Physical Cosmology 6/4/2017
Alessandro Melchiorri
alessandro.melchiorri@roma1.infn.it slides can be found here:
oberon.roma1.infn.it/alessandro/cosmo2016
Angular Diameter Distance
We can measure the distance of an object by measuring its angular size and knowing its size (standard ruler).
In the comoving reference frame we have:
The angular diameter distance of an object at redshift z is:
Angular Diameter Distance
In cosmology, the angular diameter distance and the
luminosity distance of the same object can be completely different !
Angular Diameter
Distance:SZ+X ray clusters
The hot gas in a cluster of galaxies produces a distortion in the blackbody spectrum of the Cosmic Microwave
Background that is frequency dependent.
(Inverse compton scattering, photons are shifted to higher energies).
SZ Effect in CMB maps Abel 2319
44 GHz 70 GHz 100 GHz 143 GHz
217 GHz 353 GHz 545 GHz
X ray emission from Clusters
Cluster of galaxies also emit X-ray radiation due to bremsstrahlung of ionized hot (10-100 megakelvins) intracluster gas
Angular diameter distance
To put it simply we have that:
SZ: absorption X-ray: emission
Integral over the cluster volume Free electrons density If the cluster is almost spherical we have:
By measuring absorption and emission we measure the size of the cluster and we can get its angular distance !
Angular distance from clusters
Bonamente et al.,
http://arxiv.org/pdf/astro-ph/0512349.pdf
Useful for
measuring the
Hubble constant.
Etherington’s distance duality
In principle, we can use standard candles and standard rulers at the same redshift to test this relation.
It is a fundamental prediction of an expanding universe.
Test of distance duality
Assuming eta as a constant:
http://arxiv.org/pdf/gr-qc/0606029.pdf
Lookback time
The time that a photon emitted at redshift z has spent to reach us is given by (omitting radiation):
This time is clearly the difference between the
age of the universe minus the age of the object that sent the photon and the age of the universe at the redshift of formation of the object:
Age of the universe
Age of the object
Age of the Universe at z of object’s formation
http://arxiv.org/pdf/astro-ph/0410268.pdf
H(z) from cosmic chronometers
The Hubble parameter depends on the differential age of the universe in function of redshift.
Differential Age
Differential redshift
If we measure the age and redshift of different objects for close enough redshifts and ages we could estimate the derivative and so H(z).
H(z) from ages
Left Panel: age of passively evolving galaxies obtained from stellar population synthesis models in function of z.
Right Panel: H(z) obtained from differential ages from the
same catalog. See http://arxiv.org/pdf/astro-ph/0412269.pdf
Constraints on w
Open: just CMB Filled: CMB+H(z) (from cluster ages)
http://arxiv.org/pdf/0907.3149.pdf
Cosmic Chronometers
New, recent, article on cosmic chronometers.
http://arxiv.org/abs/1604.00183
Costante di Hubble: valore sperimentale
Nonostante il progresso sperimentale il valore della costante di Hubble non e’ancora ben stabilito.
Riess (2016) propone un valore pari a:
H0=73.03 ± 1.79 km/s/Mpc Efstathiou propone invece:
H0=70.6±3.3 km/s/Mpc
Le misure di CMB da parte del satellite Planck forniscono:
H0=67.3±1 km/s/Mpc
queste ultime pero’ sono “model dependent”.
Hubble constant
New determination of the Hubble constant from local (z<0.2) measurements. The discrepancy with CMB data is even
more present!
http://arxiv.org/abs/1604.01424
Benchmark Model
Despite tensions as in the case of the Hubble constant,
the standard cosmological model provides a good fit to the data if we fix its parameters to:
So far we have often neglected the contribution from radiation since we have considered “low"redshift
probes (z<2).
But what is this term ? how do we know that is negligible today ?
Radiation Energy density
No matter what is its present value, the radiation content will dominate the expansion of the universe for sufficient high redshifts.
It must exist a redshift, called redshift of equality, when the matter and radiation energy densities contributed equally:
Radiation Energy Density
For redshifts much larger than the redshift of equality we can therefore write:
Knowing the amount of radiation can help us in determining the value of the Hubble parameter in the early universe.
As we will see, this is extremely important for the Big Bang Nucleosynthesis since the value of the Hubble parameter is crucial in determining the interaction rates between
particles and the amount of light elements (Helium, Deuterium, Lithium, etc) produced.
Radiation Energy Density
In the standard model, only two type of particles contribute to the total radiation energy density.
Photons Neutrinos
Radiation: Photons
The contribution from photons is essentially provided by the photons of the Cosmic Microwave Background.
Photons emitted by objects (galaxies, stars, gas, etc) are completely negligible since these objects forms at small redshifts (z<15) and their contribution is anyway small.
Statistics
If we consider a gas of particles at temperature T, the
probability P of having a particle with momentum between p and p+dp and position between x and x+dx is given
by the distribution function f:
We can consider for the moment an homogeneous universe (so we neglect the spatial and directional dependence). We have:
+ fermions - bosons
where m is the rest mass of the particle, mu the chemical potential, g the spin.
Statistics
we can therefore write down the following equations:
Energy Density Number Density
Pressure
Statistics
When the particles are highly relativistic and non-degenerate we have:
CMB photons
Let us apply the following two formulae to the CMB (photons, spin=2):
Using the CMB temperature of 2.73 K we get:
Entropy Density
We can define, given a gas, as entropy density the quantity:
For relativistic particles we have:
Bosons
Fermions
Entropy Conservation
It is possible to show that during the Universe evolution:
The temperature of a gas of relativistic particles during the expansion of the universe will therefore scale as:
Note that if g varies with time then the
T will not scale simply as 1/aˆ3 !!
Photons increase energy by going to higher z
We can use cosmology
as an accelerator !
Strongly coupled particles
Let us consider a gas made of several kind or particles.
If the particles are relativistic the total energy density is:
If the particles are strongly interacting then they will share the same temperature and:
Strongly coupled particles
We can write down the same equations for the entropy density:
If particles are coupled and share the same temperature we have that q=g.
Plasma temperature
scales mainly as 1/a but it can vary if g changes.
(for example, one
component annihilates in photons).
Criteria for coupled particles
Let suppose that two particles interacts.
We can write down an interaction rate as:
where n is the numerical density of target particles, v the relative velocity and s the cross section.
The particles will be coupled (and will share the same temperature) if the interaction rate is faster than the
rate of expansion of the universe.
Particles are coupled, they share same T
Particles start to uncouple. They keep sharing same T until some process can change it in one of them.
Example: electrons
Free electrons (not in atoms) are coupled to photons by Thomson scattering:
Comparing with the Hubble rate (see Dodelson eq. 3.46) we get:
where Xe is the free electron fraction.
Example: electrons
The free electron fraction drops substantially after
z=1300. Most of the electrons recombine to form neutral
hydrogen.
This epoch is called epoch of recombination and we will study the process in a future lecture.
Because of recombination, the electrons decouple from the photons at redshift z=1100. Epoch of decoupling.
Example: electrons
Without recombination and fixing Xe=1 constant, the decoupling redshift would be:
i.e. a decoupling redshift around:
That is clearly in strong disagreement with current
observation of CMB anisotropies (we should not see
them in this case, we will discuss this in a future lecture).
Photons and electrons
are coupled.
They form a plasma
Photons and electrons are uncoupled.
The Universe is trasparent to
radiation
At T>0.5 MeV photons temperature is
high enough to create electron-positrons
pairs.
Neutrinos
Neutrinos are usually treated as massless particles.
This is not true: neutrinos must have a mass.
In reality what has been
measured is 2 mass differences.
Since the current CMB temperature is about 7 10ˆ-4 eV, at least
one neutrino state should be non-relativistic.
However, for the moment, we neglect the neutrino mass.
At T> 1 Mev Neutrinos are coupled to the electron-positron
plasma. They share the same temperature with it.
At T >1 Mev the universe is dominated by a radiation
component made of
photons, electrons, positrons and neutrinos all coupled and
at the same temperature.
e+ e- pairs
annihilate heating
photons
CNB temperature
At T>1 MeV, the universe is made by electrons, photons, neutrinos and anti-neutrinos all at the same temperature.
We can write at a time 1 at T>1 Mev:
Photons
spin 1, -1 Electrons Positrons 3 neutrinos
with one spin state+
3 antineutrinos
CNB temperature
At a time T< 0.5 Mev, after electron-positron annihilation we have:
e+/e- term disappears.
Neutrino background has a different
temperature.
CNB temperature
Now we should use:
Entropy conservation Neutrino are uncoupled their temperature scales as 1/a !
CNB temperature CNB temperature
gives:
Using we have:
And, finally:
The neutrino background has a temperature
that is 0.71 lower
than the CMB temperature!
CNB temperature
If we assume that all neutrinos are still relativistic today (wrong but we assume it anyway for the moment) both CMB and CNB temperature scales as 1/a and we have today:
If relativistic,
we should have
a neutrino background surrounding us
at 1.9 K
Radiation Component
Assuming that neutrinos are relativistic today, we have that the total energy in radiation is given by:
The above formula is however not perfect.
We assumed that the neutrino decoupling is before electron-positron annihilation and instantaneous.
In reality is not instantaneous and some decoupling still happens during e-e+ annihilation.
We keep the formula but we write it now as:
With:
Massive neutrinos
After electron-positron annihilation the neutrino number density is given by
The neutrino number density must satisfy:
Today
This is true even if neutrino are non relativistic today ! Neutrino number per
species. (Multiply by 6 to have the total)
Massive neutrinos
If neutrinos are non relativistic today they contribute to matter (not radiation!) with an energy density given by
where mi are the 3 mass eigenstates.
An useful formula is given by:
Energy density
in NON relativistic neutrinos.
