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Symbolic computation methods
     in cosmology and
      general relativity
   Part II - Maple+GrTensorII programs for cosmology



      Dumitru N. Vulcanov
    West University of Timişoara
     Theoretical and Applied Physics Dept.-“Mircea Zăgănescu”




                                                                1
Introduction
Why we need new CA packages for cosmology ?

New facts from astrophysical measurements show that the Universe is in
an accelerated expansion - quot;cosmic accelerationquot; !

The standard model of the Universe must be reformulated !

One of the solutions is to introduce one or more scalar fields minimally
coupled with the gravity to trigger this expansion.

New models are proposed daily, demanding new specific computational
tools, computer algebra packages included.

This is the purpose of our Cosmo library :
to provide CA tools for theoretical cosmology, in an environment having
all the cosmological parameters and functions defined !
Is entirely done using Maple + GrTensorII !

We will describe Cosmo library in the next slides !!!
                                                                           2
Cosmology

The Cosmo library is processing Einstein equations :




 for the Friedmann-Robertson-Walker metric




  where k=1,-1,0 and R(t) is the scale factor of the Universe




                                                                3
Cosmology

The matter content of the Universe is described by the stress-energy tensor :




 as a scalar field coupled minimally with the gravity and other matter fields
 separately. Thus we have :



  where, as for a perfect fluid :




                                                                                4
Cosmology
For other matter content than the scalar field, the Cosmo library is providing
again the stress-energy tensor as a perfect fluid one :




 with corresponding pressure and density variables.

The main cosmological parameters and functions, namely the Hubble
quot;constantquot; and the deceleration function are :




                                                                                 5
The Cosmo library
 A sequence of GrTensorII and Maple commands for all this follows :

> restart;grtw();qload(rob_sons);
> grdef(`Scal := Phi(t)`);
> grdef(`T1{ i j } := Scal{ ,i } * Scal{ ,j } -
                       g{ i j } * (g{ ^a ^b }*Scal{ ,a }*Scal{ ,b }+ V(t) ) / 2 `);
> pphi(t):=diff(Phi(t),t)^2/2/c^2-V(t)/2;
  epsilonphi(t):=diff(Phi(t),t)^2/2/c^2+V(t)/2;
  grdef(`u{ i } := -c*kdelta{ i $t}`);
  > grdef(`T2{ i j } := (epsilon(t) + p(t))*u{ i }*u{ j } + p(t) * g { i j } `);
> grdef(`T{ i j } :=T1{ i j } + T2{ i j }`);
> grdef(`cons{ i }:= T{ i ^j ;j }`); grcalc(cons(dn));
> EcuKG:=grcomponent(Box[Scal],[]) -DV(t)/2;
> grdef(`Ein{ i j } := G{ i j } - 8*Pi*G*T{ i j }/ c^4`);
> grcalc(Ein(dn,dn)); gralter(Ein(dn,dn),expand);

Where pphi(t) , epsilonphi(t) , p(t) and epsilon(t) are the pressure and
density of the scalar field and matter field, respectively. With cons()
we denoted the conservation law components for the total stress-energy
tensor. The Klein-Gordon equation is EcuKG and DV(t) is the derivative
of the potential in terms of the scalar field !!!                                     6
The Cosmo library
Then follows a sequence of boring grcomponent and gralter commands to
extract the main cosmological equations. Finally we substitute the cosmological
functions in the resulting MAPLE expressions:

> Ecunr1:=expand(simplify(subs(k=K(t)*RR(t)^2,Ecunr1)));
> Ecunr2:=expand(simplify(subs(k=K(t)*RR(t)^2,Ecunr2)));
> Ecunr1:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr1);
> Ecunr22:=subs(diff(RR(t),t,t)=-2*H(t)^2*RR(t)*Q(t),Ecunr2);
> Ecunr22:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr22);
> Ecunr2:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr2);
> Ecunr2:=expand(Ecunr2);
> Ecunr2:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr2);
> Ecunr3:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr3);
> EcuKG:=subs(diff(RR(t),t)=H(t)*RR(t),EcuKG);

    Actually all these commands are transferred in Maple expressions/objects
    the main comological functions, defined and calculated in GrTensorII.
    As a result the user will not use anymore GrTensorII !!!


                                                                               7
The Cosmo library
Thus the program is providing the Klein-Gordon and the conservation law
for the scalar field, namely :




  and the Friedmann equations :




                                                                          8
The Cosmo library
All this stuff is then saved in a library called cosmo.m and can be loaded for
fast processing. As a result all the facilities of MAPLE + GrTensorII are available
together with the cosmology environment described above.

The library is kept in the main directory of GrTensorII (Grtii(6) for Windows
versions or grii in Unix implementations. Then it can be loaded in a Maple
 worksheet using read or load commands


Note : intermediate components of the Ein(i,j) tensor are kept also (apart from
Ecunr1...Ecun3 and EcuKG objects) for later use and processing.
This makes easy the development of the library for more applications


  A simple example of how to use and developp the Cosmo library follows !


       But first let’s have a short demonstration …



                                                                                9
A simple example


We shall ilustrate the use and development of
the Cosmo library in a specificmodel using the
quot;reverse - engineeringquot; method for designing
potentials of the scalar field.




In the standard cosmology the Friedmann eqs. are solved for a specific potential
of the scalar field, initially prescribed from certain physical arguments, and then
the time function R(t) is obtained and compared with the astrophysical
measurements.

In the quot;reverse-engineeringquot; method, the function R(t) is initially prescribed, as
much as possible close to the measurements, and then the potential V(t) is
obtained from Friedmann eqs, if it is possible !

                                                                              10
A simple example

 First step : simply solve two of the above Friedmanneqs. to get :




Here we have the scalar field as the only matter content of the Universe and
geometrical units : c=G=1
For a DeSitter exponential expansion, namely



we have, after a sequence of subs and simplify commands :




 Note : we used an intermediate Maple function for D2Phi(t) !

                                                                               11
A simple example

   Then, from the last expression we have :




The result : equations Ecunr1...Ecunr3 are automatically satisfied and the
Klein-Gordone equation becomes :




This one is used to check the calculations, solving it for the object DV(t).
Therefore DV(t) must fit with the one obtained directly from the scalar field
expression above, after eliminating the time...


                                                                                12
A simple example

Final step : expressing the potential and DV(t) in terms of the scalar field, again
after a boring sequence of subs and simplify commands :




 We need the last expression for DV for our numerical investigations we done,
 As initial data for numerical eolving the Einsten equations for the scalar field.
 But this is for the moment out of the goals of this lecture.

This was a simple example, just to ilustrate how the things are working with
Cosmo library and the reverse engineering method.

Other exmples, more complicated, were processed, even with two scalar fields
included to simulate different components of the dark-matter and energy !

                                                                                  13
Here are some of examples we processed …




    Ellis –Madsen potentials. Here we denoted with R0 the scale factor
    at the actual time t0 and with α the quantity φ(t) – φ0

                                                                         14
Some references




                  15
End of part II




But before the break let’s have aagain a
            demonstration !!!
                                           16

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D. Vulcanov: Symbolic Computation Methods in Cosmology and General Relativity [2]

  • 1. Symbolic computation methods in cosmology and general relativity Part II - Maple+GrTensorII programs for cosmology Dumitru N. Vulcanov West University of Timişoara Theoretical and Applied Physics Dept.-“Mircea Zăgănescu” 1
  • 2. Introduction Why we need new CA packages for cosmology ? New facts from astrophysical measurements show that the Universe is in an accelerated expansion - quot;cosmic accelerationquot; ! The standard model of the Universe must be reformulated ! One of the solutions is to introduce one or more scalar fields minimally coupled with the gravity to trigger this expansion. New models are proposed daily, demanding new specific computational tools, computer algebra packages included. This is the purpose of our Cosmo library : to provide CA tools for theoretical cosmology, in an environment having all the cosmological parameters and functions defined ! Is entirely done using Maple + GrTensorII ! We will describe Cosmo library in the next slides !!! 2
  • 3. Cosmology The Cosmo library is processing Einstein equations : for the Friedmann-Robertson-Walker metric where k=1,-1,0 and R(t) is the scale factor of the Universe 3
  • 4. Cosmology The matter content of the Universe is described by the stress-energy tensor : as a scalar field coupled minimally with the gravity and other matter fields separately. Thus we have : where, as for a perfect fluid : 4
  • 5. Cosmology For other matter content than the scalar field, the Cosmo library is providing again the stress-energy tensor as a perfect fluid one : with corresponding pressure and density variables. The main cosmological parameters and functions, namely the Hubble quot;constantquot; and the deceleration function are : 5
  • 6. The Cosmo library A sequence of GrTensorII and Maple commands for all this follows : > restart;grtw();qload(rob_sons); > grdef(`Scal := Phi(t)`); > grdef(`T1{ i j } := Scal{ ,i } * Scal{ ,j } - g{ i j } * (g{ ^a ^b }*Scal{ ,a }*Scal{ ,b }+ V(t) ) / 2 `); > pphi(t):=diff(Phi(t),t)^2/2/c^2-V(t)/2; epsilonphi(t):=diff(Phi(t),t)^2/2/c^2+V(t)/2; grdef(`u{ i } := -c*kdelta{ i $t}`); > grdef(`T2{ i j } := (epsilon(t) + p(t))*u{ i }*u{ j } + p(t) * g { i j } `); > grdef(`T{ i j } :=T1{ i j } + T2{ i j }`); > grdef(`cons{ i }:= T{ i ^j ;j }`); grcalc(cons(dn)); > EcuKG:=grcomponent(Box[Scal],[]) -DV(t)/2; > grdef(`Ein{ i j } := G{ i j } - 8*Pi*G*T{ i j }/ c^4`); > grcalc(Ein(dn,dn)); gralter(Ein(dn,dn),expand); Where pphi(t) , epsilonphi(t) , p(t) and epsilon(t) are the pressure and density of the scalar field and matter field, respectively. With cons() we denoted the conservation law components for the total stress-energy tensor. The Klein-Gordon equation is EcuKG and DV(t) is the derivative of the potential in terms of the scalar field !!! 6
  • 7. The Cosmo library Then follows a sequence of boring grcomponent and gralter commands to extract the main cosmological equations. Finally we substitute the cosmological functions in the resulting MAPLE expressions: > Ecunr1:=expand(simplify(subs(k=K(t)*RR(t)^2,Ecunr1))); > Ecunr2:=expand(simplify(subs(k=K(t)*RR(t)^2,Ecunr2))); > Ecunr1:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr1); > Ecunr22:=subs(diff(RR(t),t,t)=-2*H(t)^2*RR(t)*Q(t),Ecunr2); > Ecunr22:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr22); > Ecunr2:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr2); > Ecunr2:=expand(Ecunr2); > Ecunr2:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr2); > Ecunr3:=subs(diff(RR(t),t)=H(t)*RR(t),Ecunr3); > EcuKG:=subs(diff(RR(t),t)=H(t)*RR(t),EcuKG); Actually all these commands are transferred in Maple expressions/objects the main comological functions, defined and calculated in GrTensorII. As a result the user will not use anymore GrTensorII !!! 7
  • 8. The Cosmo library Thus the program is providing the Klein-Gordon and the conservation law for the scalar field, namely : and the Friedmann equations : 8
  • 9. The Cosmo library All this stuff is then saved in a library called cosmo.m and can be loaded for fast processing. As a result all the facilities of MAPLE + GrTensorII are available together with the cosmology environment described above. The library is kept in the main directory of GrTensorII (Grtii(6) for Windows versions or grii in Unix implementations. Then it can be loaded in a Maple worksheet using read or load commands Note : intermediate components of the Ein(i,j) tensor are kept also (apart from Ecunr1...Ecun3 and EcuKG objects) for later use and processing. This makes easy the development of the library for more applications A simple example of how to use and developp the Cosmo library follows ! But first let’s have a short demonstration … 9
  • 10. A simple example We shall ilustrate the use and development of the Cosmo library in a specificmodel using the quot;reverse - engineeringquot; method for designing potentials of the scalar field. In the standard cosmology the Friedmann eqs. are solved for a specific potential of the scalar field, initially prescribed from certain physical arguments, and then the time function R(t) is obtained and compared with the astrophysical measurements. In the quot;reverse-engineeringquot; method, the function R(t) is initially prescribed, as much as possible close to the measurements, and then the potential V(t) is obtained from Friedmann eqs, if it is possible ! 10
  • 11. A simple example First step : simply solve two of the above Friedmanneqs. to get : Here we have the scalar field as the only matter content of the Universe and geometrical units : c=G=1 For a DeSitter exponential expansion, namely we have, after a sequence of subs and simplify commands : Note : we used an intermediate Maple function for D2Phi(t) ! 11
  • 12. A simple example Then, from the last expression we have : The result : equations Ecunr1...Ecunr3 are automatically satisfied and the Klein-Gordone equation becomes : This one is used to check the calculations, solving it for the object DV(t). Therefore DV(t) must fit with the one obtained directly from the scalar field expression above, after eliminating the time... 12
  • 13. A simple example Final step : expressing the potential and DV(t) in terms of the scalar field, again after a boring sequence of subs and simplify commands : We need the last expression for DV for our numerical investigations we done, As initial data for numerical eolving the Einsten equations for the scalar field. But this is for the moment out of the goals of this lecture. This was a simple example, just to ilustrate how the things are working with Cosmo library and the reverse engineering method. Other exmples, more complicated, were processed, even with two scalar fields included to simulate different components of the dark-matter and energy ! 13
  • 14. Here are some of examples we processed … Ellis –Madsen potentials. Here we denoted with R0 the scale factor at the actual time t0 and with α the quantity φ(t) – φ0 14
  • 16. End of part II But before the break let’s have aagain a demonstration !!! 16