Systematic uncertainties in the light-element abundances and their evolution
make a rigorous statistical assessment difficult. However, using Bayesian
methods we show that the following statement is robust: the predicted and
measured abundances are consistent with 95\% credibility only if the
baryon-to-photon ratio is between $2\times 10^{-10}$ and $6.5\times 10^{-10}$
and the number of light neutrino species is less than 3.9. Our analysis
suggests that the $^4$He abundance may have been systematically underestimated.Comment: 7 pages, LaTeX(2.09), 6 postscript figures (attached). A postscript
  version with figures can be found at
  ftp://astro.uchicago.edu/pub/astro/copi/assessing_BBN . (See the README file
  for details

A. Songaila

C.J. Copi

Craig J. Copi

D. Sasselov

D. York

David N. Schramm

E.D. Skillman

F. Hoyle

H. Reeves

J. Yang

J.A. Thorburn

J.L. Linsky

K.A. Olive

L. M. Krauss

M. Spite

M.H. Pinsonneault

Michael S. Turner

N. Hata

P.M. Lee

R. Malaney

R.F. Carswell

R.I. Epstein

S.G. Ryan

T.P. Walker

English

arXiv

Crossref

Assessing Big-Bang Nucleosynthesis

Systematic uncertainties in the light-element abundances and their evolution make a rigorous statistical assessment difficult. However, using Bayesian methods we show that the following statement is robust: the predicted and measured abundances are consistent with 95\% credibility only if the baryon-to-photon ratio is between 2\times 10^{-10} and 6.5\times 10^{-10} and the number of light neutrino species is less than 3.9. Our analysis suggests that the ^4He abundance may have been systematically underestimated

Copi, C J

Schramm, David N

Turner, M S

CERN Document Server

FERMILAB{Pub{95/257-A
astro-ph/9508029
submitted to Physical Review Letters
Assessing Big-Bang Nucleosynthesis
Craig J. Copi,
1;2
David N. Schramm,
1;2;3
and Michael S. Turner
1;2;3
1
Department of Physics
Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637-1433
2
NASA/Fermilab Astrophysics Center
Fermi National Accelerator Laboratory, Batavia, IL 60510-0500
3
Department of Astronomy & Astrophysics
The University of Chicago, Chicago, IL 60637-1433
ABSTRACT
Systematic uncertainties in the light-element abundances and their evolution make a
rigorous statistical assessment dicult. However, using Bayesian methods we show that the
following statement is robust: the predicted and measured abundances are consistent with
95% credibility only if the baryon-to-photon ratio is between 2 10
 10
and 6:5 10
 10
and
the number of light neutrino species is less than 3.9. Our analysis suggests that the
4
He
abundance may have been systematically underestimated.
Big-bang nucleosynthesis occurred seconds after the bang and for this reason oers the
most stringent test of the standard cosmology. Comparison of the predicted and measured
light-element abundances has evolved dramatically over the past thirty years, beginning with
the observation that there was evidence for a signicant primeval abundance of
4
He which
could be explained by the big bang [1] to the present where the abundances of D,
3
He,
4
He
and
7
Li are all used to test the big bang.
The predictions of big-bang nucleosynthesis depend upon the baryon-to-photon ratio
( ) as well as the number of light (
<

1MeV) particle species, often quantied as the
equivalent number of massless neutrino species ( N

). For a decade it has been argued
that the abundances of all four light elements can be accounted for provided  is between
2:5  10
 10
and 6  10
 10
and N

< 3:1   4 [2, 3, 4]. The \consistency interval" provides
the best determination of the baryon density and is key to the case for the existence of
nonbaryonic dark matter. The limit to N

provides a crucial hurdle for theories that aspire
to unify the fundamental forces and particles.
However, these conclusions were not based upon a rigorous statistical analysis. Because
the dominant uncertainties in the light-element abundances are systematic such an analysis
is dicult and previous work focussed on concordance intervals. Given the importance of
big-bang nucleosynthesis it is worthwhile to try to use more rigorous methods. Here we
apply two standard methods, goodness of t and Bayesian likelihood, and identify the the
conclusions which are insensitive to the systematic errors.
We begin by reviewing the general situation. The predictions of standard big-bang nu-
cleosynthesis are shown in Fig. 1. The theoretical uncertainties are statistical, arising from
imprecise knowledge of the neutron lifetime and certain nuclear cross sections. Because of 10
Gyr or so of \chemical evolution" since the big bang (nuclear reactions in stars and elsewhere
which modify the light-element abundances) determining primeval abundances is not simple
and the dominant uncertainties are systematic.
The chemical evolution of
4
He is straightforward: stars make additional
4
He. Stars also
make metals (elements heavier than
4
He); by measuring the
4
He abundance in metal-poor,
extragalactic HII (ionized hydrogen) clouds as a function of some metal indicator (e.g.,
C, N or O) and extrapolating to zero metallicity a primeval abundance has been inferred:
Y
P
= 0:232  0:003 (stat)  0:005 (sys) [5]. Systematic uncertainties arise from trying to
convert line strengths to abundances by modeling. The range Y
P
= 0:221   0:243 allows
for 2 statistical + 1 systematic uncertainty and is consistent with the big-bang prediction
provided  ' (0:8   4)  10
 10
[2]. Others have argued that the systematic uncertainty is
a factor of two or even three larger [6]; taking Y
P
' 0:21   0:25 increases the concordance
range signicantly,  ' (0:6  10) 10
 10
, reecting the logarithmic dependence of big-bang
4
He production upon  [2].
There is a strong case that the
7
Li abundance measured in metal-poor, old pop II halo
stars,
7
Li=H = (1:50:3)10
 10
, reects the big-bang abundance [7]. However, it is possible
that even in these stars the
7
Li abundance has been reduced by nuclear burning, perhaps by
a factor of two (the presence of
6
Li in some of these stars, which is more fragile, provides an
upper limit to the amount of astration). Allowing for a 2 statistical uncertainty and for up
1
to a factor of two astration,
1
leads to the consistency interval  = (1  6)  10
 10
[2].
The interpretation of D is particularly challenging because it is burned in virtually all
astrophysical situations and its abundance has only been accurately measured in the solar
vicinity. Because D is destroyed and not produced [8] and because its abundance is so
sensitive to  (D/H / 
 1:7
), a rm upper limit to  can be obtained by insisting that
big-bang production account for the D observed locally, D/H
>

(1:6 0:1) 10
 5
[9]. This
leads to the two-decade old bound, 
<

9  10
 10
, which is the linchpin in the argument
that baryons cannot provide closure density [10]. Because D is readily destroyed, it is not
possible to use the D abundance to obtain a lower bound to . The sum of D and
3
He is
more promising: D is rst burned to
3
He, and
3
He is much more dicult to burn. On the
assumption that the
3
He survival fraction is greater than 25% the lower limit 
>

2:510
 10
has been derived [4]. The D,
3
He concordance interval,  ' (2:5   9) 10
 10
.
The overlap of the concordance intervals (see Fig. 1), which occurs for  = (2:5 6)10
 10
,
is the basis for concluding that the light-element abundances are consistent with the big-bang
predictions [2].
The dominant uncertainties in comparing the predicted and measured light-element abun-
dances are systematic: the primeval abundance of
4
He; the chemical evolution of D and
3
He;
and whether or not
7
Li in the oldest stars has been reduced signicantly by nuclear burning.
Systematic error is dicult to treat as it is usually poorly understood and poorly quantied.
(If it were understood and were well quantied it wouldn't be systematic error!) This is
especially true for astronomical observations, where the observer has little control over the
object being observed.
There are at least three kinds of systematic error. (1) A denitive, but unknown, oset
between what is measured and what is of interest. (2) A random source of error whose
distribution is poorly known. (3) An important source of error that is unknown. The rst
kind of systematic error is best treated as an additional parameter in the likelihood function.
The second kind of systematic error is best treated by use of a distribution, or by several
candidate distributions. The third type of systematic error is a nightmare.
The data themselves can clarify matters. Consider
7
Li; its measured abundance in old
pop II stars is equal to the primeval abundance with a small statistical error and a larger
systematic uncertainty due to astration. This could be a systematic error of the rst kind|
if all stars reduce their
7
Li abundance by the same factor|or of the second kind|if the
7
Li abundance in dierent stars were reduced by dierent amounts. In the latter case, the
measured
7
Li abundance should show a large dispersion|which it does not [7]. Thus, we
treat astration by considering two limiting possibilities:
7
Li/H= (1:5  0:3)  10
 10
(no
astration); and second,
7
Li/H= (3:0 0:6)  10
 10
(astration by a factor of two).
Several sources of systematic error for
4
He have been identied|dust absorption, neu-
tral
4
He, stellar absorption, and theoretical emissivities|which can reduce or increase the
1
Additional smaller, but important, systematic uncertainties arise from the modeling of stellar atmo-
spheres, which is needed to convert line strengths to abundances, and possible enhancement of
7
Li by
cosmic-ray production [2]. We use astration to illustrate the eects of systematic uncertainty in the
7
Li
abundance, but it serves to illustrate the point for the other eects too.
2
measured abundance [6]. If the same eect dominates in each measurement use of an oset
parameter in the
4
He abundance would be appropriate. On the other hand, if dierent eects
dominate dierent measurements enlarging the statistical error would be appropriate. We
allow for both: the statistical error 
Y
is permitted to be larger than 0.003, and an oset in
the
4
He abundance, Y , is a parameter in the likelihood function (Y
P
= 0:232 + Y ).
Finally, there is the systematic uncertainty associated with the chemical evolution of D
and
3
He. Based upon a recent study of the chemical evolution of D and
3
He [11] we consider
three models that encompass the broadest range of possibilities: Model 0 is the plain, vanilla
model; Model 1 is characterized by extreme
3
He destruction
2
(average
3
He survival factor
of about 15%); and Model 2 is characterized by minimal
3
He destruction. The likelihood
functions for these three models are shown in Fig. 2.
First consider the 
2
test for goodness of t. This technique is best suited to situations
where the errors are gaussian and well understood and there are many degrees of freedom;
neither apply here. Nonetheless, in Fig. 3 we show 
2
() for eight dierent assumptions about
the systematic uncertainties: (1,5) 
Y
= 0:003, Y = 0; (2,6) 
Y
= 0:01, Y = 0; (3,7)

Y
= 0:003, Y = 0:01; (4,8) 
Y
= 0:01, Y = 0:01. In (1){(4),
7
Li=H = (3 0:6) 10
 10
;
and in (5){(8)
7
Li=H = (1:5  0:3)  10
 10
. For clarity, only the results for Model 0 are
shown, the results for Models 1 and 2 are similar.
Several conclusions can be drawn from Fig. 3. First, the goodness of t depends sensi-
tively upon assumptions made about the systematic errors, with the minimum 
2
ranging
from 6 to much less than 1; it is smallest when a systematic shift in
4
He is allowed and/or

Y
is increased. Second, in all cases the the  interval dened by 
2
= 3 (from the mini-
mum 
2
) has a lower bound no lower than about 1:5  10
 10
(in all but (5), no lower than
2:5 10
 10
) and an upper bound no higher than about 6 10
 10
.
Next, we turn to Bayesian likelihood, which is best suited to determining parameters of
a theory or assessing the relative viability of two or more theories. Since there is no well
developed, alternative to the standard theory of nucleosynthesis at present, likelihood is of no
use in assessing relative viability. Systematic errors of the rst kind are treated as additional
(nuisance) parameters in the likelihood function which can be determined by the experiment
itself or can be eliminated by marginalization; we treat Y as such. We also allow 
Y
to vary
to study how results depend upon the assumed uncertainty in the
4
He abundance. Because
we are interested in setting a limit to N

, it too is taken to be a parameter. Values of N

greater than three describe extensions of the standard model with additional light degrees
of freedom.
In Fig. 4 we show contours of L(N

;Y; 
Y
= 0:003). The contours are diagonal lines
because Y and N

are not independent parameters|the primary eect of an increase in
N

is an increase in the predicted
4
He abundance (Y
P
 0:01N

). A likelihood function
that is not compact must be treated with care, because no information about the parameters
(here, N

and Y ) can be inferred independently of what was already known (the priors).
For example, the likelihood function L(N

), which is needed to set limits to N

, is
2
Because the stars that destroy
3
He also make metals, it is not possible to destroy
3
He to an arbitrary
degree without overproducing metals [11].
3
jY j Model 0 Model 1 Model 2
0 3.1/3.1 3.2/3.3 3.1/3.1
0.005 3.2/3.1 3.3/3.3 3.2/3.1
0.010 3.3/3.2 3.5/3.3 3.3/3.2
0.015 3.5/3.4 3.7/3.4 3.4/3.4
0.020 3.7/3.6 3.9/3.7 3.7/3.6
Table 1: Limits to N

for Models 0, 1, 2 and Li=H = (1:5 0:3)  10
 10
(rst number) and
Li=H = (3:0 0:6) 10
 10
(second number).
obtained by integrating over Y and depends upon the limits of integration. To derive
limits to N

we do the following: integrate from  jY j to jY j; normalize L(N

) to have
unit likelihood from N

= 3 to 1; the limit is the value of N

beyond which 5% of the total
likelihood accumulates. The dependence of the limit upon jY j is shown in Table 1.
An aside; in a recent paper the likelihood function L(N

) obtained by integrating from
Y =  0:005 to 0:005 was used in an attempt to assess the viability of the standard theory
of nucleosynthesis [12]. This likelihood function is peaked at N

= 2 and is approximately
gaussian with 
N

= 0:3. On this basis it was claimed that the standard theory of nu-
cleosynthesis is ruled out with 99.7% condence. By so doing equal weight was implicitly
given to all values of N

(uniform priors). The prior for N

= 3 (the standard model of
particle physics) is certainly orders of magnitude greater than that for N

= 2 (for which no
well developed model exists). The likelihood function L(N

) which properly included prior
information would certainly not be peaked at N

= 2:0.
In Figs. 5 and 6 we show the 95% contours of the likelihood functions L(; 
Y
) and
L(;Y ) for Models 0, 1, and 2 and both values of the central
7
Li abundance. Both gures
suggest the same thing: the uncertainty in the primordial
4
He abundance has been underes-
timated. In the 
Y
   plane 
Y
= 0:003 does not intersect the 95% credibility contour, and
in the Y    plane Y = 0 does not intersect the 95% credibility region (except for Model
1, where they barely do). The 95% credibility contour in the 
Y
   plane becomes inde-
pendent of 
Y
for 
Y
>

0:008, with 95% credibility interval  ' (3  6:5)  10
 10
(allowing
both for the uncertainty in the astration of
7
Li and in the chemical evolution of D and
3
He).
The 95% credibility contours in the Y  plane suggest that the primeval
4
He abundance
has been systematically underestimated, by an amount Y  +0:01. (Though it should be
noted that Model 1 and the lower
7
Li abundance are just consistent with Y = 0 at 95%
credibility.) Put another way, D,
3
He, and
7
Li are concordant and
4
He is the outlayer. (This
can also be seen in Fig. 2.) When the likelihood function is marginalized with respect to
Y , the 95% credibility interval is  ' (2  6:5) 10
 10
(allowing again for the uncertainty
both in astration of
7
Li and in the chemical evolution of D and
3
He).
To conclude, the fact that systematic uncertainties dominate precludes crisp statistical
statements. The lack of a viable alternative to the standard theory of nucleosynthesis com-
4
plicates matters further as the most powerful statistical techniques assess relative viability.
However, the rigorous techniques that we have applied point to several conclusions that are
insensitive to assumptions made about systematic uncertainty:
 The predictions of the standard theory of primordial nucleosynthesis are only consistent
with the extant observations with 95% credibility provided  ' (2  6:5)  10
 10
.
 Our analysis suggests that the primordial
4
He abundance has been systematically un-
derestimated (Y  +0:01) or that the random errors have been underestimated
(
Y
 0:01). Only for Model 1 (extreme destruction of
3
He) are Y = 0 and

Y
= 0:003 in the 95% credibility region (cf., Figs. 5 and 6).
 The limit to N

depends upon the systematic uncertainties in the
4
He abundance (cf.,
Table 1); taking jY j  0:02, which is four times the estimated systematic error and
also encompasses the 95% likelihood contour in the Y    plane, leads to the 95%
credible limit N

< 3:9.
This more rigorous analysis provides additional support for the conclusions reached pre-
viously about the concordance interval for  [2]. The limit to the number of neutrino species
is less stringent than previously quoted bounds [2, 3] because we allowed a chemical evolution
model with the most extreme destruction of
3
He (which permits low values of  where
4
He
production is lower) as well as a large systematic oset in the
4
He abundance.
Finally, there are two measurements that should reduce the systematic uncertainties sig-
nicantly, permitting a sharper test of big-bang nucleosynthesis. The rst is a determination
of the primeval D abundance by measuring D-Ly absorption due to high-redshift hydrogen
clouds. The second is a determination of the primeval
7
Li abundance by studying short
period, tidally locked pop II halo binaries; astration is believed to involve rotation-driven
mixing astration and is minimized in these stars because they rotate slowly [13]. At the
moment, there are conicting measurements and upper limits for the primeval D abundance
seen in high-redshift hydrogen clouds [14], and there is one study which indicates that the
7
Li abundance in short-period binaries is no higher (evidence against signicant astration)
and another that nds weak evidence that the
7
Li abundance is higher [15].
Acknowledgments. We thank Donald Q. Lamb for many valuable discussions and com-
ments. This work was supported by the DoE (at Chicago and Fermilab) and by the NASA
(at Fermilab by grant NAG 5-2788 and at Chicago by a GSRP (CJC)).
References
[1] F. Hoyle and R.J. Tayler, Nature 203, 1108 (1964).
[2] C.J. Copi, D.N. Schramm, and M.S. Turner, Science 267, 192 (1995).
5
[3] T.P. Walker et al., Astrophys. J. 376, 51 (1991); L.M. Krauss and P.J. Kernan, Phys.
Lett. B 347, 347 (1995).
[4] J. Yang et al., Astrophys. J. 281, 493 (1984).
[5] K.A. Olive and G. Steigman, Astrophys. J. Suppl. 97, 49 (1995).
[6] D. Sasselov and D.S. Goldwirth, Astrophys. J., in press (1995); E.D. Skillman, R. Ter-
levich, and D.R. Garnett, ibid, in press (1995).
[7] M. Spite et. al., Astron. Astrophys. 141, 56 (1984); J.A. Thorburn, Astrophys. J. 421,
318 (1994).
[8] R.I. Epstein, J.M. Lattimer, and D.N. Schramm, Nature 263, 198 (1976).
[9] J.L. Linsky et al., Astrophys. J. 402, 694 (1993).
[10] H. Reeves et al., Astrophys. J. 179, 909 (1973).
[11] C.J. Copi, D.N. Schramm, and M.S. Turner, astro-ph/9506094.
[12] N. Hata et al., hep-ph/9505319.
[13] M.H. Pinsonneault, C.P. Deliyannis, and P. Demarque, Astrophys. J. Suppl. 78, 179
(1992).
[14] D. York et al., Astrophys. J. 276, 92 (1984); A. Songaila et al., Nature 368, 599 (1994);
R.F. Carswell et al., Mon. Not. R. astr. Soc. 268, L1 (1994); D. Tytler et al. 1995, in
preparation.
[15] M. Spite et al., Astron. Astrophys. 290, 217 (1994); S.G. Ryan and C.P. Deliyannis,
Astrophys. J., in press (1995).
6
Figure Captions
Figure 1: The predicted light-element abundances (with 2 theoretical errors); rectangles
indicate consistency intervals, which all overlap for  = (2:5   6) 10
 10
.
Figure 2: Likelihood functions for D and
3
He (lower solid curves, from left to right: Models
1, 0, and 2),
4
He (dotted curves, from left to right: 
Y
= 0:01, 
Y
= 0:003, and Y = 0:01),
and
7
Li (broken = high
7
Li, solid = low
7
Li).
Figure 3: Reduced 
2
as function of  for eight dierent sets of assumptions about the
systematic uncertainties (see text for details).
Figure 4: The 5% of maximum likelihood contours for L(N

;Y; 
Y
= 0:003) (solid curves
= low
7
Li, broken curves = high
7
Li). Because N

and Y are not independent parameters,
the contours of likelihood are diagonal lines and the likelihood function is not compact.
Figure 5: The likelihood function L(
Y
; ;Y = 0) (solid curves = low
7
Li, broken curves
= high
7
Li).
Figure 6: Same as Fig. 5 for L(Y; ; 
Y
= 0:003).
7


Assessing big-bang nucleosynthesis

Assessing Big-Bang Nucleosynthesis

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