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recfast.f90
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recfast.f90
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!Recombination module for CAMB, using RECFAST
!cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
!C Integrator for Cosmic Recombination of Hydrogen and Helium,
!C developed by Douglas Scott (dscott@astro.ubc.ca)
!C based on calculations in the paper Seager, Sasselov & Scott
!C (ApJ, 523, L1, 1999).
!and "fudge" updates in Wong, Moss & Scott (2008).
!C
!C Permission to use, copy, modify and distribute without fee or royalty at
!C any tier, this software and its documentation, for any purpose and without
!C fee or royalty is hereby granted, provided that you agree to comply with
!C the following copyright notice and statements, including the disclaimer,
!C and that the same appear on ALL copies of the software and documentation,
!C including modifications that you make for internal use or for distribution:
!C
!C Copyright 1999-2010 by University of British Columbia. All rights reserved.
!C
!C THIS SOFTWARE IS PROVIDED "AS IS", AND U.B.C. MAKES NO
!C REPRESENTATIONS OR WARRANTIES, EXPRESS OR IMPLIED.
!C BY WAY OF EXAMPLE, BUT NOT LIMITATION,
!c U.B.C. MAKES NO REPRESENTATIONS OR WARRANTIES OF
!C MERCHANTABILITY OR FITNESS FOR ANY PARTICULAR PURPOSE OR THAT
!C THE USE OF THE LICENSED SOFTWARE OR DOCUMENTATION WILL NOT INFRINGE
!C ANY THIRD PARTY PATENTS, COPYRIGHTS, TRADEMARKS OR OTHER RIGHTS.
!C
!cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
!
!CN Name: RECFAST
!CV Version: 1.5.2
!C
!CP Purpose: Calculate ionised fraction as a function of redshift.
!CP Solves for H and He simultaneously, and includes
!CP H "fudge factor" for low z effect, as well as
!CP HeI fudge factor.
!C
!CD Description: Solves for ionisation history since recombination
!CD using the equations in Seager, Sasselov & Scott (ApJ, 1999).
!CD The Cosmological model can be flat or open.
!CD The matter temperature is also followed, with an update from
!CD Scott & Scott (2009).
!CD The values for \alpha_B for H are from Hummer (1994).
!CD The singlet HeI coefficient is a fit from the full code.
!CD Additional He "fudge factors" are as described in Wong, Moss
!CD and Scott (2008).
!CD Extra fitting function included (in optical depth) to account
!CD for extra H physics described in Rubino-Martin et al. (2010).
!CD Care is taken to use the most accurate constants.
!C
!CA Arguments:
!CA Name, Description
!CA real(dl) throughout
!CA
!CA z is redshift - W is sqrt(1+z), like conformal time
!CA x is total ionised fraction, relative to H
!CA x_H is ionized fraction of H - y(1) in R-K routine
!CA x_He is ionized fraction of He - y(2) in R-K routine
!CA (note that x_He=n_He+/n_He here and not n_He+/n_H)
!CA Tmat is matter temperature - y(3) in R-K routine
!CA f's are the derivatives of the Y's
!CA alphaB is case B recombination rate
!CA alpHe is the singlet only HeII recombination rate
!CA a_PPB is Pequignot, Petitjean & Boisson fitting parameter for Hydrogen
!CA b_PPB is Pequignot, Petitjean & Boisson fitting parameter for Hydrogen
!CA c_PPB is Pequignot, Petitjean & Boisson fitting parameter for Hydrogen
!CA d_PPB is Pequignot, Petitjean & Boisson fitting parameter for Hydrogen
!CA a_VF is Verner and Ferland type fitting parameter for Helium
!CA b_VF is Verner and Ferland type fitting parameter for Helium
!CA T_0 is Verner and Ferland type fitting parameter for Helium
!CA T_1 is Verner and Ferland type fitting parameter for Helium
!CA Tnow is the observed CMB temperature today
!CA Yp is the primordial helium abundace
!CA fHe is He/H number ratio = Yp/4(1-Yp)
!CA Trad and Tmat are radiation and matter temperatures
!CA epsilon is the approximate difference (=Trad-Tmat) at high z
!CA OmegaB is Omega in baryons today
!CA H is Hubble constant in units of 100 km/s/Mpc
!CA HO is Hubble constant in SI units
!CA bigH is 100 km/s/Mpc in SI units
!CA Hz is the value of H at the specific z (in ION)
!CA G is grvitational constant
!CA n is number density of hydrogen
!CA Nnow is number density today
!CA x0 is initial ionized fraction
!CA x_H0 is initial ionized fraction of Hydrogen
!CA x_He0 is initial ionized fraction of Helium
!CA rhs is dummy for calculating x0
!CA zinitial and zfinal are starting and ending redshifts
!CA zeq is the redshift of matter-radiation equality
!CA zstart and zend are for each pass to the integrator
!CA C,k_B,h_P: speed of light, Boltzmann's and Planck's constants
!CA m_e,m_H: electron mass and mass of H atom in SI
!CA not4: ratio of 4He atomic mass to 1H atomic mass
!CA sigma: Thomson cross-section
!CA a_rad: radiation constant for u=aT^4
!CA Lambda: 2s-1s two photon rate for Hydrogen
!CA Lambda_He: 2s-1s two photon rate for Helium
!CA DeltaB: energy of first excited state from continuum = 3.4eV
!CA DeltaB_He: energy of first excited state from cont. for He = 3.4eV
!CA L_H_ion: level for H ionization in m^-1
!CA L_H_alpha: level for H Ly alpha in m^-1
!CA L_He1_ion: level for HeI ionization
!CA L_He2_ion: level for HeII ionization
!CA L_He_2s: level for HeI 2s
!CA L_He_2p: level for HeI 2p (21P1-11S0) in m^-1
!CA Lalpha: Ly alpha wavelength in SI
!CA Lalpha_He: Helium I 2p-1s wavelength in SI
!CA mu_H,mu_T: mass per H atom and mass per particle
!CA H_frac: follow Tmat when t_Compton / t_Hubble > H_frac
!CA CDB=DeltaB/k_B Constants derived from B1,B2,R
!CA CDB_He=DeltaB_He/k_B n=2-infinity for He in Kelvin
!CA CB1=CDB*4. Lalpha and sigma_Th, calculated
!CA CB1_He1: CB1 for HeI ionization potential
!CA CB1_He2: CB1 for HeII ionization potential
!CA CR=2*Pi*(m_e/h_P)*(k_B/h_P) once and passed in a common block
!CA CK=Lalpha**3/(8.*Pi)
!CA CK_He=Lalpha_He**3/(8.*Pi)
!CA CL=C*h_P/(k_B*Lalpha)
!CA CL_He=C*h_P/(k_B*Lalpha_He)
!CA CT=(8./3.)*(sigma/(m_e*C))*a
!CA Bfact=exp((E_2p-E_2s)/kT) Extra Boltzmann factor
!CA b_He= "fudge factor" for HeI, to approximate higher z behaviour
!CA Heswitch=integer for modifying HeI recombination
!CA Parameters and quantities to describe the extra triplet states
!CA and also the continuum opacity of H, with a fitting function
!CA suggested by KIV, astro-ph/0703438
!CA a_trip: used to fit HeI triplet recombination rate
!CA b_trip: used to fit HeI triplet recombination rate
!CA L_He_2Pt: level for 23P012-11S0 in m^-1
!CA L_He_2St: level for 23S1-11S0 in m^-1
!CA L_He2St_ion: level for 23S1-continuum in m^-1
!CA A2P_s: Einstein A coefficient for He 21P1-11S0
!CA A2P_t: Einstein A coefficient for He 23P1-11S0
!CA sigma_He_2Ps: H ionization x-section at HeI 21P1-11S0 freq. in m^2
!CA sigma_He_2Pt: H ionization x-section at HeI 23P1-11S0 freq. in m^2
!CA CL_PSt = h_P*C*(L_He_2Pt - L_He_2st)/k_B
!CA CfHe_t: triplet statistical correction
!CA Hswitch is an boolean for modifying the H recombination
!CA AGauss1 is the amplitude of the 1st Gaussian for the H fudging
!CA AGauss2 is the amplitude of the 2nd Gaussian for the H fudging
!CA zGauss1 is the ln(1+z) central value of the 1st Gaussian
!CA zGauss2 is the ln(1+z) central value of the 2nd Gaussian
!CA wGauss1 is the width of the 1st Gaussian
!CA wGauss2 is the width of the 2nd Gaussian
!CA tol: tolerance for the integrator
!CA cw(24),w(3,9): work space for DVERK
!CA Ndim: number of d.e.'s to solve (integer)
!CA Nz: number of output redshitf (integer)
!CA I: loop index (integer)
!CA ind,nw: work-space for DVERK (integer)
!C
!CF File & device access:
!CF Unit /I,IO,O /Name (if known)
!C
!CM Modules called:
!CM DVERK (numerical integrator)
!CM GET_INIT (initial values for ionization fractions)
!CM ION (ionization and Temp derivatives)
!C
!CC Comments:
!CC none
!C
!CH History:
!CH CREATED (simplest version) 19th March 1989
!CH RECREATED 11th January 1995
!CH includes variable Cosmology
!CH uses DVERK integrator
!CH initial conditions are Saha
!CH TESTED a bunch, well, OK, not really
!CH MODIFIED January 1995 (include Hummer's 1994 alpha table)
!CH January 1995 (include new value for 2s-1s rate)
!CH January 1995 (expand comments)
!CH March 1995 (add Saha for Helium)
!CH August 1997 (add HeII alpha table)
!CH July 1998 (include OmegaT correction and H fudge factor)
!CH Nov 1998 (change Trad to Tmat in Rup)
!CH Jan 1999 (tidied up for public consumption)
!CH Sept 1999 (switch to formula for alpha's, fix glitch)
!CH Sept 1999 modified to CMBFAST by US & MZ
!CH Nov 1999 modified for F90 and CAMB (AML)
!CH Aug 2000 modified to prevent overflow erorr in He_Boltz (AML)
!CH Feb 2001 corrected fix of Aug 2000 (AML)
!CH Oct 2001 fixed error in hubble parameter, now uses global function (AML)
! March 2003 fixed bugs reported by savita gahlaut
! March 2005 added option for corrections from astro-ph/0501672.
! thanks to V.K.Dubrovich, S.I.Grachev
! June 2006 defined RECFAST_fudge as free parameter (AML)
! October 2006 (included new value for G)
! October 2006 (improved m_He/m_H to be "not4")
! October 2006 (fixed error, x for x_H in part of f(1))
!CH January 2008 (improved HeI recombination effects,
!CH including HeI rec. fudge factor)
! Feb 2008 Recfast 1.4 changes above added (AML)
! removed Dubrovich option (wrong anyway)
!CH Sept 2008 (added extra term to make transition, smoother for Tmat evolution)
! Sept 2008 Recfast 1.4.2 changes above added (AML)
! General recombination module structure, fix to make He x_e smooth also in recfast (AML)
!CH Jan 2010 (added fitting function to modify K
!CH to match x_e(z) for new H physics)
!AL June 2012 updated fudge parameters to match HyRec and CosmoRec (AML)
!AL Sept 2012 changes now in public recfast, version number changed to match Recfast 1.5.2.
!! ===============================================================
module RECDATA
use constants
implicit none
real(dl) Lambda,DeltaB,DeltaB_He,Lalpha,mu_H,mu_T,H_frac
real(dl) Lambda_He,Lalpha_He,Bfact,CK_He,CL_He
real(dl) L_H_ion,L_H_alpha,L_He1_ion,L_He2_ion,L_He_2s,L_He_2p
real(dl) CB1,CDB,CR,CK,CL,CT,CB1_He1,CB1_He2,CDB_He,fu
real(dl) A2P_s,A2P_t,sigma_He_2Ps,sigma_He_2Pt
real(dl) L_He_2Pt,L_He_2St,L_He2St_ion
real(dl), parameter :: bigH=100.0D3/Mpc !Ho in s-1
real(dl), parameter :: sigma = sigma_thomson
real(dl), parameter :: not4 = mass_ratio_He_H !mass He/H atom
real(dl) Tnow,HO
integer :: n_eq = 3
!The following only used for approximations where small effect
real(dl) OmegaK, OmegaT, z_eq
!Fundamental constants in SI units
! ("not4" pointed out by Gary Steigman)
data Lambda /8.2245809d0/
data Lambda_He /51.3d0/ !new value from Dalgarno
data L_H_ion /1.096787737D7/ !level for H ion. (in m^-1)
data L_H_alpha /8.225916453D6/ !averaged over 2 levels
data L_He1_ion /1.98310772D7/ !from Drake (1993)
data L_He2_ion /4.389088863D7/ !from JPhysChemRefData (1987)
data L_He_2s /1.66277434D7/ !from Drake (1993)
data L_He_2p /1.71134891D7/ !from Drake (1993)
! 2 photon rates and atomic levels in SI units
data A2P_s /1.798287D9/ !Morton, Wu & Drake (2006)
data A2P_t /177.58D0/ !Lach & Pachuski (2001)
data L_He_2Pt /1.690871466D7/ !Drake & Morton (2007)
data L_He_2St /1.5985597526D7/ !Drake & Morton (2007)
data L_He2St_ion /3.8454693845D6/ !Drake & Morton (2007)
data sigma_He_2Ps /1.436289D-22/ !Hummer & Storey (1998)
data sigma_He_2Pt /1.484872D-22/ !Hummer & Storey (1998)
! Atomic data for HeI
end module RECDATA
module Recombination
use constants
use AMLUtils
implicit none
private
real(dl), parameter :: zinitial = 1e4_dl !highest redshift
real(dl), parameter :: zfinal=0._dl
integer, parameter :: Nz=10000
real(dl), parameter :: delta_z = (zinitial-zfinal)/Nz
integer, parameter :: RECFAST_Heswitch_default = 6
real(dl), parameter :: RECFAST_fudge_He_default = 0.86_dl !Helium fudge parameter
logical, parameter :: RECFAST_Hswitch_default = .true. !include H corrections (v1.5, 2010)
real(dl), parameter :: RECFAST_fudge_default = 1.14_dl !1.14_dl
real(dl), parameter :: RECFAST_fudge_default2 = 1.105d0 + 0.02d0
!fudge parameter if RECFAST_Hswitch
real(dl) :: AGauss1 = -0.14D0 !Amplitude of 1st Gaussian
real(dl) :: AGauss2 = 0.079D0 ! 0.05D0 !Amplitude of 2nd Gaussian
real(dl) :: zGauss1 = 7.28D0 !ln(1+z) of 1st Gaussian
real(dl) :: zGauss2= 6.73D0 !ln(1+z) of 2nd Gaussian
real(dl) :: wGauss1= 0.18D0 !Width of 1st Gaussian
real(dl) :: wGauss2= 0.33D0 !Width of 2nd Gaussian
!Gaussian fits for extra H physics (fit by Adam Moss , modified by Antony Lewis)
type RecombinationParams
real(dl) :: RECFAST_fudge
real(dl) :: RECFAST_fudge_He
integer :: RECFAST_Heswitch
logical :: RECFAST_Hswitch
!0) no change from old Recfast'
!1) full expression for escape probability for singlet'
!' 1P-1S transition'
!2) also including effect of contiuum opacity of H on HeI'
!' singlet (based in fitting formula suggested by'
!' Kholupenko, Ivanchik & Varshalovich, 2007)'
!3) only including recombination through the triplets'
!4) including 3 and the effect of the contiuum '
!' (although this is probably negligible)'
!5) including only 1, 2 and 3'
!6) including all of 1 to 4'
end type RecombinationParams
character(LEN=*), parameter :: Recombination_Name = 'Recfast_1.5.2'
real(dl) zrec(Nz),xrec(Nz),dxrec(Nz), Tsrec(Nz) ,dTsrec(Nz), tmrec(Nz),dtmrec(Nz)
real(dl), parameter :: Do21cm_mina = 1/(1+900.) !at which to start evolving Delta_TM
logical, parameter :: evolve_Ts = .false. !local equilibrium is very accurate
real(dl), parameter :: Do21cm_minev = 1/(1+400.) !at which to evolve T_s
real(dl), parameter :: B01 = 3*B10
real(dl) :: NNow, fHe
logical :: Do21cm = .false.
logical :: doTmatTspin = .false.
real(dl) :: recombination_saha_z !Redshift at which saha OK
real(dl) :: recombination_saha_tau !set externally
public RecombinationParams, Recombination_xe, Recombination_tm,Recombination_ts ,Recombination_init, &
Recombination_ReadParams, Recombination_SetDefParams, Recombination_Validate, Recombination_Name, &
kappa_HH_21cm,kappa_eH_21cm,kappa_pH_21cm, &
Do21cm, doTmatTspin, Do21cm_mina, dDeltaxe_dtau, &
recombination_saha_tau, recombination_saha_z
contains
subroutine Recombination_ReadParams(R, Ini)
use IniFile
Type(RecombinationParams) :: R
Type(TIniFile) :: Ini
R%RECFAST_fudge_He = Ini_Read_Double_File(Ini,'RECFAST_fudge_He',RECFAST_fudge_He_default)
R%RECFAST_Heswitch = Ini_Read_Int_File(Ini, 'RECFAST_Heswitch',RECFAST_Heswitch_default)
R%RECFAST_Hswitch = Ini_Read_Logical_File(Ini, 'RECFAST_Hswitch',RECFAST_Hswitch_default)
R%RECFAST_fudge = Ini_Read_Double_File(Ini,'RECFAST_fudge',RECFAST_fudge_default)
AGauss1 = Ini_REad_Double_File(Ini,'AGauss1',AGauss1)
AGauss2 = Ini_REad_Double_File(Ini,'AGauss2',AGauss2)
zGauss1 = Ini_REad_Double_File(Ini,'zGauss1',zGauss1)
zGauss2 = Ini_REad_Double_File(Ini,'zGauss2',zGauss2)
wGauss1 = Ini_REad_Double_File(Ini,'wGauss1',wGauss1)
wGauss2 = Ini_REad_Double_File(Ini,'wGauss2',wGauss2)
if (R%RECFAST_Hswitch) then
R%RECFAST_fudge = R%RECFAST_fudge - (RECFAST_fudge_default - RECFAST_fudge_default2)
end if
end subroutine Recombination_ReadParams
subroutine Recombination_SetDefParams(R)
type (RecombinationParams) ::R
R%RECFAST_fudge = RECFAST_fudge_default
R%RECFAST_fudge_He = RECFAST_fudge_He_default !Helium fudge parameter
R%RECFAST_Heswitch = RECFAST_Heswitch_default
R%RECFAST_Hswitch = RECFAST_Hswitch_default
if (R%RECFAST_Hswitch) then
R%RECFAST_fudge = RECFAST_fudge_default2
end if
end subroutine Recombination_SetDefParams
subroutine Recombination_Validate(R, OK)
Type(RecombinationParams),intent(in) :: R
logical, intent(inout) :: OK
if (R%RECFAST_Heswitch<0 .or. R%RECFAST_Heswitch > 6) then
OK = .false.
write(*,*) 'RECFAST_Heswitch unknown'
end if
end subroutine Recombination_Validate
function Recombination_tm(a)
use RECDATA, only : Tnow
real(dl) zst,a,z,az,bz,Recombination_tm
integer ilo,ihi
if (.not. doTmatTspin) call MpiStop('RECFAST: Recombination_tm not stored')
z=1/a-1
if (z >= zrec(1)) then
Recombination_tm=Tnow/a
else
if (z <=zrec(nz)) then
Recombination_tm=Tmrec(nz)
else
zst=(zinitial-z)/delta_z
ihi= int(zst)
ilo = ihi+1
az=zst - int(zst)
bz=1-az
Recombination_tm=az*Tmrec(ilo)+bz*Tmrec(ihi)+ &
((az**3-az)*dTmrec(ilo)+(bz**3-bz)*dTmrec(ihi))/6._dl
endif
endif
end function Recombination_tm
function Recombination_ts(a)
!zrec(1) is zinitial-delta_z
real(dl), intent(in) :: a
real(dl) zst,z,az,bz,Recombination_ts
integer ilo,ihi
z=1/a-1
if (z.ge.zrec(1)) then
Recombination_ts=tsrec(1)
else
if (z.le.zrec(nz)) then
Recombination_ts=tsrec(nz)
else
zst=(zinitial-z)/delta_z
ihi= int(zst)
ilo = ihi+1
az=zst - int(zst)
bz=1-az
Recombination_ts=az*tsrec(ilo)+bz*tsrec(ihi)+ &
((az**3-az)*dtsrec(ilo)+(bz**3-bz)*dtsrec(ihi))/6._dl
endif
endif
end function Recombination_ts
function Recombination_xe(a)
real(dl), intent(in) :: a
real(dl) zst,z,az,bz,Recombination_xe
integer ilo,ihi
z=1/a-1
if (z.ge.zrec(1)) then
Recombination_xe=xrec(1)
else
if (z.le.zrec(nz)) then
Recombination_xe=xrec(nz)
else
zst=(zinitial-z)/delta_z
ihi= int(zst)
ilo = ihi+1
az=zst - int(zst)
bz=1-az
Recombination_xe=az*xrec(ilo)+bz*xrec(ihi)+ &
((az**3-az)*dxrec(ilo)+(bz**3-bz)*dxrec(ihi))/6._dl
endif
endif
end function Recombination_xe
subroutine Recombination_init(Recomb, OmegaC, OmegaB, Omegan, Omegav, h0inp,tcmb,yp, nnu)
!Would love to pass structure as arguments, but F90 would give circular reference...
!hence mess passing parameters explcitly and non-generally
!Note recfast only uses OmegaB, h0inp, tcmb and yp - others used only for Tmat approximation where effect small
!nnu currently not used here
use RECDATA
use AMLUtils
implicit none
Type (RecombinationParams) :: Recomb
real(dl), save :: last_OmB =0, Last_YHe=0, Last_H0=0, Last_dtauda=0, last_fudge, last_fudgeHe
real(dl) Trad,Tmat,Tspin,d0hi,d0lo
integer I
real(dl), intent(in) :: OmegaB,OmegaC, Omegan, Omegav, h0inp, yp
real(dl), intent(in), optional :: nnu
real(dl) z,n,x,x0,rhs,x_H,x_He,x_H0,x_He0,H
real(dl) zstart,zend,tcmb
real(dl) cw(24)
real(dl), dimension(:,:), allocatable :: w
real(dl) y(4)
real(dl) C10, tau_21Ts
real(dl) fnu
integer ind,nw
! --- Parameter statements
real(dl), parameter :: tol=1.D-5 !Tolerance for R-K
real(dl) dtauda
external dtauda, dverk
! ===============================================================
if (Last_OmB==OmegaB .and. Last_H0 == h0inp .and. yp == Last_YHe .and. &
dtauda(0.2352375823_dl) == Last_dtauda .and. last_fudge == Recomb%RECFAST_fudge &
.and. last_fudgeHe==Recomb%RECFAST_fudge_He) return
!This takes up most of the single thread time, so cache if at all possible
!For example if called with different reionization, or tensor rather than scalar
Last_dtauda = dtauda(0.2352375823_dl) !Just get it at a random scale factor
Last_OmB = OmegaB
Last_H0 = h0inp
Last_YHe=yp
last_fudge = Recomb%RECFAST_FUDGE
last_fudgeHe = Recomb%RECFAST_FUDGE_He
if (Do21cm) doTmatTspin = .true.
! write(*,*)'recfast version 1.0'
! write(*,*)'Using Hummer''s case B recombination rates for H'
! write(*,*)' with fudge factor = 1.14'
! write(*,*)'and tabulated HeII singlet recombination rates'
! write(*,*)
n_eq = 3
if (Evolve_Ts) n_eq=4
allocate(w(n_eq,9))
recombination_saha_z=0.d0
Tnow=tcmb
! These are easy to inquire as input, but let's use simple values
z = zinitial
! will output every 1 in z, but this is easily changed also
!Not general, but only for approx
OmegaT=OmegaC+OmegaB !total dark matter + baryons
OmegaK=1.d0-OmegaT-OmegaV !curvature
! convert the Hubble constant units
H = H0inp/100._dl
HO = H*bigH
! sort out the helium abundance parameters
mu_H = 1.d0/(1.d0-Yp) !Mass per H atom
mu_T = not4/(not4-(not4-1.d0)*Yp) !Mass per atom
fHe = Yp/(not4*(1.d0-Yp)) !n_He_tot / n_H_tot
Nnow = 3._dl*HO*HO*OmegaB/(8._dl*Pi*G*mu_H*m_H)
n = Nnow * (1._dl+z)**3
fnu = (21.d0/8.d0)*(4.d0/11.d0)**(4.d0/3.d0)
! (this is explictly for 3 massless neutrinos - change if N_nu.ne.3; but only used for approximation so not critical)
z_eq = (3.d0*(HO*C)**2/(8.d0*Pi*G*a_rad*(1.d0+fnu)*Tnow**4))*(OmegaB+OmegaC)
z_eq = z_eq - 1.d0
! Set up some constants so they don't have to be calculated later
Lalpha = 1.d0/L_H_alpha
Lalpha_He = 1.d0/L_He_2p
DeltaB = h_P*C*(L_H_ion-L_H_alpha)
CDB = DeltaB/k_B
DeltaB_He = h_P*C*(L_He1_ion-L_He_2s) !2s, not 2p
CDB_He = DeltaB_He/k_B
CB1 = h_P*C*L_H_ion/k_B
CB1_He1 = h_P*C*L_He1_ion/k_B !ionization for HeI
CB1_He2 = h_P*C*L_He2_ion/k_B !ionization for HeII
CR = 2.d0*Pi*(m_e/h_P)*(k_B/h_P)
CK = Lalpha**3/(8.d0*Pi)
CK_He = Lalpha_He**3/(8.d0*Pi)
CL = C*h_P/(k_B*Lalpha)
CL_He = C*h_P/(k_B/L_He_2s) !comes from det.bal. of 2s-1s
CT = Compton_CT / MPC_in_sec
Bfact = h_P*C*(L_He_2p-L_He_2s)/k_B
! Matter departs from radiation when t(Th) > H_frac * t(H)
! choose some safely small number
H_frac = 1D-3
! Fudge factor to approximate for low z out of equilibrium effect
fu=Recomb%RECFAST_fudge
! Set initial matter temperature
y(3) = Tnow*(1._dl+z) !Initial rad. & mat. temperature
Tmat = y(3)
y(4) = Tmat
Tspin = Tmat
call get_init(z,x_H0,x_He0,x0)
y(1) = x_H0
y(2) = x_He0
! OK that's the initial conditions, now start writing output file
! Set up work-space stuff for DVERK
ind = 1
nw = n_eq
do i = 1,24
cw(i) = 0._dl
end do
do i = 1,Nz
! calculate the start and end redshift for the interval at each z
! or just at each z
zstart = zinitial - real(i-1,dl)*delta_z
zend = zinitial - real(i,dl)*delta_z
! Use Saha to get x_e, using the equation for x_e for ionized helium
! and for neutral helium.
! Everything ionized above z=8000. First ionization over by z=5000.
! Assume He all singly ionized down to z=3500, then use He Saha until
! He is 99% singly ionized, and *then* switch to joint H/He recombination.
z = zend
if (zend > 8000._dl) then
x_H0 = 1._dl
x_He0 = 1._dl
x0 = 1._dl+2._dl*fHe
y(1) = x_H0
y(2) = x_He0
y(3) = Tnow*(1._dl+z)
y(4) = y(3)
else if(z > 5000._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
rhs = exp( 1.5d0 * log(CR*Tnow/(1._dl+z)) &
- CB1_He2/(Tnow*(1._dl+z)) ) / Nnow
rhs = rhs*1._dl !ratio of g's is 1 for He++ <-> He+
x0 = 0.5d0 * ( sqrt( (rhs-1._dl-fHe)**2 &
+ 4._dl*(1._dl+2._dl*fHe)*rhs) - (rhs-1._dl-fHe) )
y(1) = x_H0
y(2) = x_He0
y(3) = Tnow*(1._dl+z)
y(4) = y(3)
else if(z > 3500._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
x0 = x_H0 + fHe*x_He0
y(1) = x_H0
y(2) = x_He0
y(3) = Tnow*(1._dl+z)
y(4) = y(3)
else if(y(2) > 0.99)then
x_H0 = 1._dl
rhs = exp( 1.5d0 * log(CR*Tnow/(1._dl+z)) &
- CB1_He1/(Tnow*(1._dl+z)) ) / Nnow
rhs = rhs*4._dl !ratio of g's is 4 for He+ <-> He0
x_He0 = 0.5d0 * ( sqrt( (rhs-1._dl)**2 &
+ 4._dl*(1._dl+fHe)*rhs )- (rhs-1._dl))
x0 = x_He0
x_He0 = (x0 - 1._dl)/fHe
y(1) = x_H0
y(2) = x_He0
y(3) = Tnow*(1._dl+z)
y(4) = y(3)
else if (y(1) > 0.99d0) then
rhs = exp( 1.5d0 * log(CR*Tnow/(1._dl+z)) &
- CB1/(Tnow*(1._dl+z)) ) / Nnow
x_H0 = 0.5d0 * (sqrt( rhs**2+4._dl*rhs ) - rhs )
call DVERK(Recomb,3,ION,zstart,y,zend,tol,ind,cw,nw,w)
y(1) = x_H0
x0 = y(1) + fHe*y(2)
y(4)=y(3)
else
call DVERK(Recomb,nw,ION,zstart,y,zend,tol,ind,cw,nw,w)
x0 = y(1) + fHe*y(2)
end if
Trad = Tnow * (1._dl+zend)
Tmat = y(3)
x_H = y(1)
x_He = y(2)
x = x0
zrec(i)=zend
xrec(i)=x
if (doTmatTspin) then
if (Evolve_Ts .and. zend< 1/Do21cm_minev-1 ) then
Tspin = y(4)
else
C10 = Nnow * (1._dl+zend)**3*(kappa_HH_21cm(Tmat,.false.)*(1-x_H) + kappa_eH_21cm(Tmat,.false.)*x)
tau_21Ts = line21_const*NNow*(1+zend)*dtauda(1/(1+zend))/1000
Tspin = Trad*( C10/Trad + A10/T_21cm)/(C10/Tmat + A10/T_21cm) + &
tau_21Ts/2*A10*( 1/(C10*T_21cm/Tmat+A10) - 1/(C10*T_21cm/Trad+A10) )
y(4) = Tspin
end if
tsrec(i) = Tspin
tmrec(i) = Tmat
end if
! write (*,'(5E15.5)') zend, Trad, Tmat, Tspin, x
end do
d0hi=1.0d40
d0lo=1.0d40
call spline(zrec,xrec,nz,d0lo,d0hi,dxrec)
if (doTmatTspin) then
call spline(zrec,tsrec,nz,d0lo,d0hi,dtsrec)
call spline(zrec,tmrec,nz,d0lo,d0hi,dtmrec)
end if
deallocate(w)
end subroutine Recombination_init
! ===============================================================
subroutine GET_INIT(z,x_H0,x_He0,x0)
! Set up the initial conditions so it will work for general,
! but not pathological choices of zstart
! Initial ionization fraction using Saha for relevant species
use RECDATA
implicit none
real(dl) z,x0,rhs,x_H0,x_He0
if(z > 8000._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
x0 = 1._dl+2._dl*fHe
else if(z > 3500._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
rhs = exp( 1.5d0 * log(CR*Tnow/(1._dl+z)) &
- CB1_He2/(Tnow*(1._dl+z)) ) / Nnow
rhs = rhs*1._dl !ratio of g's is 1 for He++ <-> He+
x0 = 0.5d0 * ( sqrt( (rhs-1._dl-fHe)**2 &
+ 4._dl*(1._dl+2._dl*fHe)*rhs) - (rhs-1._dl-fHe) )
else if(z > 2000._dl)then
x_H0 = 1._dl
rhs = exp( 1.5d0 * log(CR*Tnow/(1._dl+z)) &
- CB1_He1/(Tnow*(1._dl+z)) ) / Nnow
rhs = rhs*4._dl !ratio of g's is 4 for He+ <-> He0
x_He0 = 0.5d0 * ( sqrt( (rhs-1._dl)**2 + 4._dl*(1._dl+fHe)*rhs )- (rhs-1._dl))
x0 = x_He0
x_He0 = (x0 - 1._dl)/fHe
else
rhs = exp( 1.5d0 * log(CR*Tnow/(1._dl+z)) &
- CB1/(Tnow*(1._dl+z)) ) / Nnow
x_H0 = 0.5d0 * (sqrt( rhs**2+4._dl*rhs ) - rhs )
x_He0 = 0._dl
x0 = x_H0
end if
end subroutine GET_INIT
subroutine ION(Recomb,Ndim,z,Y,f)
use RECDATA
implicit none
integer Ndim
Type (RecombinationParams) :: Recomb
real(dl) z,x,n,n_He,Trad,Tmat,Tspin,x_H,x_He, Hz
real(dl) y(Ndim),f(Ndim)
real(dl) Rup,Rdown,K,K_He,Rup_He,Rdown_He,He_Boltz
real(dl) timeTh,timeH
real(dl) a_VF,b_VF,T_0,T_1,sq_0,sq_1,a_PPB,b_PPB,c_PPB,d_PPB
real(dl) tauHe_s,pHe_s
real(dl) a_trip,b_trip,Rdown_trip,Rup_trip
real(dl) Doppler,gamma_2Ps,pb,qb,AHcon
real(dl) tauHe_t,pHe_t,CL_PSt,CfHe_t,gamma_2Pt
real(dl) epsilon
integer Heflag
real(dl) dtauda
real(dl) C10, dHdz
external dtauda
! the Pequignot, Petitjean & Boisson fitting parameters for Hydrogen
a_PPB = 4.309d0
b_PPB = -0.6166d0
c_PPB = 0.6703d0
d_PPB = 0.5300d0
! the Verner and Ferland type fitting parameters for Helium
! fixed to match those in the SSS papers, and now correct
a_VF = 10.d0**(-16.744d0)
b_VF = 0.711d0
T_0 = 10.d0**(0.477121d0) !3K
T_1 = 10.d0**(5.114d0)
! fitting parameters for HeI triplets
! (matches Hummer's table with <1% error for 10^2.8 < T/K < 10^4)
a_trip = 10.d0**(-16.306d0)
b_trip = 0.761D0
x_H = y(1)
x_He = y(2)
x = x_H + fHe * x_He
Tmat = y(3)
! Tspin = y(4)
n = Nnow * (1._dl+z)**3
n_He = fHe * Nnow * (1._dl+z)**3
Trad = Tnow * (1._dl+z)
Hz = 1/dtauda(1/(1._dl+z))*(1._dl+z)**2/MPC_in_sec
! Get the radiative rates using PPQ fit, identical to Hummer's table
Rdown=1.d-19*a_PPB*(Tmat/1.d4)**b_PPB &
/(1._dl+c_PPB*(Tmat/1.d4)**d_PPB)
Rup = Rdown * (CR*Tmat)**(1.5d0)*exp(-CDB/Tmat)
! calculate He using a fit to a Verner & Ferland type formula
sq_0 = sqrt(Tmat/T_0)
sq_1 = sqrt(Tmat/T_1)
! typo here corrected by Wayne Hu and Savita Gahlaut
Rdown_He = a_VF/(sq_0*(1.d0+sq_0)**(1.d0-b_VF))
Rdown_He = Rdown_He/(1.d0+sq_1)**(1.d0+b_VF)
Rup_He = Rdown_He*(CR*Tmat)**(1.5d0)*exp(-CDB_He/Tmat)
Rup_He = 4.d0*Rup_He !statistical weights factor for HeI
! Avoid overflow (pointed out by Jacques Roland)
if((Bfact/Tmat) > 680.d0)then
He_Boltz = exp(680.d0)
else
He_Boltz = exp(Bfact/Tmat)
end if
! now deal with H and its fudges
if (.not. Recomb%RECFAST_Hswitch) then
K = CK/Hz !Peebles coefficient K=lambda_a^3/8piH
else
!c fit a double Gaussian correction function
K = CK/Hz*(1.0d0 &
+AGauss1*exp(-((log(1.0d0+z)-zGauss1)/wGauss1)**2.d0) &
+AGauss2*exp(-((log(1.0d0+z)-zGauss2)/wGauss2)**2.d0))
end if
! add the HeI part, using same T_0 and T_1 values
Rdown_trip = a_trip/(sq_0*(1.d0+sq_0)**(1.0-b_trip))
Rdown_trip = Rdown_trip/((1.d0+sq_1)**(1.d0+b_trip))
Rup_trip = Rdown_trip*dexp(-h_P*C*L_He2St_ion/(k_B*Tmat))
Rup_trip = Rup_trip*((CR*Tmat)**(1.5d0))*(4.d0/3.d0)
! last factor here is the statistical weight
! try to avoid "NaN" when x_He gets too small
if ((x_He.lt.5.d-9) .or. (x_He.gt.0.98d0)) then
Heflag = 0
else
Heflag = Recomb%RECFAST_Heswitch
end if
if (Heflag.eq.0)then !use Peebles coeff. for He
K_He = CK_He/Hz
else !for Heflag>0 !use Sobolev escape probability
tauHe_s = A2P_s*CK_He*3.d0*n_He*(1.d0-x_He)/Hz
pHe_s = (1.d0 - dexp(-tauHe_s))/tauHe_s
K_He = 1.d0/(A2P_s*pHe_s*3.d0*n_He*(1.d0-x_He))
! if (((Heflag.eq.2) .or. (Heflag.ge.5)) .and. x_H < 0.99999d0) then
if (((Heflag.eq.2) .or. (Heflag.ge.5)) .and. x_H < 0.9999999d0) then
!AL changed July 08 to get smoother Helium
! use fitting formula for continuum opacity of H
! first get the Doppler width parameter
Doppler = 2.D0*k_B*Tmat/(m_H*not4*C*C)
Doppler = C*L_He_2p*dsqrt(Doppler)
gamma_2Ps = 3.d0*A2P_s*fHe*(1.d0-x_He)*C*C &
/(dsqrt(Pi)*sigma_He_2Ps*8.d0*Pi*Doppler*(1.d0-x_H)) &
/((C*L_He_2p)**2.d0)
pb = 0.36d0 !value from KIV (2007)
qb = Recomb%RECFAST_fudge_He
! calculate AHcon, the value of A*p_(con,H) for H continuum opacity
AHcon = A2P_s/(1.d0+pb*(gamma_2Ps**qb))
K_He=1.d0/((A2P_s*pHe_s+AHcon)*3.d0*n_He*(1.d0-x_He))
end if
if (Heflag.ge.3) then !include triplet effects
tauHe_t = A2P_t*n_He*(1.d0-x_He)*3.d0
tauHe_t = tauHe_t /(8.d0*Pi*Hz*L_He_2Pt**(3.d0))
pHe_t = (1.d0 - dexp(-tauHe_t))/tauHe_t
CL_PSt = h_P*C*(L_He_2Pt - L_He_2st)/k_B
if ((Heflag.eq.3) .or. (Heflag.eq.5).or.(x_H.gt.0.99999d0)) then !Recfast 1.4.2 (?)
! if ((Heflag.eq.3) .or. (Heflag.eq.5) .or. x_H >= 0.9999999d0) then !no H cont. effect
CfHe_t = A2P_t*pHe_t*dexp(-CL_PSt/Tmat)
CfHe_t = CfHe_t/(Rup_trip+CfHe_t) !"C" factor for triplets
else !include H cont. effect
Doppler = 2.d0*k_B*Tmat/(m_H*not4*C*C)
Doppler = C*L_He_2Pt*dsqrt(Doppler)
gamma_2Pt = 3.d0*A2P_t*fHe*(1.d0-x_He)*C*C &
/(dsqrt(Pi)*sigma_He_2Pt*8.d0*Pi*Doppler*(1.d0-x_H)) &
/((C*L_He_2Pt)**2.d0)
! use the fitting parameters from KIV (2007) in this case
pb = 0.66d0
qb = 0.9d0
AHcon = A2P_t/(1.d0+pb*gamma_2Pt**qb)/3.d0
CfHe_t = (A2P_t*pHe_t+AHcon)*dexp(-CL_PSt/Tmat)
CfHe_t = CfHe_t/(Rup_trip+CfHe_t) !"C" factor for triplets
end if
end if
end if
! Estimates of Thomson scattering time and Hubble time
timeTh=(1._dl/(CT*Trad**4))*(1._dl+x+fHe)/x !Thomson time
timeH=2./(3.*HO*(1._dl+z)**1.5) !Hubble time
! calculate the derivatives
! turn on H only for x_H<0.99, and use Saha derivative for 0.98<x_H<0.99
! (clunky, but seems to work)
if (x_H > 0.99) then !don't change at all
f(1) = 0._dl
!! else if (x_H > 0.98_dl) then
else if (x_H.gt.0.985d0) then !use Saha rate for Hydrogen
f(1) = (x*x_H*n*Rdown - Rup*(1.d0-x_H)*dexp(-CL/Tmat)) /(Hz*(1.d0+z))
recombination_saha_z = z
!AL: following commented as not used
! for interest, calculate the correction factor compared to Saha
! (without the fudge)
! factor=(1.d0 + K*Lambda*n*(1.d0-x_H))
! /(Hz*(1.d0+z)*(1.d0+K*Lambda*n*(1.d0-x)
! +K*Rup*n*(1.d0-x)))
else !use full rate for H
f(1) = ((x*x_H*n*Rdown - Rup*(1.d0-x_H)*exp(-CL/Tmat)) &
*(1.d0 + K*Lambda*n*(1.d0-x_H))) &
/(Hz*(1.d0+z)*(1.d0/fu+K*Lambda*n*(1.d0-x_H)/fu &
+K*Rup*n*(1.d0-x_H)))
end if
! turn off the He once it is small
if (x_He < 1.e-15) then
f(2)=0.d0
else
f(2) = ((x*x_He*n*Rdown_He &
- Rup_He*(1-x_He)*exp(-CL_He/Tmat)) &
*(1 + K_He*Lambda_He*n_He*(1.d0-x_He)*He_Boltz)) &
/(Hz*(1+z) &
* (1 + K_He*(Lambda_He+Rup_He)*n_He*(1.d0-x_He)*He_Boltz))
! Modification to HeI recombination including channel via triplets
if (Heflag.ge.3) then
f(2) = f(2)+ (x*x_He*n*Rdown_trip &
- (1.d0-x_He)*3.d0*Rup_trip*dexp(-h_P*C*L_He_2st/(k_B*Tmat))) &
*CfHe_t/(Hz*(1.d0+z))
end if
end if
if (timeTh < H_frac*timeH) then
! f(3)=Tmat/(1._dl+z) !Tmat follows Trad
! additional term to smooth transition to Tmat evolution,
! (suggested by Adam Moss)
dHdz = (HO**2/2.d0/Hz)*(4.d0*(1.d0+z)**3/(1.d0+z_eq)*OmegaT &
+ 3.d0*OmegaT*(1.d0+z)**2 + 2.d0*OmegaK*(1.d0+z) )
! dH/dz
epsilon = Hz*(1.d0+x+fHe)/(CT*Trad**3*x)
f(3) = Tnow &
+ epsilon*((1.d0+fHe)/(1.d0+fHe+x))*((f(1)+fHe*f(2))/x) &
- epsilon* dHdz/Hz + 3.0d0*epsilon/(1.d0+z)
else
f(3)= CT * (Trad**4) * x / (1._dl+x+fHe) &
* (Tmat-Trad) / (Hz*(1._dl+z)) + 2._dl*Tmat/(1._dl+z)
end if
! print *, z, f(3)*(1+z)/Tmat
if (Do21cm .and. evolve_Ts) then
! follow the matter temperature once it has a chance of diverging
if (timeTh < H_frac*timeH) then
f(4) = Tnow !spin follows Trad and Tmat
else
if (z< 1/Do21cm_minev-1) then
Tspin = y(4)