Physical Cosmology 18/4/2016
Alessandro Melchiorri
alessandro.melchiorri@roma1.infn.it slides can be found here:
oberon.roma1.infn.it/alessandro/cosmo2016
Galaxy Rotation Curve
If we look at a spiral galaxy with an angle i between the line of sight and a line
perpendicular to the disk, given the
two apparent semi-axis a and b of the projected galaxy, assuming the galaxy to be in reality perfectly circular,
we have:
radial velocity of the galaxy as a whole
Orbital speed at a distance R from the center of the disk
21 cm neutral H line
When the spins are parallel, the magnetic
dipole moments are antiparallel (because the electron and proton have opposite charge), thus one would naively expect this
configuration to actually have lower energy just as two magnets will align so that the
north pole of one is closest to the south pole of the other. This logic fails here because the electron is not spatially displaced from the proton, but encompasses it, and the
magnetic dipole moments are best thought of as tiny current loops. As parallel currents
attract, it is clear why the parallel magnetic dipole moments (i.e. antiparallel spins) have lower energy.The transition has an energy difference of 5.87433 µeV that when applied in the Plank equation gives:
This transition is highly forbidden with an extremely small rate of 2.9×10−15 s−1, and a lifetime of around 10 million (107) years.
Galaxy Rotation Curve
Here is a doppler image of the M33 (Pinwheel) galaxy. It
is observed with radio
telescopes receiving the 21 cm Hydrogen line. This shows the distribution of hydrogen gas throughout the
galaxy. The colour coding shows the relative radial
velocities and clearly
demonstrates the rotation of the galaxy as a whole. Blue is moving towards us and red
is moving away from us.
Keplerian motion
By Newton law of gravity an object orbiting on the disk should have a speed following:
The surface brightness of a galaxy follows a law:
With Rs of few kpc For MW Rs = 4 kpc For M31 Rs = 6 kpc
Spiral Galaxies don’t follow
Keplerian motion !!!
Rotation curves and Dark Matter
If we postulate the existence of a dark matter component, in order to have a flat velocity curve we need:
This means a mass/luminosity ratio of
And a density of:
DM in Galaxies
From:
Solving for M and deriving:
But we can also write:
Combining the two equations we get:
DM in Galaxies
At large distances, if we want v=constant we need:
Single Isothermal Sphere Profile
At smaller distances v grows as r and
This can be obtained by a Pseudo-Isothermal profile:
DM in Galaxies: NFW profile
Assuming Cold Dark Matter (matter made of non-relativistic particles practically since the Big Bang) Navarro, Frenk
and White in 1997 found the following profile from numerical simulations:
where the central density, ρ0, and the scale radius, Rs, are parameters that vary from halo to halo.
Cusp vs Core
The NFW profile, for small values of r, goes as:
i.e. it has a “cusp" when going to small values of r.
More recent CDM simulations of NFW have:
While the pseudo-isothermal model has:
i.e. you expect a "core".
In principle, then, if you see some indications for a cusp this could be an hint for CDM.
Cusp vs Core
This started a big debate in cosmology in the past 15 years, i.e. if galaxies at the center have
a “cusp” or a “core”.
Current observations in dwarf galaxies
prefer a “core” over a cusp.
Is this a problem for CDM ? Baryons expelled from
the center by Supernovae could solve this.
Problems for CDM (on galactic scales)
A part from the cusp/core problem there are the following discrepancies between CDM simulations and observations:
- Missing Satellites: The simulated and observed satellite distributions around the Milky Way are inconsistent, in the sense that simulations predict many more satellites.
Problems for CDM (on galactic scales)
Models that best reproduce observed satellite luminosity function - and hence best `solves’ the missing satellite problems, predicts
that all satellites have significantly larger rotational velocities!
Solutions
1- Feedback from baryons not included in the simulations.
2- Observations not correct (many satellites too faint to be detected).
3- New physics ? warm dark matter ? self interacting dark matter ?
MOND
Modified Newtonian Dynamics (!) proposed by Milgrom in 1983. We change Newton law with:
where:
for example, we can use:
MOND
For very small accelerations (a << a0):
For a particle in circular motion, assuming Newton gravitational law, we have:
Flat rotation curves !!!
The best fit to galaxy curves is provided by
(???? just a coincidence ?)
Dark Matter in Clusters of Galaxies
Let us suppose that a cluster of galaxies is comprised of N
galaxies, each of which can be approximated as a point mass,
with a mass , a position , and a velocity . .
Clusters of galaxies are gravitationally bound objects, not
expanding with the Hubble flow. They are small compared to the horizon size; the radius of the Coma cluster is
The galaxies within a cluster are moving at non-relativistic speeds; the velocity dispersion within the Coma cluster is
Dark Matter in Clusters of Galaxies
Coma cluster (view in the optical)
Dark Matter in Clusters of Galaxies
The acceleration for each galaxy in the cluster is given by:
The gravitational potential energy of the system of N galaxies is
The factor of 1/2 in front of the double summation ensures that each pair of galaxies is only counted once in computing the
potential energy.
Dark Matter in Clusters of Galaxies
The potential energy of the cluster can also be written in the form
Where:
is the total mass of all the galaxies in the cluster.
is a numerical factor of order unity which depends on the density profile of the cluster. and is the half-mass radius of the cluster: the radius of a sphere centered on the cluster’s center of mass and containing a mass M/2.
Dark Matter in Clusters of Galaxies
The kinetic energy associated with the relative motion of the galaxies in the cluster is
The kinetic energy K can also be written in the form
Dark Matter in Clusters of Galaxies
The virial theorem in the case of a a system in steady state, with a constant moment of inertia (the system is neither
expanding nor contracting) gives:
Using the previous equations, we get:
We can derive the total mass of the cluster from the velocity of galaxies.
Dark Matter in Coma
From measurements of the redshifts of hundreds of galaxies in the Coma cluster, the mean redshift of the cluster is found to
be
which can be translated into a radial velocity
and a distance
The velocity dispersion of the cluster along the line of sight is found to be
If we assume that the velocity dispersion is isotropic:
Dark Matter in Coma
If we assume that the mass-to-light ratio is constant with
radius, then the sphere containing half the mass of the cluster will be the same as the sphere containing half the luminosity of the cluster. If we further assume that the cluster is
intrinsically spherical, then the observed distribution of
galaxies within the Coma cluster indicates a half-mass radius
Moreover, for observed clusters of galaxies, it is found that α ≈ 0.4 gives a good fit to the potential energy.
Dark Matter in Coma
After all these assumptions and approximations, we may estimate the mass of the Coma cluster to be
F. Zwicky
Dark Matter in Coma
In the Coma cluster 2% of the matter is in stars and about 10% is in the Intracluster Medium, visible
in the X-rays.
ICM
the intracluster medium (ICM) is the superheated plasma present at the center of a galaxy
cluster. This gas is heated to temperatures of the order of 10ˆ11-10ˆ10 K and composed mainly of ionized hydrogen and helium, containing most of the baryonic material in the cluster. The ICM
strongly emits X-ray radiation.
The ICM is heated to high temperatures by the gravitational energy released by the formation of the cluster from smaller structures. Kinetic energy gained from the gravitational field is converted to
thermal energy by shocks. The high temperature ensures that the elements present in the ICM are ionised. Light elements in the ICM have all the electrons removed from their nuclei.
Although the ICM on the whole contains the bulk of a cluster's baryons, it is not very dense, with typical values of 10−3 particles per cubic centimeter. The mean free path of the particles is roughly 1016 m, or about one lightyear.
Although the ICM on the whole contains the bulk of a cluster's baryons, it is not very dense, with typical values of 10−3 particles per cubic centimeter. The mean free path of the particles is roughly 1016 m, or about one lightyear.
ICM
Dark Matter in Coma (Gas)
If the hot intracluster gas is supported by its own pressure against gravitational infall, it must obey the equation of
hydrostatic equilibrium:
The pressure of the gas is given by the perfect gas law
where T is the temperature of the gas, and μ is its mass in units of the proton mass (mp).
! M is the TOTAL mass
Dark Matter in Coma (Gas)
The mass of the cluster, as a function of radius, is found by combining the previous equations:
Starting from an x-ray spectrum, it is possible to fit models to the emission and thus compute the temperature, density, and chemical composition of the gas.
We can derive the mass of the cluster by using X-ray measurements.
The mass of the Coma cluster, assuming hydrostatic
equilibrium, is computed to be (3 → 4)×10ˆ14 M⊙ within 0.7 Mpc of the cluster center and (1 → 2)×10ˆ15 M⊙ within 3.6 Mpc of the center. Consistent with previous results.
MOND and Clusters of Galaxies
MOND fails on galaxy clusters !
The Mass-Temperature relation disagrees with observational data. Using the virial theorem, we need a MOND parameter a0 that is about 3 times larger than the one obtained from galaxy rotation curves.
Lensing
In GR relativity gravity can deflect photons by an angle:
For example, for a photon passing by the Sun we expect:
1919 telegram from Eddington to Einstein
Lensing from MACHO
The image of the star should appear as a ring (Einstein
ring) if the star, the MACHO and the observer are perfectly aligned with angular extension given by:
where M is the mass of the lensing MACHO, d is the distance from the observer to the lensed star, and xd (where 0 < x < 1)
is the distance from the observer to the lensing MACHO
Lensing from MACHO
For x=0.5 we have:
The typical time scale for a lensing event is the time it takes a MACHO to travel through an angular distance equal to θE as seen from Earth; for a MACHO halfway between here and the Large Magellan Cloud, is
where v is the relative transverse velocity of the MACHO and the lensed star as seen by the observer on Earth.
Results from MACHO searches
The research groups which searched for MACHOs found a scarcity of short duration lensing events, suggesting that there is not a significant population of brown dwarfs (with M < 0.08M⊙) in the dark halo of our
Galaxy. The total number of lensing events which they detected suggest that as much as 20% of the halo mass could be in the form of MACHOs.
The long time scales of the observed lensing events, which have
∆t > 35days, suggest typical MACHO masses of M > 0.15 M⊙. (Perhaps the MACHOs are old, cold white dwarfs, which would have the correct mass.) Alternatively, the observed lensing events could be due, at least in part, to lensing objects within the LMC itself.
In any case, the search for MACHOs suggests that most of the matter in the dark halo of our galaxy is due to a smoothly distributed component, instead of being congealed into MACHOs of roughly stellar mass.
Lensing from Clusters
Gravitational lensing occurs at all mass scales. Suppose, for instance, that a cluster of galaxies, with M ∼ 10ˆ14 M⊙, at a distance ∼ 500 Mpc from our Galaxy, lenses a background galaxy at d ∼ 1000Mpc. The
Einstein radius for this configuration will be
Lensing from Clusters
The Figure shows an image of the cluster Abell 2218, which has a redshift z = 0.18, and hence is at a proper distance d = 770Mpc. The elongated arcs are not oddly shaped galaxies within the cluster; instead, they are background galaxies, at redshifts z > 0.18, which are gravitationally lensed by the cluster mass.
