Testing the interaction of dark energy to dark matter through the analysis of virial relaxation of clusters Abell Clusters A586 and A1689 using realistic density profiles

0.2 MB·English

by  Orfeu Bertolami

Noname manuscript No.
(will be inserted by the editor)
Testing the interaction of dark energy to dark matter
through the analysis of virial relaxation of clusters Abell
Clusters A586 and A1689 using realistic density profiles
Orfeu Bertolami · Francisco. Gil Pedro ·
Morgan Le Delliou
2
1
Received: date/Accepted: date
0
2

n Abstract Interactionbetweendarkenergyanddarkmatterisprobedthrough
a
deviationfromthevirialequilibriumfortworelaxedclusters:A586andA1689.
J
Theevaluationofthevirialequilibriumisperformedusingrealisticdensitypro-
9
files.Thevirialratiosfoundforthemorerealisticdensityprofilesareconsistent
1
with the absence of interaction.

]
O Keywords Cosmology Gravitation Dark Matter Equivalence Principle
· · · ·
C Field-theory
.
h PACS 98.80.-k 98.80.Cq
·
p
-
o
r 1 Introduction
t
s
a Searchingforevidenceandpossiblestrategiesfordetectionofdarkenergyand
[
darkmatterareamongthemostpressingissuesofcontemporaryphysics.Dark

2 energy(DE)anddarkmatter(DM)arethefundamentalbuildingblocksofthe
v
3 O.Bertolami
3 Departamento deF´ısicaeAstronomia,FaculdadedeCiˆenciasdaUniversidadedoPorto
0 RuadoCampoAlegre687,4169-007Porto,Portugal
3 E-mail:[email protected]@fisica.ist.utl.pt
. O.B. Also at: Instituto de Plasmas e Fusa˜o Nuclear, Instituto Superior T´ecnico, Avenida
5
RoviscoPais1,1049-001Lisboa,Portugal
0
1 F.GilPedro
1 RudolfPeierlsCentreforTheoretical Physics
: 1KebleRoadOxfordOX13NP,UK
v E-mail:[email protected]
i
X M.LeDelliou
Instituto deFsicaTericaUAM/CSIC, Facultad de Ciencias, C-XI,UniversidadAut´onoma
r
a deMadrid,Cantoblanco, 28049MadridSPAINE-mail:[email protected]
Also at: Centro de de Astronomia e Astrof´ısica da Universidade de Lisboa, Faculdade de
Ciˆencias da Universidadede LisboaEdif´ıcioC8, CampoGrande P-1749-016 LISBOA Por-
tugal
2
cosmological standard model based on general relativity. If on one hand, the
interaction of the dark sector with the standard model such as neutrinos [1]
and the Higgs field [2] lead to well defined experimental implications, such as
for instance, a possible deviation from the Rutherford-Soddy radiative decay
law [4] (see eg. Ref. [5] for general discussions), on another hand, interaction
withinthedarksectoritselfshouldnotbeexcluded.Indeed,arecentstudy [6]
suggeststhataputativeinteractionmightalsobedetectableusingforinstance
gamma-raybursts. The interactionbetween darkenergy and dark matter can
beintroducedeitherbyadhocarguments[7]orbecausedarkenergyanddark
matter are unified in the context of some framework [8,9,10]. Of course it is
quite relevant to search for observational evidence of this interaction.
In a previous work [11] we have analysed the effect of the interaction be-
tween dark energy and dark matter on the virial equilibrium of clusters. Our
method is based on the generalisation of the Layzer-Irvine equation, the cos-
micvirialtheoremequation,whichwasthenappliedtotheAbellclusterA586.
Thisclusterwaschosengiventhatitispresumablyinequilibriumandhasnot
undergone interactions for quite a long time [12]. We have also argued that,
based on the analysis of the bias parameter, that the dark sector interactions
dosignalapossibleviolationoftheEquivalencePrincipleatcosmologicalscale
[11,13]. This proposal has been extended for other clusters in Refs. [14], and
as in our A586 analysis, evidence for DE-DM interaction was encountered.
Inthis workwereexaminethe clusterA586andextendouranalysisto the
Abell cluster A1689 considering that mass distribution within the cluster is
not constant. The additional cluster A1689 is chosen as it shares with A586
the featureofshowinga negligibleamountofmergers,asinferredfromits low
X-ray substructure [15]. We consider four distinct density profiles: Navarro,
Frenk and White (NFW) profile [16], isothermal sphere (ISO) profile [17,18,
19,20,21,22], Moore’s (M) profile [23] and Einasto profile [24,25,26]. We also
obtain results for up to four different techniques of evaluating the velocity
dispersion σ of the haloes. We argue that the most relevant technique for
v
determination of the velocity dispersion is through the X-ray temperature.
Our analysis reveals that the use of more realistic mass profiles brings the
virial ratio close to its canonical value in the absence of interaction.
This work is organized as follows: in section 2 we describe the techniques
used for fitting the data with the various density profiles; section 3 presents
our analysis of the virialratios for the previously used cluster A586;section 4
considers the A1689 cluster; in section 5 we interpret our results in terms of
DE-DM interaction. Finally, in section 6 we present our conclusions.
3
2 Computations of the Virial ratio for a given density profile
In this section we present our method and summarise the results of applying
realistic density profiles to estimate the interaction between DE and DM in
relaxed systems as discussed in Refs. [11,13]. We will focus on two relaxed
galaxy clusters:A586, which was already analysedin Ref. [11] with a top-hap
density profile, and A1689.
We start by defining the basic quantities and deriving the necessary rela-
tionsbetweenthem.Inparticularwefocusonobtainingvariousdensityprofile
fits for the haloes, and using the photometric coordinates of the galaxies, we
compute the averagedistance between a galaxy and the centre of the cluster,
R ,atwhichpointweareinconditiontoevaluatethevariousdensityprofiles
h i
parameters and the corresponding errorsfor the cluster A586. For the cluster
A1689, we use weak lensing surface density data. To compute the virial ratio
ρ /ρ we must alsofind the velocitydispersionσ .FromRefs. [12,28,27,29]
K W v
we obtain the weak lensing and photometric data necessary to compute the
differentdensityprofilesaswellasvaluesforσ ,asgivenbydifferentmethods;
v
notethatweapproximatethevelocitydispersionbyitsaveragevaluetherefore
considering it to be constantover the cluster ; we can then compute the ratio
of kinetic energy to potential energy using different data sources (see Table 1
for A586 and Table 3 for A1689).
InRef.[11]wehaveusedthesimplestapproachofaTop-Hatdensityprofile
and used the velocity dispersion estimated from weak lensing measurements.
For the purpose of comparison, we present in Table 2 the outputs of such
profilewiththesamevelocitydispersionasfortheothermorerealisticprofiles.
Although that analysis allows for claiming detection [11], a more accurate
estimate ofthe density leads to anunderestimationofthe potentialenergyby
placing more mass far fromthe centre than a realistic decreasing profile.This
implies an overestimate of the magnitude of the virial ratio, as can be seen
for central values in Table 2. This should be corrected with the use of more
realistic density profiles.
It is important to stress that the methods used vary slightly for the two
clusterswearestudyinghere.FortheA586weproceedinthesamespiritasin
Ref. [11], estimating the density and shape parameters for the severaldensity
profilesfromtotal3Dmassandgalaxycoordinates.Ontheotherhand,forthe
A1689 cluster we fit directly the profiles to the 2D surface density obtained
from lensing measurements [28].
2.1 NFW profile
Given its prominence in discussions about the density profile emerging from
realistic N-body simulations in the context of the ΛCDM model, the NFW
4
density profile [16] will be used. It reads
ρ0
ρ(r)= , (1)
2
r 1+ r
r0 r0
(cid:16) (cid:17)
where ρ0 andr0 are the density and shape parametersrespectively.Note that
sinceρonlydependsonr,sphericalsymmetryisbuiltintothesecomputations
from the very start.
In the analysis of the cluster A1689, the 2D surface density from lensing
measurements is used to perform a fit in order to obtain the estimates for ρ0
andr0.FortheclusterA586weproceedbycomputingthemassbyintegrating
Eq. (1) over the volume:
M ≡4πˆ0R r 1ρ0+r2r 2dr =4πr03ρ0(cid:20)ln(cid:18)1+ rR0(cid:19)− R+Rr0(cid:21). (2)
r0 r0
(cid:16) (cid:17)
The mean radius R can then be defined with
h i
MhRi≡4πˆ0R r 1ρ0+r3r 2dr =4πr04ρ0(cid:20)rR0 −2ln(cid:18)1+ rR0(cid:19)+ R+Rr0(cid:21),
r0 r0
(cid:16) (cid:17) (3)
which after considering Eq. (2) can be written as follows:
R 2ln 1+ R + R
r0 − r0 R+r0
hRi=r0h ln 1+(cid:16) R (cid:17) R i. (4)
r0 − R+r0
h (cid:16) (cid:17) i
Wecanthengetthe shapeparameterr0 forA586bynumericallyinverting
R .WethenuseittodefinetheNFWdensityparameterρ0fromtheobserved
h i
mass M contained within the radius R by inverting Eq. (2):
M
ρ0 = . (5)
R
4πr3 ln 1+ R r0
0(cid:20) (cid:16) r0(cid:17)− 1+rR0(cid:21)
The kinetic energy density is defined in terms of the average velocity dis-
persion, σ , to be
v
1 3 9 r3 R r
ρK = 2V ˆ ρσv2dV = 2R03ρ0ˆ0 [r+r0]2σv2dr. (6)
Assuming a constant average velocity dispersion, it reads
1M 9 r3 R R 9 M
ρK = 2 V 3σv2 = 2R03ρ0 ln 1+ r0 − R+r0 σv2 = 8πR3σv2, (7)
(cid:20) (cid:18) (cid:19) (cid:21)
as used in Ref. [11].
5
The potential energy is given by
4π R ρ(r)GM(r)
ρ = r2dr (8)
W −4πR3/3ˆ r
0
3GM2 1+ rR0 21 1+ rR0 −ln 1+ rR0 − 12
= . (9)
−4πR3r0h(cid:16) (cid:17)1n+(cid:16)R ln (cid:17)1+ R(cid:16) R 2(cid:17)o i
r0 r0 − r0
h(cid:16) (cid:17) (cid:16) (cid:17) i
As discussedin Ref.[11],the existence ofinteractionbetweenDE andDM
is estimated by comparing the ratio ρ /ρ with the expected 1/2 value
K W
−
arising from the virial theorem. Taking the ratio of Eqs. (7) and (9) we find:
2
3 r0 σ2 1+ R ln 1+ R R
ρK 2GM v r0 r0 − r0
= . (10)
ρW − 1+ R h1(cid:16) 1+ R(cid:17) (cid:16)ln 1+(cid:17)R i 1
r0 2 r0 − r0 − 2
h(cid:16) (cid:17)n (cid:16) (cid:17) (cid:16) (cid:17)o i
Throughout our analysis, the errors of the ratios are computed according to
the following equation:
2
ρ ∂ρ /ρ
K K W
∆ ( o ) = ∆(p ) , (11)
i i
(cid:18)ρW { } (cid:19) vuupiX∈{oi}(cid:18) ∂pi (cid:19)
t
where o are the parameters carrying measurement errors and are defined
i
{ }
for each cluster as oi = σv,M for A586 and as oi = σv,r0,ρ0 for
{ } { } { } { }
A1689.
2.2 Isothermal profile
Introduced first as the natural outcome of spherically symmetric DM self-
gravitatinginfall [17,18,19,20,21,22], the isothermaldensity profile is simpler
than the NFW one:
ρ0
ρ= . (12)
2
r
r0
(cid:16) (cid:17)
Here the fiducial radius r0 and density ρ0, or mass M0 = 4π/3ρ0r03, are ar-
bitrary as the profile is self-similar: there is no characteristic scale, so we can
chose the mass and total radius of the halo as the fiducial values,
M0 =M, (13)
r0 =R. (14)
The mass is found by integrating Eq. (12) over the volume:
M ≡4πˆ0R ρ0rr22dr =4πr02Rρ0 =M0rR0. (15)
r0
(cid:16) (cid:17)
6
The mean radius R can then be defined with
h i
MhRi≡4πˆ R ρ0r32dr =2πr02ρ0R2, (16)
0 r
r0
(cid:16) (cid:17)
which after considering Eq. (15) can be written as follows:
R
R = . (17)
h i 2
The new kinetic energy density is now defined to be
ρK = 21V3 ˆ ρσv2dV = 29Rr023ρ0ˆ0Rσv2dr = 89πRM3r00 ˆ0Rσv2dr. (18)
Assuming a constant average velocity dispersion, it reads
ρ = 1M3σ2 = 9 M0 Rσ2 = 9 Mσ2, (19)
K 2 V v 8πR3r0 v 8πR3 v
again as in Ref. [11].
The potential energy is given by
4π R ρ(r)GM(r) 3GM2 3GM2
ρ = r2dr = 0 = . (20)
W −4πR3/3ˆ r −4πR2r2 − 4πR4
0 0
Taking the ratio of Eqs. (19) and (20) we find:
ρK = 3 r0 σ2 = 3 R σ2 . (21)
ρW −2GM0 v −2GM v
2.3 Moore’s profile
We consider now the Moore density profile that also arises as a suitable dark
halo profile from N-body simulations [23]. It is believed to be quite accurate
to describe galaxy size halo formation [30]:
ρ0
ρ= , (22)
3 3
r 2 1+ r 2
r0 r0
(cid:16) (cid:17) (cid:20) (cid:16) (cid:17) (cid:21)
whereρ0 andr0 arethe newmassdensityandshapeparameters,respectively.
The mass is found by integrating Eq. (22) over the volume:
R r2 8π R 32
M =4πρ0ˆ0 r 23 1+ r 23 dr = 3 r03ρ0ln 1+(cid:18)r0(cid:19) !. (23)
r0 r0
(cid:16) (cid:17) (cid:20) (cid:16) (cid:17) (cid:21)
7
The mean radius R can then be defined with
h i
3
r 2
R
MhRi= 4πr03ρ0ˆ0 1+(cid:16)r0(cid:17)r 23dr, (24)
r0
(cid:16) (cid:17)
and considering Eq. (23), it can be written as:
1 R + R
3 R π 1 − r0 r0
hRi= 2r0r0 + 3√3 + 3ln 1+q R 2 
  r0 
  (cid:16) q (cid:17) 
2 R 1 / 3
2 r0 − R 2
arctan ln 1+ . (25)
− √3  q√3  (cid:18)r0(cid:19) !

 
We can get the shape parameter r0 numerically by inverting R , which
h i
canbethenusedtodefinetheMooredensityparameter,ρ0,withtheobserved
mass, M, contained within the radius, R, by inverting Eq. (23):
3M
ρ0 = 3 . (26)
8πr3ln 1+ R 2
0 r0
(cid:18) (cid:16) (cid:17) (cid:19)
Thus, the kinetic energy density is defined to be
ρ = 1 3 ρσ2dV = 9ρ0r03 R r12 σ2dr. (27)
K 2V ˆ v 2 R3 ˆ0 r32 +r032 v
Assuming a constant average velocity dispersion, it reads
3
ρK = 21MV 3σv2 =3ρ0(cid:16)rR0(cid:17)3ln 1+(cid:18)rR0(cid:19)2!σv2 = 89πRM3σv2, (28)
still as in Ref. [11].
The potential energy is given by
ρ = 4π R ρ(r)GM(r)r2dr = 8πGr05ρ20 rR0 ln 1+X32 dX
W −4πR3/3ˆ r − R3 ˆ 1(cid:16) 3(cid:17)
0 0 X2 1+X2
8πGr5ρ2 R (cid:16) (cid:17)
= 0 0F . (29)
− R3 r0
(cid:18) (cid:19)
Taking the ratio of Eqs. (28) and (29) we find:
3
ln2 1+ R 2
ρρWK =−GrM0 (cid:18)F R(cid:16)r0(cid:17) (cid:19)σv2 (30)
r0
(cid:16) (cid:17)
8
Table 1 Velocitydispersionsfromthevariousobservations ofA586asgivenbyRef.[12].
Method σ (Km/s)
X-rayLuminosity 1015±500
X-rayTemperature 1174±130
Weaklensing 1243±58
Velocitydistribution 1161±196
2.4 Einasto’s profile
The Einasto density profile [24,25,26] was originally used to describe the in-
ternal density profiles of galaxiesand has been proposed as a better fit model
for ΛCDM haloes [31,32,33]:
1 1
r n r n
ρ=ρeexp dn 1 =ρ0exp 2n , (31)
"− (cid:18)re(cid:19) − !# "− (cid:18)r−2(cid:19) #
where ρ0 =ρeedn is the central density and r−2 the radius at which the slope
dlnρ/dlnr = 2, the isothermal value. The radius r is defined such that it
e
−
containshalfthe totalmass andd is anintegrationboundaryto ensurethat.
n
n givesthe strengthofthe density fall. FromEq.(31) the correspondingmass
then reads:
R r n1
M =4π ρ0exp 2n r2dr. (32)
ˆ0 "− (cid:18)r−2(cid:19) #
The mean radius R then becomes defined with
h i
R r n1
M R =4π ρ0exp 2n r3dr. (33)
h i ˆ0 "− (cid:18)r−2(cid:19) #
We can get the shape parameter r−2 from observing the mean intergalactic
distance and numerically solving Eq. (33) together with Eq. (32).
Wecanthenuseittocomputethecentraldensityparameterρ0 bymaking
use of Eq.(32).The kinetic and potentialenergydensities arecomputed from
their definition
1 3 4π R ρ(r)GM(r)
ρ = ρσ2dVρ , ρ = r2dr, (34)
K 2V ˆ v W W −4πR3/3ˆ r
0
through numerical integration. Note that it is also possible to perform the
integrations analytically as done for the previous cases and find expressions
relating the profile parameters and the virial ratio to the data M, R and σ.
h i
However, since these expressions for the Einasto profile are not particularly
illuminating we will omit them.
9
Table2 VirialratiofromthevariousobservationsofA586obtainedfromdataofRef.[12]
usingdifferentdensityprofiles.
Method TopHat NFW Isothermal
X-rayLuminosity −0.516±0.516 −0.408±0.407 −0.353±0.352
X-rayTemperature −0.691±0.190 −0.545±0.150 −0.472±0.130
Weaklensing −0.774±0.145 −0.611±0.115 −0.529±0.099
Velocitydistribution −0.676±0.253 −0.533±0.200 −0.461±0.173
Moore Einaston=1 Einaston=6
−0.316±0.315 −0.416±0.415 −0.405±0.404
−0.423±0.116 −0.556±0.153 −0.542±0.149
−0.474±0.089 −0.624±0.117 −0.608±0.114
−0.413±0.155 −0.544±0.204 −0.530±0.199
3 Analysis of A586 cluster
3.1 Data Analysis
In the analysis of A586 we used the same data used in the original analysis
performed in [11], that is:
– member galaxy coordinates
– total mass inside a radius R, obtained from weak lensing measurementes
[12]:
M =(4.3 0.7) 1014M R=422Kpc (35)
⊙
± ×
– velocity dispersion from different sources, table 1.
Using the coordinatesof the galaxies that compose A586 we cancompute the
averagedistancebetweenagalaxyandthe centreofthe cluster, R .We start
h i
bydeterminingthecentreofthecluster.Thisisdonebycomputingtheaverage
declination and right ascension from the coordinates of the 31 galaxies in the
cluster.
The distance of a galaxy i, with coordinates (α ,δ ), to the centre of the
i i
cluster (α ,δ ) is given by:
c c
r2 =2d2[1 cos(α α )cos(δ )cos(δ ) sin(δ )sin(δ )]. (36)
i − i− c c i − c i
Using Eq. (36) to compute r for all galaxies in the reduced sample and then
i
taking the average:
R =223.6Kpc. (37)
h i
We can then proceed to compute the shape parameters of the various
densityprofilesasdescribedintheprevioussectionandthroughthemcompute
the virial ratio and the interaction between DE and DM.
10
Table 3 SummaryofA1689availabledata
Parameter Value Reference
ρ0 (10−25h2gr/cm3) 9.6±1.8 [28]
r0(h−1kpc) 175±18 [28]
kTx(KeV) 9.2+−00..43 [27]
σdyn(Km/s) 1172+−19293 [29]
Rdyn(h−1Mpc) 2.26 [29]
Table 4 NFWfittoA1689lensingdata.
Parameter Value Error
ρ0 (10−25h2gr/cm3) 4.513 0.330
r0(h−1kpc) 271.6 12.5
R2 0.999 −
3.2 Comments on the results for A586
We present our results in table 2. One sees that the analysis of this cluster
with a top hat profile is in agreement with the findings of Ref. [11]. We ob-
serveconsistentlythatusingweaklensingvelocitydispersion(asperformedin
Ref. [11]) yields a higher virial ratio. We stress that the weak lensing velocity
dispersion is not the most adequate data source as it introduces correlations
between mass and velocity estimation. It is crucialto avoidthese correlations
as the aim of this work is to detect deviations to the virial equilibrium and
interpret these as an effect due to DE-DM interaction.
4 Analysis of the A1689 cluster
In table 3 we gather the data available for A1689. As already mentioned, for
this cluster we find the density and shape parametersfor the different profiles
by directly fitting them to the 2D surface density obtained from weak and
strong lensing. The relation between the 2D surface density κ(R) and the
total 3 dimensional mass density ρ(r) is given by
2 ∞ ρ(r)rdr
κ(R)= , (38)
Σcrit ˆR √r2 R2
−
where Σ is the critical density for lensing, which depends on the back-
crit
ground cosmology and also on the lens and source sample. For the data we
are considering, Σcrit = 1.0122hg/cm2 [28], for H0 = 100hkm/s/Mpc and
observationally h 0.7. Then, by letting ρ(r) be NFW, isothermal sphere,
≃
Moore, or Einasto, we estimate the parameters of these four density profiles.
Intheensuinganalysis,wewillconsidertwomethods forestimatingveloc-
ity dispersion: galaxy dynamics and X-ray temperature, as shown in table 3.
While the velocity dispersion from the galaxy dynamics is a direct measure-
ment, converting the X-ray temperature into a velocity dispersion involves
assumptions that must be explained. Defining β = σ2(kT/µm )−1, where µ
v p