Download CosmologyL2

NCPP
Primorsko
June 2007
Topics in Cosmology-2
Daniela Kirilova
Institute of Astronomy, BAS
Outline
Introduction to Cosmology
The Universe Dynamics
The Expanding Universe – observational status
Universe Parameters
H constant
Universe age
The Expansion History of the Universe
Cosmological Principle is exact at large scales
> 200 Mpc (containing mlns of galaxies)
It is a property of the global Universe.
The RW Metric
In case CP holds the most general expression for a space-time metric
which has a (3D) maximally symmetric subspace of a 4D space-time is
the Robertson-Walker metric:
c = 1 assumed. By rescaling the radial coordinate the curvature
constant k may have only the discrete values +1, −1, or 0
corresponding to closed, open, or spatially flat geometries.
The observed homogeneity and isotropy enable us to describe the overall
geometry and evolution of the Universe in terms of two cosmological
parameters accounting for the spatial curvature and the overall
expansion (or contraction) of the Universe.
2.1.The Universe Dynamics
Dynamics is provided by GR.
The Einstein field quations read:
,
Finding a general solution to a set of equations as complex as the Einstein field
equations is a hopeless task. The problem is simplied greatly by considering mass
distributions with special symmetries.
The matter content is usually modelled as a perfect fluid with a stress-energy
tensor in the rest frame of the fluid:
Friedmann equations of
Motion
Differentiating Eq. (1) and subtracting Eq. (2) we obtain an equation for the
energy momentum conservation
or
Friedmann expansion driven by an ideal fluid is isentropic, dS=0
Frequently used relation between the scale factor and
temperature in an expanding Universe :
R(t)~1/T
Number of relativistic degrees of freedom is a
function of T.
Thermodinamic relations for the energy density and number densities n:
These relations are a simple consequence of the integration of the BoseEinstein or Fermi-Dirac distributions:
The Friedmann equation, Eq. (1), can be interpreted within Newtonian
mechanics. It takes the form of energy conservation for test particles
bounded in the gravitational potential created by mass
k=1 corresponds to negative binding energy, recollapse and over-critical
density, where
H2
k=-1 positive binding energy, expansion, under-critical density
Three cases should be distinguished which foreordain the type geometry of the
universe:
  cr
Flat, an open universe, having Euclidean geometry, infinite in space
and time.
  cr Spherical, a closed universe, finite but unbounded in space and finite
in time.
  cr Hyperbolic, again an open universe, infinite in space and in time,
but curved.
Possible scenarios:
green - a flat, critical density universe in which
the expansion is continually slowing down;
blue - an open, low density universe,
expansion is slowing down, but not as much
because the pull of gravity is not as strong.
red - a universe with a large fraction of
matter in a form of dark energy, causing
an accelerated expansion .
According to Einstein's theory, the force law is modied. Not only does
mass gravitate, but the pressure, too, makes its contribution to the
gravitational force. This is a very important modication, since pressure can
be negative, leading to anti-gravity and to accelerated expansion.
The present value of this parameter H is called the Hubble constant. It
describes the rate of expansion of the Universe, and can be related to
observations. Consider two points with a fixed comoving distance
The physical distance is
the relative velocity is
This is the famous Hubble’s law
2.2. Expansion History of the Universe
Matter Content in the Universe
To solve the Friedmann equations, one has to specify the Universe matter content
and the equation of state for each of the constituents. Current observations point to at
least four components:
Radiation (relativistic degrees of freedom).
Today this component consists of the photons and neutrino and gives negligible contribution into
total energy density. However, it was a major fraction at early times.
Baryonic matter.
Dark matter.
Was not directly detected yet, but should be there.
Constitutes major matter fraction today.
Dark energy.
It provides the major fraction of the total energy density.
Was not anticipated and appears as the biggest surprise and
challenge for particle physics, though conceptually it can be
very simple, being just a `cosmological constant' or vacuum energy.
Equations of state
Law of expansion
RD universe:
MD universe:
Vacuum dominated universe:
Expansion History of the Universe
2.3.The Expanding Universe
Observational status
Hubble's Law
1912- Slipher: spiral nebula are receding 1920's- Hubble: v-d
proportionality
Distance-Velocity Relationship
Distances to Galaxies:
Step by step approach (the distance ladder) based on the assumption that
cepheids, RR Lyrae stars have the same properties in other galaxies.
The same for the SN explosions. These assumptions are supported by
essentially the same spectra and light curves.
variable stars: up to 20 Mpc;
SN I (had nearly the same peak luminosity ): up to 400 Mpc;
brightest Sc I spirals, which have about the same luminosity
Tully-Fisher relation, between the rotational velocity of a spiral galaxy
and its luminosity.
Galaxies Velocities
The shift of emission lines with respect to the frequency measurements
by the local observer is related to velocity, and is used as an observable
instead of the velocity.
Apparent, absolute magnitudes and
photometric distance
If we know the apparent magnitude m and the absolute magnitude M using
we can evaluate d (photometric distance):
where d is measured in parsecs.
m ~ 2.5log f
M ~ 2.5log L
The Redshift
Systematic recession of objects, or cosmological expansion, leads
to redshift. Note that cosmological redshift is not entirely due to the
Doppler effect, but, rather, can be interpreted as a mixture of the Doppler
effect and of the gravitational redshift.
z
d  e v

e
c
zc=v, for nonrelativistic velocities z<0.2, otherwise
v
c 1
z
v
1
c
1
Hubble’s Original Diagram
From the Proceedings
of the National Academy of Sciences
Volume 15 : March 15, 1929 : Number 3
A RELATION BETWEEN DISTANCE AND RADIAL VELOCITY
AMONG EXTRA-GALACTIC NEBULAE
By Edwin Hubble
Mount Wilson Observatory, Carnegie Institution of Washington
Communicated January 17, 1929
………………………………
…..The results establish a roughly linear relation between velocities and
distances among nebulae
…………………… The outstanding feature, is the possibility that the velocitydistance relation may represent the de Sitter effect, and hence that numerical
data may be introduced into discussions of the general curvature of space. In
the de Sitter cosmology, displacements of the spectra arise from two sources,
an apparent slowing down of atomic vibrations and a general tendency of
material particles to scatter. ……. ….. the linear relation found in the present
discussion is a first approximation representing a restricted range in distance.
The Hubble Law
cz = H d
v measured in [km/s], d in [Mpc], hence H is measured in [km/s/Mpc].
H0 = 100h km/s/Mpc, 0.4 < h < 1.0
Corresponds to a homogeneous expanding universe (, T decrease)
Space itself expands
• The Hubble law provides a scheme to find the distance to a distant galaxy
by measuring its redshift.
• Applicable for distances higher than those corresponding to peculiar
velocities.
• d=3000h-1 z Mpc
• dH(t) =3t=2/H(t) at MD, dH(t) =2t=1/H(t) at RD
• Hubble age 1/H0
• If (t) and H(t) at any moment t, then  (t ) and H (t )
• Not applicable for gravitationally bound systems.
Contemporary Hubble Diagrams
2.4. Universe Parameters
The Hubble Constant
One of the "key projects" of the Hubble
Space Telescope is the
Edwin Hubble's program of
measuring distances to nearby galaxies.
The current CMB results show the
Hubble Constant to be
H=73 +3/-4 (km/sec)/Mpc.
How do we know that the expansion of the
universe is speeding up?
by comparing its expansion today with how fast it was expanding in the
distant past: By observing the motions of galaxies at different
distances, astronomers can tell how fast the universe was expanding at
different times in the past.
determine the distance to a galaxy: The technique rests on the happy
accident that when a certain type of star dies, it explodes with a
spectacular flash whose inherent brightness is known (SNI).
Sometime around 5 billion years ago, the universe began accelerating - its
expansion getting faster and faster, rather than gradually slowing
down.
Age of the Universe
If we assume that the rate of expansion is uniform we can calculate
the oldest age of the Universe, since it is more realistic that the
velocity of expansion decreased with time:
Consider the distance between any two galaxies. For the time of
existence the distance increased from zero to its current value of d.
d=v.t , then t=d/v=d/Hd
t= 1/Ho
If the universe is flat and composed mostly of matter, then the age of the
universe is 2/(3 Ho)
If the universe has a very low density of matter, then its extrapolated age
is larger: 1/Ho
If the universe contains a form of matter similar to the cosmological
constant, then the inferred age can be even larger.
In general in the case matter density is less than 1:
The universe is at least as old as the oldest globular
clusters that reside in it.
Life cycle of a star depends upon its
mass
4
LM
All of the stars in a globular cluster
LM formed at roughly the same time:
they can serve as cosmic clocks.
The oldest globular clusters contain
only stars less massive than 0.7 M.
Observation suggests that the
oldest globular clusters are
between 11 and 13 billion years
old.
4
H-R diagrams for clusters: Turnoff
points
Structure of CMB fluctuations depend on the current density, the
composition and the expansion rate.
WMAP data with complimentary observations from other CMB
experiments (ACBAR and CBI), we are able to determine an age for the
universe closer to an accuracy of 1%.
Current estimate of age fits well with what we know from other
kinds of measurements: the Universe is about 13.7 billion years
old!