Warm Dark Matter as a solution to the small scale crisis: new constraints from high redshift Lyman-alpha forest data
aa r X i v : . [ a s t r o - ph . C O ] A ug Warm Dark Matter as a solution to the small scale crisis: new constraints from highredshift Lyman- α forest data Matteo Viel , , George D. Becker , James S. Bolton , Martin G. Haehnelt INAF - Osservatorio Astronomico di Trieste,Via G.B. Tiepolo 11, I-34131 Trieste, Italy INFN/National Institute for Nuclear Physics,Via Valerio 2, I-34127 Trieste, Italy Kavli Institute for Cosmology and Institute of Astronomy,Madingley Road, Cambridge,CB3 0HA, United Kingdom School of Physics and Astronomy,University of Nottingham, University Park,Nottingham, NG7 2RD, United Kingdom (Dated: August 27, 2013)We present updated constraints on the free-streaming of warm dark matter (WDM) particlesderived from an analysis of the Lyman- α flux power spectrum measured from high-resolution spectraof 25 z > α flux power with conservative error estimates. By using a method that samplesthe multi-dimensional astrophysical and cosmological parameter space, we obtain a lower limit m WDM ∼ > . σ ) for warm dark matter particles in the form of early decoupled thermal relics.Adding the Sloan Digital Sky Survey (SDSS) Lyman- α flux power spectrum does not improve thislimit. Thermal relics of masses 1 keV, 2 keV and 2.5 keV are disfavoured by the data at about the9 σ , 4 σ and 3 σ C.L., respectively. Our analysis disfavours WDM models where there is a suppressionin the linear matter power spectrum at (non-linear) scales corresponding to k = 10 h/ Mpc whichdeviates more than 10% from a ΛCDM model. Given this limit, the corresponding “free-streamingmass” below which the mass function may be suppressed is ∼ × h − M ⊙ . There is thus verylittle room for a contribution of the free-streaming of WDM to the solution of what has been termedthe small scale crisis of cold dark matter. PACS numbers: 98.80.Cq,98.62.Ra,95.35.+d
I. INTRODUCTION
The ΛCDM paradigm, in many respects, has provento been an immensely successful cosmological model.ΛCDM is based on a cosmological constant plus “cold”dark matter, i.e. dark matter particles whose streamingvelocities are negligible for most astrophysical consider-ations. On large scales, the
Planck mission has just de-livered another ringing endorsement of this model withits first-year cosmology results [1]. On scales below afew (comoving) Mpc, however, the matter power spec-trum is still difficult to probe, and it has been repeatedlysuggested that dark matter is perhaps “warm”, with afree-streaming length that affects the properties of low-mass (dwarf) galaxies. Warm dark matter (WDM) couldalleviate the apparent difficulties of ΛCDM models in re-producing some observations related to the matter powerspectrum on scales of a few Mpc and below. The mostnotable possible tensions under ΛCDM are: the excess ofthe number of galactic satellites, the cuspiness and high(phase space) density of galactic cores, the luminosities ofthe Milky Way’s satellites and the properties of galaxies filling voids (e.g. [2–5]).The main effect of the larger velocities of WDM parti-cles, and the resulting significant free-streaming length,would be to suppress structures on Mpc scales and below.The last few years have seen a re-intensified discussion ofthis possibility, particularly in light of improvements innumerical models and observations of the mass and in-ternal structure of Local Group satellites [6]. It has beensuggested that a free-streaming length corresponding tothat of a thermal relic WDM particle with a mass of 1-2 keV (and in some cases as low as 0.5 keV) providesbetter agreement between the most recent data and nu-merical simulations [7, 8]. The difficulties associated withthe cold dark matter paradigm, however, arise on scaleswhere the matter spectrum is highly non-linear at z ∼ α absorption produced by intergalac-tic neutral hydrogen in the spectra of distant quasars(QSOs)–the so called “Lyman- α forest”–provides a pow-erful alternative tool for constraining dark matter prop-erties, particularly the free-streaming of dark matter par-ticles on the scales in question. The Lyman- α forestprobes the matter power spectrum in the mildly non-linear regime over a large range of redshifts ( z = 2 − − h − Mpc) [12, 13]. On large scales the SDSS-IIIBOSS collaboration has recently measured the BaryonicAcoustic Oscillations (BAOs) scale in the 3D correlationfunction of the Lyman- α forest in ∼ z ∼ . α forest as a tracer of cosmological large-scale structure. Constraints on the matter power spec-trum from Lyman- α forest data on small scales are onlylimited by the thermal cut-off in the flux power spectrumintroduced by pressure and thermal motions of baryonsin the photo-ionized intergalactic medium (IGM). TheIGM has a characteristic temperature of ∼ K. Whilenot trivial, modeling the relevant physics with numeri-cal simulations is reasonably straightforward; the powerspectrum at the relevant redshifts ( z ∼ −
5) and scalesis only mildly non-linear and stellar feedback effects aremuch less important than at lower redshifts [15, 16].The basic property of WDM, which impacts on bothlarge scale structure formation and the internal structureof dark matter haloes and the galaxies they are hosting,is the significant “thermal” velocities of the WDM parti-cles (see [2]). The resulting “free-streaming” eliminatesdensity fluctuations on scales below a characteristic co-moving wavenumber: k FS ∼ . h Mpc (cid:16) m WDM (cid:17) / (cid:18) . D M h (cid:19) / , (1)and leads to a very distinctive cut-off in the matterpower spectrum at a corresponding scale. For example,the wavenumber at which the linear WDM suppressionreaches 50% in terms of matter power, k / , w.r.t. theΛCDM case can be approximated as: k / ∼ . h Mpc (cid:16) m WDM (cid:17) . (cid:18) Ω D M . (cid:19) − . (cid:18) h . (cid:19) . , (2)where this equation uses the numerical results ofRef. [17]. For standard thermal relics, the shape of thecut-off is therefore well characterized in the linear regimeand there is an unambiguous relation between the massof the thermal relic WDM particle and a well-definedfree-streaming length (e.g. [17]). Note that we will alsoquote a free-streaming mass, which is the mass at themean density enclosed in a half-wavelength mode corre-sponding to k / .For the sake of simplicity, the analysis here is presentedin terms of the mass of a thermal relic dark matter par-ticle, for which there is a one-to-one correspondence be-tween the free-streaming length and the particle mass.We should point out, however, that in recent years ster-ile neutrinos and other non-thermal particles have be-come popular WDM candidates. Some of these modelsare actually more similar to mixed dark matter modelswith cold and warm dark matter components. The shape of the free-streaming “cut-off” can then be quite differ-ent from that of a thermal relic, and may instead corre-spond to a downward step in the power spectrum ratherthan a cut-off (see [8, 18]). There is also no universalrelation between free-streaming length and mass of theWDM particles in these models, and the normalizationand functional form of this relation varies greatly be-tween different non-thermal WDM candidates. Unfortu-nately, this has led to considerable confusion in the liter-ature when WDM models, characterized by their model-dependent WDM particle masses, are compared betweeneach other and/or thermal relic models and in particularwith Lyman- α forest data. For example Ref. [8] quotea sterile neutrino mass of 2keV for their thermal relicWDM model which corresponds, however, to a thermalrelic mass of 1.4 keV. For convenience and ease of compar-ison with the literature, in this work we therefore consideronly matter power spectra with cut-off shapes expectedfor thermal relic WDM particles and quote the unam-biguous thermal relic masses (and corresponding cut-offscale) to characterize our WDM models.The Lyman- α forest, due to its spectral nature, probesthe matter power spectrum in velocity space. With in-creasing redshift the ratio of a given (comoving) free-streaming length in velocity space to the thermal cut-off length scale at a given temperature increases as ∝ (1 + z ) / . There is furthermore strong observationalevidence that the temperature of the IGM decreases to-ward higher redshift over the range 3 < z < α forest data to constrain WDM were first usedin Ref. [20] where a limit of 750 eV was obtained by usingN-body simulations only. In previous work, Ref. [17], weused instead two samples of high-resolution QSO Lyman- α forest spectra at z ∼ . ∼
4. As already noted,however, care has to be taken in the correct modelingof the free-streaming properties of “non-thermal” candi-date WDM particles of a given model-dependent mass,such as the popular sterile neutrino. In Ref. [18] theauthors have focused on constraints on a range of suchmodels. Because of a non-zero mixing angle between ac-tive and sterile flavour states, X-ray flux observationscan also constrain the abundance and decay rate of suchWDM particles (e.g. [23]). The joint constraints fromLyman- α forest data and those from the X-ray fluxesof astrophysical objects now put considerable tension onthe parameter space allowed for a sterile neutrino particlewith the phase-space distribution proposed by Dodelson& Widrow [24, 25], although other, possibly more phsyi-cal scenarios should be explored [18, 26].In Ref. [27] we presented the most stringent Lyman- α forest limits up to that date on the free-streamingof dark matter, m WDM > σ ). That analysiswas based on an (at the time) unrivaled sample of high-quality, high resolution QSO absorption spectra extend-ing to z ∼ .
5. The limit is in obvious conflict withmany of the recent suggestions for alleviating the diffi-culties encountered by numerical models in reproducingthe observed properties of Local Group satellite galaxieswithin the cold dark matter paradigm. These models of-ten assume dark matter to be made up by thermal relicWDM with masses in the range 0.5-2 keV (e.g. [8]).Since our study in [27], the size of our high-quality,high-redshift QSO absorption spectra sample, the qual-ity and size (in particular the dynamic range and resolu-tion) of our numerical simulations and our knowledge ofthe thermal and ionization state of the IGM at the rele-vant redshifts have all significantly improved. Motivatedby these improvements, and in light of the lively debateof dark matter possibly being warm with masses in therange 0.5-2 keV, we present here a new and much moreextensive study of the high-redshift Lyman- α forest con-straints on the free-streaming properties of dark matter.The new study is based on an improved data set, fur-ther refined modeling of the flux power spectrum and alarge suite of new numerical hydrodynamical simulations.We also perform a comprehensive investigation of the thesystematic uncertainties related to this measurement.Finally, it is worth highlighting that WDM would haveprofound implications in many astrophysical and cos-mological contexts. In this respect, IGM constraintsare highly complementary to other probes based, forexample, on the properties of dark matter haloes [28],the number of satellites and their luminosities [3, 29],strong lensing, the velocity function in the local en-vironment [30], phase-space density constraints [31],the formation of the first stars [32], the high-redshiftquasar luminosity function [33], decays of WDM parti-cles in the high redshift universe [34], reionization [35],gamma ray-bursts [36], galaxy formation aspects [37]using N-body/hydrodynamical simulations [38, 39] oranalytical/semi-analytical methods [40–43].The paper is organized as follows. In Section II wepresent our new data set. The simulations are describedin Section III. The mock quasar sample, which will beimportant for estimating error amplitude and covariance,is introduced in Section IV. Section V discusses the effectof the most important physical parameters on the fluxpower spectrum, while most of the remaining nuisanceparameters and the impact they have in terms of fluxpower are discussed in an Appendix. Our main resultsare reported in Section VI, together with a descriptionof the Monte Carlo sampling of the likelihood space. Wesummarize our findings and conclude in Section VIII. II. DATA
Our analysis is based on high-resolution spectra of 25quasars with emission redshifts 4 . ≤ z em ≤ .
42. Com-pared to our previous analysis in Ref. [27] the number ofQSO spectra, at these redshifts, has improved by nearly afactor two. Spectra for fourteen of the objects were takenwith the Keck High Resolution Echelle Spectrometer(HIRES) [44], and the remaining eleven were taken withthe Magellan Inamori Kyocera Echelle (MIKE) spectro-graph on the Magellan Clay telescope [45]. Most of thedata have been presented elsewhere [19, 46–48]. Here webriefly review the relevant features of the spectra, anddescribe how the flux power spectra were calculated.The majority of spectra were reduced using a cus-tom set of idl routines based on optimal sky subtrac-tion [49] and optimal extraction [50] techniques, while asmall subset of the HIRES spectra (PSS 0248+1802 andBR 1202 − makee softwarepackage. The HIRES and MIKE spectra have spectralresolutions of 6.7 and 13.6 km s − (FWHM), and thespectra were extracted using 2.1 and 5.0 km s − spec-tral bins, respectively. The one-dimensional relative flux-calibrated spectra were then continuum normalized usingspline fits based on power-law extrapolations of the con-tinuum redward of the Lyman- α emission line. Mediancontinuum signal-to-noise ratios within the Lyman- α for-est of each object are typically in the range of 10 −
20 perpixel. The continuum estimates are necessarily crude dueto the high levels of absorption in the Lyman- α forest atthese redshifts. We estimate that typical uncertaintiesin the continuum are of the oder ∼ α forest in each quasar spectrum into two regionsof equal redshift length. We then computed the powerspectrum of the fractional transmission, δ F ( z ), in eachregion separately, where δ F ( z ) = F ( z ) − h F (¯ z ) ih F (¯ z ) i . (3)Here, h F (¯ z ) i is the mean transmitted flux calculated atthe mean redshift of each region. We used fixed relationsfor the mean flux given by h F ( z ) i = exp [ − τ eff ( z )], where τ eff ( z ) = ( . (cid:0) z . (cid:1) . − . , z ≤ . . (cid:0) z . (cid:1) . , z > . . (4)The fit to τ eff at z ≤ . z > . τ eff ( z ) presented by [52]. We note that our anal-ysis is not sensitive to our choice of using a fixed relationfor h F (¯ z ) i . In tests where we instead divided each regionby the mean flux in that region alone we obtained verysimilar power spectrum estimates on average. When cal-culating the flux power spectrum we do not attempt tomask metal lines (see discussion below). We do, however,mask regions of strong telluric absorption (6275 − − − − − k (s / km) = [ − . , − .
1] with 0.2 dex spacing. Thepower spectra from all regions were then further aver-aged according to instrument and the mean redshift ineach region. We used median redshifts z = 4 . , . , , . k (s / km) < − .
3, which might be affected by con-tinuum fitting uncertainties. The final data set used inthe present analysis thus consists of 49 data points.Preliminary estimates of the error in the power spectrawere calculated using a bootstrap approach. It is known,however, that bootstrapping typical underestimates thetrue errors (e.g. [53]). To be conservative, we thereforedecided to add an additional 30% uncertainty to our es-timates of the errors of the observed flux power spectrumfor our standard analysis. We will also quote the tighterlimits that would be obtained without this increase ofthe error estimate. As a further check, we used the setof mock QSO spectra described in Section IV to deter-mine what the expected covariance in the power spectrashould be (within the limits of our finite simulation box)at each redshift for a sample of similar size and quality tothe one used here. These estimate were used to correct afew error estimates in the real data that appeared to betoo small. With these corrections, the final flux powerspectra used here have error bars that are larger than σ ( P F ) /P F > . III. COSMOLOGICAL HYDRODYNAMICALSIMULATIONS
We model the flux power spectrum based on a set ofhydrodynamical simulations performed with a modifica-tion of the publicly available
GADGET-II code. This codeimplements a simplified star formation criterion [54] thatturns all gas particles that have an overdensity above1000 and a temperature below 10 K into star parti-cles. This has been first used and extensively tested inRef. [55].The reference model, hereafter referred to as (20 , h − comoving Mpc with 2 × gas and cold DM particles (with a gravitational soft-ening length of 1.3 h − kpc) in a flat ΛCDM universewith cosmological parameters Ω m = 0 . b = 0 . n s = 0 . , H = 70 . − Mpc − and σ = 0 . m WDM = 1 , , k / ∼ . , . , h /Mpc, respectively. The initial con-dition power spectra are generated with CAMB [57] andthe suppression and velocity for the WDM particles areimplemented using the approach outlined in Ref. [17]. Inorder to assess convergence and evaluate resolution cor-rections (which are model dependent), we also performfour additional (20 , , , α forest by modifying the Ultra Vio-let (UV) background photo-heating rate in the simula-tions (e.g. [58]). A power-law temperature-density re-lation, T = T (1 + δ ) γ − , arises in the low densityIGM (1 + δ <
10) as a natural consequence of theinterplay between photo-heating and adiabatic cooling[59]. We consider a range of values for the tempera-ture at mean density, T , and the power-law index ofthe temperature-density relation, γ , based on the obser-vational measurements presented recently by Ref. [19].These consist of a set of 3 different indices for thetemperature-density relation, γ ( z = 4 . ∼ . , . , . z = [4 . − .
6] and 3 different temperatures atmean density, T ( z = 4 . ∼ , , T ( z = 4 . , γ ( z = 4 . , . z re (i.e.the redshift at which the optically thin UV backgroundis switched on in the simulations) which is chosen to be z re = 12 for the reference case and z re = 8 ,
16 for twoadditional models; the Hubble constant, with two ex-tra simulations with H = 66 . , . − Mpc − ; thescalar spectral index, with n s = 0 . , . m = 0 . , .
30 and the r.m.s. ampli-tude of the matter power spectrum, with σ = 0 . , . α forest constraints was also considered inRef. [61] using this parameterisation. Overall, a totalof 54 hydrodynamical simulations have been performed.Approximately 4000 core hours were required for each(20,512) run to reach z = 2, with the higher resolutionsimulations requiring around 5 times longer.During the simulation runs, we extract the non-linearmatter power spectra in order to compare with [10]. Forthe reference case only, we additioanlly extract the po-sition of the haloes with a friends-of-friends halo findingalgorithm for our model of the impact of spatial fluctu-ations in the UV background on the flux power (see theAppendix for further details). P ( k ) W D M / P ( k ) Λ C D M z=5.4z=4.2z=3WDM 1 keVWDM 2 keVWDM 4 keV SDSS HIRES + MIKE
FIG. 1: Ratio between the 3D non-linear matter power spec-trum of 3 different WDM models (1, 2 and 4 keV, black, blueand orange curves) at 3 different redshifts ( z = 3 , . , . m WDM = 2 keV model obtained usingEq. 6 of Ref. [17]. The arrows in the bottom part of the figureindicate the maximum value of the wavenumbers probed bythe SDSS data and by the data set used in the present anal-ysis. This figure refers to the reference (20,512) simulations.
Lastly, we note that the physical properties of theLyman- α forest obtained from the TreePM/SPH code
GADGET-II are in very good agreement at the per-cent level with those inferred from the moving-mesh code
AREPO [62] and with the Eulerian code
ENZO [63].
IV. THE MOCK QSO SAMPLE
The simulated Lyman- α forest spectra are extractedalong 5000 random line-of-sights (LOSs) after interpola-tion of the relevant physical quantities along the LOSs us-ing the SPH formalism. Box-size effects on the flux powerare estimated with (60 , , a posteriori , after havingextracted the spectra, by reproducing 0 . , , . τ eff (see Appendix). At the end ofthe procedure the four-dimensional parameter space in( m WDM , τ eff , T , γ ) is explored fully by means of quadri-linear interpolation performed over the set of 36 hydro-dynamical simulations and 108 (36 × i ) we consider the total redshift path ineach redshift bin and combine the short simulated spec-tra (20 Mpc /h in length) to match the total length of anobserved QSO spectrum (approximately 40 spectra areused); ii ) we allow for an optical depth evolution alongthe LOS (which is absent since our simulated spectra arefrom snapshots at fixed redshifts) following the scalingexpected from the fluctuating Gunn-Peterson approxi-mation, τ ∝ (1 + z ) . (see e.g. [64]); iii ) for each shortsimulated spectrum we consider a ±
20% error on thequasar continuum placement (the continuum is drawnrandomly from a Gaussian distribution around the value1 with a σ = 0 . iv ) we smooth the flux with a Gaussianat a given FWHM corresponding to the spectrograph res-olution and rebin the spectra with the observed pixel-size; v ) we add Gaussian-distributed noise on top of the flux,matching the signal-to-noise of the observational data.We demonstrate in the Appendix that the (instrumental)effects of noise and finite resolution, which are scale andredshift dependent, are below 20% (6%) at the smallestscales for MIKE (HIRES).The (co)variance properties of this mock sample are inreasonable agreement with those of our observed sample,both as a function of redshift and wavenumber. Thereare only 5 data points that appear to have error bars thatare smaller than those obtained from the mock sample: 4data points from the MIKE sample (log k (s/km)= − . z = 4 .
2, log k (s/km)= − . z = 4 .
6, log k (s/km)= − . , − . z = 5) and one data point from the HIRESsample (log k = − . z = 4 . α forest spectra do f l u x Λ CDMWDM 2 keVWDM 1 keV
FIG. 2: Transmitted flux along a set of random LOSs for the ΛCDM (green curve) and WDM 1 keV (black curve) and WDM 2keV (blue curve) models at z = 4 .
6. This figure refers to the reference (20,512) simulation without adding instrumental noise.The ΛCDM flux is clearly showing more substructure as compared to the WDM models. not incorporate chemical elements other than hydrogenand helium. We have therefore estimated how uniden-tified, lower redshift metal lines in the Lyman- α forestmay bias our result, and in particular how these nar-row absorption lines may alter the flux power spectrumat small scales. Furthermore, the simulations also as-sume a spatially uniform UV background. We thereforealso estimate the impact of spatial fluctuations in theUV background on the Lyman- α forest at z = 4 . . II ionisingbackground at lower redshift, to which we refer the readerfor further details. These two systematics effects will alsobe discussed more extensively in the Appendix. V. THE FLUX POWER SPECTRUMA. The WDM and thermal cut-offs
In this Section we demonstrate the distinctively differ-ent effects that the thermal (i.e. due to the temperatureof the photo-ionized IGM) and WDM cut-off have onthe flux power spectrum. We will also check for possi-ble effects due to the limited numerical resolution of oursimulations. Firstly, however, it is instructive to con-sider Fig. 1, where we show the ratio of the non-linearmatter power spectra in the WDM and ΛCDM simula-tions. The results are shown at three different redshifts, z = (3 , . , . z = 4 . . z = 3 to show the evolution of the non-linear power atlower redshift. The three WDM models are reported asorange, blue and black curves for masses of 4,2 and 1keV, respectively, while the green curve shows the linear suppression for the 2 keV case taken from Ref. [17]. Thepower spectra are already clearly somewhat non-linear athigh redshift; the blue and green curves start to differ sig-nificantly at small scales at k > h/ Mpc. In the bottom part of the panel we show the approximate wavenumberranges that are probed by SDSS and the HIRES+MIKEdata set used in our analysis. Note that the non-linearmatter suppression is in good agreement with the fittingformula presented in Ref. [10].In Figure 2 we qualitatively compare a set of noise-less Lyman- α forest spectra extracted from the ΛCDM,WDM 1 keV and WDM 2 keV models, represented by thegreen, black and blue curves respectively. It is clear thatthe amount of small-scale substructure in the transmit-ted flux in the ΛCDM is more prominent with respectto the WDM cases. In the rest of this Section we willquantify these differences in terms of the 1D flux powerspectrum.We now turn to Figure 3, which shows the ratio be-tween the 1D flux power of the WDM and ΛCDM mod-els for the four different redshift bins used in the presentanalysis (note that we compute the power spectrum ofthe quantity δ F = F/ h F i −
1, and we refer to this as theflux power). The suppression of the flux power is largerthan that seen in the matter power spectrum. This isdue to the fact that the 1D matter power spectrum is anintegral of the 3D power spectrum and therefore very sen-sitive to the small scale cut-off. As expected, the largestdifferences exist between the 1 keV (black curves) andthe ΛCDM model. Note that the flux power also changesat large scales; the requirement of reproducing the sameobserved mean flux value (given by Eq. 4) results in anincrease of the power at those scales (the power spectrumof the WDM flux F , not δ F , does show suppression overall scales when compared to ΛCDM). Furthermore, wealso note that there is a substantial redshift evolution ofthe flux power between z = 5 . z = 4 . , , P F , W D M ( k ) / P F , Λ C D M ( k ) FIG. 3: The ratio of the 1D flux power spectrum for 3 differentWDM models (1, 2 and 4 keV represented in black, blue andorange) at 4 different redshifts ( z = 4 . , . , , . The agreement between the different resolution simula-tions is very good, typically at the percent level. Thedifferences are largest for the 1 keV case at the smallestscales probed by our data in the current analysis, wherethey reach the 10% level. The simulated flux power spec-tra for both ΛCDM and WDM models have thereforebeen corrected for resolution effects by multiplying theraw power spectra by the ratio of the results from the(20 , , T and γ , on the flux power spec-trum. As discussed earlier, the T − ρ relation is usu-ally parameterized as a power-law, T ( z ) = T (1 + δ ) γ − .In both of these figures we also plot the WDM 2keVmodel in order to emphasize the very distinct differ-ences between the thermal and WDM cut-offs, both inthe dependence on wavenumber and redshift. A hotter(colder) T value produces a suppression (enhancement)in the flux power spectrum with a redshift dependentcut-off. The WDM cut-off is instead more pronouncedand steeper than the cut-off induced by a hotter IGM. P F , W D M ( k ) / P F , Λ C D M ( k ) FIG. 4: The ratio of the 1D flux power spectrum for 2 dif-ferent WDM models (1 and 2 keV, represented in black andblue) at 2 different redshifts ( z = 4 . , . The dependence of the thermal cut-off on the slope ofthe temperature-density γ is Fig. 6 is also very differ-ent from the wavenumber and redshift dependence of theWDM cut-offs, and is much flatter over the wavenumberrange considered here.The effects due to to changing the mean flux level arediscussed in detail in the Appendix (Fig. 17). We conser-vatively assume a range of ±
20% for the observed effec-tive optical depth. We further note that the dependenceof changing the mean flux level on wavenumber is evenflatter than that obtained for variations of γ , but showsa weak scale dependence in the highest redshift bin. B. Systematic uncertainties
In this Section we now briefly discuss the followingsystematic effects: instrumental resolution; noise; spatialfluctations in the UV background and metal line contam-ination. In the Appendix there is a more detailed descrip-tion of these nuisance effects and how they are modeled.We only summarize the main quantitative results here.Instrumental resolution, which is different for the twosub-data sets, suppresses the flux power spectrum by atmost 20% and 5% for MIKE and HIRES, respectively, P F , Λ C D M c o l d , h o t ( k ) / P F , Λ C D M ( k ) Λ CDM hotWDM 2 keV Λ CDM coldz=5.4z=5z=4.6z=4.2
FIG. 5: The ratio of the 1D flux power spectrum for twoΛCDM models with different temperatures(HOT, roughlyhotter by 3000 K with respect to the reference simulation,in orange and COLD, roughly colder by 3000 K with respectto the reference simulation, in black) and at four differentredshifts ( z = 4 . , . , , . at the smallest scales probed, with a negligible redshiftdependence. The signal-to-noise ratio impacts at aboutthe 2-3% level at the smallest scales for z ≤ P F , Λ C D M γ ( k ) / P F , Λ C D M γ . ( k ) Λ CDM γ =1.6WDM 2 keV Λ CDM γ =1z=5.4z=5z=4.6z=4.2 FIG. 6: The ratio of the 1D flux power spectrum for twoΛCDM models with different slopes of the temperature-density relation ( γ = 1 . γ = 1 . z = 4 . , . , , .
4, represented bythe dot-dashed, dashed, dotted and continuous curves, respec-tively) to the corresponding ΛCDM simulations. The WDM2 keV model is also shown in blue. The mean flux is thesame for all models, and the shaded area shows the range ofwavenumbers used in the present analysis. mock QSO spectra. The remaining parameters are fullymarginalized over in our likelihood analysis.
TABLE I: Summary of the estimates of the relative errors inthe flux power spectrum due to a range of nuisance effects: theresolution of the observational data (the MIKE and HIRESdata sets have different resolutions); the signal-to-noise ra-tio of the observational data; the numerical resolution of thesimulations; contamination by metal absorbers at lower red-shift; the mean flux level; the thermal history of the IGM andthe fluctuations in the UV background. The table reports es-timates over the wavenumber range considered and the lastthree effects are properly marginalized over in the likelihoodprocedure.syst. eff. σ ( P F ) /P F Notesdata res. < −
15% correcteddata S/N <
3% correctednum. res. <
5% correctedmetals <
1% neglectedmean flux ∼
30% marginalizedthermal history ∼
30% marginalizedUV <
10% marginalized ∆ F ( k ) z=4.2 0.001 0.010 0.1000.11.0 z=4.60.001 0.010 0.100 k (s/km)0.11.0 ∆ F ( k ) z=5 0.001 0.010 0.100 k (s/km)0.11.0 z=5.4 HIRESMIKEREF (best guess) FIG. 7: The flux power spectrum in dimensionless units, P F ( k ) × k/ ! Mp !3, used in the analysis performed. There are a totalof 70 data points at 4 different redshifts. The reference ΛCDM model, which is our best guess starting point for the MonteCarlo Markov Chains, is also shown as orange curves. Only data points in the range log k (s / km) = [ − . , − .
1] are used inthe analysis (shaded area).
VI. METHOD AND RESULTS
We now turn to our analysis of the data and discuss ourresults. For all 108 flux models considered in our analy-sis, we compute the ratio of these models with respect tothe reference model. We then bin this ratio at the samewavenumbers as the data. In practice, this means wehave a 4D parameter matrix with ( m WDM , h F i , T , γ )that summarizes all the results obtained from the hydro-dynamical simulations, plus some further parameters forwhich we have established the effect on the flux powerfor the reference model only. We parameterize the effectof UV background fluctuations on the flux power with afactor f UV that multiplies the flux power spectrum cor- rections shown in Fig. 16, constrained to be in the range[0 ,
1] and applied in addition to the corrections discussedin the previous Section ( f UV = 1 means that the powerspectrum is corrected exactly by the amount shown in thelower panel of Fig. 16). We decide to neglect the effectof metal contamination since it is, as we discuss furtherin the Appendix, very small. We then perform secondorder Taylor interpolations for the following remainingparameters: z reio , Ω m , σ , H , n s , as in Refs. [27, 67].In Figure 7 we compare our best-guess model (thereference simulation), represented by the orange curves,with the observational data. This best-guess model willbe the reference point of our likelihood code that willbe described below. Note that there is already rough“visual” agreement with the data, albeit with a poor χ WDM T A ( z = . ) K WDM σ FIG. 8: The two-dimensional 1 and 2 σ contours for mean (incolour) and marginalized (solid black curves) likelihoods forthe parameters 1 /m WDM against T A ( z = 4 .
5) and 1 /m WDM against σ obtained from the MIKE+HIRES data sets. Theseresults assume a power-law evolution for T and γ , but with γ ( z ) constrained to be in the [0.7,1.7] range, and refer to a runfor which some Planck-like priors on σ , n s and Ω m have beenapplied. Note, however, that our results are not sensitive tothis choice of prior. We use a modified version of the code
COSMOMC [68] to derive parameter likelihoods from the Lyman- α forest data. For the HIRES+MIKE data, we havea set of 15 parameters: 6 cosmological parameters( σ , Ω m , n s , H , z reio , m WDM ); 4 parameters describingthe thermal state of the IGM, using a power-law pa-rameterization of the temperature-density relation, T = T ( z )(1 + δ ) γ ( z ) − , with parameters T A ( z ) = T A [(1 + z ) / . T S and γ A ( z ) = γ A [(1 + z ) / . γ SA ; 4 param-eters describing the evolution of the effective opticaldepth with redshift, since a single power-law has beenshown to be a poor approximation over this wide red-shift range (see [46]) and one parameter describing thespatial fluctuations in the UV background f UV . Weapply strong Gaussian priors to σ , Ω m , n s in order tomimic Planck constraints: Ω m = 0 . ± . , σ =0 . ± . , n s = 0 . ± . T A in the range [1000 , γ A in the range [0 . − . χ if γ ( z = 4 . , . , , .
4) is outside the physical range[0 . − . H and z reio are not constrained WDM L i k e li h ood meanmarginalized FIG. 9: The one-dimensional mean (dotted curve) andmarginalized (continuous curve) likelihoods for the parameter1 keV/ m WDM . These results refer to a run for which somePlanck-like priors on σ , n s and Ω m have been applied. Note,however, that our results are not sensitive to this. by the data and they are prior dependent: the rangechosen are H = [50 , z reio = [5 , cov d ( i, j ) = r s ( i, j ) p cov d ( i, i ) cov d ( j, j ) with r s ( i, j ) = cov s ( i, j ) / p cov s ( i, i ) cov s ( j, j ), where cov d and cov s arethe covariance matrices of the observed and simulatedspectra, respectively.Our results are summarized in Table II. We obtain a 2 σ upper limit on the parameter 1keV /m WDM of 0.3,whichtranslates into the following constraints: m WDM > . σ C.L. and m WDM > .
33 keV at the 1 σ C.L., with a best-fit value of m WDM = 33 keV. For a ∼ k / = 22 h/ Mpc, whilethe suppression at k = 10 h/ Mpc is about 10%. If wedrop the 30% additional error applied to the observedflux power spectrum (see Section II) we get a tighterlower limit of m WDM > . σ ). The χ gets worseby ∆ χ = 14, but still has a probability of 11% of beingthis large for the present number of degrees of freedom.We also took a “frequentist” approach and fixed thevalues of m WDM to 2.5 and 3.3 keV and found the results1
TABLE II: Marginalized estimates (1 and 2 σ C.L.) and best-fit values for a fit to MIKE+HIRES data using power-law fitsfor the evolution γ ( z ) and T ( z ). Planck priors on σ , n s andΩ m have been applied. The best fit χ is 34 for 37 d.o.f. (49data points - 12 free parameters) which has a probability of39% of being larger than this value.parameter (1 σ ) (2 σ ) best fit n s [0 . , .
97] [0 . , . σ [0 . , . . , . m [0 . , . . , . τ A eff ( z = 4 .
2) [1 . , .
16] [0 . , .
22] 1.16 τ A eff ( z = 4 .
6) [1 . , .
33] [1 . , .
4] 1.32 τ A eff ( z = 5) [1 . , .
96] [1 . , .
05] 1.91 τ A eff ( z = 5 .
4) [2 . , .
06] [2 . , .
21] 3.09 γ A ( z = 4 .
5) [1 . , .
54] [1 . , .
65] 1.64 γ S ( z = 4 .
5) [ − . , .
1] [ − , .
3] -0.15 T A ( z = 4 . ) K [9 . , .
4] [7 . , .
6] 9.2 T S ( z = 4 . ) K [ − , − .
05] [ − , − .
1] -2.5 f UV [0 −
1] [0 −
1] 0.18 z reio [5 −
11] [5 − .
4] 11.21 keV/ m WDM [0 − .
12] [0 − .
3] 0.03 in terms of the other parameters: in this case the χ isof course higher than in the ΛCDM model (with ∆ χ =5 . , .
8, respectively) but nevertheless compatible withthe results obtained in our standard analysis. This issimilar to the approach used in Ref. [70], in which ananalysis of mixed cold and warm models was performedin both a Bayesian and in a frequentist approach.The degeneracies between the parameter 1keV /m WDM and the other parameters are very weak. In Fig. 8 weshow the 2D contour plots for the mean likelihood (incolour) and the marginalized likelihood (black curves) for T A0 and σ versus 1 keV/ m W DM . In Fig. 9 we report the1D mean and marginalized likelihoods for 1keV /m WDM (continuous and dotted curves, respectively).We obtain the following evolution for the temperature-density relation T ( z ) = 9200 [(1 + z ) / . − . K and γ ( z ) = 1 .
64 [(1 + z ) / . − . . The inferred temperatureis decreasing with increasing redshift, while the redshiftevolution of γ is weak. We stress that the IGM ther-mal state is just one of several nuisance parameter in ourlikelihood analysis over which we marginalise. We dis-cuss it here in the context of a consistency check ratherthan as a measurement. With this in mind, in Figure10 we show the recovered redshift evolution for T com-pared to three input thermal histories used in the simu-lations and measurements obtained from high resolutionLyman- α forest data from Refs. [19, 71] (note that thepower-law index of the temperature-density relation, γ ,has not yet been measured directly at z > . ± σ of the tempera-tures obtained by our standard likelihood analysis aftermarginalization. The inferred temperature evolution (pa-rameterized as a power law in redshift) is in good agree-ment with the measurements from Ref. [19]. We alsotested a model where the IGM temperature is left to vary T / [ K ] power-law ev.z-binned ev. Becker+11 ( γ =1.0)Becker+11 ( γ =1.3)Becker+11 ( γ∼ FIG. 10: The redshift evolution of the temperature at meandensity, T , used in our reference model is shown as continuousblack curve, while the the two dashed line display our cold andhot models. Recent measurements of the IGM temperatureat mean density obtained by Ref. [19] are also shown fordifferent values of γ . The measurement at z ∼ ± σ ranges), and with orange triangles for model where weleft the temperature free in the four redshift bins (1 σ errorbars). In both cases the temperature values reported are themarginalized results. These results refer to a run for whichsome Planck-like priors on σ , n s and Ω m have been applied. freely in the four redshift bins, shown by the orange datapoints with error bars in Figure 10. In the two highestredshift bins this analysis returns temperatures that arerather cold and are disfavored by the data with an un-reasonably large temperature jump between z = 5 and z = 4 . m WDM (2 σ C.L.) which is about 1 keV lower than forour standard analysis and also returns an unreasonablylow reduced χ value. Finally we have also performed alikelihood analysis where the IGM temperature is fixedto be unrealistically cold throughout (3000K, indepen-dent of redshift) to allow for a maximum contribution ofthe free-straming of WDM to the observed cut-off in theflux power. Again just for completeness, for this modelthe constraint on m WDM (2 σ C.L.) is lower by about 0.5keV compared to our standard analysis.The recovered effective optical depth values at each2 ∆ F ( k ) MIKE z=4.2 0.01 0.10 k (s/km)0.1 z=4.6 Λ CDM b.f. Λ CDM b.f. with WDM 2.5keVWDM fixed to 2.5keV Λ CDM b.f. with ∆ T=3000 K
FIG. 11: The best fit model for the MIKE data set (black crosses) used in the present analysis, shown as the green curves andlabelled as “ΛCDM b.f.”. This model is very close to ΛCDM. We also show for qualitative purposes a few other models: aWDM model that has the same parameters as the best fit model except for the WDM mass (red curves) which is chosen tobe 2.5 keV; a model that has a hotter temperature (orange curves) and a model for which the mass of the WDM is fixed to m WDM = 2 . z = 5 . redshift bin are usually within 20% of the measured opti-cal depth evolution used as the input into the likelihoodcalculation. The inferred values for the amplitude andslope of the matter power spectrum and for the mattercontent do not show biases with respect to the Planck-like priors we used. Overall the χ for the best fit modelis 34 for 37 d.o.f. which has a reasonably high probabilityof about 60% of being larger than this value.Lastly, in Figures 11 and 12 we show our final best-fitmodel compared to the data obtained with MIKE andHIRES, respectively. The best fit model is shown as thegreen curves. We also overplot, for comparison purposesonly, three other models that are excluded with very highsignificance by the present analysis: a model which hasa WDM mass of m WDM = 2 . m WDM = 2 . z >
4. Secondly, both thesimulations and the analysis have been refined by: in-creasing the number of hydrodynamical simulations andtheir resolution; improving the method in a way that al-lows a full sampling of the most relevant parameter space(thermal parameters, WDM cutoff and mean flux) com-pared to a poorer sampling of the parameter space made in Ref. [27]. When considering only the high-resolutiondata set, we improve the limits by nearly a factor threefrom 1.2 keV to 3.3 keV at the 2 σ C.L., this is due toboth the data and the modelling of the flux power.
VII. JOINT ANALYSIS WITH SDSS DATA
In this Section we present the joint analysis with theSloan Digital Sky Survey (SDSS) 1D flux power spec-trum data of Ref. [72] where the authors have presentedthe flux power spectrum of a sample of 3035 QSO ab-sorption in the redshift range 2 < z < ∼ α absorption features, which have a velocity width of ∼
30 km s − , are not resolved. The wide redshift range,however, makes this data set very constraining in termsof cosmological parameters. As a final result of theiranalysis they present an estimate of flux power spectrum P F ( k, z ) at 12 wavenumbers in the range 0 . < k (s/km) < . k = 0 . z = 2 . , . , . , . , , . , . , . , . , , . f UV and the two pa-rameters describing the effective optical depth evolutionat z = 5) plus 13 noise-related parameters: 1 parameterwhich accounts for the contribution of Damped-Lyman- α systems and 12 parameters modeling the resolution andthe noise properties of the SDSS data set (see [72]). We3 ∆ F ( k ) HIRES z=4.2 0.01 0.100.1 z=4.60.01 0.10 k (s/km)1.0 ∆ F ( k ) z=5 0.01 0.10 k (s/km)1 z=5.4 Λ CDM b.f. Λ CDM b.f. with WDM 2.5keVWDM fixed to 2.5keV Λ CDM b.f. with ∆ T=3000 K
FIG. 12: The best fit model for the HIRES data set (blue diamonds) used in the present analysis, shown as green curves andlabelled as “ΛCDM b.f.”. As in Fig. 11 we also show a few other models: a WDM model that has the same parameters as thebest fit model except for the WDM mass (red curves) which is chosen to be 2.5 keV; a model that has a hotter temperature(orange curves) and a model for which the mass of the WDM is fixed to m WDM = 2 . z ≤
5, is excluded at more than 2 σ confidence level. do not consider the possible effect of different reionisa-tion scenarios on the SDSS flux power. The covariancematrix of the SDSS flux power is provided by the authorsof [74]. The 2 σ lower limit on m WDM is unchanged at 3.3keV but now with a χ = 183 . VIII. DISCUSSION AND CONCLUSIONS
We have presented a comprehensive analysis of thetransmitted Lyman- α flux power spectrum extractedfrom a set of 25 high-resolution QSO spectra taken withthe HIRES and MIKE spectrographs. This representsan improved and extended version of the sample origi-nally analysed in [27]. The Lyman- α forest is an excel-4 ∆ F ( k ) z=2.2z=2.4z=2.6z=2.8z=3z=3.2z=3.4z=3.6z=3.8z=4.0z=4.2z=4.6z=5z=5.4 cosmic time: 1.1-3.1 Gyrcosmic scales: 0.5/h-50/h com. MpcSDSS MIKE&HIRESbest fit Λ CDMWDM 2.5 keV
FIG. 13: Best fit model for the data sets used in the present analysis (SDSS+HIRES+MIKE) shown as green curves. We alsoshow a WDM model that has the best fit values of the green model except for the WDM mass (red dashed curves). These dataspan about two orders of magnitude in scale and the period 1.1-3.1 Gyrs after the Big Bang.TABLE III: The final summary of the marginalized estimates(1 and 2 σ C.L.) and best fit values for m WDM . Planck priorson σ , n s and Ω m have been applied. The REF. model refersto our reference conservative analysis; REF. w/o 30% refers tothe case in which we do not add an extra 30% uncertainty onthe data to account for underestimated bootstrap error bars;REF. w/o covmat refers to the case in which we use only thediagonal terms of the covariance matrix; REF+SDSS is thejoint analysis of our reference model and SDSS flux power.model (1 σ ) (2 σ ) best fit χ /d.o.f.REF. > . > . > . > . > . > . > . > . lent probe of the matter distribution at intermediate andhigh redshift in the mildly non-linear regime, from sub- Mpc up to BAO scales. In this work we have focused onconstraining any possible suppression of the total mat-ter power spectrum which could be induced by the free-streaming of WDM particles in the form of a thermalrelic. Due to the non-linear nature of the the relation-ship between the observed Lyman- α flux and underlyingmatter density, departures from the standard ΛCDM caseare expected over a range of scales that span at least onedecade in wavenumber space and can be constrained bythe data used in the present analysis. We model thissuppression by using a set of high-resolution hydrody-namical simulations and by marginalizing over a largerange of physically motivated thermal histories.The WDM cut-off exhibits a distinctive behavior whichwe demonstrate is not degenerate with other physical ef-fects due to its different redshift and scale dependence.We consider possible sources of systematic errors includ-ing metal line contamination, spatial fluctuations in the5UV background intensity and uncertainties in the meanflux level estimation. Galactic feedback either in the formof supernova driven galactic winds or Active Galactic Nu-clei (AGN) feedback should not impact the flux powerspectra at the high redshift considered in this analysis[16].Our final results are obtained by means of a MonteCarlo Markov Chain likelihood analysis around a best-guess reference model. The constraints quoted for m WDM have been calculated after marginalization over the otherastrophysical and cosmological parameters. Our analysisis conservative in the following sense: we have droppedthe estimates of the power spectrum at the largest scalesprobed by our sample in order not to be sensitive to con-tinuum fitting uncertainties; we add an additional errorof about 30% to our error estimates obtained by boot-strapping to account for the expected underestimationof the real error; and we allow for large fluctuations inthe UV background fluctuations, which appear to be themost important nuisance factor. Furthermore, we createa mock QSO sample which resembles as closely as possi-ble the real data including noise and resolution and usethe covariance matrix of this mock sample as an estimateof the error properties of the real data. Our final result ofthis analysis is m WDM > . σ C.L., wherethe mass refers to that of a thermal relic. This massimplies that WDM models for which there is a suppres-sion in terms of the 3D linear matter power at scales k = 10 h /Mpc ( k = 22 h /Mpc) larger than 10% (50%)when compared to the ΛCDM case, are disfavoured bythe present data sets. The corresponding value of the“free-streaming” mass is ∼ × M ⊙ /h . A model witha 2.5 keV thermal relic mass is disfavoured by the dataat about 3 σ C.L., a 2 keV mass at about 4 σ C.L., anda m WDM = 1 keV model at about 9 σ C.L. Our finalmarginalized estimates and best fit values for m WDM aresummarized in Table III.Overall, the final results presented are similar to thosewe have obtained in our previous analysis Ref. [27] [3.3(4.5) keV vs 4 keV previously if we include (do not in-clude) an additional 30% error to account for a possi-ble underestimate of the statistical error from a boot-strapping analysis). We emphasize, however, that thepresent analysis is considerably more robust. It uses alarger data set, a much improved analysis based on abroader suite of significantly improved simulations andas well as an extensive analysis of the systematic uncer-tainties.Further improvement of the constraints on the free-streaming of dark matter particles from Lyman- α forestdata could come mainly from an enlarged set of high-quality, high resolution spectra, especially at the highestredshifts where the flux power spectrum is most sensi-tive to the free-streaming of dark matter. An increaseof the dynamical range of the simulations and improvedindependent constraints on the thermal state and ther-mal history of the IGM are next on the list as require-ments for further corroborating and perhaps pushing the constraints to even larger thermal relic masses. In thefuture, considerably stronger constraints on WDM maybe derived using a baryonic tracer which is colder thanthe photo-ionized IGM, thus moving the thermal cutoffto smaller scales in the flux power spectrum. Studies of21 cm absorption/emission by neutral hydrogen gas be-fore reionization, for example, could eventually fulfill thisrequirement.However, with a lower limit of 10 M ⊙ h − for the massof dark matter haloes whose abundance could still besignificantly affected, the Lyman- α forest data appearsto leave already very little room for a contribution ofthe free-streaming of warm dark matter to the solutionof what has been termed the small scale crisis of colddark matter. In particular, recent suggestions for modelswith relic masses of 0.5-2keV are significantly disfavouredby our analysis. We finally note that our analysis alsosuggests that it is unlikely that sterile neutrinos couldact in that role. Acknowledgements P F , W D M , ( k ) / P F , W D M ( k ) WDM 2 keV (20,768)WDM 2 keV (20,896)z=5.4z=4.2
FIG. 14: The ratio of the 1D flux power spectrum for twoWDM 2 keV simulations at different resolutions, (20,768) and(20,896) represented by the black and blue curves respectively,to the reference WDM 2 keV run (20,512) at two redshifts( z = 4 . , . Appendix: Systematics1. Numerical Convergence
In Figure 14 we compare the 1D flux power spectrumextracted from the WDM 2 keV (20 , , , k ∼ .
08 s/km) when the flux power spectra from the(20 , , σ ) statistical error of the data at the same wavenum-ber. P F ( k ) F W H M = . , . k m / s / P F ( k ) F W H M = z=4.2z=4.6z=5z=5.4 HIRESMIKE P F ( k ) S / N = / P F ( k ) S / N = i n f t y z=4.2z=4.6z=5z=5.4 FIG. 15:
Upper panel:
The ratio of the 1D flux power spec-trum for instrument resolutions corresponding to HIRES (6.7km s − ) and MIKE (13.6 km s − ) spectrographs at 4 differ-ent redshifts ( z = 4 . , . , , .
4) to the data with the nativeresolution of the simulation.
Lower panel:
The ratio of the1D flux power spectrum for a S/N value per pixel equal to 15to a model S/N= ∞ . The mock data have been corrected toincorporate both of these effects. The mean flux is the samein all the models shown, and the shaded area indicates therange of wavenumbers used in the present analysis.
2. Instrumental Effects on the Flux Power
In Figure 15 we show the effect that instrumental res-olution and a given signal-to-noise (S/N) ratio has onthe Lyman- α flux power spectrum in the four differentredshift bins of our data sample. These results havebeen obtained from our mock QSO spectra sample. Notethat both of these effects have been incorporated into theLyman- α forest spectra used in our analysis. The S/N ra-tio results in an increase in the flux power of less than 5%over the range of wavenumbers considered in this work,while instrumental resolution effects are particularly im-portant for the MIKE sample. There is a 20% correctionat the smallest scales for the MIKE data (for the HIRESdata this value is 5%).
3. Systematic Effects on the Flux Power inducedby Metals and UV background fluctuations
We also consider two important astrophysical nuis-sance effects in this work: unwanted contamination frommetal lines in the Lyman- α forest and the effect of spa-tial fluctuations in the UV background intensity on theobserved Lyman- α forest transmission.In the redshift range we consider in this work, z = 4 . .
4, the most common metal lines in the Lyman- α for-est will arise from absorbers at lower redshifts. Wetherefore consider the effect of absorption from threeprominent absorption line doublets; C IV ( λλ IV ( λλ II ( λλ z CIV = 3 . . z SiIV =3 . .
58 and z MgII = 1 . .
78, respectively.We add these metal lines to Lyman- α forest spec-tra drawn from our reference ΛCDM hydrodynamicalmodel using the following prodcedure. We firstly inte-grate fits to the column density distribution functions(CDDFs) presented by [75] from a set of 19 high res-olution VLT/UVES quasar spectra at z qso = 2 . . N/ cm − ) = 12-15,for all three species. We then multiply the results bythe redshift path length of our Lyman- α forest dataset to provide an estimate of the number of metal lineabsorbers in the Lyman- α forest. Note that this ap-proach will likely overestimate the number of C IV andSi IV absorbers due to the somewhat lower redshift cov-erage of the [75] dataset relative to this work. Our metalcontamination estimates are therefore likely to be conser-vative in this regard. Next, we Monte Carlo sample col-umn densities and line widths for the appropriate numberof lines from the [75] CDDF fits and the Doppler param-eter distribution given by Ref. [76], with a chosen value b σ = 10 km s − for all three species. Finally, we randomlyinsert these absorption features into the sight-lines in ourmock Lyman- α forest data set.In order to estimate the impact of spatial fluctuationsin the UV background on the Lyman- α forest at z = 4 . . P F ( k ) Z / P F ( k ) z=4.2z=4.6z=5z=5.40.001 0.010 0.100 k (s/km) 0.91.01.11.2 P F ( k ) U V / P F ( k ) z=4.2z=4.6z=5z=5.4 FIG. 16:
Upper panel:
The ratio of the 1D flux power spec-trum for a model including metal line contamination to thereference model at four different redshifts.
Lower panel:
Theratio of the 1D flux power spectrum for a model including theeffect of spatial fluctuations of the UV background at four dif-ferent redshifts. Note the different y-axis in the two figures.The mean flux is the same in all the models shown, and theshaded area indicates the range of wavenumbers used in thepresent analysis. in [65], which was used to examine fluctuations in the8He II ionising background at lower redshift. We refer thereader to Ref. [65] for further details. The key differencein this work is that we compute the spatially varyingH I photo-ionisation rate along our simulated sight-lines.We achieve this by computing the specific intensity of theionising background between 1–4 Ry (replacing equation3 in [65]) by solving: J ( r , ν ) = 14 π N X i =1 L i ( r i , ν )4 π | r i − r | e − | r i − r | λ HI (cid:16) νν HI (cid:17) − β − , (A.1)where ν HI is the frequency of the H I ionisation edge, λ HI is the mean free path for ionising photons presented by[77] and β = 1 . I CDDF[80]. The summation in eq. A.1 is over all quasars withluminosities, L , drawn from the [79] B-band quasar lumi-nosity function. In order to be conservative, in this workwe adopt an extreme model which maximises the effect ofthe UV fluctuations on the Lyman- α forest by assumingall ionising photons in the IGM at z = 4 . . M B < −
22. We therefore ignorethe significant contribution to the UV background fromthe more numerous, fainter star-forming galaxies at theseredshifts, which effectively smooth out the large-scalefluctuations produced by the rarer quasars. The spa-tially fluctuating photo-ionisation rates are then obtainedby integrating the specific intensity with respect to fre-quency, weighted by the photo-ionisation cross-section.In Figure 16 we show the effect that UV fluctuationsand metal contamination have on the flux power spec-trum. In the wavenumber range considered here, theUV background fluctuations have an effect at around the10% level at the largest scales, dropping to 5% at small-est scales considered in this work. The effect is largerat high redshift (z=5,5.4) than in the two other redshiftbins. Metal contamination affects the flux power verylittle (at the ±
1% level) and is at a level below the sta-tistical error bars of our data.
4. Mean flux level uncertainties
The mean flux level h F i , or alternatively the effectiveoptical depth τ eff = − ln h F i , is a key ingredient in ourflux modeling procedure and a quantity that needs to be marginalized over in the Monte Carlo Markov Chainlikelihood estmation. In Figure 17 we demonstrate the ef-fect that a different mean flux level has on the flux powerspectrum. We choose two different mean flux levels with τ eff
20% higher and lower than the reference value (whichcorresponds to the observed mean flux). A higher (lower)value for τ eff will result in more (less) power relative tothe reference value. The trends are similar to those foundfor the evolution of γ , i.e. rather flat in wavenumberspace, with some weak scale-dependence which is onlypresent in the highest and lowest redshift bins. Our finalresults are not sensitive to the actual choice of the ef- P F , Λ C D M , < F > ( k ) / P F , Λ C D M ( k ) Λ CDM 1.2 τ eff WDM 2 keV Λ CDM 0.8 τ eff z=5.4z=5z=4.6z=4.2 FIG. 17: The ratio of the 1D flux power spectrum for 2 dif-ferent ΛCDM models ( τ eff = 1 . τ eff , obs in orange and τ eff =0 . τ eff , obs in black) at 4 different redshifts ( z = 4 . , . , , . fective optical depth values, since these are marginalizedover in the likelihood estimation. [1] Planck 2013 results. XVI. Ade, P. A. R., eprintarXiv:1303.5076[2] P. Bode, J. P. Ostriker and N. Turok, Astrophys. J. , 93 (2001); B. Moore, T. 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