Constraining dark matter properties with Cosmic Microwave Background observations

1.4 MB·

by  Daniel B. Thomas

Draft version December 13, 2016
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
CONSTRAINING DARK MATTER PROPERTIES WITH COSMIC MICROWAVE BACKGROUND
OBSERVATIONS
Daniel B. Thomas, Michael Kopp, and Constantinos Skordis
DepartmentofPhysics,UniversityofCyprus
1,PanepistimiouStreet,2109,Aglantzia,Cyprus
(Dated: December 13, 2016)
Draft version December 13, 2016
ABSTRACT
6
We examine how the properties of dark matter, parameterised by an equation of state parameter w
1 and two perturbative Generalised Dark Matter (GDM) parameters c2 (the sound speed) and c2 (the
0 s vis
viscosity), are constrained by existing cosmological data, particularly the Planck 2015 data release.
2
We find that the GDM parameters are consistent with zero, and are strongly constrained, showing

c no evidence for extending the dark matter model beyond the Cold Dark Matter (CDM) paradigm.
e The dark matter equation of state is constrained to be within −0.000896<w <0.00238 at the 99.7%
D
confidencelevel,whichisseveraltimesstrongerthanconstraintsfoundpreviouslyusingWMAPdata.
Theparametersc2 andc2 areconstrainedtobelessthan3.21×10−6 and6.06×10−6 respectivelyat
2 s vis
the 99.7% confidence level. The inclusion of the GDM parameters does significantly affect the error
1
barsonseveralΛCDMparameters, notablythedimensionlessdarkmatterdensityω andthederived
g
parameters σ and H . This can be partially alleviated with the inclusion of data constraining the
] 8 0
O expansion history of the universe.
C
h. 1. INTRODUCTION nous matter, in a way that is naturally explained within
the CDM paradigm. Since the discovery of the original
p TheLambdaColdDarkMatter(ΛCDM)cosmological
- model provides a good phenomenological fit to current system, further examples have been observed (Harvey
o et al. 2015), providing more evidence in support of the
cosmologicaldata. Itmatchesthecurrentexpansionhis-
r CDM component of the standard cosmological model.
t tory of the universe including Supernovae (Perlmutter
s Despite the promising concordance of these results, no
et al. 1999; Riess et al. 1998) Baryon Acoustic Oscilla-
a dark matter candidate has been found experimentally.
[ tion (BAO) (Anderson et al. 2014; Ross et al. 2015) and
localmeasurements(Riessetal.2011),thematterpower Moreover, many physical models of dark matter are not

2 spectrum (Gil-Mar´ın et al. 2015) and the Cosmic Mi- as idealised as the pure CDM model. Thus, it seems
v crowaveBackground(CMB)data fromthe Plancksatel- timely, with the precision cosmological data currently
7 available,toexamineconstraintsondeparturesfromthis
lite (Planck Collaboration et al. 2015a).
9 idealised model.
In this standard picture the CDM is a crucial compo-
0 Work has been carried out to constrain the properties
nent and is described as a non-interacting initially pres-
5 of dark matter using astrophysical systems, particularly
sureless perfect fluid. This perfect fluid has an equa-
0 galaxy rotation curves, see e.g. Faber and Visser (2006);
tion of state parameter w identically zero as well as zero
1. sound speed and zero viscosity. These CDM characteris- Serra and Dom´ınguez Romero (2011); Barranco et al.
0 tics make it an extremely simple system to model phys- (2013); Boehm et al. (2014). Interestingly, it has been
6 ically, either perturbatively or through cosmological N- alsopointedoutthathalopropertiesdeviatefromexpec-
1 body simulations. tations of ΛCDM (Moore 1994; Jee et al. 2014; Boylan-
: Kolchin et al. 2011; Papastergis et al. 2015). As we are
v Astrophysicalsystemsalsoprovideimportantevidence
onlyinterestedinthepropertiesofdarkmatteronlinear
i for dark matter. Galaxy rotation curves (Persic et al.
X scales in this paper, we will not discuss halos further.
1996) were one of the earliest suggestions of dark mat-
Cosmological constraints on properties of dark mat-
r ter’sexistence(RubinandFord1970;Zwicky1933). Fur-
a thermore, gravitational-lensing measurements have been ter beyond CDM have been investigated within the con-
textofspecificmodels. InYang(2015), Planckdatawas
used to infer “mass maps” (see for example Lefor et al.
used to constrain the properties of dark matter, focus-
(2013)), that show the distribution of the CDM under
ing on the dark matter mass, decay rate and the ther-
the assumption that it traces the gravitational-lensing
mallyaveragedcrosssectionforannihilation. CMBdata
potential. These maps are consistent with what is ex-
from Planck as well as large scale structure observations
pected from theory (Massey et al. 2007). One of the
have been used to place constraints on interactions of
most-studied gravitational-lensing systems is the “bul-
dark matter with other Standard Model (SM) particles,
let cluster” (Clowe et al. 2006), a system consisting of
for instance, a possible dark matter-photon (Wilkinson
two colliding galaxy clusters. The “bullet cluster” sys-
et al. 2014a) or a dark matter-neutrino (Wilkinson et al.
tem shows a clear displacement between the map of the
2014b) interaction. The elastic scattering of dark mat-
gravitational-lensing potential and the map of the lumi-
ter with other SM particles also leaves an imprint on
spectral distortions on the CMB, as investigated lately
[email protected]
[email protected] in Ali-Hamoud et al. (2015), where constraints were put
[email protected] on the interaction cross-section. Dark matter may also
2 Thomas, Kopp, Skordis
interact with a dark radiation component (Buen-Abad arethetwoscalarmetricperturbationsinthisgaugeand
et al. 2015; Lesgourgues et al. 2015) which may further D =∇(cid:126) ∇(cid:126) −1∇(cid:126)2γ isatracelessspatialoperator. The
have its own dark recombination and dark atomic struc- GDijM hais bjack3grounijd density ρ¯ and isotropic pressure
g
ture (Cyr-Racine and Sigurdson 2013). Other well moti- P¯ , related by an equation of state w, such that
g
vatedmodelsareaxions(Hlozeketal.2015),collisionless
warm dark matter (Armendariz-Picon and Neelakanta
2014;Piattellaetal.2015)andcollisionlessmassiveneu- ρ¯˙g =−3H(1+w)ρ¯g (2)
trinos (Shoji and Komatsu 2010). P¯
In this paper we are interested in constraining the w = g . (3)
ρ¯
properties of dark matter by modelling the dominant g
component of structure formation beyond a simple pres- Unlike CDM, the GDM is allowed to have a pressure
surelessperfectfluid. Thiscouldeitherbeduetoamore perturbation Π and shear perturbation Σ (Hu 1998),
g g
complicated dark matter model, or just a more precise in addition to the usual density (δ ) and velocity (θ )
g g
modelling of CDM, as in the case of the Effective Field perturbations. The perturbations obey the Euler and
Theory of Large Scale Structure (EFTofLSS) (Baumann continuity equations, as well as two postulated closure
et al. 2012). For convenience throughout this paper, we equations for the pressure perturbation and the shear
consider “CDM” to be the modelling of the dark matter (Hu 1998), see also Kopp et al. (2016) for an extended
component as a pressureless perfect fluid. Thus, both discussionandanexplanationofournotation. Theequa-
more complicated models or a more precise modelling tionsthattheGDMobeysinaspatiallyflatUniverseare
of the evolution of CDM are considered to be “beyond as follows
CDM”, in the sense that they are a change to how the (cid:18) (cid:19)
1
dark matter is usually modelled. δ˙ =3H(wδ −Π )−(1+w) h˙ −∇(cid:126)2θ (4)
We parameterise the departures from CDM accord- g g g 2 g
ing to the Generalised Dark Matter (GDM) model (Hu Π 2
1998). As we discuss in more detail in a companion pa- θ˙g =(3c2a−1)Hθg+ 1+gw + 3∇(cid:126)2Σg (5)
per (Kopp et al. 2016), the GDM parameterisation nat-
4 1
urally arises in more realistic models, for instance, the Σ˙ =−3HΣ + c2 (θ − ν˙) (6)
EFTofLSS (Baumann et al. 2012), non-equilibrium ther- g g (1+w) vis g 2
modynamics (Landau and Lifshitz 1987), the effective Π =c2δ +3H(1+w)(c2−c2)θ , (7)
g s g s a g
theory of fluids (Ballesteros 2015), tightly coupled fluids
and scalar fields. In Kopp et al. (2016) we also study wherewehaveintroducedtwosoundspeedsandaviscos-
more closely the physical effects and the interpretation ity (the equivalent equations in a spatially curved Uni-
of the GDM parameters. verse and in a general gauge can be found in Kopp et al.
Several authors have used the GDM model to put cos- (2016)). The sound speed c2s and viscosity c2vis are pa-
mological constraints on dark matter properties (Mu¨ller rameters of the GDM model, and the adiabatic sound
2005; Calabrese et al. 2009; Li and Xu 2014; Wei et al. speed c2a is defined as
2013; Kumar and Xu 2012; Xu 2014; Xu and Chang P¯˙ w˙
2013), as we also do in this paper. We perform a more c2 = g =w− . (8)
detailedcomparisontotheseworksinsection4.3. Apart a ρ¯˙g 3H(1+w)
fromdescribingdarkmatter,theGDMparameterisation
In the present work we consider only constant equation
or a subset of it, has been used in some form in the lit-
of state w so that c2 =w.
eratureforseveraldifferentpurposesincludingneutrinos a
Ingeneral,thesoundspeedc2 causesoscillationsinthe
(Trotta and Melchiorri 2005), dark energy (Weller and s
densityperturbationδ belowtheJeanslength,although
Lewis 2003; Bean and Dor´e 2004) and unified dark mat- g
if c2 and c2 become comparable in size then there are
ter/dark energy models (Kumar and Xu 2012; Yang and s vis
Xu 2013). no sound waves. For convenience, we refer to c2 as the
s
Thispaperisorganisedasfollows. Section2containsa sound speed, although this is only true for c2 (cid:29) c2
s vis
brief introduction to the GDM model. Section 3 details (Kopp et al. 2016). The viscosity c2 damps the density
vis
themethodanddatathatwereusedinouranalysis,and perturbations. For details of the model and investiga-
our results are presented in section 4. We conclude in tions into which physical models map to GDM, see Hu
section 5. (1998); Kopp et al. (2016).
To summarize, replacing CDM with GDM amounts to
2. ASHORTOVERVIEWOFTHEGDMMODEL introducingthreeparametricfunctionstothemodel: the
In this section we present the basic ingredients of the GDM equation of state w(η), the sound speed c2(η,(cid:126)x)
s
GDM model (Hu 1998). We work in the synchronous and the viscosity c2 (η,(cid:126)x). In this work we shall assume
vis
gaugeasthisisthegaugemostcommonlyusedinnumer- that all three take constant values, i.e. with no time or
ical Boltzmann codes. We are interested only in scalar spacedependence. Whilstthisisnotthecaseformanyof
perturbations, so the metric takes the form thephysicalmodelsthatGDMrelatesto, theseconstant
parameterscanbeconsideredasanulltestastowhether
(cid:26) (cid:20) (cid:21) (cid:27)
1
ds2 =a2 −dη2+ (1+ h)γ +D ν dxidxj (1) there is any evidence for departures from CDM.
3 ij ij
3. METHODANDDATA
where a is the scale factor in conformal time η, γij is a Inordertoperformouranalysis, wemodifiedtheCos-
flat spatial metric with covariant derivative ∇(cid:126), h and ν mic Linear Anisotropy Solving System (CLASS) code
Planck constraints on GDM 3
TABLE 1
Constraints on the GDM parameters for the two types of models and different combinations of experiments, for the
95.5% and 99.7% credible regions.
Λ-wDM Λ-GDM
102w 102w 106c2,upperbounds 106c2 ,upperbounds
s vis
Likelihoods 95.5% 99.7% 95.5% 99.7% 95.5% 99.7% 95.5% 99.7%
PPS 0.007+0.463 0.007+0.676 −0.040+0.473 −0.040+0.700 3.31 6.31 5.70 11.3
−0.466 −0.673 −0.468 −0.701
PPS+Lens 0.087+0.439 0.087+0.662 0.066+0.434 0.066+0.654 1.92 3.44 3.27 5.99
−0.448 −0.648 −0.427 −0.642
PPS+Lens+HST 0.256+0.217 0.256+0.322 0.259+0.216 0.259+0.321 1.87 3.38 3.11 5.56
−0.217 −0.323 −0.218 −0.326
PPS+Lens+BAO 0.063+0.108 0.063+0.163 0.074+0.111 0.074+0.164 1.91 3.21 3.30 6.06
−0.112 −0.164 −0.110 −0.163
(Lesgourgues 2011). CLASS numerically solves the sionless density ω ,2 the baryon dimensionless density
g
Boltzmann equation for each relevant component cou- ω , 100×θ whereθ istheratioofthesoundhorizonto
b s s
pled to the Einstein equations and calculates the CMB the angular diameter distance at decoupling, the optical
and matter power spectra given a set of model param- depth τ, ln(1010A ) where A is the amplitude of scalar
s s
eters. The CLASS code already includes an additional perturbations and the spectral index of scalar perturba-
dark energy fluid component with an equation of state tions n (6 parameters total). For the Λ-wDM model we
s
and sound speed (Lesgourgues and Tram 2011) which varied in addition to the Λ-CDM parameters the GDM
we further modified to include the viscosity c2 and to equation of state w (7 parameters total) and for the Λ-
vis
allow this fluid to work as a replacement for dark mat- GDM model we further varied the sound speed c2 and
s
ter rather than for dark energy. We also independently the viscosity c2 (9 parameters total). In addition to
vis
modified a different Boltzmann code (DASh) (Kapling- the main MCMC parameters we considered three more
hat et al. 2002) to include the full GDM parameterisa- derived parameters: the Hubble constant H (in units
0
tion. Weperformedafullcomparisonbetweenthecodes, of km/s/Mpc), the cosmological constant relative den-
including the background evolution, perturbation evolu- sity Ω and the RMS matter density fluctuation σ . We
Λ 8
tion, the Cls, matter power spectrum and lensing poten- used flat priors on all of the parameters and restricted
tial. Thenumericaldifferenceofthetwocodesinthecase c2 and c2 to be non-negative as dictated by theoreti-
s vis
oftheGDMmodelissimilartothecorrespondingdiffer- cal models (Kopp et al. 2016), and τ to be greater than
ence in the case of ΛCDM, within ∼ 0.1%. This level 0.01. The remaining fiducial cosmology was set as fol-
of agreement holds for all quantities in both the syn- lows. The spatial curvature was set to zero with a cos-
chronous gauge and the conformal Newtonian gauges. 1 mological constant Ω making up the remainder of the
Λ
We investigated the constraints on the GDM parame- matter content and the primordial helium fraction was
ters using a Markov Chain Monte Carlo (MCMC) ap- set to Y = 0.24773. We used two massless and one
He
proach, carried out using the publicly available Mon- massiveneutrinowithmass0.06eVkeepingtheeffective
tePython code (Audren et al. 2013), which implements number of neutrinos to N =3.046. 4
eff
the Metropolis-Hastings algorithm. The MontePython The main dataset that we used was the Planck 2015
code calls the CLASS code through a Python wrapper. data release (Planck Collaboration et al. 2015b) of the
Our covariance matrix for the final runs was generated CMB anisotropies. We used the low-l likelihood and the
using the standard methodology: first some initial runs full TT, EE and TE high-l likelihood with the complete
werecarriedoutwithasimpleestimateddiagonalcovari- “not-lite” set of nuisance parameters. The low-l likeli-
ancematrix,andthesewereusedtogenerateacovariance hood consists of the TT, EE, TE and BB spectra up
matrix that was used for the next run. After several it- to l = 29, whereas the high-l spectra are from l = 30
erations of this we had a covariance matrix that gave an upwards. See the Planck papers and wiki5 for full de-
appropriate acceptance rate for the steps in the chains tails of these likelihoods. As we always used the high-
and was suitable for the final runs. l and low-l likelihoods together, this combination will
We considered three types of models: the standard be referred to simply as Planck Power Spectrum (PPS).
ΛCDM model (used for comparison), the Λ-wDM model WeincludedGaussianpriorsonthenuisanceparameters
where CDM is replaced with GDM but the speed of
sound and viscosity are set to zero, and the Λ-GDM
model where all three GDM parameters are included. 2 ForΛCDM,theparameterωg isequaltoωc theCDMdimen-
sionlessdensity,asinthatcasew=c2=c2 =0.
Note that in all three models, the dark energy compo- 3 The value for YHe is the (roundsed) vvaislue quoted by Planck
nentisalwaysmodelledasacosmologicalconstantΛand 2013 (Planck Collaboration et al. 2014). It was verified by a pre-
only the dark matter component is modified. liminaryFisher-matrixanalysisthatincludingYHe asaparameter
In all models we varied the standard ΛCDM parame- would lead to minimal changes to the constraints on the GDM
parameters.
terswiththeMCMCalgorithm: thedarkmatterdimen-
4 In CLASS, this required us to set the effective number of
massless neutrinos parameter to Nur = 2.0328 and the neutrino
temperature parameter to T = 0.71611. Note that this is
ncdm
1 TheactualdifferencebetweenthecodesinthecaseofΛCDM slightly larger than the instantaneous decoupling value ratio to
rangesfromaround10−4 onsmallscalestoaround10−3 onlarge thephotontemperatureof(4/11)1/3,seetheCLASSexplanatory
scales. However, as DASh is an older code and not as optimized parameterfileorManganoetal.(2005)fordetails.
asCLASSwebelieveCLASStobemoreaccurate. 5 http://wiki.cosmos.esa.int/planckpla2015/index.php/
4 Thomas, Kopp, Skordis
2 3 4 5 10 15 25 (cid:230) (cid:76)CDMparametersadjusted
8 88
(cid:230) (cid:230) (cid:76)CDMbestfit
(cid:230) (cid:230)
(cid:45)TT52(cid:64)(cid:72)(cid:76)(cid:68)10T(cid:68)0l46 (cid:230) (cid:230) (cid:45)TT52(cid:64)(cid:72)(cid:76)(cid:68)10T(cid:68)0l46(cid:230)(cid:230)(cid:230)46(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) Planwwc(cid:61)(cid:61)k(cid:43)(cid:45)21(cid:137)(cid:137)1100(cid:45)(cid:45)33
2 22 (cid:230) (cid:230)
22 3(cid:230)3 44 55 1(cid:230)100(cid:230)1(cid:230)(cid:230)155(cid:230)(cid:230)2255(cid:230)(cid:230) (cid:230) 500 10(cid:230)(cid:230)0(cid:230)(cid:230)0(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:76)(cid:230)(cid:230)1C(cid:230)(cid:230)5(cid:230)D(cid:230)0(cid:230)M(cid:230)(cid:230)0(cid:230)(cid:230)p(cid:230)(cid:230)a(cid:230)(cid:230)r(cid:230)a(cid:230)m(cid:230)2(cid:230)(cid:230)e(cid:230)0(cid:230)t(cid:230)e0(cid:230)r(cid:230)0(cid:230)s(cid:230)(cid:230)f(cid:230)i(cid:230)x(cid:230)(cid:230)e(cid:230)d(cid:230)(cid:230)2(cid:230)(cid:230)5(cid:230)(cid:230)(cid:230)00
8 (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)88(cid:230)l(cid:230)(cid:230)
(cid:45)52(cid:76)(cid:68)0T06 (cid:230)(cid:230) (cid:45)52(cid:76)(cid:68)0T0(cid:230)(cid:230)66(cid:230)(cid:230)(cid:230) (cid:230)
GDFiMg.m1o.—delT. hTehe1dgrpeeonst(edroiotr-sdafsohretdh)ecuwrvpeairsafmorettehrewPiltahnicnktPhoewΛer- TT(cid:64)(cid:72)1(cid:68)l4 (cid:230) TT(cid:64)(cid:72)1(cid:68)l4(cid:230)(cid:230)(cid:230)4(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)
Spectrum (PPS) dataset, the black (solid) curve is for the combi- 2 22 (cid:230) (cid:230)
niHsaSftToiornpthroieofrPPaPPnSSdaatnnhddeLPbellunaesncd(kdaatlaeshnseestidn)cgocm(uLrbevinnesai)ts,ioftnohrewtrihteehdPt(hdPeoSta,tdeLddei)ntiscounarnvodef 2 3(cid:230) 45 1(cid:230)0(cid:230)1(cid:230)(cid:230)5(cid:230)(cid:230)25(cid:230)(cid:230) 500 10(cid:230)(cid:230)0(cid:230)(cid:230)0(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)1(cid:230)(cid:230)5(cid:230)(cid:230)0(cid:230)(cid:230)(cid:230)0(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)2(cid:230)(cid:230)(cid:230)0(cid:230)(cid:230)0(cid:230)(cid:230)0(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)2(cid:230)(cid:230)5(cid:230)(cid:230)(cid:230)00
BAOdatasetcombination. (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230) l
(cid:230) (cid:230)
(also varied as MCMC parameters) as recommended by Fig. 2.—TheΛCDMbest-fitCsalongwithCsplottedfortwo
l l
the Planck collaboration and implemented in the Mon- valuesofw,closetoour99.7%CLvalues. Inthelowerpanel,the
tePython code. We ran further chains that included standard cosmological parameters are held at the ΛCDM best-fit
values when w is varied, and in the upper panel they are var-
the Planck lensing potential likelihood (hereby referred
ied in order to mask the effect of w. All curves as well as the
to as “Lens”) in addition to the low-l and high-l likeli- Planckdatapointsandtheirerrorbarshavebeentransformedby
hoods, and found that this made a significant difference C →C +10(CΛCDM−C ),effectivelyenlargingtheresidualsby
(cid:96) (cid:96) (cid:96) (cid:96)
totheconstraintsonc2 andc2 . Finally,wealsoinvesti- 10 and shifting the mean values accordingly, to make them more
s vis visible. Comparisonoftheupperandlowerplotsshowhowthede-
gated the effect on the constraints of other cosmological
generacies that w has with the standard cosmological parameters
datasets that constrain the expansion history of the uni- acttolimittheconstraintsonw.
verse. The two datasets used for this were the HST key
project (Riess et al. 2011) and Baryon Acoustic Oscil- 4. RESULTS
lation (BAO) data (Anderson et al. 2014; Beutler et al. 4.1. Constraints on the GDM parameters
2011). The HST likelihood was implemented by a Gaus-
Our main results are displayed in table 1, where we
sian prior on H around H = 73.8±2.4, whereas the
0 0 tabulate the 95.5% and 99.7% credible regions for w, c2
BAO data constrains the distance combination s
and c2 for the Λ-wDM and Λ-GDM models for each
vis
D (z)=(cid:2)cz(1+z)2D2(z)H−1(z)(cid:3)31 , (9) of the different choices of datasets: PPS, PPS+Lens,
V A PPS+Lens+HSTandPPS+Lens+BAO.Theconstraints
whereD (z)istheangulardiameterdistancetoredshift on the common parameters to ΛCDM are displayed in
A
z, H(z) is the Hubble parameter and c is the speed of table 2.
light. The first result to note from table 1 is that all three
For the final results we generated a set of 8 chains for GDM parameters are consistent with zero in all cases.
eachdatasetcombinationwithineachofthethreemodels The constraints on the GDM parameters are strong re-
considered. Convergence of the chains was tested using gardless of the dataset combination used. The strongest
the Gelman-Rubin 1−R (Andrew Gelman 1992) test, constraints come from the PPS+Lens+BAO combina-
which compares the variance within each chain to the tion where −0.00109 < w < 0.00225 at the 99.7% CL
variance between the chains in order to assess whether in the case of Λ-wDM and −0.000896 < w < 0.00238 at
the individual chains have converged to the same pos- 99.7% CL in the case of Λ-GDM. For the latter model
terior distribution. Convergence was determined by re- theparametersc2 andc2 areconstrainedtobelessthan
s vis
quiring 1−R to be smaller than 0.01 for all parameters. 3.21×10−6 and6.06×10−6 respectivelyatthe99.7%CL.
Note that GDM is defined for linear perturbations The 1-D posteriors for the w parameter in the Λ-GDM
only, thus we have made no use of halofit to model non- model are plotted in Fig. 1 for all four combinations of
linearities. InΛCDM,halofitmakesasmalldifferenceto datasets. We now examine more closely how these con-
thelensingpotentialandthusalsotothelensedCls6. For straintsvarybetweenthedifferentchoicesofexperiment.
parametervaluesaroundthe99.7%confidencelevel(CL)
of our constraints, the GDM effects are approximately Consider first the constraints from the Planck Power
four times larger than the effect of halofit for ΛCDM, Spectra. As discussed in Hu (1998) and Kopp et al.
but in the opposite direction. Thus, we expect that the (2016), the main effect of increasing (decreasing) w is to
constraints would not undergo a significant change if a shifttheradiation-matterequalitytoearlier(later)times
full non-linear analysis was performed. which in turn decreases (increases) the acoustic driving
effectcausedbythetime-variationofthepotentialwells.
6 WethankSteffenHogstatzforbringingthistoourattention. The result is a decrease (increase) of the anisotropies
Planck constraints on GDM 5
2.(cid:180)10(cid:45)6 2.(cid:180)10(cid:45)6
4.(cid:180)10(cid:45)6
2cs 2cs
1.(cid:180)10(cid:45)6 1.(cid:180)10(cid:45)6
3.(cid:180)10(cid:45)6
s
2vi
0.1 0.11 0.12 0.13 (cid:45)0.004 0 0.004 c
4.(cid:180)10(cid:45)6 Ωg 4.(cid:180)10(cid:45)6 w 2.(cid:180)10(cid:45)6
3.(cid:180)10(cid:45)6 3.(cid:180)10(cid:45)6
2cvis 2cvis
2.(cid:180)10(cid:45)6 2.(cid:180)10(cid:45)6 1.(cid:180)10(cid:45)6
1.(cid:180)10(cid:45)6 1.(cid:180)10(cid:45)6
1.(cid:180)10(cid:45)6 2.(cid:180)10(cid:45)6
0.1 0.11 0.12 0.13 (cid:45)0.004(cid:45)0.002 0.000 0.002 0.004 0.006
Ωg w c2
s
Fig. 3.— The68.3%and95.5%credibleregionsforcombinations Fig. 4.—The68.3%and95.5%credibleregionsforc2 andc2,
ofparameterswithintheΛ-GDMmodelforthePSS+Lensdataset vis s
showingthestrongnegativecorrelationbetweentheseparameters.
combination. The displayed combinations are: contours for the
ThesecontoursarefortheΛ-GDMrunforPlanck+Plancklensing.
ωg-c2s combination(upperleft); contoursfortheωg-c2vis combina- Thered(dashed)linesdenotelinesofconstantc2+ 8 c2 ,showing
tion(lowerleft);contoursforthew-c2 combination(upperright); s 15 vis
s thatthedegeneracyiswellfitbythek parameterasexplainedin
contoursforthew-c2 combination(lowerright). Thelackofcor- d
vis thetext.
relation shown in these plots indicates a clear split between the
perturbativeGDMparametersandbackgroundGDMparameters.
compensated as for the temperature C s in figure 2.
l
around the first and second peak. As further explained The addition of the data related to the expansion his-
in Kopp et al. (2016) potential decay after recombina- tory does have a strong affect on w, as expected. The
tion is related only to a time varying total equation of 95.5% and 99.7% CL constraints on w are tighter by up
state (that is, the combined equation of state from all to a factor of two when the HST prior is included. In
species), hence, changing w has only a very mild effect addition, the mean value is also increased to a value just
on the early Integrated Sachs-Wolfe (ISW) term and the outsideofthe68.3%credibleregion, althoughthisisnot
lensingpotential. ThismeansthattheISWeffectsonthe strongly significant. Note that in this case, w = 0 is
CMB TT and TE power spectra and the lensing effects excluded at the 95.5% CL. There is a known tension be-
on all the CMB power spectra are of similar strength to tween the H value from Planck and the HST prior, so
0
ΛCDM and do not drive the constraints on w. Finally, that one cannot yet safely conclude that this is a signa-
changing w results in a change to θ that induces a lat- tureofnewphysics. Wehaverepeatedthisrunusingthe
s
eralshiftinthelocationoftheCMBpeaks,althoughthis alternativevalueH =70.6±3.3fromtheanalysisinEf-
0
is overshadowed by the changes of the peak heights. stathiou (2014), with the result that the posterior shifts
Infigure2,weplotthetemperatureC sforthebest-fit to the left and widens slightly compared to the standard
l
ΛCDMparameters,plustwocurveswithwvaluesaround HST value. In this case w is consistent with zero.
our 99.7% CL limits. In the bottom panel, the standard The change to the mean value does not appear when
6 ΛCDM parameters are held fixed when w is varied. BAOdataisusedinsteadoftheHSTprior. Furthermore,
However,intheupperpanelalloftheΛCDMparameters with the addition of the BAO data to the Planck combi-
were varied in addition to w, guided by the results from nation of PPS+Lens, the constraints on w are improved
our chains. Comparing the top and bottom plots, the by a factor of four. These are the tightest constraints on
importance of the degeneracies of w with the standard w presented in this paper. The 1-D posteriors for w for
cosmological parameters can be seen: these degeneracies theΛ-GDMmodelcanbeseeninFig.1,wherethegreen
act to mask the effects of w. Thus, if the standard cos- (dot-dashed)showstheposteriorforPPSonly,theblack
mological parameters were known, then the constraints (solid)curveforthePPS+Lenscombinationandthered
on w would be significantly tighter. Note that in these (dotted) and blue (dashed) curves show the posteriors
plots the residuals of the data points and the w curves for the addition of the HST and BAO data respectively.
with respect to the ΛCDM best-fit curve have been mul- Thechangestotheconstraints, andtheshiftofthepeak
tiplied by 10 in order to make the differences clearer, by when the HST data is included, are all clearly visible.
using the transformation C →C +10(CΛCDM−C ). We now turn to the Λ-GDM model and in particular
(cid:96) (cid:96) (cid:96) (cid:96)
The addition of CMB lensing to the temperature and on the constraints to the GDM perturbative parameters
polarisation C s has little effect on the posterior distri- c2 and c2 . Firstly, the constraints on w are not signifi-
l s vis
bution for w, other than a slight shift of the mean to cantlyaffectedbytheinclusionoftheotherGDMparam-
positive values without changing the width of the distri- eters, c2 and c2 , as can be seen by the similarity of the
s vis
bution. This is not surprising as the lensing is generally constraintsonw betweentheΛ-wDMandΛ-GDMmod-
less constraining than the other C s from Planck due to els for each dataset combination. This is to be expected
l
the larger noise. Furthermore, although w does have as our discussion above shows that w affects the CMB
someeffectsonthelensingpotential,suchascausingthe differently than the two perturbative GDM parameters.
gravitational potentials to freeze out at different values In the right panels of Fig. 3 we show the 2d-contours in
(see Kopp et al. (2016)), these effects can be partially thew-c2 (upperright)andw-c2 (lowerright)planesre-
s vis
6 Thomas, Kopp, Skordis
2 3 4 5 10 15 25 1.4
8 88 1.2 (cid:230) (cid:76)CDMbestfit
c2(cid:61) 3(cid:137)10(cid:45)6
7 1.0 s
0 (cid:230)
(cid:45)TT52(cid:64)(cid:72)(cid:76)(cid:68)10T(cid:68)0l46 (cid:45)TT52(cid:64)(cid:72)(cid:76)(cid:68)10T(cid:68)0l4646 (cid:76)ccv2s2Ci(cid:61)s(cid:61)D M36(cid:137)(cid:137)11b00es(cid:45)(cid:45)t66fit ΦΦ(cid:72)(cid:43)(cid:76)(cid:137)ll11(cid:68)l000...468 (cid:230) (cid:230) (cid:230)(cid:230) Pcv2laisn(cid:61)c(cid:230)k6(cid:137)10(cid:45)6(cid:230)
0.2 (cid:230)
2 22
0.0
0 100 200 300 400
l
2 3 45 1015 25 500 1000 1500 2000 2500
l Fig. 6.— The lensing potential power spectrum plotted for the
Fig. 5.—TheTTspectrumplottedfortheΛCDMbest-fitvalues ΛCDM best-fit values and for values of c2s and c2vis that are close
and for values of c2 and c2 that are close to our 99.7% CL. The toour99.7%CL.TheredpointsshowthePlanckdatawitherror
s vis bars.
residuals of the c2,c2 (cid:54)=0 curves with respect to the best-fit Cs
s vis l
havebeenmultipliedbyafactorof50tomakethemmorevisibleas
inFig.2. ThekeydifferencebetweentheGDMandΛCDMcurves is a good fit to the direction of the contours, providing
isthereducedsmoothingofthepeaksintheGDMmodel,caused
further evidence that this is the cause of the degeneracy
bythereducedlensingpotential.
between these parameters.
spectively for the PPS+Lens dataset combination. The Since c2s and c2vis do not affect the expansion history,
other dataset combinations give similarly looking con- the inclusion of either HST or BAO data has little effect
tours. Theseplotsshowthatw isnotstronglycorrelated ontheirconstraintsandthisispreciselywhatisobserved
with either of the other GDM parameters. This lack of in table 1. However, the inclusion of the CMB lensing
correlation indicates a clear split between the perturba- potentialdata(Lens)doeshaveasignificanteffectonthe
tiveGDMparametersandbackgroundGDMparameters. c2 andc2 constraints,asopposedtowhatwasfoundfor
s vis
MovingontotheperturbativeGDMparametersthem- theconstraintsonw. Inparticular,sincepotentialdecay
selves, the main effect of non-zero c2 and c2 , as found inΛ-GDMafterrecombinationleadstoshallowerlensing
s vis
inKoppetal.(2016),istocausethegravitationalpoten- potentials, the result is a smaller lensing potential auto-
tial to decay. Potential decay results in two main effects correlation power spectrum plus a larger temperature-
ontheCMBpowerspectrumwhichdrivetheconstraints lensing potential cross-correlation due to the larger ISW
on c2 and c2 . The first is a continuous ISW effect af- contribution to the temperature spectrum. As the lens-
s vis
ter recombination until the present time, which becomes ing potential probes low redshifts, and the effects of the
strongerwithincreasingvaluesoftheperturbativeGDM GDM parameters are cumulative over time, the effect
parameters. Thesecondeffectisonthelensingpotential, on the lensing potential is stark. This is why the con-
alsoexaminedinKoppetal.(2016). Sincethelensingpo- straints on these parameters greatly improve when the
tential is directly sourced by the gravitational potential, Planck lensing data is included.
potentialdecayleadstoasmallerCMBlensingsignalon In figures 5 and 6, we show the TT and lensing poten-
the CMB power spectra. As the CMB lensing decreases tialClsforthebest-fitΛCDMvalues,andalsoforvalues
thepeakheightsandincreasesthepeaktroughs(without of c2 and c2 that are close to our 99.7% CL. In the TT
s vis
changingtheirlocation),thereducedlensingpotentialin plot, the residuals (differences between the GDM mod-
Λ-GDM results in higher peak heights and lower peak els and the best-fit ΛCDM model) have been multiplied
troughs compared to either Λ-wDM or ΛCDM models. by 50 in order to make them visible as in Fig. 2. The
lensingpotentialplotshowsthesignificantchangeinthe
Thescalek (η)atconformaltimeη atwhichpotential
d lensing potential due to GDM parameters. In the TT
decayhappensinapureGDMuniverseisapproximately
plot, the differences to ΛCDM also arise from lensing:
(cid:114) 8 the smaller lensing potential results in reduced smooth-
k−1(η)=η c2+ c2 . (10) ing of the peaks. These plots confirm that, for constant
d s 15 vis
values of these parameters, it is predominantly the lens-
Forlengthscaleslargerthank−1,theeffectsofthesetwo ing that is generating the constraints. This could be dif-
d
parameters on the gravitational potential are indistin- ferentiftheparameterswerenotconstant. Forexample,
guishable. For the scales relevant to the CMB, this in- the effect of the lensing would be smaller if the param-
ducesadegeneracybetweenthetwoparameters. Naively, eters scaled as a−2 as is the case for warm dark matter
the expression for k suggests a negative correlation be- (Lesgourgues and Tram 2011). The values of c2 and c2
d s vis
tween the parameters such that the errors on c2s should herewerechosentoreflectthekd degeneracy,andindeed
be about half the size of those on c2 . This is approxi- there is little difference between the two GDM curves.
mately what is found for the errorsviins table 1. In addi- We can translate the upper bound on c2s and c2vis into
tion, see Fig. 4 where we show the 2-D contours in the an upper bound on the ratio of k−1 to the Hubble scale.
d
c2-c2 plane. In this figure, we have plotted lines that Atthe99.7%CL,thisisapproximately2.13×10−3(using
s vis
correspond to constant k−1. The direction of these lines η ≈H−1),sothelargestcurrentlyallowedscaleonwhich
d
Planck constraints on GDM 7
0.004 0.004
0.002 0.002
w w
0.000 0.000
(cid:45)0.002 (cid:45)0.002
(cid:45)0.004 (cid:45)0.004
0.11 0.12 0.13 60 65 70 75 80
Ω H
g 0
Fig. 7.—The68.3%and95.5%credibleregionsinthew-ωg plane
(left panel) and in the w-H0 plane (right panel). The blue (outer)
contoursarefortheΛ-GDMrunforPPS+Lensdatasetcombination,
andtheyellow(inner)contoursareforthePPS+Lens+BAOdataset
combination. Theleftpanelshowsastrongnegativecorrelationbe-
tween these parameters due to their effects on the radiation-matter
equality,whereastherightpanelshowsastrongpositivecorrelation Fig. 8.— The 1d posteriors for ωg (left panel) and H0 (right
betweenthese parameters due totheir effect on thebackgroundex- panel) in the case of ΛCDM. The black (solid) curve is with the
pansionhistory. combinationofPPS+Lens,theblue(dashed)curveiswithPPS
+Lens+BAOandthered(dotted)curveisforPPS+Lenswith
GDM can modify cosmology is significantly below the
theadditionHSTprior.
Hubble scale. In this sense we consider our constraints
on the GDM parameters to be strong. by the inclusion of either the HST or BAO data. In
Fig. 8 we also plot the 1-D posteriors for ω and H ,
g 0
4.2. Constraints on the standard ΛCDM parameters. for the combination of PPS + Lens with the inclusion of
As expected, the inclusion of the GDM parameters theHSTprior,depictedbythered(dotted)curve. Once
worsens the constraints on some of the ΛCDM param- again, there is little difference for these parameters be-
eters, notably ω , τ and the derived parameters H and tween the Λ-wDM and Λ-GDM models and we do not
g 0
σ , as seen in table 2. plot the former. We see that the addition of the HST
8
The increased error bars for ω and H are due to prior significantly improves the constraints on these pa-
g 0
the strong degeneracies with w, as all three parameters rameters. There is still an offset of the mean value of
primarily affect the background expansion. See Fig. 7 the posterior compared to ΛCDM, which is now more
for these degeneracies, which show the 2-D contours in significant due to the reduced width of the posteriors.
the w-ω and w-H planes respectively. As explained in Fig. 8 also shows the posteriors for the inclusion of
g 0
the previous subsection, ω and w shift the radiation- the BAO data rather than the HST prior, depicted by
g
matter equality which in turn affects the heights of the the blue (dashed) curve. For this combination, the con-
firstandsecondpeaksintheCMBtemperaturespectrum straintsontheparametersareeventighter,withthecon-
(seeKoppetal.(2016)). Thisresultsinanegativecorre- straintsfromtheΛ-GDMmodel(andsimilarlyfortheΛ-
lationbetweenthetwoparameters,hencethedegeneracy wDM model which is not plotted) now being almost as
seen on the left panel of Fig. 7. strong as those from the ΛCDM runs, despite the extra
As changing w also has an effect on the expansion his- parameters. Moreover, for this combination of datasets,
tory, we expect to geta degeneracy with H . Inparticu- there is no significant offset of the means compared to
0
lar,asiswellknownH0 andωg arenegativelycorrelated ΛCDM. The shrinking of the w-ωg and w-H0 contours
eveninΛCDM(Hinshawetal.2013)andsincew andω with the inclusion of BAO is also clearly visible in Fig.
g
arealsonegativelycorrelated, weexpectw andH tobe 7.
0
positively correlated. Indeed this is verified on the right In the Λ-wDM model, w has a degeneracy with σ8.
panel of Fig. 7. However, in the Λ-GDM model, this is subdominant to
The full 1-D posteriors for the PPS + Lens dataset the stronger degeneracy between σ8 and the perturba-
combination for ωg and H0 can be found in Fig. 8. In tive parameters c2s and c2vis. This is due to the strong
this plot, the black (solid) curves show the posteriors effect that these two parameters have on the growth of
for the ΛCDM model (upper panel) and Λ-GDM model structure, as found in Kopp et al. (2016). These two pa-
(lowerpanel)respectively. Inadditiontothebroadening rameters both greatly reduce the growth of the matter
of the constraints, there is also a slight shift away from perturbationsonlengthscalesbelowk−1. Sinceσ isde-
d 8
themeanvaluefoundintheΛCDMmodel,however,this finedtobetheamplitudeofthisspectrum,thisresultsin
shift remains within the 68.3% credible region for both astrongdegeneracybetweenthesetwoGDMparameters
parameters. The posterior for Λ-wDM is similar to the and σ , which is sufficiently strong to replace the usual
8
Λ-GDMcase7 asisexpectedfromthelackofcorrelation degeneracy between σ and A . In Fig. 9, we show the
8 s
oftheperturbativeGDMparameterswitheitherworω 2D contours in the σ -c2 and σ -c2 planes respectively.
g 8 s 8 vis
as shown in Fig. 3, and we choose not to plot it. Fixing either of σ and c2 (or c2 ), and increasing the
8 s vis
Astheω andH parametersprimarilyaffecttheback- other, results in A increasing. Thus we see a negative
g 0 s
groundexpansionhistory,theirconstraintsareinfluenced correlation between σ and these GDM parameters.
8
It is interesting to look more closely at σ for the the
8
7 albeitwithaslightlygreatershiftinthemean combination of Planck + Planck lensing data. The 1-
8 Thomas, Kopp, Skordis
0.9 0.9
0.8 0.8
Σ80.7 Σ80.7
0.6 0.6
0.5 0.5
1(cid:137)10(cid:45)6 2(cid:137)10(cid:45)6 1(cid:137)10(cid:45)6 2(cid:137)10(cid:45)6 3(cid:137)10(cid:45)6 4(cid:137)10(cid:45)6 Fig. 10.— The 1d posteriors for σ8 (left panel) and τ (right
c2 c2 panel)forthecombinationofPlanckPrimaryandPlanckLensing
s vis
data;theblue(dashed)curveisforΛCDM,thered(dotted)curve
Fig. 9.— The 68.3% and 95.5% credible regions of σ8 with the isfortheΛ-wDMrunandtheblack(solid)curveisfortheΛ-GDM
perturbative GDM parameters c2 (left) and c2 (right) for the run.
s vis
Λ-GDM model using the PPS + Lens dataset combination. The
strongnegativecorrelationinthesetwopairsofparametersisdue
4.3. Comparison to previous work
to the strong reduction in structure growth induced by the GDM
parameters. Although a full comparison of the GDM parameters
as a replacement for CDM has not been previously per-
dimensional posteriors are shown in Fig. 10 for ΛCDM
formed, several works have looked at including an equa-
(blue,dashed),Λ-wDM(red,dotted)andΛ-GDM(back,
tion of state for dark matter.
solid). From this plot it is easily seen that, although
One of the first works to constrain the dark mat-
the errors increase significantly for the Λ-wDM run, the
ter equation of state with cosmological data was Mu¨ller
mean value stays close to the ΛCDM value. In contrast,
(2005). In that work, two cases for w were constrained
in the Λ-GDM run, in addition to the increase in the er-
using a combination of background data and matter
rors,themeanshiftscomparedtoΛCDM.Wesawbefore
power spectrum data. Their first case corresponds to
that increasing the GDM parameters reduces the value
setting w = c2 in our notation (note that this means a
of σ for fixed cosmological parameters (including fixed s
8 negativevalueofc2wasallowedintheiranalysis,whichis
As). Since these two GDM parameters only take posi- unphysical). Fortshatcase,theyfoundstrongconstraints
tivevalues,σ canonlybereduced,i.e. itbecomesbiased
towards smal8ler numbers, relative to the ΛCDM value. onw,∼10−6 atthe99.7%CL,duetothestrongeffectof
Thus,theinclusionofthesetwoparametersresultsinthe thesoundspeedonmatterclusteringandtheinclusionof
posteriors being shifted, even though the GDM parame- the matter power spectrum data to constrain σ8. These
ters themselves are found to be consistent with zero. constraintsarecomparabletothoseobtainedinthiswork
The mean value for the parameter τ is affected by the onc2s,aswouldbeexpected. Theirsecondcaseissimilar
inclusion of the GDM parameters for the combination of to our Λ-wDM case, although not identical due to the
PPS + Lens data. The 1-dimensional posteriors for τ differenceinthedefinitionofthenon-adiabaticpressure.
fromthedifferentmodelsarealsoplottedinFig.10. We Inthatcasetheconstraintsaremuchmorenon-Gaussian
can see that for the Λ-wDM model, the mean and width than ours, with −8.78 × 10−3 < w < 1.86 × 10−3 at
of the posterior have changed little from the ΛCDM val- the 99.7% CL. Their constraints are at a similar level to
ues. For the Λ-GDM case, the width of the posterior thoseobtainedinthispaperwithPPS+Lens+HSTdata,
is very similar to the ΛCDM value, however, the mean andlessconstrainingthanthetightestconstraintsinthis
valuehasincreased,albeitbylessthanthe68.3%CLcon- paper obtained with the PPS+Lens+BAO dataset com-
straint. Since τ and A are positively correlated, while bination.
s
A and σ are anti-correlated, we expect τ and σ to be The dark matter equation of state (the Λ-wDM
s 8 8
anti-correlated. Thereforeτ isbiasedforthesamereason model) was constrained in Calabrese et al. (2009) using
as σ but in the opposite direction. WMAP(Hinshaw et al. 2009) data. They found w to be
8
Before closing this discussion let us remark that we consistent with zero, with the constraints from WMAP
also ran chains with the low-l Planck likelihood and the alone being (−0.35+1.17)×10−2 at the 95.5% CL, which
−0.98
TTonlyhigh-l likelihood,againwiththecomplete“not- is more than a factor of two worse than those we ob-
lite” set of nuisance parameters. For these runs the con- tain using PPS or PPS + Lens. The inclusion of ad-
straints on c2 and c2 become about 50% worse com- ditional datasets including extra CMB datasets, SNLS
s vis
paredtothePPSrun, asexpectedfromthereductionin supernovae data (Astier et al. 2006) and SDSS matter
information. The constraints on w worsen significantly, power spectrum data (Tegmark et al. 2006) improved
as do those of the parameters that it has its key degen- their constraints to (0.07+0.41)×10−2 at the 95.5% CL,
−0.42
eracies with (ω and H , as discussed above). This is which is very similar to the constraints we obtain from
g 0
because there is simply not enough information in the PPS + Lens. The strongest constraints in this paper,
temperature spectrum to constrain the expansion his- obtained using PPS + Lens + BAO data, are approx-
tory once the extra degeneracies introduced by w are imately 3 times tighter than the tightest constraints in
included. The extra information provided by the polari- Calabrese et al. (2009).
sationissufficienttoconstraintheΛCDMparametersto Oneofthemorewell-knownmodelsthatfitsunderthe
values that are approximately in line with Planck, with GDM framework is the Chaplygin gas, and this model
the exceptions noted above. hasbeencomparedtoPlanckdatapreviously(LiandXu
Planck constraints on GDM 9
2014). However, due to the difference between the forms geststhatincludingadditionaldataonthematterpower
oftheGDMparametersintheChaplygingasmodeland spectrum at late times, typically at even lower redshift
those adopted in this work, our constraints are not di- than that probed by the CMB lensing potential, could
rectly comparable. improve the constraints on these parameters further. In
Two works (Wei et al. 2013; Kumar and Xu 2012) ex- addition, dataonthematterpowerspectrummayprobe
aminetheequationofstateofdarkmatterusingaTaylor the region where c2 and c2 have different effects, thus
s vis
expansion around a=1 as w =w +w (1−a). In breaking degeneracies. We intend to investigate this in
m m0 ma
Wei et al. (2013), the authors constrain the equation of future work.
state of dark matter using only background quantities, We have also examined the effects of including the
obtaining error bars that are typically larger than those GDM parameters on the constraints of the standard
obtained here. In Kumar and Xu (2012), further data ΛCDM parameters. For the full Planck dataset includ-
sets including Planck and matter power spectrum mea- inglensingpotentialreconstruction, themaineffectisto
surements are also included, and the resulting error bars significantly increase the error bars on ω as well as on
g
are typically slightly smaller than those obtained here, the derived parameters σ and H . The parameters c2
8 0 s
albeit highly non-Gaussian. However, as the equation and c2 inhibit the growth of structure below the length
of state was parameterised differently to our work, the scalevki−s1, reducing the matter power spectrum on those
results are not easy to compare directly to ours. d
scales. Thus,theinclusionofc2 andc2 loosensthecon-
In a series of papers (Xu 2014; Xu and Chang 2013), s vis
straint on σ , as well as shifting the posterior to smaller
theauthorsconstraintheequationofstateofdarkmatter 8
values. Thisshiftisduetotherequirementthatthesepa-
(the Λ-wDM model) using different datasets in addition
rameters be non-negative, therefore they can only act to
to Planck, including BAO data and SNLS3 (Guy et al.
decrease σ (and not to increase it) relative to ΛCDM.
2010; Sullivan et al. 2011) data to constrain the back- 8
The parameter τ is affected as well. When only w is
ground expansion, as well as examining the effect of in-
added to the standard ΛCDM parameters, the posterior
cludingWIGGLEzmatterpowerspectrumdata(Parkin-
remainssimilartothatfortheΛCDManalysis. However,
son et al. 2012). At the 99.7% CL, the constraints are
when all three GDM parameters are included, the mean
foundtobe∼2×10−3, whichisverysimilartothecon-
of the posterior is shifted to higher values. We note that
straints on w that we find here for the combination PPS
the constraints on τ are nonetheless similar for all three
+ Lens + BAO. The inclusion of the WIGGLEz data
cases.
has only a small effect on the constraints on w, again
WeexaminedtheeffectofusingHSTandBAOdatain
similarly to what was found in this paper with the effect
additiontothePlanckdataset. Theseimprovedthecon-
on w of including the Planck Lensing (Lens) likelihood.
straints on w, with the BAO data having a larger effect
than the HST prior. For both the Λ-wDM and Λ-GDM
5. CONCLUSIONANDDISCUSSION models these additional datasets significantly improved
The main results of this paper comprise tables 1 and the constraints on ω and H . Again, the inclusion of
g 0
2. These show the results of an MCMC analysis of the BAO data has a larger effect, such that in this case
the Planck satellite data, where the standard ΛCDM H and ω are nearly as well constrained as in a ΛCDM
0 g
modelhasbeenextendedtoincludethethree(constant) cosmology without the additional GDM parameters.
GDM parameters, namely the equation of state w, the Wehaveonlyconsideredthecaseofconstantvaluesof
sound speed c2 and viscosity c2 of the dark matter the GDM parameters in this work. This should act as a
s vis
fluid. We found that these additional parameters are null test for whether there are any significant effects on
all consistent with zero. and are strongly constrained the CMB from dark matter properties. Nonetheless, it
from the combination of high-l and low-l Planck likeli- is possible that a more sophisticated parameterisation,
hoods. The inclusion of Planck lensing data, and either perhaps following a specific model, may result in non-
BAO data or the HST prior, further tightens the con- zero values of the GDM parameters being preferred. In
straints. The equation of state w is constrained to be addition, we note that in our analysis we have not in-
−0.000896 < w < 0.00238 at the 99.7% CL and the cluded many of the additional parameters that are un-
parameters c2 and c2 are constrained to be less than necessary in a ΛCDM analysis, such as the curvature
s vis
3.21×10−6 and 6.06×10−6 respectively at the 99.7% Ωk and isocurvature perturbations. It may be that the
CL.WefoundthattheCMBlensinglikelihoodisimpor- inclusion of these parameters would allow for a non-zero
tantforconstrainingtheparametersc2 andc2 ,whereas valueoftheGDMparameters,which,combinedwiththe
s vis
theinclusionofanadditionaldatasetthatconstrainsthe effects on σ8, may allow the tension between Planck and
background expansion of the universe, either BAO or observationsofthelateuniversetoberesolved. Weleave
HST, is important for constraining w. this to future work.
Weuncoveredadegeneracybetweenthetwoperturba-
tive GDM parameters c2 and c2 which remains for all
s vis
types of datasets used in this work. This is due to the WethankC.BœhmforusefuldiscussionsandT.Tram
way these two parameters affect the decay of the grav- and B. Audren for help regarding the CLASS and Mon-
itational potential on length scales smaller than k−1 as tePythoncodes. Wealsothankalloftheauthorsofthese
d
in (10). However, this degeneracy is broken well below codes for making them publicly available. The research
the scale k−1. In this regime, for a fixed k , c2 causes leading to these results has received funding from the
d d s
a faster decay of the potential as well as resulting in EuropeanResearchCouncilundertheEuropeanUnion’s
oscillations of the matter power spectrum that are not SeventhFrameworkProgramme(FP7/2007-2013)/ERC
present in a universe with c2 =0 and c2 (cid:54)=0. This sug- GrantAgreementn. 617656“TheoriesandModelsofthe
s vis
10 Thomas, Kopp, Skordis
Dark Sector: Dark Matter, Dark Energy and Gravity”.
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