standard cosmological models [for background, read chapters 10 & 11 from hawley & holcomb]
Post on 16-Jan-2016
216 Views
Preview:
TRANSCRIPT
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
top related