stuck with the thought the universe must be static he could have predicted expansion/contraction-his mistake was assuming must have a value needed to make the universe static

• Anyway, the universe is expanding, and we require a non-static solution to GR

• Need to calculate total matter-energy content of the universe and find a spacetime geometry consistent with it

• Need to consider a way to parameterize the expanding spacetime we know about, and tie it to our other known properties of the universe

Cosmological Principle + GR -> several possible metrics

Minkowski metric is valid for flat geometries (Euclidean) with no time -dependence

s2= ct)2 -(x2 +y2 +z2 )

Adding scale function which varies with time

s2= ct)2 -R2(t)(x2 +y2 +z2 )

R(t) is the scale factor describing the expansion/contraction of space

The scale factor, R• Scale factor, R, is a central concept of models!

– R tells you how “big” the universe is…

– Allows you to talk about expansion and contraction of the universe (even if universe is infinite).

• Simplest example is a (sphere)– Scale factor is just the radius of the sphere

R=1 R=2R=0.5

• If two galaxies maintain a constant separation once the overall expansion has been accounted for, then they have fixed co-moving coordinates.

• The whole coordinate system scales with time

• Consider two galaxies that have fixed co-moving coordinates.

DtR )(

DttR )( +

If scale factor increases with time

Useful concept - allows us to separate changes relative to everything from changes due to universe expansion

R R

RR

t t

tt

…also allows us to describe how the universe changes with time

expanding:expansion slowing expanding:

expansion const

expanding:expansion increasing

contracting:contractionincreasing

s2= ct)2 -R2(t)(x2 +y2 +z2 )

…only valid for flat geometry, again, we need to generalize to a metric valid for any geometry

Recall our valid Geometries for our universe

These three forms of curvature the "closed" spherethe "flat" casethe "open" hyperboloid

Geometries

Recall-

These three geometries have the properties of making space homogeneous and isotropic

-as is the observed universe (later) so these three are the subset which are possible geometries for space in the universe

Expanding to a general geometry gives the more complex form of the metric which incorporates the scale factor, R(t)

The Robertson-Walker metric

s2= ct)2 -R2(t){r2(1-kr2)-1 + r22 +r2 sin22 }

Again, R(t) is some unspecified function of R with time

The new thing is “k” the curvature constant

Revisit our 3 Geometries in terms of k

Spherical=closed, k=+1

> 1, i.e. av > crit -analogy is a ball thrown up in the air which doesn’t reach Earths escape vel

Given a line and a point (not on the line) -no parallel line can be drawn through the point

2. Flat spaces (open; k=0)

3. Hyperbolic spaces (open; k=-1)

< 1, av is lower than crit - galaxy separation slows but expansion continues forever-many parallels can be drawn through the point

= 1, av = crit galaxy separation slows, approaching zero

Euclidean geometry-given a line and a point (not on the line) -only 1 unique parallel line can be drawn through the point

1. Closed k=+1

2. Flat/open k=0

3. Hyperbolic/open k=-1

k is the same everywhere, also, R is only a function of time

Means universe has same geometry and scaling throughout

All parts expand the same way according to RW metric- consistent with homogeneity

Expansion the same in all directions-isotropic

Important features of standard models…

• All models begin with R=0 at a finite time in the past– This time is known as the BIG BANG– Space-time curvature is infinite at the big bang– Space and time come into existence at this

moment… there is no time before the big bang!– The big-bang happens everywhere in space…

but scale factor is zero

• There is a connection between the geometry and the dynamics– Closed (k=+1) universes re-collapse– Open (k=-1) universes expand forever– Flat (k=0) universe expand forever (but only

just… they almost grind to a halt).

– Separation between galaxies is given by the three cases shown

Fates of the Universe

CONNECTING STANDARD MODELS & HUBBLE’S LAW

• Recall – Cosmological Redshift is not due to velocity of

galaxies– Galaxies are (approximately) stationary in

space… – Galaxies get further apart because the space

between them is physically expanding!– The expansion of space also effects the

wavelength of light… as space expands, the wavelength expands and so there is a redshift

Relation between z and R(t)• Recall - redshift of a galaxy given by

• Using scale factor to define the expansion in space which causes the wavelength to be longer we can write

em

emobszλ

λλ −=

emgalaxy

Earthobs R

R λλ ⎟⎟⎠

⎞⎜⎜⎝

⎛= no

wthen

Relation between z and R(t)

• Using scale factor to define the expansion in space which causes the wavelength to be longer we can write

• So, we have…

emgalaxy

Earthobs R

R λλ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1−=galaxy

Earth

RR

z

now

now

then

then

em

emobszλ

λλ −=

Relation between z and R(t)

• So, we have…

• and thus redshift can be used to derive a ratio of scale factors at two different epochs

1−=galaxy

Earth

RR

z nowthen

R(t)Up to now, R has been some unspecified function of time

Lets look at dependence on time

Friedmann Equation• Where do we stand ?

• We have the RW metric which describes geometry and allows scaling with time

• Can we evaluate R and k?

• Something called the Friedmann Equation governs the evolution of the scale factor R(t), for the case of a universe described by the RW metric, ie a universe which is homogeneous & isotropic

Friedmann Equation• We know mass and energy determine geometry

• We know that as universe is homogeneous over large scales we can consider average mass properties, like average density

• Universe may also be filled with energy from sources other than rest mass-energy, these other forms can be characterized as energy/unit volume or energy density

• Can use these average values to simplify Einsteins eqn of GR -get rid of dependence of densities on location and consider only gross properties

Friedmann Equation• Full solution via GR is quite complex, suprisingly we can get a feel for

what happens via Newtonian physic because: -its adequate over small bits of flat spacetime

-adequate when expansion of universe happening at v< c-we’re averaging out density etc so detailed curvature of space not an

issue

Results from consideration of this special-condition part of the universe can still tell us some things which hold for the whole universe, so it’s a useful exercise

Friedmann EquationConsider a finite, spherical portion of universe, radius R

Place a test particle at edge of sphere with mass mtp

Recall FR=-GmtpM/R2 as the force on the particle

Sphere can expand or contractand R is radius of sphere, scalefactor & location of test particle

R

mtp

M

Friedmann Equation

As sphere expands velocity of particle given byv=R/t

also recall escape velocity eqn

vesc=(2GM/R) =R/t 2GM/R= (R/t)2

If sphere has precisely the vel toavoid grav. collapse then the expansion speed will equal vesc for every ROther expansion rates are v> vesc v< vesc

R

mtp

M

Friedmann Equation 2GM/R= (R/t)2

If sphere has precisely the vel toavoid grav. collapse then the expansion speed will equal vesc for every ROther expansion rates are v> vesc v< vesc

(R/t)2 =2GM/R +constant

k.e. of particle is Ek=mv2/2

k.e./unit mass =v2 /2

R

mtp

M

Friedmann Equation (R/t)2 = 2GM/R +constant

k.e. of particle is Ek=mv2/2

k.e./unit mass =v2 /2 2=v2

2= (R/t)2 = 2GM/R +constant

at R= constant= 2

is the k.e./unit mass remaining when sphere has expanded to infinite size

R

mtp

M

Friedmann Equation 2= (R/t)2 = 2GM/R +constant

constant= 2 k.e./unit mass remaining @ R=

(R/t)2 = 2GM/R + 2

< 0 then sphere has negative net energy. Will stop expanding before it reaches R=. It will then recollapse

= 0 sphere has zero net energyExactly right vel to keep expanding forever, vel will drop to zero as t/R ->

> 0 net positive energy. Will expand forever and reach R= with some velocity remaining

R

mtp

M

Friedmann Equation (R/t)2 = 2GM/R + 2

Now convert sphere to the universe

Mass = 4/3πR3

(R/t)2 = (8πGR2)/3 + 2

as sphere expands, mass is conserved, so R3 is constant

R

mtp

M

Friedmann Equation(R/t)2 = (8πGR2)/3 + 2

as universe expands, mass is conserved, so R3 is constant

1) negative energy -> will collapse on itself from gravity2) zero energy -> expand forever but vel->0 as R-> 3) positive energy-> expand forever -then “2” is related to the fate of the universe

If we had worked through GR we would have gotten

222

3

8kcR

G

dt

dR−=⎟

⎠⎞

⎜⎝⎛ π

Friedmann EquationIf we had worked through the GR we would have gotten

k is the curvature constant

As we can chose coordinate systems, we adjust to have k=0,+1,-1 corresponding to flat, spherical or hyperbolic geometries

This version from GR is the Friedmann Equation

222

3

8kcR

G

dt

dR−=⎟

⎠⎞

⎜⎝⎛ π

Friedmann Equation• When we go through the GR stuff, we get the Friedmann Equation… this

is what determines the dynamics of the Universe

The Friedmann Equation governs the evolution of the scale factor R(t), for the case of a universe described by the RW metric, ie a universe which is homogeneous & isotropic

222

3

8kcR

G

dt

dR−=⎟

⎠⎞

⎜⎝⎛ π

THE CRITICAL DENSITYA soln, for a given k, is a model of the universe

also recall

Dividing by R2

222

3

8kcR

G

dt

dR−=⎟

⎠⎞

⎜⎝⎛ π

2

22

3

8

R

kcGH −= π

dt

dR

Rt

R

RH

11distance

velocity =

==

Friedmann equation

• Let’s examine this equation…

• H2 must be positive… so the RHS of this equation must also be positive.

• Suppose density is zero (=0)– Then, we must have negative k (i.e., k=-1)

– So, empty universes are open and expand forever

– Flat and spherical Universes can only occur in presence of matter.

2

22

3

8

R

kcGH −= π

Friedmann Equation

• Now, suppose the Universe is flat (k=0)– Friedmann equation then gives

– So, this case occurs if the density is exactly equal to the critical density…

π3

82 GH =

G

Hcrit π

83 2

==

Critical density

• Recall the density parameter

• Can now rewrite Friedmann’s equation yet again using this… we get

c

=

22

2

1RH

kc+=

• Can now see a very important result… within context of the standard model: <1 means universe is hyperbolic and will expand forever (k=-1) =1 means universe is flat and will (just manage to) expand

forever (k=0) >1 means universe is spherical and will recollapse (k=+1)

• Physical interpretation… if there is more than a certain amount of matter in the universe, the attractive nature of gravity will ensure that the Universe recollapses.

22

2

1RH

kc+=

The deceleration parameter, q

• The deceleration parameter measures how quickly the universe is decelerating

• For those comfortable with calculus, actual definition is:

• Turns out that its value is given by

• This gives a consistency check for the standard models… we can attempt to measure in two ways:– Direct measurement of how much mass is in the Universe

– Measurement of deceleration parameter

2

2

2

1

dt

Rd

RHq −=

=2

1q

Deceleration Parameter

• Deceleration shows up as a deviation from Hubble’s law…

• A very subtle effect – have to detect deviations from Hubble’s law for objects with a large redshift