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 ...
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 n i ; 2. The " shift " vector, b i describing how coordinates move between hypersurfaces . Latin indices will run from 1 to 3 !!!
Theoretical background Numerical relativity Two adjacent spacelike hypersurfaces. The figure shows the definitions of the lapse function a and the shift vector b i Foliation of spacetime into three-dimensional spacelike hypersurfaces.
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
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