The effect of neutrinos on the matter distribution as probed by the Intergalactic Medium
aa r X i v : . [ a s t r o - ph . C O ] J un Preprint typeset in JHEP style - HYPER VERSION
The effect of neutrinos on the matter distribution asprobed by the Intergalactic Medium
Matteo Viel , , Martin G. Haehnelt , , Volker Springel INAF-Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy INFN sez. Trieste, Via Valerio 2, 34127 Trieste, Italy Institute of Astronomy, Madingley Road, CB3 0HA, UK KICC-Kavli Insitute of Cosmology, Cambridge, UK Max-Planck Institut f¨ur Astrophysik, Karl-Schwarzschild Str. 1, 85748 Garching,GermanyEmail: [email protected] , [email protected] , [email protected] Abstract:
We present a suite of full hydrodynamical cosmological simulations that quan-titatively address the impact of neutrinos on the (mildly non-linear) spatial distribution ofmatter and in particular on the neutral hydrogen distribution in the Intergalactic Medium(IGM), which is responsible for the intervening Lyman- α absorption in quasar spectra. Thefree-streaming of neutrinos results in a (non-linear) scale-dependent suppression of powerspectrum of the total matter distribution at scales probed by Lyman- α forest data whichis larger than the linear theory prediction by about 25 % and strongly redshift dependent.By extracting a set of realistic mock quasar spectra, we quantify the effect of neutrinos onthe flux probability distribution function and flux power spectrum. The differences in thematter power spectra translate into a ∼ .
5% (5%) difference in the flux power spectrumfor neutrino masses with Σ m ν = 0 . α forest data and nearly perfectly degenerate with the over-all amplitude of the matter power spectrum as characterised by σ . If the results of thenumerical simulations are normalized to have the same σ in the initial conditions, thenneutrinos produce a smaller suppression in the flux power of about 3% (5%) for Σ m ν = 0 . m ν < . σ C.L.), comparable to constraints obtained from thecosmic microwave background data or other large scale structure probes. ontents
1. Introduction 12. The simulations 53. The matter power spectrum 9
4. The effect of neutrinos on statistics of the flux distribution in the Lyman- α forest 21 vs particle based simulations of the effect of neutrinos on Lyman- α forest data 25
5. An upper limit on the neutrino mass from the SDSS Lyman- α forestdata 27
6. Summary and Discussion 31
1. Introduction
One of the most exciting results in particle physics in the last decade has been that neutrinoshave been established to be massive particles. Solar, atmospheric, reactor and acceleratorneutrino experiments have confirmed the existence of flavour oscillations of active neutrinos,implying that neutrinos have non-zero mass (see Ref. [1] and references therein). This isgenerally considered as definite evidence for new physics beyond the Standard Model.The neutrino oscillation experiments do, however, not pin down the absolute neutrinomasses. The experiments instead provide a lower limit for the sum of the neutrino masses of0 . − . . β -decay experiment KATRIN is themost ambitious current direct detection experiment and is expected to probe an electronneutrino mass of ∼ . .
05 eV ≤ Σ m ν ≤ . ≥ z ≥ v th ∼
150 (1 + z ) (cid:20) m ν (cid:21) km / s . (1.1)As a result present-day velocities (of the most massive neutrino species) range between 100km/s for the upper and and 3000 km/s for the lower end of the still allowed range of thesum of the neutrinos masses. Dark matter particles with such a high velocity dispersion areusually called hot dark matter. A dominant contribution of hot dark matter to the totaldark matter content would be at odds with current observations. Neutrinos in the stillallowed mass range instead constitute a sub-dominant contribution complementing colddark matter comprised of some other elementary particle, such as neutralinos or axions.The effect of cosmological neutrinos on the evolution of density perturbation in thelinear regime is well understood. Neutrinos affect both the cosmic expansion rate and thegrowth of structure ([3, 1]). The neutrino contribution in terms of energy density can beexpressed as: f ν = Ω ν / Ω , Ω ν = Σ m ν . h eV , (1.2)where h is the present value of the Hubble constant in units of 100 km/s/Mpc and Ω isthe matter energy density in terms of the critical density.When neutrinos become non relativistic in the matter dominated era, there is a mini-mum wavenumber k nr ∼ .
018 Ω / (cid:20) Σ m ν (cid:21) / h/ Mpc , (1.3)above which the physical effect produced by neutrino free-streaming damps small-scaleneutrino density fluctuations, while modes with k < k nr evolve according to linear theory.The free-streaming leads to a suppression of power on small scales which in linear theorycan be approximated by ∆ P/P ∼ − f ν for f ν < .
07. With increasing energy contentin neutrinos (corresponding to increasing neutrino mass) the suppression becomes largerand its shape and amplitude depends mainly on Σ m ν and weakly on redshift [4]. At ∼ katrin – 2 –cales k > . h/ Mpc the suppression is constant while at scales 0 . < k ( h/ Mpc) < . α forest data, galaxy surveys, and CMB experiments, respectively.A large number of studies have used the effect of neutrinos on the matter powerspectrum (or perhaps better the lack thereof) to put upper limits on the energy contentand therefore the masses of neutrinos. Unfortunately, there is no single data set yet whichfully covers the characteristic imprint of neutrinos on the matter power spectrum andthe reliability of these limits therefore depends strongly on the somewhat questionableassumption that there are no systematic offsets between measurements of the matter powerspectrum with different methods which are not reflected in the quoted measurement errors.The Lyman- α forest data thereby plays a special role in probing the effect of the free-streaming of neutrinos on the matter power spectrum as it allows us to measure the matterpower spectrum on the scales where the suppression due to neutrinos is most pronouncedwhile still being in the mildly non-linear regime ([5, 6], see Ref. [7] for a more generalreview of the IGM). Ref. [8] have used high resolution spectra to obtain an early stillrather weak limit of Σ m ν < . α forest data alone. Ref. [9] have claimeda rather extreme limit of Σ m ν < .
17 eV (2 σ C.L.) based on the Sloan Digital Sky Survey(SDSS) quasar data set, combined with other large scale structure probes. This is thetightest limit obtained so far from cosmological data. Other measurements using cosmicmicrowave background data, galaxy redshift surveys and growth of clusters of galaxies areusually a factor three to six larger than this (e.g. [10, 11, 12, 13, 14, 15]). Note, however,also the rather low upper limit of Σ m ν < .
28 eV (2 σ C.L.) obtained by [16] based onLuminous Red Galaxies (LRG) in the SDSS Data Release 7 combined with data on thescale of Baryonic Acoustic Oscillations and the luminosity distance of distant supernovae.Forecasts for future CMB, weak lensing and Lyman- α forest data obtained by Planck ,the Baryon Oscillation Spectroscopic Survey and other surveys are presented e.g. in[17, 1, 18, 19].We would like, however, to stress again that the validity of current limits dependsstrongly on the assumption that there are no systematic offsets between estimates of thematter power spectrum obtained with different methods which are not reflected in thequoted measurement errors. To make further progress it will be very important to identifythe characteristic signatures of the effect of neutrinos on the detailed shape of the matterpower spectrum and its evolution with redshift. The Lyman- α forest data has here againparticular potential as it covers on its own a reasonably wide redshift range. With theLyman- α forest Baryonic Acoustic Oscillations (BAO) survey planned as part of the SloanDigital Sky Survey (SDSS-3) it should be possible to reach the scales where the suppressiondue to neutrinos becomes scale dependent.While linear theory is sufficient to quantify the impact of neutrinos on large scales andon the cosmic microwave background, the non-linear evolution of density fluctuations hasto be taken into account on smaller scales at lower redshift. A range of numerical studies – 3 –f the effect of neutrinos on the distribution of (dark) matter has been performed somewhile ago (e.g.[4, 20, 21]) with a renewed interest in the last couple of years ([22, 23, 24]).These numerical studies of the non-linear evolution have been complemented by analyticalestimates based on the renormalization group time-flow approach [25, 26], perturbationtheory [27, 28] or the halo model [29, 30].The use of Lyman- α forest data for accurate measurements of the matter powerspectrum benefits tremendously from the careful modeling of quasar absorption spectrawith hydrodynamical simulations (e.g. [31]). No such modeling has yet been performedincluding the effect of neutrinos. We will be closing this gap here and present results of themodeling of Lyman- α forest data in the non-linear regime including the effect of neutrinosby using a modified version of the hydrodynamical code GADGET-3 .Modeling the effect of neutrinos in the mass range of interest is non-trivial due to theirrather large thermal velocities. We mainly focus here on an implementation of the neutrinosas a separate set of particles. Ref. [23] have recently suggested to model the neutrinos witha grid based approach as a neutrino fluid instead of neutrino particles. In this approachthe gravitational force due to neutrinos is calculated based on the linearly evolved densitydistribution of the neutrinos in Fourier space. This approach has the advantage that itdoes not suffer from the significant shot noise on small scales introduced by the particlerepresentation of the fast moving neutrinos yielding higher accuracy at scales and redshiftswhere the effect of the non-linear evolution of the neutrinos is still moderate especially forsmall neutrino masses.In addition to our particle based neutrino simulations we have also experimented withsuch a grid based implementation of neutrinos. In this implementation the linear growth ofthe perturbation in the neutrino component is followed by interfacing the hydrodynamicalcode with the public available Boltzmann code
CAMB .Further advantages of such a grid based implementation of neutrinos, aside from elim-inating the Poisson noise, are the reduced requirements with regard to memory (there areno neutrino positions and velocities to be stored) and computational time. However, as wewill demonstrate in Section 3 for the scales and redshift of interest for the Lyman- α forestdata, non-linear effects are important. Taking their effect into account with the particle-based implementation actually offers a somewhat higher accuracy despite the reduction ofthe shot noise at the smallest scales offered by a grid based implementation of the linearevolution of the neutrino density.The main improvement of our work presented here compared to previous studies arethe use of full hydrodynamical simulations in a regime in which baryons are expected tosignificantly impact on the matter power (e.g. [32]), the focus on small scales and highredshift and the estimate of statistical properties of the Lyman- α flux distribution.The outline of the paper is as follows. In Section 2 we describe the numerical methodsand how we generate the initial conditions for the different simulations. Section 3 quantifiesthe impact of the neutrino component on the matter power spectrum. In this section wealso address the role of numerical parameters such as the initial redshift, number of neutrino http://camb.info/ – 4 –articles, Poisson noise and velocities in the initial conditions. Section 4 focuses on theimpact of neutrinos on two statistics of the flux distribution in Lyman- α forest spectra,the flux probability distribution function and the flux power spectrum. Section 5 presentsthe upper limit on the sum of the neutrino masses that we have obtained from the SDSSflux power spectrum alone. Section 6 summarizes our conclusions.We recall that the scales of interest for the Lyman- α forest low-resolution SDSSspectra are k ∈ [0 . − h/ Mpc, or k ∈ [0 . − .
02] s/km. High-resolution spectra asthe UVES/Large Programme LUQAS sample reach k max = 3 h/ Mpc [33]. The results forneutrinos and matter power spectra will be presented as a function of wavenumber k inunits of h/ Mpc, while those that refer to (one-dimensional) flux power spectrum will becast in terms of s/km. The conversion between wavenumbers expressed in s/km and h/ Mpcis redshift dependent and is given by the factor H ( z ) / (1 + z ) which for the cosmology usedbelow is 99, 111.5 and 123.6 km/s/Mpc, at redshifts z = 2, 3, and 4, respectively.
2. The simulations
In order to facilitate a straightforward comparison with the findings of [22], we have usedthe following cosmological model based on cold dark matter and a cosmological constant(ΛCDM): n s = 1, Ω = 0 .
3, Ω = 0 .
05, Ω cdm + Ω ν = 0 . , Ω = 0 . h = 0 . H = 100 h km/s). For all our simulations, we use the hydrodynamical TreePM-SPH (TreeParticle Mesh-Smoothed Particle Hydrodynamics) code GADGET-3 , which is an improvedand extended version of the code described in Ref. [34]. We have modified the code in orderto simulate the evolution of the neutrino density distribution. The neutrinos are treated asa separate collisionless fluid, just like the dark matter. In order to save computational time,most of our simulations assume however that the clustering of neutrinos on small scalesis negligible and the short-range gravitational tree force in
GADGET ’s TreePM scheme isnot computed for the neutrino particles. This means that the spatial resolution for theneutrino component is only of order the grid resolution used for the PM force calculation,while it is about an order of magnitude better for the dark matter, star and gas particlescalculated with the Tree algorithm. We also implemented memory savings such that thenumber of neutrino particles can be made (significantly) larger than the number of darkmatter particles, which helps to reduce the Poisson noise present in the sampling of the(hot) neutrino fluid.In the grid based implementation the power spectra of the neutrino density componentis interpolated in a table produced via
CAMB of one hundred redshifts in total spanninglogarithmically the range z = 0 −
49. The gravitational potential is calculated at the meshpoints and the neutrino contribution is added when forces are calculated by differentiatingthis potential. We have checked that we have reached convergence with this number ofpower spectrum estimates and also explicitly checked that increasing the linear size of thePM grid by a factor two has an impact below the 1% level on the total matter power forthe wavenumbers k < h/ Mpc. For the grid simulations the starting redshift has beenchosen as z = 49, well in the linear regime.– 5 –inear size (Mpc /h ) Ω m N / − gas N / ν PM / Σ m ν (eV) Ω ν (%) z IC
60 0.3 512 512 – 0.15 0.325 760 0.3 512 512 – 0.3 0.65 760 0.3 512 512 – 0.6 1.3 760 0.3 512 512 – 1.2 2.6 760 0.3 512 1024 – 0.6 1.3 760 0.3 512 1024 – 0.15 0.325 760 0.3 512 – – – – 760 0.3 512 – – – – 460 0.3 512 – – – – 4960 0.3 384 – – – – 760 0.3 512 512 – 0.15 1.3 460 0.3 512 512 – 0.15 1.3 4960 0.2 512 512 – 0.6 1.3 760 0.4 512 512 – 0.6 1.3 760 0.6 512 512 – 0.6 1.3 7512 0.3 512 – – – – 7512 0.3 512 – – – – 49512 0.3 512 512 – 0.6 1.3 7512 0.3 512 512 – 0.6 1.3 4960 0.3 512 – 512 0.6 1.3 4960 0.3 512 – 512 0.6 1.3 760 0.3 512 – 512 0.15 0.325 760 0.3 512 – 1024 0.6 1.3 49512 0.3 512 – 512 0.6 1.3 49
Table 1:
Summary of the most important parameters of the hydrodynamical simulations. Thesimulation with box size 60 Mpc /h and 512 Mpc /h (comoving) have been stopped at z = 1 . z = 0, respectively. The gravitational softening is 4 h − comoving kpc for all the different matterspecies for the small boxes and 30 h − comoving kpc for the large boxes. The particle-mesh grid ischosen to be equal to N / ν for which the parameter PM (Particle Mesh, see text) is also reported.The bottom part of the table describes the grid based simulations. Several other simulations, notreported in this table, have been used to test the dependence on the particle-mesh grid, thermalneutrino velocities in the initial conditions, different r.m.s. values for the power spectrum amplitude,total matter content, time-stepping, box-size issues, number of neutrino particles in the initialconditions and the method proposed (the particle based and grid based methods are extensivelydiscussed in the text). The initial conditions were generated based on linear matter power spectra separatelycomputed for each component (dark matter, gas and neutrinos) with
CAMB [35]. The totalmatter power spectrum was normalized such that its amplitude (expressed in terms of σ )matched the prediction by CAMB at the same redshift. After some testing the startingredshift for most of our runs was chosen as a rather low z = 7 to reduce the shot noise due– 6 –o the neutrino particles. When generating the initial conditions, we picked random phasesfor the modes in k -space but eliminated the Rayleigh sampling of the mode amplitudesin order to more accurately match the mean power expected in each mode, especially onlarge scales. This (artificially) reduces cosmic variance on the scale of the box, but sincewe are mainly interested in comparing power spectra at two different redshifts (i.e. in therelative growth), we do expect this effect to have a negligible impact on our main results.Initial neutrino velocities are drawn randomly from a Fermi-Dirac distribution (Eq. [1.1]).We have also tested a momentum pairing scheme in the initial conditions, as originallysuggested in [21, 20], by splitting each neutrino particle into two particles, giving themhalf the original mass and equal but opposite thermal velocities. However, we found thisto have no influence on our results.We have used the Zel´dovich approximation [36] to generate initial conditions. Weacknowledge that the use of a second-order Lagrangian perturbation theory scheme asproposed by [37] and used in [22] should improve the accuracy for simulations with lowstarting redshifts. As we are mainly interested here in the relative effect due to the free-streaming of neutrinos and not an absolute measurement of the overall amplitude of thematter power spectrum at a given redshift this should, however, not be a concern.We employ a simplified criterion for star formation to avoid spending most of ourcomputational time on the small-scale dynamics of compact galaxies that form in oursimulations. All gas particles whose overdensity with respect to the mean is above 1000 andwhose temperature is less than 10 K are turned into star particles immediately. We haveshown previously that such a star formation recipe has very little impact on Lyman- α fluxstatistics [6, 38], but speeds up the simulations considerably. As in our previous simulations[6] the heating rates have been multiplied by a factor ∼ z = 2 − m ν =0.6 eV case, the mass per simulation particle at our default resolution is2 . × , 10 and 5 . × M ⊙ /h for gas, dark matter and neutrinos, respectively.In Table 1 we summarize the most important parameters of the main hydrodynamicalsimulations that we use in this study. We stress that the scales and redshifts probed bymost of these simulations are very different from those explored in Refs. [22, 23, 24] butwe have also run a few simulation with the same large box size to facilitate a comparison.Most of the simulations run for this work are moderately time consuming. For example,the f ν =0.13 neutrino simulation took about 12 hours on 200 CPUs to reach z = 2, whileincreasing the number of neutrino particles by a factor eight for the same setup required10 hrs on 512 CPUs, meaning that it has become about two times slower in terms of totalcomputational expense. For comparison, the N gas = N dm = 512 simulation of the f ν =0.13model with the same amplitude of the matter power spectrum took 12 hrs on 160 CPUs, soincluding neutrino particles slows down the code only by ∼
20% (all the above numbers referto runs performed on the HPCS system DARWIN at Cambridge University). However, thememory requirements for storing a large number of neutrino particles are quite demanding,and are in fact the limiting factor for simulations with the particle based implementation ofthe neutrino density. Note that the grid based simulation of the small box size simulationshas taken about 1.6 times less CPU time to run than the corresponding particle based– 7 –imulation. The total CPU consumption for simulations with the smallest neutrino massΣ m ν =0.15 eV is thereby about 10% larger than that for the largest mass we investigatedΣ m ν =0.6 eV. y [ h - M p c ] z=3, DM (blue) + GAS (red) z=3, DM (blue) + GAS (red) + ν (green)
20 40 60 80 100
GAS
20 40 60 80 100 DM ν Figure 1:
Density slices of thickness 6 h − comoving Mpc at z = 3 extracted from two 60 h − Mpchydrodynamical simulations with gas and dark matter and no neutrinos. The right column showsa simulation that includes neutrinos with Σ m ν =1.2 eV. The presence of neutrinos (bottom panel,green) clearly affects both the gas (red) and the dark matter (blue) distribution. In Figure 1 we show illustrative slices of the density distribution of thickness 6 /h comoving Mpc extracted from two 60 /h comoving Mpc simulations at z = 3 with and– 8 –ithout neutrinos (for the particle based method). The left column shows a simulation withdark matter and gas but without neutrinos, while the right column shows the correspondingslices for the dark matter, gas and neutrinos for a three-component simulation with thesame initial phases and Σ m ν =1.2 eV. The distribution of the neutrino density (in green,bottom panel) has been smoothed to eliminate spurious Poisson noise at the smallest scalesin order to highlight that the genuine cosmological density fluctuations of the neutrinosoccur only on large scales due to the free-streaming of the neutrinos. The growth ofstructure is clearly less evolved in the simulation with neutrinos (the voids are for exampleless empty), since the suppressed clustering of the neutrinos slows down the growth of theperturbations in the overall matter density. Typical neutrino fluctuations at the largestscales are about 10% around the mean, while fluctuations of the gas and dark matterdensity are usually much larger than this.
3. The matter power spectrum
In a series of papers Refs. [22, 23, 24] have recently discussed the relative benefits anddrawbacks of implementing the effect of neutrinos in the form of particles taking intoaccount the non-linear evolution of the gravitationally coupled neutrino, dark matter andgas components of the matter density and a grid based implementation accounting onlyfor the effect of the linearly evolved neutrino density distribution. Here, we will primarilyfocus on modeling Lyman- α forest data and are therefore interested in different scalesand redshifts than those probed by other authors. However, in order to compare our workwith that of Ref. [23] we performed some simulations with our grid and particle basedimplementation of neutrinos with a large box size of 512 h/ Mpc. These should correspondto simulations C1 and C3 of Ref. [23].We measure the total matter power spectrum from the simulations by performing aCIC (Cloud-In-Cell) assignment to a grid of the same size as the PM grid used to computethe long-range gravitational forces. The smoothing effect of the CIC kernel is deconvolvedwhen the density field at the grid points is obtained. Power spectra are computed foreach component separately (gas, dark matter, stars and neutrinos), as well as for the totalmatter distribution.In the left panel of Figure 2 we compare the matter power spectra at z = 3 in dimen-sionless units for simulations with neutrino mass Σ m ν =0.6 eV for the grid based (thickorange curve) and particle based (thick blue line) implementations with that of a simula-tion without neutrinos (thick black curve) and the prediction of linear theory (thin blackcurve). We also show the neutrino power spectrum (thin blue curves) and the (redshiftindependent) Poisson contribution (red dashed curve). The Poisson contribution (for the N ν = 512 case) exceeds the neutrino power at k > . h/ Mpc. Only results for simulationswith the large box size are shown. In the right panel we show the fractional difference ofthe matter power spectrum of simulations with the particle and grid based implementationof neutrinos at different redshifts (in percent). At large scales k < . h/ Mpc the differencesare largest at z = 0, of the order of 2 % while at z = 3 are smaller and around 1%. The– 9 – .01 0.10 1.00 10.00 k (h/Mpc)10 -6 -4 -2 ∆ ( k ) Poisson ν lin. theory matter with ν lin. theory matter no ν matter with ν part. matter with ν grid ( P ( k ) g r i d - P ( k ) pa r t . ) / P ( k ) pa r t . z=3z=1z=0 Figure 2:
Left:
Dimensionless matter power spectrum at z = 3. We show the following quantities:linear matter power spectrum for a model with massive neutrinos with Σ m ν =0.6 eV (thin blackline); non-linear matter power spectrum obtained with the particle implementation (thick bluecurve) and with the grid implementation (thick orange curve); non-linear matter power spectrumfor a model without neutrinos (thick black line); linear neutrino power spectrum (thin blue curve);Poisson contribution due to neutrinos (dashed red curve). All results are for simulations with boxsize 512 Mpc /h . N ν = 512 for the particle based and P M = 512 for the grid based implementationof neutrinos. Right:
Fractional difference of the matter power spectrum for simulations with thegrid and particle based implementation of neutrinos at different redshifts ( z = 0 , , z = 49. results can be directly compared to those obtained by [23] for the same Σ m ν = 0.6 eV(figure 1 in their paper) but note that despite our attempt to choose similar parametersthere may be still small differences in some of the parameters and that the simulations in[23] do not contain baryons. The discrepancies between the two implementations albeitsmall on large scales appear to be somewhat larger in our simulations.In Figure 3 we compare results from the two methods in terms of neutrino suppressionfor the large simulation box with results for a box size nearly ten times smaller (60 /h Mpc), more appropriate for the modeling of Lyman- α forest data. Large boxes are shownas thin curves which are red dashed in the grid implementation and black continuous inthe particle one, respectively. Smaller boxes are reported as thick curves only at z = 3.At the smaller scales, that are not fully resolved by the large box-size simulation, non-linear effects are already important at the redshifts probed by Lyman- α forest data andthis is clearly demonstrated by the discrepancies between large and small scales. In fact,at ( k = k max , z = 3) ∆ ∼ (see left panel of Figure 2) and this non-linearevolution is missed in the large box simulations. We have checked that we get numericalconvergence in terms of non-linear matter power spectra between the N dm , gas = 512 andthe N dm , gas = 384 cases, so our results can be trusted at a quantitative level. We interpret– 10 – .1 1.0 10.0 k (h/Mpc)0.40.50.60.70.80.9 P ( k ) m a tt e r f ν / P ( k ) m a tt e r f ν =
512 Mpc/h part. z=0 and z=3 512 Mpc/h grid. z=0 and z=3 60 Mpc/h grid z=3 60 Mpc/h part. z=3 linear theory
Figure 3:
Comparison between the particle based and grid based implementation of neutrinos forsimulations with large and small box size . Ratio of matter power spectra for simulations with andwithout neutrinos as described in the text. The thin curves refer to simulations with a large linearbox size (512 /h Mpc): the grid based neutrino implementation (thin red dashed curves) and theparticle based neutrino implementation (thin black continuous curves) at z = 0 and z = 3. Thethick curves refer to simulations with the default linear box size of 60 /h Mpc: with the grid based(thick red dashed curve) and the particle based implementation of neutrinos, (black continuouscurve). The dotted curves show the predictions of linear theory at z = 0 and z = 3. The shadedarea indicates approximately the scales that are probed by the SDSS flux power spectrum data set. this discrepancy as due to the fact that in simulations with the grid-based implementationthe enforced linear evolution of the neutrinos with the same phases prevents a properresponse to the dark matter growth. At the small scales Fourier mode mixing is importantfor the phase association and can alter the linear theory picture significantly. This appearsto result in a significantly larger discrepancy between simulations with the grid and particlebased implementations on the scales and redshifts relevant for Lyman- α forest data. Thedifferences between should thereby be mainly due to the fact that the non-linear evolutionat small scales is not properly reproduced by the grid method. We will therefore focusmainly on simulations with the particle based implementation in the rest of the paper,keeping in mind that our results are affected by Poisson noise in the neutrino componentsat the smallest scales probed.Note that increasing the accuracy of the simulations with the particle based neutrinoimplementation further by pushing up the number of neutrino particles in order to decrease– 11 –he Poisson contribution to the matter power spectrum is rather demanding in terms ofparallel computing resources. Increasing our default number of neutrino particles (512 )by a factor of eight is still doable on the machine we had available for this (DARWIN) andrequires a factor ∼ In this Section we first contrast the effect of the free-streaming of neutrinos on the powerspectrum of the total matter density for a range of neutrino masses in full numerical sim-ulation with that predicted by linear theory. We will only refer to results from simulationswith the particle based implementations of neutrinos here, unless explicitly stated. In orderto quantify the suppression of structure growth induced by neutrinos, we divide the matterpower spectra of the neutrino simulations by the corresponding matter power spectrumextracted from the ΛCDM simulation without neutrinos. P f ν / P f ν = z=2 z=3 z=4 Σ m ν =0.15 eV Σ m ν =0.3 eV Σ m ν =0.6 eV Σ m ν =1.2 eV Figure 4:
Effect of different f ν on the matter power and comparison with linear prediction . Ratiobetween matter power spectra for simulations with and without neutrinos for four different valuesof the neutrino mass, Σ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (red dashed), z = 3 (black continuous) and z = 4 (blue dotted). Thepredictions of linear theory are shown as the thick curves. An estimate of the overall suppressionbased on the hydrodynamical simulations is shown as a thick short green line, ∆ P/P ∼ − . f ν . In Figure 4, we compare the non-linear power spectra from the numerical simulationswith the results predicted by linear theory, shown as thick curves. The suppression ofthe matter power spectrum increases with increasing Σ m ν (recall that these simulationsare normalized at the CMB scale). Note the plateau of constant suppression predictedby linear theory, which is approximately described by ∆ P/P ∼ − f ν , and depends onlyvery weakly on redshift. Linear theory provides a good description of the matter powerspectrum at z = 2 − k ∼ . h/ Mpc, and the agreement is more– 12 –ccurate for the smaller neutrino masses. The non-linear matter power spectrum does, onthe other hand, depend strongly on redshift and the dependence on scale becomes steeperwith decreasing redshift. For Σ m ν =0.6 eV, a good fit to the suppression at z = 3 inthe range that deviates from linear theory, k ( h/ Mpc) ∈ [0 . , P f ν /P f ν =0 = T ν ( k ) ∝ log ( k ) − . , − . , − . at z = 2 , ,
4, respectively; while for Σ m ν =0.3 eV, we find T ν ( k ) ∝ log ( k ) − . , − . , − . at the same redshifts. We also note that the maximumreduction of power shifts to larger scales with decreasing redshift.The maximum of the non-linear suppression can be described by ∆ P/P ∼ − . f ν (green thick curves in Fig. 4) for neutrino masses Σ m ν =0.15, 0.3, 0.6 eV, respectively. Forthe most massive case we considered the suppression is about ∆ P/P ∼ − f ν . Our resultsdiffer somewhat from those of Ref. [22], who reported ∆ P/P ∼ − . f ν (at z = 0) while wemeasure ∆ P/P ∼ − . f ν , apart from the most massive case in which the suppression issmaller, ∆ P/P ∼ − f ν . We must remind, however, that the above linear approximationstarts to break down for large neutrino masses and is already very poor for Σ m ν =1 eV(e.g. [1]). P f ν / P f ν = z=3 z=2 z=4 σ -> Σ m ν =0.15 eV 0.1 1.0 10.0k (h/Mpc)0.20.40.60.81.0 σ -> Σ m ν =0.3 eV 0.1 1.0 10.0 k (h/Mpc)0.20.40.60.81.0 σ -> Σ m ν =0.6 eV 0.1 1.0 10.0 k (h/Mpc)0.20.40.60.81.0 σ -> Σ m ν =1.2 eV Figure 5:
Effect of a different r.m.s. value for the amplitude of the matter power spectrum.
Ratiobetween matter power spectra with different values of σ . Four different cases are presented thathave exactly the same σ at z = 7 as those of the models with Σ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (red dashed), z = 3 (blackcontinuous), and z = 4 (blue dotted). Note that there is an up-turn in the suppression at scales of about 5, 8, 10 h /Mpcfor z = 2, 3, 4, respectively. A similar upturn was found by Ref. [22], but at a scale of1 h/ Mpc at z = 0. We have checked that this feature does not depend on the number ofneutrino particles in the simulation. It does depend weakly on the value of f ν (or Ω cdm ),moving to larger scales when f ν is decreased. The upturn appears to be related to the non-linear collapse of haloes, which decouple from the large scale modes slightly differently insimulations with neutrinos than in simulations that have a different value for the amplitudeof the power spectrum and no neutrinos. This suggests that the virialization of halos isslightly modified by the smoothly distributed neutrino component, in a similar fashion asdone by dark energy where this is a well-known effect (see [39] for a recent study).– 13 – .1 1.0 10.0k (h/Mpc)0.20.40.60.81.0 P f ν / P f ν = z=3 z=2 z=4 σ -> Σ m ν =0.15 eV 0.1 1.0 10.0k (h/Mpc)0.20.40.60.81.0 σ -> Σ m ν =0.3 eV 0.1 1.0 10.0 k (h/Mpc)0.20.40.60.81.0 σ -> Σ m ν =0.6 eV 0.1 1.0 10.0 k (h/Mpc)0.20.40.60.81.0 σ -> Σ m ν =1.2 eV Figure 6:
Effect of σ vs. effect of f ν . Ratio between matter power spectra with different values of σ (no neutrinos, thin curves) and different neutrino energy density (with neutrinos, thick curves).Four different cases are presented that have exactly the same σ at z = 7 as the models includingneutrinos with Σ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (dashed), z = 3 (continuous), and z = 4 (dotted). For clarity we show the threedifferent redshifts only for the most massive case (rightmost panel). In the other panels we showthe z = 3 results only. The main effect of the free-streaming of neutrinos is a reduction of the amplitude of thematter power spectrum on small scales. This results in a well known degeneracy betweenthe values of σ and Σ m ν . In order to explore this degeneracy in more detail we haverun four further hydrodynamical simulations without neutrinos that have the same valueof σ at z = 7 as the four different neutrino simulations, namely: σ = 0.137, 0.132, 0.122,and 0.103, mimicking the simulations with Σ m ν = 0 .
15, 0 .
3, 0 .
6, and 1 . σ = 0 .
141 at z = 7). The corresponding z = 0values are σ = 0 . . . . σ = 0 .
878 for the default simulation. Thedifferences in terms of the amplitude of density fluctuations range from 3% (0.15 eV) and36% (1.2 eV). For the 0.6 eV simulations the difference is 15% which is very close to thecorresponding 1 σ uncertainty in the linear matter power spectrum amplitude at z = 3 atscales k = 0 .
009 s/km, as derived from SDSS Lyman- α observations by [40].The results of the simulations without neutrinos but a decreased power spectrumamplitude are shown in Figures 5 and 6, where we can see that the effects of neutrinosand a overall suppression of the matter power spectrum amplitude are very similar. ForΣ m ν =0.15, 0 . m ν =1.2 eV) , where the suppression is largest, we show the results of theneutrino simulations of Figure 4 for all three redshifts. On the small scales consideredhere the effect of the neutrinos on the non-linear matter power spectrum is to a highdegree degenerate with an overall reduction of the matter power spectrum amplitude.The scale and redshift dependent differences to a simulation with an overall decrease ofthe power spectrum amplitude are small but nevertheless noticeable and increase with– 14 –ncreasing neutrino mass. For neutrino masses with Σ m ν > . σ at z = 7. Note that Figure 6 is meant tohighlight the small differences in the shape of the matter power spectra due to neutrinos ifa normalization of the matter power spectrum at small scales is assumed (same σ , i.e. thefluctuations are effectively normalized at the Lyman- α forest scale). In the last section we had investigated the effect of varying the neutrino mass at fixed totalmatter content Ω m . In linear theory the effect of neutrinos is well parameterized by the ratioof mass content in neutrinos to total matter content, f ν . We test here how well this holdsfor full non-linear simulations including neutrinos by varying Ω m at fixed neutrino mass.We have run simulations with and without neutrinos with Ω m = 0 . , . , . m ν =0.6eV. The results are shown in Figure 7. To further test the degeneracy with simulations withadapted value of σ we also run two simulations without neutrinos but with the same σ value as the simulations with Ω m = 0 . , . P f ν / P f ν = Ω m =0.2 Ω ν =0.013 0.1 1.0 10.0k (h/Mpc)0.20.40.60.81.0 Ω m =0.3 Ω ν =0.013 0.1 1.0 10.0 k (h/Mpc)0.20.40.60.81.0 Ω m =0.4 Ω ν =0.013 0.1 1.0 10.0 k (h/Mpc)0.20.40.60.81.0 Ω m =0.6 Ω ν =0.013 z=3 z=2 z=4 Figure 7:
Effect of varying Ω m at fixed Σ m ν . Ratio between matter power spectra with differentvalues of Ω m at fixed neutrino contribution to the energy density (Σ m ν =0.6eV). Four differentcases are shown with Ω m = 0 . , . , . , .
6, from left to right. Different line-styles refer to differentredshifts: z = 2 (dashed), z = 3 (continuous), and z = 4 (dotted). The thick curves in the first threepanels are for simulations without neutrinos but with the same σ as the simulations with neutrinos.The green thick curves in the right-most panel are for simulation with (Ω m = 0 . , Ω ν = 0 . f ν as the (Ω m = 0 . , Ω ν = 0 . We note the same trends as before. The presence of neutrinos results in an additionalsuppression of the matter power that, is well parameterized by the quantity f ν = Ω ν / Ω m also in the non-linear regime. At fixed neutrino mass the suppression is therefore larger for– 15 –maller values of Ω m . For example, for Ω m = 0 . m = 0 . − . f ν that we found for the fixed Ω m = 0 . m is also well captured by the parameter f ν . This is also demonstrated bythe green line in the right-most panel of Figure 7 where the non-linear power spectra for the(Ω m = 0 . , Ω ν = 0 . z = 3 result obtained for (Ω m = 0 . , Ω ν =0 . f ν parameter is the relevant quantity to describe the effect ofneutrinos. We therefore conclude that on scales relevant for the Lyman- α forest data theparameter f ν is sufficient to characterize the effect of Ω m and the neutrino mass also onthe non-linear power spectrum. although these results should be confirmed by using largerbox size simulations, where possibly one starts to be more sensitive to the overall shape ofthe matter power spectrum. In this subsection we investigate several numerical effects that impact the power spectrameasurements presented in the previous section: the number of neutrino particles, thevelocities in the initial conditions, the sampling of the initial conditions with neutrino pairsto balance momentum, and the starting redshift. Here we are not discussing resolutioneffects with regard to the number and mass of dark matter and gas particles since thesehave already been discussed extensively elsewhere with respect to the scales probed byLyman- α data (e.g. [6, 40, 41]). Furthermore, our results are mainly presented in terms ofratios of power spectra extracted from simulations with the same resolution. This stronglyreduces the sensitivity to the dark matter and gas resolution.First, we will take a closer look at the power spectrum of the neutrino component ofthe matter density. In Figure 8, we compare the non-linear neutrino power spectrum withpredictions from linear theory for some of the simulations in Table 1. At scales of about ∼ h/ Mpc the power spectrum starts to deviate strongly from linear theory and followsinstead the expectation for Poisson noise, P ( k ) ∝ k L /N particles . The Poisson contri-bution to the power spectrum depends as expected on the number of neutrino particlesused. This is demonstrated in the first and third panels where we also show results forsimulations with Σ m ν =0.15, 0 . N ν = 1024 instead of 512 neutrino particles.Doubling the number of neutrino particles for each spatial dimension shifts the Poissoncontribution to the matter power spectrum by a factor of roughly two to smaller scales.When modeling the Lyman- α forest flux power spectrum one ideally would like tosample the neutrino power spectrum properly on scales between 0.1 and 2 h/ Mpc. Asevident from Figure 8 this will be difficult as the neutrino distribution is affected by shotnoise at the smallest relevant scales. Reducing this shot noise to negligible levels requiresa number of neutrino particles with memory requirements beyond our current capabilities.In the following, we will see that despite the fact that the neutrino power spectrum isaffected by shot noise at the smallest scales relevant for Lyman- α studies, the impact onthe one-dimensional flux power is still very small.– 16 – .1 1.0k (h/Mpc)10 -6 -5 -4 -3 -2 n e u t r i n o po w e r s p ec t r u m z=2 z=3 z=4 Σ m ν =0.15 eV 0.1 1.0k (h/Mpc)10 -6 -5 -4 -3 -2 Σ m ν =0.3 eV 0.1 1.0 k (h/Mpc)10 -6 -5 -4 -3 -2 Σ m ν =0.6 eV 0.1 1.0 k (h/Mpc)10 -6 -5 -4 -3 -2 Σ m ν =1.2 eV Figure 8:
Effect of different f ν on the neutrino power spectrum and comparison with predictionof linear theory . Power spectra (dimensionless units) for the neutrino component as a function ofwavenumber. Four different cases are presented with Σ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (red dashed), z = 3 (black continuous),and z = 4 (blue dotted ). The prediction of linear theory are represented by the continuous thincurves. In the first and third panel, for the Σ m ν = 0 .
15, 0 . N ν = 1024 instead of N ν = 512 particles (thick curves). P f ν N = / P f ν N = z=2 z=3 z=4 Σ m ν = 0.15 eV Σ m ν = 0.6 eV Figure 9:
Resolution test for simulations with neutrinos: effect on the matter power spectrum .Ratio between matter power spectra with different number of neutrino particles (512 and 1024 )for simulations with Σ m ν = 0 .
15 eV (left panel) and Σ m ν = 0 . z = 2 (red dashed), z = 3 (black continuous), and z = 4 (bluedotted). In Figure 9, we show the ratios of the (total) matter power spectra for a simulationwith N ν = 1024 neutrino particles to that of our default simulations with 512 neutrinoparticles, for Σ m ν = 0.15 eV (left panel) and Σ m ν = 0.6 eV (right panel). The increased– 17 – .1 1.0 10.0 k (h/Mpc)0.900.951.001.051.10 P f ν n o ve l . i n I C s / P f ν z=2 z=3 z=4 Figure 10:
Effect of neutrino velocities on matter power spectra . Ratio between matter powerspectra with and without considering the velocities in the initial conditions at z = 7 (Σ m ν = 0 . z = 2 (dashed red), z = 3 (continuous black), and z = 4 (dotted blue). number of neutrino particles results in an additional suppression of about 5-10% at scalesabove k ∼ h/ Mpc, while at the scales probed by the Lyman- α forest the effect on thetotal matter power is of the order of 1% or less. While the Poisson contribution to theneutrino power spectrum is significant, its effect on the the total matter power spectrumis still small at small scales. The suppression is thereby slightly larger for Σ m ν = 0.6 eV,where the neutrinos constitute a larger fraction of the overall matter density. However, weshould stress here that the Lyman- α data is primarily sensitive the one-dimensional matterdistribution along the line-of-sight (although cross-correlating information in the transversedirection is a promising tool for future observational data sets). The one-dimensional powerspectrum, being a projection of the three-dimensional information, will be affected out tolarger scales than the three-dimensional power spectrum by the suppression (or increase)of power at a given scale [42, 43].Next we will investigate the impact of the neutrino velocities assigned in the initialconditions. In Figure 10 we show for our default simulations with z IC = 7, the ratio ofthe matter power spectrum for a simulation without the thermal velocities relative to thematter power spectrum of a simulation where the velocities have been included in the initialconditions. Without the velocities in the initial conditions the power is less suppressed at k ∼ h/ Mpc by roughly 3%, 2% and 1% at z = 2, 3, and 4, respectively, with a very weak– 18 – .1 1.0 10.0 k (h/Mpc)0.20.40.60.81.01.2 n e u t r i n o pa i rs P ν / P ν Σ m ν =0.6eV z=2 z=3 z=4 n e u t r i n o pa i rs P m a tt e r / P m a tt e r Figure 11:
Effect of a momentum conserving sampling of the initial distribution using neutrinopairs . Ratio between neutrino power spectra (left panel) and matter power spectra (right panel),with and without pairing of neutrinos in the initial conditions at z = 7 (Σ m ν = 0 . z = 2 (dashed red), z = 3 (continuous black), and z = 4(dotted blue). dependence on the wavenumber considered. These values are in good agreement with the z = 0 results reported by Ref. [22].Another effect that could potentially affect our results are the details of the samplingof the initial phase-space density distribution of the neutrinos. Ref. [21] suggested that itwould be advantageous to conserve momentum by creating pairs of neutrinos with equaland opposite thermal velocities. To test this, we modified the initial condition code toproduce neutrino pairs with mass m p = m ν,p /
2, instead of a single neutrino particle withmass m ν,p . The two neutrino particles are then assigned the same velocities in oppositedirections in order to conserve momentum. The results are shown in Figure 11, where wereport the ratios of neutrino power spectra in the left panel and that of the matter powerspectra in the right panel. The impact is very small and is fully accounted for by thedifferent number of neutrino particles used, which decreases the Poisson contribution inthe case of neutrino pairs by a factor of two at k > − h/ Mpc.The last effect that we examine here is the dependence of the matter power spectrumon the initial redshift of the simulation [44, 45]. For this purpose we have performed fouradditional simulations with initial redshifts z = 4 and z = 49 for the simulation withΣ m ν = 0.6 eV and the ΛCDM simulation without neutrinos. In Figure 12, we plot thesuppression due to the effect of neutrinos for the matter power spectrum at z = 2 and 3(black and red curves) for the three different values of the starting redshift. At k ∼ h/ Mpcthere are differences of the order of 10% (3%) at z = 3 ( z = 2). As expected the results forour default simulation lie between those for the low and the high starting redshift. For the– 19 – .1 1.0 10.0 k (h/Mpc)0.50.60.70.80.9 P f ν z I C = , , / P f ν = z I C = , , Σ m ν =0.6 eV z=2 z IC =4 z=3 z IC =4 z=2 z IC =7 z=3 z IC =7 z=2 z IC =49 z=3 z IC =49 Figure 12:
Effect of the initial redshift on the matter power spectrum . Comparison of the ratiobetween the matter power spectra of simulations with different initial redshift z IC . We show thefollowing quantities at z = 2 (red dashed) and z = 3 (black continuous): P ( k, f ν , z IC = 4) /P ( k, f ν =0 , z IC = 4) (thin curves), P ( k, f ν , z IC = 7) /P ( k, f ν = 0 , z IC = 7) (very thick curves and defaultcase) P ( k, f ν , z IC = 49) /P ( k, f ν = 0 , z IC = 49) (thick curves). All simulations are for Σ m ν = 0 . early starting redshift ( z = 49) the neutrino component becomes effectively Poissonian evenat the largest scales, since the neutrino power spectrum as computed by CAMB will be ingeneral much smaller at high redshift than the Poisson contribution (which is independentof redshift): this translates into a larger overall suppression of the matter power spectrumat small scales. Note that the different amount of Poisson power with respect to physicalneutrino clustering in the initial conditions has an impact also on the subsequent clusteringof the neutrino and matter components. For the low starting redshift ( z = 4), the relevantscales are already affected by mildly non-linear growth. Note further that the relativelystrong dependence on the initial redshift at z = 2 − z = 0between simulations with different initial redshifts. This can probably be attributed atleast partially to their use of second-order Lagrangian corrections in the initial conditions,which reduces the errors introduced when a low starting redshift is used. In addition, theseauthors studied much larger scales and by z = 0 the non-linear evolution tends to largelyerase the memory of differences in the initial conditions on small scales. For the purposesof our study z = 7 appears to be an acceptable compromise. We provide further supportfor this in the next sections where we quantify the impact of the starting redshift on theLyman- α flux power spectrum. – 20 – . The effect of neutrinos on statistics of the flux distribution in theLyman- α forest In this Section we focus on the effect of neutrinos on the matter distribution as probedby the IGM, and in particular on the transmitted Lyman- α flux and its one-point fluxprobability distribution function (PDF), and its two-point statistics (the flux power spec-trum). To perform our analysis we have extracted 1000 mock quasar absorption spectrafrom the simulations at many different redshifts. All spectra are constructed in redshiftspace, taking into account the effect of the peculiar velocities of the IGM v pec , k along theline-of-sight. The flux at redshift-space coordinate u (in km/s/Mpc) can be written as F ( u ) = exp[ − τ ( u )] with τ ( u ) = σ ,α cH ( z ) Z ∞−∞ dx n HI ( x ) G h u − x − v IGMpec , k ( x ) , b ( x ) i d x , (4.1)where σ ,α = 4 . × − cm is the hydrogen Lyman- α cross-section, H ( z ) is the Hubbleconstant at redshift z , x is the real-space coordinate (in km s − ), b = (2 k B T /mc ) / is thevelocity dispersion in units of c , G = ( √ πb ) − exp[ − ( u − y − v IGMpec , k ( y )) /b ] is a Gaussianprofile that approximates the Voigt profile well in the regime considered here.The neutral hydrogen density in real-space in the equation above is approximatelyrelated to the underlying gas density (e.g. [46]) as, n HI ( r , z ) ≈ − n IGM ( z ) (cid:18) Ω b h . (cid:19) (cid:18) Γ − . (cid:19) − × (cid:18) T ( r , z )10 K (cid:19) − . (cid:18) z (cid:19) (1 + δ IGM ( r , z )) , (4.2)where Γ − is the hydrogen photoionization rate in units of s − , T is the IGM temperature, n IGM ( z ) is the mean IGM density as a function of redshift and r is the real-space coordinate.As we have the benefit of a full hydrodynamical simulations there is, however, no needto make the approximations underlying equation (4.2). We calculate the integral in eq. (4.1)to obtain the Lyman- α optical depth along each simulated line-of-sight using directly therelevant hydrodynamical quantities from the numerical simulations: δ IGM , T, v pec , n HI . Fur-ther details on how to extract a mock quasar spectrum from an hydrodynamical simulationusing the SPH formalism can be found in [47]. We have added noise typical for observedspectra and convolved the spectra with the instrumental resolution corresponding to ob-served high-resolution spectra. Note that the resolution has a larger effect on the flux PDFthan the flux power spectrum. The ensemble of all our spectra have then been normalizedby adjusting the assumed Ultra Violet background such that the observed mean flux levelin high-resolution quasar spectra [48] at a given redshift, < F ( z ) > = exp( − τ eff ( z )) with τ eff ( z ) = 0 . z ) . is reproduced.The Lyman- α flux PDF is very well measured especially from high resolution quasarspectra (the statistical errors are at the percent level). We recently performed a carefulanalysis of the systematic uncertainties and were able to extract interesting astrophysicaland cosmological constraints from the flux PDF [38]. In Figure 13, we show the ratio be-tween the flux probability distribution functions of simulations with and without neutrinos:– 21 – .0 0.2 0.4 0.6 0.8 1.0 flux 0.00.51.01.52.0 f l u x pd f f ν / f l u x pd f f ν = z=3 z=2 z=4 Σ m ν =0.15 eV 0.0 0.2 0.4 0.6 0.8 1.0 flux 0.00.51.01.52.0 Σ m ν =0.3 eV 0.0 0.2 0.4 0.6 0.8 1.0 flux 0.00.51.01.52.0 Σ m ν =0.6 eV 0.0 0.2 0.4 0.6 0.8 1.0 flux 0.00.51.01.52.0 Σ m ν =1.2 eV Figure 13:
Effect of f ν on the flux probability distribution function for four different neutrinomasses, Σ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (red dashed), 3 (black continuous) and 4 (blue dotted). from left to right, the cases for Σ m ν = 0 .
15, 0 .
3, 0 .
6, and 1 . z = 2,3, and 4. The larger the neutrino masses Σ m ν , the more peaked the flux distribution be-comes at intermediate flux values. The reason for this trend is that the growth of structureis suppressed in the simulation with neutrinos. As a result, voids (flux ∼
1) are less emptyand clustered regions are less dense than in the simulation without neutrinos and the effectis stronger at high redshift than at low redshift. f l u x pd f σ / f l u x pd f σ d e f a u lt z=2 z=3 z=4 σ -> Σ m ν =0.15 eV 0.0 0.2 0.4 0.6 0.8 1.0 flux 0.00.51.01.52.0 σ -> Σ m ν =0.3 eV 0.0 0.2 0.4 0.6 0.8 1.0 flux 0.00.51.01.52.0 σ -> Σ m ν =0.6 eV 0.0 0.2 0.4 0.6 0.8 1.0 flux 0.00.51.01.52.0 σ -> Σ m ν =1.2 eV Figure 14:
Effect of different r.m.s. values for the amplitude of the matter power on the fluxprobability distribution . Four cases are presented that have exactly the same σ at z = 7 as thoseof the models Σ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (red dashed), 3 (black continuous) and 4 (blue dotted). Analogous to our discussion in the previous section we also compute the flux propertiesfor simulations without neutrinos but with a reduced overall amplitude of the matterpower spectrum normalized to the same σ at z = 7. The results are shown in Figure– 22 –4. The trends with neutrino mass are similar to those seen in Figure 13, but slightly lesspronounced. f l u x po w e r f ν / f l u x po w e r f ν = z=2 z=3 z=4 Σ m ν =0.15 eV 0.001 0.010 k (s/km)0.60.81.01.21.41.6 Σ m ν =0.3 eV 0.001 0.010 k (s/km)0.60.81.01.21.41.6 Σ m ν =0.6 eV 0.001 0.010 k (s/km)0.60.81.01.21.41.6 Σ m ν =1.2 eV Figure 15:
Effect of f ν on the flux power spectrum . Ratio between flux power spectra with andwithout neutrinos as a function of wavenumber in s/km. Four different cases are presented withΣ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (red dashed), z = 3 (black continuous), and z = 4 (blue dotted). The shaded area indicatesthe range of wavenumbers probed by the SDSS flux power spectrum. We now turn to the flux power spectrum, a quantity which is more closely relatedto the underlying matter power and has been extensively used to constrain cosmologicaland astrophysical parameters (e.g. [5]). The Lyman- α flux power spectrum providesa more direct link to the matter power spectrum than the flux PDF: it is sensitive tocosmological parameters, the thermal state of the IGM, instrumental effects (signal tonoise and resolution), the presence of metal lines and the nature of dark matter at smallscales, etc. (see for example [33, 40]). The flux power spectrum has been measured overa wide redshift range, z = 2 − .
5, using both high and low-resolution data. The growthof cosmic structures can thus be constrained over a significant fraction of the cosmic time,lifting the degeneracies between astrophysical and cosmological parameters that presentdifferent redshift and scale dependencies in this range.We show the measured flux power spectra for our different simulations in Figures 15and 16. Note that the results have not been smoothed. We recall that the useful range ofhigh resolution spectra reaches to k = 0 .
03 s/km while we can reach to k ∼ m ν =0.15 eV, the only effect of neutrinos on the flux power is a <
5% suppressionat z = 4. As expected the effect becomes larger with increasing neutrino mass. At thelargest scales the flux power in the simulations with neutrinos is suppressed by 5, 7 and 15%for Σ m ν = 0.3, 0.6 and 1.2 eV, respectively. There is some dependence of the suppressionon wavenumber with an upturn at small scales of about 0.01 s/km and a bump at k ∼ .
05 s/km. – 23 –he relationship between one-dimensional flux power spectrum and three-dimensionalmatter power is non-trivial, not only because of the fact that the one-dimensional matterpower is an integral of the three-dimensional spectrum, but also due to non-linearities in theflux-density relation. As clearly demonstrated in Ref. [49], systems with column densities ∼ cm − contribute most to the flux power at k ∼ .
05 s/km, and these absorbers areproduced by gas which is close to the mean density [50]. The differences in the flux powerspectrum of simulations with and without neutrinos reflect the differences in the spatialdistribution of gas in models which have experienced different amounts of growth of struc-ture: at z <
3, a model with a reduced amplitude of the matter power spectrum has morestructure at mean density than a high- σ model for which the gas probability distributionfunction is more skewed. Note that the suppressions for the simulations without neutri-nos in Figure 16 are very similar to those with a reduced amplitude of the matter powerspectrum with the same value of σ . The small differences visible in the three dimensionalmatter power spectra are thus even smaller in the flux power spectrum. f l u x po w e r σ / f l u x po w e r σ , d e f a u lt z=2 z=3 z=4 σ −>Σ m ν =0.15 eV 0.001 0.010 k (s/km)0.60.81.01.21.41.6 σ −>Σ m ν =0.3 eV 0.001 0.010 k (s/km)0.60.81.01.21.41.6 σ −>Σ m ν =0.6 eV 0.001 0.010 k (s/km)0.60.81.01.21.41.6 σ −>Σ m ν =1.2 eV Figure 16:
Effect of different r.m.s. power spectrum amplitudes on the flux power spectrum .Four different cases are presented that have exactly the same σ at z = 7 as those of the modelsΣ m ν = 0 .
15, 0.3, 0.6, 1 . z = 2 (red dashed), z = 3 (black continuous), and z = 4 (blue dotted). In this subsection we explore the sensitivity of our results for the flux power spectrumon a number of numerical effects. In Figure 17, we show the ratios of the flux powerspectrum for simulations with Σ m ν =0.15 eV (left panel) and Σ m ν =0.6 eV (right panel)with N ν = 512 and N ν = 1024 neutrino particles. There is an opposite trend here towhat we found in the corresponding plot for the matter power shown in Figure 9. Thereis a bump at k ∼ .
05 s/km where the ratio rises above unity. The ratio of the matterpower spectrum at a similar scale of 5 h/ Mpc does not change or is mildly suppressed. Atthese scales the matter power spectrum is affected by Poisson noise due to the neutrinos.– 24 –he larger Poisson contribution to the matter power spectrum in the simulation with thesmaller number of neutrinos should result in less diffuse small scale absorbers and lowerthe amplitude of the flux power spectrum at scales > .
01 s/km.We have also checked the effect of a different number of mesh points on the PM gridby running a simulation with N ν = 1024 and a PM grid of 512 mesh points. We findthat the impact is negligible in the range of scales of interest, about ±
5% ( ± k > . .
06) s/km. Note however that these scales are much smaller thanthose we are interested in. P F N ν = / P F N ν = z=2 z=3 z=4 Σ m ν =0.15 eV Σ m ν =0.6 eV Figure 17:
Effect of different number of neutrino particles on the flux power . Comparison ofsimulations with N ν = 512 and N ν = 1024 neutrino particles with masses of Σ m ν = 0 .
15 eV (leftpanel) and Σ m ν = 0 . z = 2, 3, and 4. In Figure 18 we show the effect of varying the starting redshifts on the flux powerspectrum ratios. The differences are significantly smaller than those for the matter powerspectra (Figure 12). For neutrino masses with Σ m ν =0.6 eV the differences at z = 2 and z = 3 are at the level of 2% or less over the whole range of relevant wavenumbers (notethat curves of the same color should be compared with each other).The flux power spectrum of our simulation at z = 2 − k ∈ [0 . , .
03] s/km. This level of numerical convergence shouldbe sufficient for the analysis of presently available high- or low-resolution Lyman- α forestdata. The SDSS Lyman- α flux power spectrum has statistical errors of the order of 3% ormore at the smallest scales, and up to 10-15% at the largest scales, while the statisticalerrors of the high-resolution data are about two times larger than these. vs particle based simulations of the effect of neutrinos on Lyman- α forest data We now have a closer look at the relative merits of grid based and particle based simulationon scales relevant for Lyman- α forest data especially with regard to the Lyman- α flux power– 25 – .001 0.010 k (s/km)0.80.91.01.11.21.3 P F f ν z I C = , , / P F f ν = z I C = , , z=2 z IC =4 z=3 z IC =4 z=2 z IC =7 z=3 z IC =7 z=2 z IC =49 z=3 z IC =49 Σ m ν =0.6 eV Figure 18:
Effect of different initial redshifts on the flux power spectrum . We show the followingquantities at z = 2 (red dashed) and z = 3 (black continuous): P ( k, f ν , z IC = 4) /P ( k, f ν =0 , z IC = 4) (thin curves), P ( k, f ν , z IC = 7) /P ( k, f ν = 0 , z IC = 7) (very thick curves and defaultcase) P ( k, f ν , z IC = 49) /P ( k, f ν = 0 , z IC = 49) (thick curves). All simulations shown are forΣ m ν = 0 . spectrum. We will show how the differences shown in Figure 2 for the matter power spectrapropagate to the flux power spectrum. In the left panel of Figure 19 we show the suppres-sion of the matter power spectrum due to the free-streaming of neutrinos with Σ m ν =0.6eV model on the matter power for simulations with the grid based implementation of theneutrino density. The suppression is larger than that of the corresponding simulation withthe particle based neutrino implementation by about 10% (the horizontal green thick lineindicates a value of − f ν ). In the middle panel, we directly compare the two implemen-tations and while it is evident that at still linear scales, k < . h /Mpc, the agreement isat the 2% level, at smaller scales the differences are larger, about 7% at k ∼ k max . In therightmost panel, we compare the two implementations in terms of (one-dimensional) fluxpower. In this case the differences are of the order of <
4% for the scales considered inhere. On the scales relevant for the Lyman- α forest data the non-linear evolution of thematter distribution is more important than the effect of the Poisson contribution to theneutrino power spectrum justifying our choice of the particle based implementation for aquantitative analysis. – 26 – .1 1.0 10.0 k (h/Mpc)0.20.40.60.81.0 P ( k ) g r i d f ν / P ( k ) f ν = z=2 z=3 z=4 0.1 1.0 10.0 k (h/Mpc)0.900.951.001.051.10 P ( k ) g r i d f ν / P ( k ) pa r t . f ν P F ( k ) g r i d f ν / P F ( k ) pa r t . f ν Σ m ν =0.6 eV Figure 19:
Effect of neutrinos on the matter and flux power for a grid based implementation .Suppression induced by a Σ m ν =0.6 eV model (left panel) at z = 2 , , − f ν is shownas a thick green line. Ratio between the grid and the particle based implementations (middle panel).Impact on the flux power (right panel).
5. An upper limit on the neutrino mass from the SDSS Lyman- α forestdata We now turn to deriving an upper limit on the neutrino mass from the SDSS Lyman- α forest data for which the flux power spectrum has been measured by [51]. This uniquedata set consists of 3035 absorption spectra of quasars in the redshift range 2 < z < R ∼ α absorption features with a width of ∼
30 km/s are not resolved. Thesignal-to-noise of the individual spectra is rather low, S/N ∼ P F ( k, z ) at 12 wavenumbers inthe range 0 . < k (s/km) < . k = 0 . z = 2 .
2, 2.4,2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0, and 4.2, for a total of 132 data points. They alsoprovide the covariance matrix. Here, we will use this flux power spectrum together withthe recommended corrections to the data and the recommended treatment of the errors of– 27 –hese corrections.We note that Ref. [40] has interpreted this flux power spectrum previously based ona set of numerical simulations giving a measurement of the linear matter power spectrumat z = 3 and k = 0 .
009 s/km. Due to the wide redshift range sampled, many degeneraciesbetween cosmological and astrophysical parameters can be broken, allowing for a highprecision measurement of the linear power spectrum at small scales.
In order to explore the multi-dimensional astrophysical and cosmological parameter spacewe will use an improved version of a method based on a Taylor expansion of the fluxpower spectrum around a fiducial model as presented in Ref. [31]. Note that this is anapproximate approach that assumes a physically motivated best-guess model and allowsan exploration of the likelihood function around it. If we denote with p an arbitrary vectorof astrophysical, cosmological and noise-related parameters close to the best-guess modeldescribed by p , we assume that: P F ( k, z ; p ) = P F ( k, z ; p )+ N X i ∂P F ( k, z ; p i ) ∂p i (cid:12)(cid:12)(cid:12)(cid:12) p = p ( p i − p i )+ N X i ∂ P F ( k, z ; p i ) ∂p i (cid:12)(cid:12)(cid:12)(cid:12) p = p ( p i − p i ) , (5.1)where p i are the N components of the vector p , which represent the astrophysical andcosmological parameters. We perform here the Taylor expansion to second order for eachparameter independently (i.e. neglecting cross derivatives). To obtain the derivatives of theflux power spectrum we run a suite of hydrodynamical simulations changing one parameterat a time with respect to those of the best-guess model and keeping all other parametersfixed. We then calculate the first and second order coefficients according to equation (5.1).This procedure is performed for the astrophysical parameters describing the thermal stateof the IGM, T and γ (both described by a power-law at z = 3 with three parameters each:amplitude at z = 3, and power-law indices for z < z > n s , σ , H , Ω , Σ m ν ) that affect the growth of structure.We then use a Monte-Carlo Markov Chain technique based on a suitable modification ofthe publicly available COSMOMC code [52]. We thereby allow for parameters that describenoise, resolution and those that model the contribution of Damped Lyman- α systems tothe flux power spectrum. In total a set of 28 parameters are allowed to vary. For a moreextensive discussion on the use of the MCMC method in this context we refer to [53, 54, 55].The main difference to our previous work in this respect is the addition of the effect ofthe neutrino mass Σ m ν on the flux power spectrum. Our analysis here does thereby not relyon any additional data which independently constrains the amplitude of the matter powerspectrum on large scales. The results obtained in the previous sections refer to simulationsthat have a different σ and Ω than our best-guess model ( σ = 0 . , Ω m = 0 . , n S =0 . σ , P F ( k, z )[ f ν , σ = 0 .
85] = P F ( k, z )[ f ν , σ ] × P F ( k, z )[ f ν = 0 , σ = 0 . /P F ( k, z )[ f ν = 0 , σ ].– 28 – .01k (s/km)0.010.10 ∆ F Σ m ν =0.02 eV Σ m ν =1.2 eV Figure 20:
Effect of different values of Σ m ν on the flux power spectrum for simulations normalizedat Lyman- α forest scales . The solid blue curves show the the best-fit model to the SDSS flux powerspectrum (data points with error bars), a model which is ruled out by the data at > σ level is alsoreported (red dashed curves). The effect of the free-streaming of the neutrinos Σ m ν > σ is asmall additional scale dependent suppression of the flux power which depends on redshiftand mass of the neutrinos. The difference between the thick and thin curves in Figure 6shows the effect for the matter power spectrum. Comparison of Figure 15 with Figure 16shows that the corresponding effect on the flux power spectrum is somewhat smaller.In order to demonstrate the effect of different Σ m ν values on the flux power spectrumwe compare the theoretical predictions directly with the SDSS data points in Figure 20.Two different models are shown: the best fit model to the data with Σ m ν =0.02 eV andthe same model (i.e. all the parameters fixed to the same values) but with a differentΣ m ν =1.2 eV. The constraining power is largest for the data points at small scale andhigh redshift, similar to studies which constrain the free-streaming by warm dark matterparticles.Our results with regard to the upper limit on the mass of neutrinos are summarized inFigure 21, where the one-dimensional marginalized (continuous curves) and mean (dashedcurves) likelihoods for Σ m ν and σ are shown in the left and right panels, respectively.The constraints are σ = 0 . ± .
04 (1 σ error bars) and Σ m ν < .
86 eV (2 σ confidence– 29 – igure 21: One-dimensional mean and marginalized likelihoods for the r.m.s. value of the matterpower spectrum amplitude and Σ m ν . The likelihoods computed from the Monte-Carlo MarkovChains from the SDSS flux power spectrum using the simulations including neutrinos shown forΣ m ν (left panel) and σ (right panel). The blue curves represent the results obtained if the effect ofneutrinos is approximated by changing σ in simulations without neutrionos. Mean likelihoods arerepresented by the dotted curves, while marginalized likelihoods are shown as continuous curves. level). The value for σ is in good agreement with previous analyses (e.g. [31]). We stressagain that the constraint on Σ m ν is obtained from the SDSS flux power spectrum alone without considering other external data sets. The χ -value for the best fit model is 138.3for 129 degrees of freedom, which should occur with a probability of 11%. We regard thisupper limit as a conservative one, since having used the results obtained with the gridbased approach would have produced a lower value than the one presented here, being theneutrino induced suppression larger for grid based simulations.We note that the likelihoods in Figure 21 stay somewhat flat even for Σ m ν values ∼ σ level. In this Figurethe mean likelihood is represented by the dotted curve, while the marginalized one is shownas a continuous curve. Note that this is very different from the result of the analysis in [9],where a very low upper limit of Σ m ν =0.17 eV was obtained. There are, however a numberof important differences to our study here. The study of [9], is based on: i ) inferring thelinear dark matter power spectrum with a suite of approximate hydrodynamical simulationsthat do not incorporate neutrinos; and ii ) combining this measurement with other largescale structure probes. As extensively discussed in [9] the tension between the high r.m.s.values for the amplitude of the matter power spectrum suggested by Lyman- α data and thelower values inferred from cosmic microwave background experiments is the main reasonfor the very low limit on Σ m ν .It is also interesting to compare our findings with those that would be obtained bynot considering the additional non-linear effects in simulations including neutrinos on the shape of the Lyman- α flux power spectrum. The effect on the flux power spectrum canin this case be captured by the degenerate effect of changing the value of σ . In orderto investigate this, we first add an extra parameter to represent neutrino’s mass fraction– 30 –n the markov chains, which has exactly the same flux derivatives of the corresponding σ value, and then vary it independently from the others obtaining Σ m ν < .
75 eV (2 σ C.L). The results are shown in Figure 21 and one can see that the hold method results ina tighter upper bound because the likelihood is less flat for large Σ m ν values than in thenew method (the 1 σ upper limits are 0.38 eV and 0.18 eV for the old and new method,respectively). With this approximation the constraints are thus only slightly tighter thanif we use the results from simulations including neutrinos which take the full non-lineareffects on the shape of the flux power spectrum into account. The upper limit obtained inthis way is also comparable to the constraint derived by just mapping (a-posteriori) the 2 σ lower limit on σ ∼ .
75, into an upper limit on Σ m ν using the code CAMB . In this lattercase, however, σ and Σ m ν are not treated as independent parameters as they should ifthe matter power spectrum is normalized at the Lyman- α forest scales. We think that thesomewhat tigther constraints obtained without the results from the numerical simulationsincluding neutrinos is due to the method we used. With the Taylor-expansion method wemodel small departures from a best-guess case and this is more accurately described byimplementing the exact results on the flux power from the numerical simulations ratherthan treating the effect of Σ m ν and σ independently (in this second case the flux power isless likely to depart significantly from the reference case than in the first case). However,given the small difference between the two methods we would not regard this discrepancyas particularly significant.We have focused here instead on investigating the impact of neutrinos on the (non-linear) spatial distribution of the neutral hydrogen in the IGM and the resulting flux powerspectrum and obtaining a consistent upper limit on neutrino masses from the SDSS fluxpower spectrum alone. The rather small but scale dependent and redshift dependent impactof neutrinos with Σ m ν = 0 . −
6. Summary and Discussion
Lyman- α forest data in combination with cosmic microwave background measurementsprovide presently the lowest upper limits on the masses of neutrinos. A careful assessmentof the ability of Lyman- α forest data and in particular the Lyman- α flux power spectrumto put limits on the effect of the free-streaming of neutrinos on the matter distributionis therefore important. The use of Lyman- α forest data for measurements of the matterpower spectrum relies heavily on the accurate modeling of the spatial distribution of neutralhydrogen. We have presented here for the first time a study of a large suite of hydrody-namical cosmological simulations that allow a quantification of the impact of neutrinos onthe non-linear matter distribution as probed by the Intergalactic Medium. The simulationswere performed with a modified version of GADGET-3 in which we have incorporated theeffect of neutrinos by using a particle and a grid based method.We have investigated a wide range of numerical issues relevant to simulating the ef-fect of the free-streaming of neutrinos on the matter distribution. We find that with aparticle based implementation the spatial distribution of the fast moving neutrinos suffers– 31 –ignificantly from Poisson noise. On scales of interest for the use of Lyman- α forest datathe corresponding errors are, however, still smaller than the measurement errors of theflux power spectrum. The less CPU and memory demanding grid based implementationof neutrinos on the other hand does not suffer from Poisson noise but results in errors inthe matter/flux power spectrum due to the assumption of linear theory for the growthof perturbations in the neutrino density. The error in the flux power spectrum in thiscase is as large as 4%, larger or comparable to the measurement errors. At linear scales k < . h/ Mpc simulations with the two different neutrino implementations agree at the2% level at z = 0 and at the 1% level at z = 3. The impact of other numerical effectsinvestigated (starting redshift, velocities of the neutrino particles in the initial conditions)is also smaller or comparable to the statistical errors of the SDSS flux power spectrum.By extracting a set of realistic mock quasar spectra, we quantify the effect of neutrinoson the flux probability distribution function and flux power spectrum. The free-streamingof neutrinos results in a (non-linear) scale-dependent suppression of the power spectrumof the total matter distribution at scales probed by Lyman- α forest data which is largerthan the linear theory prediction by about 25 % and strongly redshift dependent. Thedifferences in the matter power spectra translate into a ∼ .
5% (5%) difference in the fluxpower spectrum for neutrino masses with Σ m ν = 0 . σ . Breaking this degeneracy is important for a reliable assessment of the robustness ofthe upper limits on neutrino masses from Lyman- α forest data as the lowest upper limitsare based on combining measurements of the matter power spectrum on different scaleswith very different methods.We find that the differences in the flux power spectrum and the flux probability distri-bution function between simulations including neutrinos and simulations without neutrinoswith a reduced overall amplitude of the matter power spectrum are small but noticeable.Motivated by our findings, we then investigated whether the present SDSS data set alone can give constraints on the neutrino masses. We have explored the multi-dimensional like-lihood space using the flux derivative method proposed by [31]. We found a conservativeupper limit of Σ m ν = 0.9 eV at the 2 σ level, obtained from SDSS quasar spectra alone,which is comparable to limits obtained with other probes of large scale structure. Thislimit is of course much weaker than published constraints obtained by combining Lyman- α forest data with information from large scales, because the latter leverages the differentr.m.s. value for the amplitude of the matter power spectrum suggested by small-scaleand large-scale observables and turns this into a tight constraint for the absolute neutrinomasses. The robustness of these recently published relatively low upper limits depends,however, strongly on the somewhat questionable assumption that there are no systematicoffsets between the measurements obtained on small and large scales with these very dif-– 32 –erent methods which are not yet fully understood or not correctly taken into account inthe error analysis.We have demonstrated here that a quantitative investigation of the effect of the free-streaming of neutrinos on the non-linear matter distribution as probed by the IGM struc-tures can be efficiently performed with numerical hydrodynamical simulations. Reachingan accuracy below the one percent level at scales relevant for Lyman- α forest or weaklensing data will still be challenging but should be doable and will be an important step inturning the exciting prospect of an actual measurement of neutrino masses into reality. Acknowledgments
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