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semiclassical_collapse_Stewart.c
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///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/*
This code was written by Benjamin Berczi as part of the PhD project titled "Simulating Semiclassical Black Holes" from the University of Nottingham.
It is a self-contained C file that simulates a massless quantum scalar field coupled to Einstein gravity in the double null formulation.
Details may be found in Benjamin Berczi's publications and PhD thesis.
*/
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Initialising the libraries and constants for the code */
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <omp.h>
#include <gsl/gsl_sf.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_bessel.h>
#include <gsl/gsl_sf_gamma.h>
#include <complex.h>
/* CONSANTS */
#define PI 3.1415926535897932384626433832795028841971693993751058209749445923078164062
#define M_P 1.0 //sqrt(8*PI)
#define c 1.0 //speed of light
/* GRID PARAMETERS */
#define u_size 513 // size of grid in u direction
#define v_size 513 // size of grid in v direction
#define du 0.02 // step size in u direction
#define dv 0.02 // step size in v direction
#define u_max du*(u_size-1)
#define v_max dv*(v_size-1)
#define N_HALF 4
#define r0 0.0
/* SCALAR FIELD PARAMETERS */
#define amplitude 0.0 // initial amplitude
#define mass 0.0 // initial mass
#define initial_width 1.0 // initial width of the gaussian scalar field
#define initial_radius 5.0 // initial radius of the gaussian scalar field
/* QUANTUM OR CLASSICAL SIMULATION */
#define hbar 0 // set to 1 for quantum, 0 for classical. This just sets the backreaction, and is in set_bi_linears.c, the quantum modes are still evolved
#define coherent_state_switch 1 // set to 0 to just have the mode functions
/* QUANTUM GHOST FIELD PARAMETERS */
#define number_of_q_fields 6 // number of quantum fields, 1 real, 5 ghosts for regularisation
#define muSq 0.0 // mass of scalar field
#define mSqGhost 1.0 // base mass of the Pauli-Villars regulator fields
double massSq[number_of_q_fields] = { muSq, mSqGhost, 3.0 * mSqGhost, mSqGhost, 3.0 * mSqGhost, 4.0 * mSqGhost }; // masses of the ghost fields
double ghost_or_physical[6] = { 1 , -1 , 1 , -1 , 1, -1 }; // distinguishing between the real and ghost fields
/* QUANTUM MODE PARAMETERS */
#define k_min 1.0*PI/15.0 //PI/6.0;//5.0*2.0*PI/(lattice_size*dr); // minimum value of k, also =dk
#define dk k_min
#define number_of_k_modes 1 // number of k modes
#define number_of_l_modes 1 // number of l modes
#define k_start 0
#define l_start 0 //the range of l is l_start, l_start+l_step, l_start+2l_step...
#define l_step 1
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Creating a structure for the variables */
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
struct fields {
double* phi;
double* sigma;
double* r;
double* D_phi;
double* D_sigma;
double* K_phi;
double* K_r;
double* D_r;
double* mu;
double* lambda;
};
typedef struct fields Fields;
struct q_fields {
__complex__ double*** phi;
__complex__ double*** D_phi;
__complex__ double*** K_phi;
__complex__ double*** mu;
};
typedef struct q_fields Q_Fields;
struct initial_fields_u0 {
double* D_phi_u0;
double* D_sigma_u0;
};
typedef struct initial_fields Initial_Fields;
struct bi_linears {
double* phi_phi;
double* Dphi_Dphi;
double* Kphi_Kphi;
double* Dphi_Kphi;
double* del_theta_phi_del_theta_phi;
};
typedef struct bi_linears Bi_Linears;
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Making the points for the spatial grid*/
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void make_points(double N, double v[v_size]) {
double v_min = (pow(2.0, N) - 1.0) / pow(2.0, N) * v_max;
double dvv = dv / pow(2.0, N);
for (int i = 0; i < v_size; ++i) {
v[i] = v_min + i * dvv;
}
}
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Functions to save things */
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void save_points(double v[v_size]) {
FILE* pointsout;
pointsout = fopen("vpoints2.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(pointsout, "%.20f ", v[m]);
}
fclose(pointsout);
}
void save_horizonv(double hor[u_size]) {
FILE* pointsout;
pointsout = fopen("horizonv.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(pointsout, "%.20f ", hor[m]);
}
fclose(pointsout);
}
void save_horizonu(double hor[u_size]) {
FILE* pointsout;
pointsout = fopen("horizonu.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(pointsout, "%.20f ", hor[m]);
}
fclose(pointsout);
}
void save_upoints() {
FILE* pointsout;
pointsout = fopen("upoints2.txt", "w");
for (int m = 0; m < u_size; ++m) {
fprintf(pointsout, "%.20f ", m*du);
}
fclose(pointsout);
}
void save_Dphiq(__complex__ double* field) {
FILE* finout;
finout = fopen("Dphiq.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.100f ", __real__ field[m]);
}
fclose(finout);
}
void save_Kphiq(__complex__ double* field) {
FILE* finout;
finout = fopen("Kphiq.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.100f ", __real__ field[m]);
}
fclose(finout);
}
void save_muphiq(__complex__ double* field) {
FILE* finout;
finout = fopen("muphiq.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.100f ", __real__ field[m]);
}
fclose(finout);
}
void save_phi(double* field) {
FILE* finout;
finout = fopen("phi_null2.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", field[m]);
}
fclose(finout);
}
void save_Dphi(double* field) {
FILE* finout;
finout = fopen("Dphi.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", field[m]);
}
fclose(finout);
}
void save_Kphi(double* field) {
FILE* finout;
finout = fopen("Kphi.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", field[m]);
}
fclose(finout);
}
void save_muphi(double* field) {
FILE* finout;
finout = fopen("mu.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", field[m]);
}
fclose(finout);
}
void save_r1(double* field) {
FILE* finout;
finout = fopen("r_null2.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", field[m]);
}
fclose(finout);
}
void save_sigma(double* field) {
FILE* finout;
finout = fopen("sigma_null2.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", field[m]);
}
fclose(finout);
}
void save_constr(double constr[v_size]) {
FILE* finout;
finout = fopen("constr_null2.txt", "w");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", constr[m]);
}
fclose(finout);
}
void save_r(double** field) {
FILE* finout;
finout = fopen("phiphi_matrix100.txt", "w");
for (int n = 0; n < N_HALF *(u_size - 1) / 2; n++) {
fprintf(finout, "\n");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.100f ", field[n][m]);
}
}
fclose(finout);
}
void save_phi0(double* field) {
FILE* finout;
finout = fopen("phi01.txt", "w");
for (int n = 0; n < N_HALF * (u_size - 1) / 2; n++) {
fprintf(finout, "%.20f ", field[n]);
}
fclose(finout);
}
void save_vmat() {
double v[v_size];
FILE* finout;
finout = fopen("v_matrix2.txt", "w");
for (int n_halved = 0; n_halved < N_HALF; n_halved++) {
make_points(n_halved, v);
for (int n = 0; n < (u_size - 1) / 2; n++) {
fprintf(finout, "\n");
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.100f ", v[m]);
}
}
}
fclose(finout);
}
void save_u() {
//double u[N_HALF * (u_size - 1) / 2];
double T = -dv;
//u[0] = 0.0;
FILE* finout;
finout = fopen("u_points.txt", "w");
for (int n_halved = 0; n_halved < N_HALF; n_halved++) {
for (int n = 0; n < (u_size - 1) / 2; n++) {
fprintf(finout, "\n");
T = T + dv / pow(2.0, n_halved);
for (int m = 0; m < v_size; ++m) {
fprintf(finout, "%.20f ", T);
}
}
}
fclose(finout);
}
void save_u0(double* u) {
double T = 0.0;
FILE* finout;
finout = fopen("u0.txt", "w");
for (int n = 0; n < N_HALF * (u_size - 1) / 2; n++) {
fprintf(finout, "%.20f ", u[n]);
}
fclose(finout);
}
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Derivative function for real fields */
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
double first_deriv_vo2(double dvv, double x1,double x2,double x3,double x4) {
double der = 0.0;
return der = (x2 - x1 + x4 - x3) / (2.0 * dvv);
}
double first_deriv_uo2(double dvv, double x1, double x2, double x3, double x4) {
double der = 0.0;
return der = (x3 - x1 + x4 - x2) / (2.0 * dvv);
}
double first_deriv_vo1(double dvv, double x1, double x2) {
double der = 0.0;
return der = (x2 - x1) / (1.0*dvv);
}
double first_deriv_uo1(double dvv, double x1, double x3) {
double der = 0.0;
return der = (x3 - x1) / (1.0*dvv);
}
double first_derivv(double dvv, int m, double* field) {
double der = 0.0;
if (m == 0) {
return der = 0.0;
}
if (m == 1) {
return der = 0.0;// (-field[m + 2] + 8.0 * field[m + 1] - 8.0 * field[m - 1] + field[1]) / (12.0 * dvv);
}
if (m == v_size - 1) {
return der = (3.0 * field[m - 4] - 16.0 * field[m - 3] + 36.0 * field[m - 2] - 48.0 * field[m - 1] + 25.0 * field[m]) / (12 * dvv);
}
if (m == v_size - 2) {
return der = (-field[m - 3] + 6.0 * field[m - 2] - 18.0 * field[m - 1] + 10.0 * field[m] + 3.0 * field[m + 1]) / (12 * dvv);
}
if (2 <= m < v_size - 2) {
return der = (-field[m + 2] + 8.0 * field[m + 1] - 8.0 * field[m - 1] + field[m - 2]) / (12.0 * dvv);
}
}
double first_derivu(double dvv, int m, double* field) {
double der = 0.0;
if (m == 0) {
return der = (field[m + 1] - field[m]) / (dvv);
}
if (m == u_size - 1) {
return der = (field[m] - field[m - 1]) / (dvv);
}
if (m != 0 && m != u_size - 1) {
return der = (field[m + 1] - field[m - 1]) / (2.0 * dvv);
}
}
double second_derivv(double dvv, int m, double* field) {
double der = 0.0;
if (m == 0) {
return der = (field[m] - 2.0 * field[m+1] + field[m +2]) / (dvv * dvv);
}
if (m == v_size - 1) {
return der = (field[m - 2] - 2.0 * field[m - 1] + field[m]) / (dvv * dvv);
}
if (m != 0 && m != v_size - 1 ) {
return der = (field[m + 1] - 2.0 * field[m] + field[m - 1]) / (dvv * dvv);
}
}
double second_derivu(double dvv, int m, double* field) {
double der = 0.0;
if (m == 0) {
return der = (field[m] - 2.0 * field[m+1] + field[m +2]) / (dvv * dvv);
}
if (m == u_size - 1) {
return der = (field[m - 2] - 2.0 * field[m - 1] + field[m]) / (dvv * dvv);
}
if (m != 0 && m != u_size - 1 ) {
return der = (field[m + 1] - 2.0 * field[m] + field[m - 1]) / (dvv * dvv);
}
}
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* First derivative function for complex fields, uses 20 neighbouring points at the moment*/
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
__complex__ double sixth_deriv_comp(int i, int n, double dvv, __complex__ double* field) {
__complex__ double der = 0.0;
if (3 < i < v_size - 3) {
der = (field[i + 3] - 6.0 * field[i + 2] + 15.0 * field[i+1] - 20.0 * field[i] + 15.0 * field[i - 1] - 6.0 * field[i - 2] + field[i - 3]) / pow(dvv, 6.0);
}
if (i < n+3) {
der = (field[i] - 6.0 * field[i + 1] + 15.0 * field[i + 2] - 20.0 * field[i + 3] + 15.0*field[i + 4] - 6.0*field[i+5] + 1.0*field[i+6]) / pow(dvv, 6.0);
}
if (i>v_size-4) {
der = 0.0;
}
return der;
}
__complex__ double fourth_deriv_comp(int i, int n, double dvv, __complex__ double* field) {
__complex__ double der = 0.0;
if (2 < i < v_size - 2) {
der = (field[i + 2] - 4.0 * field[i + 1] + 6.0 * field[i] - 4.0 * field[i - 1] + field[i - 2]) / pow(1.0 * dvv, 4.0);
}
if (i < n + 2) {
der = (field[i] - 4.0 * field[i + 1] + 6.0 * field[i + 2] - 4.0 * field[i + 3] + field[i + 4]) / pow(dvv, 4.0);
}
if (i > v_size - 3) {
der = 0.0;
}
return der;
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Function that initialises the classical variables */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void charac_initial_conditions(Fields* fields) {
double v[v_size];
make_points(0, v);
/* initial v slice */
for (int i = 0; i < v_size; ++i) {
fields->D_sigma[i] = 0.0;
fields->phi[i] = amplitude * v[i] * v[i] * exp(-1.0 / 2.0 * pow((v[i] - initial_radius) / initial_width, 2.0));
fields->D_phi[i] = -((v[i] - initial_radius) / (initial_width * initial_width)) * fields->phi[i] + 2.0 * amplitude * v[i] * exp(-1.0 / 2.0 * pow((v[i] - initial_radius) / initial_width, 2.0));
}
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* This function provides a version of gsl's Bessel function that ignores any underflow error */
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
double gsl_sf_bessel_jl_safe(int l, double x) {
gsl_sf_result answer;
gsl_error_handler_t* old_error_handler = gsl_set_error_handler_off(); // turn off the error handler
int error_code = gsl_sf_bessel_jl_e(l, x, &answer); //compute the answer, and construct an error code
gsl_set_error_handler(old_error_handler); //reset the error handler
if (error_code == GSL_SUCCESS) { //if there's no error then return the correct answer
return answer.val;
}
else {
//printf ("error in gsl_sf_bessel_jl_safe: %s\n", gsl_strerror (error_code));
//exit(1);
if (error_code == GSL_EUNDRFLW) {
return 0.0;
}
else {
printf("error in gsl_sf_bessel_jl_safe: %s\n", gsl_strerror(error_code));
exit(1);
}
}
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* These functions provides the initial profile functions */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
__complex__ double phi_mode_profile_0(double k, int l, double vv) {
double v = vv / 2.0;
return (sqrt(k / PI) * cexp(-I*k*v)*gsl_sf_bessel_jl_safe(l, k * v) / pow(v, l));
}
__complex__ double phi_mode_profile_0_prime(double k, int l, double vv) {
double v = vv / 2.0;
return 0.5*( (-k * sqrt(k / PI) * cexp(-I * k * v) * gsl_sf_bessel_jl_safe(l + 1, k * v) / pow(v, l)) - I *k* (sqrt(k / PI) * cexp(-I * k * v) * gsl_sf_bessel_jl_safe(l, k * v) / pow(v, l)));
}
__complex__ double phi_mode_profile_0_dot(double k, int l, double vv) {
double v = vv / 2.0;
return 0.5 * ((k * sqrt(k / PI)* cexp(-I * k * v) * gsl_sf_bessel_jl_safe(l + 1, k * v) / pow(v, l)) - I * k * (sqrt(k / PI) * cexp(-I * k * v) * gsl_sf_bessel_jl_safe(l, k * v) / pow(v, l)));
}
__complex__ double phi_mode_profile_massive(double msq, double k, int l, double vv) {
double v = vv / 2.0;
return (k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I *sqrt(k * k + msq) * v) * gsl_sf_bessel_jl_safe(l, k * v) / pow(v, l));
}
__complex__ double phi_mode_profile_massive_prime(double msq, double k, int l, double vv) {
double v = vv / 2.0;
return 0.5 * ((-k * k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * v) * gsl_sf_bessel_jl_safe(l + 1, k * v) / pow(v, l)) - I *sqrt(k * k + msq) * (k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * v) * gsl_sf_bessel_jl_safe(l, k * v) / pow(v, l)));
}
__complex__ double phi_mode_profile_massive_dot(double msq, double k, int l, double vv) {
double v = vv / 2.0;
return 0.5 * ((k * k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * v) * gsl_sf_bessel_jl_safe(l + 1, k * v) / pow(v, l)) - I * sqrt(k * k + msq) * (k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * v) * gsl_sf_bessel_jl_safe(l, k * v) / pow(v, l)));
}
__complex__ double phi_mode_profile_0_u(double k, int l, double vv, double uu) {
double r = (vv-uu) / 2.0;
double t = (vv+uu) / 2.0;
return sqrt(k / PI) * cexp(-I * k * t) * gsl_sf_bessel_jl_safe(l, k * r) / pow(r, l) ;
}
__complex__ double phi_mode_profile_0_prime_u(double k, int l, double vv, double uu) {
double v = vv / 2.0;
double r = (vv - uu) / 2.0;
double t = (vv + uu) / 2.0;
return 0.5 * ( -k * sqrt(k / PI) * cexp(-I * k * t) * gsl_sf_bessel_jl_safe(l+1, k * r) / pow(r, l) - I * k * sqrt(k / PI) * cexp(-I * k * t) * gsl_sf_bessel_jl_safe(l, k * r) / pow(r, l) );
}
__complex__ double phi_mode_profile_0_dot_u(double k, int l, double vv, double uu) {
double v = vv / 2.0;
double r = (vv - uu) / 2.0;
double t = (vv + uu) / 2.0;
return 0.5 * ( k * sqrt(k / PI) * cexp(-I * k * t) * gsl_sf_bessel_jl_safe(l + 1, k * r) / pow(r, l) - I * k * sqrt(k / PI) * cexp(-I * k * t) * gsl_sf_bessel_jl_safe(l, k * r) / pow(r, l) );
}
__complex__ double phi_mode_profile_massive_u(double msq, double k, int l, double vv, double uu) {
double v = vv / 2.0;
double r = (vv - uu) / 2.0;
double t = (vv + uu) / 2.0;
return k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * t) * gsl_sf_bessel_jl_safe(l, k * r) / pow(r, l);
}
__complex__ double phi_mode_profile_massive_prime_u(double msq, double k, int l, double vv, double uu) {
double v = vv / 2.0;
double r = (vv - uu) / 2.0;
double t = (vv + uu) / 2.0;
return 0.5 * ( -k * k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * t) * gsl_sf_bessel_jl_safe(l + 1, k * r) / pow(r, l) - I * sqrt(k * k + msq) * k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * t) * gsl_sf_bessel_jl_safe(l, k * r) / pow(r, l));
}
__complex__ double phi_mode_profile_massive_dot_u(double msq, double k, int l, double vv, double uu) {
double v = vv / 2.0;
double r = (vv - uu) / 2.0;
double t = (vv + uu) / 2.0;
return 0.5 * ( k * k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * t) * gsl_sf_bessel_jl_safe(l + 1, k * r) / pow(r, l) - I * sqrt(k * k + msq) * k / sqrt(PI * sqrt(k * k + msq)) * cexp(-I * sqrt(k * k + msq) * t) * gsl_sf_bessel_jl_safe(l, k * r) / pow(r, l) );
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Initial conditions for the quantum fields */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void charac_initial_conditions_quantum(Q_Fields** q_fields) {
double v[v_size];
make_points(0, v);
double k_wavenumber, omega_phi;
int l_value;
/* initial v slice */
for (int k = 0; k < number_of_k_modes; ++k) {
k_wavenumber = (k_start + (k + 1)) * k_min;
for (int l = 0; l < number_of_l_modes; ++l) {
l_value = l_start + l * l_step;
for (int which_q_field = 0; which_q_field < number_of_q_fields; ++which_q_field) {
omega_phi = sqrt(k_wavenumber * k_wavenumber + massSq[which_q_field]);
if (massSq[which_q_field] == 0) {
for (int i = 1; i < v_size; ++i) {
q_fields[which_q_field]->phi[k][l][i] = phi_mode_profile_0 (k_wavenumber, l_value, v[i]);
q_fields[which_q_field]->D_phi[k][l][i] = phi_mode_profile_0_prime(k_wavenumber, l_value, v[i]);
q_fields[which_q_field]->K_phi[k][l][i] = phi_mode_profile_0_dot(k_wavenumber, l_value, v[i]);
}
}
else {
for (int i = 1; i < v_size; ++i) {
q_fields[which_q_field]->phi[k][l][i] = phi_mode_profile_massive (massSq[which_q_field], k_wavenumber, l_value, v[i]);
q_fields[which_q_field]->D_phi[k][l][i] = phi_mode_profile_massive_prime(massSq[which_q_field], k_wavenumber, l_value, v[i]);
q_fields[which_q_field]->K_phi[k][l][i] = phi_mode_profile_massive_dot (massSq[which_q_field], k_wavenumber, l_value, v[i]);
}
}
q_fields[which_q_field]->phi[k][l][0] = k_wavenumber / sqrt(PI * omega_phi) * pow(k_wavenumber, l_value) / gsl_sf_doublefact(2 * l_value + 1);
q_fields[which_q_field]->D_phi[k][l][0] = -I * 0.5 * omega_phi * q_fields[which_q_field]->phi[k][l][0];
q_fields[which_q_field]->K_phi[k][l][0] = -I * 0.5 * omega_phi * q_fields[which_q_field]->phi[k][l][0];
}
}
}
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/*Initial conditions for the functions at the centre */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void set_initial_conds_v0(int n, Fields* fields){
fields->sigma[n] = fields->sigma[n + 1];
fields->r[n] = 0.0;
fields->D_r[n] = 0.5 * exp(fields->sigma[n] / 2.0);
fields->phi[n] = fields->phi[n + 1];
fields->K_r[n] = -0.5 * exp(fields->sigma[n] / 2.0);
fields->K_phi[n] = fields->D_phi[n];
}
void set_initial_conds_v0_quantum(int n, Q_Fields** q_fields, Q_Fields** q_fields_old, Q_Fields** q_fields_old_old) {
for (int k = 0; k < number_of_k_modes; ++k) {
for (int l = 0; l < number_of_l_modes; ++l) {
for (int which_q_field = 0; which_q_field < number_of_q_fields; ++which_q_field) {
q_fields[which_q_field]->phi[k][l][n] = q_fields[which_q_field]->phi[k][l][n+1];
q_fields[which_q_field]->K_phi[k][l][n] = q_fields[which_q_field]->D_phi[k][l][n];
}
}
}
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Set Minkowski spacetime values for the initial fields */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void set_minkowski(Fields* fields) {
for (int i = 0; i < v_size; ++i) {
fields->r[i] = 0.5*i*dv;
fields->D_r[i] = 0.5;
fields->K_r[i] = -0.5;
fields->sigma[i] = 0.0;
fields->mu[i] = 0.0;
fields->lambda[i] = 0.0;
}
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Functions to find the v derivative of fields to find them at each time step */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void find_sigma(int i, int n, double dvv, Fields* fields) {
double sigma_prime;
sigma_prime = fields->D_sigma[i];
fields->sigma[i + 1] = fields->sigma[i] + dvv * sigma_prime;
}
void find_r_D_r(int i, int n, double dvv, Fields* fields, Bi_Linears* bi_linear) {
double r_prime, D_r_prime;
r_prime = fields->D_r[i];
D_r_prime = fields->D_r[i] * fields->D_sigma[i] -0.5 * fields->r[i] * bi_linear->Dphi_Dphi[i];
fields->r[i + 1] = fields->r[i] + dvv * r_prime;
fields->D_r[i + 1] = fields->D_r[i] + dvv * D_r_prime;
}
void find_phi(int i, int n, double dvv, Fields* fields) {
double phi_prime;
phi_prime = fields->D_phi[i];
fields->phi[i + 1] = fields->phi[i] + dvv * phi_prime;
}
void find_K_r(int i, int n, double dvv, Fields* fields) {
double K_r_prime;
K_r_prime = -(i != n ? fields->lambda[i] / fields->r[i] : 0.0);
fields->K_r[i + 1] = fields->K_r[i] + dvv * K_r_prime;
}
void find_K_phi(int i, int n, double dvv, Fields* fields) {
double K_phi_prime;
fields->mu[i] = fields->D_r[i] * fields->K_phi[i] + fields->D_phi[i] * fields->K_r[i];
K_phi_prime = -(i != n ? fields->mu[i] / fields->r[i] : 0.0);
fields->K_phi[i + 1] = fields->K_phi[i] + dvv * K_phi_prime;
}
void find_q_phi(int i, int n, double dvv, Q_Fields** q_fields, Q_Fields** q_fields_old) {
__complex__ double q_field_prime=0.0;
for (int k = 0; k < number_of_k_modes; ++k) {
for (int l = 0; l < number_of_l_modes; ++l) {
for (int which_q_field = 0; which_q_field < number_of_q_fields; ++which_q_field) {
if (i == n) {
q_field_prime = 0.5 * (q_fields[which_q_field]->D_phi[k][l][i] + q_fields[which_q_field]->D_phi[k][l][i + 1]);
q_fields[which_q_field]->phi[k][l][i + 1] = q_fields[which_q_field]->phi[k][l][i] + dvv * (q_field_prime - 0.0 * pow(dvv, 3.0) * fourth_deriv_comp(i, n, dvv, q_fields_old[which_q_field]->phi[k][l]));
}
if (i == n + 1) {
q_field_prime = 0.5 * (q_fields[which_q_field]->D_phi[k][l][i] + q_fields[which_q_field]->D_phi[k][l][i + 1]);
q_fields[which_q_field]->phi[k][l][i + 1] = q_fields[which_q_field]->phi[k][l][i] + dvv * (q_field_prime - 0.0 * pow(dvv, 3.0) * fourth_deriv_comp(i, n, dvv, q_fields_old[which_q_field]->phi[k][l]));
}
if (v_size-1 > i > n + 1) {
q_field_prime = 1.0/24.0 * (9.0*q_fields[which_q_field]->D_phi[k][l][i+1] + 19.0*q_fields[which_q_field]->D_phi[k][l][i] - 5.0 * q_fields[which_q_field]->D_phi[k][l][i-1] + 1.0 * q_fields[which_q_field]->D_phi[k][l][i-2]);
q_fields[which_q_field]->phi[k][l][i + 1] = q_fields[which_q_field]->phi[k][l][i] + dvv * (q_field_prime - 0.0 * pow(dvv, 3.0) * fourth_deriv_comp(i, n, dvv, q_fields_old[which_q_field]->phi[k][l]));
}
if (i == v_size - 1) {
q_field_prime = q_fields[which_q_field]->D_phi[k][l][i];
q_fields[which_q_field]->phi[k][l][i + 1] = q_fields[which_q_field]->phi[k][l][i] + dvv * (q_field_prime - 0.0 * pow(dvv, 3.0) * fourth_deriv_comp(i, n, dvv, q_fields_old[which_q_field]->phi[k][l]));
}
}
}
}
}
void find_q_K_phi(int i, int n, double dvv, Fields* fields, Q_Fields** q_fields, Q_Fields** q_fields_old, Bi_Linears* bi_linears) {
fields->lambda[i] = fields->D_r[i] * fields->K_r[i] + 1.0 / 4.0 * exp(fields->sigma[i]) * (1.0 -bi_linears->del_theta_phi_del_theta_phi[i]);
for (int k = 0; k < number_of_k_modes; ++k) {
for (int l = 0; l < number_of_l_modes; ++l) {
for (int which_q_field = 0; which_q_field < number_of_q_fields; ++which_q_field) {
double k_wavenumber = (k_start + (k + 1)) * k_min;
double l_value = l_start + l * l_step;
__complex__ double q_field_prime, q_field_primef, phi, Kphi, Dphi, mu, K_f_prime_n, F_f;
__complex__ double fk1, fk2, fk3, fk4;
__complex__ double k1, k2, k3, k4;
double r;
fields->lambda[i + 1] = fields->D_r[i + 1] * fields->K_r[i + 1] + 1.0 / 4.0 * exp(fields->sigma[i + 1]) * (1.0 - bi_linears->del_theta_phi_del_theta_phi[i + 1]);
if (i < v_size -1 ) {
fields->lambda[i + 1] = fields->D_r[i + 1] * fields->K_r[i + 1] + 1.0 / 4.0 * exp(fields->sigma[i + 1]) * (1.0 - bi_linears->del_theta_phi_del_theta_phi[i + 1]);
q_fields[which_q_field]->mu[k][l][i] = (i != n ? fields->D_r[i] * q_fields[which_q_field]->K_phi[k][l][i] + fields->K_r[i] * q_fields[which_q_field]->D_phi[k][l][i] : 0.0);
K_f_prime_n = (i != n ? -(l_value + 1.0) / fields->r[i] * q_fields[which_q_field]->mu[k][l][i]
- pow(l_value / fields->r[i], 2.0) * fields->lambda[i] * q_fields[which_q_field]->phi[k][l][i] : 0.0)
- 0.25 * massSq[which_q_field] * exp(fields->sigma[i]) * q_fields[which_q_field]->phi[k][l][i];
F_f = pow(l_value / fields->r[i + 1], 2.0) * fields->lambda[i + 1] * q_fields[which_q_field]->phi[k][l][i + 1] + 0.25 * massSq[which_q_field] * q_fields[which_q_field]->phi[k][l][i + 1];
q_fields[which_q_field]->K_phi[k][l][i + 1] = (q_fields[which_q_field]->K_phi[k][l][i] + dvv * (-fields->K_r[i + 1] * (l_value + 1.0) / fields->r[i + 1] * q_fields[which_q_field]->D_phi[k][l][i + 1] - F_f - 0.0 * pow(dvv, 3.0) * fourth_deriv_comp(i, n, dvv, q_fields_old[which_q_field]->K_phi[k][l]))) / (1.0 + dvv * fields->D_r[i + 1] * (l_value + 1.0) / fields->r[i + 1]);
}
else {
q_fields[which_q_field]->mu[k][l][i] = fields->D_r[i] * q_fields[which_q_field]->K_phi[k][l][i] + fields->K_r[i] * q_fields[which_q_field]->D_phi[k][l][i];
q_field_prime = (i != n ? -(l_value + 1.0) / fields->r[i] * q_fields[which_q_field]->mu[k][l][i]
- pow(l_value / fields->r[i], 2.0) * fields->lambda[i] * q_fields[which_q_field]->phi[k][l][i] : 0.0)
- 0.25 * massSq[which_q_field] * exp(fields->sigma[i]) * q_fields[which_q_field]->phi[k][l][i];
q_fields[which_q_field]->K_phi[k][l][i + 1] = q_fields[which_q_field]->K_phi[k][l][i] + dvv * (q_field_prime -0.0 * pow(dvv, 3.0) * fourth_deriv_comp(i, n, dvv, q_fields_old[which_q_field]->K_phi[k][l]));
}
}
}
}
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Function that calculates the norm */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
double norm(__complex__ double number) {
double nor = 0.0;
nor = (pow((__real__ number), 2.0) + pow((__imag__ number), 2.0));
return nor;
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Function that calculates the bilinears of the matter field */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void set_bi_linears(double cosm_const, int i, int n, double dvv, Bi_Linears* bi_linears, Fields* fields, Q_Fields** q_fields) {
double r, r_l, rprime, rdot;
double phi_phi, chi_chi, chi_pi, pi_pi, del_theta_phi_del_theta_phi_over_r_sq;
_Complex double Phi_mode, Phi_mode_plus, Chi_mode, Pi_mode;
int l_value;
double v[v_size];
make_points(0, v);
phi_phi = 0.0;
chi_chi = 0.0;
chi_pi = 0.0;
pi_pi = 0.0;
del_theta_phi_del_theta_phi_over_r_sq = 0.0;
if (coherent_state_switch != 0) {
phi_phi = fields->phi[i] * fields->phi[i];
chi_chi = fields->D_phi[i] * fields->D_phi[i];
pi_pi = fields->K_phi[i] * fields->K_phi[i];
chi_pi = fields->D_phi[i] * fields->K_phi[i];
del_theta_phi_del_theta_phi_over_r_sq = 0.0;
}
//note that these modes are actually modes of phi, where Phi = r^l phi
//Phi = r^l phi
//Pi = lr^{l-1} dr/du u + r^l pi
//Psi = lr^{l-1} dr/dv u + r^l psi
if (hbar != 0) {
//#pragma omp parallel for
for (int k = 0; k < number_of_k_modes; ++k) {
for (int l = 0; l < number_of_l_modes; ++l) {
for (int which_q_field = 0; which_q_field < number_of_q_fields; ++which_q_field) {
l_value = l_start + l * l_step;
r = fields->r[i];
r_l = pow(r, l_value);
rprime = fields->D_r[i];
rdot = fields->K_r[i];
/* PHI MODE */
Phi_mode = r_l * (q_fields[which_q_field]->phi[k][l][i]);
if (i == n) {
Phi_mode_plus = pow(fields->r[i+1], l_value) * (q_fields[which_q_field]->phi[k][l][i + 1]);
}
/* CHI MODE */
if (l_value == 0) {
Chi_mode = q_fields[which_q_field]->D_phi[k][l][i];
}
else if (l_value == 1) {
Chi_mode = rprime * q_fields[which_q_field]->phi[k][l][i] + r * q_fields[which_q_field]->D_phi[k][l][i];
}
else {
Chi_mode = l_value * pow(r, l_value - 1) * rprime * q_fields[which_q_field]->phi[k][l][i] + r_l * (q_fields[which_q_field]->D_phi[k][l][i]);
}
/* PI MODE */
if (l_value == 0) {
Pi_mode = q_fields[which_q_field]->K_phi[k][l][i];
}
else if (l_value == 1) {
Pi_mode = rdot * q_fields[which_q_field]->phi[k][l][i] + r * q_fields[which_q_field]->K_phi[k][l][i];
}
else {
Pi_mode = l_value * pow(r, l_value - 1) * rdot * q_fields[which_q_field]->phi[k][l][i] + r_l * (q_fields[which_q_field]->K_phi[k][l][i]);
}
/* ACTUAL BILINEARS */
phi_phi = phi_phi + hbar * ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * norm(Phi_mode); // instead of norm
chi_chi = chi_chi + hbar * ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * norm(Chi_mode);
pi_pi = pi_pi + hbar * ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * norm(Pi_mode);
chi_pi = chi_pi + hbar * ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * 0.5*((__real__(Pi_mode * conj(Chi_mode))) + (__real__(Chi_mode * conj(Pi_mode))));
if (i != n) {
del_theta_phi_del_theta_phi_over_r_sq = del_theta_phi_del_theta_phi_over_r_sq + hbar * ghost_or_physical[which_q_field] * dk / (4.0 * PI) * 0.5 * l_value * (l_value + 1.0) * (2.0 * l_value + 1.0) * norm(Phi_mode) / (r * r);
}
else {//use the data at r=dr to estimate the r=0 case. This is only relevant for l=1
del_theta_phi_del_theta_phi_over_r_sq = del_theta_phi_del_theta_phi_over_r_sq + hbar * ghost_or_physical[which_q_field] * dk / (4.0 * PI) * 0.5 * l_value * (l_value + 1.0) * (2.0 * l_value + 1.0) * norm(Phi_mode_plus) / (fields->r[i+1] * fields->r[i+1]);
}
}
}
}
}
//printf("\n %.100f, ", norm(chi_mode));
bi_linears->phi_phi[i] = phi_phi -2* cosm_const;
bi_linears->Dphi_Dphi[i] = chi_chi;
bi_linears->Kphi_Kphi[i] = pi_pi;
bi_linears->Dphi_Kphi[i] = chi_pi + cosm_const/2.0;
bi_linears->del_theta_phi_del_theta_phi[i] = del_theta_phi_del_theta_phi_over_r_sq - cosm_const;
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Function that calculates the cosmological constant */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
double set_cosm_constant(Fields* fields, Q_Fields** q_fields, Bi_Linears* bi_linears) {
double rho, S_A, A, B;
int i, n;
double phi_phi, chi_chi, pi_pi, chi_pi, del_theta_phi_del_theta_phi_over_r_sq;
A = 1.0;
B = 1.0;
i = 0;//buff_size+0;
n = 0;
double cosm_const = 0.0;
double r, r_l, rprime, rdot;
//double phi_phi, chi_chi, chi_pi, pi_pi, del_theta_phi_del_theta_phi_over_r_sq;
_Complex double Phi_mode, Phi_mode_plus, Chi_mode, Pi_mode;
int l_value;
double v[v_size];
make_points(0, v);
phi_phi = 0.0;
chi_chi = 0.0;
chi_pi = 0.0;
pi_pi = 0.0;
del_theta_phi_del_theta_phi_over_r_sq = 0.0;
if (coherent_state_switch != 0) {
phi_phi = fields->phi[i] * fields->phi[i];
chi_chi = fields->D_phi[i] * fields->D_phi[i];
pi_pi = fields->K_phi[i] * fields->K_phi[i];
chi_pi = fields->D_phi[i] * fields->K_phi[i];
del_theta_phi_del_theta_phi_over_r_sq = 0.0;
}
//note that these modes are actually modes of phi, where Phi = r^l phi
//Phi = r^l phi
//Pi = lr^{l-1} dr/du u + r^l pi
//Psi = lr^{l-1} dr/dv u + r^l psi
//#pragma omp parallel for
for (int k = 0; k < number_of_k_modes; ++k) {
for (int l = 0; l < number_of_l_modes; ++l) {
for (int which_q_field = 0; which_q_field < number_of_q_fields; ++which_q_field) {
l_value = l_start + l * l_step;
r = fields->r[i];
r_l = pow(r, l_value);
rprime = fields->D_r[i];
rdot = fields->K_r[i];
/* PHI MODE */
Phi_mode = r_l * (q_fields[which_q_field]->phi[k][l][i]);
if (i == n) {
Phi_mode_plus = pow(fields->r[i + 1], l_value) * (q_fields[which_q_field]->phi[k][l][i + 1]);
}
/* CHI MODE */
if (l_value == 0) {
Chi_mode = q_fields[which_q_field]->D_phi[k][l][i];
}
else if (l_value == 1) {
Chi_mode = rprime * q_fields[which_q_field]->phi[k][l][i] + r * q_fields[which_q_field]->D_phi[k][l][i];
}
else {
Chi_mode = l_value * pow(r, l_value - 1) * rprime * q_fields[which_q_field]->phi[k][l][i] + r_l * (q_fields[which_q_field]->D_phi[k][l][i]);
}
/* PI MODE */
if (l_value == 0) {
Pi_mode = q_fields[which_q_field]->K_phi[k][l][i];
}
else if (l_value == 1) {
Pi_mode = rdot * q_fields[which_q_field]->phi[k][l][i] + r * q_fields[which_q_field]->K_phi[k][l][i];
}
else {
Pi_mode = l_value * pow(r, l_value - 1) * rdot * q_fields[which_q_field]->phi[k][l][i] + r_l * (q_fields[which_q_field]->K_phi[k][l][i]);
}
/* ACTUAL BILINEARS */
phi_phi = phi_phi + ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * norm(Phi_mode); // instead of norm
chi_chi = chi_chi + ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * norm(Chi_mode);
pi_pi = pi_pi + ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * norm(Pi_mode);
chi_pi = chi_pi + ghost_or_physical[which_q_field] * dk / (4.0 * PI) * (2.0 * l_value + 1.0) * 0.5 * ((__real__(Pi_mode * conj(Chi_mode))) + (__real__(Chi_mode * conj(Pi_mode))));
if (i != n) {
del_theta_phi_del_theta_phi_over_r_sq = del_theta_phi_del_theta_phi_over_r_sq + ghost_or_physical[which_q_field] * dk / (4.0 * PI) * 0.5 * l_value * (l_value + 1.0) * (2.0 * l_value + 1.0) * norm(Phi_mode) / (r * r);
}
else {//use the data at r=dr to estimate the r=0 case. This is only relevant for l=1
del_theta_phi_del_theta_phi_over_r_sq = del_theta_phi_del_theta_phi_over_r_sq + ghost_or_physical[which_q_field] * dk / (4.0 * PI) * 0.5 * l_value * (l_value + 1.0) * (2.0 * l_value + 1.0) * norm(Phi_mode_plus) / (fields->r[i + 1] * fields->r[i + 1]);
}
}
}
}
return del_theta_phi_del_theta_phi_over_r_sq; //-chi_pi * 2.0;//
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Calculates initial configuration of the fields that have to be integrated in the v direction */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void v_evolution_initial(double cosm_const, Fields* fields, Q_Fields** q_fields, Bi_Linears* bi_linears) {
double dvv = dv;
// integrate out first order radial equations (sigma, r, D_r, phi, K_r, K_phi)
for (int i = 0; i < v_size; ++i) {
bi_linears->phi_phi[i] = fields->phi[i] * fields->phi[i];
bi_linears->Dphi_Dphi[i] = fields->D_phi[i] * fields->D_phi[i];
bi_linears->Kphi_Kphi[i] = fields->K_phi[i] * fields->K_phi[i];
bi_linears->Dphi_Kphi[i] = fields->D_phi[i] * fields->K_phi[i];
bi_linears->del_theta_phi_del_theta_phi[i] = 0.0;
///////////////////////////////////////////////////////////////////
find_r_D_r(i, 0, dvv, fields, bi_linears);
find_K_r(i, 0, dvv, fields);
fields->lambda[i] = fields->D_r[i] * fields->K_r[i] + 1.0 / 4.0 * exp(fields->sigma[i]) * (1.0 - bi_linears->del_theta_phi_del_theta_phi[i]);
for (int k = 0; k < number_of_k_modes; ++k) {
for (int l = 0; l < number_of_l_modes; ++l) {
for (int which_q_field = 0; which_q_field < number_of_q_fields; ++which_q_field) {
q_fields[which_q_field]->mu[k][l][i] = fields->D_r[i] * q_fields[which_q_field]->K_phi[k][l][i] + fields->K_r[i] * q_fields[which_q_field]->D_phi[k][l][i];
}
}
}
}
}
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Function that calculates the fields that have to be integrated in the v direction at each time (u) step */
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
void v_evolution(double cosm_const, int n, int n_halved, Fields* fields, Q_Fields** q_fields, Q_Fields** q_fields_old, Q_Fields** q_fields_old_old, Bi_Linears* bi_linears) {
double dvv = dv / pow(2.0, n_halved);
// integrate out first order radial equations (sigma, r, D_r, phi, K_r, K_phi)
set_initial_conds_v0 (n, fields);
set_initial_conds_v0_quantum(n, q_fields, q_fields_old, q_fields_old_old);
for (int i = n; i < v_size; ++i) {
//////////////////////////////////////////////////////////////////////
set_bi_linears(cosm_const, i, n, dvv, bi_linears, fields, q_fields);
//////////////////////////////////////////////////////////////////////
find_sigma(i, n, dvv, fields);
find_phi (i, n, dvv, fields);
find_q_phi(i, n, dvv, q_fields, q_fields_old);
find_r_D_r(i, n, dvv, fields, bi_linears);