---
abstract: 'The lithium abundance in 62 halo dwarfs is determined from accurate equivalent widths reported in the literature and an improved infrared flux method (IRFM) temperature scale. The Li abundance of 41 plateau stars (those with [T$_{\rm eff}$ ]{}$>$ 6000 K) is found to be independent of temperature and metallicity, with a star-to-star scatter of only 0.06 dex over a broad range of temperatures (6000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6800 K) and metallicities ($-3.4 <$ \[Fe/H\] $< -1$), thus imposing stringent constraints on depletion by mixing and production by Galactic chemical evolution. We find a mean Li plateau abundance of $A_{Li}$ = 2.37 dex ($^7$Li/H = 2.34 $\times 10^{-10}$), which, considering errors of the order of 0.1 dex in the [*absolute*]{} abundance scale, is just in borderline agreement with the constraints imposed by the theory of primordial nucleosynthesis and WMAP data (2.51 $< A_{Li}^{\rm WMAP}$ $<$ 2.66 dex).'
author:
- Jorge Meléndez and Iván Ramírez
title: 'Reappraising the Spite Lithium Plateau: Extremely Thin and Marginally Consistent with WMAP '
---

= ==1=1=0pt =2=2=0pt =2=2=0pt

Introduction
============

The Li plateau was discovered by Spite & Spite (1982), who showed that the $^7$Li abundance obtained from the Li doublet at 6708 Å in F and early G halo dwarfs is independent of temperature and metallicity, suggesting that the Li abundance determined in halo stars represents the primordial abundance from Big Bang nucleosynthesis (BBN). The standard theory of BBN predicts the abundance of light elements, in particular $^7$Li, as a function of the baryon-to-photon ratio $\eta$. Thus, if the plateau is primordial, an inverse analysis could be used to determine $\eta$ (Spite & Spite 1982).

On the other hand, Ryan and collaborators (Ryan et al. 1996, 1999, 2001; hereafter jointly referred as R01) suggested that the Li abundance in halo stars is postprimordial, with a 0.3 dex increase from \[Fe/H\] = $-$3.5 to \[Fe/H\] = $-$1.[^1] Those results rely on the temperature scale of Magain (1987), which was calibrated with only 3 metal-poor (\[Fe/H\] $<$ -2) stars. Considering the metallicity dependence of the Li abundances, R01 extrapolated a primordial Li abundance of $A_{Li}$ $\approx$ 2.0 - 2.1 dex.[^2]

Recently, precise measurements of the cosmic background anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP, Spergel et al. 2003) have been used to constrain $\eta$, and employing the standard theory of BBN, the primordial Li abundance was predicted to be $A_{Li}^{\rm WMAP}$ = 2.58$_{-0.07}^{+0.08}$ dex (Cyburt et al. 2003), a value that is much higher than the primordial abundance proposed by R01. The large difference of about 0.5 dex between the results of Ryan et al. and WMAP has stimulated theoretical work on non-standard BBN (e.g. Feng et al. 2003; Ichikawa & Kawasaki 2004).

Bonifacio & Molaro (1997, hereafter BM97) found that when the [T$_{\rm eff}$ ]{}scale based on the infrared flux method (IRFM) is employed (that of Alonso et al. 1996), the Li abundance in metal-poor stars does not depend on metallicity, and $A_{Li}$ = 2.24 dex is obtained, higher than the primordial abundance proposed by R01, but still lower by a factor of 2 than the abundance suggested by the WMAP data. Furthermore, Bonifacio et al. (2002) analyzed 5 metal-poor field stars and 12 turn-off stars in the globular cluster NGC 6397, obtaining a Li abundance 0.05 and 0.1 dex higher than in BM97, respectively.

Ford et al. (2002) have determined the Li abundance in a few metal-poor stars employing the extremely weak ($\approx$ 2-4 mÅ) Li lines at 6104 Å, resulting in $A_{Li}$ $\approx$ 2.5 dex, but note that the observational errors are very large (about 30%), not allowing them to draw any firm conclusions. In fact, when the Li lines at 6708 Å were used by Ford et al. (2002), $A_{Li}$ $\approx$ 2.2 dex was found.

The temperature is the most important atmospheric parameter required to obtain the Li abundance, and since metal-poor stars are too distant to measure their angular diameters, indirect methods have to be used to estimate their temperatures. The closest (and less model-dependent) approach to the direct method is the IRFM, where the temperature is found by a comparison between the observed and theoretical ratio of the bolometric flux to the infrared flux. In a pioneer work, Magain (1987) calibrated the [T$_{\rm eff}$ ]{}scale of metal-poor dwarfs (mostly [T$_{\rm eff}$ ]{}$<$ 6000 K and \[Fe/H\] $> -2$) employing IRFM temperatures of 11 dwarfs. The titanic work by Alonso et al. (1996) improved the [T$_{\rm eff}$ ]{}scale including a larger number of metal-poor stars (down to \[Fe/H\] $\approx$ -3), employing spectroscopic metallicities (when available) determined before 1992. In the meantime, new spectroscopic and photometric surveys have appeared in the literature (a milestone was the completion of 2MASS), providing the necessary input data for a new revision of the [T$_{\rm eff}$ ]{}scale (Ramírez & Meléndez 2004b,c, hereafter RM04b,c). As an example, only from January 2001 to June 2003, more than 1500 new spectroscopic metallicity determinations for field FGK stars have appeared in the literature, increasing significantly the number of calibrating stars with reliable \[Fe/H\] values, and hence allowing to better define the metallicity dependence of the [T$_{\rm eff}$ ]{}scale. We have determined the IRFM [T$_{\rm eff}$ ]{}of more than 10$^3$ stars (RM04b) using updated input data, and metallicity-dependent [T$_{\rm eff}$ ]{}vs. color calibrations were derived in the metallicity range $-3.5 <$ \[Fe/H\] $< +0.4$ (RM04c).

Employing our improved IRFM temperature scale, we show that the Spite plateau has essentially zero scatter and that the derived abundances are now consistent, although only marginally, with WMAP data.

Data and Atmospheric Parameters
===============================

The sample was selected from relatively unevolved halo stars (log $g \geq$ 3.5, \[Fe/H\] $<$ $-$1) with previous Li observations and having either an IRFM effective temperature determined by us (RM04b) and/or with a previous determination of metallicity as reported in the “2003 updated” version of the Cayrel de Strobel et al. (2001, hereafter C03) catalog (see $\S$2 in Ramírez & Meléndez 2004a).

For most works previous to 1997, the equivalent widths ($W_\lambda$) of the Li doublet at 6708 Å  were taken from the compilation of Ryan et al. (1996), adopting a minimum error of 1.5 mÅ for the individual measurements. Other $W_\lambda$ considered are those by Molaro et al. (1995), Spite et al. (1996), Nissen & Schuster (1997), Ryan & Deliyannis (1998), Smith et al. (1998), Gutiérrez et al. (1999), Hobbs et al. (1999), King (1999), Ryan et al. (1999, 2001), Bonifacio et al. (2002), and Ford et al. (2002). The weighted averages of the equivalent widths and errors ($W_\lambda$ and $\delta W_\lambda$) were computed, as well as the standard errors ($\sigma_m$). The adopted error is $\sigma_W \equiv$ max($\delta W_\lambda$, $\sigma_m$, 1 mÅ). A star was rejected if $\sigma_W \geq 0.1 W_\lambda$. Stars with E(B-V) $>$ 0.05 dex were also discarded. Sixty two dwarfs satisfied our selection criteria.

The key parameter for a reliable Li abundance determination in metal-poor stars is T$_{\rm eff}$; errors in the other atmospheric parameters (log $g$, \[Fe/H\], $v_t$) result in abundance errors of the order of 0.01 dex (e.g. Ryan et al. 1996). Surface gravities and metallicities were taken from the C03 catalog, and a microturbulence $v_t$ = 1 km s$^{-1}$ was adopted. For 3 stars not included in the C03 catalog, we adopted metallicities from a photometric calibration (RM04b) that is based on C03 \[Fe/H\] values. The effective temperatures are based on our IRFM [T$_{\rm eff}$ ]{}scale. Individual IRFM temperatures were taken from RM04b or recomputed employing new reddening corrections, as well as calculated employing the IRFM [T$_{\rm eff}$ ]{}calibrations of RM04c. We used up to 17 colors in the [*BVRI*]{}(Johnson-Cousins), Tycho-Hipparcos, Strömgren, DDO, Vilnius, Geneva and 2MASS systems. Most of the optical photometry was taken from the General Catalogue of Photometric Data (Mermilliod et al. 1997) and the Tycho catalog, while the infrared photometry was taken from 2MASS. E(B-V) values were computed employing several extinction maps, giving a higher weight to the maps by Schlegel et al. (1998) for high latitude stars (RM04b).

The use of several colors improves the determination of the effective temperature, since it alleviates the impact of photometric errors. Following RM04c, the adopted [T$_{\rm eff}$ ]{} was the mean of the IRFM determination and the temperature from the IRFM calibrations (T$_{\rm eff}^{\rm cal}$):

$${\rm T_{\rm eff}} = \case{1}{n+m} (n {\rm T_{eff}^{IRFM}} + m {\rm T_{eff}^{cal}})$$

where the weights we adopted were $n$ = 1 and $m$ equal to the square root of the number of colors employed (typically 10, thus $m \approx 3$). T$_{\rm eff}^{\rm cal}$ was calculated taking into account the errors of each IRFM color calibration. The mean internal error of the [T$_{\rm eff}$ ]{}derived in this work is $\delta$[T$_{\rm eff}$ ]{}$\approx$ 66 K. It is important to note that the use of eq. (1) results in a [T$_{\rm eff}$ ]{}that is only $\approx$ 22 K lower than the temperature directly determined from the IRFM. If only T$_{\rm eff}^{\rm IRFM}$ were used, A$_{Li}$ would be about 0.015 dex higher.

Li Abundances
=============

In order to determine the Li abundances, we computed a grid of theoretical equivalent widths of the Li doublet in the parameters space: [T$_{\rm eff}$ ]{}= \[5250 K, 5500 K, ..., 6750 K\], log $g$ = \[3.5, 4.0, 4.5\], \[Fe/H\] = \[-1.0, -1.5, ..., -3.5\], $A_{Li}$ = \[1.7, 1.9, ..., 2.5\]. We took into account the fine and hyperfine structure of the doublet, with wavelengths and $gf$-values as given in Andersen et al. (1984). Note that Smith et al. (1998) and Hobbs et al. (1999) reported new $gf$-values based on theoretical calculations, but the difference with our adopted (laboratory) values is only $\approx$ 1% (Smith et al. 1998). The collisional broadening constant was obtained from Barklem et al. (2000). A grid of 630 synthetic spectra was computed employing the 2002 version of the LTE spectral synthesis code MOOG (Sneden 1973) and Kurucz overshooting models. The theoretical $W_\lambda$ were measured by integration.

Using the stellar parameters and observed $W_\lambda$ as input, the Li abundance was determined from a full 4-dimensional interpolation in the synthetic grid $A_{Li}$([T$_{\rm eff}$ ]{}, log $g$, \[Fe/H\], log $W_\lambda$). Note that in some previous studies (Thorburn 1994; Ryan et al. 1996, 1999) a more restricted approach was adopted, considering only two fixed log $g$ values (no interpolation) and metallicity interpolation only for models within $-$2 $<$ \[Fe/H\] $<$ $-$1 (Ryan et al. 1996) or $-$3 $<$ \[Fe/H\] $<$ $-$2 (Thorburn 1994). Our full interpolation results in lower internal abundance uncertainties, which is very important to determine the true scatter of the Li abundances.

NLTE effects are small for the Li plateau stars. We applied the NLTE corrections ($\Delta x \equiv$ NLTE - LTE) by Carlsson et al. (1994), which are $\Delta x$ $\approx$ $-$0.04 and 0.0 dex for the hottest and coolest plateau dwarfs, respectively. Although these NLTE corrections were not specifically calculated for our set of model atmospheres, their relative dependence with [T$_{\rm eff}$ ]{}and \[Fe/H\] is still useful, and should minimize the scatter of the derived abundances. Since the [*absolute*]{} NLTE corrections may be different, we prefer to preserve the mean $A_{Li}$(LTE) of the plateau stars by adopting a NLTE correction of $\Delta x$ + 0.02 dex. The final Li abundances are reported in Table 1.

In Fig. 1 are shown the Li abundances obtained in this work. In the lower panel $A_{Li}$ vs. [T$_{\rm eff}$ ]{}is plotted, where it is clearly seen that stars with [T$_{\rm eff}$ ]{}$>$ 6000 K have approximately the same Li abundance (plateau stars), as no trend with [T$_{\rm eff}$ ]{}(slope of $4 \pm 5 \times 10^{-3}$ per 100 K) is observed. In the upper panel are plotted 41 plateau stars ([T$_{\rm eff}$ ]{}$>$ 6000 K) as a function of \[Fe/H\], showing an impressive constancy in their Li abundance from \[Fe/H\] $\approx$ $-$3.4 to $-$1, with an amazingly small scatter and essentially zero slope (0.013 $\pm$ 0.018 per dex). This figure strongly supports the primordial origin of the Li plateau (Spite & Spite 1982), and imposes stringent constraints on stellar mixing and Galactic chemical evolution.

The very small star-to-star [*observed*]{} scatter ($\sigma_{obs}$ = 0.06 dex) is fully explained by errors in [T$_{\rm eff}$ ]{} and $W_\lambda$. Adding in quadrature the typical errors of $\delta$T = 66 K ($\delta A_{Li} \approx$ 0.045 dex) and $\delta W_\lambda$ = 5.5% ($\delta A_{Li} \approx$ 0.025 dex) (as well as errors of 0.3 dex in log $g$ and \[Fe/H\]), a predicted error $\sigma_{pred}$ = 0.053 dex is obtained, in excellent agreement with the observed scatter ($\sigma_{obs}$ = 0.06 dex). Note that if the 2 plateau stars with the highest Li abundances are discarded, then $\sigma_{obs}$ = 0.05 dex.

Discussion
==========

The abundances obtained in this work for plateau stars show a very small star-to-star-scatter. This is a consequence of employing very accurate temperatures based on the IRFM and their photometric calibrations. RM04b,c have shown that the IRFM [T$_{\rm eff}$ ]{}scale is in almost perfect agreement (at the 10 K level) with interferometric measurements of dwarfs and giants with \[Fe/H\] $> -0.6$. Provided that the plateau is truly primordial, the constancy of the Li abundance in our plateau stars indicates that our IRFM [T$_{\rm eff}$ ]{}scale is also highly reliable when we approach low metallicities, and this would be consistent with the small dependence of the IRFM on the detailed assumptions of model atmospheres.

Although the Li abundances derived in this work have very small internal errors, systematic errors dominate the [*absolute*]{} abundances. For example, the use of Kurucz models with no overshooting (NOVER, Castelli et al. 1997) results in smaller $A_{Li}$ by about 0.08 dex. Note that the Kurucz solar overshooting model reproduces more solar observables than the NOVER model (Castelli et al. 1997). Interestingly, the [*(R-I)*]{}$_{C}$ synthetic colors computed from overshooting models are in better agreement with the IRFM [T$_{\rm eff}$ ]{}scale than colors computed with NOVER models, at least for solar metallicity dwarfs (6000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6800 K, Meléndez & Ramírez 2003) and giants (4000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6300 K, Ramírez & Meléndez 2004a).

The mean Li abundance obtained for the plateau stars, $A_{Li}$ = 2.37 dex, is considerably higher than the primordial Li abundance proposed by R01 ($A_{Li} \approx$ 2.0 - 2.1 dex). As shown in Fig. 2, the reason for this are the lower temperatures adopted by R01. The temperature difference (between our [T$_{\rm eff}$ ]{}and those adopted by R01) is metallicity dependent, and at very low metallicities (\[Fe/H\] $< -3)$ the [T$_{\rm eff}$ ]{}adopted by R01 is about 400 K lower than ours. When the temperatures adopted by R01 are used, we also find a trend of Li abundance with metallicity. If we use the same [T$_{\rm eff}$ ]{}(and $W_\lambda$) as those employed by R01, $<A_{Li}>$ = 2.13 dex is found for stars with [T$_{\rm eff}$ ]{}$>$ 6000 K and \[Fe/H\] $<$ -3, whereas R01 found $<A_{Li}>$ = 2.07 dex for the same stars (the small difference is due to the different model atmospheres adopted). Note that when Magain (1987) calibration is used in its validity range ([T$_{\rm eff}$ ]{}$<$ 6000 K, \[Fe/H\] $> -2$), the agreement with our [T$_{\rm eff}$ ]{}scale is satisfactory (open circles, Fig. 2); outside that range it may not be appropriate to use it.

Most (0.07 dex) of the 0.13 dex difference between the present Li plateau abundance and that obtained by BM97 is due to different model atmospheres, and the remaining difference is because our [T$_{\rm eff}$ ]{}scale is hotter for metal-poor F dwarfs. We have compared our IRFM temperatures with those given by Alonso et al. (1996) (employing the same reddening adopted by Alonso et al.) The agreement is good for metal-poor stars with [T$_{\rm eff}$ ]{}$<$ 6500 K, and our temperatures are just $\approx$ 50 K higher in the range 6000 $<$ [T$_{\rm eff}$ ]{}$<$ 6500. For hotter stars the differences are larger. Since only 5 stars with [T$_{\rm eff}$ ]{}$>$ 6500 K are included in the present work, their exclusion does not alter our results (Fig. 1), but restricts our claim of the constancy of the plateau to $-3 <$ \[Fe/H\] $< -1$ and 6000 K $<$ [T$_{\rm eff}$ ]{}$<$ 6500 K.

Ryan et al. (1999) have criticized Alonso’s calibrations for turn-off stars because of their strong metallicity dependence. Nevertheless, this effect is predicted by both Kurucz and MARCS model atmospheres. In fact, for metal-poor (\[Fe/H\] = -3) F dwarfs, our ($V-K$) IRFM calibration is in good agreement with $V-K$ colors predicted from MARCS models (Houdashelt et al. 2000). At \[Fe/H\] = -3, our calibration (RM04c) gives 6250 and 6500 K for $V-K_{2MASS}$ = 1.31 and 1.21, while Houdashelt calibration results in [T$_{\rm eff}$ ]{}[*higher*]{} by +76 and +7 K, respectively. The same exercise employing Kurucz models at $-2.5$ (M. S. Bessell 2004, private communication) gives a similar result.

Asplund et al. (2003) have shown that 3D hydrodynamical models with NLTE computations give similar Li abundances than 1D + NLTE. However, that result was obtained neglecting collisions with H, and taking them into account, the Li abundance is lowered (Barklem et al. 2003). It is possible to compare 1D calculations (considering relative NLTE corrections) with 3D NLTE results by using the plateau star HD 84937, which has predicted equivalent widths in 3D NLTE (Barklem et al. 2003). Employing the same atmospheric parameters and Li abundance as given by Barklem et al. (2003), we obtain a 1D $W_\lambda$ = 11.4 mÅ, which is very close to the 3D NLTE results, both without collisions (11.1 mÅ) and when collisions are included (12.6 mÅ). Within 0.05 dex, both 3D NLTE calculations are in (probably fortuitous) agreement with the 1D calculation. Nevertheless, this result is by any means conclusive, as the 3D results are based on only one snapshot (Barklem et al. 2003).

The observational star-to-star scatter found in this work ($\sigma_{obs}$ $\approx$ 0.05-0.06 dex) is considerably lower than those found in previous analyses of large samples: 0.13, 0.06-0.08, 0.089 and 0.094 dex according to Thorburn (1994), Spite et al. (1996), Ryan et al. (1996) and BM97, respectively. This extremely thin scatter of the plateau stars is fully explained by small errors in [T$_{\rm eff}$ ]{}and $W_\lambda$, leaving little room (if any) for mixing or Galactic evolution. Considering the constancy of the Li abundance of the plateau stars, depletion is constrained probably at a level of 0.05 dex, and if rotational mixing plays a role, then our mean $A_{Li}$ should be slightly increased.

Within the [*absolute*]{} abundance uncertainties involved (of the order of 0.1 dex) in our simple 1D modeling, we conclude that our results are just in marginal agreement with the constraint imposed by WMAP, and if new physics or non-standard Big Bang nucleosynthesis are invoked (e.g. Feng et al. 2003; Ichikawa & Kawasaki 2004), the present work constraints their effects to a level of 0.1-0.15 dex.

We thank the referee for his constructive comments, and JG Cohen for her suggestions. JM acknowledges support from NSF grant AST-0205951 to JGC.

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[lccccrcrrrrl]{} \[tab1\] BD+002058 & 6092 & 4.19 & -1.16 & 41.9 & 2.49\
BD+023375 & 6124 & 3.97 & -2.29 & 32.5 & 2.34\
BD+030740 & 6443 & 3.76 & -2.68 & 19.5 & 2.31\
BD+090352 & 6075 & 4.38 & -2.03 & 34.0 & 2.34\
BD+092190 & 6487 & 4.11 & -2.05 & 17.3 & 2.28\
BD+174708 & 6154 & 3.93 & -1.64 & 25.8 & 2.28\
BD+203603 & 6248 & 4.15 & -2.12 & 26.2 & 2.32\
BD+241676 & 6510 & 3.91 & -2.32 & 25.3 & 2.46\
BD+262606 & 6157 & 4.20 & -2.37 & 30.5 & 2.33\
BD+282137 & 6229 & 3.50 & -2.20 & 27.2 & 2.33\
BD+342476 & 6433 & 3.92 & -2.10 & 22.6 & 2.37\
BD+422667 & 6086 & 4.13 & -1.51 & 30.6 & 2.32\
BD+592723 & 6121 & 4.00 & -1.83 & 30.5 & 2.33\
BD+710031 & 6331 & 4.05 & -1.96 & 31.0 & 2.44\
BD-043208 & 6404 & 3.77 & -2.36 & 25.4 & 2.40\
BD-100388 & 6240 & 3.61 & -2.37 & 27.0 & 2.33\
BD-133442 & 6484 & 3.98 & -2.87 & 20.5 & 2.35\
CD-2417504 & 6507 & 4.12 & -3.36 & 18.6 & 2.31\
CD-3018140 & 6336 & 4.08 & -1.95 & 30.3 & 2.44\
CD-3301173 & 6674 & 4.19 & -2.99 & 16.3 & 2.35\
CD-7101234 & 6375 & 4.43 & -2.35 & 26.0 & 2.38\
G059-024 & 6191 & 4.36 & -2.38 & 36.0 & 2.42\
G064-012 & 6682 & 4.10 & -3.31 & 23.4 & 2.53\
G064-037 & 6775 & 4.05 & -3.13 & 15.6 & 2.39\
G075-031 & 6036 & 4.07 & -1.04 & 48.0 & 2.54\
G192-043 & 6184 & 4.30 & -1.61 & 34.5 & 2.42\
G201-005 & 6370 & 3.79 & -2.51 & 25.0 & 2.37\
G206-034 & 6310 & 4.18 & -2.53 & 27.2 & 2.37\
HD016031 & 6216 & 4.07 & -1.82 & 27.0 & 2.33\
HD074000 & 6392 & 4.27 & -1.96 & 23.9 & 2.36\
HD084937 & 6345 & 3.96 & -2.13 & 24.8 & 2.35\
HD102200 & 6104 & 4.14 & -1.21 & 32.8 & 2.38\
HD108177 & 6099 & 4.30 & -1.64 & 31.4 & 2.33\
HD160617 & 6099 & 3.68 & -1.79 & 40.0 & 2.43\
HD166913 & 6096 & 4.08 & -1.58 & 37.0 & 2.40\
HD181743 & 6038 & 4.47 & -1.81 & 38.0 & 2.38\
HD218502 & 6222 & 3.94 & -1.79 & 27.1 & 2.34\
HD284248 & 6133 & 4.21 & -1.61 & 29.9 & 2.33\
HD338529 & 6504 & 3.90 & -2.27 & 23.9 & 2.43\
LP0056-0075 & 6197 & 4.36 & -2.67 & 25.5 & 2.27\
LP0831-0070 & 6521 & 4.23 & -3.07 & 23.8 & 2.42\

[^1]: \[A/B\] $\equiv$ log ($N_A/N_B$) - log($N_A/N_B$)$_\odot$

[^2]: $A_{X}$ $\equiv$ log ($N_X$/$N_H$) + 12
