Cosmological and astrophysical constraints from the Lyman-alpha forest flux probability distribution function

Abstract

We use the probability distribution function (PDF) of the lya forest flux at z=2-3, measured from high-resolution UVES/VLT data, and hydrodynamical simulations to obtain constraints on cosmological parameters and the thermal state of the intergalactic medium (IGM) at z 2-3. The observed flux PDF at z=3 alone results in constraints on cosmological parameters in good agreement with those obtained from the WMAP data, albeit with about a factor two larger errors. The observed flux PDF is best fit with simulations with a matter fluctuation amplitude of sigma_8=0.8-0.85 \pm 0.07 and an inverted IGM temperature-density relation (gamma ~ 0.5-0.75), consistent with our previous results obtained using a simpler analysis. These results appear to be robust to uncertainties in the quasar (QSO) continuum placement. We further discuss constraints obtained by a combined analysis of the high-resolution flux PDF and the power spectrum measured from the Sloan Digital Sky Survey (SDSS) lya forest data. The joint analysis confirms the suggestion of an inverted temperature-density relation, but prefers somewhat higher values (sigma_8 ~ 0.9) of the matter fluctuation amplitude than the WMAP data and the best fit to the flux PDF alone. The joint analysis of the flux PDF and power spectrum (as well as an analysis of the power spectrum data alone) prefers rather large values for the temperature of the IGM, perhaps suggesting that we have identified a not yet accounted for systematic error in the SDSS flux power spectrum data or that the standard model describing the thermal state of the IGM at z ~ 2-3 is incomplete. Comment: 5+ pages, MNRAS Letters in press

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arXiv:0907.2927v1 [astro-ph.CO] 16 Jul 2009
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 16 July 2009(MN LATEX style file v2.2)
Cosmological and astrophysical constraints from the
Lyman-α forest flux probability distribution function
Matteo Viel1,2, James S. Bolton3& Martin G. Haehnelt4,5
1INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy (viel@oats.inaf.it)
2INFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy
3Max-Planck Institut f¨ ur Astrophysik, Karl-Schwarzschild Str. 1, 85748 Garching, Germany
4Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA
5KICC-Kavli Institute of Cosmology, Cambridge
16 July 2009
ABSTRACT
We use the probability distribution function (PDF) of the Lyman-α forest flux at
z = 2 − 3, measured from high-resolution UVES/VLT data, and hydrodynamical
simulations to obtain constraints on cosmological parameters and the thermal state
of the intergalactic medium (IGM) at z ∼ 2 − 3. The observed flux PDF at z = 3
alone results in constraints on cosmological parameters in good agreement with those
obtained from the WMAP data, albeit with about a factor two larger errors. The
observed flux PDF is best fit with simulations with a matter fluctuation amplitude
of σ8 = 0.8 − 0.85 ± 0.07 and an inverted IGM temperature-density relation (γ ∼
0.5 − 0.75), consistent with our previous results obtained using a simpler analysis.
These results appear to be robust to uncertainties in the quasar (QSO) continuum
placement. We further discuss constraints obtained by a combined analysis of the
high-resolution flux PDF and the power spectrum measured from the Sloan Digital
Sky Survey (SDSS) Lyman-α forest data. The joint analysis confirms the suggestion
of an inverted temperature-density relation, but prefers somewhat higher values (σ8∼
0.9) of the matter fluctuation amplitude than the WMAP data and the best fit to
the flux PDF alone. The joint analysis of the flux PDF and power spectrum (as
well as an analysis of the power spectrum data alone) prefers rather large values for
the temperature of the IGM, perhaps suggesting that we have identified a not yet
accounted for systematic error in the SDSS flux power spectrum data or that the
standard model describing the thermal state of the IGM at z ∼ 2 − 3 is incomplete.
Key words: cosmology: theory – methods: numerical – galaxies: intergalactic medium
1INTRODUCTION
The Lyman-α forest is an important cosmological observable
that probes matter density fluctuations in the IGM over a
unique range of redshifts, scales and environments. Many
attempts have been made to measure physical properties of
the IGM using Lyman-α forest data. The two most common
approaches are either based on decomposing the information
encoded in the transmitted flux via Voigt profile fitting or
treating the flux as a continuous field with directly mea-
surable statistical properties (e.g. Rauch et al. 1997; Rauch
1998; Theuns et al. 1998; Croft et al. 2002; Meiksin 2007).
In the second approach, measurement of the zero, one, two-
point or three-point probability distribution functions (i.e.
the mean flux level, the flux PDF, the flux power and bispec-
trum) enable a variety of physical properties to be explored.
The mean flux level for example, is sensitive to the am-
plitude of the meta-galactic UV background (Tytler et al.
2004; Bolton et al. 2005) while the flux PDF is sensitive
to the thermal evolution of the IGM (McQuinn et al. 2009;
Bolton et al. 2009b). The flux power spectrum has been used
to constrain cosmological parameters and the behaviour of
dark matter at small scales (Viel et al. 2004; Seljak et al.
2006; Viel et al. 2008) and the flux bispectrum can be used
to search for signatures of non-gaussianities in the matter
distribution (Viel et al. 2009). Ideally, a given IGM model
described by a set of cosmological and astrophysical param-
eters should agree with all these statistics including the re-
sults from Voigt profile decomposition at the same time. In
practice, the interpretation of the data is complex and is
heavily dependent on numerical simulations that incorpo-
rate the relevant physical ingredients, but have a limited
dynamic range.
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2M. Viel, J.S. Bolton & M.G. Haehnelt
The data used for these investigations consist mainly
of two kinds of sets of QSO spectra: the SDSS low–
resolution, low signal–to–noise sample and UVES/VLT or
HIRES/KECK samples of high–resolution spectra. The
characteristics of the low and high-resolution data sets are
very different (the number of SDSS spectra is about a fac-
tor ∼ 200 larger than that of high-resolution samples, but
the latter probes smaller scales due to the higher spectral
resolution). Measurements based on Lyman-α forest data
have reached a level of accuracy where an understanding of
systematic uncertainties at the percent level or below (the
magnitude of statistical errors associated with the SDSS
sample) has become important. In this Letter, we will re-
visit the flux PDF, which has been investigated previously
by several authors either on its own (e.g. McDonald et al.
2000; Jena et al. 2005; Becker et al. 2007; Lidz et al. 2006;
Kim et al. 2007; Bolton et al. 2008), or jointly with the flux
power spectrum (Meiksin et al. 2001; Zaroubi et al. 2006;
Desjacques et al. 2007). We shall focus on the flux PDF of
the UVES/VLT data as recently measured by Kim et al.
(2007) (hereafter K07); the systematic and statistical er-
rors for this sample have been addressed in detail. Recently,
both Bolton et al. (2008) (from the K07 UVES/VLT data)
and Becker et al. (2007) (from independent HIRES/KECK
spectra) have found evidence for a density-temperature re-
lation of the IGM which appears inverted if approximated
as a power–law (γ < 1 for T = T0(1 + δ)γ−1). Here, we will
improve on the analysis performed in Bolton et al. (2008)
(hereafter B08) and check its robustness by fully exploring
the cosmological and astrophysical parameter space. In ad-
dition, we briefly discuss a joint analysis of the flux PDF
and SDSS flux power spectrum, and the possible implica-
tions for constraints on cosmological parameters describing
the linear matter power spectrum and the thermal history
of the IGM.
2 METHOD
We use simulations performed with the parallel hydrody-
namical (TreeSPH) code GADGET-2 (Springel 2005) to cal-
culate the flux statistics for models with a wide range of
cosmological and astrophysical parameters by expanding
around a reference model. For the reference model we choose
here the 20-256 simulation of B08. We refer the reader to
this paper for further details, including resolution and box
size convergence tests (see Bolton & Becker (2009) for re-
cent convergence tests on SPH simulations). We will com-
pare these simulations to improved measurements of the
PDF made by K07 in three redshift bins at ?z? = 2.07,
?z? = 2.52 and ?z? = 2.94 based on a set of 18 high reso-
lution (R ∼ 45 000), high signal–to–noise (S/N ? 30 − 50)
VLT/UVES spectra. Further details regarding the observa-
tional data and its reduction, with particular emphasis on
metal removal and continuum fitting errors, may be found
in K07. In all instances the mock QSO spectra have been
processed to have the same instrumental properties as the
observed data: i.e. the same signal–to–noise, resolution and
pixel size.
We explore the following cosmological and astrophys-
ical parameters: Ωm, ns, H0, σ8 for the cosmological part
and TA,S
0
(z = 3) and γA,S(z = 3) for the IGM thermal
Param.(20,256)0.1−0.8
(20,256)0−0.9
(20,256)0−1
σ8
ns
Ω0m
H0
T0
Ts1
0
γA
γs1
τA
eff
τs1
eff
fc× 100
0.81 ± 0.07 (0.80)
0.96 ± 0.03 (0.95)
0.23 ± 0.07 (0.19)
82 ± 7 (80)
19 ± 6 (15)
0.6 ± 1.4 (−0.6)
0.75 ± 0.21 (0.72)
−1.0 ± 1.0 (−1.7)
0.312 ± 0.012 (0.312)
3.17 ± 0.18 (3.20)
0 ± 1 (0)
0.85 ± 0.07
0.96 ± 0.03
0.22 ± 0.06
84 ± 6
24 ± 8
1.3 ± 1.2
0.51 ± 0.13
−1.6 ± 0.9
0.321 ± 0.010
3.16 ± 0.13
−0.8 ± 0.4
0.86 ± 0.06
0.96 ± 0.03
0.22 ± 0.06
85 ± 8
26 ± 7
1.5 ± 1.0
0.51 ± 0.13
−1.3 ± 1.0
0.324 ± 0.010
3.25 ± 0.14
−1 ± 0.4
χ2/d.o.f.35.2/36 45.6/4864/54
Table 1. Marginalised cosmological and astrophysical parameters
derived from fitting the flux PDF at z = 2.07,2.52,2.94 in the
flux ranges F=[0.1-0.8], F=[0.1-0.9] and F=[0-1] (left, middle and
right columns respectively): T0is measured in units of 103K, H0
in km/s/Mpc. The probabilities of having a χ2larger than the
obtained values are 50, 57 and 17%, respectively. The values in
parentheses are best-fit values for (20,256)[0.1−0.8].
history, where A and S indicate the amplitude and slope
for the temperature and γ relations normalised at z = 3
(y = A[(1+z)/4]S). The amplitude and slope of the effective
optical depth evolution, τeff = −?F?, are varied assuming a
power–law evolution with redshift in order to conservatively
span the observed range suggested by high-resolution and
low-resolution data sets. We furthermore varied the reion-
ization redshift (zre = 9 in our reference model) but found
this had no impact on the flux PDF at z < 3 (although
the differences in “Jeans smoothing” will be important at
redshifts close to zre, e.g. Pawlik et al. (2009)). For the ef-
fect on the flux power we refer to McDonald et al. (2005)
and Viel & Haehnelt (2006). We also consider the effect of
a misplaced continuum level by adding an extra parameter
fc (the flux following a continuum correction is assumed to
be F × (1 + fc)).
We compute derivatives of the flux statistics from the
20-256 model at second order using between two and four
simulations for each cosmological and astrophysical param-
eter. For the thermal history we explore a wide range of
possible T0 and γ values by extending the original grid of
simulations presented in B08. The 20-256 model has γ ∼ 1.3
below z = 3 and the temperature at mean density in the
three PDF redshift bins are T0 = 14.8,17.6,20.8 × 103K.
This model was shown to be a poor fit to the K07 data in the
simple analysis performed by B08. Here we will calculate the
χ2of our models varying all the parameters that affect the
flux PDF and not just the effective optical depth, enabling us
to expand around this model. This simple Taylor–expansion
method was introduced in Viel & Haehnelt (2006) in order
to explore constraints for the SDSS flux power spectrum. It
has the advantage of enabling the exploration of the param-
eter space close to the reference model with an accurate set
of hydrodynamical simulations. However, the full parameter
space cannot be probed in this way with the same high ac-
curacy (see McDonald et al. 2005 for a different approach).
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Constraints from the Lyman-α forest flux PDF3
0.00.20.40.60.81.0
0.1
1.0
10.0
z=2.07
0.1
1.0
10.0
z=2.52
0.1
1.0
10.0
z=2.94
flux
flux pdf
Figure 1. Best fit to the flux PDF for model (20-256)0.1−0.8
(continuous blue) and (20-256)0−1(dashed green) in the three
redshift bins at ?z? = (2.07, 2.52,2.94).
3RESULTS FOR THE FLUX PDF
We obtain the best fit to the observed flux PDF for three flux
intervals F = [0.1−0.8], F = [0.1−0.9], F = [0−1]. Different
flux levels are subject to different systematic effects, such
as the presence of noise and strong absorption systems at
F ∼ 0 and the effect of continuum fitting errors at F ∼ 1
(see K07 for details). Since the PDF error bars are correlated
we expect these systematic errors to nevertheless impact on
the PDF over the full flux range. The level of consistency
between the fits to these three flux intervals should indicate
to what extent these systematic errors may or may not affect
the results.
For the flux range F = [0.1 − 0.8] we have a total of
45 data points to fit and a set of 9 free parameters that
will be varied in the Markov Chain Monte Carlo routines.
We use the following priors on the effective optical depth,
τA
eff= 3.65 ± 0.21, based on the ob-
servational results obtained by K07. Note, however, that the
final results are affected very little by the choice of these pri-
ors; the constraints on the effective optical depth amplitude
at z = 3 are in fact much tighter than these priors assume.
The results of this analysis are summarised in Table 1 and
Figs. 1 and 2. We obtain a very good fit to the flux PDF
for the flux range F = [0.1 − 0.8] (reduced χ2= 0.98, a
χ2larger than this has 50 per cent probability). With the
data points at F < 0.1 we obtain slightly larger values for
σ8 and T0, but the results are in agreement at the 1σ level.
Adding the flux range at F > 0.9 results in a poor fit unless
the error bars on the last two data points are increased by a
factor of four. Note that the covariance properties of these
data points are strongly influenced by the choice of the con-
tinuum level. Increasing the error bars by this factor would
account for a misplacement of the continuum level by a few
percent. We also find evidence (2σ) that the data prefer a
non-zero continuum offset, fc, when we add the regions at
eff= 0.36 ± 0.11 and τS
τeff
A (z=3)
τeff
S (z=3)
0.280.290.30.310.320.33 0.34
2.8
3
3.2
3.4
3.6
τeff
A (z=3)
γA (z=3)
0.280.290.30.310.320.330.34
0.2
0.4
0.6
0.8
1
1.2
T0
A (z=3) K x 104
γA (z=3)
123
x 104
0.2
0.4
0.6
0.8
1
1.2
τeff
A (z=3)
σ8
0.280.29 0.3 0.310.32 0.330.34
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
γA (z=3)
σ8
0.20.61
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
T0
A (z=3) K x 104
σ8
123
x 10
4
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Figure 2. 2D marginalized and mean likelihood contours for
cosmological and astrophysical parameters for flux PDF model
(20,256)0.1−0.8(continuous curves and filled contours). Marginal-
ized likelihoods for the model (20,256)0−1are also shown in green.
low and high transmissivity, but note that realistic errors in
the assumed continuum level should depend on the flux and
noise level and are expected to vary along the spectrum.
In order to further explore the sensitivity to the last two
data points (F = 0.95,1) in each redshift bin, we performed
an additional analysis by combining the two data points for
the highest flux levels into one point at F = 0.975. We then
recomputed the data covariance matrix and the PDF deriva-
tives (without multiplying the covariance values of this data
point by four). In this instance the results are consistent
with those for the F = [0 − 0.9] and F = [0 − 1] flux ranges
range to within 1σ. In this case we obtain χ2= 60.6 for
51 d.o.f., which indicates a reasonable fit (the probability
for a value larger than this is 17% ). We therefore conclude
from the results in Table 1 that our findings are robust to
continuum fitting uncertainties and that the impact of con-
tinuum uncertainties is mainly restricted to the flux range
F = 0.975 − 1.025.
The effective optical depth is constrained very well by
the data with a best-fit value of τeff(z = 3) = 0.31 ±
0.01, consistent with observational measurements from high-
resolution spectra. The constraints on the temperature den-
sity relation at z = 3 are (TA
in agreement with the findings of B08, and no significant
evolution in the equation of state below z < 3 is inferred (in
agreement with Schaye et al. (2000); Ricotti et al. (2000)).
The PDF alone provides interesting constraints on cosmo-
logical parameters describing the evolution of the linear
power spectrum; σ8 and ns are constrained to be in the
0,γA) = (19 ± 6,0.75 ± 0.21),
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4M. Viel, J.S. Bolton & M.G. Haehnelt
range σ8 = 0.8 − 0.85 ± 0.07 and ns = 0.96 ± 0.03. The de-
rived cosmological parameters are in good agreement with
the results of other large-scale structure probes such as
WMAP and weak lensing data, albeit with about a factor
two larger errors than those from the WMAP data (e.g.
Lesgourgues et al. (2007)).
Our results corroborate the suggestion of an inverted
temperature-density relation at z = 3. As for the analysis
of the full flux range the statistical significance of the data
favouring an inverted temperature-density relation γ < 1 is
about 3σ at z ∼ 3. At z < 3 the data is consistent with
an isothermal (γ ∼ 1) temperature-density relation The re-
sults regarding the thermal state of the IGM do not change
significantly if we omit the flux range F > 0.9. If we dis-
card both the flux ranges at low and high emissivity and
consider only the flux range F = [0.1−0.8], there is still ev-
idence for an inverted T −ρ relation, but at a reduced level
of significance (1 − 1.5σ C.L.). The likelihood contours in
Fig. 2 indicate that a value of γ ∼ 1.3 suggested recently by
the He II reionization simulations of McQuinn et al. (2009)
is between 2 and 3 − 3.5σ discrepant with the marginalized
value we obtain when fitting the flux range F = [0.1 − 0.8]
and the full flux range, respectively.
4ADDING THE FLUX POWER SPECTRUM
In this section we first revisit constraints from the SDSS
flux power spectrum (PS) alone before proceeding to com-
bine this data set with the flux PDF of the UVES/VLT data
for a joint analysis. The SDSS flux power spectrum is based
on 3035 QSO spectra with low resolution and low signal-
to-noise, spanning the redshift range z = 2.2 − 4.2 (mea-
surements are made at 11 wavenumbers in 12 redshift bins).
Dealing with the systematic uncertainties of low resolution
and low signal–to–noise QSO spectra and extracting the flux
power is a difficult task. We refer to McDonald et al. (2005)
for a comprehensive study of the removal of continuum fluc-
tuations, metal line contamination, damped Lyman-α sys-
tems and dealing with the resolution of the spectrograph
and noise level in each of the redshift bins. All these effects
need to be properly taken into account, as a poor treatment
would impact the obtained flux power in a non-trivial way.
In the following, we will use the flux power provided by the
SDSS collaboration, introducing “nuisance parameters” for
the resolution and noise in each redshift bin as suggested by
McDonald et al. (2005), and implicitly assuming that all the
contaminants above have been either removed or properly
modelled.
We compute the constraints from the SDSS flux power
spectrum in a similar way to Viel & Haehnelt (2006) with
the notable difference that we calculate the predicted flux
statistics by expanding around a model with γ ∼ 1 in the
redshift range z = [2 − 4], while the original analysis was
based on simulations with γ ∼ 1.6. Furthermore, the flux
power is computed using a Taylor expansion to second in-
stead of first order. The parameters of the fiducial cosmolog-
ical simulation are those of the B2 model in Viel & Haehnelt
(2006). As before the flux statistics have been corrected for
box-size and resolution effects. We compute the derivatives
required for the Taylor expansion by performing between
four and six hydrodynamical simulations for every cosmo-
Param.
B2(lowz)
1
B2(allz)
1
B2(lowz)
1
+(20,256)0.1−0.8
σ8
ns
Ω0m
H0
T0
Ts1
0
Ts2
0
γA
γs1
γs2
τA
eff
τs1
eff
zre
0.85 ± 0.05
0.93 ± 0.03
0.25 ± 0.04
78 ± 7
38 ± 7
−0.6 ± 1.3
−2.3 ± 1.3
0.63 ± 0.50
−0.7 ± 2.6
−1.2 ± 2.1
0.326 ± 0.028
3.19 ± 0.25
11.9 ± 3.8
0.86 ± 0.04
0.96 ± 0.02
0.26 ± 0.03
77 ± 7
42 ± 6
−0.3 ± 1.1
−3.9 ± 1.3
0.79 ± 0.51
0.4 ± 2
−1.4 ± 1.5
0.322 ± 0.028
3.25 ± 0.23
9.1 ± 2.7
0.90 ± 0.02
0.95 ± 0.02
0.25 ± 0.03
80 ± 5
26 ± 4
1.4 ± 0.5
−3.1 ± 1.5
0.50 ± 0.20
−2.1 ± 1.6
−1.4 ± 1.6
0.320 ± 0.007
3.12 ± 0.10
14.3 ± 3.6
χ2/d.o.f.78.9/85139.7/121136/130
Table 2. Cosmological and astrophysical parameters derived
from the B21 model (γ ∼ 1) for the flux power spectrum: (lowz)
flux PS fitted in the range z = [2.2 − 3.6]; (allz) flux PS fit-
ted in the range z = [2.2 − 4.2]. The probability of having a χ2
value larger than this for model B2(lowz)
B2(allz)
1
it is 12 %. The constraints for a joint analysis of flux PS
and PDF (F = 0.1 − 0.8) are shown in the 3rdcolumn (prob. is
35%).
1
is 64 %, while for model
logical and astrophysical parameter considered. In addition,
we now allow for the effect of the reionization redshift, zre,
and introduce this as an extra parameter. We interpolate be-
tween two very different reionization histories with zre ∼ 15
and zre ∼ 7. We also introduce two extra parameters de-
scribing the redshift evolution of the thermal state of the
IGM, the power–law index of the T and γ relations at z > 3
(a redshift range which is not probed by the PDF).
The results for the power spectrum only analysis are
summarised in the first two columns of Table 2 for a low
redshift only sample, z = [2.2 − 3.6], and the full SDSS
data set, z = [2.2 − 4.2]. We decided to perform a sepa-
rate analysis which omits the highest redshift bins following
Viel & Haehnelt (2006), who obtained a somewhat poorer
fit for the high redshift (z > 3.6) PS estimates. Despite the
fact that the flux statistics were calculated by expanding
around a reference model with very different thermal his-
tory, in both instances the analysis still gives constraints on
the cosmological parameters that are in agreement with the
previous analysis of Viel & Haehnelt (2006). This is rather
reassuring. However, there are some aspects of the results
that need scrutiny. First, we note that for the flux power
only the temperature at mean density, T0, is significantly
higher than that preferred by the PDF (and higher than ex-
pected for the photoionized IGM). Secondly, the value of σ8
is now somewhat on the lower end of values allowed by the
previous analysis of Lyman-α data and thus in better agree-
ment with the CMB data (e.g. Komatsu et al. 2008). This
is due to the degeneracy between σ8 and γ discussed in B08;
allowing for γ < 1 means that the flux power can now be
fitted by a slightly lower σ8 (but note that other parameters
also have a a significant influence on the inferred σ8, most
notably the mean flux level).
Overall the results from the new flux PS analysis are
consistent with those inferred from the PDF alone except
for the value of T0 at z = 3, which is several σ above that
inferred from the best fit to the flux PDF. In the last col-
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Constraints from the Lyman-α forest flux PDF5
umn of Table 2 we show the constraints for a joint analysis
of flux PDF and PS. Somewhat surprisingly the joint analy-
sis prefers a larger value of σ8 = 0.9±0.02 with rather small
errors. We explicitly checked that this large value of σ8 is
related to the rather different TA
PS favour. If we artificially remove the constraint of the tem-
perature being simultaneously consistent with the somewhat
discrepant temperatures favoured by the flux PDF and PS,
the joint analysis gives σ8 = 0.86 ± 0.03 with an improve-
ment of ∆χ2= 12. A not yet accounted for systematic error
in the measurement of the flux PDF and/or PS appears to
be a possible explanation for this discrepancy. Alternatively
the inconsistencies may suggest that a power–law T − ρ re-
lation is a poor approximation to the thermal state of the
IGM and a wider range of physically motivated relations
should be considered in future simulations.
0 values that the PDF and
5DISCUSSION
We have presented cosmological and astrophysical con-
straints derived from the K07 Lyman-α flux PDF measured
from a set of 18 high-resolution QSO spectra whose statisti-
cal and systematic errors have been carefully estimated. The
Lyman-α flux PDF on its own provides tight constraints on
the thermal state of the IGM and on cosmological parame-
ters describing the linear dark matter PS. The results have
been obtained by fitting the flux PDF at three different red-
shift bins in the range 2 < z < 3 and for three different
flux ranges F = [0.1 − 0.8], F = [0 − 0.9], and F = [0 − 1].
There is good agreement between the analyses for the full
flux range and the two restricted flux ranges and the re-
sults are consistent with those derived in the simpler analy-
sis made by B08. An inverted temperature-density relation
is favoured at the ∼ 3σ level (at z ∼ 3) if we consider the
PDF for the full flux range, but the significance is reduced
to 1 − 1.5σ if we restrict the analysis to F = [0.1 − 0.8].
The constraints for other parameters are in agreement with
those presented in the literature (e.g. the SDSS flux power
in McDonald et al. 2005). We have also refined the method
used by Viel & Haehnelt (2006) and updated the constraints
from the SDSS flux power spectrum. The constraints from
the flux power spectrum are consistent with those from the
flux PDF, with the exception that the flux power spectrum
prefers a significantly larger temperature at mean density,
T0. A joint PS–PDF analysis gives a reasonable fit to the
data but results in a larger σ8 than an analysis of CMB
data alone. This discrepancy appears to be related to the
higher T0 that the flux power spectrum prefers.
Recent simulations
He II reionization indicate that an inverted T − ρ re-
lation is very difficult to achieve within the standard
model describing the thermal state of the IGM even
if radiative transfer effects are taken into account, at
least if He II reionization is driven primarily by QSOs
(McQuinn et al. 2009; Bolton et al. 2009a). It therefore
appears difficult to reproduce the observed flux PDF with-
out invoking a not yet identified source of IGM heating,
or additional systematic errors which impact on the flux
PDF. Similarly, the high T0 values preferred by the PS are
very difficult to reconcile with constraints from the widths
of thermally broadened absorption lines (e.g. Schaye et al.
of photo-heatingduring
(2000); Ricotti et al. (2000)) and our understanding of the
heating of the photoionized IGM.
As the results from the flux PDF and the CMB data
agree very well, the rather high values of T0 preferred by
the flux power spectrum suggests perhaps instead that we
have identified a not yet accounted for systematic error in
the SDSS flux power spectrum data. Independent analy-
sis based on line statistics, on new data sets at medium
and high–resolution and further progress in incorporating
He II reionization models into high resolution hydrodynam-
ical simulations will hopefully allow us to further improve
our understanding of the systematic uncertainties of Lyman-
α forest data and resolve these small but statistically signif-
icant inconsistencies.
ACKNOWLEDGMENTS.
Numerical computations were performed on the COSMOS
supercomputer at DAMTP and on the High Performance
Computer Cluster Darwin (HPCS) in Cambridge (UK).
COSMOS is a UK-CCC facility which is supported by
HEFCE, PPARC and Silicon Graphics/Cray Research. Part
of the analysis was also performed at CINECA (Italy) with
CPU time assigned thanks to an INAF-CINECA grant. We
thank A. Lidz and P. McDonald for suggestions and useful
criticism.
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