Effects of Unstable Dark Matter on Large-Scale Structure and Constraints from Future Surveys

1.7 MB·English

by  Mei-Yu Wang

Effects of Unstable Dark Matter on Large-Scale Structure and Constraints from
Future Surveys
Mei-Yu Wang and Andrew R. Zentner
Department of Physics and Astronomy & Pittsburgh Particle physics,
Astrophysics, and Cosmology Center (PITT PACC),
The University of Pittsburgh, Pittsburgh, PA 15260 ∗
(Dated: January 13, 2012)
In this paper we explore the effect of decaying dark matter (DDM) on large-scale structure and
possible constraints from galaxy imaging surveys. DDM models have been studied, in part, as a
waytoaddressapparentdiscrepanciesbetweenthepredictionsofstandardcolddarkmattermodels
2
and observations of galactic structure. Our study is aimed at developing independent constraints
1
on these models. In such models, DDM decays into a less massive, stable dark matter (SDM)
0
particle and a significantly lighter particle. Thesmall mass splitting between theparent DDMand
2
thedaughter SDMprovides theSDM with a recoil or “kick” velocity vk, inducing a free-streaming
n suppressionofmatterfluctuations. Thissuppressionmaybeprobedviaweaklensingpowerspectra
a measuredbyanumberofforthcomingimagingsurveysthataimprimarilytoconstraindarkenergy.
J
Using scales on which linear perturbation theory alone is valid (multipoles ℓ < 300), surveys like

1 Euclid or LSST can be sensitive to vk >∼ 90 km/s for lifetimes τ ∼ 1−5 Gyr. To estimate more
1 aggressiveconstraints,wemodelnonlinearcorrectionstolensingpowerusingasimplehaloevolution
modelthatisingoodagreementwithnumericalsimulations. Inourmostambitiousforecasts,using
O] multipoles ℓ < 3000, we find that imaging surveys can be sensitive to vk ∼ 10 km/s for lifetimes
τ <∼ 10 Gyr. Lensing will provide a particularly interesting complement to existing constraints in
C that they will probe the long lifetime regime (τ ≫H−1) far better than contemporary techniques.
0
. Acaveat totheseambitious forecasts isthat theevolution of perturbationson nonlinear scales will
h
needtobewellcalibratedbynumericalsimulationsbeforetheycanberealized. Thisworkmotivates
p
the pursuit of such a numerical simulation campaign to constrain dark matter with cosmological
-
o weak lensing.
r
t
s PACSnumbers: 95.35.+d,98.80.-k,98.62.Gq,
a
[

I. INTRODUCTION and the dark matter halos in which they reside. Never-
1
theless, these potential shortcomingsof CDM may point
v
6 towardnovelpropertiesofdarkmatterandmanyalterna-
Many and various astronomical observations indicate
2 tivestoCDMhavebeenconsidered,includingwarmdark
that 5/6 of the mass density of the Universe is non-
4 ∼ matter (WDM) [10–14] and self-interacting dark matter
baryonic dark matter (reviews include Refs. [1–3]). The
2 (SIDM) [15–17].
. simplest model of so-calledcold dark matter (CDM) can
1
be successfully appliedto interpretanenormousamount As one alternative to CDM, unstable dark matter has
0
of observational data, particularly those characterizing been considered in a number of recent studies [15, 17–
2
1 the large-scale (> a few Mpc) structure of the Universe 25]. In such models, a dark matter particle of mass
: and the gross pr∼operties of galaxies. In particular, the M decays into a less massive daughter particle of mass
v
CDM model is consistent with the cosmic microwave m = (1 f)M and a significantly lighter, relativistic
i −
X background (CMB) anisotropy spectrum measured by particle, with a lifetime on the order of the age of the
r the Wilkinson Microwave Anisotropy Probe (WMAP) Universe. In decaying dark matter models, cosmolog-
a and observations of the large-scale (k < 0.1 h/Mpc) ical structure growth is altered in a time- and scale-
galaxyclusteringspectrummeasuredbyth∼eSloanDigital dependent manner [20, 26]. We explored models with
SkySurvey(SDSS)[4]. Onsmallerscales,thesituationis f 1 in a previous paper [26]. If f 1, the sta-
≃ ≪
murkier. Several observations indicate possible discrep- ble dark matter (SDM) daughter particle will receive a
anciesbetweenCDMtheoryandobservationsonsmaller non-relativistic kick velocity, vk fc. This kick can be
≃
scales. Amongthesearethewell-knownmissingsatellites sufficient to alter small-scale structure growth, modify-
problem[5,6]andthesteeprotationcurvesoflow-surface ing the predicted structures and abundances of the dark
brightnessgalaxies[7–9]. Exploringsmall-scalestructure matter halos that host galaxies. Previous studies have
ischallengingbothobservationallyandtheoretically. On explored just this possibility and placed limits on these
the theoretical side, it is necessary to model highly non- scenariosby demanding that the alterationsto structure
linear phenomena to predict the properties of galaxies growth not be so severe as to destroy the successes that
CDM theory enjoys on large scales. An upper bound
on the lifetime of the decaying dark matter (DDM) par-
ticle of τ > 30 40 Gyr for v > 100 km/s can be de-
k
∗Electronicaddress: [email protected],[email protected] rivedfrom∼theo−bservedgalaxy-cl∼ustermassfunctionand
2
galaxy mass-concentration relation [15, 17]. Lifetimes Lyman-α forest, in a forthcoming follow-up paper.
τ < 30 Gyr with 20 km/s < v < 200 km/s may be in In the following sections, we will show that the im-
k
ten∼sion with the observed p∼roper∼ties of the Milky Way printofDDMonlensingpowerspectraissufficientlydis-
satellite galaxy population, but uncertainties associated tinct from other cosmological parameters, such as neu-
with the details of nonlinear structure growth in these trinomass,thatonecandisentangledegeneraciesamong
models as well as the star formation histories of Local them. Indeed,forthcomingdatawillbeabletodisentan-
Group galaxies are significant [22]. Significantly tighter glethetwoparametersofDDMmodels,lifetimeandkick
constraints can be obtained if the daughter particles are velocity (or equivalently f). Finally, large-scale lensing
standard model particles [23]. surveys will provide, at minimum, competitive and in-
With a number of forthcoming large-scale galaxy sur- dependent constraints on DDM models exploiting only
veys being undertaken this decade, such as the Dark scales on which linear cosmological perturbation theory
EnergySurvey(DES)1,the PanoramicSurveyTelescope is appropriate. These probes will have the particular
andRapidResponseSystem(PanSTARRS)2,theLarge advantage of probing significantly larger DDM lifetimes
Synoptic Survey Telescope (LSST)3 [27], Euclid4 [28], than small-scale structure studies and will not require
and WFIRST5, it will be possible to investigate possi- detailedmodeling ofgalaxyformation. Moreover,arela-
ble subtle imprints of decaying dark matter on the rel- tively modest numerical simulation programmay enable
atively large-scale structure of the Universe. In DDM onetousemildlynonlinearscalesk <1h/Mpctoobtain
models, the kick velocity at decay imparts upon the sta- constraintsonDDM that exceedcur∼rentconstraintsand
bledaughterparticlestheabilitytosmoothgravitational arerobusttouncertaintiesassociatedwithstarformation
potentialperturbationsonscalessmallerthantheclassic and feedback in small galaxies.
free-streaming scale. This behavior is similar to the cos- The outline of this paper is as follows. In II, we de-
§
mological influence of massive neutrinos or WDM, and scribe cosmological weak lensing observables. In III,
§
numerousstudies haveshownthatthesefeaturesmaybe we discuss the perturbation evolution of both the par-
detectableinlarge-scalestructurethroughgalaxycluster- ent and daughter particles in DDM models. In IV,
§
ing [29], Lyman-α forest [30–32], and cosmological weak we study the effect of DDM on dark matter halo den-
lensing [33–35]. sity profiles and explore possible influences of DDM on
nonlinear structure. In V, we describe the methods we
In this paper, we explore the effect of DDM model
§
use to compute constraints on DDM model parameters.
on large-scale structure and study the possible indepen-
We present our results in VI, beginning with the gen-
dent constraintsonthese models fromforthcomingweak
§
eral effects of DDM on large-scale structure, including a
lensingsurveys. Therearedistinctadvantagestothisap-
detailed discussion of the free-streaming of the daughter
proach. First,DDMmodelsareexploredlargelyinorder
SDMparticles. We alsogiveourforecastsforconstraints
to mitigate the possible small-scale problems of CDM,
onDDM fromlarge-scaleweak lensing surveysandcom-
so it is necessaryto exploreindependent predictions and
parepossiblefutureboundswithexistinglimits. In VII
probesinordertotestsuchmodels. Second,theeffectsof
§
we summarize our work.
DDMonlarge-scalestructure,andonweaklensingpower
spectra in particular, are simpler to model in that they
do notrequiredetailedknowledgeofgalaxyformationin
II. WEAK GRAVITATIONAL LENSING
the highlynonlinearregimeincludingstarformationhis-
OBSERVABLES
tories, star formation and active galaxy feedback, scale-
dependent galaxy clustering bias and numerous other
Weak lensing as a cosmological probe has been dis-
complicatedphenomenathat areknownto be important
cussed at length in numerous papers (a recent review is
on small scales. Third, many surveys already have the
Ref.[37]). Wegiveabriefdescriptionofourmethodsbe-
goal of measuring cosmological weak lensing as a probe
low, which are based on the conventions and notation in
of darkenergy [36], so this test canbe performedlargely
Ref. [38]. The most robust forecasts derive from consid-
with the observationalinfrastructure used to study dark
erationsofpossibleweaklensingmeasurementsrestricted
energyatnoadditionalcost. Lyman-αforestspectrapro-
toscaleswherelinearperturbativeevolutionofthemetric
videanadditional,promisingmethodtoconstrainDDM,
potentials remains useful. However, we attempt to esti-
but also introduce complications associated with model-
matepossibleimprovementstotheconstrainingpowerof
ingtheclusteringofneutralhydrogenalonglinesofsight
weaklensing observables,providedthatmildly nonlinear
to high redshift. We will present Lyman-α constraints
evolution can be modeled robustly.
on DDM, and forecasts for future measurements of the
We consider the set of observables that may be avail-
able from large-scale galaxy imaging surveys to be the
auto- and cross-spectra of lensing convergence from sets
1 http://www.darkenergysurvey.org of galaxies in NTOM redshift bins. The NTOM(NTOM +
2 http://pan-starrs.ifa.hawaii.edu/ 1)/2 distinct convergence spectra are
3 http://www.lsst.org
4 http://sci.esa.int/euclid Pij(ℓ)= dzWi(z)Wj(z)P (k =ℓ/D ,z), (1)
5 http://wfirst.gsfc.nasa.gov κ H(z)D2(z) m A
Z A
3
where i and j label the redshift bins of the source galax- where γ2 isthenoisefromintrinsicellipticitiesofsource
h i
ies. We take N = 5 and evenly space bins in red- galaxies, and n is the surface density of galaxies in the
TOM i
shift from a minimum redshift of z = 0 to a maximum ith tomographic bin. We follow recent convention and
redshift of z = 3. Increasing the number of bins beyond set γ2 = 0.2, subsuming additional errors on galaxy
h i
N =5addsonlynegligiblytotheconstrainingpower shape measurements into aneffective meannumber den-
TOM
p
of lensing data, in accord with an analogous statement sity of galaxies, n¯. Assessments of intrinsic shape noise
for dark energy constraints [39]. per galaxy may be found in, for example [27, 46, 47].
In Eq. (1), H(z) is the Hubble expansion rate, D (z) AssumingGaussianityofthelensingfield,thecovariance
A
is the comovingangulardiameter distance, andP (k,z) between observables P¯ij and P¯kl is
m κ κ
is the matter powerspectrumat wavenumberk andred-
shiftz. Inthefollowingsection,wedescribeouruseofthe CAB =P¯κikP¯κjl+P¯κilP¯κjk (7)
publicly-available CMBFASTcode to calculate P (k) from
m
cosmological perturbation evolution. In this case it will where the i and j map to the observable index A, and
be more natural to work in the synchronous gauge, and k and l map to B such that CAB is a square covariance
transformingbetween different gaugesystems canbe ac- matrix with NTOM(NTOM+1)/2 rowsand columns. We
complished straightforwardly by following, for example, assume Gaussianity throughout this work and even in
the methods described in Ref. [40]. ourmostaggressiveforecasesweconsideronlymultipoles
The W are the lensing weight functions for source ℓ < 3000, at which point the Gaussian assumption and
i
galaxiesinredshiftbini. Inpractice,thegalaxieswillbe severalweaklensingapproximationsbreakdown[48–52].
binnedbyphotometricredshift,sothatthebinswillhave
non-trivial overlap in true redshift (see Refs. [39, 41] for
detaileddiscussions). Definingthe trueredshiftdistribu- III. DECAYING DARK MATTER MODELS
tionofsourcegalaxiesintheithphotometricredshiftbin
as dn /dz, the weights are A. Evolution of the Average Properties of
i
Unstable Dark Matter
3 D (z,z′)dn
W (z)= Ω H2(1+z)D (z) dz′ A i (2)
i 2 M 0 A D (z′) dz′
Z A We consider dark matter decays into another species
whereD (z,z′)istheangulardiameterdistancebetween of stable dark matter with a small mass splitting,
A
redshift z and z′ and H is the present Hubble rate. DDM SDM+L,whereLdenotesa“massless”daugh-
0 →
We model the uncertainty induced by utilizing photo- terparticle,SDMisthestabledarkmatterwithmassm,
metric galaxy redshifts with the probability function of andDDMisthedecayingdarkmatterwithmassM. The
assigning an individual source galaxy photometric red- masslossfractionfofDDMisdirectlyrelatedtothekick
sdhisifttrizbpugtiiovnenofastoruurecreesdisnhtihftezi,thP(pzhpo|zto).mTethreictrruedesrheidftshbiifnt veneleorcgiyty-mdoempoesnittuedmtcootnhseerSvDatMionp.arTthicelefoblylofwi≃ngvkre/lcatfiroonms
is arevalidintherestframeofDDMparticleswiththekick
velocity of SDM being the velocity relative to the DDM
dni(z) zp(h,iigh) dn(z) rest frame.
= dz P(z z) (3)
dz Zzp(l,oiw) p dz p| anFdoirnvaegrseenedreacladyse,catyh,enreagtleecotfincghaPnaguelii-nbltohcekiDngDfMactdoirss-
Here we take the true redshift distribution to be tribution function is
dnd(zz) =n¯ 42zπ2z3 exp[−(z/z0)2] (4) f˙DDM(qDDM)=−EaMΓ fDDM(qDDM), (8)
0 DDM
with z0 ≃ 0.92, so thapt the median survey redshift to where f˙ denotes the partial derivative of the distribu-
z = 1, and n¯ as the total density of source galaxies
med tion function with respectto conformaltime, dτ =dt/a,
per unit solid angle [42–44]. We assume that uncertain
Γ is the decay rate, a is the cosmological scale fac-
photometric redshifts can be approximated by taking
tor, q is the comoving momentum, and E =
DDM DDM
1 (z z)2 q2 +M2a2. Specializing to two-body decays, one
p DDM
P(zp|z)= √2πσz exp(cid:20)− 2−σz2 (cid:21) (5) ctprainbusthioown tfuhnactttihonecwoirlrlebspeo[2n0d,in5g3]changetotheSDMdis-
whereσ (z)=0.05(1+z)[39]. Complexityinphotomet-
z
ric redshift distributions is an issue that must be over- aM2Γ Ef
f˙ (p )= dEf (p),
come to bring weak lensing constraints on cosmology to SDM SDM 2E p p DDM
fruition (e.g., Ref. [41, 45]). SDM SDM CM ZEi (9)
Observed convergence power spectra P¯κij(ℓ), contain where
both signal and shot noise,
1
P¯κij(ℓ)=Pκij +niδijhγ2i (6) Ef,i = 2ESDMm20±pSDMpCMM/m2SDM,
4
the quantity p is the center-of-mass momentum, and respectively. Given the DDM energy density, the decay
CM
m2 M2+m2. productenergydensityρ =ρ +ρ canbeobtained
0 ≡ d SDM L
We define the average distribution function, f0(q,τ), using the first law of thermodynamics [54, 55] from
i
and the perturbation to the distribution function,
Ψi(~x,~q,τ), for each species of particle labeled by i, ac- da3ρd = P da3 d(a3ρDDM). (18)
d
cording to dτ − dτ − dτ
This implies that the energy density evolution of the
f (~x,~q,τ)=f0(q,τ)[1+Ψ (~x,~q,τ)] (10)
i i i massless daughter particle L is
where i can be the DDM, SDM, or L. Since DDM parti- a˙ a˙ a(M2 m2)
clesarenon-relativistic,theirzeroorderphase-spacedis- ρ˙L+3a(ρL+PL)=ρ˙L+4aρL =Γ 2M−2 ρDDM
tribution is the Maxwell-Boltzmann function. The zero (19)
orderphase-spacedistributionfunctionofSDMis[20,25]
ΓΩMρcrit B. Perturbations
f (q,a)= exp( Γt )Θ(ap q) (11)
0,SDM Mq3H(a′) − q CM−
To compute the contemporary lensing power spectra,
whereq isthecomovingmomentumoftheSDMparticle,
it is necessary to compute the perturbations to the dark
a′ = q/p , and t = t(a′). This can be derived from
CM q matter distributions and the metric. Our treatment of
the fact that the decay always generates SDM particles
perturbations follows the conventions established in Ma
with the same physical momentum p . In the SDM
CM and Bertschinger [56]. We will present our results in
distributionfunction,the spectrumofdifferentmomenta
the synchronous gauge, because this choice lends itself
arises from decays at different times, designated by the
to numerical evaluation. In the synchronous gauge, the
cosmic scale factor a′ so that q = p a′. The Heavi-
CM FouriertransformoftheBoltzmannequationcanbewrit-
side step function Θ(ap q) (see Eq. 11) enforces a
CM − ten
cut-off q = ap at a given redshift a. This maxi-
max CM
mummomentum stems fromthe factthatthe maximum ∂Ψ q dlnf h˙ +6η˙ 1 ∂f
momentum at a given redshift is from decay processes +i (~k nˆ)Ψ+ 0 η˙ (kˆ nˆ)2 =
happeningatthattime, whileSDMwithlowermomenta ∂τ E · dlnq " − 2 · # f0 (cid:18)∂τ(cid:19)C
(20)
are from the earlier decays. To be explicit, the average
The DDM perturbation equations are the same as the
comovingnumberdensityofSDMparticlesistheintegral
well-knownequations describing CDM (see Ref. [56]), so
of f over momentum space,
0
we will not describe them any further. The term on the
right-handsideofEq.(20)istheso-calledcollisionalterm
n = q2dqdΩf (q) (12)
SDM 0,SDM representingthe changeinthedistributionfunctions due
Z to interactions (decays in our case). In the absence of
dn (q)=4πq2dqf (q) (13)
→ SDM 0,SDM non-gravitational interactions, (∂f/∂τ)C = 0. For the
decay products, the collision terms are
Thus f can be written as
0,SDM
∂f am2Γ
f (q)= dnSDM(q) = dnSDM = 1 dnSDM ∂SτDM = 2M0Ef0,DDM(1+ΨDDM) (21)
0,SDM q2dq q2p da′ H(a′)q3 dt′ (cid:18) (cid:19)C
CM
(14) and
→f0,SDM(q)= MH(1a′)q3d(ρDdDtM′ a′3) (15) ∂∂fτL = a(M22M−Em2)Γf0,DDM(1+ΨDDM). (22)
(cid:18) (cid:19)C
This then implies Eq. (11) after enforcing the maximum The factors m2/(2M2) and (M2 m2)/(2M2) that ap-
momentum at qmax =apCM. pear in the SD0M and L collision t−erms can be easily un-
The evolution equations for the mean energy densi- derstood. Consider a two-body decay in the rest frame
ties in the two dark matter components are givenby the of the DDM particle, A B +C, with corresponding
integrals of Eq. 8 and Eq. 9 using the unperturbed dis- masses m , m , and m →. The energies of B and C in
A B C
tribution function. They read the rest frame of A are E = (m2 +m2 m2)/(2m )
and E = (m2 +m2 Bm2)/(2mA ). BSo−theCse factoArs
a˙ C A C − B A
ρ˙ +3 ρ = aΓρ (16) represent the ratios of energy that have been deposited
DDM DDM DDM
a − into different daughter particle species.
The perturbations for the massless relativistic daugh-
and
ter particles may be treated in a manner analogous to
a˙ am2 thatofmasslessneutrinos,saveforthe peculiardistribu-
ρ˙SDM +3a(ρSDM +PSDM)=Γ2M02ρDDM (17) tion function of the L. We integrate out the momentum
5
∂Ψ qk E f
dependence in the distribution function by defining (in 1 1 DDM,0
= (Ψ 2Ψ ) aΓ Ψ , (34)
0 2 1
Fourier space) ∂τ 3E − − M fSDM,0
q2dqqf0(q,τ)Ψ (~k,q,nˆ,τ)
F (~k,nˆ,τ)= L L (23) ∂Ψ qk 1 2 df
L q2dqqf0(q,τ) 2 = (2Ψ 3Ψ ) ( h˙ + η˙) SDM,0
R L ∂τ 5E 1− 3 − 15 5 dlnq
An expansion of FL in aRseries of Legendre polynomials E1fDDM,0
Pl(kˆ·nˆ) has the form −aΓ E fSDM,0Ψ2, (35)
FL(~k,nˆ,τ)=Σ∞l=0(−i)l(2l+1)FL,l(~k,τ)Pl(kˆ·nˆ). (24) and
The standard synchronous gauge perturbations in den- ∂Ψl qk E1fDDM,0
sity, velocity, and anisotropic stress are ∂τ = (2l+1)E(lΨl−1−(l+1)Ψ(l+1))−aΓM f Ψl.
SDM,0
(36)
δ =F , (25)
L L,0 for l 3.
≥
Ifwerestrictattentiononlytocasesinwhichthemass
3 differencebetweenthe DDMandSDMparticlesissmall,
θ = kF , (26)
L L,1
4 f = 1 m/M 1, the SDM particle will receive an
− ≪
extremely non-relativistic kick velocity v fc. As we
and k
≃
should expect, SDM behaves similarly to CDM, aside
1
σ = F (27) from the fact that it is endowed with a non-negligible
L L,2
2 distribution of momentum due to the DDM decays. In
respectively. Evaluating the Boltzmann equationfor our this limit, the SDM perturbations evolve as for a stan-
Legendre polynomial expansion in Eq. (24) yields the dard non-relativistic dark matter species,
evolutionofthemultipolecoefficientsintheconventional
1 E ρ
notation, δ˙ = θ h˙ +aΓ 1 DDM(δ δ )
SDM SDM DDM SDM
− − 2 M ρ −
SDM
2 E ρ (37)
δ˙ = (h˙ +2θ )+aΓ 2 DDM(δ δ ), (28)
L −3 L M ρ DDM− L and
L
a˙ δP E ρ
θ˙ = θ + SDMk2δ aΓ 1 DDMθ ,
θ˙ =k2(δL σ ) aΓE2ρDDMθ , (29) SDM −a SDM δρSDM SDM− M ρSDM SDM
L 4 − L − M ρ L (38)
L
where
σ˙ = 2 (2θ +h˙ +6η˙ 9kF ) aΓE2ρDDMσ , (30) δP 4πa−4 q2dqq2f (q)Ψ
L 15 L −4 L,3 − M ρ L c2 = SDM = 3 E 0 0 (39)
L s δρ 4πa−4 q2dqEf (q)Ψ
SDM R 0 0
and
ThehighermultipoletermsbecoRmenegligibleinthenon-
k E ρ
F˙L,l = [lFL,l−1 (l+1)FL,l+1] aΓ 2 DDMFL,l, l 3,relativisticastheyareproportionaltopowersoftheratio
2l+1 − − M ρL ≥ of the kinetic energy to the total energy, q/ǫ.
(31)
Thoughwesolvethe completeequationsfortheevolu-
HerewehavedefinedE =(M2+m2)/(2M)=m2/(2M)
1 0 tion of the SDM perturbations, the non-relativistic kick
and E =(M2 m2)/(2M).
2 velocity approximation is valid in most of our calcula-
−
The SDM must be treated differently to account for
tions. The most interesting constraints from future sur-
their finite mass and non-trivial velocity kicks. We ex- veys are relevant for models with v < 10−3c and rela-
k
pandthe perturbationtothe distributionfunction,Ψ,in tivistickickshavealreadybeenruledou∼tforawiderange
a Legendre series
of lifetimes [22, 57].
∞ Perturbation growth is suppressed on scales smaller
Ψ(~k,nˆ,q,τ)= ( ı)l(2l+1)Ψ (~k,q,τ)P (kˆ nˆ). (32) than the free-streaming scale. The free-streaming scale
l l
− ·
l=0 is, in turn, determined by an integralof the sound speed
X
c . We defer a detailed discussion of the free-streaming
We have dropped the “SDM” subscript on Ψ for brevity s
scaleinourdecayingdarkmattermodelsanditsimprint
as there should be no cause for confusion in this con-
on the matter and lensing power spectra to VI.
text. Evaluating the Boltzmann evolution equation on §
this expansion, we obtain for the different multipoles
IV. NONLINEAR CORRECTIONS TO
∂Ψ qk 1 dlnf E f
0 = Ψ + h˙ SDM,0+aΓ 1 DDM,0Ψ STRUCTURE GROWTH
1 DDM,0
∂τ −E 6 dlnq E f
SDM,0
E f
aΓ 1 DDM,0Ψ , (33) Our most robust constraints stem from perturbations
0
− E fSDM,0 on linear scales. However, it is interesting to estimate
6
the level of constraints that may be achieved by ex-
ploiting mildly nonlinear scales as is common practice
in the established framework for exploring dark energy
withlensingandgalaxyclusteringstatistics[36]. Includ-
ing mildly nonlinear scales improves constraints because
it increases the signal-to-noise of lensing measurements
andbecause it includes informationregardingthe effects
of DDM on the abundance and internal structures of
cluster-sized dark matter halos. We explore constraints
including mildly nonlinear scales as a means of estimat-
ing the level of constraints that may be achievable after
an exhaustive numerical simulation program, similar to
what is being performed for dark energy [58].
We implement the nonlinear correctionsto the matter
andlensingpowerspectrausingthehalomodel[59]. The
halo model is known to exhibit mild systematic offsets
comparedtonumericalsimulationsandthenonlinearcor-
rection of Ref. [60]. However,we use the halo model be-
cause it provides a convenient framework for estimating
the alterations to nonlinear structure induced by DDM
beforeperforminganexhaustivenumericalinvestigation.
We combinethe standardaspectsofthe halomodelwith
ananalyticalmodelproposedbySa´nchez-Salcedo[18]for
the alterations to dark matter halo structure due to the
FIG.1: Darkmatter densityprofiles timesradius, rρ(r),as
kick velocities generated in the decay process.
a function of radius and time. The dark matter halo mass
Forrelevantlifetimes (Γ−1 >H0−1), dark matterhalos is Mh = 1012M⊙. In the absence of dark matter decays,
begin with the same density∼profiles as in the standard the halo concentration is c = 5. The halo has a virial speed
CDM model. Their density distributions can be well de- vvir ≡ pGMh/Rvir ≈ 130 km/s. Different panels are for
scribed by Navarro et al. [61] (NFW) profiles. As the differentchoicesofkickvelocity andlifetime aslabeled along
DDM decays,the kinetic energyofdark matter particles the top and right axes respectively. In each panel the solid
will change because SDM particles receive a small kick lines show the initial NFW profile. The short-dashed line,
long-dashedline,dash-dottedline,anddash-double-dotedline
velocity from their parent particles. Assuming that we
represent density profiles after 2.5, 5, 7.5, and 10 Gyr. This
only consider decay processes with f 1, the mass of
≪ figureisdesignedtobedirectlycomparabletothesimulation
the parent and daughter particles will be nearly iden-
results displayed in Fig. 1 of Ref. [15].
tical. As discussed in Sa´nchez-Salcedo [18], on average
the net effect of decays is to impart an amount of en-
ergy ∆E mv2/2 on the dark matter, independent of average position of the daughter is similar to the radius
≈ k
the initial velocity. The changes in average kinetic en- of a circular orbit at the new value of the orbitalenergy.
ergy will result in changes in particle orbits, causing an This is conservative in the present context, because cir-
expansion of dark matter halos and a shallowing of dark cular orbits in equilibrium are least susceptible to such
matter profiles. expansion[18]. Inthis assumption,the radialpositionof
To demonstrate the effect of density profile modifica- the daughter particles, r′, will be
tion,weadoptatwo-stepcalculation. AssumethatDDM
2 β
particles in halos follow circular orbits prior to any sig- 1 v (R)
r′ = r1/β + k r1/β . (40)
nificant DDM decays. The particles orbitin the gravita- 2β+1 v 0
(cid:18) 0 (cid:19) !
tional potential of the NFW halo, which can be approx-
imately described by a power law v (r) = v (r/r )1/2β The model we have described is not self-consistent, so
c 0 0
overanysufficiently smallrangeof r. In a giventime in- it is important to validate the basic predictions of the
terval,asmallfractionofDDMparticlesdecayandtheir model against more complete calculations. To check the
daughter SDM particles gain a small amount of energy validity of this model, we compare our analytical calcu-
∆E mv2/2. In general, the daughter particles will lation results with the N-body simulation results from
≈ k
move from circular orbits to elongated orbits, character- [15]. InFigure1, weplotdensityprofilesfora darkmat-
izedby the new energyrelativeto the halopotentialand ter halo with mass Mh = 1012M⊙ and an initial NFW
anapocentric radius r. Orbits in the NFW potentialare concentration parameter c = 5 for several different life-
not closed, rendering it a numericalproblem to compute times and kick velocities. Peter et al. [15] computed the
the time-averaged value of the radial coordinate of the profiles of dark matter halos in the same model using
daughter particle. To obtain a simplistic estimate of the N-body simulations that accounted for the dark matter
new radii the particles move to, we assume that the new decays. Fig. 1 above is the same as Figure 1 in Peter
7
Mh 1013 1014M⊙ [38]. Such halos have significantly
large≈r viria−l velocities than the 1012M⊙ halos consid-
ered above. Typical virial velocities of these larger halos
lie in the range v 280 600 km/s. This suggests
vir
≈ −
that our model can be used at larger v than the value
k
v 200 km/s that we arrived at by comparing to sim-
k
≈
ulations of a 1012M⊙ halo above, because these kicks
represent a smaller fraction of the potential well depth.
Forinstance,Peteretal.[15]pointedoutthatthecluster
mass function is insensitive to v < 500 km/s, because
k
the typical virial speeds clusters ar∼e v > 600 km/s. For
k
completeness, we show the correspondin∼g density profile
modifications for these group- and cluster-sized halos in
Figure 2. We will show in VI that our calculations are
§
only sensitive to DDM parameters that result in density
profiles with mild changes.
We include this effect in our nonlinear halo model cal-
culation by giving all recomputing halo profiles as de-
scribed above prior to computing lensing correlations.
We modified halo profiles by assuming initial halos with
the same profiles, including concentrations,as their con-
cordance ΛCDM counterparts and implementing the
above model on these halos. The remainder of our halo
model implementation follows the approach we used in
FIG. 2: Similar to Figure 1 but for halos with Mh = 5× Ref.[26], sowedo notrepeatit. Ideally,onewouldtreat
1013M⊙ and NFW concentration c=5. The halo virial speed nonlinear corrections to structure growth using program
is vvir ≈ 477 km/s. ofcosmologicalnumericalsimulations. However,weplace
such a study outside the scope of the present work as
our initial aim is to estimate the constraining power of
et al. [15] save for the fact that we have computed mod-
forthcoming surveys. In this manner, we estimate the
ified halo profiles according to the analytic model de-
fruit that a computationally-intensive numerical simula-
scribed in this section. A comparison of the two figures
tion program may bear on the problem of unstable dark
reveals that the analytic model and the numerical simu-
matter.
lations are in remarkable agreement for all models with
v <200km/sandΓ−1 >10Gyr. Thereareseveralpos-
k
sibl∼e explanationsforth∼e inconsistenciesthatarisewhen
V. FORECASTING METHODS
v > 200 km/s and Γ−1 < 10 Gyr. One is that when
k
changes to the gravitational potential are not small, the
final gravitational potential is sufficiently different from The Fisher Information Matrix provides a simple es-
the initial gravitational potential that the initial poten- timate of the parameter covariance given data of spec-
tial cannot be used to approximate the new positions of ified quality. The Fisher matrix has been utilized in
theSDMparticles. Anotherpossibilityisthattypicalcir- numerous, similar contexts in the cosmology literature
cular orbits no longer provide useful approximations for [26, 38, 45, 62–69], so we give only a brief review of im-
the degreeofhaloexpansion. Asdiscussedin[15],where portantresultsandthe caveatsinourparticularapplica-
they look atvelocity anisotropyof their simulatedhalos, tion. WehaveconfirmedthevalidityoftheFishermatrix
they found that the orbits become radially biased at the approximation in models of unstable dark matter using
halo outskirts. Moreover, v =200 km/s is considerable Monte Carlo methods as described in Ref. [26].
k
compared to the virial velocity of a Mh =1012M⊙ halo, TheFishermatrixofobservablesinEq.(1),subjectto
so itis notsurprisingthatthose halos arenotin dynam- covariance as in Eq. (7), can be written as
ical equilibrium for large v and small lifetime. These
k
simulationresults showthatthe assumptions ofour sim- ℓmax ∂P ∂P
F = (2ℓ+1)f κ,A[C−1] κ,B +FP
ple modelareviolatedinthe regimeofhighkickvelocity ij sky ∂p AB ∂p ij
i j
and low lifetime. As we show in VI, our primary re- ℓ=Xℓmin AX,B
§ (41)
sults in which the nonlinear model is used correspondto
v < 200 km/s and lifetimes Γ−1 > 100 Gyr, so our use where the indices A and B run over all NTOM(NTOM +
k
1)/2 spectra and cross spectra, the p are the parame-
ofthis modelforafirstforayintothis regimeis justified. i
ters of the model, f is the fraction of the sky imaged
Unlike Peter et al. [15], we are interested in cos- sky
−1/2
mological weak lensing as our observable. The halo by the experiment, and ℓmin = 2fsky is the smallest
mass most relevant to weak lensing lie in the range multipole constrained by the experiment. FP is a prior
ij
8
Fishermatrixincorporatingpreviousknowledgeofviable vant lifetimes the dark matter decays should cause only
regions of parameter space. We set ℓ = 300 for lin- subtle alterations to the cosmic microwave background
max
ear forecasts and ℓ = 3000 in our most ambitious anisotropyspectrumsothisanalysisshouldapproximate
max
nonlinear forecasts. On smaller scales (higher ℓ), vari- a self-consistent simultaneous analysis of all data.
ous assumptions, such as the Gaussianity of the lensing In some cases, we will estimate nonlinear power spec-
field, break down [38, 48–52, 70]. To be conservative,we tra in models with significant neutrino masses. In such
explore modest priors on each parameter independently, cases, we follow the empirical prescription established in
so that FP = δ /(σP)2, where σP is the 1σ prior on previous studies (e.g., Refs. [33, 34, 68]) and take
ij ij i i
parameter p . The forecast, 1σ, marginalized constraint
i
on parameter p is σ(p )= [F−1] . 2
i i ii P (k)= f Plin(k)+f PNL (k) (42)
Our DDM model has two independent parameters, m ν ν b+DM b+DM
namely decay rate Γ (or lpifetime, Γ−1) and mass loss (cid:20) q q (cid:21)
fraction f (which is related to v via v = fc). Either where
k k
one of these parameters can independently be tuned to
Ω
ν
render the effects of DDM negligible, so it is not useful fν = , (43a)
Ω
to marginalize over one parameter to derive constraints m
Ω +Ω
onthe other. Inwhatfollows,wechooseto illustratethe f = DM b, (43b)
b+DM
effectiveness of lensing to constrain DDM by fixing life- Ωm
time and quoting possible constraints on f. Other than
Plin(k) is the linear power spectrum of neutrinos, and
the mass loss fraction f, we also consider six cosmolog- ν
PNL (k)is the nonlinearpowerspectrumevaluatedfor
ical parameters that we expect to modify weak lensing b+DM
baryons and dark matter only. However, we note that
power spectra at significant levels and to exhibit partial
recent work has questioned the robustness of this treat-
degeneracy with our model parameters. We construct
mentofneutrinomassusingdirectnumericalsimulations
our forecasts for DDM lifetime bounds after marginaliz-
[72] and perturbationtheory [73], so it may become nec-
ing over the remaining parameters. Our six additional
essary to revisit this aspect of the modeling of power
parameters and their fiducial values (in parentheses) are
spectra prior to the availability of observational data.
the dark energy density Ω (0.74), the present-day dark
Λ
matter density, ω = Ω h2 (0.11), the baryon den- Weexplorepossibleconstraintsfromavarietyofforth-
DM DM
sityω =Ω h2 (0.023),tiltparametern (0.963),thenat- coming data sets. We consider the Dark Energy Sur-
b b s
vey (DES) as a near-term imaging survey that could
ural logarithm of the primordial curvature perturbation
normalizationln(∆2) ( 19.94),and the sum of the neu- provide requisite data for this test. We model DES
R − by taking a fractional sky coverage of f = 0.12 and
trino masses m (0.05 eV). This choice of fiducial sky
modelimpliesaismaνill-scale,low-redshiftpowerspectrum with n¯ = 15/arcmin2. Second, we consider a class of
normalizationPof σ 0.82. The optical depth to reion- future “Wide” surveys as may be carried out by the
8
ization has a negligi≃ble effect on the lensing spectra on Large Synoptic Survey Telescope (LSST)[27] or Euclid
scales of interest, so we do not vary it in our analysis. [28]. We model these Wide surveys with fsky = 0.5 and
n¯ = 50/arcmin2. Lastly, we consider a comparably nar-
We take priors on our cosmological parameters of
σ(ω ) = 0.007, σ(ω ) = 1.2 10−3, σ(ln∆2) = 0.1, row, deep imaging survey. We refer to such a survey
σ(nm) = 0.015, and σb(Ω ) = ×0.03. We assumRe no pri- as a “Deep” survey and model it with fsky = 0.05 and
s Λ n¯ = 100/arcmin2. Such a survey may be more typical
ors on DDM model parameters or neutrino mass. Our
of a space-based mission similar to the proposed Wide-
fiducial model is motivated by the WMAP seven-year
FieldInfraRedSurveyTelescope(WFIRST).Inallcases,
result and our priors represent marginalized uncertain-
tiesontheseparametersbasedontheWMAPseven-year we take γ2 = 0.2 and assume particular shape mea-
h i
data [4]. These priors are very conservative and allow surement errors from each experiment are encapsulated
p
for weaker constraints on DDM than would be expected in their effective number densities, in accordwith recent
fromfuturedata,wherestrongerpriorsmaybeavailable. conventional practice in this regard. Our results are rel-
Toestimatethepotentialpoweroflensingconstraintson atively insensitive to number density because shot noise
DDM when stronger cosmological constraints are avail- does not dominate cosmic varianceon the scales we con-
able, we also explore prior constraints on these parame- sider, and our linear constraints are completely insensi-
ters at the level expected from the Planck mission6 us- tive to the choice of galaxy number density over a wide
ing the entire Planck prior Fisher matrix of Ref. [71]. range.
Of course, using published priors from other analyses is
not self-consistent because these priors were derived in
analyses that assume stable dark matter, but for rele- VI. RESULTS
There are several effects of DDM on lensing power
spectra at low redshift. First, decays change the cosmo-
6 http://www.esa.int/planck logicalenergy density. This change alters both structure
9
which can be defined as
3 (a)
k (a)= H , (44)
FS
2c (a)
r s
where (a)=ada/dτ and −1 is the comoving horizon
H H
scale.
We show the evolution of free-streaming scale of SDM
particles as a function of scale factor in Figure 3 for
several mass loss fractions f and lifetimes. As dis-
cussed in [25], the behavior of the free-streaming scale
of DDM can be divided into two regimes. When the
decay process is still occurring, corresponding to cosmo-
logicaltimeslessthanthedecaylifetime,daughterparti-
cles with the same physical momentum are continuously
created so that the sound speed stays approximately
the same. In this case, the evolution of free-streaming
scale will simply trace the evolution of horizon. If de-
cays have ceased, which will happen when Γ−1 < H−1,
0
the sound speed will decrease as c a−1. The free-
FIG. 3: Free-streaming scale as a function of scale factor. s ∝
streaming scale shrinks as the initial velocities are red-
The blue lines show free-streaming scales for lifetime much
shifted away. This effect alsohappens to massiveneutri-
greater than the age of universe ( > 100 Gyr) for several
different mass loss fractions. The dash-dotted magenta lines nos as they become non-relativistic. At early times the
are for f = 10−2 and three lifetimes. From top-to-bottom neutrinofree-streamingscaletracesthehorizonsolongas
at right, these are 0.01 Gyr, 0.1 Gyr, and 1 Gyr. The green the neutrinos have relativistic velocities. In Figure 3 we
lineisthefree-streamingscaleformassiveneutrinowithmν,i can see that after neutrinos become non-relativistic, at
=0.4eV.Structuregrowsonscalesbetweenthefree-streaming a 1.3 10−3(0.4eV/m ), their free-streaming scale
nr ν
scaleandhorizon. Onscalessmallerthanfree-streamingscale varie≃s as k× a1/2 during matter domination, whichis
FS
(k>kFS),structure growth is suppressed. identical to fr∝ee-streaming in the small lifetime limit of
DDM.
growth and distance. Further, decaying dark matter re-
sults in significant free-streaming of daughter SDM par- B. Weak lensing Power Spectrum
ticles. While each of these effects can be important, for
modelsnearthelimitofwhatmaybeconstrainedbylens- As we mentioned above, DDM affects lensing power
ing surveys, it is the effect of free-streaming that largely spectra in two respects. First, the power spectra for po-
determines the lensing power spectra. In this case, tential and density fluctuations are modified by the free
the free-streaming velocity of SDM suppresses structure streaming of the daughter SDM particles. At k > k ,
FS
growth on scales smaller than free-streaming scale, an structure growthis suppressed. Second, the matt∼er den-
effect similar to that caused by massive neutrinos. In sity is reduced as decays occur, slightly suppressing the
VIA we show the behavior of the free-streaming scale late-time growth of structure. In the left panel in Fig-
§
inourmodelandcompareitwithmassiveneutrinos. We ure4,weshowthatsignificantdecrementsinpoweroccur
follow this by showing the alterations to lensing power at roughly the same scale, k > 10−2hMpc−1 for a vari-
spectra in VIB. Our final results are the forecast con- ety of lifetimes, so long as the∼lifetime Γ−1 H−1 (the
straints on§DDM, which we give in VIC. regime most relevant to our work). This su≫ppre0ssion is
§
due to free streaming and indeed, the scale on which the
suppression occurs agrees with the estimates of the free-
streamingscaleshowninFig.3. TherightpanelofFig.4
illustrates that the scale of suppression is determined by
A. Free-streaming Scale
the mass-loss fraction f, in the limit that Γ−1 H−1.
≫ 0
In models with larger f, the velocities of the daughter
In the standard cosmological scenario, matter density SDM particles are higher, so at fixed lifetime, they free-
fluctuations at a particular scale grow once the scale en- stream greater distances. Both panels in Fig. 4 show a
ters the horizon (k > H) during the matter-dominated smallincrementin power onlargescales for models with
epoch. However, species with non-negligible primordial smalllifetimes(Γ−1 <50Gyr)andlargermass-lossfrac-
velocities will be able to escape the potential wells and tions(f >0.1). This∼delineatestheparameterregimefor
suppress the formation of structure. The scale that cor- which th∼e overall change in the energy budget begins to
responds to this effect is the free-streaming scale k , have a non-negligible effect on fluctuation growth. The
FS
10
FIG.4: FractionaldifferencebetweenmatterpowerspectrumforstandardΛCDMandadecayingdarkmattermodelevaluated
at z = 0. Left: The effect of varying the DDM lifetime at fixed mass-loss fraction, f = 10−1. Solid curves show the linear
theorypredictions,anddashordash-dotlinesshowpredictionsthatincludethenonlinearcorrectionsimplementedviathehalo
model. The green lines show the spectrum in a ΛCDM with massive neutrinos, Σmν =0.5 eV, for comparison. Right: The
effect of varyingmass-loss fraction f, at a fixed lifetime of Γ−1 =50 Gyr.
small increment on large scales in these cases enforces a thechangesinmatterpowerspectrumareapproximately
fixed observed CMB normalization. 40 60%for k >2 10−2 hMpc−1). The corresponding
Notice in the left panel of Fig. 4 that with f 10−1, cha−ngesinangu∼lard×iameterdistancesarelessthan0.1%.
∼ The constraining power of weak lensing comes primarily
the free-streaming suppressionis similar to that induced
from the suppression of structure growth. Incidentally,
bymassiveneutrinoswiththesumoftheneutrinomasses
this is a promising feature because the primary informa-
Σm 0.5 eV. This suggests that neutrinos may be de-
ν
≈ tion used to constrain dark energy using lensing surveys
generatewithDDM,andthiswouldbethecaseifitwere
is carried by the geometric piece of the lensing signal
not possible to probe a wide range of length scales and
[74, 75].
redshifts. In practice,we find thatmassive neutrinosare
distinguishable from DDM for two reasons. First, the Aoyama et al. [25] have considered constraints on un-
differences in scale dependence exhibited in Fig. 4 give stabledarkmatterstemmingfromcontemporarydataon
a possible handle with which to separate the two. More the cosmological distance-redshift relations. Using con-
importantly, the redshift dependence of the power spec- straints on the Hubble parameter, the baryon acoustic
trum differs in the two models. This is most easily seen oscillation scale, and the angular positions of the cos-
in Fig. 3. The evolution of the free-streaming scale of mic microwave background anisotropy spectrum peaks,
massive neutrinos and the free-streaming scale of DDM they have placed competitive constraints on such mod-
differs significantly. Deep, large-scale survey data that els. Aoyama et al. [25] find that Γ−1 > 0.1 Gyr with
enable probes of structure at a variety of redshifts be- f <3.5 10−2 and that Γ−1 >30 Gyr w∼hen f 1.
tween 0 < z < 3, as is expected of forthcoming surveys, ∼The D×ash-dotted lines in F∼igure 5 exemplify∼the al-
break th∼e po∼tential degeneracy between massive neutri- terations to the small-scale lensing convergence power
nos and DDM. spectra incurred when we account for the altered halo
Theobservedstrengthofgravitationallensingalsohas profiles that result from dark matter decays. As f in-
adependenceupongeometry,sodifferencesinangulardi- creases,kickvelocitiesincrease,andthe fractionalpower
ameterdistancemayleadtomodifiedlensingpowerspec- decrement increases, as we should expect. This addi-
tra. Changes in relative partitioning of energy among tional suppression is confined to relatively small scales
relativistic and non-relativistic species will change the (large multipoles, ℓ > 300) for most of the parameter
evolution of the angular diameter distance. However, space of interest (vk <∼200 km/s for Γ−1 <100 Gyr).
∼ ∼
we consider small mass loss fractions (f 1) and large As we pointed out in right panel in Figure 4, DDM
lifetimes (Γ−1 >> H−1), so angular diam≪eter distances may partially mimic massive neutrinos if redshift evolu-
0
are altered only by negligible amounts and, although we tioninformationinnotaccessible. InFigure6,weshowa
account for these changes, they do not provide lever- comparisonoftheredshiftevolutionofDDMandmassive
age on constraining DDM. For example, for a lifetime neutrinolensingpowerspectrainthreetomographicred-
of Γ−1 = 50 Gyr and f = 10−1, Figure 4 shows that shift bins. Other than the difference in shapes, it is also