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

Dumitru N. Vulcanov The West University of Timisoara October 2009 Introduction to cosmology and numerical cosmology (with Cactus code) Second part

Plan of the presentation
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)
Introduction about scalar fields and cosmic acceleration
Cactus code – short introduction
Cosmo and RealSF thorns
Numerical results with cosmological models using Cactus code

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 Cactus code - This is the point of view of an user of Cactus code ! - But, those who are using the Cactus code, became slowly developpers . - So now, my point of view is totally changed ! - You will see why, in a minute !

The Cactus code Cactus is a freely avaiable , modular, portable and manageable environment for collaboratively developing parallel, high performance multidimensional simulations. So, what we mean ?

The Cactus code
Cactus was developed as a general, computational framework for solving PDEs (originally in numerical relativity and astrophysics).
Modular... for easy development, maintenance and collaboration. Users suply "thorns" which plug - into compact core "flesh"
Configurable... thorns register parameters, variables and scheduling information with " runtime function registry" ( RFR ). Object oriented inspired features.
Scientist friendly... thorns written in F77, F90, C, C++ and even in perl ...

The Cactus code
Accessible parallelism... driver layer (thorn) is hidden from physics thorns by a fixed flesh interface
Standard interfaces... Interpolation, reduction, IO, coordinates. Actual routines supplied by thorns
Portable... Cray T3E, Origin, Linux, O2, Dec Alpha, Exemplar, SP2, BSD, FreeBSD and OpenBSD ...
Free... distributed under GNU GPL . Uses as much as free software as possible
Up-to-date... new computational developments and/or thorns immediately avaiable to users

The Cactus code
Collaborative... thorn structure make it possible for large number of people to use and develop toolkits
New version... almost at every 6 month !!! There are avaiable allways a stable and a development version.
Latest version avaiable Cactus 4.0 beta 16 !

The Cactus code
On the website : http://www.cactuscode.org
Here one can find everything !! Namely :
How to download (via CVS mainly)
Documentation (general, about the thorns and arrangements)
Latest news !
The community !
Let's see it !

The Cactus code

History of Cactus code Developed at AEI Potsdam- Golm , starting from 1996 under the leadership of Ed Seidel Today the Cactus team is located in Louisiana, at LSU in USA ! The group leader is now Gabrielle Allen .

History of Cactus code Cactus team in 1999..

Cactus team in 2003

Structure of the Cactus code The arrangements of Cactus thorns that are part of the distribution : CactusBase Contains general utility thorns, providing for example coordinates, boundary conditions, and basic IO functionality . CactusBench Contains thorns for benchmarking different architectures, message passing layers, and IO capabilities. CactusConnect Contains thorns for interactive monitoring and steeing of Cactus simulations. CactusDoc Contains documentation for Cactus. CactusEinstein Contains thorns for solving the Einstein equations of general relativity. CactusElliptic Contains the infrastructures and solvers for elliptic equations. CactusExamples Contains example applications. CactusExternal Contains external packages which can be compiled as thorns for use with Cactus. CactusIO Contains thorns providing general IO methods, independent of a particular driver. Note: "Standard" IO methods are also provided by thorns in CactusBasic.

Structure of the Cactus code Oher Cactus thorns : CactusPUGH Contains the default MPI based driver thorn and utility thorns which are closely tied to it. CactusPUGHIO Contains thorns with IO methods which use internal structures and funciton from the PUGH (MPI) driver thorn. CactusTest Contains thorns to test different aspects of the Cactus infrastructure. CactusUtils Contains utility thorns for helping with debugging or optimization. CactusWave Contains simulations of a 3D scalar field. This is standard example application, available in all supported thorn programming languages.

Structure of the Cactus Base arrangment Boundary This thorn defines a generic interface for boundary conditions, which allows, among other things, symmetry boundary conditions associated with the grid to be applied without other thorns having knowledge of them. CartGrid3D This thorn sets up a Cartesian grid, for a given domain. It also provides a method for registering symmetries of Grid Functions across the grid axes, and a call for applying symmetry boundary conditions. CartGrid3d also registers a coordinate system spher3d, under the old API, but this is deprecated. Spherical coordinate systems will be provided by another thorn in the future. CoordBase Provides generic handling of coordinates for grid arrays. … IOASCII, IOBasic,IOUtil, InitBase, LocalInterp LocalReduce, SymBase, Time …

Numerical relativity and the Cactus code Purpose : using supercomputers and Cactus to study black holes, gravitational waves, and other phenomena predicted by Einstein's Theory of General Relativity ! Numerical relativity is the most established application making use of Cactus, and the framework was originally designed by this community to satisfy their requirements: large-scale simulations, collaborative development and use, large data sets, advanced visualization, etc. For more information on Numerical Relativity applications please see the Albert Einstein Institute in Potsdam, germany numerical relativity pages (see at http://numrel.aei.mpg.de)

Numerical relativity and the Cactus code CactusEinstein is that application of the Cactus code which transpose the ADM formalism presented in the previous lecture in numericla codes. Cactus is released with an arrangement of thorns called CactusEinstein, providing core infrastructure for numerical relativity. These thorns provide evolution methods, analysis tools, initial data, etc. However, more importantly, the Einstein arrangement defines and uses a set of standards, which enables better co-working of thorns, and hence enables collaborations between different groups.

Numerical relativity and the Cactus code The thorn structure of CactusEinstein arrangement

The Cactus code : new applications How to compose a specific application for Cactus code ?
Write a new thorn (or an arrangement of thorns)
Thus :
We developped Cosmo thorn for numerically solving Einstein eqs . for comology
Cosmo thorn is using Friedman-Robertson-Walker metric - FRW (see later)
We simulated pure FRW cosmologies (no matter)
We simulated FRW cosmologies with cosmological constant, with scalar field
New routines for dealing with boundaries, specific for FRW were added
Cosmo at output : Hubble function, scale factor...
All these are in gr- qc /0210006

What Cosmo thorn is doing ?
Provides Cactus code the metric components as are for FRW metric - at the initial time and lapse (one) and shift (zero) at every time step
Provides Cactus code the initial values of the cosmological constants and functions (H0,R0...)
Computes at every step the Hubble function H(t) and the scale factor R(t), using :

What Cosmo thorn is doing ?
Fixes the boundaries for the metric and the extrinsic curvature ! How ? Again using
for calculating H(t) and R(t) on all the grid, then imposing flat boundary conditions on these two functions (do to the fact that they are spatially homogeneous). Then, going back, recomputing the extrinsic curvature K and the metric components on the boundaries, using again the above relations.

What RealSF thorn is doing ?
Provides initial data for the scalar field and the potential
Solves numerically the Klein-Gordon equation and evolving the scalar field through it
Here a small problem arised : while the scalar field itself is spatially homogeneous ("flat" boundaries are
appropriate) for the rpi variable (a mixture of metric components and the time derivative of the scalar field) we need first to extract that derivative, impose flat boundary condition on it, then recompute the rpi . Same trick we used in Cosmo thorn !

A special problem : the ADMConstraints and the constraints values at the boundaries
There are no correct boundaries for the constraints, and this is sometime affecting the L2-norm value, used for checking the convergence
An example will be shown later. We avoided this problem, by defining (in Cosmo thorn) a new function, which containes only the inner values of the hamiltonian constraint
Thus now we have good results and an easy way to check the convergence, even through L2-norm

Models with cosmolgical constant First a small trick : to include the cosmological constant in the CalcTmunu files we transferred the term with L in the right hand of the Einstein eqs . as a "fake" matter term. Now comes the initial data prescription. In this purpose we shall rescale all the parameters and variables we are using in term of the initial value of the Hubble parameter H ! We can define the density factor as : and at the initial time we have :

Models with cosmolgical constant We are using this time the gaussian curvature : being at the initial time. Then the stress-energy tensor becomes where and

Models with cosmolgical constant Numerical results

Numerical results with Ellis- Madsen potentials
We tested all 5 Ellis- Madsen potentials for :
accuracy of the time evolution for R(t) and the scalar field
second order convergence
long term evolution - stable and accurate
We also outputed the Hubble function time behaviour - more cosmological parameters at the output are in view
We used Cosmo+ RealSF and ADM_ BSSN thorns

Numerical results...

Hamiltonian constraint...

Convergence tests

Scalar field evolution

Hubble function

Conclusion. Further developments
We prouved the correct evolution of Friedman eqs . for gravity coupled minimally with a scalar field through Cactus code
More potentials are in study. Some are with matter and scalar field (quintessence and dark-matter)
More cosmological parameters to add (as density factor or deceleration)
More than one scalar field - or maybe a complex one - are in view
See soon the article on gr- qc - in a couple of days, I sincerely hope - and Cosmo thorn at CVS ...

References
Ellis GFR , Madsen MS - Class.Quantum Grav ., 8 (1991) 667
Padmanabhan T, PRD 66 (2002), 021301(R)
Cardenas VH , del Campo S, astro - ph /0401031
Ellis GFR , et . al ., Clas . Quantum Grav ., 21 (2004), 233
Guzman F., Scalar Fields Thorn, documentation of the RealSF thorn
D.N. Vulcanov, IJMPC , vol. 6, p. 803, july 2002
D.N. Vulcanov, gr- qc /0210006

Einstein and the … Cactus code

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

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