cosmotool/src/powerSpectrum.cpp

#include <vector>
#include <algorithm>
#include <iostream>
#include <cmath>
#include <fstream>
#include <gsl/gsl_integration.h>
#include "powerSpectrum.hpp"

using namespace std;

#define USE_GSL
#define TOLERANCE 1e-6
#define NUM_ITERATION 8000

#define POWER_EFSTATHIOU 1
#define HU_WIGGLES 2
#define HU_BARYON 3
#define OLD_POWERSPECTRUM 4
#define POWER_BARDEEN 5
#define POWER_SUGIYAMA 6
#define POWER_BDM 7
#define POWER_TEST 8

#define POWER_SPECTRUM POWER_EFSTATHIOU

namespace Cosmology {

double n = 1.0;
double K0 = 1;
double V0 = 627;

  double CMB_VECTOR[3] = {
    56.759,
    -540.02,
    313.50
  };


// WMAP5
double h = 0.719;
double SIGMA8 = 0.77;
double OMEGA_B = 0.043969;
double OMEGA_C = 0.21259;


// WMAP5-modification
//double h = 0.719;
//double SIGMA8 = 0.77;
//double OMEGA_B = 0;
//double OMEGA_C = 0.21259+0.043969;

// LCDM STRAUSS ?
//double h = 0.67;
//double SIGMA8 = 0.67;
//double OMEGA_B = 0;
//double OMEGA_C = 0.30; 

// SCDM STRAUSS
//double h = 0.5;
//double SIGMA8= 1.05;
//double OMEGA_B = 0;
//double OMEGA_C = 1;

// Sugiyama test
//double h = 0.5;
//double SIGMA8= 0.5;//1.05;
//double OMEGA_B = 0.0125*4;
//double OMEGA_C = 0.1-OMEGA_B;

// HU TEST
//double h = 0.5;
//double SIGMA8 = 0.5;
//double OMEGA_B = 0.09;
//double OMEGA_C = 0.21;

// HDM STRAUSS
//double h = 0.5;
//double SIGMA8 = 0.86;
//double OMEGA_B = 0;
//double OMEGA_C = 1;

// FOR "BEST FIT"
//double h = 0.82;
//double SIGMA8 = 0.76;
//double OMEGA_B = 0.043969;
//double OMEGA_C = 0.15259;


// FOR JUSZKIEWICZ CHECKING (CDM) ! WARNING ! He smoothes 
// with a gaussian filter the density field, i.e. one has
// to multiply P(k) by exp(-k^2 R^2) with R the radius
// of the filter. Dammit !
//double h = 0.5;
//double SIGMA8=1/2.5;
//double OMEGA_B=0.;
//double OMEGA_C=1;
//#define JUSZKIEWICZ_PATCH
//#define RJUSZ 6.0

// (BDM)
//double h = 0.5;
//double SIGMA8=1;
//double OMEGA_B=0.0;
//double OMEGA_C=0.4;


// FOR HU CHECKING
//double h = 0.5;
//double SIGMA8= 1;
//double OMEGA_B=0.09;
//double OMEGA_C=0.21;


double OMEGA_0 = OMEGA_B+OMEGA_C;
double Omega = OMEGA_0;
double Theta_27 = 2.728 / 2.7;
double beta = pow(OMEGA_0, 5./9);
double OmegaEff = OMEGA_0 * h * h;
double Gamma0 = OMEGA_0 * h * h;


/*
 * This is \hat{tophat}
 */
double tophatFilter(double u)
{
  if (u != 0)    
    return 3 / (u*u*u) * (sin(u) - u * cos(u));
  else
    return 1;
}

/*
 * This is \tilde{w}
 */
double externalFilter(double u)
{
  if (u != 0)
    return 1 - sin(u)/u;
  return 0.;
}

double powC(double q, double alpha_c)
{
  return 14.2 / alpha_c + 386 / (1 + 69.9 * pow(q, 1.08));
}

double T_tilde_0(double q, double alpha_c, double beta_c)
{
  double a = log(M_E + 1.8 * beta_c * q);
  return a / ( a  + powC(q, alpha_c) * q * q);
}

double j_0(double x)
{
  if (x == 0)
    return 1.0;
  return sin(x)/x;
}

double powG(double y)
{
  double a = sqrt(1 + y);

  return y * (-6 * a + (2 + 3 * y) *log((a + 1)/(a - 1)));
}

/*
 * This function returns the power spectrum evaluated at k (in Mpc, not in Mpc/h).
 */
double powerSpectrum(double k, double normPower)
{
#if POWER_SPECTRUM == POWER_EFSTATHIOU
  double a = 6.4/Gamma0;
  double b = 3/Gamma0;
  double c = 1.7/Gamma0;
  double nu = 1.13;
  
  double f = (a*k) + pow(b*k,1.5) + pow(c*k,2);

  // EFSTATHIOU  ET AL. (1992)
  return normPower * pow(k,n) * pow(1+pow(f,nu),(-2/nu));
#endif

  // EISENSTEIN ET HU (1998)
  // FULL POWER SPECTRUM WITH BARYONS AND WIGGLES

#if POWER_SPECTRUM == HU_WIGGLES
  // EISENSTEIN ET HU (1998)
  // FULL POWER SPECTRUM WITH BARYONS AND WIGGLES

  double k_silk = 1.6 * pow(OMEGA_B * h * h, 0.52) * pow(OmegaEff, 0.73) * (1 + pow(10.4 * OmegaEff, -0.95));
  double z_eq = 2.50e4 * OmegaEff * pow(Theta_27, -4);
  double s = 44.5 * log(9.83 / OmegaEff) / (sqrt(1 + 10 * pow(OMEGA_B * h * h, 0.75)));
  double f = 1 / (1 + pow(k * s / 5.4, 4));
  double k_eq = 7.46e-2 * OmegaEff * pow(Theta_27, -2);
  double a1 = pow(46.9 * OmegaEff, 0.670) * (1 + pow(32.1 * OmegaEff, -0.532));
  double a2 = pow(12.0 * OmegaEff, 0.424) * (1 + pow(45.0 * OmegaEff, -0.582));
  double alpha_c = pow(a1, -OMEGA_B/ OMEGA_0) * pow(a2, -pow(OMEGA_B / OMEGA_0, 3));  

  double q = k / (13.41 * k_eq);
  double b1_betac = 0.944 * 1/(1 + pow(458 * OmegaEff, -0.708));
  double b2_betac = pow(0.395 * OmegaEff, -0.0266);
  double beta_c = 1/ ( 1 + b1_betac * (pow(OMEGA_C / OMEGA_0, b2_betac) - 1)   );
  double T_c = f * T_tilde_0(q, 1, beta_c) + (1 - f) * T_tilde_0(q, alpha_c, beta_c);

  double b1_zd = 0.313 * pow(OmegaEff, -0.419) * (1 + 0.607 * pow(OmegaEff, 0.674));
  double b2_zd = 0.238 * pow(OmegaEff, 0.223);
  double z_d = 1291 * pow(OmegaEff, 0.251) / (1 + 0.659 * pow(OmegaEff, 0.828)) * (1 + b1_zd * pow(OmegaEff, b2_zd));
  double R_d = 31.5 * OMEGA_B * h * h * pow(Theta_27, -4) * 1e3 / z_d;

  double alpha_b = 2.07 * k_eq * s * pow(1 + R_d, -0.75) * powG((1 + z_eq)/(1 + z_d));
  double beta_b = 0.5 + OMEGA_B / OMEGA_0 + (3 - 2 * OMEGA_B / OMEGA_0) * sqrt(pow(17.2 * OmegaEff, 2) + 1);
  double beta_node = 8.41 * pow(OmegaEff, 0.435);
  double s_tilde = s * pow(1 + pow(beta_node / (k * s), 3), -1./3);

  double T_b = (T_tilde_0(q, 1, 1) / (1 + pow(k * s / 5.2, 2)) + alpha_b / (1 + pow(beta_b / (k * s), 3)) * exp(-pow(k/k_silk, 1.4))) * j_0(k * s_tilde);  

  double T_k = OMEGA_B/OMEGA_0 * T_b + OMEGA_C/OMEGA_0 * T_c;  

  return normPower * pow(k,n) * T_k * T_k;
#endif

  // EISENSTEIN ET AL. (2008), SHAPED POWER SPECTRUM WITH BARYON, WITHOUT WIGGLES
#if POWER_SPECTRUM == HU_BARYON
  double s = 44.5 * log(9.83 / OmegaEff) / (sqrt(1 + 10 * pow(OMEGA_B * h * h, 0.75)));
  double alpha_Gamma = 1 - 0.328 * log(431 * OmegaEff) * OMEGA_B / OMEGA_0 + 0.38 * log(22.3 * OmegaEff) * pow(OMEGA_B / OMEGA_0, 2);
  double GammaEff = OMEGA_0 * h * (alpha_Gamma + (1 - alpha_Gamma)/(1 + pow(0.43 * k * s, 4)));
  double q = k/(h*GammaEff) * pow(Theta_27, 2);
  double L_0 = log(2 * M_E + 1.8 * q);
  double C_0 = 14.2 + 731 / (1 + 62.5 * q);
  double T0 = L_0 / (L_0 + C_0 * q * q);
  
  return normPower * pow(k,n) * T0 * T0;
#endif

#if POWER_SPECTRUM == OLD_POWERSPECTRUM
  // OLD FUNCTION: 
  static const double l = 1/(Omega * h*h);
  static const double alpha = 1.7 * l, beta = 9.0 * pow(l, 1.5), gamma = l*l;
  return normPower * pow(k,n) * pow(1 + alpha * k +  beta * pow(k,1.5) + gamma *k*k,-2);
#endif

#if POWER_SPECTRUM == POWER_SUGIYAMA
  double q = k * Theta_27*Theta_27 / (OmegaEff * exp(-OMEGA_B - sqrt(h/0.5)*OMEGA_B/OMEGA_0));
  double L0 = log(2*M_E + 1.8 * q);
  double C0 = 14.2 + 731 / (1 + 62.5 * q);
  double T_k = L0 / (L0 + C0 * q*q);

      //  double poly = 1 + 3.89 * q + pow(16.1*q,2) + pow(5.46*q,3) + pow(6.71*q,4);
      //  double T_k = log(1+2.34*q)/(2.34*q) * pow(poly,-0.25);

  return normPower * pow(k,n) * T_k * T_k;

#endif

#if POWER_SPECTRUM == POWER_BARDEEN
  double q = k / (OmegaEff);

  double poly = 1 + 3.89 * q + pow(16.1*q,2) + pow(5.46*q,3) + pow(6.71*q,4);
  double T_k = log(1+2.34*q)/(2.34*q) * pow(poly,-0.25);

  return normPower * pow(k,n) * T_k * T_k;

#endif

#if POWER_SPECTRUM == POWER_BDM

  k /= h*h;
  double alpha1 = 190;
  double Gmu = 4.6;
  double alpha2 = 11.5;
  double alpha3 = 11;
  double alpha4 = 12.55;
  double alpha5 = 0.0004;
  return normPower*k*alpha1*alpha1*Gmu*Gmu/(1+(alpha2*k)+pow(alpha3*k,2)+pow(alpha4*k,3))*pow(1+pow(alpha5/k,2), -2);
#endif

#if POWER_SPECTRUM == POWER_TEST
  return 1/(1+k*k);
#endif
}

/*
 * This function computes the normalization of the power spectrum. It requests
 * a sigma8 (density fluctuations within 8 Mpc/h)
 */
double gslPowSpecNorm(double k, void *params)
{
  double f = tophatFilter(k*8.0/h);
  return powerSpectrum(k, 1.0)*k*k*f*f;
}

double computePowSpecNorm(double sigma8)
{
  int Nsteps = 30000;
  double normVal = 0;

#ifndef USE_GSL
  for (int i = 1; i <= Nsteps; i++)
    {
      double t = i * 1.0/(Nsteps+1);
      // Change of variable !
      double k = (1-t)/t * K0;

      // The filter
      double filter_val = tophatFilter(k*8.0/h);
      // The powerspectrum
      double powVal = powerSpectrum(k, 1.0);
      // Multiply by the tophat filter
      powVal *= filter_val*filter_val;
     
      powVal *= k*k;
      
      // Account for change of variable
      powVal /= (t*t);
  
      // Integrate !
      normVal += powVal;
    }

  normVal /= 2*M_PI*M_PI;

  // The dt element
  normVal *= 1.0/(Nsteps+1) * K0;
#else
  double abserr;
  gsl_integration_workspace *w = gsl_integration_workspace_alloc(NUM_ITERATION);
  gsl_function f;

  f.function = gslPowSpecNorm;
  f.params = 0;

  gsl_integration_qagiu(&f, 0, 0, TOLERANCE, NUM_ITERATION, w, &normVal, &abserr);

  gsl_integration_workspace_free(w);

  normVal /= (2*M_PI*M_PI);
#endif

  return sigma8*sigma8/normVal;
}

/*
 * This function computes the variance of the Local Group velocity components
 * for a survey which depth is topHatRad1 (in Mpc/h). This variance should
 * be multiplied by (H \beta)^2 to be equal to a velocity^2.
 */
double gslVariance(double k, void *params)
{
  double R1 = *(double *)params;
  double f = externalFilter(k * R1 / h);

  double a = f*f;

#ifdef JUSZKIEWICZ_PATCH
  a *= exp(-k*k*(RJUSZ*RJUSZ/(h*h)));
#endif
  a *= powerSpectrum(k, 1.0);

  return a;
}

double computeVariance(double powNorm,  double topHatRad1)
{
  int Nsteps = 100000;
  double varVal = 0;

#ifndef USE_GSL
  
  for (int i = 1; i <= Nsteps; i++)
    {
      double t = i * 1.0/(Nsteps+1);
      // Change of variable !
      double k = (1-t)/t * K0;
      double powVal = powerSpectrum(k, powNorm);

      double filter1Val = externalFilter(k*topHatRad1/h);

#ifdef JUSZKIEWICZ_PATCH
      powVal *= exp(-k*k*(RJUSZ*RJUSZ/(h*h)));
#endif
      powVal *= filter1Val*filter1Val;
      
      powVal /= (t*t);

      varVal += powVal;
    }

  varVal *= 1.0/(Nsteps) * K0;
  varVal *= 1.0/(6*M_PI*M_PI);

#else

  double abserr;
  gsl_integration_workspace *w = gsl_integration_workspace_alloc(NUM_ITERATION);
  gsl_function f;
  
  f.function = gslVariance;
  f.params = &topHatRad1;

  gsl_integration_qagiu(&f, 0, 0, TOLERANCE, NUM_ITERATION, w, &varVal, &abserr);

  gsl_integration_workspace_free(w);

  varVal *= powNorm/(6*M_PI*M_PI);

#endif

  return varVal;
}

/*
 * This function computes the same quantity as computeVariance but
 * for a survey infinitely deep.
 */
double gslVarianceZero(double k, void *params)
{
  double a = 1.0;

#ifdef JUSZKIEWICZ_PATCH
  a *= exp(-k*k*(RJUSZ*RJUSZ/(h*h)));
#endif
  a *= powerSpectrum(k, 1.0);

  return a;
}

double computeVarianceZero(double powNorm)
{
  int Nsteps = 100000;
  double varVal = 0;
  
#ifndef USE_GSL

  for (int i = 1; i <= Nsteps; i++)
    {
      double t = i * 1.0/(Nsteps+1);
      // Change of variable !
      double k = (1-t)/t * K0;
      double powVal = powerSpectrum(k, powNorm);
      
#ifdef JUSZKIEWICZ_PATCH
      powVal *= exp(-k*k*(RJUSZ*RJUSZ/h*h));
#endif

      powVal /= (t*t);

      varVal += powVal;
    }

  varVal *= 1.0/(Nsteps+1) * K0;
  varVal *= 1.0/(6*M_PI*M_PI);

#else
 
  double abserr;
  gsl_integration_workspace *w = gsl_integration_workspace_alloc(NUM_ITERATION);
  gsl_function f;
  
  f.function = gslVarianceZero;
  f.params = 0;

  gsl_integration_qagiu(&f, 0, 0, TOLERANCE, NUM_ITERATION, w, &varVal, &abserr);

  gsl_integration_workspace_free(w);

  varVal *= powNorm/(6*M_PI*M_PI);
#endif

  return varVal;
}

/*
 * This function computes the correlation between the infinitely deep
 * velocity of the Local Group and the one estimated from a survey
 * which depth is topHatRad1.
 * This corresponds to \gamma.
 * This quantity must be multiplied by H \beta to be equal to a velocity^2.
 */
double gslCorrel(double k, void *params)
{
  double R1 = ((double *)params)[0];
  double a = externalFilter(k * R1 / h);// * externalFilter(k * R2 / h);

#ifdef JUSZKIEWICZ_PATCH
  a *= exp(-k*k*(RJUSZ*RJUSZ/(h*h)));
#endif
  a *= powerSpectrum(k, 1.0);

  return a;
}

double gslCorrelBis(double t, void *params)
{
  double k = (1-t)/t;
  double v = gslCorrel(k, params);

  return v/(t*t);
}

double gslCorrel2(double k, void *params)
{
  double R1 = ((double *)params)[0];
  double R2 = ((double *)params)[1];
  double a = externalFilter(k * R1 / h) * externalFilter(k * R2 / h);        
  
#ifdef JUSZKIEWICZ_PATCH
  a *= exp(-k*k*(RJUSZ*RJUSZ/(h*h)));
#endif
  a *= powerSpectrum(k, 1.0) ;
  
  return a;
}

double gslCorrel2bis(double t, void *params)
{
  double k = (1-t)/t;
  double v = gslCorrel2(k, params);

  return v/(t*t);
}


double computeCorrel(double powNorm,  double topHatRad1)
{
  int Nsteps = 100000;
  double varVal = 0;
  
#ifndef USE_GSL

  for (int i = 1; i <= Nsteps; i++)
    {
      double t = i * 1.0/(Nsteps+1);
      // Change of variable !
      double k = (1-t)/t * K0;
      double powVal = powerSpectrum(k, powNorm);

      double filter1Val = externalFilter(k*topHatRad1/h);

#ifdef JUSZKIEWICZ_PATCH
      powVal*=exp(-k*k*(RJUSZ*RJUSZ/h*h));
#endif

      powVal *= filter1Val;
      
      powVal /= (t*t);

      varVal += powVal;
    }

  varVal *= 1.0/(Nsteps) * K0;
  varVal *= 1.0/(6*M_PI*M_PI);
#else

  double abserr;
  gsl_integration_workspace *w = gsl_integration_workspace_alloc(NUM_ITERATION);
  gsl_function f;
  
  f.params = &topHatRad1;

#if 1
  f.function = gslCorrelBis;
  gsl_integration_qag (&f, 0, 1, 0, TOLERANCE, NUM_ITERATION, GSL_INTEG_GAUSS61, w, &varVal, &abserr);
#else
  f.function = gslCorrel;
  gsl_integration_qagiu(&f, 0, 0, TOLERANCE, NUM_ITERATION, w, &varVal, &abserr);
#endif

  gsl_integration_workspace_free(w);

  varVal *= powNorm/(6*M_PI*M_PI);

#endif
  return varVal;
}

/*
 * This function computes the correlation between the infinitely deep
 * velocity of the Local Group and the one estimated from a survey
 * which depth is topHatRad1.
 * This corresponds to \gamma.
 * This quantity must be multiplied by H \beta to be equal to a velocity^2.
 */
double computeCorrel2(double powNorm,  double topHatRad1, double topHatRad2)
{
  int Nsteps = 100000;
  double varVal = 0;
#ifndef USE_GSL
  
  for (int i = 1; i <= Nsteps; i++)
    {
      double t = i * 1.0/(Nsteps+1);
      // Change of variable !
      double k = (1-t)/t * K0;
      double powVal = powerSpectrum(k, powNorm);

      double filter1Val = externalFilter(k*topHatRad1/h);
      double filter2Val = externalFilter(k*topHatRad2/h);

      powVal *= filter1Val * filter2Val;
      
      powVal /= (t*t);

      varVal += powVal;
    }

  varVal *= 1.0/(Nsteps) * K0;
  varVal *= 1.0/(6*M_PI*M_PI);

#else

  double abserr;
  gsl_integration_workspace *w = gsl_integration_workspace_alloc(NUM_ITERATION);
  gsl_function f;
  double rads[] = {topHatRad1, topHatRad2 };

  f.params = rads;

#if 1  
  f.function = gslCorrel2bis;
  gsl_integration_qag (&f, 0, 1, 0, TOLERANCE, NUM_ITERATION, GSL_INTEG_GAUSS61, w, &varVal, &abserr);
#else
  f.function = gslCorrel2;
  gsl_integration_qagiu(&f, 0, 0, TOLERANCE, NUM_ITERATION, w, &varVal, &abserr);
#endif

  gsl_integration_workspace_free(w);

  varVal *= powNorm/(6*M_PI*M_PI);

#endif

  return varVal;
}


  void updateCosmology()
  {
    OMEGA_0 = OMEGA_B+OMEGA_C;
    Omega = OMEGA_0;
    Theta_27 = 2.728 / 2.7;
    beta = pow(OMEGA_0, 5./9);
    OmegaEff = OMEGA_0 * h * h;
    Gamma0 = OMEGA_0 * h * h;

#if 0
    cout << "Cosmology is :" << endl
	 << "  O0=" << OMEGA_0 << " Theta=" << Theta_27 << " beta=" << beta << " h=" << h << " G0=" << Gamma0 << endl
	 << "  OmegaB=" << OMEGA_B << " Omega_C=" << OMEGA_C << endl;
#endif
  }

};