The Statefinder hierarchy: An extended null diagnostic for concordance cosmology

0.19 MB·English

by  Maryam Arabsalmani

The Statefinder hierarchy: An extended null diagnostic for concordance cosmology
Maryam Arabsalmania and Varun Sahnia
a Inter University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune, 411007, India
(Dated: January 19, 2011)
Weshowhowhigherderivativesoftheexpansionfactorcanbedevelopedintoanulldiagnosticfor
concordancecosmology (ΛCDM).Itis well knownthattheStatefinder–thethirdderivativeof the
expansion factor written in dimensionless form, a(3)/aH3, equals unity for ΛCDM. We generalize
this result and demonstrate that the hierarchy, a(n)/aHn, can be converted to a form that stays
pegged at unity in concordance cosmology. This remarkable property of the Statefinder hierarchy
enables it to beused as an extendednull diagnostic for the cosmological constant. The Statefinder
1
hierarchycombinedwiththegrowthrateofmatterperturbationsdefinesacompositenulldiagnostic
1
which can distinguish evolving dark energy from ΛCDM .
0
2
PACSnumbers: 98.80.Es,98.65.Dx,98.62.Sb

n
a
J INTRODUCTION hierarchy can be expressed in terms of elementary func-

8 tions of the deceleration parameter q (equivalently the
1 Despite itsradicalconnotations,thereismountingob- density parameter Ωm). This property singles out the

servationalevidence in support of a universe that is cur- cosmological constant from evolving DE models and al-
]
O rently accelerating [1]. Theoretically there appear to be lows the Statefinder hierarchyto be usedas anextended
C two distinct ways in which the universe can be made null diagnostic for ΛCDM.
. to accelerate [2]: (i) through the presence of an addi-
h
tional component in the matter sector, which, following
p THE STATEFINDER HIERARCHY
- [3], we call physical dark energy. Physical DE models
o possess large negative pressure and lead to the violation
r The expansion factor of the universe can be Taylor
t of the strong energy condition, ρ+3P ≥0, which forms
s expanded around the present epoch t as follows
a necessary condition for achieving cosmic acceleration. 0
a
[ Prominent examples of this class of models include the ∞
1 cosmological constant ‘Λ’, Quintessence, the Chaplygin (1+z)−1 := aa(t) =1+ Ann(!t0)[H0(t−t0)]n (1)
v gas, etc. (ii) The universe can also accelerate because of 0 nX=1
6 changes in the gravitational sector of the theory. These where
3 models(sometimesreferredtoasgeometricalDEormod-
4 ified gravity) include f(R) theories, extra-dimensional a(n)
A := , n∈N, (2)
3 Braneworldmodels, etc. n aHn
.
1 Due to its elegance and simplicity the cosmological a(n) is the nth derivative of the scale factor with respect
0
constant, with P = −ρ, occupies a privileged place in
totime. Historicallydifferentlettersofthealphabethave
1
the burgeoning pantheon of DE models. Although the
1 beenusedtodescribevariousderivativesofthescalefac-
: reasons behind the extremely small value of Λ remain tor. Thus q ≡ −A is the deceleration parameter, while
v 2
unclear,concordancecosmology(ΛCDM)doesappearto
A , which was first discussed in [7], has been called the
i 3
X provideaverygoodfittocurrentdata(althoughpossible
Statefinder ‘r’ [8] as well as the jerk ‘j’ [9], A is the
4
r departuresfromP =−ρhavealsobeennoted[4]). Given snap ‘s’, A is the lerk ‘l’, etc. (See for instance [9–11]
a 5
the successandsimplicityofconcordancecosmologyitis
and references therein.)
perhapsnaturaltodiscussdiagnosticmeasureswhichcan
It is quite remarkable that, in a spatially flat universe
beusedtocompareagivenDEmodelwithΛCDM.‘Null
consistingofpressurelessmatterandacosmologicalcon-
measures’ of concordance cosmology proposed so far in-
stant (henceforth referred to as concordance cosmology
clude the Om diagnostic [5, 6], and the Statefinders [8].
or ΛCDM), all the A parameters can be expressed as
n
While Om involves measurements of the expansion rate,
elementary functions of the deceleration parameter q, or
H(z), the Statefinders are related to the third derivative the density parameter Ω . For instance 1
m
of the expansion factor. In a spatially flat ΛCDM uni-
verse,the Statefinders and Om remainpeggedat a fixed
value during our recent expansion history (z < 103).
∼
Inthisletterweintroducethenotionofthe‘Statefinder [1]Itiseasytoseethatsuccessivedifferentiationsofa¨=a(H˙ +H2)
hierarchy’whichincludeshigherderivativesoftheexpan- and R = 6(H˙ +2H2) in conjunction with (3), can be used to
sion factor dna/dtn,n ≥ 2. We demonstrate that, for expresshigherderivativesoftheHubbleparameter,H(n)/Hn+1
, and the Ricci scalar, R(n)/6Hn+2, as polynomial expansions
concordance cosmology, all members of the Statefinder
inq (orΩm),forconcordance cosmology.
2
6
The Statefinder stays pegged at unity for ΛCDM
A7 A5
S =1, (5)
4 n
(cid:12)ΛCDM
(cid:12)
(cid:12)
during the entire course(cid:12)of cosmic expansion !
2 Equations (5) define a null diagnostic for concordance
A
3 cosmology, since some of these equalities are likely to
n
A be violated by evolving DE models for which one might
A expect one (or more) of the Statefinders to depend upon
0 2
time, see figure 2.
A
4
-2 A
6 Fractional Statefinders
Interestingly, for n > 3 there is more than one way in
-4
0.0 0.2 0.4 0.6 0.8 1.0 which to define a null diagnostic. Using the relationship
W Ω = 2(1+q),validinΛCDM,itiseasytoseethatthe
m m 3
Statefinders can also be written in the alternate form
FIG. 1: An are plotted against Ωm(z) for ΛCDM .
S(1) :=A +3(1+q)
4 4
S(1) :=A −2(4+3q)(1+q) , etc. (6)
5 5
Clearly both methods yield identical results for ΛCDM
3
A =1− Ω , : S(1) := S = 1, S(1) := S = 1. For other DE models
2 2 m 4 4 5 5
however, the two alternate definitions of the statefinder,
A3 =1, Sn & Sn(1), are expected to give different results.
In [8] it was shown that a second Statefinder could be
32 (1)
A4 =1− 2 Ωm, constructed from the Statefinder S3 :=S3, namely
33 (1)
A5 =1+3Ωm+ 2 Ω2m, S3(2) = 3S(q3−−1/12) . (7)
33 34
A =1− Ω −34Ω2 − Ω3 , etc, (3)
6 2 m m 4 m In concordance cosmology S(1) = 1 while S(2) stays
3 3
pegged at zero. Consequently the Statefinder pair
whereΩm =Ω0m(1+z)3/h2(z)andΩm = 23(1+q)incon- {S3(1),S3(2)} = {1,0} provides a model independent
cordance cosmology. It is interesting to note that while
means of distinguishing evolving dark energy models
A3 remains pegged at unity, the remaining An param- from the cosmological constant [8]. In analogy with (7)
eters evolve with time with even A2n (odd A2n+1) re- wedefinethesecondmemberoftheStatefinderhierarchy
maining smaller (larger) than unity, see figure 1. All as follows2
A approach unity in the distant future: A → 1 when
n n
dΩemfin→e t0heanSdtaΩteΛfi→nde1r.hTiehrearacbhoyveSnex:pressionsallow us to Sn(2) = αS(qn(1−)−1/12), (8)
3 where α is an arbitrary constant. In concordance cos-
S :=A + Ω ,
2 2 2 m mology S(2) =0 and
n
S :=A ,
3 3 {S(1),S(2)}={1,0} . (9)
n n
32
S :=A + Ω ,
4 4 m
2
33
SS5 ::==AA5−+33Ω3Ωm−+23Ω4Ω2m2, + 34Ω3 , etc. (4) [2]o{AtS4h3()e1/r)9,nΩSum3(l2l,)t}Σes2≡ts:={inr,Acsl4u}d+iinn3g([A:8]3O.−mEA1q.2=),(31w)−hhΩce2a0rmneOb+emΩu1ms=e(dz1)t,,oΣΣd11,e2:fi==ne21(s1tfoi−lrl
6 6 2 m m 4 m ΛCDM.
3
LCDM LCDM
1.4
w=-0.8 w=-0.8
w=-1.2 2.0 w=-1.2
1.2 Κ=1 Κ=1
Κ=2 Κ=2
DGP DGP
1.5
1.0
L
2 1
S H
0.8 3
S
1.0
0.6
0.5
0.4
0.2 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
W W
m m
FIG.2: Theleft(right)panelshowstheStatefinderS2 (S3(1) ≡S3)plottedagainstΩm ≡Ω0m(1+z)3/h2. LargevaluesΩm →1
correspond to the distant past (z ≫ 1), while small values Ωm → 0 correspond to the remote future (z → −1). The models
are: DE with w = −0.8 (blue), phantom with w = −1.2 (green), Chaplygin gas (purple), DGP (red). The horizontal black
line shows ΛCDM. The vertical band centered at Ω = 0.3 roughly corresponds to the present epoch. Note that the near
0m
degeneracy seen in S between DGP, w=−0.8 and κ=1 Chaplygin gas, at Ω ≃0.3, is absent in S(1).
2 0m 3
The second statefinder S(2) serves the useful purpose of A+B(1+z)6. The expansionrateofa universe
n
breaking some of the degeneracies present in S(1). For containingtheChaplygingasandpressurelessmat-
n p
DE with a constant equation of state w ter is given by
9w
S3(1) = 1+ 2 (w+1)ΩDE H(z) Ω A 1/2
= Ω (1+z)3+ 0m +(1+z)6 ,
S3(2) = w+1 H0 " 0m κ rB #
27 7 27
S(1) = 1− w(w+1)(w+ )Ω − w2(w+1)Ω2
4 2 6 DE 4 DE
where κ defines the ratio between the density in
7 1
S(2) = −(w+1)(w+ )− w(w+1)Ω cold dark matter and the Chaplygin gas at the
4 6 2 DE
commencement of the matter-dominated stage of
where S(2) = S4(1)−1 and q−1/2= 3wΩ . expansion and
4 9(q−1) 2 DE
2
Aswedemonstrateinfigures2,3,4,5,theStatefinder
hierarchy{Sn(1),Sn(2)}provideuswithanexcellentmeans A=B κ2 1−Ω0m 2−1 .
ofdistinguishingdynamicalDEmodelsfromΛCDM.Our ( (cid:18) Ω0m (cid:19) )
discussion focuses on the following models:
1. Dark energy with a constant equation of state
3. The Braneworld model suggested by Dvali,
H(z) = Ω (1+z)3+Ω (1+z)3(1+w) 1/2 , Gabadadze and Porrati [13]
0m DE
H
0
h i
where concordance cosmology (ΛCDM) corre- 2
H(z) 1−Ω 1−Ω
sponds to w = −1. For simplicity we neglect the = 0m + Ω0m(1+z)3+ 0m .
presence of spatial curvature and radiation. H0 (cid:18) 2 (cid:19) s (cid:18) 2 (cid:19) 
 
2. The Chaplygin gas [12] has the interesting EOS
p = −A/ρ while its density evolves as ρ =
c c c
4
3 LCDM
w=-0.8 2.0 LCDM
w=-1.2 w=-0.8
Κ=1 w=-1.2
2 Κ=2 1.5 Κ=1
Κ=2
DGP
DGP
1.0
L
1 L
H 1 1
4 H 0.5
S 4
S
0.0
0
-0.5
-1 -1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
W
z
m
FIG. 3: The left (right) panelshows theStatefinderS4(1) plotted against Ωm (left panel) andz (right panel). Thedark energy
models are: DE with w =−0.8 (blue), phantom with w = −1.2 (green), Chaplygin gas (purple), DGP (red). The horizontal
black line shows ΛCDM. The vertical band centered at Ω =0.3 in the left panel roughly corresponds to the present epoch.
0m
Ω =0.3 is assumed for DEmodels in the right panel.
0m
LCDM LCDM
0.4
w=-0.8 w=-0.8
0.5
w=-1.2 w=-1.2
0.2 Κ=1 Κ=1
Κ=2 Κ=2
0.0 DGP DGP
0.0
L2 -0.2 L2
H H
4 3
S S
-0.4
-0.5
-0.6
-0.8
-1.0 -1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
W W
m m
FIG. 4: The left (right) panel shows the Statefinder S4(2) (S3(2)) plotted against Ωm (left panel). The vertical band centered
at Ω =0.3 in the left panel corresponds to the present epoch. Comparing figure 4 with figure 3, we find that S(2) does not
0m 4
appear to perform as well as S(1), or even S(2), in distinguishing between theDE models considered in this paper.
4 3
GROWTH RATE OF PERTURBATIONS where f(z)= dlogδ/dlogz describes the growth rate of
linearized density perturbations [14]
The Statefinders can be usefully supplemented by the
f(z) ≃ Ω (z)γ (10)
fractional growth parameter ǫ(z) [18] m
3 3 (1−w)(1− 3w)
γ(z) = + 2 (1−Ω (z))
f(z) 5− w 125 (1− 6w)3 m
ǫ(z):= . 1−w 5
fΛCDM(z) + O[(1−Ω (z))]2 . (11)
m
5
3 LCDM DGP cosmology but with γ ≃2/3 [17].)
w=-0.8 For this reason, the fractional growth parameter ǫ(z),
w=-1.2 can be used in conjunction with the Statefinders to de-
Κ=1
fine a composite null diagnostic: CND ≡{S ,ǫ}, where
2 Κ=2 n
{S ,ǫ}={1,1} for ΛCDM . Figure 6 demonstrates how
DGP n
DEmodelsaredistinguishedbymeansofCND.(Onecan
L1 also use the Om diagnostic in place of the Statefinder in
H 1
4 CND.)
S
0 CONCLUSIONS
In this paper we have shown that a simple series of
-1 relationships links the Statefinder hierarchy in concor-
0.5 1.0 1.5 2.0 dance cosmology with the deceleration/density param-
S H1L eters. These relationships can be used to define null
3
tests for the cosmological constant. Including informa-
tion pertaining to the growth rate of perturbations in-
FIG. 5: The statefinders S(1) and S(1) ≡ S are shown for creasestheeffectivenessofthishierarchyofnulldiagnos-
4 3 3
the DE models discussed in the previous figures. The fixed tics. Our results demonstrate that lower order members
pointat{1,1}isΛCDM.Thearrowsshowtimeevolutionand of the Statefinder hierarchy already differentiate quite
the present epoch in the different models is shown as a dot.
well between concordance cosmology on the one hand,
Ω =0.3 is assumed.
0m and Braneworld models and the Chaplygin gas, on the
other. Since, for n ≥ 3, the nth Statefinder S contains
n
terms proportional to w(n−2)/Hn−2, higher members of
1.1 LCDM
the hierarchy will contain progressively greater informa-
w=-0.8
1.0 tion about the evolution of the equation of state of DE.
DGP
From the observational perspective, however, one might
0.9 note that a determination ofthe S statefinders involves
n
priorknowledgeofthe(n−1)th derivativeofH(z). Thus
L 0.8 only lower order Statefinders, S , n ≤ 4, together with
1 n
H
3 the Om diagnostic,mayprovecompatiblewiththe qual-
S 0.7
ity of observational data expected in the near future.
0.6
Acknowledgments
0.5
0.4 We acknowledge useful discussions with Ujjaini Alam
0.80 0.85 0.90 0.95 1.00 1.05 1.10
ΕHzL and Alexei Starobinsky.
FIG. 6: The composite null diagnostic {S(1),ǫ} is plotted for
3
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