New Constraints on the free-streaming of warm dark matter from intermediate and small scale Lyman-$α$ forest data
Vid Irši?, Matteo Viel, Martin G. Haehnelt, James S. Bolton, Stefano Cristiani, George D. Becker, Valentina D'Odorico, Guido Cupani, Tae-Sun Kim, Trystyn A. M. Berg, Sebastian López, Sara Ellison, Lise Christensen, Kelly D. Denny, Gábor Worseck
NNew Constraints on the free-streaming of warm dark matter from intermediate andsmall scale Lyman- α forest data Vid Irˇsiˇc , , , ∗ Matteo Viel , , , † Martin G. Haehnelt , James S. Bolton , Stefano Cristiani , ,George D. Becker , , Valentina D’Odorico , Guido Cupani , Tae-Sun Kim , Trystyn A. M. Berg ,Sebastian L´opez , Sara Ellison , Lise Christensen , Kelly D. Denny , and G´abor Worseck University of Washington, Department of Astronomy,3910 15th Ave NE, WA 98195-1580 Seattle, USA Institute for Advanced Study,1 Einstein Drive, NJ 08540 Princeton, USA The Abdus Salam International Centre for Theoretical Physics,Strada Costiera 11, I-34151 Trieste, Italy SISSA-International School for Advanced Studies,Via Bonomea 265, 34136 Trieste, Italy INAF - Osservatorio Astronomico di Trieste,Via G. B. Tiepolo 11, I-34143 Trieste, Italy INFN - National Institute for Nuclear Physics,via Valerio 2, I-34127 Trieste, Italy Institute of Astronomy and Kavli Institute of Cosmology,Madingley Road, Cambridge CB3 0HA, UK School of Physics and Astronomy,University of Nottingham, University Park,Nottingham, NG7 2RD, UK Space Telescope Science Institute,3700 San Martin Drive, Baltimore, MD 21218, USA Department of Physics and Astronomy,University of Victoria, Victoria, BC V8P 1A1, Canada Departamento de Astronom´ıa,Universidad de Chile, Casilla 36-D, Santiago, Chile Dark Cosmology Centre, Niels Bohr Institute,University of Copenhagen, Juliane Maries Vej 30,DK-2100 Copenhagen, Denmark Department of Astronomy,The Ohio State University, 140 West 18th Avenue,Columbus, OH 43210, USA Max-Planck-Institut f¨ur Astronomie,K¨onigstuhl 17, D-69117 Heidelberg, Germany
We present new measurements of the free-streaming of warm dark matter (WDM) from Lyman- α flux-power spectra. We use data from the medium resolution, intermediate redshift XQ-100 sampleobserved with the X-shooter spectrograph ( z = 3 − .
2) and the high-resolution, high-redshift sampleused in Viel et al. (2013) obtained with the HIRES/MIKE spectrographs ( z = 4 . − . α flux-power spectrum on the free-streaming of dark matter, cosmological parameters, as well as the thermal history of the intergalacticmedium (IGM) with hydrodynamical simulations, we obtain the following limits, expressed as theequivalent mass of thermal relic WDM particles. The XQ-100 flux power spectrum alone gives alower limit of 1.4 keV, the re-analysis of the HIRES/MIKE sample gives 4.1 keV while the combinedanalysis gives our best and significantly strengthened lower limit of 5.3 keV (all 2 σ C.L.). Thefurther improvement in the joint analysis is partly due to the fact that the two data sets havedifferent degeneracies between astrophysical and cosmological parameters that are broken when thedata sets are combined, and more importantly on chosen priors on the thermal evolution. Theseresults all assume that the temperature evolution of the IGM can be modelled as a power lawin redshift. Allowing for a non-smooth evolution of the temperature of the IGM with suddentemperature changes of up to 5000K reduces the lower limit for the combined analysis to 3.5 keV.A WDM with smaller thermal relic masses would require, however, a sudden temperature jump of5000 K or more in the narrow redshift interval z = 4 . − .
8, in disagreement with observations ofthe thermal history based on high-resolution resolution Lyman- α forest data and expectations forphoto-heating and cooling in the low density IGM at these redshifts. PACS numbers: ∗ E-mail: [email protected] (VI) † E-mail: [email protected] (MV) a r X i v : . [ a s t r o - ph . C O ] J un I. INTRODUCTION
The intergalactic medium (IGM) and its main observ-able manifestation, the Lyman- α forest (see [1]), havebeen used as unique tools to address key cosmologicalissues: the free-streaming of dark matter and in particu-lar departures from cold dark matter, generally labelledas warm dark matter (WDM) [2–8]; measuring the lin-ear power spectrum at small scales [9–14]; probing theeffect of the free-streaming and thus the masses of neu-trinos [14, 15], and placing (high-redshift) geometricalconstraints on our Universe from Baryonic Acoustic Os-cillations measurements [16, 17].At present, constraints on the matter power spec-trum are either derived from moderate size sampleswith tens of high-resolution, high signal-to-noise spectra(VLT, HIRES/KECK, [7, 9, 12]) or large samples withthousands of low-resolution, low signal-to-noise spectra(SDSS-II, SDSS-III/BOSS, [13, 18, 19]). The XQ-100 [20]sample bridges the gap between these two regimes withits homogeneous set of intermediate resolution and in-termediate signal-to-noise QSO absorption spectra, withthe additional benefit that the flux power spectrum in-ferred from medium resolution QSO absorption spectrais subject to quite different systematic and statisticaluncertainties. Here, we will provide constraints on thefree-streaming length of dark matter from modelling theXQ-100 flux-power spectrum as well as from a combinedanalysis with new modelling of the HIRES/MIKE fluxpower spectrum presented in [7]. These two data setshave a small redshift overlap and it can be expectedthat a combined analysis will further break degenera-cies with remaining uncertainties in the parameters de-scribing the thermal evolution of the IGM, the evolutionof the mean flux and cosmological parameters. Push-ing the constraints on the free-streaming length of darkmatter as far as possible is very relevant for the widerastrophysics community given that considerable tensionswith the Cold Dark Matter (CDM) model on small scalescontinue to persist for a range of astrophysical observa-tions, especially with regard to the dynamical propertiesof Milky Way satellites (see e.g [21]). From a particlephysics point of view, small scale modifications of thecold dark matter power spectrum can e.g. arise fromthe free-streaming of sterile neutrinos [22–24] or ultra-light bosons [25], alternatives to the more generic ther-mal relics on which we concentrate our modelling here[26]. For the purpose of our analysis here the differentDM candidates differ in the exact shape of the suppres-sion of the small scale power spectrum they cause dueto free-streaming. However, apart from changing the na-ture of the DM several studies have shown that baryonicphysics could help in alleviating or even solving the smallscale tensions [27, 28].In Section II we briefly describe the data sets used; Sec-tion III presents the hydrodynamical simulations grid; the method is briefly outlined in Section IV while Sec-tion V contains all the new results (with an Appendix fo-cusing on degeneracies between the various parameters).We conclude with a summary in Section VI. II. DATA SETS
We make use of two different and complementary datasets: the XQ-100 and HIRES/MIKE samples. XQ-100consists of 100 medium resolution and signal-to-noiseQSO spectra obtained as part of the XQ-100 survey withemission redshifts 3 . < z < . − − (FWHM) depending onwavelength and the pixel size used for calculating the fluxpower spectrum for the two spectral arms is 20 (UVB)and 11 km s − (VIS), respectively (see [29] for details).The spectral resolution sets the smallest scales probedby the data. The flux power spectrum extraction hasbeen extensively modeled with mock data sets built fromhydrodynamic simulations which allows an accurate esti-mation of statistical and systematic uncertainties of theflux power at z = 3 , . , . , . , . , , . k − space in the range 0 . .
057 s km − . In [29] the co-variance matrix was multiplied with a constant factor 1.1,to correct for the underestimation of variance throughthe bootstrap method. We use the same correction fac-tor here unless otherwise noted. We further assume thatthe cross-correlations between different redshift bins iszero. A total of 133 ( k, z ) points are thus used in ourmodelling of the XQ-100 flux power spectrum.We also combine the new data with measurements ofthe power spectrum used in [7], measured at higher red-shift z = 4 . , . , . , .
4, for 10 k − bins in the range0 . .
08 s km − . The QSO absorption spectra of thissample have been obtained with the HIRES/KECK andthe MIKE/Magellan spectrographs, with resolution 6.7km s − and 13.6 km s − , respectively. Following [7] aconservative cut is imposed on the MIKE and HIRESdata, such that wavenumbers with k < .
005 s km − areremoved, due to possible systematic uncertainties on thelarge scales of those measurements that used only a smallnumber of QSO sightlines. Moreover, we also do not con-sider the highest redshift bin for the MIKE data, wherethe flux power spectrum measurements have large errorbars. The analysis in [7] used a correction factor of 1 . σ F /P F < .
075 should be set to 0.075,which we also do in our analysis here (see [7] for moredetails). We further regularize the covariance matrix ofthe HIRES/MIKE data following the procedure of [7].A total of 49 ( k, z ) points is used in the HIRES/MIKEanalysis.
III. SIMULATIONS
Our analysis of the flux power spectrum is based ona set of hydrodynamical simulations that is significantlyextended compared to that used in [7]. The hydrody-namical simulations were performed with the GADGET-3 code, which is a modified version of the publicly avail-able GADGET-2 code [30]. A simplified star formationcriterion is applied for which gas particles above an over-density 1000 and temperature below T= 10 K are con-verted into stars (e.g. [31]). The reference model simula-tion has a box length of 20 /h comoving Mpc with 2 × gas and (cold) dark matter particles (with gravitationalsoftening 1.04 /h comoving kpc) in a flat ΛCDM universewith cosmological parameters Ω m = 0 . b = 0 . n s = 0 . H = 70 . − Mpc − and σ = 0 .
829 inagreement with [32]. Three different WDM models withmasses m WDM = 2 , , α forest by modifying the photo-heating rates inthe simulations as in [33]. The low density IGM (∆ =1+ δ <
10) is well described by a power-law temperature-density relation, T = T ∆ γ − . We consider a range ofvalues for the temperature at mean density T and theslope of the T − ρ relation, γ , based on the previous anal-ysis of the Lyman- α forest and recent observations [34].These consist of a set of three different temperatures atmean density, T ( z = 3 .
6) = 7200 , , T − ρ relation: γ ( z = 3 .
6) = 1 . , . , . T ( z = 3 . , γ ( z = 3 . , . σ , the slope of the initial power spectrum n s and Ω m as in [7], we exploit the fact that these three parametersare tightly connected in ΛCDM (and ΛWDM) modelsand impact on the flux power spectrum only in termsof the amplitude and the (effective) slope of the matter power spectrum at scales that are probed by the Lyman- α forest . We therefore use instead only two parametersdescribing cosmology, σ and n eff = d ln P m ( k ) /d ln k ,evaluated at k = 0 .
005 s km − , similarly to what wasdone in [13]. Five different values are considered forboth σ = 0 . , . , . , . , . n eff = − . , − . , − . , − . , − . σ , n eff , n s ) = (0 . , − . , . n eff is implemented with small changes of n s .We also vary the redshift of reionization z rei which ischosen to be z rei = 9 for the reference model as well as z rei = 7 ,
15 for two additional models. The z rei = 7 modelhas also been simulated for all 3 values of the WDM ther-mal relic mass, since the redshift of reionization has animpact on the Jeans smoothing scale and could affect thecutoff scale of the flux power spectrum. We note here,however, that the effect is large enough for the degener-acy between free-streaming and Jeans smoothing to bebroken (although see [35] in the context of CDM models)and that the data are not constraining this parameterwell (see B for details).The final parameter we explored characterizes the pos-sible effect of ultraviolet (UV) background fluctuations.A model has been chosen where the spatial fluctuationsof the meta-galactic UV background are dominated byrare QSOs, which has a strong scale dependent effecton the flux power spectrum particularly at high redshiftand at large scales. The model of UV fluctuations usedhere is an update of the model presented in [7] (see ap-pendix there). The updated model uses the more re-cent mean free path measurements of [36] and parame-terizes the effect of UV fluctuations on the flux powerspectrum as f UV – defined as the fraction of the volumeaveraged hydrogen photo-ionisation rate that arises froma fluctuating QSO component. The remaining fraction,1 − f UV is attributed to a spatially uniform UV back-ground arising from faint galaxies with a typical separa-tion much less than the mean free path of ionising pho-tons. The flux power spectrum template is built froma set of 3 models variations with f UV = 0 , . , f UV = 0 corresponds to a spatially uniform UV back-ground. Note, however, that a comprehensive treatmentof spatial UV (and temperature, which we neglect here)fluctuations would require computationally prohibitiveradiative transfer calculations in large volumes. As dis-cussed in [25] spatial variations in the IGM temperature[37], mean free path [38] and fluctuations from bright Ly-man break galaxies at high redshift [39] (particularly at z >
5) may also have an uncertain impact on the fluxpower.Last but not least, we also vary the mean flux(or equivalently amplitude of the UV background) byrescaling τ eff = − ln ¯ F . We use three different values(0 . , , . × τ obs , eff , with the reference values of τ obs , eff chosen to be those of the SDSS-III/BOSS measurements[19]. The mean flux evolution derived from the SDSS-III/BOSS analysis has values that are 5-8% lower com-pared to those measured by [34], but note that the rangeof values considered in our analysis brackets the observedvalues by [34] as well.Finally, a few lower resolution simulations have alsobeen run to check convergence and a single 1 keV WDMmodel has been considered to check the validity of themethod described below. Each simulation used about20,000 CPU hours. The total grid consists of 23 simu-lations at the reference resolution and 10 simulations atlower resolution. IV. METHOD
Using the models of the transmitted flux ob-tained from the simulations we establish a grid ofpoints for each redshift, in the parameter space of( ¯ F ( z ) , T ( z ) , γ ( z ) , σ , z rei , n eff , f UV , m WDM ). We thenperform a linear interpolation between the grid pointsin this multidimensional parameter space. The interpo-lation is done in the P F ( k, z ) space directly, rather thanfor ratios of flux power spectra as in [7]. We performseveral tests of the interpolation scheme (by predictingthe value of the flux power at a given grid point whereexact values are known, without using that grid point inthe interpolation) and conclude that while a small sys-tematic error due to interpolation exists ( <
5% of theflux power spectrum), it does not bias the results. Addi-tional tests were done when including this correction inthe error budget of the likelihood estimation and resultswere unchanged. This reflects the fact that the inter-polation error is small compared to the statistical errorand sub-dominant in the systematic error budget of thecurrent data. A Gaussian likelihood estimation was thenused to evaluate a Monte Carlo Markov Chain (MCMC)algorithm to obtain the set of parameters that minimizesthe likelihood for a given data set.To estimate the convergence, four independent chainswere run from randomly chosen initial set of parame-ters with different seed values for pseudo-random numbergenerators. Using the Gelman-Rubin test on all of thechains we concluded that the chains have converged suf-ficiently (for each of the parameters the Gelman-Rubinmeasure of convergence was required to be less than 1 . V. RESULTS
We performed a detailed MCMC analysis for threedifferent data sets: XQ-100 (the new data set),HIRES/MIKE (as in [7]) and the combined data sets.For the reference analysis case we model the mean fluxparameters independently for each redshift bin, the num-ber of which varies for each data set (XQ-100 has 7,MIKE/HIRES has 4 and the combined analysis has 10redshift bins). We complement these parameters withan additional 9 parameters: 5 parameters describing ei-ther cosmology or astrophysics ( σ , n eff , z rei , m WDM , f UV ) and 4 parameters describing the thermal stateof the IGM, using a power-law T − ρ relation, T = T ∆ γ − . Unless otherwise noted we model the redshiftevolution of the parameters T and γ as power-laws,such that T ( z ) = T A [(1 + z ) / (1 + z p )] T S and γ ( z ) = γ A [(1 + z ) / (1 + z p )] γ S . The pivot redshift is differentfor each data set and roughly corresponds to the redshiftat which most of the Lyman- α forest pixels are coming from ( z p = 3 . , . , . F ) in each redshift bin, according tothe τ eff fit to the data presented in [7], with 0 .
04 stan-dard deviation (1 σ ). These priors account for the factthat different continuum treatments and different mea-surements give a slightly different normalization for themean flux. The chosen fit roughly represents the medianvalues of the observations (see [29]), with the 1 σ standarddeviations capturing the uncertainty in the normalizationgiven by different measurements.For all three data sets, the preferred ranges of otherparameters are in agreement with independent observa-tions. In particular the values of cosmological parameters σ and n eff are consistent with the latest Planck resultswithin 1 σ for XQ-100 and HIRES/MIKE and within 2 σ for the combined analyses of XQ-100 and HIRES/MIKE.We have furthermore verified that the moderate 2 σ dis-crepancy in σ (and to lesser extent in n eff ) can be al-leviated by using additional priors on the above param-eters. The applied priors were Gaussian on σ and n eff of ± σ ) around Planck values. Our measurementsof the cosmological parameters are consistent with thosemeasured by SDSS/BOSS collaboration [19], and more-over also show a similar tendency towards slightly highervalues of σ and slightly lower values of n eff .It is also important to emphasise that the redshift cov-erage of XQ-100 and the higher resolution HIRES/MIKEdata sets is mostly complementary (covering lower andhigher redshifts respectively) and thus different con-straints and degeneracies are expected in each. Eventhough XQ-100 covers a similar redshift range as theSDSS-II and SDSS-III Lyman- α power spectrum mea-surements, it extends to significantly smaller scales andshould carry more information from the thermal cut-offin the flux power-spectrum. Note that the thermal cut-offis fixed in comoving co-ordinates in real space, while thecut-off in the observed transmitted flux power spectrumscales as H ( z ) / (1 + z ) in velocity space. At a fixed veloc-ity scale this means smaller comoving length scales (andthus free streaming lengths) are probed with increasingredshift. As a result, higher redshift data are more sensi-tive to the equivalent larger WDM relic mass than lowerredshift data, where the effect of the thermal motionsdominates already at larger comoving length scales. Mea-surements of WDM from lower redshift data, like thoseobtained from SDSS/BOSS flux power spectra, are thusmostly sensitive to the change of the power spectrum am-plitude on the large scales, instead of probing the shapeand redshift evolution of the free-streaming cut-off. Inour analysis, this is supported by the fact that large de-generacies are found in our MCMC analysis for XQ-100between the WDM mass and the values of the mean fluxat each redshift (see Fig. A.1 in the appendix). Fur-thermore, since XQ-100 consists of fewer QSO spectra, . . . . . z . . . . . . . T [ K ] sims T z-binsXQ-100HIRES/MIKE XQ-100 + HIRES/MIKEBecker+11 γ ∼ . Becker+11 γ = 1 . FIG. 1: Temperature measurements (2 σ ) as a function of redshift: reference simulation (black curve), XQ-100 (shaded bluearea), HIRES/MIKE (shaded red area), joint constraints (shaded green area); points (same color coding) represent measure-ments obtained using T in redshift bins, limiting temperature variations to ∆ T = 5000 K between adjacent bins, rather thanassuming a power-law evolution. Cyan and orange points with error bars are the IGM temperature measurements from [34]for two values of the slope of the temperature-density relation, γ = 1 . . the error bars are larger than that of the SDSS measure-ments, which is why we do not expect the results fromXQ-100 alone to constrain the WDM mass as tightly asvarious SDSS measurements.Before discussing our new free-streaming constraints,in Fig. 1 we show the temperature estimates from ourMCMC analysis of the flux power spectrum for the dif-ferent data sets as 2 σ shaded regions, assuming the tem-perature of the IGM varies smoothly with redshift as apower-law. In addition, we show individual points with2 σ errorbars that are obtained by allowing the tempera-ture to float freely from bin to bin, but with a maximumtemperature jump between bins of ∆ T = 5000 K (dis-cussed further below). Both results are in good agree-ment with the measurements of [34] obtained from thecurvature of the transmitted flux, shown as orange andcyan points for two different assumptions for the power-law slope γ of the temperature-density relation (but notethat these measurements were calibrated with hydrody-namical simulations where the dark matter was assumedto be cold). While our measurements are consistentwith no evolution in temperature in the redshift range3 < z < .
4, the preferred slope is negative (temperatureincreasing with decreasing redshift), which is in agree-ment with HeII reionisation occuring somewhere aroundredshifts 3 − α forest data [40–43]. Earlier measurements of the IGM temperature at z =3–4 . T (cid:39) , ± σ ) at z = 3 . m WDM > . σ ).While the peak of the likelihood is not at 0, the peak isnot statistically significant (not even at 1 σ ). Moreover,the exact position of the peak is strongly dependent onthe choice of priors. However, the 2 σ upper limit for1 /m WDM is nearly independent of prior choice, and con-stitutes a very robust measurement. We also show thecase where a correction factor of 1.3 has been applied tothe covariance matrix and with weak priors on cosmo-logical parameters ( σ and n eff have Gaussian priors of ± σ ) around Planck values and the assumed tem-perature T A is 10 , ± σ )). When we moveto the model with with freely floating T ( z ) bins ratherthan a power-law evolution of the temperature the free-streaming length inferred from the XQ-100 sample doesnot change.Constraints on the WDM mass using theHIRES/MIKE sample were first presented in [7].Compared to the analysis presented in [7] the mainimprovements in this work are as follows: the referencesimulations have higher resolution and better coverageof parameter space, the model of spatial UV fluctuationshas been extended, the interpolation scheme is basedon the prediction of flux power rather than flux powerratios, and the (now one) cosmological parameterdescribing the slope of the power spectrum is closer towhat is constrained by the data (n eff rather than Ω m and n s ).We furthermore explored more physical priors for theevolution of the temperature that do not allow sudden,large jumps in the temperature. For reference, we haverepeated the analysis in [7]) with the reference priors (andthermal history parameterized as a power law) used hereand found that the the lower limit on m WDM increases,from > . > . σ and n s as used by [7]) and > . ± . σ ) around Planck values for σ and n eff . The resultappears thus quite robust to changes in the choice of priorin cosmological parameters and details of the analysis butis sensitive to the assumed thermal priors. If we drop theassumption of a power-law evolution for the temperatureof the IGM we get a lower bound of m WDM > . z = 0 . . T = 5000 K. Anychange in the (volume averaged) IGM temperature over3 < z < . T = 5000–10000 K which aregradual and occur over ∆ z (cid:38) T ∝ (1 + z ) . Larger values of ∆ T are not readilyachievable within physically motivated reionization mod-els. For this reason, [7] strongly disfavored the binnedanalysis they performed for completeness that weakenedtheir constraint by around 1 keV. Our analysis of the combined data sets also gives sig-nificantly strengthened constraints on the WDM mass,driven mostly again by the high redshift HIRES/MIKEdata set. However, unlike combining the above dataset with SDSS (as in [7] where the inclusion of SDSS-II data did not impact on the free-streaming con-straints), our combined analysis of the high-redshift,high-resolution data with the XQ-100 sample gives a sig-nificantly stronger lower limit on m WDM . This is againmostly due to the more physical temperature evolutionthat we assumed when combining the two data sets, thatdoes not allow for sudden jumps in the temperature evo-lution. We expect that were such a prior on the tem-perature evolution used also in the case when combiningSDSS-II with HIRES/MIKE, that a stronger bound onthe WDM mass would also be obtained.Much like in our analysis of the HIRES/MIKE onlydata set, the results for the combined data sets is – apartfrom the thermal priors – largely independent of thechoices of prior and thus robust. For the combined datasets the 2 σ C.L. lower limit is 5 . . T = 5000 K. Parameter XQ-100 HIRES/MIKE Combined m WDM [keV] > . > . > . σ [0 . , .
92] [0 . , .
32] [0 . , . n eff [ − . , − .
25] [ − . , − .
11] [ − . , − . T A ( z p ) [10 K] [0 . , .
27] [0 . , .
12] [0 . , . T S ( z p ) [ − . , .
89] [ − . , − .
80] [ − . , − . γ A ( z p ) [1 . , .
45] [1 . , .
52] [1 . , . γ S ( z p ) [ − . , .
17] [ − . , .
77] [ − . , . z rei [6 . , .
66] [6 . , .
88] [6 . , . f UV [0 . , .
96] [0 . , .
96] [0 . , . χ /d.o.f. /
124 33 /
40 185 / z p = 3 . , . , . VI. CONCLUSIONS
We have presented new constraints on the free-streaming of WDM based on an MCMC analysis of theXQ-100 and HIRES/MIKE Lyman- α forest data sets.The new constraints in terms of the mass of a thermalrelic WDM particle, m WDM > . σ , are thestrongest to date, and thus imply significantly colder darkmatter than the 2 − . . . . . . . /m WDM [keV − ] L ref C f = 1 . Weak priors XQ-100HIRES/MIKEXQ-100 + HIRES/MIKE
FIG. 2: One dimensional posterior likelihood distributions for the WDM mass for XQ-100, HIRES/MIKE and the combineddata (blue, red and green solid curves). We also show how the results change by using a larger value for the correction factorof the XQ-100 covariance matrix (dotted curves) and using weak priors (see text) on the thermal history and cosmologicalparameters (dot-dashed curves). Vertical lines show the corresponding 2 σ confidence limits. Previous analysis of the same high-resolution Lyman- α forest data had given constraints of m WDM > . > .
95 keV ([8]; using SDSS-III/BOSS). Adding the new data from the XQ-100 surveywhich has similar redshift coverage as SDSS, but extendsto significantly smaller scales, has strengthened the con-straints to a significantly smaller free-streaming lengthand corresponding larger values of the mass of a thermalrelic WDM particle. Another important aspect of ournew analysis was the assumption of more physical priorson the gas temperature evolution with redshift. Whilethe results of our analysis for the new XQ-100 data alonegive relatively weak constraints, the combined analysis isvery robust to different choices of priors and also givesa largely consistent picture with independent, more di-rect measurements of the the thermal history of the IGMover a wide redshift range z = 3 − . T ( z ) bins (but with limited temperature jumps betweenadjacent bins) in our analysis. In this case the limit weak-ens to 3 . α forest data.Secondly, [25] have recently suggested that tempera-ture fluctuations could compensate for the WDM cutoffby providing an increase of power at small scales (but seealso [46]). This is potentially an important systematic ef-fect that should be better quantified by performing tem-plate fitting based on more accurate modelling of spatialfluctuations of the meta-galactic UV background, as wellas the residual temperature fluctuations from hydrogenreionization with radiation hydrodynamical simulationsthat incorporate radiative transfer effects rather than an-alytical modeling.Thirdly, it should (at least in principle) be possible tofurther (moderately) strengthen the limits on the free-streaming of warm dark matter by reducing the statisti-cal errors of the high-redshift, small scale flux power spec-trum obtainable with high-resolution spectrographs andfurther constraining the thermal and reionization historyof the IGM. Appendix A: Parameter degeneracies
Degeneracies between the parameters play an impor-tant role in how well a specific parameter (e.g. freestreaming length/WDM mass) can be estimated usingdifferent data sets. In the bottom row of Fig. A.1, it isclear that for low redshift data (XQ-100; blue colouredcontours), there are strong degeneracies between themass of a thermal relic WDM particle and the temper-ature (at a given redshift). This is not surprising, sinceboth the temperature and WDM effects change the powerspectrum on large as well as small scales. At lower red-shifts the effects are small in both cases and thus harderto distinguish within the observational errorbars. Thedegeneracy with temperature is an anti-correlation thatis expected; the data prefers either higher temperaturesand lower masses of WDM, or lower temperatures withhigher WDM masses.However, whereas the temperature degeneracy withthe free streaming length comes as no surprise, the meanflux degeneracy might not be naively expected (bottomleft panel of Fig. A.1). Since this degeneracy has a sim-ilar anti-correlation with the mass of the WDM as seenfor the temperature (also shown as positive correlationbetween mean flux and temperature - top left 2D panelin Fig. A.1), it means the sensitivity of the XQ-100 datato the WDM mass comes mostly from the overall ampli-tude of the flux power spectrum, rather than its shape inthe cutoff regime at smaller scales. A possible solution(apart from measuring different statistics, and increasingthe precision of the current measurements) would be toincrease the maximum scale up to which the flux powerspectrum is measured. If the thermal/Jeans smoothingand smoothing due to a high WDM mass are differentenough a feature (kink) should be observable on some(arbitrarily) small scales where the flux power spectrumcutoff transitions from being dominated by the ther-mal/Jeans smoothing to being dominated by the mass ofthe WDM. This would, however, only work if the WDMmass is large enough.The above degeneracies almost disappear when usingthe higher redshift data in the analysis (HIRES/MIKE;red coloured contours). Fig. A.1 shows no appreciabledegeneracy between mass of the WDM and any otherparameters. This is because at higher redshifts, for theWDM masses we consider here the cutoff scale by the free-streaming of the WDM becomes more and more im-portant and this scale will show no redshift evolutionand will be thus easier to pick up in the data. This isthe reason why the higher redshift data becomes sucha powerful tool for constraining the free-streaming. Toincrease the constraining power, more observations to de-crease the statistical errors would be more beneficial thanpushing to smaller scales (although the latter would behelpful as well). This is because the MCMC analysisshows that the constraints on the free-streaming lengthare largely independent of the different assumed valuesof priors, meaning that the resulting lower bound on themass of the WDM is driven by the statistical error.Lastly we draw attention to a slight discrepancy in themeasurement of the slope of the T − ρ relation (third rowof Fig. A.1). The value of γ ( z = 4 .
2) measured from thelow and high redshift data sets are in modest (1 − σ )tension. While this could be a statistical fluke, we wouldlike to point out that this might actually be an expectedresult, if HeII reionization happens somewhere betweenredshift 3 −
4. The high redshift data (HIRES/MIKE)measures the thermal history above a redshift of z = 4 . γ at these red-shifts is thus expected to slowly increase and approachthe asymptotic value of around 1 . z = 3 − γ , whereits value falls to γ = 1 . . z ∼ γ measured from high redshift data set, and lower overallamplitude of γ at lower redshifts. We note, however, thata more detailed model of γ ( z ) evolution may be necessaryat lower redshift to capture possible HeII reionization ef-fects. Dropping the assumption of a simple power-lawdescribing the evolution of T ( z ) and γ ( z ), and allowingfor the power-law evolution to have different slope belowand above z p = 4 .
2, relaxes this tension considerably, asis shown in the third row of Fig. A.1 (magenta colour– double powerlaw). The WDM constraints in this caseare slightly weaker compared to the reference case of theanalysis of the combined data sets, and exclude WDMmasses above m WDM > . T in independent redshift bins, eventhough γ ( z ) is still described as a single power-law insuch a case. The WDM limits derived from this case aredescribed in the main text. keV /m WDM T ( z = . ) γ ( z = . ) σ n e ff F ( z =4 . k e V / m W D M T ( z =4 . γ ( z =4 . σ n eff XQ-100HIRES/MIKEXQ-100 + HIRES/MIKEXQ-100 + HIRES/MIKE + double powerlaw
FIG. A.1: Two dimensional posterior (marginalized) likelihood distributions for the main parameters for the XQ-100,HIRES/MIKE and combined data sets (blue, red and green curves), respectively. We also show contours when using a doublepower-law evolution of the thermal parameters (cyan curves), as described in more details in the text. Instead of 4 parametersdescribing the thermal history only values evaluated at a specific redshift were chosen. The redshift chosen is where differentdata sets overlap.
Appendix B: Degeneracy between WDM mass andredshift of reionization
Due to the fact that smoothing from both WDM ther-mal relic as well as pressure smoothing act on the 3D matter power spectrum a certain amount of degeneracybetween the parameters is expected. However, in thissection we show that this degeneracy is largely brokenby the long redshift range considered in the data analy-sis.0 .
003 0 .
006 0 .
01 0 .
03 0 . k [km − s] . . . . . . . P F ( k ) / P F , r e f ( k ) z = 4 . z = 4 . z = 5 . z = 5 . z rei = 7 z rei = 15 m WDM = 2 keV m WDM = 4 keV
FIG. B.2: The flux power spectrum for different models vary-ing mass of the WDM ( m WDM ) and redshift of reionization( z rei ). The colours show two values of m WDM - 2 keV in blueand 4 keV in black - and two values of z rei - 7 in green and15 in red. The reference model against which the flux poweris compared, was ΛCDM model with z rei = 9. Different linestyles show the redshift evolution of the flux power: full line( z = 4 . z = 4 . z = 5 . z = 5 . Fig. B.2 shows the flux power ratio when we varyWDM and z rei models compared to the reference ΛCDMcase. The plot nice nicely illustrates how the differentredshift evolution of the effect of reionization redshift andfree-streaming of the dark matter on the flux power spec-trum makes it possible to separate the two effects. Tofully capture the effect these two degenerate parametershave, we used used a grid of simulations that samples theparameter plane of 1 /m WDM and z rei .To illustrate the effect of redshift evolution further, weshow a 2D plot of the posterior likelihood distributionin the parameter plane of m WDM and z rei (Fig. B.3).The degeneracy between the two parameters is increasedwhen only three redshift bins are considered in the anal-ysis. These redshift bins also do not span the wholelength of the redshift range the combined data set tracks,but are centered around the pivot redshift of z = 4 . z = 4 . , . , . λ F = 2 π/k F . We also show the Jeans smooth- z rei k e V / m W D M
10 z-bins3 z-bins
FIG. B.3: The 2D posterior likelihood contours in the param-eter plane of mass of WDM particle and redshift of reioniza-tion. Different colours represent different subsets of the com-bined data set used. In particular, the blue colour shows thefull analysis of the combined data (XQ-100 + HIRES/MIKE)which used 10 redshift bins. In red we show the results whenonly 3 redshift bins were used in the analysis, centered around z = 4 . z = 4 . , . , . ing scale since it has been argued in [47] that the filter-ing scale λ F will always be smaller than the Jeans scale λ J . Thus the Jeans scale plays a role of a (conservative)upper limit on the amount of pressure smoothing. Theplot shows that thermal and filtering (or Jeans) scaleshave a very different redshift evolution compared to thefree-streaming scale of the warm dark matter, which isthe only scale slowly increasing with redshift in velocityspace. Fig. B.4 is meant to be of illustrative purposeonly, to show that different scales evolve differently withredshift. We would also like to caution the reader that,while the pressure smoothing scale (Jeans or filtering)and WDM free-streaming scale, are acting on the 3Dmatter density field, the thermal scale is a 1D smoothingscale that operates on the optical depth field.While Fig. B.4 shows that the redshift evolution differsbetween different smoothing scales, the MCMC boundsderived in this paper make use of the full shape of theflux power spectrum. Furthermore, the flux power spec-trum traces the integral over the 3D matter power, and isthus sensitive to small scales at any given parallel wave-number. Indeed, this is why lower resolution surveys arealso able to put bounds on the WDM free-streaming scale[8]. The effect of redshift evolution on the shape of theflux power spectrum is shown in Fig. B.5. Even on large1 . . . . . . z λ s m oo t h [ k m / s ] thermalJeansFiltering m WDM = 2 . m WDM = 5 keV
FIG. B.4: The redshift evolution of the different smoothingscales in units of km s − : thermal (red), Jeans (blue), fil-tering (green) and free-streaming from WDM thermal relic(magenta). The two line-styles show different values of theWDM mass for 2.5 keV (full line) and 5 keV (dashed line)respectively. scales ( k < .
01s km − ), models with varying amount ofthermal or WDM free-streaming smoothing have quitedistinct shapes. Combining the shape with the redshiftevolution helps break the degeneracies among the IGMparameters and the mass of the WDM. Appendix C: The effect of different priors on thelimits of m WDM
In this section we show an extended table of how thelimits on the mass of the WDM change when imposingdifferent priors. The priors for reference case and weakpriors are the same as the ones plotted in Fig. 2.The reference priors consist of weak priors on the val-ues of mean flux in each redshift bin. These priors werechosen to be Gaussian with mean value as predicted bythe empirical fit by [19] with ± σ ). Further,the reference priors include bounds on some of the pa-rameters that are physically motivated: m WDM ≥ ≤ z rei ≤
16, 0 K ≤ T A ≤ − ≤ T S ≤ ≤ γ ( z i ) < . z i . We have alsochecked that the exact values for upper and lower boundson z rei and the temperature amplitude ( T A ) and slope( T S ) do not have an impact on the final constraints ofthe WDM. The bounds for γ at each redshift are physi-cally motivated for the time of HeII reionization [1, 48].The weak priors, as already described in the mainbody of the text, add the following priors to the refer-ence values: σ and n eff have Gaussian priors of ± σ ) around Planck values and the assumed temperature T A is 10000 ± σ ). .
003 0 .
006 0 .
01 0 .
03 0 . k [km − s] . . . . . . . P F ( k ) / P F , r e f ( k ) z = 4 . z = 4 . z = 5 . z = 5 . CDM cold Λ CDM hot m WDM = 2 keV m WDM = 4 keV
FIG. B.5: The flux power spectrum for different models vary-ing mass of the WDM ( m WDM ) and the amplitude of the IGMtemperature at the mean density ( T A ). The colours show twovalues of m WDM - 2 keV in black and 4 keV in green - andtwo ΛCDM models with different temperatures - hot IGM inred (roughly 3000K hotter) and cold in blue (roughly 3000Kcolder). The reference model against which the flux poweris compared, was ΛCDM model with T ( z = 3 .
6) = 11000K.Different line styles show the redshift evolution of the fluxpower: full line ( z = 4 . z = 4 . z = 5 .
0) and dotted line ( z = 5 . > . > . > . > . > . > . > . > . > . > . > . > . m WDM inthe units of keV. Different priors used are the reference case,weak priors, Planck priors on the cosmological parameters andphysical priors on thermal evolution where T varies freelywith redshift bins. Compared to the result shown in Table Imore decimal points are shown in the result. Furthermore, priors on cosmological parameters wereadded to the reference once (Planck priors), such that: σ and n eff have Gaussian priors of ± σ ) aroundPlanck values.And lastly, we also considered a temperature evolutionwhere the temperature T was allowed to vary freely ineach of the redshift bins. In this case we have used refer-ences values for priors to which we have added additionalconstraint on the change of the temperature between red-shift bins, such that the change in temperature jumpsbetween redshift bins of ∆ z = 0 . . T = 5000 K. Appendix D: Bestfit and confidence levels tables
In this section we show the full tables of the best-fitparameters (and their 1 and 2 σ confidence intervals) forthe MCMC analysis of the three different data sets: XQ-100 (Table S1), HIRES/MIKE (Table S2) and combinedXQ-100 + HIRES/MIKE (Table S3). Parameter (1 σ ) (2 σ ) Best fit¯ F ( z = 3 .
0) [0 . , .
68] [0 . , .
70] 0 . F ( z = 3 .
2) [0 . , .
62] [0 . , .
64] 0 . F ( z = 3 .
4) [0 . , .
56] [0 . , .
57] 0 . F ( z = 3 .
6) [0 . , .
51] [0 . , .
53] 0 . F ( z = 3 .
8) [0 . , .
45] [0 . , .
46] 0 . F ( z = 4 .
0) [0 . , .
39] [0 . , .
41] 0 . F ( z = 4 .
2) [0 . , .
36] [0 . , .
38] 0 . T A ( z = z p ) [10 K] [0 . , .
12] [0 . , .
27] 1 . T S ( z = z p ) [ − . , .
73] [ − . , . − . γ A ( z = z p ) [1 . , .
31] [1 . , .
45] 1 . γ S ( z = z p ) [ − . , − .
97] [ − . , . − . σ [0 . , .
86] [0 . , .
92] 0 . z rei [9 . , .
47] [6 . , .
66] 11 . n eff [ − . , − .
32] [ − . , − . − . /m WDM [keV − ] [0 , .
63] [0 , .
77] 0 . f UV [0 . , .
72] [0 . , .
96] 0 . z p = 3 . σ ) (2 σ ) Best fit¯ F ( z = 4 .
2) [0 . , .
38] [0 . , .
46] 0 . F ( z = 4 .
6) [0 . , .
29] [0 . , .
37] 0 . F ( z = 5 .
0) [0 . , .
17] [0 . , .
21] 0 . F ( z = 5 .
4) [0 . , .
06] [0 . , .
08] 0 . T A ( z = z p ) [10 K] [0 . , .
91] [0 . , .
12] 0 . T S ( z = z p ) [ − . , − .
11] [ − . , − . − . γ A ( z = z p ) [1 . , .
38] [1 . , .
52] 1 . γ S ( z = z p ) [ − . , .
81] [ − . , .
77] 0 . σ [0 . , .
01] [0 . , .
32] 0 . z rei [8 . , .
52] [6 . , .
88] 10 . n eff [ − . , − .
33] [ − . , − . − . /m WDM [keV − ] [0 , .
17] [0 , .
28] 0 . f UV [0 . , .
70] [0 . , .
96] 0 . z p = 4 .
5. Parameter (1 σ ) (2 σ ) Best fit¯ F ( z = 3 .
0) [0 . , .
70] [0 . , .
71] 0 . F ( z = 3 .
2) [0 . , .
63] [0 . , .
64] 0 . F ( z = 3 .
4) [0 . , .
57] [0 . , .
58] 0 . F ( z = 3 .
6) [0 . , .
52] [0 . , .
53] 0 . F ( z = 3 .
8) [0 . , .
46] [0 . , .
47] 0 . F ( z = 4 .
0) [0 . , .
39] [0 . , .
40] 0 . F ( z = 4 .
2) [0 . , .
35] [0 . , .
36] 0 . F ( z = 4 .
6) [0 . , .
27] [0 . , .
29] 0 . F ( z = 5 .
0) [0 . , .
14] [0 . , .
17] 0 . F ( z = 5 .
4) [0 . , .
04] [0 . , .
06] 0 . T A ( z = z p ) [10 K] [0 . , .
94] [0 . , .
06] 0 . T S ( z = z p ) [ − . , − .
99] [ − . , − . − . γ A ( z = z p ) [1 . , .
63] [1 . , .
69] 1 . γ S ( z = z p ) [0 . , .
42] [ − . , .
81] 1 . σ [0 . , .
89] [0 . , .
95] 0 . z rei [8 . , .
37] [6 . , .
43] 9 . n eff [ − . , − .
35] [ − . , − . − . /m WDM [keV − ] [0 , .
13] [0 , .
22] 0 . f UV [0 . , .
65] [0 . , .
94] 0 . z p = 4 . Acknowledgments
VI is supported by US NSF grant AST-1514734. VIalso thanks M. McQuinn for useful discussions, and IAS,Princeton, for hospitality during his stay where partof this work was completed. MV and TSK are sup-ported by ERC-StG ”cosmoIGM”. SL has been sup-ported by FONDECYT grant number 1140838 and par-tially by PFB-06 CATA. VD, MV, SC acknowledge sup-port from the PRIN INAF 2012 ”The X-Shooter sampleof 100 quasar spectra at z ∼ .
5: Digging into cosmol-ogy and galaxy evolution with quasar absorption lines.GB is supported by the NSF under award AST-1615814.SLE acknowledges the receipt of an NSERC DiscoveryGrant. MH acknowledges support by ERC ADVANCEDGRANT 320596 ”The Emergence of Structure during theepoch of Reionization”. LC is supported by YDUN DFF4090-00079. KDD is supported by an NSF AAPF fel-lowship awarded under NSF grant AST-1302093. JSBacknowledges the support of a Royal Society UniversityResearch Fellowship. Based on observations collected atthe European Organisation for Astronomical Research inthe Southern Hemisphere under ESO programme 189.A-0424. This work made use of the DiRAC High Perfor-mance Computing System (HPCS) and the COSMOSshared memory service at the University of Cambridge.These are operated on behalf of the STFC DiRAC HPCfacility. This equipment is funded by BIS National E-3infrastructure capital grant ST/J005673/1 and STFC grants ST/H008586/1, ST/K00333X/1. [1] M. McQuinn. The Evolution of the IntergalacticMedium.
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