Physical Cosmology 31/3/2017
Docente: Alessandro Melchiorri
alessandro.melchiorri@roma1.infn.it
Suggested textbooks
http://www.astro.caltech.edu/~george/ay21/readings/Ryden_IntroCosmo.pdf
Barbara Ryden, Introduction to Cosmology
Suggested textbooks
An introduction to General Relativity, Sean Carroll
Suggested textbooks
Modern Cosmology, Scott Dodelson
Suggested textbooks
T. Padmanabhan, structure formation in the universe
Cosmological Constant
- Current cosmological data suggest the presence of a cosmological constant at high significance.
Assuming a flat universe, as confirmed by CMB observations (we will see this in a future lecture), SN-Ia (JLA) data gives:
- But a cosmological constant is of extremely difficult theoretical interpretation !
- 123 orders of magnitude difference (smaller) with the vacuum fluctuations energy expected in Quantum Field Theory !
- Why now problem ? why we live with a cosmological constant today ?
Major goal of modern cosmology
- Do we really need a cosmological constant ?
- Maybe data could be explained by a different component ?
- We need to falsify a cosmological constant !
- A cosmological constant has:
Constant with time
(redshift) energy density Constant with time (redshift) equation of state and equal to -1 !
We need to test these two things !
Dark Energy
As a first step we can fit the data with a component with a generic equation of state w constant with redshift.
From the continuity equation.
As we can see if
w is different from -1 energy density is
evolving with z !
Assuming a flat universe, current SN-Ia(JLA)+CMB data gives:
Very close to a
cosmological constant !!!!
No indication for w different from -1 !!!
Quintessence
But do we have physical models different from a cosmological constant that can lead to an accelerated universe ?
If we consider a scalar field minimally coupled to gravity the action can be written as:
where and is the field potential.
Varying the action respect to the field we have the equation of motion:
Quintessence
The energy momentum tensor can be written as:
Energy and pressure densities of the field are given by:
Leading to the Friedmann equations:
Quintessence
The equation of state can be written as:
We therefore have an accelerating universe (w<-1/3) if
And we expect a time-evolving equation of state ! Example:
And we have acceleration with p>1
Note: w is
always larger than -1 !
Tracking Quintessence
Several quintessence models have been proposed.
One interesting property of some of them is to follow the dominant energy component (tracking).
This helps in alleviating the Why Now ? problem.
http://arxiv.org/pdf/astro-ph/0403324v3.pdf
Quintessence Tracking
Most of these models show a “tracking”
behaviour.
Quintessence
- There are plenty of Quintessence models.
- Quintessence tracks the dominant energy component, this helps in solving the why now problem.
- The transition to an accelerating universe is often connected to the radiation-matter equality.
- Problems with Quintessence: energy scale too low, long range forces not observed.
Phantom models
Models with w <-1 are compatible and also slightly preferred by current SN-Ia data.
These models are called “Phantom"and have quite dramatic consequences.
for w<-1 in the
future (z<0) this term
could diverge in a finite time.
Phantom models
In these models the scale factor grows as:
(teq is the time of dark energy-matter equality)
And diverges in a finite amount of time !
Phantom models
For w=-1.1 …
Dynamical Dark Energy
One could try a different parametrization introducing an equation of state that evolves with time.
A possible (old) parametrization is:
(not good, diverges at high redshifts!)
In this case the luminosity distance is (try at home):
Results from
SN-Ia from Riess et al, 2004 plus prior
on matter density.
Black dot
is a cosmological constant.
Parametrization of Chevallier-Polarski-Linder (CPL) Is, in practice, a Taylor expansion in a at first order:
At high redshift, small a, converges to w0+wa The continuity equation can be written as:
Integrating, we have that the energy density follows:
This parametrization could seem trivial…
…but sometimes trivial things work well, these CP+L papers are extremely well cited !!!
Constraints on CPL
Gray region is SN-Ia (JLA) +
CMB (Planck+WP) +
Galaxy Clustering (BAO)
Constraints are weaker on w0 respect to w
constant.
Constraints on wa are very weak !!!
Cosmological constant (two dashed lines) is ok.
Betoule et al., 2014
Recent
constraints from Planck 2015
Again, no evidence for something different from a
cosmological constant. But constraints on the evolution of w are weak !
Quintessence
Several model of quintessence (and even of modified gravity as DGP) are well mapped
by the CPL parametrization.
The current models of
quintessence that provide the best fit to observations are of the type of
“thawing”quintessence.
These models have w=-1 at high redshifts.
For thawing models actually one parameter is enough, fixing:
The ESA Euclid satellite experiment,
expected to launch in 2020, by measuring
galaxy clustering and
Lensing should determine these parameters with
the following accuracy:
i.e. more than one order of magnitude better than what we have now.
More General Parametrizations
Another possible parametrization is the following one:
Constraints inside the bins are correlated.
With current data, an increase in the number of bins
does not change the result.
BSH is BAO, SN-Ia and
Hubble constant constraint.
http://arxiv.org/pdf/1502.01590v1.pdf
Other parametrizations…
Chaplygin gas.
Introduced in aerodynamics in 1904.
Assuming , we get from the continuity equation:
This is a first example of Unified Dark Energy - Dark Matter model. At high redshift behaves as matter, at small redshift as a cosmological constant.
Another extremely well cited
paper…
Chaplygin Gas
Chaplygin gas with
alfa=1 is excluded from structure formation.
Excluded
Excluded
Excluded Excluded
http://arxiv.org/pdf/astro-ph/0301308.pdf
Cardassian Universe
We can modify the Friedmann equation by hand:
and we get acceleration even if we have just ordinary matter.
Modified Gravity
On the other hand, one could consider cosmic acceleration as a failure of General Relativity at cosmic scales.
One possibility to “modify gravity” is to include a function of the Ricci Scalar in the action:
New term Energy Content:
ordinary matter !
Modified Gravity - f(R)
This brings to new Friedmann equations:
In practice, there are 2 workable f(R) models:
Hu W., Sawicki I., 2007, arXiv:0705.1158v1
Starobinsky A.A., 2007, JETP Lett., 86, 157
Hu and Sawicki
Starobinsky
When compared with observations, the best fit parameters of the models produce an acceleration very close to lambda.
Hu-Sawicki
If fitted as a dark energy
component, the Hu-Sawicki model provides an equation
of state that varies with redshift and crosses w=-1.
Current constraints on this model are weak.
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
