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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 ([email protected])
!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 Recombination
use constants
use classes
use DarkAge21cm
use MathUtils
use results
use MpiUtils, only : MpiStop
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
Type RecombinationData
real(dl) :: Recombination_saha_z !Redshift at which saha OK
real(dl), private :: NNow, fHe
real(dl), private :: zrec(Nz),xrec(Nz),dxrec(Nz), Tsrec(Nz) ,dTsrec(Nz), tmrec(Nz),dtmrec(Nz)
real(dl), private :: DeltaB,DeltaB_He,Lalpha,mu_H,mu_T
real(dl), private :: HO, Tnow, fu
integer, private :: n_eq = 3
logical :: doTspin = .false.
!The following only used for approximations where small effect
real(dl) :: OmegaK, OmegaT, z_eq
class(CAMBdata), pointer :: State
end Type RecombinationData
type, extends(TRecombinationModel) :: TRecfast
real(dl) :: RECFAST_fudge = RECFAST_fudge_default2
real(dl) :: RECFAST_fudge_He = RECFAST_fudge_He_default
integer :: RECFAST_Heswitch = RECFAST_Heswitch_default
logical :: RECFAST_Hswitch = RECFAST_Hswitch_default
!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'
!fudge parameter if RECFAST_Hswitch
!Gaussian fits for extra H physics (fit by Adam Moss , modified by Antony Lewis)
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
Type(RecombinationData), allocatable :: Calc
contains
procedure :: ReadParams => TRecfast_ReadParams
procedure :: Validate => TRecfast_Validate
procedure :: Init => TRecfast_init
procedure :: x_e => TRecfast_xe
procedure :: xe_Tm => TRecfast_xe_Tm !ionization fraction and baryon temperature
procedure :: T_m => TRecfast_tm !baryon temperature
procedure :: T_s => TRecfast_ts !Spin temperature
procedure :: Version => TRecfast_version
procedure :: dDeltaxe_dtau => TRecfast_dDeltaxe_dtau
procedure :: get_Saha_z => TRecfast_Get_Saha_z
procedure, nopass :: SelfPointer => TRecfast_SelfPointer
end type TRecfast
character(LEN=*), parameter :: Recfast_Version = 'Recfast_1.5.2'
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 :: 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), parameter :: B01 = 3*B10
!Fundamental constants in SI units
! ("not4" pointed out by Gary Steigman)
real(dl), parameter :: Lambda = 8.2245809d0
real(dl), parameter :: Lambda_He = 51.3d0 !new value from Dalgarno
real(dl), parameter :: L_H_ion = 1.096787737D7 !level for H ion. (in m^-1)
real(dl), parameter :: L_H_alpha = 8.225916453D6 !averaged over 2 levels
real(dl), parameter :: L_He1_ion = 1.98310772D7 !from Drake (1993)
real(dl), parameter :: L_He2_ion = 4.389088863D7 !from JPhysChemRefData (1987)
real(dl), parameter :: L_He_2s = 1.66277434D7 !from Drake (1993)
real(dl), parameter :: L_He_2p = 1.71134891D7 !from Drake (1993)
! 2 photon rates and atomic levels in SI units
real(dl), parameter :: A2P_s = 1.798287D9 !Morton, Wu & Drake (2006)
real(dl), parameter :: A2P_t = 177.58D0 !Lach & Pachuski (2001)
real(dl), parameter :: L_He_2Pt = 1.690871466D7 !Drake & Morton (2007)
real(dl), parameter :: L_He_2St = 1.5985597526D7 !Drake & Morton (2007)
real(dl), parameter :: L_He2St_ion =3.8454693845D6 !Drake & Morton (2007)
real(dl), parameter :: sigma_He_2Ps = 1.436289D-22 !Hummer & Storey (1998)
real(dl), parameter :: sigma_He_2Pt = 1.484872D-22 !Hummer & Storey (1998)
! Atomic data for HeI
! Set up some constants so they don't have to be calculated later
real(dl), parameter :: Lalpha = 1.d0/L_H_alpha
real(dl), parameter :: Lalpha_He = 1.d0/L_He_2p
real(dl), parameter :: DeltaB = h_P*C*(L_H_ion-L_H_alpha)
real(dl), parameter :: CDB = DeltaB/k_B
real(dl), parameter :: DeltaB_He = h_P*C*(L_He1_ion-L_He_2s) !2s, not 2p
real(dl), parameter :: CDB_He = DeltaB_He/k_B
real(dl), parameter :: CB1 = h_P*C*L_H_ion/k_B
real(dl), parameter :: CB1_He1 = h_P*C*L_He1_ion/k_B !ionization for HeI
real(dl), parameter :: CB1_He2 = h_P*C*L_He2_ion/k_B !ionization for HeII
real(dl), parameter :: CR = const_twopi*(m_e/h_P)*(k_B/h_P)
real(dl), parameter :: CK = Lalpha**3/(const_eightpi)
real(dl), parameter :: CK_He = Lalpha_He**3/(const_eightpi)
real(dl), parameter :: CL = C*h_P/(k_B*Lalpha)
real(dl), parameter :: CL_He = C*h_P/(k_B/L_He_2s) !comes from det.bal. of 2s-1s
real(dl), parameter :: CT = Compton_CT / MPC_in_sec
real(dl), parameter :: 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
real(dl), parameter :: H_frac = 1D-3
procedure(obj_function), private :: dtauda
public TRecfast, CB1
contains
subroutine TRecfast_ReadParams(this, Ini)
use IniObjects
class(TRecfast) :: this
class(TIniFile), intent(in) :: Ini
this%RECFAST_fudge_He = Ini%Read_Double('RECFAST_fudge_He', RECFAST_fudge_He_default)
this%RECFAST_Heswitch = Ini%Read_Int('RECFAST_Heswitch', RECFAST_Heswitch_default)
this%RECFAST_Hswitch = Ini%Read_Logical('RECFAST_Hswitch', RECFAST_Hswitch_default)
this%RECFAST_fudge = Ini%Read_Double('RECFAST_fudge', RECFAST_fudge_default)
call Ini%Read('AGauss1',this%AGauss1)
call Ini%Read('AGauss2',this%AGauss2)
call Ini%Read('zGauss1',this%zGauss1)
call Ini%Read('zGauss2',this%zGauss2)
call Ini%Read('wGauss1',this%wGauss1)
call Ini%Read('wGauss2',this%wGauss2)
if (this%RECFAST_Hswitch) then
this%RECFAST_fudge = this%RECFAST_fudge - (RECFAST_fudge_default - RECFAST_fudge_default2)
end if
end subroutine TRecfast_ReadParams
subroutine TRecfast_Validate(this, OK)
class(TRecfast),intent(in) :: this
logical, intent(inout) :: OK
if (this%RECFAST_Heswitch<0 .or. this%RECFAST_Heswitch > 6) then
OK = .false.
write(*,*) 'RECFAST_Heswitch unknown'
end if
end subroutine TRecfast_Validate
function TRecfast_tm(this,a)
class(TRecfast) :: this
real(dl), intent(in) :: a
real(dl) zst,z,az,bz,TRecfast_tm
integer ilo,ihi
z=1/a-1
associate( Calc => this%Calc)
if (z >= Calc%zrec(1)) then
TRecfast_tm=Calc%Tnow/a
else
if (z <=Calc%zrec(nz)) then
TRecfast_tm=Calc%Tmrec(nz)
else
zst=(zinitial-z)/delta_z
ihi= int(zst)
ilo = ihi+1
az=zst - ihi
bz=1-az
TRecfast_tm=az*Calc%Tmrec(ilo)+bz*Calc%Tmrec(ihi)+ &
((az**3-az)*Calc%dTmrec(ilo)+(bz**3-bz)*Calc%dTmrec(ihi))/6._dl
endif
endif
end associate
end function TRecfast_tm
function TRecfast_ts(this,a)
class(TRecfast) :: this
!zrec(1) is zinitial-delta_z
real(dl), intent(in) :: a
real(dl) zst,z,az,bz,TRecfast_ts
integer ilo,ihi
z=1/a-1
associate(Calc => this%Calc)
if (z.ge.Calc%zrec(1)) then
TRecfast_ts=Calc%tsrec(1)
else
if (z.le.Calc%zrec(nz)) then
TRecfast_ts=Calc%tsrec(nz)
else
zst=(zinitial-z)/delta_z
ihi= int(zst)
ilo = ihi+1
az=zst - ihi
bz=1-az
TRecfast_ts=az*Calc%tsrec(ilo)+bz*Calc%tsrec(ihi)+ &
((az**3-az)*Calc%dtsrec(ilo)+(bz**3-bz)*Calc%dtsrec(ihi))/6._dl
endif
endif
end associate
end function TRecfast_ts
function TRecfast_xe(this,a)
class(TRecfast) :: this
real(dl), intent(in) :: a
real(dl) zst,z,az,bz,TRecfast_xe
integer ilo,ihi
z=1/a-1
associate(Calc => this%Calc)
if (z.ge.Calc%zrec(1)) then
TRecfast_xe=Calc%xrec(1)
else
if (z.le.Calc%zrec(nz)) then
TRecfast_xe=Calc%xrec(nz)
else
zst=(zinitial-z)/delta_z
ihi= int(zst)
ilo = ihi+1
az=zst - ihi
bz=1-az
TRecfast_xe=az*Calc%xrec(ilo)+bz*Calc%xrec(ihi)+ &
((az**3-az)*Calc%dxrec(ilo)+(bz**3-bz)*Calc%dxrec(ihi))/6._dl
endif
endif
end associate
end function TRecfast_xe
subroutine TRecfast_xe_Tm(this,a, xe, Tm)
class(TRecfast) :: this
real(dl), intent(in) :: a
real(dl), intent(out) :: xe, Tm
real(dl) zst,z,az,bz
integer ilo,ihi
z=1/a-1
associate(Calc => this%Calc)
if (z.ge.Calc%zrec(1)) then
xe=Calc%xrec(1)
Tm = Calc%Tnow/a
else
if (z.le.Calc%zrec(nz)) then
xe=Calc%xrec(nz)
TM = Calc%Tmrec(nz)
else
zst=(zinitial-z)/delta_z
ihi= int(zst)
ilo = ihi+1
az=zst - ihi
bz=1-az
xe=az*Calc%xrec(ilo)+bz*Calc%xrec(ihi)+ &
((az**3-az)*Calc%dxrec(ilo)+(bz**3-bz)*Calc%dxrec(ihi))/6._dl
Tm=az*Calc%Tmrec(ilo)+bz*Calc%Tmrec(ihi)+ &
((az**3-az)*Calc%dTmrec(ilo)+(bz**3-bz)*Calc%dTmrec(ihi))/6._dl
endif
endif
end associate
end subroutine TRecfast_xe_Tm
function TRecfast_version(this) result(version)
class(TRecfast) :: this
character(LEN=:), allocatable :: version
version = Recfast_Version
end function TRecfast_version
subroutine TRecfast_init(this,State, WantTSpin)
use MiscUtils
implicit none
class(TRecfast), target :: this
class(TCAMBdata), target :: State
real(dl) :: Trad,Tmat,Tspin
integer :: I
Type(RecombinationData), pointer :: Calc
logical, intent(in), optional :: WantTSpin
real(dl) :: z,n,x,x0,rhs,x_H,x_He,x_H0,x_He0,H, Yp
real(dl) :: zstart,zend,z_scale
real(dl) :: cw(24)
real(dl), dimension(:,:), allocatable :: w
real(dl) :: y(4)
real(dl) :: C10, tau_21Ts
integer :: ind, nw
real(dl), parameter :: tol=1.D-5 !Tolerance for R-K
procedure(TClassDverk) :: dverk
if (.not. allocated(this%Calc)) allocate(this%Calc)
Calc => this%Calc
select type(State)
class is (CAMBdata)
Calc%State => State
Calc%doTspin = DefaultFalse(WantTSpin)
! 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(*,*)
Calc%n_eq = 3
if (Evolve_Ts) Calc%n_eq=4
allocate(w(Calc%n_eq,9))
Calc%Recombination_saha_z=0.d0
Calc%Tnow = State%CP%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
H = State%CP%H0/100._dl
!Not general, but only for approx
Calc%OmegaT=(State%CP%omch2+State%CP%ombh2)/H**2 !total dark matter + baryons
Calc%OmegaK=State%CP%omk !curvature
! convert the Hubble constant units
Calc%HO = H*bigH
Yp = State%CP%Yhe
! sort out the helium abundance parameters
Calc%mu_H = 1.d0/(1.d0-Yp) !Mass per H atom
Calc%mu_T = not4/(not4-(not4-1.d0)*Yp) !Mass per atom
Calc%fHe = Yp/(not4*(1.d0-Yp)) !n_He_tot / n_H_tot
Calc%Nnow = 3._dl*bigH**2*State%CP%ombh2/(const_eightpi*G*Calc%mu_H*m_H)
n = Calc%Nnow * (1._dl+z)**3
Calc%z_eq = State%z_eq
! Fudge factor to approximate for low z out of equilibrium effect
Calc%fu=this%RECFAST_fudge
! Set initial matter temperature
y(3) = Calc%Tnow*(1._dl+z) !Initial rad. & mat. temperature
Tmat = y(3)
y(4) = Tmat
Tspin = Tmat
call get_init(Calc,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 = Calc%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
z_scale = Calc%Tnow/COBE_CMBTemp * (1+z) -1
if (z_scale > 8000._dl) then
x_H0 = 1._dl
x_He0 = 1._dl
x0 = 1._dl+2._dl*Calc%fHe
y(1) = x_H0
y(2) = x_He0
y(3) = Calc%Tnow*(1._dl+z)
y(4) = y(3)
else if(z_scale > 5000._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
rhs = exp( 1.5d0 * log(CR*Calc%Tnow/(1._dl+z)) &
- CB1_He2/(Calc%Tnow*(1._dl+z)) ) / Calc%Nnow
rhs = rhs*1._dl !ratio of g's is 1 for He++ <-> He+
x0 = 0.5d0 * ( sqrt( (rhs-1._dl-Calc%fHe)**2 &
+ 4._dl*(1._dl+2._dl*Calc%fHe)*rhs) - (rhs-1._dl-Calc%fHe) )
y(1) = x_H0
y(2) = x_He0
y(3) = Calc%Tnow*(1._dl+z)
y(4) = y(3)
else if(z_scale > 3500._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
x0 = x_H0 + Calc%fHe*x_He0
y(1) = x_H0
y(2) = x_He0
y(3) = Calc%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*Calc%Tnow/(1._dl+z)) &
- CB1_He1/(Calc%Tnow*(1._dl+z)) ) / Calc%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+Calc%fHe)*rhs )- (rhs-1._dl))
x0 = x_He0
x_He0 = (x0 - 1._dl)/Calc%fHe
y(1) = x_H0
y(2) = x_He0
y(3) = Calc%Tnow*(1._dl+z)
y(4) = y(3)
else if (y(1) > 0.99d0) then
rhs = exp( 1.5d0 * log(CR*Calc%Tnow/(1._dl+z)) &
- CB1/(Calc%Tnow*(1._dl+z)) ) / Calc%Nnow
x_H0 = 0.5d0 * (sqrt( rhs**2+4._dl*rhs ) - rhs )
call DVERK(this,3,ION,zstart,y,zend,tol,ind,cw,nw,w)
y(1) = x_H0
x0 = y(1) + Calc%fHe*y(2)
y(4)=y(3)
else
call DVERK(this,nw,ION,zstart,y,zend,tol,ind,cw,nw,w)
x0 = y(1) + Calc%fHe*y(2)
end if
Trad = Calc%Tnow * (1._dl+zend)
Tmat = y(3)
x_H = y(1)
x_He = y(2)
x = x0
Calc%zrec(i)=zend
Calc%xrec(i)=x
Calc%tmrec(i) = Tmat
if (Calc%doTspin) then
if (Evolve_Ts .and. zend< 1/Do21cm_minev-1 ) then
Tspin = y(4)
else
C10 = Calc%Nnow * (1._dl+zend)**3*(kappa_HH_21cm(Tmat,.false.)*(1-x_H) &
+ kappa_eH_21cm(Tmat,.false.)*x)
tau_21Ts = line21_const*Calc%NNow*(1+zend)*dtauda(State,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
Calc%tsrec(i) = Tspin
end if
! write (*,'(5E15.5)') zend, Trad, Tmat, Tspin, x
end do
call spline_def(Calc%zrec,Calc%xrec,nz,Calc%dxrec)
call spline_def(Calc%zrec,Calc%tmrec,nz,Calc%dtmrec)
if (Calc%doTspin) then
call spline_def(Calc%zrec,Calc%tsrec,nz,Calc%dtsrec)
end if
class default
call MpiStop('Wrong state type')
end select
end subroutine TRecfast_init
! ===============================================================
subroutine GET_INIT(Calc,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
Type(RecombinationData) :: Calc
real(dl) z,x0,rhs,x_H0,x_He0, z_scale
z_scale = Calc%Tnow/COBE_CMBTemp*(z+1)-1
if(z_scale > 8000._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
x0 = 1._dl+2._dl*Calc%fHe
else if(z_scale > 3500._dl)then
x_H0 = 1._dl
x_He0 = 1._dl
rhs = exp( 1.5d0 * log(CR*Calc%Tnow/(1._dl+z)) &
- CB1_He2/(Calc%Tnow*(1._dl+z)) ) / Calc%Nnow
rhs = rhs*1._dl !ratio of g's is 1 for He++ <-> He+
x0 = 0.5d0 * ( sqrt( (rhs-1._dl-Calc%fHe)**2 &
+ 4._dl*(1._dl+2._dl*Calc%fHe)*rhs) - (rhs-1._dl-Calc%fHe) )
else if(z_scale > 2000._dl)then
x_H0 = 1._dl
rhs = exp( 1.5d0 * log(CR*Calc%Tnow/(1._dl+z)) &
- CB1_He1/(Calc%Tnow*(1._dl+z)) ) / Calc%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+Calc%fHe)*rhs )- (rhs-1._dl))
x0 = x_He0
x_He0 = (x0 - 1._dl)/Calc%fHe
else
rhs = exp( 1.5d0 * log(CR*Calc%Tnow/(1._dl+z)) &
- CB1/(Calc%Tnow*(1._dl+z)) ) / Calc%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(this,Ndim,z,Y,f)
class(TRecfast), target :: this
integer Ndim
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) C10, dHdz, z_scale
type(RecombinationData), pointer :: Recomb
Recomb => this%Calc
! 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 + Recomb%fHe * x_He
Tmat = y(3)
! Tspin = y(4)
n = Recomb%Nnow * (1._dl+z)**3
n_He = Recomb%fHe * Recomb%Nnow * (1._dl+z)**3
Trad = Recomb%Tnow * (1._dl+z)
Hz = 1/dtauda(Recomb%State,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. this%RECFAST_Hswitch) then
K = CK/Hz !Peebles coefficient K=lambda_a^3/8piH
else
!c fit a double Gaussian correction function
z_scale = this%Calc%Tnow/COBE_CMBTemp*(1+z)-1
K = CK/Hz*(1.0d0 &
+this%AGauss1*exp(-((log(1.0d0+z_scale)-this%zGauss1)/this%wGauss1)**2.d0) &
+this%AGauss2*exp(-((log(1.0d0+z_scale)-this%zGauss2)/this%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 = this%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*Recomb%fHe*(1.d0-x_He)*C*C &
/(dsqrt(const_pi)*sigma_He_2Ps*const_eightpi*Doppler*(1.d0-x_H)) &
/((C*L_He_2p)**2.d0)
pb = 0.36d0 !value from KIV (2007)
qb = this%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 /(const_eightpi*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*Recomb%fHe*(1.d0-x_He)*C*C &
/(dsqrt(const_pi)*sigma_He_2Pt*const_eightpi*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+Recomb%fHe)/x !Thomson time
timeH=2./(3.*Recomb%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))
Recomb%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/Recomb%fu+K*Lambda*n*(1.d0-x_H)/Recomb%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 = (Recomb%HO**2/2.d0/Hz)*(4.d0*(1.d0+z)**3/(1.d0+Recomb%z_eq)*Recomb%OmegaT &
+ 3.d0*Recomb%OmegaT*(1.d0+z)**2 + 2.d0*Recomb%OmegaK*(1.d0+z) )
epsilon = Hz*(1.d0+x+Recomb%fHe)/(CT*Trad**3*x)
f(3) = Recomb%Tnow &
+ epsilon*((1.d0+Recomb%fHe)/(1.d0+Recomb%fHe+x))*((f(1)+Recomb%fHe*f(2))/x) &
- epsilon* dHdz/Hz + 3.0d0*epsilon/(1.d0+z)
else
f(3)= CT * (Trad**4) * x / (1._dl+x+Recomb%fHe) &
* (Tmat-Trad) / (Hz*(1._dl+z)) + 2._dl*Tmat/(1._dl+z)
end if
if (evolve_Ts) then
! follow the matter temperature once it has a chance of diverging
if (timeTh < H_frac*timeH) then
f(4) = Recomb%Tnow !spin follows Trad and Tmat
else
if (z< 1/Do21cm_minev-1) then
Tspin = y(4)
C10 = n*(kappa_HH_21cm(Tmat,.false.)*(1-x_H) + kappa_eH_21cm(Tmat,.false.)*x)
f(4) = 4*Tspin/Hz/(1+z)*( (Tspin/Tmat-1._dl)*C10 + Trad/T_21cm*(Tspin/Trad-1._dl)*A10) - f(1)*Tspin/(1-x_H)
else
f(4)=f(3)
end if
end if
end if
end subroutine ION
function TRecfast_dDeltaxe_dtau(this,a, Delta_xe,Delta_nH, Delta_Tm, hdot, kvb,adotoa)
!d x_e/d tau assuming Helium all neutral and temperature perturbations negligible
!it is not accurate for x_e of order 1
class(TRecfast) :: this
real(dl) TRecfast_dDeltaxe_dtau
real(dl), intent(in):: a, Delta_xe,Delta_nH, Delta_Tm, hdot, kvb,adotoa
real(dl) Delta_Tg