The Astrophysical Journal, 580:29–35, 2002 November 20 # 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A. COSMOLOGICAL RECOMBINATION OF LITHIUM AND ITS EFFECT ON THE MICROWAVE BACKGROUND ANISOTROPIES Phillip C. Stancil,1 Abraham Loeb,2 Matias Zaldarriaga,3 Alexander Dalgarno,2 and Stephen Lepp4 Received 2002 January 11; accepted 2002 July 22 ABSTRACT The cosmological recombination history of lithium, produced during big bang nucleosynthesis, is presented using updated chemistry and cosmological parameters consistent with recent cosmic microwave background (CMB) measurements. For the popular set of cosmological parameters, about a fifth of the lithium ions recombine into neutral atoms by a redshift z $ 400. The neutral lithium atoms scatter resonantly the CMB at 6708 A˚ and distort its intensity and polarization anisotropies at observed wavelengths around $300 lm, as originally suggested by Loeb. The modified anisotropies resulting from the lithium recombination history are calculated for a variety of cosmological models and found to result primarily in a suppression of the power spectrum amplitude. Significant modification of the power spectrum occurs for models that assume a large primordial abundance of lithium. While detection of the lithium signal might prove difficult, it offers the possibility of inferring the lithium primordial abundance and is the only probe proposed to date of the largescale structure of the universe for z $ 500–100. Subject headings: atomic processes — cosmic microwave background — cosmology: theory — early universe — nuclear reactions, nucleosynthesis, abundances 1. INTRODUCTION Lithium is the heaviest stable element produced during big bang nucleosynthesis (Burles, Nollett, & Turner 2001 and references therein). However, because of its low primordial abundance relative to hydrogen, XLi $ 10À10 to 10À9, its significance in the early universe was thought to be restricted to only the formation of rare molecules such as LiH (Stancil, Lepp, & Dalgarno 1996, hereafter SLD96, and references therein). Hence, the recombination history of lithium itself was calculated only as an intermediate step in a chain of chemical reactions and under a set of simplifying assumptions (Palla, Galli, & Silk 1995; SLD96; Stancil, Lepp, & Dalgarno 1998, hereafter SLD98). These preliminary calculations indicated that a substantial fraction (e20%) of the singly charged lithium ions formed neutral atoms by recombination in the redshift interval z $ 400– 500. Recently, Loeb (2001) has shown that the formation of neutral lithium can strongly modify the anisotropy maps of the cosmic microwave background (CMB) through the absorption and reemission at its resonant 6708 A˚ transition from the ground state. Despite the exceedingly low lithium abundance5 left over from the big bang, the resonant optical depth after lithium recombination is expected to be as high as Li i $ 0:4ðXLi=3:8  10À10Þ at a redshift z $ 400 if half of the lithium ions recombine by then. The scattering refers to an observed wavelength of ðzÞ ¼ ð6708 GÞð1 þ zÞ ¼ ð268:3 lmÞ½ð1 þ zÞ=400Š, where XLi % 3:8  10À10 is the latest estimate of the lithium-to-hydrogen number density ratio (Burles et al. 2001). Loeb (2001) argued that resonant scattering would suppress the original anisotropies by a factor of expðÀLi iÞ but would generate new anisotropies in the CMB temperature and polarization on subdegree scales primarily through the Doppler effect. Observations at different far-infrared wavelengths could then probe different thin slices of the early universe. Zaldarriaga & Loeb (2002) calculated in detail the expected anisotropies in both the temperature and polarization of the CMB and assessed their detectability relative to the far-infrared background (FIB). They concluded that the modified polarization signal could be comparable to the expected polarization anisotropies of the FIB on subdegree angular scales (le100). However, these calculations assumed a range of trial values for the neutral fraction of lithium in the redshift range z ¼ 400–500 lacking knowledge of its full redshift dependence. In this paper, we calculate rigorously the recombination history of lithium and its subsequent effect on the CMB anisotropies. In x 2 we discuss the physics responsible for the recombination history of lithium, and in x 3 we obtain the resulting optical depth for the 6708 A˚ transition. Section 4 describes the effects of the lithium optical depth on the CMB power spectra. Finally, x 5 summarizes this work. 1 Department of Physics and Astronomy and Center for Simulational Physics, University of Georgia, Athens, GA 30602-2451; stancil@ physast.uga.edu. 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138; aloeb@cfa.harvard.edu, adalgarno@ cfa.harvard.edu. 3 Physics Department, New York University, 4 Washington Place, New York, NY 10003; matiasz@physics.nyu.edu. 4 Department of Physics, University of Nevada, Las Vegas, NV 89154- 4002; lepp@nevada.edu. 5 Note that by the redshift of interest, all the 7Be produced during big bang nucleosynthesis has been converted to 7Li. 2. THE LITHIUM RECOMBINATION HISTORY In order to assess the importance of the optical depth of lithium on the CMB anisotropies, an accurate determination of the neutral lithium abundance as a function of redshift is required. The post-recombination abundance of Li was first addressed by Lepp & Shull (1984) in their investigation of molecules formed in the early universe. While this work gave no information on the Li and Liþ abundances or the adopted rate coefficients, they were implicitly needed to obtain the LiH abundance, a main focus of their study. Puy 29 30 STANCIL ET AL. Vol. 580 et al. (1993) also investigated the LiH post-recombination abundance but neglected to give any details concerning the evolution of Li and Liþ. The first explicit results for the redshift-dependent Li and Liþ abundances were presented by Palla et al. (1995), although no information was given on the adopted chemistry or reaction rate coefficients. Their results, which varied little with the adopted cosmological model parameters, suggested that the neutral Li formation redshift was zf $ 500 (which we define here as the redshift when the neutral abundance reaches 10% of the elemental abundance nLi) and that the Li ionization fraction n(Liþ)/nLi ! 0:16 as z ! 0. The significant residual Liþ abundance, a result predicted by Dalgarno & Lepp (1987), is a consequence of the depletion of electrons following the cosmological recombination of hydrogen and helium. The related chemistry was described in more detail and improved upon in SLD96, Bougleux & Galli (1997), and Galli & Palla (1998, hereafter GP98). The primary Li formation and destruction mechanisms are radiative recombination, Liþ þ eÀ ! Li þ  ð1Þ (L5),6 and photoionization, Li þ  ! Liþ þ eÀ ð2Þ (L10), respectively. For the former, they adopted the rate coefficients of Verner & Ferland (1996, hereafter VF96), which is based on R-matrix calculations of the photoionization cross sections of Li (Peach, Saraph, & Seaton 1988) but adjusted to match experiment and smoothly fitted to higher energy Hartree-Dirac-Slater calculations (Verner et al. 1996). Through comparison of their photoionization cross sections to experiment, VF96 suggest that their case A recombination coefficients are accurate to better than 10%. The Li photoionization rate for a blackbody radiation field was then obtained by detailed balance from the radiative recombination rate coefficients (see GP98). Using the same cosmological models of Palla et al. (1995; see also model III of SLD96), Bougleux & Galli (1997) found similar results, zf $ 500 and n(Liþ)/nLi ¼ 0:18 at z ¼ 10. However, with further improvements in the chemistry, GP98 predicted zf $ 400 and n(Liþ)/nLi ¼ 0:56 at z ¼ 10. In their comprehensive chemistry, SLD96 adopted the radiative recombination coefficients of Caves & Dalgarno (1972) determined from their model potential calculations of the photoionization cross sections. SLD96 also estimated the photoionization rate by detailed balance and found zf $ 450 and n(Liþ)/nLi ¼ 0:34 at z ¼ 10 (using the same cosmological parameters; model III). Following improvements in the lithium chemistry (see Stancil & Dalgarno 1997, 1998; Stancil & Zygelman 1996), SLD98 obtained zf $ 440 and n(Liþ)/nLi ¼ 0:48 at z ¼ 10. Figure 1 displays the lithium abundances as a function of redshift from various calculations using model III of SLD96. To explore the dependencies of the models on the adopted rate coefficients, we repeated the calculations of SLD98, but using the rate coefficients for reactions (1) and (2) from GP98. The small changes shown in Figure 1 reflect the fact that the radiative recombination rate coefficients of VF96 are only about 5% larger than those determined by Caves & 6 The reaction labels (Lx) correspond to the process x in SLD96. 10−10 Li+ fractional abundance Li SLD98 Using VF96 Li recom. coeff. Using RECFAST xe GP98 10−11 1000 100 z 10 Fig. 1.—Comparison of various calculations of the Li and Liþ fractional abundance (relative to hydrogen) for the cosmological model III of SLD96 (0 ¼ 1, b ¼ 0:0367, h ¼ 0:67, bh2 ¼ 0:0165, Y ¼ 0:242, and XLi ¼ 2:3  10À10). Dalgarno (1972), which is however within the 10% accuracy claimed by VF96. Any differences in the rate coefficients of other processes result in only secondary effects, as can be seen in Figure 2, which displays both the formation and destruction rates [multiplied by the neutral Li fractional abundance nðLiÞ=nH] of the dominant processes and their timescales [per Hubble time 1=HðzÞ]. For z > 500, the CMB field at temperatures T > 1300 K efficiently photoionizes any neutral Li atom produced, but at lower redshifts the photoionization rate (reaction [2]; L10) falls precipitously, allowing the rapid increase in neutral Li through reaction (1) (L5). Figure 2 shows clearly that for z < 500, the only important process is radiative recombination (reaction [1]; L5) with the radiative charge transfer reaction Li þ Hþ ! Liþ þ H þ  ð3Þ Rate*n(Li)/nH (s−1) 10−20 10−23 10−26 10−29 10−32 10−35 1010 105 100 10−5 10−10 (a) L24 L15 L11 L10 L6 L24 L23 L5 L13 L22 L10 L6 L11 L23 L22 (b) 1000 L5 L15 100 z L13 10 Time/Hubble time Fig. 2.—Dominant formation (solid lines) and destruction (dashed lines) processes for neutral lithium for model III of SLD96. (a) Rates [multiplied by the neutral Li fractional abundance nðLiÞ=nH]. (b) Timescale per Hubble time 1=HðzÞ per Li atom. The reaction labels (Lx) correspond to the process x listed in SLD96. No. 1, 2002 COSMOLOGICAL RECOMBINATION OF LITHIUM TABLE 1 Cosmological Model Parameters Parameter 0 .............. K .............. à .............. zeq .............. bh2 ........... 10 .............. b .............. h................. Y ................ XD ............. XLi ............. Model A 0.33 0.0 0.67 1.01E4 0.02 5.479 4.734EÀ2 0.65 0.24714 3.105EÀ5 3.907EÀ10 Model B 0.33 0.0 0.67 1.35E4 0.02 5.479 3.556EÀ2 0.75 0.24714 3.105EÀ5 3.907EÀ10 Model C 0.33 0.0 0.67 1.01E4 0.03 8.219 7.101EÀ2 0.65 0.25108 1.611EÀ5 8.560EÀ10 Model D 0.33 0.0 0.67 1.35E4 0.03 8.219 5.333EÀ2 0.75 0.25108 1.611EÀ5 8.560EÀ10 Model E 0.33 0.0 0.67 1.01E4 0.037 10.14 8.757EÀ2 0.65 0.25298 1.116EÀ5 1.197EÀ9 31 (L13; Stancil & Zygelman 1996) being only marginally significant with a rate about 2 orders of magnitude smaller. Further, neutral Li is mostly made over the narrow redshift interval z $ 500 300, before its formation timescale becomes greater than the Hubble time. While the minor differences in the rate coefficients are unable to explain the discrepancies between the Li and Liþ abundances of GP98 and SLD98, it seems likely that the adopted hydrogen recombination model may be responsible for the difference. For z < 650, GP98 obtained a smaller electron fractional abundance xe, with the SLD98 result being $60% larger by z ¼ 100. This trend is in agreement with the lithium abundances displayed in Figure 1, indicating a significant dependence on electron abundance, as would be expected. Given that the two hydrogen recombination models are rather rudimentary, it would seem appropriate to use an improved model. An ameliorated hydrogen recombination calculation has been performed by Seager, Sasselov, & Scott (2000; see also Seager, Sasselov, & Scott 1999) that allowed for the H excited-state populations to depart from equilibrium. Their calculation, which consists of a 300-level model H atom including all relevant boundbound and bound-free transitions, finds a change for the electron fraction of about 10% from simple equilibrium population results. Seager et al. (1999) have provided the program RECFAST, which emulates their detailed nonequilibrium population calculation. We have coupled RECFAST to our code and obtained the lithium abundances shown in Figure 1, where we have also continued to use the GP98 rate coefficients for reactions (1) and (2) and for the remainder of this work. The neutral Li abundance has increased further since the electron abundance computed by RECFAST is larger than obtained from both GP98 and SLD98, being about 80% larger than GP98 at z ¼ 100. Henceforth, the electron abundances will be obtained from RECFAST. The cosmological parameters (model III of SLD96) used above have been superseded by new values based on analysis of the latest CMB anisotropy experiments (e.g., de Bernardis et al. 2002; Jaffe et al. 2001; Wang, Tegmark, & Zaldarriaga 2002). To explore the range of possible Li abundances allowable within the CMB uncertainties, we considered the five models listed in Table 1. Model A is considered our fiducial case since all parameters are consistent with the best-fit values of Wang et al. (2002), the third column of their Table 2. Model B has the same values, but uses h ¼ 0:75, close to h ¼ 0:72 from the Hubble Space Telescope Hubble Key Project (Freedman et al. 2001). Models A and B are also consistent with the BOOMERANG experiment (de Bernardis et al. 2002). Models C and D follow the result of Jaffe et al. (2001), bh2 ¼ 0:032þÀ00::000054, which is also just barely outside of the 95% confidence limit of Wang et al. (2002; see their Table 5). Model E is an extreme case taken as the upper limit from Jaffe et al. (2001). The corresponding primordial abundances (Y, XD, and XLi) were obtained from Burles et al. (2001). Using the electron abundances determined with RECFAST, the latest lithium chemistry model of SLD98 with some improvements listed in Lepp, Stancil, & Dalgarno (2002) but with the rate coefficients for reactions (1) (L5) and (2) (L10) replaced by the values from GP98, and the parameters listed in Table 1, we have calculated the new lithium neutral fractions7 shown in Figures 3 and 4. The neutral lithium fraction is defined as fLiðzÞ ¼ nðLiÞ nLi ¼ nðLiÞ nðLiÞ þ nðLiþÞ ; ð4Þ where nðLiÞ ¼ fLiðzÞXLinHðzÞ ; ð5Þ nHðzÞ ¼ 1:123  10À5ð1 À Y Þbh2ð1 þ zÞ3 cmÀ3 : ð6Þ The lithium neutral fractions fLi are identical for models A and B and for models C and D, while comparison of all the models reveals very little difference. In fact, for all models zf $ 440 and fLi ¼ 0:55 0:56 [or n(Liþ)/nLi ¼ 0:44 0:45] at z ¼ 10. 3. THE OPTICAL DEPTH OF NEUTRAL LITHIUM The Sobolev optical depth of neutral lithium is given by LiðzÞ ¼ X u 31uAu1 8 gu g1 nðLiÞð1 À H ðzÞ bÞ ð7Þ (see Seager et al. 2000; Bougleux & Galli 1997, eq. [A9]), 7 We do not consider the multiply charged ions Li3þ and Li2þ since they were found by Lepp et al. (2002) to have completely converted to Liþ by z $ 4000 through sequential recombination. 32 STANCIL ET AL. Vol. 580 100 fLi Optical depth or neutral fraction 10−1 τLi Model A Model B 10−2 1000 100 z 10 Fig. 3.—Lithium neutral fraction and optical depth for models A (h ¼ 0:65) and B (h ¼ 0:75); bh2 ¼ 0:02 in both cases. where  HðzÞ2 ¼ H02 1 0 þ zeq ð1 þ zÞ4 þ  0ð1 þ zÞ3 þ K ð1 þ zÞ2 þ à ; ð8Þ depths for models A–E are plotted in Figures 3 and 4. In all cases, Li is maximum near z $ 325 where fLi $ 0:275. Over the five models, the maximum Li varies from $0.1 to $0.66, which can be compared to the redshift-independent optical depths of 0.5–2 adopted by Zaldarriaga & Loeb (2002). Only the extreme model E approaches the optical depths considered by Zaldarriaga & Loeb (2002). As expected (see Loeb 2001; Zaldarriaga & Loeb 2002), the optical depth increases with XLi, but the current results also demonstrate that Li increases with bh2 and decreases with h. Making the approximation HðzÞ ¼ 10=2H0ð1 þ zÞ3=2 and taking model A as the fiducial case, we find that the optical depth reduces to    LiðzÞ ¼ 0:380fLiðzÞ XLi 3:907  10À10 1ÀY 0:75286   bh2  0:65  0:33 1=2 1 þ z 3=2 ; 0:02 h 0 300 ð9Þ which can be compared to equation (1) of Zaldarriaga & Loeb (2002). The numerical trends discussed above are evident in this relation in addition to dependencies on Y and 0. The shape of the optical depth curves arises from a competition between the increasing neutral Li fraction fLi and the expansion of the universe. As a consequence, the possible effects of lithium on the CMB power spectrum are restricted to the epoch of z $ 100 500. b accounts for the fraction of neutral Li in excited states, gu and g1 are the degeneracies of the upper and ground states, respectively, and the sum is over the excited 2p 2PJ finestructure levels J ¼ 1=2; 3=2. We took b ¼ 0 and adopted the wavelengths 6707.76 and 6707.91 A˚ for the 1=2 ! 3=2 and 1=2 ! 1=2 transitions from the NIST Atomic Spectra Database,8 which also gives the transition probability A21 ¼ 3:72  107 sÀ1 for both transitions. The optical Optical depth or neutral fraction 8 See http://aeldata.phy.nist.gov/cgi-bin/AtData/main_asd. 100 τLi fLi Model E 10−1 Model C Model D 10−2 1000 100 z 10 Fig. 4.—Lithium neutral fraction and optical depth for models C (h ¼ 0:65, bh2 ¼ 0:03), D (h ¼ 0:75, bh2 ¼ 0:03), and E (h ¼ 0:65, bh2 ¼ 0:037). 4. EFFECTS ON COSMIC MICROWAVE BACKGROUND ANISOTROPIES We have calculated the signatures of lithium resonant scattering on the CMB anisotropies following the methods described in Zaldarriaga & Loeb (2002). To isolate the effect of the scattering we have kept all cosmological parameters fixed as we varied the optical depth. As a reference model we considered the standard low-density cold dark matter (LCDM) model9 and compared it to models having optical depths equal to those of models A, C, and E of the previous section. In Figure 5 we show the temperature anisotropy power spectra predicted for a wavelength of  ¼ 268:3 lm (corresponding to scattering at a redshift z ¼ 400). In Figure 6 we show the analogous plot for polarization. The effect of the scattering increases as the optical depth increases, so it is smallest in model A and largest in model E. The changes are dominated by two contributions: (1) the Doppler anisotropies induced at the sharp lithium scattering surface and (2) the uniform expðÀLiÞ suppression of the primary anisotropies that were generated at decoupling. At large multipoles l (small angular scales), Figure 6 illustrates how the expðÀLiÞ suppression of the anisotropies reduces the amplitude of power spectra by a factor expðÀ2LiÞ. This effect leaves the shape of the power spectra unchanged. For the models under consideration, the optical depth ranges from 0.1 to 0.6, so on these scales the power spectra are suppressed by a factor between 0.3 and 0.8. As was discussed by Zaldarriaga & Loeb (2002), the Doppler effect reaches a maximum on degree scales and is 9 For LCDM we chose 0 ¼ 0:3, à ¼ 0:7, h ¼ 0:7, and bh2 ¼ 0:02. No. 1, 2002 COSMOLOGICAL RECOMBINATION OF LITHIUM 33 LCDM 1000 LCDM 100 1000 100 10 100 1000 l Fig. 5.—Temperature power spectra of the CMB anisotropies in models with optical depths equal to those of models A, C, and E together with the spectra of the standard LCDM model without lithium scattering. The figure corresponds to an observed wavelength of 268.3 lm. responsible for the change of shape of the power spectra on these scales. This effect can be noticed most dramatically in model E for the temperature and in both models C and E for the polarization. Measurements of the CMB anisotropies at long photon wavelengths, such as those that will be made by the MAP and Planck satellites, will provide a baseline for comparison against which the effect of lithium could be extracted. These 100 LCDM 10 10 100 10 LCDM 1 0.1 0.01 0.001 10 100 l 1000 Fig. 7.—Difference between a long-wavelength power spectrum (such as the one that will be produced by the MAP or Planck satellite) and power spectra at 268.3 lm when the optical depths are those of models A, C, and E. We also show the spectra for the standard LCDM model without lithium scattering. The top panel shows the results for the temperature anisotropies and the bottom one for the polarization anisotropies of the CMB. long-wavelength maps, which are unaffected by the lithium scattering, can be compared with maps obtained at shorter wavelengths. In particular, a useful statistic to study is the power spectra of the difference map (see Zaldarriaga & Loeb 2002 for details). Here we subtract the power spectra obtained at long wavelengths from that observed at 268.3 lm and consider the power spectrum of the residuals. The results for this statistic are presented in Figure 7. Figure 7 indicates that the largest signal relative to the unperturbed spectra is expected in polarization at degree angular scales ðl $ 200Þ. In this range and for both models C and E, the power in the difference spectrum is comparable to (or even larger than) the power in the long-wavelength spectrum. 1 0.1 0.01 10 100 1000 l Fig. 6.—Polarization power spectra of the CMB anisotropies for models with optical depths equal to those of models A, C, and E at an observed wavelength of 268.3 lm compared to the standard LCDM without lithium scattering. 5. DISCUSSION We have calculated the recombination history of lithium for a set of different cosmological parameters (see Table 1). The resulting opacity to the 6708 A˚ resonant transition of neutral lithium (Figs. 3 and 4) distorts both the temperature (Fig. 5) and polarization (Fig. 6) power spectra of the CMB anisotropies over a band of observed photon wavelengths around $300 lm. The distortion is most pronounced in model E, which assumes a relatively high lithium abundance of XLi ¼ 1:2  10À9. Our detailed results agree with earlier estimates by Loeb (2001) and Zaldarriaga & Loeb (2002). The predicted polarization signal is comparable to the expected polarization anisotropies of the FIB (see Zaldarriaga & Loeb 2002). A strategy for eliminating the contribution from the brightest FIB sources would be helpful in isolating the lithium signal. The relevant wavelength range overlaps with the highest frequency channel of the Planck mission (352 lm), with the 34 STANCIL ET AL. Vol. 580 Balloon-borne Large-Aperture Submillimeter Telescope (BLAST),10 which will have 250, 350, and 500 lm channels, and with the proposed balloon-borne Explorer of Diffuse Galactic Emissions (EDGE),11 which will survey 1% of the sky in 10 wavelength bands between 230 and 2000 lm with a resolution ranging from 60 to 140 (see Table 1 in Knox et al. 2001). However, in order to optimize the detection of the lithium signature on the CMB anisotropies, a new instrument design is required with multiple narrow bands (D=d0:1) at various wavelengths in the range  ¼ 250– 350 lm. The experiment should cover a sufficiently large area of the sky so as to determine reliably the statistics of fluctuations on degree scales, and the detector should be sensitive to polarization. For reference, the experiment should also measure the anisotropies at shorter wavelengths where the FIB dominates. All of these requirements, of course, will make the detection of the lithium signature a difficult experiment. The difficulty cannot be fully assessed until more is known about the FIB and the sources responsible for it. Some considerations and strategies are discussed in Zaldarriaga & Loeb (2002). Further, in order to detect the effect of lithium, high signal-to-noise ratio maps of the primordial CMB at long wavelengths should be made for the same region of the sky. Fortunately, these maps will become available from future CMB missions such as the Planck satellite. On the theoretical front, improvements on the predicted lithium recombination history can be made by performing an explicit calculation of the nonequilibrium populations of the excited states in the same level-by-level fashion as that of the hydrogen recombination calculation of Seager et al. (1999, 2000). For hydrogen it was found that the excitedstate levels were overpopulated for z < 800 since the levels fall out of equilibrium with the background radiation field. A slower photoionization rate coupled with faster cascade rates results in a faster net recombination rate, ultimately resulting in a $10% change in the ionization fraction. Seager et al. (1999, 2000) also confirmed that all Lyman transitions are optically thick during hydrogen recombination, indicating that case B recombination holds at z > 800 but not at lower redshifts. The optical depths computed here for lithium are less than 1 for the ground-state transition, suggesting that all other transitions should be optically thin, indicating case A recombination. The case A lithium equilibrium recombination rate coefficient of VF96 might result in an underestimation of the neutral lithium fraction (and the optical depth), but probably only by $10%. We also note that nonconventional recombination histories may have a substantial impact on our predicted signal. For example, it has recently been proposed that recombina- 10 See http://www.hep.upenn.edu/blast. 11 See http://topweb.gsfc.nasa.gov. tion could have been delayed by the presence of additional radiation at z $ 103, possibly from stars, active galactic nuclei, or accretion by primordial compact objects (Peebles, Seager, & Hu 2000; Miller & Ostriker 2001). The neutral fraction of lithium (and its optical depth) could be larger than we calculated based on the standard recombination history, since the neutral lithium fraction is sensitive to the residual electron abundance. However, the stronger radiation field might override any gains in the recombination of lithium because of the increase in the photoionization rate. Models with detailed radiation fields are needed to calculate the net effect that these processes have on the lithium optical depth. Finally, while experiments to observe the lithium distortion on the temperature and polarization CMB power spectra appear to require redesign of detectors and new observational strategies, the benefits as mentioned in Zaldarriaga & Loeb (2002) could potentially be significant: 1. If the lithium signal could be separated from the FIB contamination, it would offer the possibility of constraining the lithium primordial abundance. Inferring the lithium primordial abundance from stellar observations is known to be complicated by stellar lithium depletion and galactic lithium production. Further, the lithium primordial abundance is a sensitive indicator of the baryon abundance, its mean value as well as inhomogeneities. It would then provide a possible means to discriminate among big bang nucleosynthesis models. 2. The lithium signature on the CMB anisotropies is the only probe proposed so far for structure in the dark ages of the early universe. Other methods, such as the suggestion by Iliev et al. (2002) that angular fluctuations in 21 cm emission from minihalos could probe redshifts between reionization and z $ 20, rely on the existence of collapsed objects and do not reach the high redshifts (z $ 100 to d500) probed by the lithium signal (see review by Barkana & Loeb 2001). The lithium and 21 cm signals would both give measures of the baryonic density fluctuations, the latter during the era of early star formation leading to reionization and the former just before the condensation of the first objects. This work was supported in part by NASA grants NAG5-7039 and NAG5-7768, by NSF grants AST 0071019 (A. L.), AST 00-87172 (P. C. S.), AST 00-87348 (S. L.), AST 00-88213 (A. D.), and AST 00-98506 and PHY 01-16590 (M. Z.), and the David and Lucile Packard Foundation (M. Z.). We thank Sara Seager and Gary Ferland for helpful discussions and Daniel Galli for supplying his numerical results. P. C. 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