Theory Saturday PGF V2 - Oxford Physics

Cosmology
from
General Relativity
Pedro G. Ferreira
University of Oxford
Oxford, Sept 2015
“An important contribution of the general
theory of relativity to cosmology has been to
keep out theologians by a straightforward
application of tensor analysis.”
E. Schucking
2
Georges Lemaitre
Albert Einstein
http://www.bhm.ch/de/news_04a.cfm?bid=4&jahr=2006
3
Georges Lemaitre
Albert Einstein
“Your mathematics is correct,
your physics is abominable”
Einstein to Lemaitre (1927)
http://www.bhm.ch/de/news_04a.cfm?bid=4&jahr=2006
4
Newtonian Gravity
Gravitational Potential
Gravitational Force
and
fix
Density of matter
Given
solve for
, calculate
5
and then solve for .. .
General Relativity
“Geometry”= “Energy”
http://en.wikipedia.org/wiki/Spacetime
“Space-time tell matter how to move;
matter tells space-time how to curve”
John Archibald Wheeler
6
General Relativity in practice
4 by 4 symmetric matrix of functions of space-time:
The metric of space time
7
General Relativity in practice
The Ricci tensor can be determined from the metric but ...
where
8
General Relativity in practice
“Geometry”
“Energy”
and the energy momentum tensor:
mass density
pressure
9
Cosmology
... find a symmetry
and solve nonlinear equations.
When equation are
too complicated
we simplify ...
... solve a linearized
version of the full
equations
10
Cosmology
Homogeneity- at any given time, the Universe
looks exactly the same at any point in space.
Cosmological Principle:
Isotropy- at any given time, and at any point, the
Universe looks the same in any direction.
11
Cosmology
Energy is “smoothly” distributed:
10 functions of
space-time ...
and
.
...1 function of time
scale factor
12
Cosmology
Homogeneous and isotropic metric “rescales” space
over time.
13
Cosmology
Dramatic simplification of the equations ...
curvature tensor
Einstein equations
14
Cosmology
energy
Gas of massive particles:
energy density today
volume
Solution:
Universe is dynamic not static.
at
Initial singularity.
15
Cosmology
Three homogeneous and isotropic spaces:
Hyper-spherical (positive curvature)
Hyperbolic (negative curvature)
Euclidean or “flat” (no curvature)
16
Cosmology
negative curvature
zero curvature
positive curvature
17
Cosmology
A ball in a gravitational potential:
Mass in volume (
kinetic energy
Sign of
):
potential energy
determines if orbit is bound or unbound.
18
Cosmology
A ball in a gravitational potential:
negative curvature
zero curvature
positive curvature
19
Cosmology
Can the expansion accelerate?
cosmological constant
New terms in Einstein equation:
accelerated expansion
Positive acceleration term
20
Cosmology
Ball in a potential:
accelerated expansion
negative curvature
no curvature
positive curvature
21
Cosmology
Einstein’s static universe:
and
solve:
and
but ... unstable!
22
Cosmology
Universe must evolve!
23
Cosmology
How do we observe the evolution?
slope:
24
Cosmology
light ray
The redshift effect:
observed brightness
Doppler effect
Redshift
25
observed wavelength (Angstroms)
Cosmology
The further away an object the dimmer it is.
luminosity
flux
area
luminosity distance
26
Expanding Universe
Vesto Slipher 1917
21 out of 25 nebula
were redshifted ...
27
Expanding Universe
Lemaitre (1927)
Eddington’s
translation of
Lemaitre (1931)
28
Expanding Universe
Hubble (1929)
1 parsec = 3.26 light years
29
Expanding Universe
Hubble (1929)
Hubble constant:
30
Expanding Universe
The age problem.
where
Redshift effect:
Solution to Einstein equation:
Invert
to find age of the universe
Hubble’s measurement:
Age:
Universe is too young!
31
Expanding Universe
Hubble (1929)
Hubble constant:
32
Expanding Universe
Reiss et al (1995)
Hubble constant:
33
A Hot Big Bang
volume
pressure
Adiabatic expansion:
adiabatic index
Ideal gas law:
A gas of photons:
The universe was hotter in the past
34
A Hot Big Bang
High Temperature
Low Temperature
Binding energy:
35
A Hot Big Bang
number density
of protons
early times
number density
of Hydrogen
36
late times
A Hot Big Bang
Tight coupling:
Free streaming:
37
A Hot Big Bang
P. Roll & D. Wilkinson (1966)
38
A Hot Big Bang
39
An accelerating Universe
Taylor expand the scale factor:
Use
and
Hubble constant
To get:
deceleration rate
An accelerating Universe
Taylor expand the scale factor:
or
Quadratic corrections to Hubble’s
law tell us about acceleration!
An accelerating Universe
Supernovae IA
apparent magnitude
luminosity distance
Betoule et al (2014)
An accelerating Universe
An accelerating Universe
accelerated expansion
negative curvature
no curvature
positive curvature
An accelerating Universe
Small Cosmological Constant is “unnatural”
Quantum vacuum fluctuations give a cosmological
constant:
with
Electron:
Too much accelerated expansion!
An accelerating Universe
Alternatives to
modify gravity
dark energy
scalar field (like the Higgs)
corrections to the Einstein’s equations
new forces
gravitational collapse
potential energy
slow roll
“Remember that the most
beautiful things in the world
are the most useless.”
John Ruskin

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