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MNRAS 000, 1–19 (2018) Preprint 28 September 2018 Compiled using
MNRAS LATEX style file v3.0
Lyman-α forest and non-linear structure characterizationin Fuzzy
Dark Matter cosmologies
Matteo Nori,1,2,3? Riccardo Murgia,4,6 Vid Iršič,5 Marco
Baldi1,2,3 and Matteo Viel4,6,71Dipartimento di Fisica e
Astronomia, Alma Mater Studiorum - University of Bologna, Via Piero
Gobetti 93/2, 40129 Bologna BO, Italy2INAF - Osservatorio
Astronomico di Bologna, Via Piero Gobetti 93/3, 40129 Bologna BO,
Italy3INFN - Istituto Nazionale di Fisica Nucleare, Sezione di
Bologna, Viale Berti Pichat 6/2, 40127 Bologna BO, Italy4SISSA, Via
Bonomea 265, 34136 Trieste, Italy5University of Washington,
Department of Astronomy, 3910 15th Ave NE, WA 98195-1580 Seattle,
USA6INFN - Istituto Nazionale di Fisica Nucleare, Sezione di
Trieste, Via Bonomea 265, 34136 Trieste, Italy7INAF - Osservatorio
Astronomico di Trieste, via Tiepolo 11, I-34143 Trieste, Italy
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACTFuzzy Dark Matter (FDM) represents an alternative and
intriguing description of thestandard Cold Dark Matter (CDM) fluid,
able to explain the lack of direct detectionof dark matter
particles in the GeV sector and to alleviate small scales tensions
inthe cosmic large-scale structure formation. Cosmological
simulations of FDM modelsin the literature were performed either
with very expensive high-resolution grid-basedsimulations of
individual haloes or through N-body simulations encompassing
largercosmic volumes but resorting on significant approximations in
the FDM non-lineardynamics to reduce their computational cost. With
the use of the new N-body cos-mological hydrodynamical code
AX-GADGET , we are now able not only to overcomesuch numerical
problems, but also to combine a fully consistent treatment of
FDMdynamics with the presence of gas particles and baryonic
physical processes, in orderto quantify the FDM impact on specific
astrophysical observables. In particular, inthis paper we perform
and analyse several hydrodynamical simulations in order toconstrain
the FDM mass by quantifying the impact of FDM on Lyman-α forest
ob-servations, as obtained for the first time in the literature in
a N-body setup withoutapproximating the FDM dynamics. We also study
the statistical properties of haloes,exploiting the large available
sample, to extract information on how FDM affects theabundance, the
shape, and density profiles of dark matter haloes.
Key words: cosmology: theory – methods: numerical
1 INTRODUCTION
In the first half of the last century, the scientific commu-nity
consensus gathered around two crucial facts about ourUniverse, that
are now considered the pillars of modern cos-mology: firstly that
the Universe is expanding and it is do-ing so at an accelerated
rate and, secondly, that the esti-mated baryonic matter content
within it cannot account forall the dynamical matter needed to
explain its gravitationalbehaviour.
The standard cosmological framework built upon theseconcepts,
called ΛCDM, still holds today. It implies the ex-istence of dark
energy, as a source of energy for the accel-erated expansion of the
Universe, and of dark matter, as anadditional gravitational source
alongside standard matter,
? E-mail: [email protected]
without however specifying their fundamental nature thatstill
represents a major puzzle for cosmologists.
The evidence for a cold and dark form of matter (CDM)– a
not-strongly electromagnetically interacting particle ora
gravitational quid that mirrors its effect – span over dif-ferent
scales and are related to dynamical properties of sys-tems, as e.g.
the inner dynamics of galaxy clusters (Zwicky1937; Clowe et al.
2006) and the rotation curves of spi-ral galaxies (Rubin et al.
1980; Bosma 1981; Persic et al.1996), but also to the gravitational
impact on the under-lying geometry of space-time, as strong
gravitational lens-ing of individual massive objects (Koopmans
& Treu 2003)as well as the weak gravitational lensing arising
from thelarge-scale matter distribution (Mateo 1998; Heymans et
al.2013; Planck Collaboration et al. 2015; Hildebrandt et al.2017).
Further evidence is based on the relative abundanceof matter with
respect to the total cosmic energy budget
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required in order to reconcile Large Scale Structures (LSS)– as
observed through low-redshift surveys – with the angu-lar power
spectrum of CMB temperature anisotropies thatseed the early
universe density perturbations (as observede.g. from WMAP and
Planck Komatsu et al. 2011; PlanckCollaboration et al. 2016,
respectively), on the clustering ofluminous galaxies (see e.g. Bel
et al. 2014; Alam et al. 2017),on the abundance of massive clusters
(Kashlinsky 1998) andtheir large-scale velocity field (Bahcall
& Fan 1998).
Whether dark matter consists indeed of a yet unde-tected
fundamental particle or it represents an indirect ef-fect of some
modification of Einstein’s General Relativitytheory of gravity is
still widely debated. Nevertheless, it hasbeen possible to exclude
some of the proposed dark mat-ter effective models, such as e.g.
the Modified NewtonianDynamics and its variants (MOND see e.g.
Milgrom 1983;Sanders & McGaugh 2002; Bekenstein 2004), recently
ruledout (Chesler & Loeb 2017) by the implications of the
grav-itational wave event GW170817 (Abbott et al. 2017). Thelack of
detection of dark matter particles in the GeV massrange through
neither of indirect astronomical observations(see e.g. Albert et
al. 2017), direct laboratory detections(see e.g. Danninger 2017),
nor artificial production in high-energy collisions experiments
(see e.g. Buonaura 2018) hasbeen undermining the appeal of the most
massive dark mat-ter particle candidates, as e.g. the Weakly
Interactive Mas-sive Particles (WIMPs), and it is presently
shifting the sci-entific community efforts in the hunt of direct
observationsfrom such high mass ranges towards lower ones (see
e.g.Bertone et al. 2005).
A good starting point where to focus research and toclarify such
long-standing uncertainties would be the appar-ent failures of the
ΛCDM model at scales . 10 kpc – asgiven e.g. by the cusp-core
problem (Oh et al. 2011), themissing satellite problem (Klypin et
al. 1999), the too-big-to-fail problem (Boylan-Kolchin et al. 2012)
–, all arising asan apparent inconsistency between simulations and
obser-vations, the latter being more in line with less
pronounceddensity fluctuations at those scales than predicted by
theformer. However, the nature of such apparent failures hasbeen
subject of debate in the astrophysics community. Itis still
unclear, in fact, whether they should be ascribed toan imperfect
baryonic physics implementation in numericalsimulations (see e.g.
Maccio et al. 2012; Brooks et al. 2013),to an intrinsic diversity
of properties related to the forma-tion history and local
environment of each individual darkmatter halo (Oman et al. 2015),
to the fundamental natureof the dark matter particle (see e.g.
Spergel & Steinhardt2000; Rocha et al. 2013; Kaplinghat et al.
2000; Medvedev2014) or even to a combination of all these possible
causes.
Among the particle candidates that have been proposedin the
literature, Fuzzy Dark Matter (FDM) models describedark matter as
made up of very light bosonic particles (seee.g. Hui et al. 2017,
for a review on the topic), so light thattheir quantum nature
becomes relevant also at cosmologicalscales. This requires a
description of dark matter dynamicsin terms of the Schrödinger
equation, in order to take intoaccount quantum corrections, and can
be mapped in a fluid-like description where a quantum potential
(QP) enters theclassical Navier–Stokes equation (Hu et al.
2000).
The typical wave-like quantum behaviour adds to thestandard CDM
dynamics a repulsive effective interaction
that, along with creating oscillating interference
patterns,actively smooths matter over-densities below a
redshift-dependent scale that decreases with the cosmic evolution–
as confirmed by FDM linear simulations (see e.g. Marsh
&Ferreira 2010; Hlozek et al. 2015) – thus potentially
easingsome of the previously mentioned small-scale
inconsistenciesof the CDM model.
The lack of density perturbations at small scales in-duced by
the QP is represented, in Fourier space, by a sharpsuppression of
the matter power spectrum, that persists – atany given scale –
until the action range of the QP shrinksbelow such scale and cannot
balance any longer the effect ofthe gravitational potential (see
e.g. Marsh 2016b, for anotherdetailed review on the subject). As a
matter of fact, whilelinear theory predicts that perturbations at
scales smallerthan the cut-off scale never catch up with those at
largerscales – untouched by FDM peculiar dynamics –,
non-linearcosmological simulations have shown that gravity is
indeedable to restore intermediate scales to the unsuppressed
level,in a sort of healing process (Marsh 2016a; Nori &
Baldi2018).
FDM non-linear cosmological simulations have beenperformed over
the years either with highly numerically in-tensive high-resolution
Adaptive Mesh Refinement (AMR)algorithms able to solve the
Schrödinger-Poisson equationsover a grid (see e.g. Schive et al.
2010, 2017) or with stan-dard N-Body codes that, however, include
the (linear) sup-pression only in the initial conditions but
neglect the inte-grated effect of the FDM interaction during the
subsequentdynamical evolution (see e.g. Schive et al. 2016; Iršič
et al.2017a; Armengaud et al. 2017) – basically treating FDM
asstandard dark matter with a suppressed primordial powerspectrum,
similarly to what is routinely done in Warm DarkMatter simulations
(Bode et al. 2001). The former approachled to impressive results in
terms of resolution (see e.g. Woo& Chiueh 2009; Schive et al.
2014) but is extremely com-putationally demanding, thereby
hindering the possibilityof adding a full hydrodynamical
description of gas and starformation for cosmologically
representative simulation do-mains. On the other hand, the latter
allows for such pos-sibility because of its reduced computational
cost which is,however, gained at the price of the substantial
approxima-tion of neglecting QP effects during the simulation (see
e.g.Schive et al. 2016).
For these reasons, following the approach first proposedin Mocz
& Succi (2015), we devised AX-GADGET (Nori &Baldi 2018), a
modified version of the N-body hydrodynam-ical cosmological code
P-GADGET3 (Springel 2005), to in-clude the dynamical effect of QP
through Smoothed Parti-cle Hydrodynamics (SPH) numerical methods.
The explicitapproximation of the dependence on neighbouring
particlesresults in a less numerically demanding code with respect
tofull-wave AMR solvers, without compromising cosmologicalresults,
with the additional ability to exploit the gas and starphysics
already implemented in P-GADGET3 , along with itsmore advanced and
exotic beyond-ΛCDM extensions such asModified Gravity (Puchwein et
al. 2013) or Coupled DarkEnergy models (Baldi et al. 2010).
Given that gravity, as mentioned above, can restore
thesuppressed power at intermediate scales in the non-linearregime,
major observables related to the LSS at such scalesmay appear
similar in both FDM and CDM picture cosmolo-
MNRAS 000, 1–19 (2018)
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Lyman-α and LSS properties in FDM cosmologies 3
gies at sufficiently low redshifts. For this reason,
Lyman-αforest observations could play a crucial role in
distinguishingsuch radically different models of dark matter, being
one ofthe most far reaching direct astrophysical probes in terms
ofredshift of the LSS observables, sampling the redshift rangez ∼
2−5 (see e.g. Iršič et al. 2017a, for Lyman-α forest anal-ysis in
N-body simulations, with neglected QP dynamicaleffects).
In this paper, we performed several simulations with themain
goal of studying the effects of FDM on Lyman-α forestobservations
in a fully consistent FDM set-up – i.e. withoutneglecting the QP
during cosmic evolution – , in order toconstrain the FDM mass. As a
by-product of our simula-tions, we are also able to perform an
extended analysis ofthe statistical and structural properties of
haloes, exploit-ing the large statistical sample at our disposal,
to extractvaluable information about how FDM affects, among
oth-ers, the halo mass function as well as the shape and
densitydistribution of dark matter haloes.
The paper is organized as follows: in Section 2 we
brieflydescribe the FDM models under consideration, providingall
the basic equations that enter our numerical implemen-tation (2.1),
and review the theoretical background behindLyman-α forest
observations and its physical implications(2.2). In Section 3, we
then recall how FDM dynamics isimplemented in the AX-GADGET code
(3.1), we present thesimulation sets performed (3.2) and the
strategy used toextract Lyman-α information (3.3); in Sections 3.4
and 3.5the procedures to deal with numerical fragmentation and
tomatch haloes across different simulations are outlined.
Theresults are collected in Section 4 and presented in
decreasingorder of interested scale, including the matter power
spec-trum (4.1), the Lyman-α statistics (3.3) and the
structurecharacterization (4.3). Finally, in Section 5 we draw our
con-clusions.
2 THEORY
In this Section we recall the main properties of a lightbosonic
field in a cosmological framework, how it affects thegrowth of LSS
and how Lyman-α forest analysis can be usedto probe these
modifications.
2.1 Fuzzy Dark Matter models
The idea of describing dark matter and its key role in theLSS
formation in terms of a ultra-light scalar particles – i.e.a
particle with mass ∼ 10−22eV/c2 was introduced in Huet al. (2000),
in which the term Fuzzy Dark Matter was usedfor the first time and
the cosmological implications inducedby the quantum behaviour of
such light dark matter field onlinear cosmological perturbations
were outlined.
The Schrödinger equation describing the dynamics ofthe bosonic
field φ̂i associated with a single particle can bewritten as
ih̄ ∂tφ̂i =−h̄2
m2χ∇2φ̂i+mχΦφ̂i (1)
where mχ is the typical mass associated with FDM particles–
often represented in terms of m22 = mχ1022c2/eV – and
Φ is the gravitational potential, satisfying the usual
Poissonequation
∇2Φ = 4πGa2ρb δ (2)
with δ = (ρ−ρb)/ρb being the density contrast with respectto the
background field density ρb (Peebles 1980).
Under the assumption that all the particles belong to
aBose-Einstein Condensate, the many-body field φ̂ of a col-lection
of particles factorizes and the collective dynamicsfollows exactly
Eq. 1. If this is the case, it is possible thento express the
many-body field φ̂ in terms of collective fluidquantities as
density ρ and velocity v, using the Madelungformulation (Madelung
1927)
v = h̄mχ=∇φ̂φ̂
(3)
ρ=mχ|φ̂|2 (4)
which translates into the usual continuity equation and
amodified quantum Navier–Stokes equation reading
v̇ + (v ·∇)v =−∇Φ +∇Q (5)
where Q is the so-called Quantum Potential (QP)
Q= h̄2
2m2χ∇2√ρ√ρ
= h̄2
2m2χ
(∇2ρ2ρ −
|∇ρ|2
4ρ2
)(6)
also known as Quantum Pressure if expressed in the equiv-alent
tensorial form as
∇Q= 1ρ∇PQ =
h̄2
2m2χ1ρ∇(ρ
4∇⊗∇ lnρ). (7)
The additional QP term accounts for the quantum behaviourof
particles with a repulsive net effect that counteracts
gravi-tational collapse below a certain scale, related to the
Comp-ton wavelength λC = h̄/mχc identified by the boson mass(Hu et
al. 2000). This can be heuristically viewed as theresult coming
from two combined effects of quantum wave-like nature: decoherence,
originating from the Heisenberguncertainty principle, stirring
towards space-filling config-urations and interference creating
oscillatory patterns (Huiet al. 2017).
In an expanding universe described by a scale factora and the
derived Hubble parameter H = ȧ/a, the lineardensity perturbation
δk in Fourier space satisfies – in thecomoving frame – the
relation
δ̈k+ 2Hδ̇k+(
h̄2k4
4m2χa4− 4πGρb
a3
)δk = 0 (8)
that directly sets the typical scale
kQ(a) =(
16πGρba3m2χh̄2
)1/4a1/4 (9)
where the gravitational pull is balanced by the QP
repulsion,sometimes referred as quantum Jeans scale in analogy
withthe homonym classical one (Chavanis 2012).
The growing solution of Eq. 8, expressed in terms of
thedimensionless variable x(k,a) =
√6 k2/k2Q(a), is
D+(x) =[(
3−x2)
cosx+ 3 xsinx]/x2 (10)
whose time dependence is bounded from above and below
MNRAS 000, 1–19 (2018)
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4 M. Nori et al.
by the large and small scale limits, respectively, as
D+ ∝{a for k� kQ(a)1 for k� kQ(a)
(11)
thereby recovering the standard ΛCDM perturbations evo-lution at
large scales and halting growth of small scales over-densities
(Marsh 2016b).
Structures are unable to collapse until the quantumJeans scale
kQ(a) becomes so little that gravity can over-come the QP repulsive
action and, in the linear perturba-tion regime, will forever carry
information about their pastsuppressed state (Marsh 2016b). In
non-linear simulations,instead, a recovery induced by gravity of
the intermediatescales is indeed observed: in terms of matter power
spec-trum, this implies that a portion of a FDM Universe, ob-served
at a fixed scale, will eventually look like a CDM Uni-verse if a
sufficient time for gravity recovery has passed afterthe crossing
of the quantum Jeans scale (as argued also ine.g. Marsh 2016a).
All this considered, it is clear the reason why FDMmod-els
peculiar imprints on LSS are to be looked for at verysmall scales
for low redshifts, while larger scales may pro-vide relevant
information only as long as higher redshifts areavailable to
observations. In particular, FDM may reveal itspresence in the
inner part of small collapsed haloes in theform of a flat solitonic
core (see e.g. Schive et al. 2014; Marsh& Pop 2015; De Martino
et al. 2018; Lin et al. 2018) whilelarger scales may show FDM
imprints in the high-redshiftgas distribution (Iršič et al. 2017a;
Armengaud et al. 2017;Kobayashi et al. 2017).
2.2 Lyman-α forest
The Lyman-α forest is the main manifestation of the
in-tergalactic medium (IGM), the diffuse filamentary matterfilling
the space between galaxies, and it constitutes a verypowerful
method for constraining the properties of DM inthe small scale (0.5
Mpc/h . λ . 20 Mpc/h) and high red-shift regime (2 . z . 5) (see
e.g. Viel et al. 2005, 2013).The physical observable for Lyman-α
experiments is the fluxpower spectrum PF(k,z). Constraints on the
matter powerspectrum from Lyman-α forest data at small
cosmologicalscales are only limited by the thermal cut-off in the
fluxpower spectrum, introduced by pressure and thermal mo-tions of
baryons in the photo-ionised IGM. That is why thisastrophysical
observable has provided some of the tightestconstraints up-to-date
on DM scenarios featuring a small-scale power suppression (Iršič et
al. 2017c; Murgia et al.2018), including FDM models, both in the
case where theyconstitute the entire DM (Iršič et al. 2017a;
Armengaudet al. 2017), and in the case in which they are a
fractionof the total DM amount (Kobayashi et al. 2017).
Ultra-light scalar DM candidates are indeed expected tobehave
differently with respect to standard CDM on scales ofthe order of
their de Broglie wavelength, where they inducea suppression of the
structure formation, due to their wave-like nature. In particular,
for FDM particles with masses∼ 10−22eV , such suppression occurs on
(sub)galactic scales,being thereby the ideal target for Lyman-α
forest obser-vations. Moreover, as we discussed in the previous
section,Lyman-α forest observations probe a redshift and scales
range in which the difference between ΛCDM and the FDMmodels –
for the masses considered – is highly significant.
All the limits found in the literature on FDM param-eters – i.e.
the mass mχ – using Lyman-α observations (ase.g. Iršič et al.
2017a; Armengaud et al. 2017; Kobayashiet al. 2017) have been
computed by assuming that ultra-light scalars behave as standard
pressure-less CDM and bycomparing Lyman-α data with flux power
spectra obtainedfrom standard SPH cosmological simulations, which
com-pletely neglected the QP effects during the non-linear
struc-ture evolution. In other words, the non-standard nature ofthe
dark matter candidate was simply encoded in the sup-pressed initial
conditions used as inputs for performing thehydrodynamical
simulations.
One of the goals of the present work is to use AX-GADGET in
order to provide the first fully accurate con-straints on the
FDMmass, by going beyond the standard dy-namical approximation of
ignoring the time-integrated QPeffect. Including such effect in our
numerical simulations isthereby expected to tighten the limits
published so far inthe literature, since it introduces a repulsive
effect at smallscales throughout the simulation evolution that
contributesto the matter power spectrum suppression. Besides
present-ing the new constraints, we will also carry out a
meticulouscomparison with the bounds determined under the
afore-mentioned approximation, in order to exactly quantify
itsvalidity.
3 NUMERICAL METHODS
In this Section we briefly review the implementation of
theAX-GADGET code routines that are devoted to the FDMdynamics (an
in depth description featuring analytic andcosmological tests can
be found in Nori & Baldi 2018). Wethen continue presenting how
Lyman-α forest observationsare modelled and extracted from
numerical simulations. Fi-nally, we describe our approach to
discriminate spurioushaloes – which are expected to form in
particle-based simula-tions featuring a suppressed power spectrum
(see e.g. Wang& White 2007), such as Warm Dark Matter, Hot Dark
Mat-ter, or FDMmodels – from genuine ones, in order to properlytake
into account the known problem of numerical fragmen-tation,
together with the strategy we used to cross-matchhaloes in the
different simulations.
3.1 The code: AX-GADGET
AX-GADGET is a module available within the cosmologi-cal and
hydrodynamical N-Body code P-GADGET3 , a non-public extension of
the public GADGET2 code (Springel2005). It features a new type of
particle in the system– i.e. ultra-light-axion (ULA) – whose
strongly non-linearquantum dynamics is solved through advanced and
refinedSmoothed Particle Hydrodynamics (SPH) routines, used
toreconstruct the density field from the particle distributionand,
therefore, to calculate the QP contribution to
particleacceleration.
The general SPH approach relies on the concept thatthe density
field ρ underlying a discrete set of particles canbe approximated
at particle i position with the weighted
MNRAS 000, 1–19 (2018)
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Lyman-α and LSS properties in FDM cosmologies 5
sum of the mass m of neighbouring particles NN(i)
ρi =∑
j∈NN(i)
mjWij , (12)
where the mass is convolved with a kernel function Wij ofchoice,
characterized by a particle-specific smoothing lengthhi, and whose
extent is fixed imposing43πh
3i ρi =
∑j∈NN(i)
mj (13)
so that only a given mass is enclosed within it.Once the density
field is reconstructed, every observable
is locally computed through weighted sums as
Oi =∑
j∈NN(i)
mjOjρjWij (14)
and its derivatives are iteratively obtained with
∇Oi =∑
j∈NN(i)
mjOjρj∇Wij (15)
where the derivation is applied on the window function.The exact
scheme of the SPH algorithm is not fixed,
since each observable can be expressed in many
analyticallyequivalent forms that, however, translate into
different op-erative summations. For example, the QP of Eq. 6 can
becalculated using recursive derivatives of ρ,√ρ or logρ
in-termediate observables. An important consequence of
suchflexibility is that different but analytically equivalent
expres-sions will map into operative sums that carry different
nu-merical errors. Among the several strategies that have
beenemployed in the literature to reduce the residual
numericalerrors (see e.g. Brookshaw 1985; Cleary & Monaghan
1999;Colin et al. 2006), the following has proven the more
stableand accurate for the QP case (see Nori & Baldi 2018, fora
comparison between different implementations), and willtherefore be
the one of our choice:
∇ρi =∑
j∈NN(i)
mj∇Wijρj −ρi√ρiρj
(16)
∇2ρi =∑
j∈NN(i)
mj∇2Wijρj −ρi√ρiρj
− |∇ρi|2
ρi(17)
∇Qi =h̄2
2m2χ
∑j∈NN(i)
mjfjρj
∇Wij
(∇2ρj2ρj
−|∇ρj |2
4ρ2j
). (18)
AX-GADGET has undergone various stability tests andhas proven to
be not only less numerically intensive with re-spect to Adaptive
Mesh Refinement (AMR) full-wave solvers(Schive et al. 2010), due to
the intrinsic SPH local approx-imation, but also to be accurate for
cosmologically relevantscales as it agrees both with the linear
(Hlozek et al. 2015)and the non-linear results (Woo & Chiueh
2009) available inthe literature, even if a proper convergence and
code com-parison test has not yet been performed, since it would
benecessary to assess the consistency of different numericalmethods
at very small scales.
In fact, while cosmological and analytical results – ase.g. the
soliton formation – are well recovered by N-Bodysimulations,
interference patterns emerging at very small
scales seem more challenging to be represented accurately,due to
their oscillatory nature that can be overly smoothedif the
resolution – i.e. the number of particles – used is toolow. N-body
simulations at very high-resolution – i.e. to thepc level – have
yet to be performed, but as also argued inmore detail in Appendix
A, it is our opinion that whetherinterference patterns can be
observed or not is ultimately amatter of resolution.
The implementation of FDM physics in AX-GADGET includes the
possibility to simulate Universeswith multiple CDM and FDM species
or FDM particleswith self- or external interactions, as recently
included withthe merging of the AX-GADGET module with the
C-Gadgetmodule of Coupled Dark Matter models (Baldi et al.
2010).
Moreover, AX-GADGET inherits automatically all thelarge
collection of physical implementations – ranging fromgas cooling
and star formation routines to Dark Energy andModified Gravity
implementations – that have been devel-oped for P-GADGET3 by a wide
range of code developers.
All these properties allow to investigate a yet unex-plored wide
variety of extended FDM models and make ofAX-GADGET a valuable tool
– complementary to high res-olution AMR codes – to study the
effects of FDM on LSSformation and evolution. In this work, we
consider the sim-plest non-interacting case with the totality of
the dark mat-ter fluid composed by FDM.
3.2 Simulations
In this work, we performed two sets of simulations, for a
totalnumber of fourteen cosmological runs. The first set consistsin
DM-only simulations used to characterize the small scalestructures
at low redshift – i.e. down to z = 0 –, while thesecond one is
evolved to z = 2 and includes gas particles anda simplified
hydrodynamical treatment, as described in Sec-tion 3.1,
specifically developed for Lyman-α forest analyses(the so-called
"QLYA", or Quick-Lyman-alpha method, seeViel et al. 2004). Both
sets consist in three pairs of sim-ulations, one pair for each
considered FDM mass, evolvedeither including or neglecting the
effect of the QP in the dy-namics – labelling these two cases as
FDM and FDMnoQP,respectively –, in order to to assess and quantify
the entityof such approximation often employed in the
literature.
Both sets of simulations follow the evolution of 5123dark matter
particles in a comoving periodic box with sidelength of 15Mpc,
using 1Kpc as gravitational softening.The mass resolution for the
dark-matter-only simulations is2.2124× 106M�. In all cases we
generate initial conditionsat z = 99 using the 2LPTic code (Crocce
et al. 2006), whichprovides initial conditions for cosmological
simulations bydisplacing particles from a cubic Cartesian grid
followinga second-order Lagrangian Perturbation Theory based
ap-proach, according to a random realisation of the
suppressedlinear power spectrum as calculated by axionCAMB
(Hlozeket al. 2015) for the different FDM masses under
investiga-tion. To ensure a coherent comparison between
simulations,we used the same random phases to set up the initial
con-ditions. In particular, the FDM masses mχ considered hereare
2.5×10−22, 5×10−22 and 2.5×10−21eV/c2, in order tosample the mass
range preferred by the first Lyman-α con-straints in the literature
(see in particular Iršič et al. 2017a;
MNRAS 000, 1–19 (2018)
-
6 M. Nori et al.
Armengaud et al. 2017; Kobayashi et al. 2017), obtainedthrough
N-body simulations with approximated dynamics.
Cosmological parameter used are Ωm = 0.317, ΩΛ =0.683, Ωb =
0.0492 and H0 = 67.27 km/s/Mpc, As =2.20652× 10−9 and ns = 0.9645.
A summary of the simu-lation specifications can be found in Tab.
1.
3.3 Lyman-α forest
The flux power spectrum PF (k,z) is affected both by
astro-physical and cosmological parameters. It is therefore
crucialto accurately quantify their impact in any investigation
in-volving the flux power as a cosmological observable. To thisend,
our analysis is based on a set of full hydrodynamicalsimulations
which provide a reliable template of mock fluxpower spectra to be
compared with observations.
For the variations of the mean Lyman-α forest flux,F̄ (z), we
have explored models up to 20% different thanthe mean evolution
given by Viel et al. (2013).
We have varied the thermal history of the Intergalac-tic Medium
(IGM) in the form of the amplitude T0 andthe slope γ of its
temperature-density relation, generallyparameterized as T = T0(1 +
δ)γ−1, with δ being the IGMover-density (Hui & Gnedin 1997). We
have then consid-ered a set of three different temperatures at mean
density,T0(z = 4.2) = 7200,11000,14800 K, which evolve with
red-shift, as well as a set of three values for the slope of
thetemperature-density relation, γ(z = 4.2) = 1.0,1.3,1.5.
Thereference thermal history has been chosen to be definedby T0(z =
4.2) = 11000 and γ(z = 4.2) = 1.5, providing agood fit to
observations (Bolton et al. 2017). Following theconservative
approach of Iršič et al. (2017a), we have mod-elled the redshift
evolution of γ as a power law, such thatγ(z) = γA[(1 + z)/(1 +
zp)]γ
S
, where the pivot redshift zpis the redshift at which most of
the Lyman-α forest pixelsare coming from (i.e. zp = 4.2 for
MIKE/HIRES+XQ-100).However, in order to be agnostic about the
thermal historyevolution, we let the amplitude T0(z) free to vary
in eachredshift bin, only forbidding differences greater than 5000
Kbetween adjacent bins (Iršič et al. 2017c).
Furthermore, we have also explored several values forthe
cosmological parameters σ8, i.e. the normalisation of thematter
power spectrum, and neff , namely the slope of thematter power
spectrum at the scale of Lyman-α forest (0.009s/km), in order to
account for the effect on the matter powerspectrum due to changes
in its initial slope and amplitude(Seljak et al. 2006; McDonald et
al. 2006; Arinyo-i Pratset al. 2015). We have therefore considered
five different val-ues for σ8 (in the interval [0.754,0.904]) and
neff (in therange [−2.3474,−2.2674]).
We have also varied the re-ionization redshift zrei, forwhich we
have considered the three different values zrei =7,9,15, with zrei
= 9 being the reference value and, finally,we have considered
ultraviolet (UV) fluctuations of the ioniz-ing background, that may
have non-negligible effects at highredshift. The amplitude of this
phenomenon is parameter-ized by the parameter fUV: the
corresponding template isbuilt from a set of three models with fUV
= 0,0.5,1, wherefUV = 0 is associated with a spatially uniform UV
back-ground.
Based on the aforementioned grid of simulations, we
have performed a linear interpolation between the gridpoints in
such multidimensional parameter space, to obtainpredictions of flux
power for the desired models.
We have to note that the thermal history implementa-tion of
Iršič et al. (2017a) and the one used in this workare slightly
different. For this reason, since the simulationsof the grid were
performed without the introduction of theQP in the dynamics, we
mapped our results into the gridones using the ratio between FDM
and FDMnoQP simu-lations. This is, of course, not an exact
procedure but weassume that the ratio of flux power spectrum with
and with-out quantum pressure is relatively insensitive to the
thermalhistory (Murgia et al. 2018).
In order to constrain the various parameters we haveused a data
set given by the combination of intermediateand high resolution
Lyman-α forest data from the XQ-100and the HIRES/MIKE samples of
QSO spectra, respectively.The XQ-100 data are constituted by a
sample of mediumresolution and intermediate signal-to-noise QSO
spectra,obtained by the XQ-100 survey, with emission redshifts3.5≤
z ≤ 4.5 (López, S. et al. 2016). The spectral resolutionof the
X-shooter spectrograph is 30− 50km/s, dependingon the wavelength.
The flux power spectrum PF(k,z) hasbeen calculated for a total of
133 (k,z) data points in theranges z = 3,3.2,3.4,3.6,3.8,4,4.2 and
19 bins in k-space inthe range 0.003− 0.057s/km (see Iršič et al.
2017b, for amore detailed description). MIKE/HIRES data are
insteadobtained with the HIRES/KECK and the
MIKE/Magellanspectrographs, at redshift bins z = 4.2,4.6,5.0,5.4
and in 10k-bins in the interval 0.001−0.08s/km, with spectral
resolu-tion of 13.6 and 6.7km/s, for HIRES and MIKE,
respectively(Viel et al. 2013). As in the analyses by Viel et al.
(2013) andIršič et al. (2017c), we have imposed a conservative cut
onthe flux power spectra obtained from MIKE/HIRES data,and only the
measurements with k > 0.005s/km have beenused, in order to avoid
possible systematic uncertainties onlarge scales due to continuum
fitting. Furthermore, we donot consider the highest redshift bin
for MIKE data, forwhich the error bars on the flux power spectra
are verylarge (see Viel et al. 2013, for more details). We have
thusused a total of of 182 (k,z) data points. Parameter
con-straints are finally obtained with a Monte Carlo MarkovChain
(MCMC) sampler which samples the likelihood spaceuntil convergence
is reached.
3.4 Numerical Fragmentation
For cosmological models whose LSS properties depart sensi-bly
from ΛCDM only at small scales – as FDMmodels – , thethorough
analysis of the statistical overall properties and thespecific
inner structures of haloes represents the most rele-vant and often
largely unexploited source of information.InN-body simulations,
this implies the use of a suitable cluster-ing algorithm to build a
halo catalogue in order to identifygravitationally bound structures
that can then be studied intheir inner structural properties.
In this work, we rely on the SUBFIND routine alreadyimplemented
in P-GADGET3 , a two step halo-finder whichcombines a
Friends-Of-Friends (FoF) algorithm (Davis et al.1985) to find
particle clusters – that defines the primarystructures of our halo
sample – with an unbinding procedureto identify gravitationally
bound substructures within the
MNRAS 000, 1–19 (2018)
-
Lyman-α and LSS properties in FDM cosmologies 7
Table 1. Summary of the properties of the simulations set used
for structure characterization.
Model QP in dynamics mχ [10−22eV/c2] N haloes N genuine haloes
Mcut [1010M�]
LCDM × - 57666 56842 -
FDM-25 X 25 25051 13387 0.04056FDM-5 X 5 10058 2736
0.1645FDM-2.5 X 2.5 8504 1301 0.3151
FDMnoQP-25 × 25 25432 13571 0.04056FDMnoQP-5 × 5 10376 2856
0.1645FDMnoQP-2.5 × 2.5 8819 1374 0.3151
primary haloes (Springel et al. 2001). Hereafter, we use
theterms primary structures to identify the substructures ofeach
FoF group containing the most gravitationally boundparticle,
subhaloes for the non-primary structures and haloeswhen we
generally consider the whole collection of structuresfound.
However, a long-standing problem that affects N-bodysimulations,
when characterized by a sharp and resolved cut-off of the matter
power spectrum, has to be taken into ac-count in the process of
building a reliable halo sample. Thisis the so-called numerical
fragmentation, i.e. the formationof artificial small-mass spurious
clumps within filaments (seee.g. Wang & White 2007; Schneider
et al. 2012; Lovell et al.2014; Angulo et al. 2013; Schive et al.
2016).
While it has been initially debated whether the na-ture of such
fragmentation was to be considered physical ornumerical, the
detailed analysis by Wang & White (2007)showed that in Warm and
Hot Dark Matter simulations (ase.g. Bode et al. 2001) – which are
characterised by a highlysuppressed matter power spectrum – the
formation of smallmass subhaloes was resolution dependent and
related to thelarge difference between force resolution and mean
parti-cle separation (as already suggested by Melott &
Shandarin1989).
To identify spurious haloes in simulations and selecta clean
sample to study and characterize the structures ofFDM haloes in
each simulation, we take cue from the pro-cedure outlined in Lovell
et al. (2014): in particular, we usethe mass at low redshift and
the spatial distribution of par-ticles as traced back in the
initial conditions as proxies forthe artificial nature of haloes as
described below.
In fact, the more the initial power spectrum is sup-pressed at
small scales, the more neighbouring particles arecoherently
homogeneously distributed, thus facilitating theonset of
artificially bounded and small ensembles that even-tually outnumber
the physical ones. As already shown byWang & White (2007), the
dimensionless power spectrumpeak scale kpeak and the resolution of
the simulation – i.e.described through the mean inter-particle
distance d – canbe related together to get the empirical
estimateMlim = 10.1 ρb d / k2peak (19)describing the mass at which
most of the haloes have a nu-merical rather than a physical origin.
In Lovell et al. (2014),this mass is used as a pivotal value for
the mass MCUTused to discriminate genuine and spurious haloes –
lyingabove and below such threshold, respectively – which is setas
MCUT = 0.5Mlim.
In addition to the mass discriminating criterion, Lovell
et al. (2014) showed that particles that generate spurioushaloes
belong to degenerate regions in the initial conditionsand are more
likely to lie within filaments, stating that thereconstructed shape
of the halo particles ensemble in the ini-tial conditions can be
used to identify spurious structures.N-Body initial conditions are
generally designed as regularlydistributed particles on a grid from
which are displaced inorder to match the desired initial power
spectrum. Hence,numerical fragmentation originates mostly from
particles ly-ing in small planar configurations, belonging to the
samerow/column domain or a few adjacent ones.
Therefore, we need a method to quantitatively describethe shape
of subhaloes and of the distribution of their mem-ber particles
once traced back to the initial conditions of thesimulation. To
this end, we resort to the inertia tensor of theparticle
ensemble
Iij =∑
particles
m (êi · êj) |r|2− (r · êi) (r · êj) (20)
where m and r are the particle mass and position respec-tively
and ê are the unit vectors of the reference orthonormalbase. The
eigenvalues and the eigenvectors of the inertiatensor represent the
square moduli and unitary vectors ofthe three axes of the
equivalent triaxial ellipsoid with uni-form mass distribution. We
define a ≥ b ≥ c the moduli ofthe three axes and the sphericity s =
c/a as the ratio be-tween the minor and the major ones: a very low
sphericitywill characterize the typical degenerate domains of
numeri-cal fragmentation.
For these reasons, we use the combined information car-ried by
the mass and the sphericity in the initial conditionto clean the
halo catalogues from spurious ones by apply-ing independent cuts on
both quantities as will be detailedbelow.
In Fig. 1 the mass-sphericity distributions of the differ-ent
simulations are plotted at z = 0 (upper left panel) andat z = 99
(upper right panel) where each point representsa halo identified by
SUBFIND, without applying any selec-tion. Solid and dash-dotted
lines denote the median and the99th percentile of the distribution;
in the side panels we dis-play the cumulative distributions, where
the contribution ofspurious haloes is highlighted in black.
By looking at the two panels, it is possible to notice thatthe
total cumulative sphericity distribution at low redshiftis fairly
model independent, so that distinguishing spurioushaloes from
genuine ones is impossible. However, if we tracethe particle
ensembles of each halo found at z = 0 back tothe initial conditions
at redshift z = 99, using particles ID,
MNRAS 000, 1–19 (2018)
-
8 M. Nori et al.
z = 0 z = 99
mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
LCDM
mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
FDM
-25
mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
FDM
-5
10 1 100 101 102 103mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
10 3 10 2
FDM
-2.5
mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
LCDM
mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
FDM
-25
mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
FDM
-5
10 1 100 101 102 103mass [1010M ]
0.00
0.25
0.50
0.75
1.00
c/a
10 3 10 2
FDM
-2.5
10 2 10 1 100 101 102 103mass [1010M ]
70%
80%
90%
100%
110%
120%
130%
c/a
/c/a
LCD
M
LCDMFDM-25FDM-5FDM-2.5
10 2 10 1 100 101 102 103mass [1010M ]
20%
40%
60%
80%
100%
120%
140%
c/a
/c/a
LCD
M
LCDMFDM-25FDM-5FDM-2.5
Figure 1. Sphericities of all dark matter particles ensembles as
found by SUBFIND as function of their mass (upper panels) at
redshiftsz = 0 and z = 99 (left and right panels, respectively).
The black-shaded area represents the discarded region below the
different masscuts MCUT , corresponding to each model. Each black
dot represents a subhalo and the solid (dot-dashed) lines describe
the median(99th percentile) of the total distribution, which are
all gathered and contrasted with ΛCDM in the lower panel. The total
sphericitydistribution – integrated in mass – is represented in the
side panels where the contribution of the discarded sample to
medians anddistributions are portrayed in black. Lower panels
feature the median of the mass-sphericity distributions, presented
as the ratio withrespect to ΛCDM. The shaded areas, corresponding
to the ±1σ of the distribution, are colour-coded as in the upper
panels. The blackenedmedian and shaded areas represent the excluded
portion of the sphericity distributions below the corresponding
MCUT .
MNRAS 000, 1–19 (2018)
-
Lyman-α and LSS properties in FDM cosmologies 9
and we study the resulting reconstructed mass-sphericity
re-lation, the anomalous component of the distribution associ-ated
with spurious haloes clearly emerges as a low-sphericitypeak, which
is more pronounced for smaller values of theFDM particle mass.
In fact, as the mass mχ decreases, the smoothing ac-tion of the
QP becomes more efficient, inducing homogeneityat larger and larger
scales in the initial conditions and in-creasing, consequently, the
contamination of numerical frag-mentation. It clearly appears that
the population of haloesin the initial conditions is homogeneously
distributed inΛCDM while a bimodal structure emerges at lower
andlower FDM mass. In particular, an increasing number ofhaloes are
located in a small region characterized by lowmass (M . 109M�) and
low sphericity (s. 0.20).
As there is no theoretical reason why the QP shouldfavour the
collapse of ensembles with very low sphericitiesin the initial
conditions with respect to the ΛCDM case, weconsider this second
population as the result of numericalfragmentation.
As in Lovell et al. (2014), we choose to computeMCUT = 0.5Mlim
using Eq. 19 – one MCUT for each valueof the FDM mass, as reported
in Tab. 1 –, that define theupper bound of the discarded mass
regions, i.e. the black-shaded areas in all panels of Fig. 1.
It is interesting to notice that the massesMCUT appearto be very
close to the values at which the sphericity mediansof the
simulation sample – in the initial conditions – departfrom the ones
of ΛCDM, as can be seen in the lower-rightpanel of Fig. 1. As the
MCUT values we obtain are slightlylarger compared to these
departing values, we confirm thechoice of the former over the
latter, as a more conservativeoption for the mass thresholds
dividing spurious from gen-uine haloes.
In Lovell et al. (2014), the selection in terms of
initialsphericity was operationally performed discarding every
halowith sphericity lower than sCUT = 0.16, equal to the
99thpercentile of the distribution of haloes with more than
100particles in the ΛCDM simulation. In our set of simulations,a
similar value denotes the 99th percentile as measured atthe MCUT
mass in each simulations, so we adopt it as ourown threshold in
sphericity. Let us stress that the haloesthat are discarded through
sphericity selection in the initialconditions have sphericities at
z = 0 that are statisticallyconsistent with the genuine sample,
making their numericalorigin impossible to notice based only on the
sphericity dis-tribution at z = 0. However, the mass constraint is
far morerigid than the sphericity one in all models but ΛCDM,
whereno mass limit is imposed.
Finally, in Tab. 2 we have summarized the comparisonof the
number of haloes in the FDMnoQP set-up with re-spect to the
corresponding FDM set-up, presented as theratio of the total number
of haloes found by SUBFIND andthe number of genuine haloes
remaining after the exclusionof spurious ones. It is possible to
see that in the FDMnoQPsimulations, for the three FDM masses
considered, the totalnumber of haloes is overestimated by a factor
∼ 2.5% on av-erage while the genuine haloes excess becomes more
impor-tant as the FDM mass decreases, up to 5.6% for m22 = 2.5.This
means that neglecting the effects of the QP during thesimulation
leads to the formation of haloes which are notpresent when the full
QP dynamics is taken into account
Table 2. The total and genuine number of haloes, presented asthe
ratio between the simulations neglecting and considering theQP
dynamical effects.
mχ [10−22eV/c2] N haloes N genuine haloes
25 101.6% 101.4%5 103.5% 104.4%2.5 103.1% 105.6%
and that, using our à la Lovell et al. (2014) spurious
detec-tion selection, such haloes pass the numerical
fragmentationtest and contaminate any halo statistical property
charac-terization.
3.5 Inter-simulations halo matching
In FDM models, as we said in the previous sections, notonly the
initial power spectrum of matter perturbation issuppressed at small
scales, thereby preventing the forma-tion of small mass structures,
but the dynamical evolutionof density perturbations changes due to
the effect of theQP, intimately affecting the development of
structures dur-ing the whole cosmological evolution by opposing
gravita-tional collapse. The implementation of such effect in
AX-GADGET breaks the one-to-one correspondence of the spa-tial
position of collapsed structures in simulations with dif-ferent FDM
masses – especially for smaller objects –, despitethe identical
random phases used to set up the initial con-ditions.
We indeed expect bigger haloes not to change dramat-ically their
position at low redshift across different simula-tion, while this
is not the case for lighter subhaloes whichare more affected by the
evolving local non-linear balancebetween gravity and the QP of the
environment.
This makes it more difficult to identify matching col-lapsed
objects of common origin across the simulations, tostudy how FDM
models affect the inner structure of haloeson a halo-to-halo
basis.
To this end, we devise an iterative matching procedure,to be
repeated until no more couples are found, as the fol-lowing: given
a halo i at position ri and total mass mi insimulation A,
(i) select all haloes j belonging to simulation B as po-tential
counterparts if |ri− rj |/(ai+aj) < R̃ where ai andaj are the
major axes of the haloes computed through theinertia tensor of all
their member particles.(ii) within the ensemble selected at the
previous point, re-
tain only the haloes k⊆ j whose masses satisfy the
condition|mi−mk|/(mi+mk)< M̃(iii) if more than one halo l ⊆ k is
left, then choose the
one for which |ri−rl|/(ai+al) is minimum.(iv) after having
considered all the haloes in A, if more
than one are linked to the same halo l belonging to B,choose the
couple (i, l) that minimizes [|ri−rl|/(ai+al)]2
+[|mi−ml|/(mi+ml)]2, in order give the same weight to thetwo
criteria.
This method is flexible enough to account for the shiftin mass
and position we expect from simulations with differ-ent FDM mass
models, but conservative enough to ensure
MNRAS 000, 1–19 (2018)
-
10 M. Nori et al.
Table 3. Number of common matches across LCDM and
FDMssimulations, using different values of the parameter M̃
represent-ing the minimum allowed ratio between the minimum and
maxi-mum masses of each candidate couple.
M̃ mmin/mmax N matches
1/39 95% 531/19 90% 1623/37 85% 2341/9 80% 2791/7 75% 3041/3 50%
3463/5 25% 3611 0% 389
the common origin of the subhalo couples. Moreover, usingthe
combination of position and mass filters, we are able
todiscriminate couples in all mass ranges: position filtering
isweaker constraint in the case of bigger haloes – since theyoccupy
a big portion of a simulation – where instead themass filter is
very strict; vice-versa, it is more powerful forsmaller haloes for
which the mass filter select a large numberof candidates.
Operatively, we use the previous procedure to matchhaloes in
each simulation with the ΛCDM one and we referto the subset of
haloes that share the same ΛCDM compan-ion across all the
simulations as the common sample.
For geometrical reasons, we set the limit value for R̃to be 0.5:
this represents the case in which two haloes withthe same major
axis a have centres separated exactly bythe same amount a. The
configurations that are selected bypoint (i) are the ones for which
the distance between the halocentres is less or equal the smallest
major axis between thetwo. A higher value for R̃ would include
genuine small haloesthat have been more subject to dynamical QP
drifting butwould also result in a spurious match of bigger haloes.
Forthese reasons, we adopt R̃ = 0.5, checking that the
selectedsample gains or loses ∼ 5% of components if values 0.45and
0.55 are used, without modifying the overall statisticalproperties
of the sample itself.
With respect to M̃ at point (ii), instead, we appliedthe
matching algorithm using several values, each denotinga specific
threshold of the minimal value allowed for the massratio of halo
couples in order to be consider as a match. Asreported in Tab. 3,
more than 60% of all the matching haloesacross LCDM and FDMs
simulations without mass selection– M̃ = 1 case – have a mass ratio
in the 100−85% ratio rangeand almost 80% in the 100−75% range. In
order not to spoilour matching catalogue, especially with very
close but highlydifferent in mass halo couples, we choose the
limiting valueof M̃ = 1/7.
4 RESULTS
In this Section we present the results obtained from
oursimulations in decreasing order of scale involved, startingfrom
the matter power spectrum, to the simulated Lyman-α forest
observations, to the statistical characterization ofhalo properties
and their density profiles.
4.1 Matter Power spectrum
The relative difference of the matter power spectrum of
thevarious FDM models with respect to ΛCDM at four
differentredshifts is displayed in Fig. 2.
As already found in the literature (see e.g. Marsh 2016a;Nori
& Baldi 2018), the evolution of the matter power spec-trum
shows that the initial suppression – encoded in thetransfer
functions used to build up the initial conditions –is restored at
intermediate scales to the unsuppressed level,eventually, by the
non-linear gravitational evolution.
At the redshifts and scales that are relevant for Lyman-α forest
observations, however, the relative suppression withrespect to ΛCDM
is still important and ranges from 5−20%for the lowest FDM mass
considered.
The relative difference of the matter power spectrum,displayed
in Fig. 3, shows an additional suppression withrespect ΛCDM (by a
factor ≈ 1.15) when the QP is includedin the dynamical evolution
(i.e. in the comparison betweenthe FDM and the FDMnoQP
simulations). This is consistentwith the QP full dynamical
treatment contributing as anintegrated smoothing force that
contrasts the gravitationalcollapse of the otherwise purely
collisionless dynamics.
4.2 Lyman-α forest flux statistics
In order to build our simulated Lyman-α observations we
ex-tracted 5000 mock forest spectra from random
line-of-sightswithin the simulated volume. The spectra are
extracted ac-cording to SPH interpolation and the ingredients
necessaryto build up the transmitted flux are the HI-weighted
pecu-liar velocity, temperature and neutral fraction. Among
thedifferent flux statistics that can be considered, we focus onthe
flux probability distribution function (PDF) and fluxpower
spectrum. Unless otherwise stated we normalize theextracted flux
arrays in order to have the same observedmean flux over the whole
sample considered and for all thesimulations. In any case, we do
find that the scaling factorfor the optical depth arrays over the
whole simulated volumeis 1.6, 1.4 and 1.1 times higher than in the
ΛCDM case inorder to achieve the same mean flux for them22 = 2.5, 5
and25 FDM cases with negligible – between 1−2% – differencesbetween
the FDMs and FDMnoQP cases.
In Fig. 4 we show the flux (top panel) and gas (bottompanel) PDF
ratios between the simulations that include theQP and those that do
not include it – FDM and FDMnoQP,respectively– at z = 5.4, one of
the highest redshift bins inwhich Lyman-α data are available.
It is possible to see that there is a 2−6% peak at flux∼
0.6−0.8, i.e. in regions of low transmissivity that are ex-pected
to trace voids. The fact that FDM simulations dis-play a more
peaked PDF compared to FDMnoQP ones forthis range of fluxes means
that, on average, in those modelsit is more likely to sample such
void environments. In fact,the different PDFs should reflect the
underlying different gasPDFs at the same redshifts and along the
same lines-of-sight.In the bottom panel of Fig. 4, showing the
corresponding gasPDF, it is indeed apparent that in models with
FDMs thegas PDF is more skewed towards less dense regions, that
aretypically associated to high transmission. The effect due tothe
QP is thus to increase the volume filling factor of regions
MNRAS 000, 1–19 (2018)
-
Lyman-α and LSS properties in FDM cosmologies 11
z = 9 z = 5.4
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-40%
-30%
-20%
-10%
0%
P(k)
i/P
(k) LC
DM
1[%
]
FDM-2.5FDM-5FDM-25
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-20%
-15%
-10%
-5%
0%
P(k)
i/P
(k) LC
DM
1[%
]
FDM-2.5FDM-5FDM-25
z = 3.6 z = 1.8
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-12%
-10%
-8%
-6%
-4%
-2%
0%
P(k)
i/P
(k) LC
DM
1[%
]
FDM-2.5FDM-5FDM-25
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-7%
-6%
-5%
-4%
-3%
-2%
-1%
0%
1%P(
k)i/P
(k) LC
DM
1[%
]
FDM-2.5FDM-5FDM-25
Figure 2. Matter power spectrum of FDM models contrasted with
LCDM at different redshifts.
below the mean density with respect to the correspondingFDMnoQP
case.
In Fig. 5 we plot the percentage difference in terms offlux
power spectrum at three different redshifts and for theFDM models,
both compared to ΛCDM (right panels) andto the corresponding
FDMnoQP case (left panels). The in-crease of power at z = 5.4 in
the largest scales – comparedto the ΛCDM case – is due to the
imposed normalization atthe same mean flux, while the evident
suppression at smallscales is related to the lack of structures at
those scales. Thecomparison with the FDMnoQP set-ups, instead,
reveals anadditional suppression which is always below the 5%
levelfor all the masses considered. Since the flux power spectrumis
an exponentially suppressed proxy of the underlying den-sity field,
these results are consistent with the matter powerspectrum results
previously shown in Fig. 2 and Fig. 3.
Since the Lyman-α constraint are calculated by weight-ing the
contribution from all the scales, we expect the boundon m22 found
in Iršič et al. (2017a) to change comparablyto the additional
suppression introduced, that in our case is2−3%.
This is exactly what can be seen in in Fig. 6, wherethe
marginalised posterior distribution of mχ obtained in
the present work is plotted and compared with the
resultspresented in Iršič et al. (2017a). The red line refers to
ourMCMC analysis, whereas the green line corresponds to theresults
obtained by Iršič et al. (2017a). The correspondingvertical lines
show the 2σ bounds on the FDM mass. The2σ bound on the FDM mass
changes from 20.45×10−22 eVto 21.08× 10−22 eV, which matches with
our expectationand confirm that the approximation of neglecting the
QPdynamical effects in Iršič et al. (2017a) was legitimate
toinvestigate the Lyman-α typical scales. The agreement be-tween
the sets of results obtained with and without the dy-namical QP
implementation is evident and is not sensiblyaffected by varying
the assumptions on the IGM thermalhistory.
This result represents – to our knowledge – the firstFDM mass
constraint derived from Lyman-α forest obser-vations that accounts
for the full non-linear treatment ofthe QP, which introduces an
additional – albeit not big –suppression of the matter power
spectrum in the redshiftrange and comoving scales probed by the
Lyman-α forest.The agreement with previous results implies that the
non-linear evolution of the large-scale structure and the non-
MNRAS 000, 1–19 (2018)
-
12 M. Nori et al.
z = 9 z = 5.4
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-6%
-5%
-4%
-3%
-2%
-1%
0%
P(k)
i/P
(k) j
1[%
]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-2%
-1.5%
-1%
-0.5%
0%
P(k)
i/P
(k) j
1[%
]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
z = 3.6 z = 1.8
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-0.7%
-0.6%
-0.5%
-0.4%
-0.3%
-0.2%
-0.1%
0%
P(k)
i/P
(k) j
1[%
]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75log10 k [hMpc 1]
-0.6%
-0.4%
-0.2%
0%
0.2%
P(k)
i/P
(k) j
1[%
]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
Figure 3. Matter power spectrum percentage differences between
FDM simulation and their FDMnoQP counterpart at different
redshifts.The difference in power spectrum suppression of having
the QP in the dynamics result in a multiplication of ∼ 115% factor
of thesuppression with respect to LCDM of Fig. 2 at scales k ∼ 10
hMpc−1.
linear mapping between flux and density effectively makeup for
the additional suppression introduced.
4.3 Structure characterization
The statistical properties of the genuine haloes belonging
toeach simulation are summarised in Fig. 7, where we displaythe
cumulative halo mass function (top right panel), the halomass
outside R200 (top left panel) – where R200 identifiesthe distance
from the halo centre where the density is 200times the critical
density of the Universe andM200 the masscontained within a R200
radius sphere –, the subhalo massfunction (bottom left panel), and
the subhalo radial distri-bution (bottom right panel). In order to
highlight the impactof numerical fragmentation and simplify the
comparison ofthe different models to ΛCDM, relative ratios are
displayedin the bottom panels and shaded lines represent the
distri-bution of the full halo sample, i.e. including also
spurioushaloes.
The analytical fit used by Schive et al. (2016) to param-eterize
the cumulative HMF drop of the FDM models with
respect to ΛCDM
N(>M)FDM =∫ +∞M
∂MNCDM
[1 +(M
M0
)−1.1]−2.2dM
(21)
with M0 = 1.6× 1010m−4/322 M�, are plotted as reference
– one for each FDM mass – in the top left panel of Fig. 7(dotted
lines).
As expected, we find that the number of small masssubhaloes is
drastically reduced in the FDM models andthe cumulative
distributions depart from ΛCDM at higherand higher masses as the mχ
mass decreases. The valuesat which the drop occurs are
approximately 5× 1010M�,2.5×1010M� and 5×109M� for values of m22 of
2.5, 5 and25, respectively: this suggest a linear trend of the
thresholdmass
Mt ' 5×1010M�( 2.5m22
)(22)
describing the approximate mass below which the numberof haloes
starts decreasing with respect to ΛCDM.
MNRAS 000, 1–19 (2018)
-
Lyman-α and LSS properties in FDM cosmologies 13
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8flux
-2.0%
0.0%
2.0%
4.0%
6.0%
P(flu
x)i/P
(flux
) j - 1
[%]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
2.5 2.0 1.5 1.0 0.5 0.0 0.5log(1+ )
-100%
0%
100%
200%
300%
400%
500%
P(ga
s)i/P
(gas
) j - 1
[%]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
Figure 4. Relative differences of the flux PDF (top panel)
andgas PDF (bottom panel) for FDM models with respect to
theircorresponding FDMnoQP counterparts, at redshift z = 5.4.
Looking at the distribution of subhaloes masses as com-pared to
their associated primary halo M200 and the radialdistribution to
R200, it is evident how the numerous smallsubhaloes in ΛCDM, far
from the gravitational centre of themain halo, are the ones that
were not able to form in a FDMuniverse.
The haloes that have masses above Mt not only havebeen able to
survive the disrupting QP action up to red-shift z = 0, but the
cumulative distribution shows how theyalso gained extra mass, at
the smallest (sub)halo expenses.This is confirmed by the cumulative
distribution of the pri-mary structures N(>Mtot−M200),
representing the massaccumulated outside the R200 radius, which is
systemati-cally higher with respect to ΛCDM case as the FDM
masslowers – up to peaks of 200% ratio for the lowest m22 –:this is
consistent with the picture of bigger primary haloesaccreting the
mass of un-collapsed smaller subhaloes thatdid not form.
The fitting function of Eq. 21 is consistent with the scaleof
the drop of the HFM, which is indeed expected to be al-most
redshift independent, since it is predominantly givenby the initial
PS cut-off (Hu et al. 2000). However, it failsto reproduce the data
on two levels: on one hand it does not
recover the slope of the cumulative distribution – especiallyin
the mass range close to Mt where the HMF departs fromΛCDM – and, on
the other hand, does not account for themass transfer from smaller
haloes, unable to collapse dueto QP repulsive interaction, to
bigger ones, that accrete themore abundant available matter from
their surroundings.The discrepancies between the Schive et al.
(2016) fittingfunction and our results are probably due to the fact
thatthe former is based on simulations with approximated
FDMdynamics and evolved only to redshifts z = 4, thus repre-senting
a different collection of haloes that are, moreover, inan earlier
stage of evolution.
Therefore, the analysis of the aggregated data of cu-mulative
distributions of genuine haloes in each simulationlead us to
conclude that formation, the evolution and theproperties of a FDM
halo subject to the real effect of theQP – as compared to the
FDMnoQP approximation – canfollow three general paths depending on
its own mass andon the mass of the FDM boson: if the halo mass isM
�Mt,there is high chance that the halo does not form at all
sincegravitational collapse is prevented by the QP; if M
&Mt,the halo can be massive enough to form but its
propertieswill be affected by the QP – especially on its internal
struc-ture, as we will see below –, while for M �Mt the halois not
severely affected by the QP, and will simply accretemore easily
un-collapsed mass available in its surroundings.
In order to study in more detail the impact of FDMon the halo
properties and structures, we divided our com-mon sample, that by
construction collects the haloes acrossall the simulations that
share the same ΛCDM match (asdescribed in detail in Section 3.5),
in three contiguous massranges. Let us remind that matching haloes
have similar butnot necessarily equal mass, so mass intervals are
to be re-ferred to the ΛCDM halo mass; the other matching
haloesbelonging to the FDM simulations are free to have lower
andhigher mass, compatibly with the limit imposed by the
M̃parameter of the common sample selection procedure. Thecommon
sample low mass end is clearly limited by the FDM-2.5 model, since
it is the one with higher Mt, below whichhaloes have statistically
lower chance to form. The threemass ranges are [0.5− 4], [4− 100],
[100− 4000]× 1010M�,in order to be compatible with the three halo
categories de-scribed in the previous paragraph for the FDM-2.5
model,being Mt(m22 = 2.5)∼ 5×1010M�
For all the matching haloes considered, we have testedthe
sphericity distribution, the halo volume and the totalhalo mass
with respect to ΛCDM, as well as the radial den-sity profiles.
Properties of inter-simulation matching haloes are gath-ered in
Fig. 8, where the total sample is divided column-wisein the three
mass ranges. The sphericity, the volume occu-pied and the total
mass of the haloes – contrasted with thecorresponding ΛCDM match –
are shown in the first row(left panels), together with related
distribution functions(right panels). The second and the third row
represent theoverall density profiles, stacked in fractional
spherical shellsof R200 and ellipsoidal shells of the major axis a
– identifiedwith the vertical dashed lines –, respectively, .
Density pro-files are divided by the value of the density
calculated withinthe R200 and a shells and are shown both in
absolute value(top panels) and relatively to ΛCDM (bottom
panels).
MNRAS 000, 1–19 (2018)
-
14 M. Nori et al.
z = 5.4
0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]
-100%
-80%
-60%
-40%
-20%
0%
20%
40%
P(k)
i/P
(k) LC
DM
1[%
]
FDM-2.5FDM-5FDM-25
0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]
-6%
-5%
-4%
-3%
-2%
-1%
0%
1%
2%
P(k)
i/P
(k) j
1[%
]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
z = 4.0
0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]
-80%
-70%
-60%
-50%
-40%
-30%
-20%
-10%
0%
10%
P(k)
i/P
(k) LC
DM
1[%
]
FDM-2.5FDM-5FDM-25
0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]
-4%
-3%
-2%
-1%
0%
1%
P(k)
i/P
(k) j
1[%
]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
z = 3.0
0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]
-20%
-15%
-10%
-5%
0%
5%
10%
15%
P(k)
i/P
(k) LC
DM
1[%
]
FDM-2.5FDM-5FDM-25
0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2log10 k [hMpc 1]
-2.5%
-2%
-1.5%
-1%
-0.5%
0%
0.5%
1%
1.5%
P(k)
i/P
(k) j
1[%
]
FDM-2.5 / FDMnoQP-2.5FDM-5 / FDMnoQP-5FDM-25 / FDMnoQP-25
Figure 5. Flux power spectrum comparison between all simulations
and LCDM (left panels), and between FDM simulation and theirFDMnoQP
counterparts (right panels) at different redshifts.
MNRAS 000, 1–19 (2018)
-
Lyman-α and LSS properties in FDM cosmologies 15
10−3 10−2 10−1
1/mχ [1022/eV]
Pro
bab
ility
dis
trib
uti
on
FDM noQP
FDM
Figure 6. Here we plot the marginalised posterior distributionof
1/mχ from both the analyses performed by Iršič et al. (2017a)(green
lines, without QP) and ours (red lines, with QP). Thevertical lines
stand for the 2σ C.L. limits.
The sphericity distributions confirm that, in the massrange
considered, there is no statistical deviation fromΛCDM, except for
a mild deviation towards less sphericalconfigurations of the less
massive haloes, especially in them22 = 2.5 model. This is
consistent with the analysis of thesphericity distributions of the
genuine samples (see lowerpanels in Fig. 1) that reveals that
haloes appear to be sta-tistically less spherical with respect to
ΛCDM at z = 0 whenlower FDM masses are considered, down to a
maximum of∼ 10% decrease in sphericity for m22 = 2.5 and halo mass
of∼ 5×109M�.
For all the FDM models the volume occupied by thehaloes is
systematically larger, consistently with a delayeddynamical
collapse of the haloes. All mass ranges show suchproperty and it is
emphasized by lower m22 mass – i.e.stronger QP force –; however,
while bigger haloes occupy al-most systematically 20% more volume
form22 = 2.5, smallerhaloes can reach even twice the volume
occupied by theirΛCDM counterparts when the same model is
considered.
Comparing the mass of the haloes in the various modelswith the
one in ΛCDM, it is possible to see that small haloesare less
massive and big ones, on the contrary, become evenmore massive,
confirming our hypothesis of mass transferfrom substructures
towards main structures.
The stacked density profiles provide even more insighton the
underlying different behaviour between the chosenmass ranges.
Starting from the less massive one, the stackedprofiles look very
differently if plotted using the sphericalR200-based or the
ellipsoidal a-based binning. This is dueto two concurrent reasons
related to the properties of thismass range: first of all, as we
said before, the sphericityis mχ dependent and thus it is not
constant with respectto ΛCDM, so the geometrical difference in the
bin shapebecomes important when different models are
considered;secondly, since the FDM haloes have lower mass but
occupylarger volumes, the two lengths are different from each
other– being R200 related to density and a purely to geometry –so
that the actual volume sampled is different. Nevertheless,it is
possible to see that in FDM models there is an excess ofmass in the
outskirts of the halo – seemingly peaking exactlyat distance a –
and less mass in the centre.
The intermediate mass range shows also a suppressionin the
innermost regions but a less pronounced over-density
around a as expected, since the effectiveness of the
repulsiveforce induced by the QP in tilting the density
distributiondecreases as its typical scale becomes a smaller
fraction ofthe size of the considered objects. In fact, stacked
densityprofiles of the most massive haloes are very similar in the
twobinning strategies, being R200 ∼ a and sphericity constantamong
the various models, and consistent with no majordeviation from
ΛCDM, except for a central over-density. Itis our opinion, however,
that such feature in the very centreof most massive haloes could be
a numerical artefact, sinceits extension is comparable with the
spatial resolution used.
The results presented in this Section have been obtainedthrough
the detailed analysis of the statistical properties ofhaloes found
at z = 0 in the FDM simulations. The sameanalysis, repeated at z =
0, of the FDMnoQP simulationsshows very similar results which are,
therefore, not shown inthe present work. Such consistency suggests
that the proper-ties of haloes at low redshift are – at the
investigated scales –not sensible to modifications induced by the
dynamical QPrepulsive effect, which are expected to appear more
promi-nently at scales of ∼ 1Kpc with the formation solitonic
cores.
5 CONCLUSIONS
We have presented the results obtained from two sets of
nu-merical simulations performed with AX-GADGET , an exten-sion of
the massively parallel N-body code P-GADGET3 fornon-linear
simulations of Fuzzy Dark Matter (FDM) cos-mologies, regarding
Lyman-α forest observations and thestatistical detailed
characterization of the Large Scale Struc-tures.
More specifically, our main aim was to design a set
ofsimulations covering the typical scales and redshifts involvedin
Lyman-α forest analyses, in order to extract synthetic
ob-servations, compare them with available Lyman-α data, andfinally
to place a constraint on the mass of the FDM parti-cle. In the
literature, Lyman-α forest was already used forthis purpose but
only in approximated set-ups, in which thequantum dynamical
evolution of FDM was only encoded inthe initial conditions transfer
function, and neglected dur-ing the simulation (Iršič et al. 2017a;
Armengaud et al. 2017;Kobayashi et al. 2017), while the AX-GADGET
code allowsus to drop such approximation and take into account
thenon-linear effects of full FDM dynamics.
The constrain the FDM mass we find is 21.08 ×10−22 eV, which is
3% higher with respect to what was foundin Iršič et al. (2017a).
The fact that these two bounds aresimilar, despite the different
dynamical evolution consideredin these different works, implies
that the additional suppres-sion deriving from the Quantum
Potential dynamical con-tribution, at the scales and redshifts
probed by Lyman-α ,is compensated by the gravitational growth of
perturbationswhen these enter the non-linear regime, implying also
that –even if the QP does play a role in the Large Scale
Structureevolution – the approximation of Iršič et al. (2017a)
(alsoadopted by Armengaud et al. 2017; Kobayashi et al. 2017)is
valid and sufficient at these scales.
Secondly, we studied in detail the statistical propertiesof the
Large Scale Structures through the analysis of theaggregated data
on haloes regarding their mass, volumesand shapes, as well as their
individual inner structure.
MNRAS 000, 1–19 (2018)
-
16 M. Nori et al.
101
102
103
104
Nha
lo(>
Mha
lo)
LCDMSchive16-25FDM-25Schive16-5
FDM-5Schive16-2.5FDM-2.5
10 2 10 1 100 101 102 103Mhalo [1010M ]
0%
25%
50%
75%
100%
125%
N/N
LCD
M
101
102
103
104
Nha
lo(>
Mha
loM
200) LCDMFDM-25
FDM-5FDM-2.5
10 3 10 2 10 1 100 101 102Mhalo M200 [1010M ]
0%
100%
200%
N/N
LCD
M
100
101
102
103
104
Nsu
bhal
o(>
Msu
bhal
o/M20
0)
LCDMFDM-25
FDM-5FDM-2.5
10 5 10 4 10 3 10 2 10 1 100Msubhalo/M200
0%
20%
40%
60%
80%
100%
N/N
LCD
M
100
101
102
103
104
Nsu
bhal
o(>
R sub
halo
/R20
0) LCDMFDM-25
FDM-5FDM-2.5
10 1 100Rsubhalo/R200
0% 20% 40% 60% 80%100%
N/N
LCD
M
Figure 7. Properties of the halo and subhalo samples at z = 0,
with (dashed lines) and without (solid lines) including the haloes
markedas spurious as described in Sec. 3.4. In particular, the
cumulative distributions of halo mass (top left panel), the halo
mass outside R200(top right panel), the subhalo-halo relative mass
(bottom left panel) and the subhalo-halo distance (bottom right
panel) are displayed.The fitting functions of the cumulative halo
mass distribution of (Schive et al. 2016) of Eq. 21 are plotted for
reference – dotted line inthe top left panel –.
The main results regarding the effects of FDM on LSSthat we
found can be summarized as follows:
• the FDM particle mass m22 defines a typical mass scaleMt '
1.25×1011/m22 M� characterising the halo distribu-tion of different
FDM models; all halo properties can beinterpreted within the
framework of having two families ofhaloes: the small ones with M
.Mt, and the big ones withM �Mt (since the very small ones M �Mt do
not form atall);• small haloes, according to the above definition,
show
outward tilted profiles and a lower total mass, are less
spher-ical and more voluminous, so less dense overall;• big haloes
instead are almost unaffected in their internal
structure – apart from the expected solitonic inner coresthat we
cannot resolve with our simulations –, they occupya larger volume
and they also have higher total mass, mostlyaccreted outside R200,
compatible with the collection of thesubhaloes mass that were not
able to form
To conclude, we have performed for the first time a suiteof
hydrodynamical simulations of a statistically significantvolume of
the universe for Fuzzy Dark Matter models fea-
turing a fully consistent implementation of the QuantumPotential
effects on the dynamical evolution of the system.These simulations
allowed to perform for the first time afully consistent comparison
of mock Lyman-α observationswith available data and to update
existing constraints onthe allowed FDM mass range. As the new
constraints arenot significantly different from previous ones, this
representsthe first direct validation of the approximations adopted
inprevious works. Furthermore, our large halo sample allowedus to
perform an extensive characterisation of the proper-ties of dark
matter haloes in the context of FDM scenarios,highlighting the
typical mass scale below which FDM effectsstart to appear. Higher
resolution simulations will soon al-low us to explore even smaller
scales where we expect toobserve the formation of solitonic
cores.
ACKNOWLEDGEMENTS
MN and MB acknowledge support from the Italian Min-istry for
Education, University and Research (MIUR)through the SIR individual
grant SIMCODE, project num-
MNRAS 000, 1–19 (2018)
-
Lyman-α and LSS properties in FDM cosmologies 17
[0.5 − 4] × 1010M� [4 − 100] × 1010M� [100 − 4000] × 1010M�
75%
100%
125%
150%
175%
c/a
/c/a
LCD
M
M [1010M ]
100%
200%
300%
V/V
LCD
M
100M [1010M ]
80%
100%
120%
140%
M/M
LCD
M
0% 50% 100%
FDM-25 FDM-5 FDM-2.5
75%
100%
125%
150%
175%
c/a
/c/a
LCD
M
M [1010M ] 50%
100%
150%
200%
250%
300%
V/V
LCD
M
101 102M [1010M ]
80%
100%
120%
140%
M/M
LCD
M
0% 50% 100%
FDM-25 FDM-5 FDM-2.5
80%
90%
100%
110%
c/a
/c/a
LCD
M
M [1010M ]
100%
120%
140%
V/V
LCD
M
102 103M [1010M ]
90%
95%
100%
105%
110%
115%
M/M
LCD
M
0% 50% 100%
FDM-25 FDM-5 FDM-2.5
10 1
100
101
/20
0
LCDMFDM-25
FDM-5FDM-2.5
10 1 100R/R200
60%
80%
100%
120%
140%
/LC
DM
Stacked densities of 105 subhaloes, binned in spherical
shells
10 1
100
101
102
/20
0
LCDMFDM-25
FDM-5FDM-2.5
10 1 100R/R200
80%
90%
100%
110%
/LC
DM
Stacked densities of 186 subhaloes, binned in spherical
shells
10 1
100
101
102
/20
0
LCDMFDM-25
FDM-5FDM-2.5
10 2 10 1 100R/R200
80%
100%
120%
140%
/LC
DM
Stacked densities of 13 subhaloes, binned in spherical
shells
10 2
10 1
100
101
102
103
/a
LCDMFDM-25
FDM-5FDM-2.5
10 2 10 1 100R/a
40%
60%
80%
100%
120%
/LC
DM
Stacked densities of 105 subhaloes, binned in ellipsoidal
shells
10 210 1100101102103104
/a
LCDMFDM-25
FDM-5FDM-2.5
10 2 10 1 100R/a
60%
80%
100%
/LC
DM
Stacked densities of 186 subhaloes, binned in ellipsoidal
shells
10 210 1100101102103104
/a
LCDMFDM-25
FDM-5FDM-2.5
10 2 10 1 100R/a
80%
100%
120%
140%
/LC
DM
Stacked densities of 13 subhaloes, binned in ellipsoidal
shells
Figure 8. Properties of inter-simulation matching haloes. The
total sample is divided column-wise in three mass ranges. The
sphericity,the volume occupied and the total mass of the haloes –
contrasted with the corresponding ΛCDM match – are shown in the
first row(left panels), together with related distribution
functions (right panels). The second and the third row represent
the overall densityprofiles, stacked in fractional spherical shells
of R200 and ellipsoidal shells of the major axis a – identified
with the vertical dashed lines –,respectively. Density profiles are
divided by the value of the density calculated within R200 and a
and are shown both in absolute value(top panels) and relatively to
ΛCDM. (bottom panels)
ber RBSI14P4IH. The simulations described in this workhave been
performed on the Marconi supercomputer atCINECA thanks to the PRACE
allocation 2016153604. MVand RM are supported by INFN I.S.
PD51-INDARK.
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