Motivations and general formulation

3.4 New method of solution: the effective multilevel atom

3.4.1 Motivations and general formulation

We first note that the only quantity of importance for CMB power spectrum calculations is the free electron fraction as a function of redshift, xe(z). The populations of the excited states are calculated only as an intermediate step — if they are desired (for example, to calculate Hαscattering features [84] or the recombination spectrum [74, 75, 35]), the populations of the excited states can be obtained by solving Eqs. (3.12, 3.13) once the free electron fraction is known. Furthermore, only the

“interface” states 2sandnp(andns, ndwhen accounting for two-photon processes from higher-order states) are directly connected to the ground state and directly appear in the evolution equation for the free electron fraction Eq. (3.14). All other (“interior”) excited states are only connected with other excited states or with the continuum, through optically thin radiative transitions (and to a lesser extent through collisions [36]). Interior states are only transitional states: an electron in the “interior” rapidly transitions through spontaneous and stimulated decays or absorptions caused by the blackbody radiation field (or collisions with free electrons and protons), until it is either photoionized, or reaches an interface state. There can be a very large number of transitions before any of these outcomes occurs, but the passage through the “interior” is always very short compared to the overall recombination timescale, and can be considered as instantaneous (for the same reason that the steady-state approximation is valid in the standard MLA formulation).

Instead of computing the fraction of hydrogen in each interior stateK, one can rather evaluate the probabilities that an atom initially in the interior stateKultimately reaches one of the interface states or gets photoionized. Of course, after reaching an interface state, the atom may perfectly transition back to an interior state, or get photoionized. However, we consider the probability of first reaching a given interface state before any other one, which is uniquely defined. For an atom in the interior stateK, we denote byP_Kⁱ the probability of ultimately reaching the interface statei, and P_K^e the probability of ultimately being photoionized. An excited atom initially in the interior stateK has a probabilityRK,i/ΓK todirectly decay to the interface statei, where ΓK is the total width (or inverse lifetime) of the stateK:

ΓK≡ X

L6=K

RK,L+X

j

RK,j+βK. (3.15)

The atom may also first decay or get excited to another interior state L 6= K, with a probabil-

P

_K^e

K P

_Kⁱ

i e

⁻

+ p

⁺

H(1s)

“interior”

states

“interface” states

Figure 3.1: Schematic representation of the formulation of the recombination problem adopted in this work. Dotted arrows represent possibly numerous fast transitions within the “interior”.

ity RK,L/ΓK, from which it may then eventually reach the interface state i with probabilityP_Lⁱ. Mathematically, these various paths translate into the following system of linear equations for the probabilitiesP_Kⁱ :

P_Kⁱ = X

L6=K

RK,L

ΓK

P_Lⁱ +RK,i

ΓK

. (3.16)

Similarly, theP_K^e must satisfy the self-consistency relations:

P_K^e = X

L6=K

RK,L

ΓK

P_L^e+βK

ΓK

, (3.17)

We show in Appendix 3.A.2 that these linear systems are invertible and therefore uniquely determine P_Kⁱ andP_K^e. In Appendix 3.A.3 we prove the complementarity relation,

X

i

P_Kⁱ +P_K^e = 1, (3.18)

which has the simple physical interpretation that an atom in theKth interior state eventually reaches an interface state or is photoionized with unit probability.

Once these probabilities are known, it is possible to describe the large number of transitions between all the states in a simplified manner, in terms of effective rates into and out of the interface states. To clarify the explanation, we illustrate in Figure 3.1 the processes described below.

An electron and a proton can effectively recombine to the interface stateieither through a direct recombination (with coefficientαi), or following a recombination to an interior stateK (with coef- ficientαK), from which a sequence of interior transitions may ultimately lead to the interface state

iwith probability P_Kⁱ . The effective recombination coefficient to the interface stateiis therefore:

Aⁱ≡αi+X

K

αKP_Kⁱ . (3.19)

Conversely, an atom in the interface state i may effectively be ionized either through a direct photoionization (with rateβi), or after being first excited to an interior stateK (with rateRi,K), from which the atom may ultimately be photoionized after a series of interior transitions with probabilityP_K^e. The effective photoionization rate from the interface stateiis therefore:

Bⁱ≡βi+X

K

Ri,KP_K^e. (3.20)

Finally, atoms can effectively transition from an interface stateito another interface statej, either through a direct transition if it is allowed, or after first transitioning through the interior. The effective transfer rate between theith and jth interface states is therefore:

R^i,j≡Ri,j+X

K

Ri,KP_K^j (j 6=i). (3.21)

The rate of change of the population of the interface stateiis therefore:

˙

xi=x²_enHAⁱ+X

j6=i

xjR^j,i+x1sR˜1s,i−xi

Bⁱ+X

j6=i

R^i,j+ ˜Ri,1s

, (3.22)

where we have included the effective transitions described above, as well as transitions from and to the ground state.

The system of equations (3.16–3.22) is exactly equivalent to the standard MLA formulation, as we shall show in Section 3.4.2 below. Moreover, we will show that the rate of change of the free electron fraction, Eq. (3.14), can also be written in the equivalent form

˙

xe=−X

i

nHx²_eAⁱ−xiBⁱ

. (3.23)

Let us now consider the dependences of the effective rates. In the purely radiative case, the prob- abilitiesP_Kⁱ andP_K^e only depend on the radiation temperatureTr, since transitions between excited states and photoionizations only depend on the locally thermal radiation field. As a consequence, the effective recombination rates Aⁱ(Tm, Tr) are only functions of matter and radiation tempera- tures and the effective photoionization and bound-bound rates Bⁱ(Tr) and R^i,j(Tr) are functions of the radiation temperature only. When including collisional transitions, all effective rates become functions of the three variables Tr, Tm and ne. In all cases, effective rates can be easily tabulated once and for all and interpolated when needed for a recombination calculation.

Intuitively, we would expect thatAⁱ,Bⁱ, andR^i,j satisfy the detailed balance relations,

gie^−Ei/TrR^i,j(Tr) =gje^−Ej/TrR^j,i(Tr) (3.24) and

gie^−Ei/TrBⁱ(Tr) = (2πµeTr)^3/2

h³ Aⁱ(Tm=Tr, Tr). (3.25) We show in Appendix 3.A.4 that these equations are indeed valid. This means that we only need to tabulate half of theR^i,j [the other half can be obtained from Eq. (3.24)] and all theAⁱ [theBⁱ can be obtained from Eq. (3.25); in particular, we do not need to solve for theP_K^e].

We note that the probabilities P_Kⁱ, P_K^e are a generalization of the cascade matrix technique introduced by Seaton [85]. Seaton’s calculation assumed a vanishing ambient radiation field, so that electrons can only “cascade down” to lower energy states. In the context of the recombination of the primeval plasma, one cannot ignore the strong thermal radiation field, and electrons rather

“cascade up and down,” following spontaneous and stimulated decays or photon absorption events.

The spirit of our method is however identical to Seaton’s cascade-capture equations [85], where the

“cascading” process is decoupled from the particular process populating the excited states, or from the depopulation of the interface states.

We emphasize that the method presented here is a simple generalization of the effective TLA model, with the case-B recombination coefficient being replaced by the effective recombination co- efficients, which are computed individually for each interface state 2s, 2p,3p, ....