Spectral Lines - Doppler Modeling the Universe Chapter 11...
Transcript of Spectral Lines - Doppler Modeling the Universe Chapter 11...
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Modeling the UniverseChapter 11 Hawley/Holcomb
Adapted fromDr. Dennis Papadopoulos
UMCP
Spectral Lines - Doppler
obs em
em
z λ λλ−
=
1 em
obs
z λλ
+ =
Doppler Examples
Doppler Examples Expansion Redshifts
= Rnow/Rthen
z=2 three times, z=10, eleven times
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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 RedshiftInterpretation 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.
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Relation between z and R(t)• Using our relativistic interpretation of
cosmic redshifts, we write
• Redshift of a galaxy is defined by
• So, we have…em
emobszλ
λλ −=
λobs =Rpresent
Remitted
⎛
⎝ ⎜
⎞
⎠ ⎟ λem
z =Rpresent
Remitted
−1=Rpresent − Remitted
Remitted
≈ΔRR
Hubble Law for “nearby” (z<0.1) objects
• Thus
where Hubble’s constant is defined by
• But also, for comoving coordinates of two galaxies differing by space-time intervald=R(t)×Dcomoving , have
v= Dcomoving × ΔR/Δt=(d/R)×(ΔR/Δt)• Hence v= d× H for two galaxies with fixed
comoving separation
H =1R
ΔRΔt
=1R
dRdt
cz ≈cΔRR
= cΔt ×(ΔR /Δt)
R= dlight− travel × H
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:
• 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
H =1R
ΔRΔt
=1R
dRdt
R(t)
t
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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
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Stability GR vs. Newtonian
Newtonian Universe
Expanding Sphere Fates of Expanding Universe
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Spherical Universe Friedman Universes
Einstein’s Greatest Blunder
THE DYNAMICS OF THE UNIVERSE – EINSTEIN’S MODEL
• Einstein’s equations of GR
TG 48cGπ
=“G” describes the space-time 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
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• 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 Universe2 2 /
2sV GM R k
k E∞
= +
=2 2 2sGMR E
R ∞= +&
FINITE SPHERE
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
3
2
43s
s
M R
RV Rt
GMV R gt R
π ρ=
Δ= ≡
ΔΔ
≡ = = −Δ
&
&&
2 2
43
8 23
R G R
R G R E
π ρ
π ρ ∞
= −
= +
&&
&What happens when R-> ?
Explore R->
Standard Model
2 2 283
R G R kcπ ρ= −&
From Newtonian to GR
The Friedmann Equation
2 2
43
8 23
R G R
R G R E
π ρ
π ρ ∞
= −
= +
&&
&
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
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Standard Model Simplifications
2 2 283
R G R kcπ ρ= −&
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.
All models decelerate 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.
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R G Rπ ρ= −&&
0R <&&
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).
/ /H v l R R= = &Enter Hubble
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
2 2 2
2 22
2 2
83
83
R G R kc
R kcH GR R
π ρ
π ρ
= −
= = −
&
&
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?
22 002 2
0 0
8( 1)3Gkc H
R Hπ ρ
= − 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 density0
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20
0
8 13
38
M
c
Mc
GH
HG
π ρ
ρπρρ
≡ Ω =
=
Ω =
Critical Density
020
20
0
8 13
38
M
c
Mc
GH
HG
π ρ
ρπρρ
≡ Ω =
=
Ω =
• 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 Mpcgives 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
2 2 200 ( )M
RR H kcR
= Ω −&
2
RqRH
≡ −&& Deceleration Parameter. Now q0 . All
standard models decelerate q.0. Need cosmological constant to change it
12 Mq = Ω For standard models specification of q determines
geometry of space and therefore specific model
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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!
TG 48cGπ
=
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).
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Hubble timeWe 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 Universe to size of Universe:
• 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
H =1R
ΔRΔt
=1R
dRdt
R(t)
t
Hubble time for nonuniformexpansion
– 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
Slope of R(t)curve is dR/dt
NOW
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?• Back to Einstein’s eq….
• 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
• What are the terms involved?– G is Newton’s universal constant of gravitation– 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
2 2 2 28 23G GMR R kc kc
Rπ ρ= − = −&
.
TG 48cGπ
=
R&
• If we divide Friedmann equation by R2 , we get:
• 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.
22 2
2
8( )3
R G kcHR R
π ρ≡ = −&
Critical density• What are the observables for flat solution
(k=0)– Friedmann equation then gives
– So, this case occurs if the density is exactly equal to the critical density…
– “Critical” density means “flat” solution for a given value of H, which is the most easily observed parameter
ρπ3
82 GH =
GH
crit πρρ
83 2
==
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• In general, we can define the density parameter…
• Can now rewrite Friedmann’s equation yet again using this… we get
Ω ≡ρ
ρcrit
≡8πGρ3H 2
22
2
1RH
kc+=Ω
22
2
83G kcH
Rπ ρ= −
Omega in standard models
• 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!
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2
1RH
kc+=Ω
Value of critical density
• For present best-observed value of the Hubble constant, H0=72 km/s/Mpccritical density is equal toρcritical=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:
2
RqRH
= −&&
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
• 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
q =Ω2
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
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• 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=+1
k=0
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 :
– 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
R(t) = R(t0)tt0
⎛
⎝ ⎜
⎞
⎠ ⎟
2 / 3
Modified Einstein’s equation• But Einstein’s equations most generally also can include an
extra constant term; i.e. the T term in
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 notassociated 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”
TG 4
8cGπ
=
Modified Friedmann Equation• When Einstein equation is modified to include Λ, the Friedmann
equation governing evolution of R(t) changes to become:2
2 2 2 2 283 3G RR H R R kcπ ρ Λ
= = + −&
1= ΩM + ΩΛ + Ωk
• 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:
ΩM ≡ρ0
ρcrit
≡ρ0
(3H02 /8πG) Ωk ≡ −
kc 2
R02H0
2ΩΛ ≡
Λ3H0
2
• Altogether, at the present time, t0, we have
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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:
22 2 2 0
0 00
M kR RR H RR RΛ
⎡ ⎤⎛ ⎞ ⎛ ⎞= Ω + Ω + Ω⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦
&
• The only terms in this equation that vary with time are the scale factor R and its rate of change
• 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
R&
Special solutionStatic 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!
2 2 2 2/ (8 / 3) ( / 3) ( / )0
R R G kc RR
π ρ= + Λ −
=
&&
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 Rkeeps increasing
• Positive Λ can create accelerating expansion!
22 2 2 2 28
3 3G RR H R R kcπ ρ Λ
= = + −&
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“de Sitter” model: Solution with Ωk=0 (flat space!), ΩM=0 (no matter!), and Λ >0Hubble parameter is constant
Expansion is exponential
R = R0eHt / t0
THE DE SITTER UNIVERSE
• 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
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