Post on 18-Sep-2020
curvaturevacuumradiation+neutrinosbaryons+DM
The cosmologist T-shirt (if 𝞚>0):
DENSITYISN’T
DESTINY!
ΩM,0=2.0 ΩΛ,0=0.1
ΩM,0=2.0
ΩM,0=0.9 ΩΛ,0=1.6
ΩM,0=2.5
ΩR=0
TODAY
68%
27%
5%Dark Energy
Dark Matter
100%
Dark Energy
10 t0
24%
64%
12%
Radiation
Dark Matter
Baryons
Baryons
Z=1100
COSMOLOGY WITH SUPERNOVAE
ΩM+ΩΛ=1
COSMOLOGY WITH THE CMB
ECHOS FROM THE BIG BANG
−270.42 C−454.76 F
6
As the Universe expands and cools down, the energy spectrum of the CMB continues to correspond to a thermal distribution, but one with even lower temperature…..
COBE
launched Nov 89
The CMB is the most accurate Planck curve ever measured. Physicists in their labs cannot make a better blackbody!
The WMAP all sky map, after removal of the radiation coming from Milky Way disk.
“hot” spot:T=2.7266 K
“cold” spot:T=2.7262 K
Planck
How to Measure the CMB Power Spectrum
1.Measure <T> inside a circular region of diameter θ degrees centered on a random spot.
2.Repeat this for many random spots.
3.Measure the variance (`scatter’) in these average temperatures. This is the power on a scale of θ degrees.
4.Repeat steps 1-3 for different values of θ. The power as a function of θ is called the power spectrum….
Planck
This analysis reveals a series of harmonically related peaks at distances of 0.1° to 1°, indicative of gravity-driven pressure waves in the original cosmic hot plasma.
These gravity-driven pressure waves which propagated in the early universe have some of the characteristics of acoustic waves generated in the air inside the resonating cavity of a musical instrument in that they have a fundamental frequency and higher harmonics.
The power spectrum of the CMB is based on the two-dimensional distribution of pressure waves on the surface of a sphere of radius 380,000 light years. Recall that these pressure waves were propagating through a three dimensional volume until recombination, when their image was frozen onto the last scattering surface, which we now see from the inside out.
Timbre is the sound quality that allows us to distinguish a flute from a clarinet from a bagpipe from an organ pipe, even if they are all playing the same pitch at the same volume. It is the audible manifestation of the power spectrum of an instrument. The power spectrum of an instrument is a like a fingerprint which distinguishes it from other instruments, and depends on the characteristic properties of the instrument: the material it is made of, its size and shape, whether it is a wind or string, among others.
THE “MUSIC” OF THE CMB
Best cosmological model (Planck, k=0) as of 2015!
EdS
Note that, if dark energy ≠Λ, then:
phantom energy Big Rip!
If w≈−1, the coincidence between the observed vacuum energy and the current matter density appears completely unnatural, as the relative balance of vacuum and matter changes rapidly as the Universe expands:
singularity in the future!
can get acceleration with an energy density that
decreases with time!
Planck 2018: wDE=-1.03±0.03 Λ!
As a consequence, at early times the vacuum energy was negligible in comparison to matter, while at late times matter is negligible. There is only a brief epoch of the Universe's history during which it would be possible to witness the transition from domination by one type of component to another. It seems remarkable that we live during the short transitional period between these two eras. Let us compute Ωi(a) as
Ω𝜦ΩMΩR
e
WHO ORDERED THIS?
The approximate coincidence between matter and vacuum energies in the current Universe is one of several puzzling features of the composition of the total energy density. Another great surprise is the comparable magnitudes of the baryon density and the density of cold non-baryonic dark matter, and perhaps also that in massive neutrinos. In our current understanding, these components are relics of completely unrelated processes in the very early Universe, and there seems to be no good reason why they should be of the same order of magnitude. The real world seems to be a more rich and complex place than Occam’s razor might have predicted.
It is important to keep in mind, however, the crucial distinction between the coincidences relating the various matter components and that relating the matter and vacuum energy: the former are set once and for all by primordial processes and remain unchanged as the Universe evolves, while the latter holds true only during a certain era.
Classroom Problems
1. If we live in a radiation–dominated Friedmann Universe with Ω0 = 2.5, show that thescale factor a will increase to a maximum value, amax = 1.29 a0.
The Friedmann equation
H2 =8πG
3ρ −
kc2
a2
can be rewritten at t = t0 (today) as
H20 (Ω0 − 1) = kc2.
At a = amax → H(amax) = 0 and ρ = ρ0/a4max, and the Friedmann equation becomes
0 =Ω0H2
0
a4max
−
kc2
a2max
=Ω0H2
0
a4max
−
(Ω0 − 1)H20
a2max
.
Solving for amax, one gets
amax =
!
Ω0
Ω0 − 1,
and this yields amax = 1.29 for Ω0 = 2.5.
2. Using Friedmann equation show that, if ΩM,0 = 0.2 now, ΩM (z) at z = 9 (the redshiftof the most distant galaxy) was equal to 0.9492 (assume ΩΛ = 0.7).
In a open Universe, ΩK,0 = 1−ΩM,0−ΩΛ,0. Friedmann equation at redshift z (scale factora) can be written as:
H2 = H20 [ΩM,0/a3 + ΩK,0/a2 + ΩΛ,0].
The matter density parameter at a is then:
ΩM (a) =ρM (a)
ρcrit(a)=
ρM,0/a3
ρcrit(0)(H2/H20 )
=ΩM,0/a3
[ΩM,0/a3 + ΩK,0/a2 + ΩΛ,0].
For a = 0.1 and ΩM,0 = 0.2 this expression gives ΩM (a) = 0.9492.
Written in 300 BC, Euclid’s Elements is the most influential works in the history of mathematics: 13 books, studied for 24 centuries, 2nd only to the Bible in the number of editions published (>1000)!
It is impossible to derive the Parallel Postulate from the first four. The numerous (and failed) attempts to do that gave rise to a slew of statements equivalent to the postulate itself:
1. There exists a pair of similar noncongruent triangles.2. There exists a pair of straight lines everywhere equidistant
from one another.3. For any three noncollinear points, there exists a circle
passing through them.4. If three angles of a quadrilateral are right angles, then the
fourth angle is also a right angle.5. If a straight line intersects one of two parallels it will
intersect the other.6. Straight lines parallel to a third line are parallel to each
other.7. Two straight lines that intersect one another cannot be
parallel to a third line.8. There is no upper limit to the area of a triangle.
The last one seems especially intuitive. The reverse holds in non-Euclidean geometries of Lobachevsky and Riemann. Lewis Carroll (mathematicians and author of Alice in Wonderland) could not accept this assertion and considered it as a proof of the contradictory nature of non-Euclidean geometries.
Euclidean flat geometry:
1) the angles of a triangle add up to 180o.2) the circumference of a circle of radius r is equal to 2πr.
Riemann (a student of Gauss) “On the hypotheses which lie at the foundation of geometry” (1868):
1) Euclid’s final axiom was an arbitrary choice.2) founded non-Euclidean geometries which are the mathematical foundation of Einstein’s GR.
Simplest non-Euclidean geometry is spherical (or elliptical) k>0:
1) Unlike the case of a flat geometry (k=0), the spherical surface is finite in extent (its area being 4πr2), and yet there is no boundary or edge. If we draw parallel lines on the surface of the Earth, then they violate Euclid's final axiom.
3) If you draw a triangle on a sphere, we find that the angles do not add up to 180o either: start at the North Pole, draw two great circles down towards the equator, 90o apart, and then join them with a line at the equator. You have drawn a triangle in which all 3 angles are 90o.
2) The definition of a straight line is the shortest distance between two points (geodesic), which means that the straight lines in spherical geometry are segments of great circles, such as the equator or the lines of longitude.
4) The circumference of a circle of radius r is smaller than 2πr.
Hyperbolic (saddle-shaped) geometry k<0:
1) In a hyperbolic geometry, parallel lines never meet — in fact they break Euclid's axiom by diverging away from one another. Because parallel lines never meet, such a Universe must be infinite in extent, just an in the flat case.
2) The angles of a triangle add up to less than 180o.
3) The circumference of a circle of radius r is greater than 2πr.
THE WAY OF NEWTON:Matter tells gravity how to exert a force (F=-GMm/r2);
Force tells mass how to accelerate (F=ma).
THE WAY OF EINSTEIN:Matter-energy (stress-energy tensor) tells space-
time how to curve. Curved space-time tells mass-energy how to move.
Gravity according to GR is caused by a warping of space-time:
Global Positioning System. The satellites making up the GPS system contain atomic clocks which broadcast a time signal, and the rate at which those clocks “tick” has to be adjusted to take general relativity into account.
Without the relativistic correction, the clocks would drift by some +38 (+45 from GR -7 from SR) microseconds a day, corresponding to 11 km of position uncertainty. As the system works to give you your position on the Earth to within a few meters, we know that the relativistic correction works, and thus general relativity is correct.