Non-equilibrium H$_2$ formation in the early Universe: energy exchanges, rate coefficients and spectral distortions

0.33 MB·English

by  Carla Maria Coppola

Draftversion January10,2012
PreprinttypesetusingLATEXstyleemulateapjv.03/07/07
NON-EQUILIBRIUM H FORMATION IN THE EARLY UNIVERSE:
2
ENERGY EXCHANGES, RATE COEFFICIENTS AND SPECTRAL DISTORTIONS
C. M. Coppola1,2, R. D’Introno3, D. Galli4, J. Tennyson2, S. Longo1,5
Draft version January 10, 2012
ABSTRACT
EnergyexchangeprocessesplayacrucialroleintheearlyUniverse,affectingthethermalbalanceand
2 the dynamicalevolutionof the primordialgas. In the presentwork we focus onthe consequences ofa
1 non-thermaldistributionofthelevelpopulationsofH : first,wedeterminetheexcitationtemperatures
2
0 ofvibrationaltransitionsandthenon-equilibriumheattransfer;second,wecomparethemodifications
2
to chemical reaction rate coefficients with respect to the values obtained assuming local thermody-

n namic equilibrium; third, we compute the spectral distortions to the cosmic background radiation
a generated by the formation of H in vibrationally excited levels. We conclude that non-equilibrium
2
J processes cannot be ignored in cosmological simulations of the evolution of baryons, although their

9 observational signatures remain below current limits of detection. New fits to the equilibrium and
non-equilibrium heat transfer functions are provided.

] Subject headings: molecular processes; cosmology: early Universe, cosmic microwave background
O
C
. 1. INTRODUCTION (although considering specific rotational transitions the
h critical density is below the baryon one up to lower red-
Understanding the thermal evolution of the Universe
p shift z ≈ 500). In addition, several physical phenomena
- in the epoch in which atoms and molecules formed is a
o crucialsteptoproperlymodelthebirthofthefirstbound (e.g. shocks) and chemical processes can produce de-
r structures (e.g., Flower & Pineau des Forˆets 2001). In viations from LTE. In particular, most of the gas-phase
t molecularformationprocessesselectivelyproducespecies
s particular,thebalancebetweencoolingandheatingpro-
a in states that deviate significantly from LTE.
cesseshas to be takeninto accountandmodeled accord-
[ In the case of the early Universe, Coppola et al.
ingtothechemicalandphysicalprocessesoccuringinthe
(2011a) (hereafter C11)computedthe vibrationaldistri-
1 primordial plasma. It is well established that Lyman-
v α cooling is effective at gas temperature higher than bution of H2 and H+2 formed at redshifts 10<z < 1000
0 ∼ 8000 K, corresponding to redshifts z & 2700, while and found that high sovrathermal tails are present es-
2 primordial molecules, in particular H and HD formed pecially at low z. The existence of non-equilibrium fea-
2
7 at z . 1000, are the most efficient cooling agents of the tures in the level populations of H2 is important, be-
1 pristine plasma at lower temperatures. cause of the role of this species as a coolant of primor-
1. Several authors have calculated the heating and dialgas. Inthepresentwork,weextendtheworkofC11
0 cooling functions of the primordial molecular species: to study the followingphysicalquantities relevantto the
2 Palla et al. (1983), Lepp & Shull (1984), Puy et al. non-equilibrium energy exchange in the primordial Uni-
1 (1993), Le Bourlot et al. (1999), Puy & Signore (1996), verse: excitation temperatures, heat transfer functions,
: Galli & Palla 1998 (hereafter GP98), Coppola et al. reaction rates and spectral distortions of cosmic back-
v
i (2011b). One of the standard assumptions in these cal- ground radiation (CBR).
X culations is that the population of internalstates can be 2. REDUCEDMODEL:ORTHO-ANDPARA-STATES
r described by a Boltzmann distribution. This hypothesis
a is valid in many astrophysical environments where the Becausethe number ofroto-vibrationallevelsinvolved
in the kinetics of H is too high (≈ 300) for a direct ex-
density is sufficiently high to bring the internal degrees 2
tensionoftheapproachusedbyC11,inthepresentwork
offreedomtoaconditionoflocalthermodynamicequilib-
weimplementareducedmodelinordertoprovideasim-
rium(LTE).Foragivenspecies,e.g. H ,thisconditionis
2 pler starting point for more extended calculations. This
quantifiedintermsofacriticaldensityn (H)definedas
cr model is based on the assumptions that: (1) the most
the ratio between radiative and collisional de-excitation
importantchannelsdeterminingthepopulationofvibra-
coefficientsofH . Itiseasytocheckthatevenathighest
redshifts (z ≈ 12000), the ambient baryon density n is tional states are the associative detachment reaction
b
∼2ordersofmagnitudebelowthecriticaldensityncr(H) H−+H→H2+e,
and the spontaneous and stimulated radiative transi-
Electronicaddress: [email protected]
tions between roto-vibrational levels; and (2) a steady-
1Universit`a degli Studi di Bari, Dipartimento di Chimica, Via
Orabona4,I-70126Bari,Italy state approximation can be applied to the kinetics of
2DepartmentofPhysicsandAstronomy,UniversityCollegeLon- vibrational levels. Under these hypotheses, two steady-
don,GowerStreet,LondonWC1E6BT state Master Equations (for ortho- and para- states, re-
3Universit`a degli Studi di Bari, Dipartimento di Fisica, Via
spectively) can be written for the 15 vibrational levels
Amendola173,I-70126Bari,Italy
4INAF-OsservatorioAstrofisicodiArcetri,LargoE.Fermi5,I- (i=0,14) of H2:
50125Firenze,Italy
5IMIP-CNR, Section of Bari, via Amendola 122/D, I-70126 fi Rij − Rjifj =kifH−fHnb, (1)
Bari,Italy
j,i=6 j j,i=6 j
X X
2 Coppola et al.
where fi is the fractional abundance of H2 in the ith 100
level, fH− and fH those of H− and H respectively, Rij vv==10 vv==34 vv==67 v=v=190 vv==1123
is the matrix of radiative coefficients including absorp- 10-5 v=2 v=5 v=8 v=11 v=14
tion processes, calculated as in C11 averaging over the
initial rotational levels and summing over the final ones 10-10
e
the Einstein coefficients computed by Wolniewicz et al. nc
(1998). The values of fH− and fH are taken from the unda 10-15
complete kinetic model by C11. The rate coefficients k ab
for the associative detachment reaction for the ith vii- onal 10-20
brational level formation have been evaluated using the acti
cross-sections σ calculated by C˘´ıˇzek et al. (1998) sum- fr 10-25
ij
ming over all final rotational states:
10-30
jmax(i)
k (T )= k˜ (T ), (2) 10-35
i m ij m 1000 100 10
j=0 z
X
where Fig. 1.— Fractional abundances of the vibrational levels of H2
accordingtothereducedsteady-statemodeldescribedinthetext.
8 ∞
k˜ (T )= Eσ (E)e−E/(kBTm)dE,
ij m sπµ(kBTm)3 Z0 ij where Ev and nv represent the energy and fractional
(3) abundance of the vth vibrational level, respectively, and
with µ reduced mass of the system and Tm matter tem- kB is the Boltzmann constant.
perature. It should be noted that experimental results In Figure 2, the excitation temperature of the transi-
byKreckel et al.(2010)recentlyshowedverygoodagree- tions 0−v is compared to the temperatures of matter
ment with these quantum calculations. andradiationasafunctionofz. Forbetterclarity,afew
The equation for the vibrational ground state f in curves have been repeated in the two panels. The fig-
0
Eq. 1 is replaced by the normalization condition ureshowsthatvibrationdecouplesfromthetranslational
motion,andthishappensathigherzthanthedecoupling
fi =fH2, (4) of radiation and matter; for the most excited levels this
i corresponds to an epoch where the conditions are near
X
equilibrium. Later, after the decoupling of matter and
where f (z) is the fraction of H which is also taken
H2 2 radiation, T is slightly largerthan the radiationtem-
from the complete model. Figure 1 shows the fractional 0−1
peratureT ; the excitationtemperatures T for higher
abundance of vibrational levels obtained using the re- r 0−v
v are progressively higher. This result can be explained
ducedmodeldescribedinthepresentsection. Theresults
byconsideringthe reactionnetwork: the vibrationallev-
forallf aresatisfactoryclosetothoseobtainedwiththe
i
elsareformedbyassociativereactionsthatpreferentially
fully kinetic model shown in Figure 10 of C11, at least
populatehighlyexcitedlevels,whilethevibrationalman-
for not too high values of z. These results show that the
ifold as a whole is coupled to the CBR (which provides
hypothesis of steady-state can be applied to the present
a heat sink) much better than to the matter; this lat-
problemwith enoughconfidence to proceedto the study
ter coupling occurs via the relatively ineffective H /H
of the ortho- and para- states. For this we calculate two 2
VT processes. Therefore, all levels pairs are expected to
sets of rate coefficients k (T ) and k (T ) for
i,ortho m i,para m
be warmer than the radiation field, but the lowest pairs
each vibrational level i,
are closer to equilibrium with the radiation because the
k (T )= k˜ (T ), chemical heating is lower.
i,ortho m ij m
TheseparationofT andT occursatz ≈100,which
j odd 0−1 r
X (5) brings this phenomenon not far from potential indirect
k (T )= k˜ (T ).
i,para m ij m observation (e.g. effects on reaction rates of processes
j even and consequent different fractional abundances of chem-
X
The R coefficients are also thermally averaged over a ical species at lower z). Another relevant feature of our
ij
partialdistribution. The first equation of eachsystem is calculation is that T0−1 is stable at about 200 K at the
replaced by the normalization condition age of formation of the first structures. Since the 1→0
transition is an important heat radiator, our results in-
fi,ortho/para =1, (6) dicate that T0−1 is a more appropriate initial condition
Xi for the vibrational temperature of H2 in hydrodynamic
which means that we are calculating the vibrationaldis- collapse models than Tr, which is considerably lower. In
tribution of each of the two species but not the total Table 1 the values for zdec (redshift at which excitation
fraction of ortho- and para- hydrogen, which cannot be temperature decouples from radiation temperature) and
calculated by a steady-state approach. zfreeze−out (redshift at which the freeze-out temperature
is reached) are reported for each T ; the former are
0−i
3. EXCITATIONTEMPERATURES evaluated considering a relative deviation from T larger
r
For each transition 0–v, the excitation temperature is than1%,whilethelatterarecalculatedsearchingforrel-
defined as ativedeviationofT0−i ateachz fromthevalueatz =10
Ev−E0 smaller than 10−3.
T = , (7)
0−v k ln(n /n ) Figure 3 shows the values of the relative difference be-
B 0 v
Non-equilibrium H formation in the early Universe 3
2
104
0.03
T
0-1
T
0-2
T
0.02 T0-3
0-4
T
103 ence 0.01 TTT000---567
T [K] e differ 0 0-8
g
a
102 Tr ent
TTm erc -0.01
0-1 P
T
0-2
T
T0-3 -0.02
0-4
T
101 0-5
1000 100 -0.03
z 1000 100 10
z
104
0.4
0.2
103
e
c 0
n
T [K] differe -0.2
e
102 ag -0.4
nt
e
c
101 TT0T-m6r TTT000---789 TTT000---111012 TT00--1134 Per --00..86 TTTTT000---0111-0129
1000 100 -1 T0-13
0-14
z
1000 100 10
z
Fig. 2.—H2excitationtemperaturesT0−v,comparedtothetem-
peratureoftheradiation(solidcurve)andmatter(dashed curve).
Top panel: v=1tov=5;bottom panel: v=6tov=14. Fig. 3.—Ratio of excitation temperatures forortho- and para-
statesgivenas(T0−i,ortho−T0−i,para)/T0−i,ortho. Toppanel: v=
1tov=8;bottom panel: v=9tov=14.
TABLE 1 addressedaccuratelyinfuturestudies,i.e. bysolvingthe
Decoupling redshiftof excitationtemperatures
Master Equation for a full rotovibrationalmanifold.
ET0x−ci1t4ationtemperature(K) z6d9e7c z3f8r2eeze−out 4. HEATTRANSFERFUNCTION
T0−13 689 379 In this paper we discuss the role of chemical energy
T0−12 676 377 fromexothermic reaction,which is ultimately dissipated
T0−11 647 379 either into radiation or into the thermal energy of H
T0−10 627 382 atoms. This energy flow is described by heating and
T0−9 593 537 cooling functions, usually indicated with the symbols Γ
T0−8 551 524
and Λ, respectively. We consider the net molecular heat
T0−7 511 389
T0−6 457 389 transfer, defined as the sum of all radiative excitations
T0−5 391 377 ofH2 followedbycollisionalde-excitationswithHatoms
T0−4 363 346 andallcollisionalexcitationsfollowedbyradiativedecay:
T0−3 300 313
1
T0−2 198 174 Φ(T ,T )=(Γ−Λ) = ×
T0−1 108 84 m r H2 n(H2)
(n(v′,j′)(Tr)·k(v′,j′)→(v,j)(Tm) (8)
(v′,jX′)<(v,j)
tween the vibrational excitation temperatures of ortho- −n(v,j)(Tr)·k(v,j)→(v′,j′)(Tm)(Ev,j −Ev′,j′),
and para- states. As can be seen, significant deviations
between vibrational temperature of states with different where Tm and Tr are the temperatures of matter
rotational symmetry can be detected at low z and high and radiation, k(v′,j′)→(v,j) is the VT (vibrational-
i. These differences suggest that the issue of rotational translational) rate coefficient for H+H (v′,j′) →
2
non-equilibrium could be important and deserves to be H+H (v,j), and n describes the distribution of ro-
2 (v,j)
4 Coppola et al.
tovibrational levels, 10-20
cooling (equilibrium vdf)
10-21 cooling (non equilibrium vdf)
gjnv(2j+1)exp −Ev,kjB−TErv,0 -1s] 10-22 GP98
n(v,j)(Tr)= Zv(T(cid:16)r) (cid:17), (9) 3g cm 10-23
with g equal to 1/4 and 3/4 for the para- and ortho- n [er 10-24
j o
states,respectively,andZ (T )rotationalpartitionfunc- cti 10-25
v r n
u
tion for the vth level. It can be seen from Eqs. (8)-(9) er f 10-26
that the rotational energy is distributed according to ansf 10-27
ethffieciBenotltszdmeapnenndlaownwthitehmtaetmteprertaetmupreerTatruwrehiTlem.VTThcuos-, heat tr 10-28
the heattransferfunction depends ontwotemperatures, 10-29
asaconsequenceofthedifferentcouplingsoftheinternal 10-30
degrees of freedom of molecules with the radiation and 10 100 1000
thematter: afastercouplingoccuringbetweenradiation T [K]
and rotation, a slower coupling between translation and 10-20
cooling (equilibrium vdf)
vibration.
10-21 cooling (non equilibrium vdf)
caWsese, ceoxrprelosrpeonbdointhg toeqduiiffliebrreinutmvaaluneds onfonnv-;eqinuitlihberifuomr- 3-1m s] 10-22 GP98
mer case the vibrational levels are distributed following c 10-23
g
twheeaBdoolptztmthaennlepveolppuolaptuiolantieoqnusarteiosnu,ltwinhgerfreoamsinthtehekilnatetteicr on [er 10-24
modelofC11. VT ratecoefficientshavebeentakenfrom ncti 10-25
u
Esposito et al. (1999, 2001). These coefficients are given er f 10-26
asfunctions ofthe initial andfinalrotovibrationalquan- ansf 10-27
tum numbers and of temperature allowing the inclusion at tr 10-28
of the full sets of collisional transitions in our calcula- e
h
tion. The equilibrium one-temperature heat transfer us- 10-29
ing the same rate coefficients has been compared with 10-30
the analytical expression for the cooling function given 10 100 1000
byGP98. InGP98,thegaswasnotembeddedinanyra- T [K]
diationfield;consequently,onlycollisionalde-excitations
were considered. Fig. 4.—Heat transfer function(Γ−Λ)H2 as afunction of the
As it can be seen from Figure 4, the non-equilibrium radiation temperature Tr. Dotted line: GP98 fit for the cooling
heat transfer function decreases more rapidly than the function;solid line: LTEcalculationwithpresentVTcoefficients;
dashedline: non-equilibriumvibrationaldistributioncontribution.
equilibrium one at lower temperature, because of the in-
Calculation are reported both in the one-temperature case (top
creasedefficiency inthe energyexchangedue to the long panel)andthetwo-temperatures one(bottom panel).
sovrathermaltails in the vibrationaldistribution. In the
two temperatures calculation, the deviation is amplified
as a consequence of the strong decoupling in the energy TABLE 2
exchange among degrees of freedom and of the differ- H2 heattransfer function
ent trend of the gas and radiation temperatures. The Fittingcoefficients
first effect, which is due to the deviation of the vibra- equilibrium a0=−145.05
tional population from the equilibrium distribution (es- a1=136.085
sentially those of the lowest levels) is seen at z ≈ 1000, a2=−58.6885
and amounts to a factor of ∼ 2. The effect of the sep- a3=11.2688
aration of T and T occurs at low z where these two a4=−0.786142
temperaturesmare conrsiderably different. Consequently, non-equilibrium a0=−393.441
in the computation of the heat transfer function at least a1=588.474
two temperatures must be used: one is the temperature a2=−380.78
a3=123.858
of the level population and the other is the translational a4=−20.1349
temperature. This difference is evident in Eq. (8). a5=1.30753
All heat transfer functions available in the literature
are calculated assuming a single temperature, usually
set equal to T . Such usage cannot capture the second
r
non-equilibrium effect described above, since k(T ,T ) obtained in the form:
m r
is implicitly set equal to k(T ). A better solution is to
r N
useinthe contextofearlyUniversemodels heattransfer log Φ= a (log T )n. (10)
10 n 10 r
functionscalculatedincludingnon-equilibriumeffectsal-
n=0
thoughfitted later as afunction of a singletemperature. X
Fits for the non-equilibrium case using the present two- The coefficients an are listed in Table 2. For the non-
temperatures model and the equilibrium case using the equilibriumcase,thevalidityofthefitisuptoT ≈100K
newestavailabledata by Esposito et al.(1999,2001) are (additional data are available upon request).
Non-equilibrium H formation in the early Universe 5
2
10-16 10-16
LTE LTE
non-equilibrium vdf non-equilibrium vdf
10-18 10-18
3-1ms] 10-20 3-1ms] 10-20
cient [ 10-22 H2(ν)+H+--->H2+(ν’)+H cient [ 10-22 H2(ν)+e---->H+H-
effi effi
o o
e c 10-24 e c 10-24
at at
r r
10-26 10-26
10-28 10-28
1000 100 1000 100
z z
10-16
LTE LTE
non-equilibrium vdf non-equilibrium vdf
10-18 10-22
3-1ms] 10-20 3-1ms]
cient [ 10-22 H2+(ν)+H+--->H+H+H+ cient [ 10-24 H2(ν)+H+--->H+H+H+
effi effi
o o
e c 10-24 e c
at at 10-26
r r
10-26
10-28 10-28
1000 100 1000 100
z z
Fig. 5.— Rate coefficients as a function of z: LTE approxima-
tion(dashed line,fitbyCoppolaetal.2011a)andnon-equilibrium
vibrationaldistributionfunction(solid line).
5. NON-EQUILIBRIUMREACTIONRATES
Using the real non-equilibrium vibrational distribu-
tions, vibrationally resolved rate coefficients have been
recomputed and compared with the corresponding LTE
fits byC11. InFigure5the resultsforthe followingpro-
cesses introduced in the model are shown, both for H
2
andH+: (1)H (v)/H+ chargetransfer(2)H (v)/e− dis-
2 2 2
sociativeattachment(3)H+(v)dissociationbycollisions
2
withH(4)H (v)dissociationbycollisionswithH+. The
2
LTE fit for dissociative attachment of H is taken from
2
Capitelli et al. (2007). Strong deviations from the LTE
fits can be noted, due to the non-equilibrium pattern,
especially at low z, where the hypothesis of Boltzmann
distribution of the vibrational level manifold fails. The
peak at z ≈ 300 corresponds to that on H+ (and con-
2
sequently to H , via the process of charge transfer with
2
H+). It should be noted that, in the case of H disso-
2
ciative attachment, the non-equilibrium calculation fol-
lows the trend reported by Capitelli et al. (2007), where
a simplified model for the non-equilibrium distribution
was assumed.
6 Coppola et al.
6. SPECTRALDISTORTIONSOFTHECBR
10-22 CBR
The photons created in the H formation process pro-
2
duce a distortion of the black-body spectrum of the cos- 10-24
bmpyriocadsbusacotccikoiagntriovouefnHdde2traaocdchicamutriesonntin(inCviBcboRrlal)it.siioSonninasclleoyfteHhxecaimtnedadxHsimt−autaemst -1-1-1Hz s sr] 1100--2286 H+He recombination
rthedesehmiftisssiboenloowfrozvi≈bra1t0io0n(asleteraCnosiptpioonlaswetithalw.a(v2e0l1e1nag)t)h, -2g cm 10-30
λthe≈W2ieµnmpiasrtreodfshtihfteedCBtoRd.ayAnateλarl≈y e1s0t0im–2a0t0e µomf t,hiins J [erν 10-32 ∆v=1
distortion, based on the rovibrational-resolved associa- 10-34 ∆v=2
tive detachment cross sections computed by Bieniek & ∆v=3 ∆v=4
10-36
Dalgarno (1979), was made by Khersonskii (1982) while 10 100 1000
Shchekinov & E´nt´el (1984) developed a model for the ν [cm-1]
molecular hydrogen distortion due to secondary heating
Fig. 6.— CBR spectrum at z=0 together with
processes. We reconsider here the process of vibrational
spectral distortions due to H and He recombina-
emission of primordial H2 molecules with our updated tions (Chluba,Rubin˜o-Mart´ınandSunyaev (2007),
chemical network and with a fully kinetic treatment of Rubin˜o-Mart´ın,ChlubaandSunyaev (2008)) and to non-
the level populations of H . equilibrium vibrational molecular transitions for H2. The
2 contributionofmultiquantumtransitionsupto∆v=4areshown.
For each transition from an upper level v to a lower
u The difficulty of detecting spectral distortions in the
levelv ,withlevelpopulationsn andn anddegeneracy
l u l Wien side of the CBR, in the presence of an infrared
coefficients g and g , the relative perturbation in the
u l background(bothGalacticandextragalactic)severalor-
CBR at the present time is given by
dersofmagnitudebrighter,havebeendiscussedbyWong
∆J et al. (2006). While a direct detection appears challeng-
ν =[S(z )−1]τ(z ), (11)
J int int ing(see alsoSchleicheretal.2008),westressthatanex-
ν (cid:12)z=0 cessofphotonsovertheCBRatwavelengthsshorterthan
(cid:12)
where (cid:12) thepeakcouldrepresentasignificantcontributiontosev-
(cid:12)
g n (z ) −1 hν eral photodestruction processes, as shown by Switzer &
S(zint)= u l int −1 exp ul −1 Hirata (2005) for the photoionization of Li and Hirata
g n (z ) kT (z )
(cid:20) l u int (cid:21) (cid:26) (cid:20) r int (cid:21) (cid:27) &Padmanabhan(2006)forthe photodetachmentofH−.
(12)
Another possibility is fluorescence,i.e. the absorptionof
is the source function, and
the short-wavelengthsnon-thermal photons by atoms or
τ(z )= c3 A gu 1− glnu(zint) nl(zint) , (13) molecules followed by re-emission at longer wavelenghts
int 8πν3 ulg g n (z ) H (z ) in the Rayleigh-Jeans region of the CBR, as suggested
ul l (cid:20) u l int (cid:21) z int by Dubrovich & Lipovka (1995). These issues will be
is the redshift-integrated optical depth (see e.g. Ap-
addressed elsewhere.
pendix A of Bougleux & Galli 1997). In Eqs. (12) and
(13), zint is the interaction redshift, at which the ob- 7. CONCLUSIONS
served frequency ν is equal to the redshifted frequency
We have considered several vibrational non-
ν of the transition, i.e.
ul equilibrium effects on the chemistry and physics of
ν(1+z )=ν , (14) early Universe. Although our present calculations raise
int ul
several questions about the very use of the concept of
A are the Einstein coefficients, and H is the Hubble
ul z temperature in the early Universe, the most important
function
issueconcerningthermaltransferandchemicalreactivity
H =H [Ω (1+z)4+Ω (1+z)3+Ω (1+z)2+Ω ]1/2 has been addressed. Our calculations show that the
z 0 r m k Λ
(15) differences between the excitation, translation and radi-
(see Coppolaet al.(2011a)for adefinition ofthe cosmo- ationtemperaturescanaffectthe heattransferfunctions
logical constants and their adopted values). of important species, an effect here demonstrated for
Figure6showstheemissionproducedbyH transitions H . Our results underline the necessity to fully include
2 2
with ∆v =1, 2, 3 and 4 in the frequency range ν = 10– the consequences of the temperature separations which
1000 cm−1, corresponding to wavelengths λ = 10 µm– occur at different epochs in the chemical and physical
1 mm. To avoid confusion, only the first 4 transitions evolution of the early Universe.
for each ∆v are shown (i.e., v = 1 → 0, 2 → 1, Excitation temperatures appear to be higher than
3 → 2 and 4 → 3 for ∆v = 1, etc.). The figure also radiation temperature at low z; this “chemical” pre-
shows the CBR in the Wien region, and, for compar- heating should be considered while modeling the for-
ison, the spectral features produced by the cosmologi- mation of galaxies, together with virialization heating
cal recombination of H and He (e.g. Chluba & Sunyaev and other physical mechanisms usually suggested (e.g.
2007,2008,Chluba, Rubin˜o-Mart´ın and Sunyaev (2007), Mo et al. (2005), Wang & Abel (1996)). We have also
Rubin˜o-Mart´ın, Chluba and Sunyaev (2008)). The lat- assessed the hypothesis of steady-state for the vibra-
teraremainlyformedbyredshiftedLy-αandtwo-photon tional distribution presenting a reduced kinetic model,
transitions of H and the corresponding lines from He and calculated the deviations between vibrational tem-
(Chluba & Sunyaev 2010 have recently produced new peraturesofortho-andpara-statesasafirststeptowards
results for this last contribution). a full non-equilibrium rotovibrational kinetics. Heat
Non-equilibrium H formation in the early Universe 7
2
transfer functions are calculated for both equilibrium WehavecomputedthespectraldeviationstotheCBR
and non-equilibrium cases, considering also a novel two- duetothenon-equilibriumlevelpopulations,considering
temperatures approach that takes into account the dif- all the transitions. Although the present Planck exper-
ferentratesofenergyexchangeamongmoleculardegrees iment and the upcoming James Webb Space Telescope
of freedom. For pure vibrational transitions, the critical (JWST) are able to detect galaxies at high redshift, a
density is greater than the baryon density, so that the direct observation of this effect is challenging; for this
hypothesis of non-equilibrium is also valid at higher z. reason, an alternative study of the non-thermal vibra-
Resolvingrotationsandvibrationsgivesdifferentresults, tionalphotonsonthephotochemicalpathwaysofatomic
making the limit for z lower (e.g. for the (0,2)→ (0,0) and molecular kinetic should be undertaken.
transitionthe criticaldensity is about ≈2.7x107 m−3 at
z ≈500.)
We have evaluated the effects of vibrational non-
equilibrium on reaction rates. A general increase has ACKNOWLEDGMENTS
been pointed out because of the formation of long We are gratefulto Jens Chluba for havingmade avail-
sovrathermal tail in the vibrational distribution, espe- able his data and for helpful discussions. CMC and SL
cially at low z; this evidence should be added to the acknowledgefinancial support of MIUR-Universit`a degli
increase of rate coefficients due to the inclusion of the Studi di Bari, (“fondi di Ateneo 2011 ”). This work
entire vibrational manifold, that by itself can affect in a has also been partially supported by the FP7 project
deep way the fate of the system modeled (as described ”Phys4Entry”-grantagreementn. 242311. JTacknowl-
bySethi et al.(2010)forthedissociativeattachmentpro- edges support from ERC Advanced Investigator Project
cess). 267219.
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