double rec_HMLA_dxedlna(double xe, double nH, double Hubble, double TM, double TR, double energy_rate, HRATEEFF *rate_table){
double Alpha[2];
double Beta[2];
double R2p2s;
double matrix[2][2];
double RHS[2];
double det, RLya;
double x2[2];
double x1s_db;
double C_2p;
double chi_ion_H;
interpolate_rates(Alpha, Beta, &R2p2s, TR, TM / TR, rate_table);
x1s_db = (1.-xe)*exp(-E21/TR);
/******** 2s *********/
matrix[0][0] = Beta[0] + 3.*R2p2s + L2s1s;
matrix[0][1] = -R2p2s;
RHS[0] = xe*xe*nH *Alpha[0] + L2s1s *x1s_db;
/******** 2p *********/
RLya = 4.662899067555897e15 *Hubble /nH/(1.-xe); /*8 PI H/(3 nH x1s lambda_Lya^3) */
matrix[1][1] = Beta[1] + R2p2s + RLya;
matrix[1][0] = -3.*R2p2s;
RHS[1] = xe*xe*nH *Alpha[1] + 3.*RLya *x1s_db;
/**** invert the 2 by 2 system matrix_ij * xj = RHSi *****/
det = matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0];
x2[0] = (matrix[1][1] * RHS[0] - matrix[0][1] * RHS[1])/det;
x2[1] = (matrix[0][0] * RHS[1] - matrix[1][0] * RHS[0])/det;
C_2p=(RLya+R2p2s*L2s1s/matrix[0][0])/(matrix[1][1]-R2p2s*3.*R2p2s/matrix[0][0]);
//chi_ion_H = (1.-xe)/3.; // old approximation from Chen and Kamionkowski
if (xe < 1.)
chi_ion_H = 0.369202*pow(1.-pow(xe,463929),1.70237); // coefficient as revised by Galli et al. 2013 (in fact it is a fit by Vivian Poulin of columns 1 and 4 in Table V of Galli et al. 2013)
else
chi_ion_H = 0.;
return (x1s_db*(L2s1s + 3.*RLya) -x2[0]*L2s1s -x2[1]*RLya)/Hubble
+chi_ion_H/nH*energy_rate*(1./EI+(1.-C_2p)/E21)/Hubble;
}
double rec_HMLA_dxedlna(double xe, double nH, double Hubble, double TM, double TR, double energy_rate, HRATEEFF *rate_table){
double Alpha[2];
double Beta[2];
double R2p2s;
double matrix[2][2];
double RHS[2];
double det, RLya;
double x2[2];
double x1s_db;
double C_2p;
interpolate_rates(Alpha, Beta, &R2p2s, TR, TM / TR, rate_table);
x1s_db = (1.-xe)*exp(-E21/TR);
/******** 2s *********/
matrix[0][0] = Beta[0] + 3.*R2p2s + L2s1s;
matrix[0][1] = -R2p2s;
RHS[0] = xe*xe*nH *Alpha[0] + L2s1s *x1s_db;
/******** 2p *********/
RLya = 4.662899067555897e15 *Hubble /nH/(1.-xe); /*8 PI H/(3 nH x1s lambda_Lya^3) */
matrix[1][1] = Beta[1] + R2p2s + RLya;
matrix[1][0] = -3.*R2p2s;
RHS[1] = xe*xe*nH *Alpha[1] + 3.*RLya *x1s_db;
/**** invert the 2 by 2 system matrix_ij * xj = RHSi *****/
det = matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0];
x2[0] = (matrix[1][1] * RHS[0] - matrix[0][1] * RHS[1])/det;
x2[1] = (matrix[0][0] * RHS[1] - matrix[1][0] * RHS[0])/det;
C_2p=(RLya+R2p2s*L2s1s/matrix[0][0])/(matrix[1][1]-R2p2s*3.*R2p2s/matrix[0][0]);
return (x1s_db*(L2s1s + 3.*RLya) -x2[0]*L2s1s -x2[1]*RLya)/Hubble
+(1.-xe)/(3.*nH)*energy_rate*(1./EI+(1.-C_2p)/E21)/Hubble;
}
double rec_HMLA_2photon_dxedlna(double xe, double nH, double H, double TM, double TR,
HRATEEFF *rate_table, TWO_PHOTON_PARAMS *twog,
double zstart, double dlna, double **logfminus_hist, double *logfminus_Ly_hist[],
unsigned iz, double z, double energy_rate){
double xr[2];
double xv[NVIRT];
double xedot, Pib, feq;
double fplus[NVIRT], fplus_Ly[3];
unsigned b, i;
double Trr[2][2];
double matrix[2][2];
double *Trv[2];
double *Tvr[2];
double *Tvv[3];
double sr[2];
double sv[NVIRT];
double Dtau[NVIRT];
double Alpha[2], Beta[2];
double RLya;
double R2p2s;
double C_2p;
for (i = 0; i < 2; i++) Trv[i] = create_1D_array(NVIRT);
for (i = 0; i < 2; i++) Tvr[i] = create_1D_array(NVIRT);
for (i = 0; i < 3; i++) Tvv[i] = create_1D_array(NVIRT);
/* Redshift photon occupation number from previous times and higher energy bins */
fplus_from_fminus(fplus, fplus_Ly, logfminus_hist, logfminus_Ly_hist, TR,
zstart, dlna, iz, z, twog->Eb_tab);
/* Compute real-real, real-virtual and virtual-virtual transition rates */
populateTS_2photon(Trr, Trv, Tvr, Tvv, sr, sv, Dtau, xe, TM, TR, nH, H, rate_table,
twog, fplus, fplus_Ly, Alpha, Beta, z);
/* Solve for the population of the real and virtual states */
solve_real_virt(xr, xv, Trr, Trv, Tvr, Tvv, sr, sv);
/*************************************************************/
/* Dark matter annihilation*/
RLya = 4.662899067555897e15 *H /nH/(1.-xe); /*8 PI H/(3 nH x1s lambda_Lya^3) */
interpolate_rates(Alpha, Beta, &R2p2s, TR, TM / TR, rate_table);
matrix[0][0] = Beta[0] + 3.*R2p2s + L2s1s;
matrix[1][1] = Beta[1] + R2p2s + RLya;
C_2p=(RLya+R2p2s*L2s1s/matrix[0][0])/(matrix[1][1]-R2p2s*3.*R2p2s/matrix[0][0]);
/*************************************************************/
/* Obtain xe_dot */
xedot = -nH*xe*xe*(Alpha[0]+Alpha[1]) + xr[0]*Beta[0] + xr[1]*Beta[1]
+(1.-xe)/(3.*nH)*energy_rate*(1./EI+(1.-C_2p)/E21);
/* Update fminuses */
for (b = 0; b < NVIRT; b++) {
if (Dtau[b] != 0) {
Pib = (1.-exp(-Dtau[b]))/Dtau[b];
feq = -xr[0]*Tvr[0][b] - xr[1]*Tvr[1][b];
feq -= (b == 0 ? xv[1]*Tvv[2][0]:
b == NVIRT-1 ? xv[NVIRT-2]*Tvv[1][NVIRT-1]:
xv[b+1]*Tvv[2][b] + xv[b-1]*Tvv[1][b]);
feq /= (1.-xe)*(1.-Pib)*Tvv[0][b];
logfminus_hist[b][iz] = log(fplus[b] + (feq - fplus[b])*(1.-exp(-Dtau[b])));
}
else logfminus_hist[b][iz] = log(fplus[b]);
}
/* log(xnp/(3 x1s)) , n = 2, 3, 4, assuming 3p and 4p in Boltzmann eq. with 2s */
logfminus_Ly_hist[0][iz] = log(xr[1]/3./(1.-xe));
logfminus_Ly_hist[1][iz] = log(xr[0]/(1.-xe)) - E32/TR;
logfminus_Ly_hist[2][iz] = log(xr[0]/(1.-xe)) - E42/TR;
for (i = 0; i < 2; i++) free(Trv[i]);
for (i = 0; i < 2; i++) free(Tvr[i]);
for (i = 0; i < 3; i++) free(Tvv[i]);
return xedot/H;
}
void populateTS_2photon(double Trr[2][2], double *Trv[2], double *Tvr[2], double *Tvv[3],
double sr[2], double sv[NVIRT], double Dtau[NVIRT],
double xe, double TM, double TR, double nH, double H, HRATEEFF *rate_table,
TWO_PHOTON_PARAMS *twog, double fplus[NVIRT], double fplus_Ly[],
double Alpha[2], double Beta[2], double z) {
double R2p2s;
unsigned b;
double RLya, Gammab, Pib, dbfact;
double *Aup, *Adn;
double A2p_up, A2p_dn;
Aup = create_1D_array(NVIRT);
Adn = create_1D_array(NVIRT);
RLya = 4.662899067555897e15 *H /nH/(1.-xe); /*8 PI H/(3 nH x1s lambda_Lya^3) */
interpolate_rates(Alpha, Beta, &R2p2s, TR, TM / TR, rate_table);
/****** 2s row and column ******/
Trr[0][0] = Beta[0] + 3.*R2p2s
+ 3.* RLya * (1.664786871919931 *exp(-E32/TR) /* Ly-beta escape */
+ 1.953125 *exp(-E42/TR)); /* Ly-gamma escape */
Trr[0][1] = -R2p2s;
sr[0] = nH * Alpha[0] * xe*xe
+ 3.* RLya * (1.-xe) * (1.664786871919931 *fplus_Ly[1]
+ 1.953125 *exp(-E41/TR));
#if (EFFECT_A == 0)
/* Standard treatment of 2s-->1s two-photon decays */
Trr[0][0] += L2s1s;
sr[0] += L2s1s * (1.-xe) * exp(-E21/TR);
#endif
/****** 2p row and column ******/
Trr[1][1] = Beta[1] + R2p2s + RLya;
Trr[1][0] = -3.*R2p2s;
sr[1] = nH * Alpha[1] * xe*xe + 3.*RLya * (1.-xe) * fplus_Ly[0];
/***** Two-photon transitions: populating Trv, Tvr and updating Trr ******/
for (b = 0; b < NVIRT; b++) {
dbfact = exp((twog->Eb_tab[b] - E21)/TR);
Trr[0][0] -= Tvr[0][b] = -twog->A2s_tab[b]/fabs(exp((twog->Eb_tab[b] - E21)/TR)-1.);
Trv[0][b] = Tvr[0][b] *dbfact;
Trr[1][1] -= Tvr[1][b] = -exp(-E32/TR)/3. * twog->A3s3d_tab[b]/fabs(exp((twog->Eb_tab[b] - E31)/TR)-1.)
-exp(-E42/TR)/3. * twog->A4s4d_tab[b]/fabs(exp((twog->Eb_tab[b] - E41)/TR)-1.);
Trv[1][b] = Tvr[1][b] *3.*dbfact;
}
/****** Tvv and sv. Accounting for DIFFUSION ******/
populate_Diffusion(Aup, Adn, &A2p_up, &A2p_dn, TM, twog->Eb_tab, twog->A1s_tab);
/* Updating Tvr, Trv, Trr for diffusion between line center ("2p state") and two neighboring bins */
Trr[1][1] += (A2p_dn + A2p_up);
for (b = 0; b < NVIRT; b++) {
Gammab = -(Trv[0][b] + Trv[1][b]) + Aup[b] + Adn[b]; /* Inverse lifetime of virtual state b */
/*** Diffusion region ***/
if ( (b >= NSUBLYA - NDIFF/2 && b < NSUBLYA - 1)
||(b > NSUBLYA && b < NSUBLYA + NDIFF/2)) {
Tvv[1][b] = -Aup[b-1];
Tvv[2][b] = -Adn[b+1];
}
/* Bins neighboring Lyman alpha. */
if (b == NSUBLYA-1) {
Tvv[1][b] = -Aup[b-1];
Tvv[2][b] = 0.;
Tvr[1][b] -= A2p_dn;
Trv[1][b] -= Aup[b];
}
if (b == NSUBLYA) {
Tvv[1][b] = 0.;
Tvv[2][b] = -Adn[b+1];
Tvr[1][b] -= A2p_up;
Trv[1][b] -= Adn[b];
}
/*********************/
Dtau[b] = Gammab * (1.-xe) * cube(hPc/twog->Eb_tab[b]) * nH /8. /M_PI /H;
if (Dtau[b] > 1e-30) {
Pib = (1.-exp(-Dtau[b]))/Dtau[b];
Tvv[0][b] = Gammab/(1.-Pib);
sv[b] = Tvv[0][b] * (1.-xe) * fplus[b] * Pib;
}
else { /* Nearly vanishing optical depth: free streaming */
Tvv[0][b] = 1.;
Trv[0][b] = Trv[1][b] = Tvr[0][b] = Tvr[1][b] = 0;
sv[b] = (1.-xe) * fplus[b];
}
}
free(Aup);
free(Adn);
}