Introduction to cosmology and numerical cosmology (with the Cactus code) (1/2)

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)


When only a scalar field is present as a matter field, the stress-energy can be written as

where

and, as usual

with


Thus Einstein equations are

where the Hubble function and the Gaussian curvature 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.



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).



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 :


where we denoted with an "0" index all values at the initial actual time. These are the Ellis-Madsen potentials.


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 .


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.


Two adjacent spacelike hypersurfaces.

The figure shows the definitions of the

lapse function a and the shift vector bi


Then the 4-dimensional interval becomes :

Consequently, the 4-dimensional metric is split as :


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 !!


Finally, the EE are split in ADM standard form, namely into two sets of equations; first we have the dynamical equations :


And a set of constraint equations :

The Hamiltonian constraint

The Momentum constraints


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 !


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

!


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 ?


●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.


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


Waveforms : what happens in nature ...

Early simulations ...

Some recent simulations with Cactus code

Gravitational waves from the collision of two black-holes


Neutron stars collision


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 !!!


Two neutron stars colliding


Two neutron stars colliding 2


Two black-holes stars colliding

End of Part I

Introduction to cosmology and numerical cosmology (with the Cactus code) (1/2)

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Transcript of Introduction to cosmology and numerical cosmology (with the Cactus code) (1/2)