Modeling the Universe
Chapter 11 Hawley/Holcomb
Adapted from
Dr. Dennis Papadopoulos
UMCP
Spectral Lines - Doppler
1+ z =
λem
λobs
λobs − λem
z=
λem
Doppler Examples
Doppler Examples
Expansion Redshifts
= Rnow/Rthen
z=2 three times, z=10,
eleven times
Expansion Redshifts
Expansion - Example
Current Record Redshift
Hubbleology
• Hubble length DH=c/H ,
• Hubble sphere: Volume enclosed in Hubble
sphere estimates the volume of the Universe
that can be in our light-cone; it is the limit of the
observable Universe. Everything that could have
affected us
• Every point has its own Hubble sphere
• Look-back time: Time required for light to travel
from emission to observation
Gravitational Redshift
Interpretation of Hubble law in
terms of relativity
• New way to look at redshifts observed by Hubble
• 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, as R(t) in the metric equation, also
affects the wavelength of light… as space expands, the
wavelength expands and so there is a redshift.
• So, cosmological redshift is due to cosmological
expansion of wavelength of light, not the regular
Doppler shift from local motions.
Relation between z and R(t)
• Using our relativistic interpretation of
cosmic redshifts, we write
⎛ R present ⎞
λobs = ⎜
⎟ λem
⎝ Remitted ⎠
• Redshift of a galaxy is defined by
z=
λobs − λem
λem
• So, we have…
Rpresent
R present − Remitted ΔR
z=
−1 =
≈
R
Remitted
Remitted
Hubble Law for “nearby” (z<0.1) objects
• Thus
cΔR
(ΔR /Δt)
cz ≈
= cΔt ×
= dlight−travel × H
R
R
where Hubble’s constant is defined by
H=
1 ΔR 1 dR
=
R Δt R dt
• But also, for comoving coordinates of two
galaxies differing by space-time interval
d=R(t)×Dcomoving , have
v= Dcomoving × ΔR/Δt=(d/R)×(ΔR/Δt)
• Hence v= d× H for two galaxies with fixed
comoving separation
Peculiar velocities
• Of course, galaxies are not precisely at fixed
comoving locations in space
• They have local random motions, called
“peculiar velocities”
– e.g. motions of galaxies in “local group”
• This is the reason that observational Hubble
law is not exact straight line but has scatter
• Since random velocities do not overall
increase with comoving separation, but
cosmological redshift does, it is necessary to
measure fairly distant galaxies to determine
the Hubble constant accurately
Distance determinations
further away
• In modern times, Cepheids in the Virgo galaxy
cluster have been measured with Hubble Space
Telescope (16 Mpc away…)
Virgo cluster
Tully-Fisher relation
• Tully-Fisher relationship (spiral galaxies)
– Correlation between
• width of particular emission line of hydrogen,
• Intrinsic luminosity of galaxy
– So, you can measure distance by…
• Measuring width of line in spectrum
• Using TF relationship to work out intrinsic luminosity of galaxy
• Compare with observed brightness to determine distance
– Works out to about 200Mpc (then hydrogen line becomes too
hard to measure)
Hubble time
•
•
Once the Hubble parameter has been determined accurately, it
gives very useful information about age and size of the
expanding Universe…
Recall Hubble parameter is ratio of rate of change of size of
Universe to size of Universe:
1 ΔR 1 dR
H=
•
R Δt
=
R dt
If Universe were expanding at a constant rate, we would have
ΔR/Δt=constant and R(t) =t×(ΔR/Δt) ; then would have
H=
(ΔR/Δt)/R=1/t
• ie tH=1/H would be age of Universe since Big Bang
R(t)
t
Modeling the Universe
BASIC COSMOLOGICAL
ASSUMPTIONS
• 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…
• How to proceed… it’s a horribly complex
problem
How to make progress…
• Proceed by ignoring details…
– Imagine that all matter in universe is “smoothed” out
– i.e., ignore details like stars and galaxies, but deal
with a smooth distribution of matter
• Then make the following assumptions
– Universe is homogeneous – every place in the
universe has the same conditions as every other
place, on average.
– Universe is isotropic – there is no preferred direction
in the universe, on average.
• There is clearly large-scale structure
– Filaments, clumps
– Voids and bubbles
• But, homogeneous on very large-scales.
• So, we have the…
• The Generalized Copernican Principle… there
are no special points in space within the
Universe. The Universe has no center!
• These ideas are collectively called the
Cosmological Principles.
Key Assumptions
Riddles of Conventional Thinking
Stability
GR vs. Newtonian
Newtonian Universe
Expanding Sphere
Fates of Expanding Universe
Spherical Universe
Friedman Universes
Einstein’s Greatest Blunder
THE DYNAMICS OF THE
UNIVERSE – EINSTEIN’S MODEL
• Einstein’s equations of GR
8πG
G= 4 T
c
“G” describes the spacetime curvature (including its
dependence with time) of
Universe… here’s where we
plug in the RW geometries.
“T” describes the matter
content of the Universe.
Here’s where we tell the
equations that the Universe is
homogeneous and isotropic.
• Einstein plugged the three homogeneous/isotropic cases
of the FRW metric formula into his equations of GR to
see what would happen…
• Einstein found…
– That, for a static universe (R(t)=constant), only the spherical
case worked as a solution to his equations
– If the sphere started off static, it would rapidly start collapsing
(since gravity attracts)
– The only way to prevent collapse was for the universe to start off
expanding… there would then be a phase of expansion
followed by a phase of collapse
• So… Einstein could have used this to predict that the
universe must be either expanding or contracting!
• … but this was before Hubble discovered expanding
universe (more soon!)– everybody thought that
universe was static (neither expanding nor
contracting).
• So instead, Einstein modified his GR equations!
– Essentially added a repulsive component of gravity
– New term called “Cosmological Constant”
– Could make his spherical universe remain static
– BUT, it was unstable… a fine balance of opposing
forces. Slightest push could make it expand
violently or collapse horribly.
• Soon after, Hubble discovered that the
universe was expanding!
• Einstein called the Cosmological Constant
“Greatest Blunder of My Life!
• ….but very recent work suggests that he
may have been right (more later!)
Sum up Newtonian Universe
Newtonian Universe
Send r->, k is twice the kinetic
energy per unit mass remaining when
the sphere expanded to infinite size
Fates of Expanding Universe
FINITE SPHERE
V 2 = 2GM s / R + k
k = 2 E∞
Explore R->
2GM s
2
&
R =
+ 2 E∞
R
1. E <0, negative energy per unit mass; expansion stops and re-collapses
2. E =0, zero net energy; exactly the velocity required to expand forever
but velocity tends to zero as t and R go to infinity
3. E> 0, positive energy per unit mass; keeps expanding forever; reaches
infinity with some velocity to spare
BIG LEAP -> CONSIDER SPHERE THE UNIVERSE
4
M s = π R3 ρ
3
ΔR &
V=
≡R
Δt
GM
ΔV &
≡ R&= g = − 2 s
Δt
R
What happens
when R-> ?
&
&= − 4 π G ρ R
R
3
8
R&2 = π G ρ R 2 + 2 E∞
3
Standard Model
From Newtonian to GR
The Friedmann Equation
&
&= − 4 π G ρ R
R
3
8
2
&
R = π G ρ R 2 + 2 E∞
3
8
R&2 = π G ρ R 2 − kc 2
3
Robertson Walker (RW)
metric : k=0, +1, -1
In Friedmann’s equation R is the scale factor rather than the radius of an
arbitrary sphere
Gravity of mass and energy of the Universe acts on space time scale factor
much as the gravity of mass inside a uniform sphere acts on its radius
and
E replaced by curvature constant. Term retains significance as an energy at
infinity but it is tied to the overall geometry of space
Standard Model Simplifications
8
R& = π G ρ R 2 − kc 2
3
2
4
&
&
R = − π Gρ R
3
To solve we need to know how mass-energy density changes with time.
If only mass ρR3=constant.
Now need relativistic equation of mass-energy conservation and
equation of state i.e ρE =f(ρm).
Notice that here ρmR3 =constant but ρER4=constant. Why the extra R?
Mainly photons left out of Big-Bang. Red-shifting due to expansion
reduces energy density per unit volume faster than 1/R3
Photons dominant early in Universe are negligible source of space time
curvature compared with mass to day.
&
&< 0
All models decelerate R
Also now dR/dt>0 expansion.
For all models R=0 at some time. R=0 at t=0. Density -> infinity, and kc2
term negligible at early times. Great simplification.
Fate of Universe-Standard Model
While early time independent of curvature factor ultimate fate critically
dependant on value of k, since mass-energy term decreases as 1/R.
Fate of Universe in Newtonian form depended on value of E . In
Friedmann Universe it depends on value of curvature k.
All models begin with a BANG but only the spherical ends with BANG
while the other two end with a whimper.
Theoretical Observables
• Friedmann equation describes evolution of scale factor R(t) in the
Robertson-Walker metric. i.e. universe isotropic and homogeneous.
• Solution for a choice of ρ and k is a model of the Universe and gives
R(t)
• We cannot observe R(t) directly. What else can we observe to check
whether model predictions fit observations?
• Need to find observable quantities derived from R(t).
Enter Hubble
H = v / l = R&/ R
Since R and its rate are functions of time H function of time. NOT
CONSTANT. Constant only at a particular time. Now given symbol H0
8
2
&
R = π G ρ R 2 − kc 2
3
R&2
8
kc 2
2
= H = π Gρ − 2
2
R
3
R
Time evolution equation for H(t)
Replaces scale factor R by
measurable quantities H, ρ and
spatial geometry
Observing Standard Model
Average mass density critical parameter why?
kc 2
2 8π G ρ 0
= H0 (
− 1)
2
2
R0
3H 0
Measurement of H0 and ρ0 give
curvature constant k
Explore equation:
1. Empty universe ρ=0, k negative hyperbolic universe, expand forever
2. Flat or require matter or energy.
3. k=0 -> critical density
8π G ρ
3 H 02
≡ Ω
3 H 02
=
8π G
ρc
Ω
0
M
=
ρ0
ρc
M
= 1
Critical Density
8π G ρ 0
≡ ΩM = 1
2
3H 0
2
0
3H
ρc =
8π G
ρ0
ΩM =
ρc
• If H0 100 km s-1 Mpc-1 critical
density is 2x10-26 kg/m3 or 10
Hydrogen atoms per cubic meter of
space
• Scales as H2
• 50 km/sec Mpc gives ¼ density
• Current value of 72 km/sec Mpc
gives critical density 10-26 kg/m3
• Ω M=1 gives boundary between
open hyperbolic universes and
closed, finite, spherical universe
• In a flat universe Ω is constant
otherwise it changes with cosmic
time
Deceleration Parameter q
R0
2
2
&
R = H 0 Ω M ( ) − kc 2
R
&
&
R
q≡−
2
RH
1
q = ΩM
2
Deceleration Parameter. Now q0 . All
standard models decelerate q.0. Need
cosmological constant to change it
For standard models specification of q determines
geometry of space and therefore specific model
Summary - Definitions
Review
• How does R(t) (and H) change in time?
And what is the value of the curvature k?
• Need to solve Einstein’s equation!
8πG
G= 4 T
c
STANDARD COSMOLOGICAL
MODELS
•
•
•
In general Einstein’s equation relates geometry to dynamics
That means curvature must relate to evolution
Turns out that there are three possibilities…
k=-1
k=0
k=+1
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 and time come into existence at this
moment… there is no time before the big
bang!
– The big bang happens everywhere in
space… not at a point!
• There is a connection between the
geometry and the dynamics
– Closed (k=+1) solutions for universe expand
to maximum size then re-collapse
– Open (k=-1) solutions for universe expand
forever
– Flat (k=0) solution for universe expands
forever (but only just barely… almost grinds
to a halt).
Hubble time
We can relate this to observations…
•
•
Once the Hubble parameter has been determined accurately
from observations, it gives very useful information about age
and size of the expanding Universe…
Recall Hubble parameter is ratio of rate of change of size of
1 ΔR 1 dR
Universe to size of Universe:
H=
•
R Δt
=
R dt
If Universe were expanding at a constant rate, we would have
ΔR/Δt=constant and R(t) =t×(ΔR/Δt) ; then would have
H=
(ΔR/Δt)/R=1/t
• i.e. tH=1/H would be age of Universe since Big Bang
R(t)
t
Hubble time for nonuniform
expansion
Slope of R(t)
curve is dR/dt
Big
Bang
NOW
– Hubble time is tH=1/H=R/(dR/dt)
– Since rate of expansion varies, tH=1/H gives an estimate of the age
of the Universe
– This tends to overestimate the actual age of the Universe since the
Big Bang
Terminology
– Hubble distance, D=ctH (distance that light travels in a
Hubble time). This gives an approximate idea of the size of
the observable Universe.
– Age of the Universe, tage (the amount of cosmic time since
the big bang). In standard models, this is always less than
the Hubble time.
– Look-back time, tlb (amount of cosmic time that passes
between the emission of light by a certain galaxy and the
observation of that light by us)
– Particle horizon (a sphere centered on the Earth with radius
ctage; i.e., the sphere defined by the distance that light can
travel since the big bang). This gives the edge of the actual
observable Universe.
Friedmann Equation
•
•
•
Where do the three types of evolutionary solutions come
from?
8πG
Back to Einstein’s eq….
G=
T
c4
When we put the RW metric in Einstein’s equation and go
though the GR, we get the Friedmann Equation… this is what
determines the dynamics of the Universe
8π G
2GM
R&2 =
ρ R 2 − kc 2 =
− kc 2
3
R
•
What are the terms involved?
– G is Newton’s universal constant of gravitation
– R&is the rate of change of the cosmic scale factor
same as ΔR/Δt for small changes. in time
– ρ is the total matter and energy density
– k is the geometric curvature constant
• If we divide Friedmann equation by R2 , we get:
2
&
R 2
8π G
kc
2
( ) ≡H =
ρ− 2
R
3
R
• Let’s examine this equation…
• H2 must be positive… so the RHS of this equation
must also be positive.
• Suppose density is zero (ρ=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
(enough) matter.
Critical density
• What are the observables for flat solution
(k=0)
– Friedmann equation then gives
8πG
2
ρ
H =
3
– So, this case occurs if the density is exactly equal
to the critical density…
ρ = ρ crit
3H 2
=
8πG
– “Critical” density means “flat” solution for a given
value of H, which is the most easily observed
parameter
2
8
π
G
kc
H2 =
ρ− 2
3
R
• In general, we can define the density
parameter…
8πGρ
ρ
Ω≡
≡
ρ crit 3H 2
• Can now rewrite Friedmann’s equation yet
again using this… we get
kc 2
Ω = 1+ 2 2
H R
Omega in standard models
kc 2
Ω = 1+ 2 2
H R
• Thus, within context of the standard model:
– Ω<1 if k=-1; then universe is hyperbolic and will expand
forever
– Ω=1 if k=0; then universe is flat and will (just manage to)
expand forever
– Ω>1 if k=+1; then universe is spherical and will recollapse
• Physical interpretation:
If there is more than a certain amount of matter in the
universe (ρ>ρcritical), the attractive nature of gravity will
ensure that the Universe recollapses!
Value of critical density
• For present best-observed value of the
Hubble constant, H0=72 km/s/Mpc
critical density is equal to
ρcritical=10-26 kg/m3 ; i.e. 6 H atoms/m3
• Compare to:
– ρwater = 1000 kg/m3
– ρair=1.25 kg/m3 (at sea level)
– ρinterstellar gas =2 × 10-21 kg/m3
The deceleration parameter, q
• The deceleration parameter measures
how quickly the universe is decelerating
(or accelerating)
• In standard models, deceleration occurs
because the gravity of matter slows the
rate of expansion
• For those comfortable with calculus,
actual definition of q is:
&
&
R
q=−
RH 2
Matter-only standard model
• In standard model where density is from rest mass
energy of matter only, it turns out that the value of the
deceleration parameter is given by
Ω
q=
2
• This gives a consistency check for the standard,
matter-dominated models… we can attempt to
measure Ω in two ways:
– Direct measurement of how much mass is in the Universe -i.e. measure mass density and compare to critical value
– Use measurement of deceleration parameter
– Measurement of q is analogous to measurement of Hubble
parameter, by observing change in expansion rate as a
function of time: need to look at how H changes with redshift
for distant galaxies
Direct observation of q
• 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
• Newtonian interpretation is therefore:
– k=-1 is “positive energy” universe (which is why it
expands forever)
– k=+1 is “negative energy” universe (which is why it
recollapses at finite time)
– k=0 is “zero energy” universe (which is why it
expands forever but slowly grinds to a halt at infinite
time)
k=-1
k=0
k=+1
Expansion rates
• For flat (k=0, Ω=1), matter-dominated
universe, it turns out there is a simple solution
to how R varies with t :
⎛t⎞
R(t) = R(t 0 )⎜ ⎟
⎝ t0 ⎠
2/3
– This is known as the Einstein-de Sitter solution
• In solutions with Ω>1, expansion is slower
(followed by re-collapse)
• In solutions with Ω<1, expansion is faster
Modified Einstein’s equation
•
But Einstein’s equations most generally also can include an
extra constant term; i.e. the T term in
8πG
G= 4 T
c
has an additional term which just depends on space-time
geometry times a constant factor, Λ
• This constant Λ (Greek letter “Lambda”) is known as the
“cosmological constant”;
• Λ corresponds to a vacuum energy, i.e. an energy not
associated with either matter or radiation
• Λ could be positive or negative
– Positive Λ would act as a repulsive force which tends to make
Universe expand faster
– Negative Λ would act as an attractive force which tends to make
Universe expand slower
•
Energy terms in cosmology arising from positive Λ are now
often referred to as “dark energy”
Modified Friedmann Equation
•
When Einstein equation is modified to include Λ, the Friedmann
equation governing evolution of R(t) changes to become:
2
8
π
G
Λ
R
R&2 = H 2 R 2 =
ρ R2 +
− kc 2
3
3
•
•
•
Dividing by (HR)2 , we can consider the relative contributions of
the various terms evaluated at the present time, t0
The term from matter at t0 has subscript “M”;
Two additional “Ω” density parameter terms at t0 are defined:
ρ
ρ0
ΩM ≡ 0 ≡
ρ crit (3H 0 2 /8πG)
•
Λ
ΩΛ ≡
2
3H 0
Altogether, at the present time, t0, we have
1 = ΩM + ΩΛ + Ωk
kc 2
Ωk ≡ − 2 2
R0 H 0
Generalized Friedmann Equation in terms
of Ω’s
• The generalized Friedmann equation governing
evolution of R(t) is written in terms of the present
Ω’s (density parameter terms) as:
2
⎡
⎛ R0 ⎞
⎛R ⎞
2
2
2
&
R = H 0 R0 ⎢Ω M ⎜ ⎟ + Ω Λ ⎜ ⎟ + Ω k
⎝ R0 ⎠
⎝ R⎠
⎢⎣
⎤
⎥
⎥⎦
• The only terms in this equation that vary with time are
the scale factor R and its rate of change R&
• Once the constants H0, ΩM, ΩΛ, Ωk are measured
empirically (using observations), then whole future of the
Universe is determined by solving this equation!
• Solutions, however, are more complicated than when
Λ=0
Special solution
Static model (Einstein’s)
–
–
–
–
Solution with Λc=4πGρ , k=ΛcR2/c2
No expansion: H=0, R=constant
Closed (spherical)
Of historical interest only since Hubble’s discovery
that Universe is expanding!
R&2 / R 2 = (8π G ρ / 3) + (Λ / 3) − (kc 2 / R 2 )
R&= 0
Effects of Λ
• Deceleration parameter (observable) now
depends on both matter content and Λ (will
discuss more later)
• This changes the relation between evolution
and geometry. Depending on value of Λ,
– closed (k=+1) Universe could expand forever
– flat (k=0) or hyperbolic (k=-1) Universe could
recollapse
Consequences of positive Λ
• Because Λ term appears with positive power of R in
Friedmann equation, effects of Λ increase with time if R
keeps increasing R& = H R = 8π G ρ R + ΛR − kc
2
2
2
2
2
3
2
3
• Positive Λ can create accelerating expansion!
THE DE SITTER UNIVERSE
“de Sitter” model:
Solution with Ωk=0 (flat space!), ΩM
=0 (no matter!), and Λ >0
Hubble parameter is constant
Expansion is exponential
R = R0e
Ht / t 0
• Steady solution:
–
–
–
–
Constant expansion rate
Matter constantly created
No Big Bang
Ruled out by existing observations:
• Distant galaxies (seen as they were light travel time in the
past) differ from modern galaxies
• Cosmic microwave background implies earlier state with
uniform hot conditions (big bang)
• Observed deceleration parameter differs from what would be
required for steady model