introduction to cosmology and numerical cosmology (with the cactus code) (1/2)
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Dumitru N. Vulcanov
The West University of Timisoara
October 2009
Introduction to cosmology and numerical cosmology
(with the Cactus code) First lecture
Timisoara - my city
West University of
Timisoara main building and entrance
The orthodox cathedral
Bega ...
Timisoara - my city
Victory place
The Dome
Plan of the presentation ●Introduction about scalar fields and
cosmic acceleration ●Theoretical background - Klein Gordon equation, Friedmann equation, Ellis Madsen potentials ●Theoretical background of the nuumerical relativity●Tasks of the numerical relativity (with picures and movies)●Cactus code – short introduction●Cosmo and RealSF thorns ●Numerical results with cosmological models using Cactus code
Introduction : Why scalar fields ?
●Recent astrophysical observations ( Perlmutter et . al .) shows that the universe is expanding faster than the standard model says. These observations are based on measurements of the redshift for several distant galaxies, using Supernova type Ia as standard candles. ●As a result the theory for the standard model must be rewriten in order to have a mechanism explaining this ! ●Several solutions are proposed, the most promising ones are based on reconsideration of the role of the cosmological constant or/and taking a certain scalar field into account to trigger the acceleration of the universe expansion. ●Next figure (from astro - ph /9812473) contains, sintetically the results of several years of measurements ...
Introduction : Cosmic acceleration
Theoretical background - cosmology
We are dealing with cosmologies based on Friedman- Robertson-Walker ( FRW ) metric
Where R(t) is the scale factor and k=-1,0,1 for open, flat or closed cosmologies. Inserting FRW metric in Einstein equations
Greek indices run from 0 to 3, and we have geometrical units (G=c=1)
Theoretical background - cosmology
When only a scalar field is present as a matter field, the stress-energy can be written as
where
and, as usual
with
Theoretical background - cosmology
Thus Einstein equations are
where the Hubble function and the Gaussian curvature are
Theoretical background - cosmology
Thus Einstein equations are
It is easy to see that these eqs . are not independent. For example, a solution of the first two ones (called Friedman equations) satisfy the third one - which is the Klein-Gordon equation for the scalar field.
Theoretical background - cosmology
Thus Einstein equations are
The current method is to solve these eqs . by considering a certain potential (from some background physical suggestions) and then find the time behaviour of the scale factor R(t) and Hubble function H(t).
Theoretical background - cosmology
Thus Einstein equations are
Ellis and Madsen proposed another method, today considered (Ellis et . al , Padmanabhan ...) more appropriate for modelling the cosmic acceleration : consider "a priori " a certain type of scale factor R(t), as possible as close to the astrophysical observations, then solve the above eqs . for V and the scalar field.
Theoretical background - cosmology Following this way, the above equations can be rewritten as
Solving these equations, for some initially prescribed scale factor functions, Ellis and Madsen proposed the next potentials - we shall call from now one Ellis- Madsen potentials :
Theoretical background - cosmology
where we denoted with an "0" index all values at the initial actual time. These are the Ellis-Madsen potentials.
Theoretical background - cosmology
We shall test Cosmo and RealSF thorns comparing the colomns 2 and 3 of the table with the respective values as are at the numerical output ! But first we shall describe how we adapted these thorn for our purposes.
Theoretical background Numerical relativity
Basically, Numerical Relativity (NR) is dealing with the problem of solving numerically the Einstein Equations ( EE ), namely :
where greek indices runs from 0 to 3 and l is the
cosmological constant
The left hand of the above EE is the Einstein tensor; it can
be constructed from a certain metric of the space-time
The right hand contains the stress-energy tensor wich
describes the matter-fields contents of the spacetime .
Theoretical background Numerical relativity
3+1 dimensioal split of spacetime :-Spacetime is foliated into a set of non-intersecting three-dimensional
spacelike hypersurfaces having a riemannian geometry.
-Two kinematic objects describe the evolution between the
hypersurfaces :
1. The "lapse" function, a which describes the rate of advance of
time between two hypersurfaces along a timelike unit vector normal
to a surface ni ;
2. The "shift" vector, bi describing how coordinates move between
hypersurfaces .
Latin indices will run from 1 to 3 !!!
Foliation of spacetime into three-dimensional
spacelike hypersurfaces.
Theoretical background Numerical relativity
Two adjacent spacelike hypersurfaces.
The figure shows the definitions of the
lapse function a and the shift vector bi
Theoretical background Numerical relativity
Then the 4-dimensional interval becomes :
Consequently, the 4-dimensional metric is split as :
Theoretical background Numerical relativity
An important geometric object is the "extrinsic curvature" which
describes how the hypersurfaces are embedded in the four dimensional
spacetime ; it is defined as :
where :
" / " means three-dimensional covariant derivative (relative to the
riemannian geometry of the hypersurfaces ,
" ,0 " means the time derivative;
all the latin indices i,j,k,l,m,n.... runs between 1,2,3 !!
Theoretical background Numerical relativity
Finally, the EE are split in ADM standard form, namely into two sets of equations; first we have the dynamical equations :
Theoretical background Numerical relativity
And a set of constraint equations :
The Hamiltonian constraint
The Momentum constraints
Theoretical background Numerical relativity
This is the so called "ADM formalism" (Arnowitt , Deser , Misner )
but slightly different from the orginal version ! Why?
Because here the extrinsic curvature plays a real dynamical role,
instead of the ADM momenta, defined for initial use in canonical
quatization of gravity... A long unfinished dream ! Nothing to do with
Numerical Relativity (NR) !
Doing NR means to solve numerically the above equations, mainly by
finite differencing them.... Looks easy, but now comes the... real
nightmare !
Numerical relativity
First attemps : late '60's and '70's : total failure ! Why ?
Because in even the must simple case, of a head-on
collision of two identical black-holes that time computers
where too small : thousand of terms in the equations and at
least 1 GB of RAM to handle...
Only late '80's supercomputers (which became avaiable at
that time to the scientific cummunity ) done the job !!!
So we are speaking of Numerical relativity only starting
from around 1990 !
Numerical relativity
It was a gradual development, of codes, numerical
techniques and hardware too ...
Why new numerical techniques (as Bona - Masso or, more
recently ADM_ BSSN method) ?
Because, the equations are not standard ones !
What's about hardware : now we are doing NR even on
PC's ...or palmtops !!! And by remote, on internet connection
!
Numerical relativity
Early codes were uni-dimensional or bi-dimensional - see
the Grand Challange project in USA (1990-1997).
Till today, only Cactus code is a fully 3D , high performance
code for NR !!! …Cactus + Globus + Portal ....
It's an entire community - hundred of people, dozens of
institutes and groups allover the world
Grid-lab, EuroGrid , TerraGrid , and so on... all involves
Cactus !
Numerical relativity and cosmology
What’s the plan ?
We developped a new application for Cactus code to
deal with cosmology numerically (Cosmo thorn)
We used the theoretical recipes for cosmology before
introduced for providing initial data for Cactus code
Run the Cactus code for solving numerically EE in this
context
The task of Numerical Relativity
●What people are doing with Cactus code in NR ?
●Manny things, but mainly simulations on black-holes and colliding and merging black-holes !
●The question is :
Why ?
The task of Numerical Relativity
●Why we are dealing in NR with ... "black-holes" collisions ?
●Because NR is providing numerical "data" for those who are trying to detect gravitational waves ( GW ) !!!
●Gravitational waves are predicted by GR ! It is one of the greatest challanges of the modern science to detect it !
●Black-holes collisions/mergers and other violent astrophysical proceses are the most probable sources of gravitational waves to detect.
The task of Numerical Relativity
LIGO , VIRGO, GEO, LISA .... more than 1 billion $ worldwide to detect GW
We need Numerical Relativity to :
●Detect GW ... pattern matching against numerical templates to enhance signal/noise ratio ●Understand them... just what are the waves telling us
The task of Numerical Relativity
Waveforms : what happens in nature ...
Early simulations ...
Some recent simulations with Cactus code
Gravitational waves from the collision of two black-holes
Some recent simulations with Cactus code
Neutron stars collision
Some recent simulations with Cactus code
Where we one can find some of these nice vizualisations ?
On : http://jean-luc.aei.mpg.de
http://jean-luc.ncsa.uiuc.edu
Let's see now some of the movies done with
numerical simulations with Cactus code !!!
Some recent simulations with Cactus code
Two neutron stars colliding
Some recent simulations with Cactus code
Two neutron stars colliding 2
Some recent simulations with Cactus code
Two black-holes stars colliding
End of Part I