R - MSU Department of Physics and Astronomy

R - MSU Department of Physics and Astronomy

hi
Lecture 20, 03.24.2016
Cosmology 2
1
housekeeping
I got nothin’
2
serious
consideration
3
review
General Relativity and the whole universe
Cosmological Principle: the only way to solve his GR equations…assume
that the universe is homogeneous and isotropic.
Einstein wanted a permanent, spherical (in 3d) universe without an edge
GR equations? Infinite, expanding universe, so he added a term to stop
the expansion, the Cosmological Constant, Λ
de Sitter found a solution to GR without any matter
Friedman found a whole class of solutions to GR
Edwin Hubble:
combined variable star dynamics with atomic spectroscopy
to find that most galaxies are moving away from Earth
4
curvature, “k” – hypervolumes
k = +1,
positive curvature
finite, unbounded
k = 0,
no curvature
infinite, unbounded
k = –1,
negative
curvature
infinite, unbounded
Amathema2calfact:
These3aretheonly
geometriesthatcanbeboth
homogeneousandisotropic
1inch
isimpossibletovisualizethe
nega2vecurvature3dshape...
it’slikeasaddle,ormmm
mmmgood
PringlesPotatoHyperChips
}
ifa=eralong2me,you’remorethan1inch
apart...youliveinanega2vecurvaturespace
5
the
dreaded
Cosmological
Constant, ⇤
geometry
G=T
energy,
pressure,mass
headdedanega%vepressureterm...
G+⇤=T
the“CosmologicalConstant”
later:
“My biggest blunder.”
anega2vepressure-like
term...thatonlyis
relevantonhugescales
MakestheUniversesta%c...notexpandingorcontrac%ng
6
Alexander
Friedman
(1888– 1925)
Addinginsulttoinjury,anunknown
mathema2calmeteorologistfrom
RussiaopenedTheGeneralRela2vity
Pandora’sBox.
G=T
G+⇤=T
in 1922, 23
finds a whole class of
solutions!
with and without Λ
Now,themodernbasisofGRsolu2ons:
the“FriedmanSolu2ons”
29June1922,submitspaper“OnthecurvatureofSpace”to
toZeitschri9fürPhysik
Einsteindidn’ttakeitwell.
7
Hubble
used
His results:
Wavelengths shifted to longer “redshifted”
meaning all of his galaxies seemed to be
moving away from us
eg, seemingly, Doppler shifts at work:
the finger-print
tool of
spectroscopy
far away galaxy
plus
the distance
determination
tool of Cepheid
Variables
nearby star
wavelengths where elements should be
Ca
HI
Mg I
Na I
http://www.astro.ucla.edu/~wright/doppler.htm
8
Doppler
Effect
change of pitch
when source of
sound
v
fO =
fe
v ± ve
v=f
ve
=
v
O
v - speed of sound in air
+ receding
– approaching
e
e
ve
=
c
If light?
O
e
e
moves towards you
or away from you
wavelengths where elements should emit
O
>
e
so
ve > 0
“away from us”
red shifted
9
Hubble’s remarkable conclusion:
all of the galaxies are moving away from us.
1929: a stunning quantitative conclusion
Buttheactualreasonisevenmorestunning.
10
this is not that.
but Hubble presumed that it was
11
remember the Gravitational Red Shift?
12
well, this is not that
either!
13
HUBBLE’S
CONSTANT = 1/T
FROM PRESUMED
DOPPLER-SHIFTED
SPECTRA
Hubble’s Law
H: a measure of the time a galaxy has been “traveling”
v = rH
It’saliVletricky...ThinkBalloons.
FROM LEAVITT’S
CEPHEID VARIABLE
RELATION
14
balloon world
d0
d0
A
B
d0
C
D
Time1
15
balloon world
2d0
A
2d0
2d0
B
C
D
Time2
16
balloonworld
4d0
A
Time3
4d0
B
4d0
C
D
keep track of how far away everything is
from Galaxy A
= 3 d0
(A to C) = 2d0
(A to B) = d0
(A to D)
d0
d0
A
B
d0
C
D
Goingtocalculate
thespeedatwhich
BandCrecede
fromAin
Time1-2
and
Time2-3
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what we had
Time1
Time2
distance, time 1
r = (A to B)= d0
r = (A to C)= 2d0
r = (A to D)= 3d0
distance, time 2
r = 2d0
r = 4d0
r = 6d0
difference:
Δr between
time 1 and 2 = Δt
speed
Δr (A to B)
Δr = 2d0 - d0 = d0
d0/Δt = v0
Δr (A to C)
Δr = 4d0 - 2d0 = 2d0
2d0/Δt = 2v0
Δr (A to D)
Δr = 6d0 - 3d0 = 3d0
3d0/Δt = 3v0
19
plot ‘em up
5v0
slope=2v0/4d0 =v0/2d0
4v0
v 3v0
(between A and ...) 2v0
v0
D
C
B
2d0 4d0 6d0 8d0
r (from A)
v = (slope)r =
( (r
v0
2d0
supposer=5d0 ?
what’sv ?
Also:lookatthedimensionsofthatslope
2.5v0
( (
v0
velocity
m/s
1
:
2d0
distance ~ m ~ 2me
20
Hubble’s Law
a profound discovery about the Universe
v = rH
21
rela2onalert:
Hubble’sLaw
refersto:
example:
v = rH
Speedofagalaxyispropor2onaltothe
distanceawayfromanypoint.
20
galaxyNGC1832is9.57x10 kmaway,
soHubble’sLawsaysitwouldbemoving
atv=2150km/s
original results:
H = 160 km/sMly
1 light year = c ⇥ 1 year = 9.5 ⇥ 1015 m
The outstanding feature, however, is the
possibility that the velocity-distance
relation may represent the de Sitter
effect, and hence that numerical data may
be introduced into discussions of the
general curvature of space.
23
So, what does Hubble’s Law mean?
apart from the balloon...
v = rH
24
Georges
Lemaître
(1894-1966)
5v0
The father of the
Big Bang
get it?
25
1927
three kinds of education
war
seminary
physics
1923
1914
http://www.flickr.com/photos/miguelcalleja/sets/72157604962600986/detail/
1927
Lemaître’s model
published obscurely
again,Einstein
behavedbadly
he believed that
General Relativity
required an
expanding universe
"Your math is correct, but your physics is abominable."
27
Again, Einstein let’s his
prejudices
get the better of him
he’d pay for that
28
In 1927 he published a solution
“A homogeneous Universe of constant mass and growing
radius accounting for the radial velocity of extragalactic
nebulae”
Solving G = T….with spacetime geometry set free
in an obscure Belgian journal
HepredictedtheHconstant!
29
his model required the Universe
to be explicitly expanding
When Hubble’s results were announced
“brilliant”
he showed it to his old advisor, Sir Arthur Eddington who made it
public in 1930:
The Lemaître-Eddington model:
constant size, with Einstein’s value...and expands from there...
30
Lemaître was the first to realize that
Hubble had demonstrated:
1. spacetime is stretching
The entire kit and caboodle is expanding
31
Here’s what it does NOT
mean:
32
galaxies
are not
“moving
away”
universe
inside of the
universe
33
what
stretching
DOES mean
universe
is complicated!
34
what
stretching
DOES mean
universe
is complicated!
35
Lemaître was the first to realize that
Hubble had demonstrated:
1. spacetime is stretching
The entire kit and caboodle is expanding
2. But then he realized that the current Universe could
have come from something smaller
36
think about the balloon coming from a
smaller size
5v0
4v0
and still smaller
3v0
2v0
and still smaller
v0
until.
D
C
B
2d0 4d0 6d0 8d0
*blink*
r (from A)
37
“
We can compare space-time to an open,
conic cup...The bottom of the cup is
the origin of atomic disintegration;
it is the first instant at the bottom
of space-time, the now which has no
yesterday because, yesterday, there
was no space.
George Lemaitre, The Primeval Atom
Lemaître envisioned
A “primeval atom”
it was the heady times of quantum mechanics and early nuclear
physics
He envisioned a fissioning of a big, big nucleus
39
think about this.
a Catholic Priest-Theoretical Physicist
envisioning the beginning of the Universe...a “creation story”?
41
Sir Arthur Eddington states that, philosophically,
the notion of the beginning of the present order of
Nature is repugnant...I would rather be inclined to
think that the present state of quantum theory
suggests a beginning of the world very different
from the present order of Nature.
“
If the world had begun with a single quantum, the
notions of space and time would altogether fail to
have any meaning...
the beginning
of the world happened a little
before the beginning of space and
time. I think that such a beginning...is far
If this suggestion is correct,
Lemaître: Nature
Was his theology in the way of his science?
No. He was explicit in his separation of the science and
his faith
And, the respect that his colleagues held for him
did not result in accusations of him pushing his religion into
Cosmology
43
undercut
Lemaître had
been very careful
to not mix religion
and science
Imagine his panic
when 1951 “Study
Week” the Pious XIII
made a statement:
44
. . . contemporary science, with one sweep back across the
centuries, has succeeded in bearing witness to the august
“
instant of the primordial Fiat Lux, which along with
the matter there burst forth from nothing a sea of light and
radiation . . . Thus, with that concreteness which is
characteristic of physical proofs, modern science has confirmed
the contingency of the universe and also the well-founded
deduction to the epoch when the world came forth from the
hands of the creator.
Pious XIII, Un'Ora,1951
Whoa. Lemaître was stunned.
Science and religion to him: two completely different paths
46
As far as I can see, such a theory remains entirely
outside any metaphysical or religious question. It leaves
the materialist free to deny any transcendent Being. 1
We may speak of this event as of a beginning. I do not say
a creation. Physically it is a beginning in the sense that
if something happened before, it has no observable
influence on the behavior of our universe, as any feature
of matter before this beginning has been completely lost
by the extreme contraction at the theoretical zero. Any
preexistence of the universe has a metaphysical character.
Physically, everything happens as if the theoretical zero
was really a beginning. The question if it was really a
beginning or rather a creation, something started from
nothing, is a philosophical question which cannot be
settled by physical or astronomical considerations. 2
1
Solvay Conference 1958
2
quoted in: Godart and Heller, Cosmology of Lemaître, 67
Lemaître
WWII was hard on Belgium
after the war, Lemaître did not go back to first-principle
cosmology
but he pioneered scientific computing on cosmological
parameters before anyone in the 1950’s
48
Like Copernicus
Within days of his death, Lemaître learned of Penzias and
Wilson’s discovery of the cosmic microwave background
consistent with the Big Bang
June 20, 1966
49
a big “uh oh”
almost immediately after Hubble’s measurement
50
original results:
1 light year = c ⇥ 1 year = 9.5 ⇥ 1015 m
H = 160 km/cMly = 1.68 x 10-17 s-1 )
1
9
= 2 ⇥ 10 years
H
51
oops.
geologists already understood that the Earth was at least 3
By old.
That required some work!
Refinements found a number of assumptions in need of updating
for example...there are 2 kinds of Cepheid Variable stars, and other issues
This is the beginning of quantitative Cosmology.
52
Measuring the Hubble Constant is an
important cottage industry in astronomy
current best result:
H0 = 69.3 ± 0.8 km/sec/Mpc
H0 = 2.25 x 10-18 s-1
1 megaparsec (Mpc) = 106 parsec = 3.26 x 106 light years = 3.086 x 1016 m
53
How old is the universe?
How fast are you expanding from me?
can estimate!
54
constantof
nature:
Hubble“Constant”
value:
H0 = 67.8 +/- 0.9
units:
(km/s)/Mpc (1Mpc = 3 x 1022m) so
H0 = 2.26 x 10-18 s-1
usage:
fundamental measurable in
experimental cosmology
some cautionary comments
The Hubble Constant isn’t constant.
H0 = 67.8 ± 0.9 km/sec/Mpc
H0 = 2.26 x 10-18 s-1
TheinverseoftheHubble
Constantisn’tnecessarily
theageoftheuniverse
Thesubscript“0”means:“Now”
H0-1=4.42x1017s=14.2By
(staytuned)
56
what do the red shift(S) actually imply? &
what was Lemaître’s insight?
not doppler
not gravitational
here’swhatitis.
57
the “red
shift”
close
late
isn’t a Doppler
velocity
it’s geometry
early
58
The further away a galaxy is:
the more red-shifted its spectrum will be
and the faster it will appear to be receding
the older it will be
and the younger it will appear to be!
59
here’s how this is described
a little technical, but you can do it!
60
energy/mass/pressure
energy/mass/pressure
G=T
TIME
G.R.
R(t)
R0
What’s R(t)?
The “scale factor”
the stretchiness of spacetime
62
fabric of
spacetime
Spacetime Fabrics
63
Thenaspiritualmomentoccurs.
involvingspandex.
So,weneed4dei%es:
dark specter from power rangers from space
lord firth watership down
fujin from Mortal Kombat
Arceus from Pokémon
The Friedman, Walker, Robertson models
Friedman’sandLemaître’sworkwasexpandedonbyHowardPRobertson
andArthurGWalkerin1936
Theyfoundexactsolu2onstotheEinsteinequa2ons,usingtheFriedmantechniques.
Theirmodelofcosmologyisvariouslycalledthe:
FWRmodel
FLWRmodel
StandardModelofCosmology
G+⇤=T
G=T
G.R.
s, R(t)
varyingassump2ons,k,density
66
the interval again – spacetime
separation between two events.
InSpecialRela2vity...whichisflatspace2me:
s = (c t)
2
Event1
( r)
2
2
s2 = s2
Event2
s1 = [c(t2
t1 )]2
[(r2
r1 )]2
r2, t2
r1, t1
Ingeneral:
r, t = 0
s = g00 (c t) + g11 ( r)
2
2
2
ForFLRWmodel,aparameteriza2on:
a2me-dependentscale
factor
s = (c t)
2
2
R2 (t)
1+
kr 2
2(
2
r)
everypointinspacereceivesthesame
scalefactorateachinstantof2me
4
thecurvatureneverchanges
thecurvature...
can catalogue the behavior
of R
for different choices of the Cosmological Constant and k
68
Einstein’s original model
r2, t2
r1, t1
R(t )
r, t = 0
WhatdidEinsteinsaywouldbethecase?
time
r2, t2
r1, t1
r, t = 0
Sta2c...forwhichheneededapar2cularvalueof
theCosmologicalConstant
4⇡G⇢
⇤E =
c2
69
FLRW catalogue of Universes
⇤<0
R(t )
⇤ = ⇤E
⇤=0
R(t )
R(t )
k = +1
⇤>0
⇤ > ⇤E
R(t )
LemaîtreEddington
Lemaître
Einstein
time
R(t )
time
R(t )
time
time
R(t )
cri2cal
k=0
accelera2ng
time
R(t )
k=
time
R(t )
1
time
R(t )
open
accelera2ng
time
time
time
70
FLRW catalogue of Universes
⇤<0
R(t )
k = +1
⇤ = ⇤E
⇤=0
A
R(t )
⇤>0
D
⇤ > ⇤E
G
R(t )
R(t )
I
H
time
R(t )
k=0
B
time
R(t )
cri2cal
E
time
time
R(t )
J
accelera2ng
time
R(t )
k=
C
1
time
R(t )
F
time
R(t )
K
open
accelera2ng
time
time
time
71
which one is ours?
that’s the story of the last 3 decades
stay tuned
72