Observational constraints on dark energy with generalized equations of state

0.32 MB·English

by  S. Capozziello

Observational constraints on dark energy with generalized equations of state
S. Capozziello1, V.F. Cardone2, E. Elizalde3, S. Nojiri4, S.D. Odintsov3
1Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II” and INFN, Sez. di Napoli,
Compl. Univ. Monte S. Angelo, Edificio N, Via Cinthia, I-80126, Napoli, Italy,
2Dipartimento di Fisica ”E.R. Caianiello”, Universit`a di Salerno, and INFN,
Sez. di Napoli, Via S. Allende, I-84081, Baronissi (Salerno), Italy,
3Instituci`o Catalana de Recerca i Estudis Avanc¸ats (ICREA) and Institut d’ Estudis Espacials de Catalunya (IEEC/ICE),
Edifici Nexus, Gran Capit`a 2-4, 08034 Barcelona, Spain,
4Department of Applied Physics, National Defence Academy, Hashirimizu Yokosuka 239-8686, Japan
WeinvestigatetheeffectsofviscositytermsdependingontheHubbleparameteranditsderivatives
in the dark energy equation of state. Such terms are possible if dark energy is a fictitious fluid
originating from corrections to the Einstein general relativity as is the case for some braneworld
inspired models or fourth order gravity. We consider two classes of models whose singularities in
6
theearly and late time universehavebeen studied by testing themodels against thedimensionless
0
coordinatedistancetoTypeIaSupernovaeandradio-galaxies also includingpriorsontheshift and
0
2 theacousticpeakparameters. Itturnsoutthatbothmodelsareabletoexplaintheobservedcosmic
speed up without theneed of phantom (w<−1) dark energy.
n
a
PACSnumbers: 98.80.-k,98.80.Es,97.60.Bw,98.70.Vc
J

7
2 I. INTRODUCTION is an effective fluid originating from an ultralight scalar
field φ, dubbed quintessence, evolving under the action
3
ofasuitablychosenself-interactionpotentialV(φ). The
v According to the new cosmological picture emerging
choice of V(φ) is crucial and different functional expres-
0 from data (only few years ago unexpected), we live in
5 a spatially flat universe filled with a subcritical matter sionshavebeeninvestigated,frompower-law[9]toexpo-
3 nentials[10,11]andacombinationofboth[12]. Without
content and undergoing an accelerated expansion. The
8 entering into details (for which the reader is referred to
HubblediagramofTypeIaSupernovae(hereafterSNeIa)
0 [13]), we only stress that such models are able to cor-
[1]hasbeenthefirstcornerstone,butquitesoonotherob-
5
rectly reproduce the observed data, but are affected by
0 servationaldata,fromthe cosmicmicrowavebackground
finetuningproblemsandtheopenissueofunderstanding
/ (hereafter CMBR) anisotropy spectrum [2] to the large
h where the quintessence scalar field comes from.
scalestructureproperties[3],furthercorroboratethefirst
p
- impression from SNeIa. The astonishingly precision of Notwithstanding the strongefforts made up to now, it
o the CMBR spectrum measured by the WMAP satellite isnotknownwhatis the fundamentalnature ofthe dark
tr [4] and the Gold SNeIa sample of Riess et al. [5] repre- energysothatitisalsoworthinvestigatingthepossibility
s sent the lastand still moreconvincing evidences in favor that dark energy and dark matter (the other unknown
a
: of this new description of the universe. component of the universe) are two different aspects of
v
Both the cosmic speed up and the flatness of the uni- the same fluid. This is the underlying idea of models
i
X verse have posed serious problems to the cosmologist belongingtotheUDEclass. Insuchanapproach,asingle
community. Since matter alone (both visible and dark) fluid with an exotic equation of state plays the role of
r
a cannotbeenoughtoclosetheuniverse,anewcomponent dark matter at high densities and dark energy at low
was invoked as a dominant term. Moreover, in order to densities. Typical examples are the Chaplygin gas [14],
explaintheobservedcosmicspeedup,thisnewfluidmust the tachyonic fluid [15] and the Hobbit [16] model1.
have a negative pressure. Being obscure both in its ori-
In both classes described above, dark energy is ex-
ginanditsproperties,thiscomponentwasbaptizeddark
plainedin terms of a new fluid in the frameworkof stan-
energy. Understanding its nature and nurture is one of
dard general relativity. However, the Einstein theory of
the most fascinating and debated challenges of modern
gravityhas been experimentally tested only on scales up
cosmology. Although the old cosmological constant [6]
to the Solar System and hence it is far from be verified
may play naturally the role of dark energy and also fits
that its validity holds also on cosmological scales. Moti-
well the observed dataset [7, 8], it is affected by serious
vatedbytheseconsiderations,differentmodelshavebeen
theoretical shortcomings that have motivated the search
proposedwherethecosmicspeedupisexplainedinterms
for alternative candidates. It is nevertheless worth ob-
of a matter only universe regulated by dynamical equa-
serving that, to a large extent, the different models may
be broadly classified in three classes which we will refer
to as scalar fields, unified dark energy (hereafter UDE)
and modified gravity. (Sometimes, some of these models
1 Similarto the UDE models is the phenomenological inflessence
maybe rewritten as models from another class).
scenario[17]whereasinglefluidisusedtoexplainbothinflation
For models belonging to the first class, dark energy anddarkenergy.
2
tionsthataredifferentfromtheusualFriedmannonesas II. THE MODELS
a consequence of a generalizedgravity theory. Examples
are the braneworld inspired Dvali-Gabadadze-Porrati For a spatially flat homogenous and isotropic Robert-
model [18] and fourth order theories of gravity, both in son-Walker universe, the Einstein equations reduce to
the metric [19, 20, 21, 22] and Palatini [23, 24, 25] for- the usual Friedmann equations:
mulation. Although such models are able to give rise
to accelerated expansion without the need of any dark
energy,astrongdebateisstillopenabouttheircompati- H2 = κ2ρ , (1)
bilitywiththestandardtestsofgravityinthelowenergy 3
limit.
2H˙ = κ2(ρ+p) (2)
−
Fromtheshort(andfarfromexhaustive)overviewpre- whereH =a˙/a is the Hubble parameter,a the scale fac-
sented above, it is clear that there is much confusion. tor and an overdot denotes the derivative with respect
However, it is possible to put some order in this some- to cosmic time t. On the right hand side of Eqs.(1) and
whatchaoticsituationbyconsideringaparticularfeature (2), we have set κ2 = 8πG, while ρ and p are the total
of the dark energy, namely its equation of state (here- energy density and pressure respectively. Since the ra-
after EoS), i.e. the relation between the pressure and diation term is nowadays negligible, we assume that the
the energy density. Indeed, whatever is the model con- universeisfilledonlywithdustmatterandadarkenergy
sidered, it is always possible to introduce a sort of dark fluid and use subscripts M and X to denote quantities
energyfluidwhoseenergydensityandpressurearedeter- referring to the first and second component respectively.
mined by the characteristics of the given theory. While ProvidedtheEoSpi =pi(ρi)issomewhatgiven,theevo-
for scalar fields and UDE models, the EoS is a function lution of the energy density of the i-th fluid may be (at
of the energy density only, in the case of modified grav- least, in principle) determined by solving the continuity
ity theories, the effective EoS depends also on geometry, equation:
e.g. ontheHubbleparameterand/oritsderivatives. Itis
thereforetemptingtoinvestigatethepropertiesofcosmo-
ρ˙ +3H(ρ +p )=0 . (3)
logicalmodels startingfromthe EoSdirectlyandtesting i i i
whether a given EoS is able to give rise to cosmological
This may be conveniently rewritten in terms of the red-
models reproducing the available dataset. A first step
shift z = 1/a 1 (having assumed a = 1 at the present
forwardinthisdirectionhasbeenundertakeninarecent −
day) as:
paper [26] where the singularities of models assigned by
generalized EoS have been investigated. Since the main
interestwasthereinasymptoticbehaviors(i.e.,inthefar dρi 3(1+wi)ρi
= (4)
past and far future), any matter term was not included dz 1+z
inthe analysisin[26]whichmakesit possibleto analyti-
with w p /ρ the EoS parameter2. For dust mat-
callysolvethedynamicalequations. Inordertoelaborate i i i
≡
ter, w = 0, Eq.(4) is easily integrated to give ρ =
further on this originalidea, we have to include the con- M M
Ω ρ (1 +z)3 with Ω = ρ /ρ the matter den-
tribution of dust (dark/baryonic) matter. The present M crit M M crit
sity parameter, ρ = 3H2/κ2, the present day critical
work is thus complementary to [26] since i.) we consider crit 0
densityandhereonquantitieswithasubscript0areeval-
thepresentratherthantheasymptoticstatesofthemod-
uated at z =0.
els,andii.)weinvestigatetheviabilityofthemodelsand
Concerning the dark energy, almost nothing is known
constrain their parameters using the observational data.
about its EoS given the ignorance of its nature and its
fundamental properties. The simplest choice
The plan of the paper is as follows. In Sect.II, we in-
p =w ρ , (5)
troduce the models that we investigate, chosen for their X X X
interesting asymptotic behaviors and give a general the-
with w a constant, has the virtue of leading to aninte-
oreticaldiscussionongeneralizedEoS.Theobservational X
grable continuity equation thus yelding:
data,themethodusedtoconstrainthemodelparameters
and the results of the analysis are presented in Sect.III.
Some hints on the high redshift behaviour of the models ρ =Ω ρ (1+z)3(1+wX) (6)
X X crit
are presented in Sect.IV, while, in Sect.V, we qualita-
tively discuss the issue of structure formation. We sum-
marize and conclude in Sect.VI, while further details on
thetheoreticalfoundationofthegeneralizedEoSwecon- 2 In the following, as EoS we will refer both to the relation pi =
sider are given in two appendixes. pi(ρi)andtowi indifferently.
3
with Ω the dark energy density parameter. Note that,
X
because of the flatness condition, it is Ω +Ω =1.
M X
Although a constant EoS dark energy model (also re- pX =wfρX +wHH2 (7)
ferred to as quiessence or QCDM) nicely fits the avail-
with (w ,w ) two undetermined constants3. Eq.(7) is a
able data, there are two serious shortcomings in this ap- f H
particular case of the more general class of dark energy
proach. First, there are no theoreticalmodels predicting
models with EoS:
a rigorously constant EoS so that this assumption lacks
a whatever background motivation. Moreover, fitting a
largesetofdatapointstowardsw < 1asbestfit(see,
X − p = ρ Aρα BH2β (8)
e.g., [27]) sothat ρX+pX <0. Models havingwX < 1 X − X − X −
−
are collectively referred to as phantom models [28] and
that reduces to Eq.(7) by setting α = β = 1, w =
havesomedisturbingfeatures suchasviolatingthe weak f
(1 + A) and B = w . The singularities (in the far
energyconditionρX+pX ≥0andleadingtoadivergence −past and in the far futHure when the matter term may
ofthe scale factora(t) ina finite time (referredto asBig
be neglected) of this class of models have been investi-
Rip).
gatedindetailin[26]whereithasbeenshownthatawide
It is worth noting that allowing w to evolve with
X rangeofinterestingpossibilitiesmaybeachieveddepend-
z does not seem to solve the problem. On the con-
ing on the values of (α,β). In particular, for α=β =1,
trary, fitting parametrized model independent EoS to
even if w < 1, the Big Rip is avoided provided that
0
the available dataset, as wX = w0 +w1z [29] or wX = w = w +w− κ2/3 > 1 (for general classification of
w + w z/(1 + z)p with p = 1,2 [30], still points at eff f H −
0 a futureDEsingularities,see[33]). Whilethegeneralmodel
w < 1. Moreover, the results seem to suggest that
0 − in Eq.(8) is characterizedby five parameters(namely, α,
w crosses the phantom divide, i.e. the EoS changes
X β, A, B and Ω ) and is therefore quite difficult to con-
M
from w > 1 in the past to w < 1 today. Recently,
X − X − strain observationally, the case in Eq.(7) has the virtue
this feature has been recovered in exact models where
ofavoidingtheBigRipand(possibly)theneedforphan-
inflation is matched to late-time acceleration by scalar
tom fields with the further advantage of being assigned
phantom-non-phantom transitions [31]. Other realistic
byonlythreeparameters. Forthesereasons,wewillcon-
modelsareachievedbyintroducingalsodarkmatterinto
sider hereon only this particular realization of Eq.(8).
the game [32].
The EoS (7) may be written in a more significantway
Here, we investigate the possibility that the dark en- using Eq.(1) to express H2 as function of ρ and ρ .
X M
ergyEoSdependsnotonlyontheenergydensityρ ,but
X We thus get:
also on the Hubble parameter H and/or its derivatives.
We will refer to such a model as generalized EoS. As a
preliminary remark, let us note that these same models p = w ρ +w H2 Ω (1+z)3+η(z)
X f X H 0 M
werealsoreferredtoasinhomogeneous EoSin[26]. How-
ever, it is worth stressing that the term inhomogeneous (cid:2) (cid:3)
weffΩ (1+z)3
does not mean that we consider an inhomogeneous uni- = w + M M ρ (z) , (9)
DE X
verse (so that the RW metric may still be retained) nor " η(z) #
that we consider the possibility that dark energy may
cluster in the nonlinear regimeof perturbations(as, e.g., so that the dark energy EoS reads:
in [35]). We use the term inhomogeneous since assuming
w = w (ρ ,H,H˙) introduces a viscosity term in the
coXntinuitXy eqXuationsimilarto what happens influidody- w =w + wMeffΩM(1+z)3 . (10)
X DE
namics for an inhomogeneous fluid. See Appendix A for η(z)
anexplicitderivation,whilespecificexamplesofEoSdue
to time-dependent bulk viscosity are given in, e.g., [34]. In Eq.(9), in the first row, we have rewritten Eq.(1) as
H2/H2 = E2 = Ω (1+z)3+η(z) with η ρ /ρ ,
In the following, we will describe two classes of mod- 0 M ≡ X crit
while, in the second row, we have defined:
els introduced in [26] chosen because of their interesting
asymptotic properties. Actually, we slightly generalize
them and find out some degeneracies from the point of
κ2w
H
viewofobservations. Finally,wediscussthegeneraltheo- w =w +
DE f
3
reticalfeaturesthatagivengeneralizedEoSshouldfulfill  . (11)
in order to match with any observational cosmology.  weff = κ2wH
M 3

A. Increased Matter
3 Note that, whilewf isdimensionless,thedimensions ofwH are
Let us first consider the case: fixedinsuchawaythatwHH2 isapressureterm.
4
Eq.(10) nicely shows that the model we are considering and then of the true barotropic factor w in Eq.(10) as:
f
is a simple generalizationof the QCDM case to which it
reduces when w = 0. For w = 0, w (z) may still be
H H X
expressed analytically. To this a6im, let us insert Eq.(10) wf =wDE −wMeff (16)
in the continuity equation for ρ and integrate it with
X
whichshowsthat,assumingw >0forwhatsaidabove,
the initial condition ρ (z = 0) = Ω ρ . Using the H
X X crit
leads to w < w . As a consequence, if the effective
dimensionless quantity η =ρ /ρ , we find: f DE
X crit
barotropicfactorw isinthe phantomregion,thetrue
DE
one w will be deeper in the phantom. As such, the
f
η = Ω + wMeffΩM (1+z)3(1+wDE) wMeffΩM(1+z)3. model is thus not able to evade the need for phantom
X wDE ! − wDE dark energy, but it is still interesting since it makes it
(12) possible to avoid any Big Rip if wDE +wMeff is larger
Inserting now Eq.(12) into Eq.(1), we obtain: than the critical value (wX = 1).
−
There is, actually, an easy generalizationof the model
able to both eliminate future singularities and phantom
E2(z)=Ω˜M(1+z)3+(1 Ω˜M)(1+z)3(1+wDE) (13) fields. To this aim, let us consider the following EoS:
−
with
p =w ρ +w H2+w H˙ (17)
X f X H dH
weff
Ω˜M =ΩM 1− wMDE! . (14) iwnit[h26w]dtHoawnheicwhciotnrsetadnutc.esEqb.y(1s7e)ttisinignswpire=dbwy,Ewq.(47=)
f H
3(1+w)/κ2 andw = 2/κ2. Itiseasytoshowthat,
Wethusgetaquiteinterestingresult. TheEoS(10)leads dH
− −
with these positions and in absence of matter, Eq.(17)
to the Hubble parameter that has formally the same ex-
is an identity. When a matter term is added and the
pression as that of QCDM models, but at the price of
constants(w ,w ,w )areleftfree,thisisnomoretrue
shifting boththe matter density parameterandthe dark f H dH
andinterestingconsequencescomeout. Toseethis,letus
energy EoS. As such, a fitting analysis based on observ-
use the Friedmann equations to get H2 and H˙ in terms
ables depending only on E(z)as is the case of the SNeIa
of(ρ ,ρ ,p ). ItistheneasytoshowthatEq.(17)may
Hubble diagram (see later) will give biased results. In M X X
particular, if w < 0 and w > 0, then Ω˜ > Ω be rewritten as:
DE H M M
so that the best fit values (wfit,Ωfit) may erroneously
DE M
point towards a universe with a higher matter content w κ2 w κ2 w κ2
dH H dH
1+ p = w + ρ
and a phantom dark energy. Because of this peculiar- X f X
2 3 − 2
ity, we will refer in the following to this scenario as the (cid:18) (cid:19) (cid:18) (cid:19)
w κ2 w κ2
increased matter (hereafter IM) model, even if the in- + H dH ρ (18)
M
creasing of the matter content is only apparent. 3 − 2
(cid:18) (cid:19)
It is interesting to note that, fitting only the SNeIa
so that the dark energy EoS is given again by Eq.(10)
Gold dataset without any prior on Ω , Jassalet al. [36]
M provided Eqs.(11) are generalized as:
have found Ω = 0.47 as best fit, significantly higher
M
than the fiducial value Ω 0.3 suggested by cluster
M
≃
abundances. Ontheotherhand,fittingtheWMAPdata κ2w w κ2 w κ2 −1
H dH dH
only gives Ω = 0.32 as best fit value thus rising the w = w + 1+
M DE f 3 − 2 2
problem of reconciling these two different estimates. In  (cid:18) (cid:19)(cid:18) (cid:19)
tbhleeisrycstoenmclautsiiconersr,oJrassisnaloneet oafl.thaergduaetaisnetf.aEvoqrs.o(1f3p)oasnsid-  weff = κ2wH wdHκ2 1+ wdHκ2 −1
(14), however, furnish an alternative explanation. Sim- M 3 − 2 2
ply, since (as we will see more in detail later) the SNeIa  (cid:18) (cid:19)(cid:18) (cid:19) (19)
Hubble diagram only probes E(z), fitting this dataset while Ω˜ is still defined by Eq.(14) with w and weff
M DE M
with no priors gives constraints on Ω˜ rather than Ω . now given by Eqs.(19). By using this generalization,
M M
Since Ω˜ >Ω , Eq.(14) suggests that a a possible way we can still get Ω˜ > Ω , but now the conditions
M M M M
to reconcile this worrisomediscrepancy may be to resort weff/w > 0 and w < 0 does not imply w > 0.
M DE DE H
toourmodelassumingweff/w <0. Sinceitisreason- As a further consequence, the true barotropic factor w
M DE f
able to expect that w <0, w >0 have to imposed. is now given as:
DE H
Provided a suitable method is used to determine Ω ,
M
w andΩ˜ ,itispossibletogetanestimateofweff as
DE M M w κ2
w = 1+ dH (w weff) (20)
f 2 DE − M
w (Ω Ω˜ ) (cid:18) (cid:19)
weff = DE M − M (15)
M ΩM and we can get wf >−1 provided that the condition
5
whichmaybeintegratednumericallywiththeinitialcon-
ditionE(z =0)=1providedthatvaluesof(Ω ,E ,C )
2 w weff +1 M s s
w > DE − M (21) are given. To this end, it is convenient to express Cs in
dH −κ2 wDE −wMeff ! treermmesmobferathmaotrtehceodmecmeolenraqtuioanntpiatyr.amTeotetrhqis=aima,a¨l/ea˙t2uiss
is verified. Since Eqs.(9) and (18) lead to models that related to the dimensionless Hubble parameter−E(z) as:
arefullyequivalentfromthedynamicalpointofview,we
willconsiderhereonthese modelsasasingleonereferred
to as the increased matter model. 1+zdE(z)
q(z)= 1+ (25)
− E(z) dz
B. Quadratic EoS so that (dE/dz) = 1 + q . Evaluating (dE/dz)
z=0 0 z=0
from Eq.(24) and solving with respect to C , we get:
s
Let us now consider a different approach to the dark
energy EoS. As yet stated above, we may think at the
[3Ω 2(1+q )]2
EoS as an implicit relation such as: C = M − 0 (26)
s 9(1 Ω )2(1 E )
M s
− −
F(p ,ρ ,H)=0
X X
where no sign ambiguity is present. Hereon we will
which is not constrained to lead to a linear dependence characterize the model by the values of the parameters
of p on ρ . As a particularly interesting example, let (q ,E ,Ω ). It is interesting to note that C vanishes
X X 0 s M s
us consider the case: for Ω = 2(1+q )/3 which takes physically acceptable
M 0
values for 1 < q < 1. In such cases, Eq.(22) reduces
0
−
to p = ρ , i.e. we recover the usual ΛCDM model
(ρ +p )2 C ρ2 1 Hs =0 (22) for wXhich−itXis indeed q = 1+3Ω /2. On the other
X X − s X − H 0 − M
(cid:18) (cid:19) hand, Cs seems to diverge for ΩM = 1. Actually, this is
notthe case. Indeed, for Ω =1,we havea matter only
withC andH twopositiveconstants. Eq.(22)hasbeen M
s s
universe for which it is q =1/2 thus giving C =0 and
proposed in [26] where it has been shown that the cor- 0 s
E2 =Ω (1+z)3 so that Eq.(24) is identically satisfied.
responding cosmological model presents both Big Bang M
OnceEq.(24)hasbeenintegrated,onemayuseEq.(1)to
and Big Rip singularities. As such, this model may be
get ρ (z) and hence w (z) from Eq.(23).
a good candidate to explain not only the present state X X
Some important remarks are due at this point. If we
of cosmic acceleration, but also the inflationary one and
start from a non-phantom phase (1+w > 0) impos-
is therefore worth to be explored. Since its EoS is given X
ing z = 0 as initial condition, for large z (far past)
by a quadratic equation in w , we will refer to it in the
X
the universe remains in a non-phantom phase. This is
following as the quadratic EoS (hereafter QE) model.
the case which we will match with observational data.
Asafirststep,itisconvenienttorewritethecontinuity
On the other hand, starting from a phantom regime
equation for ρ as a first order differential equation for
X
(1+w < 0), we can reach a phase where 1+w = 0
E(z) = H(z)/H . To this aim, one may differentiate X X
0
both members of Eq.(1) with ρ = ρ +Ω ρ (1+z)3 or E = Es in a finite time, i.e. for finite z. In this
X M crit
case, E(z) decreases with z, reaches E for a finite z
and solve with respect to dρ /dz. On the other hand, s s
X
and after starts to increase. The sign in Eq.(23) hanges
from Eq.(22), we easily get:
at E =E , then E is a minimum for E(z). This means
s s
that, for z < z , the sign of the second term of r.h.s of
s
Es Eq.(24) is minus, while, for z > zs, the sign is plus. In
(1+w )ρ = ρ C 1 (23)
X X X s conclusion,depending onthe nature of darkenergy fluid
± s − E
(cid:18) (cid:19) (w ), the model can represent a non-phantom cosmol-
X
where the sign changes for w = 1 (phantom divide), ogy or the evolution from a phantom to a non-phantom
X
corresponding to the standard ΛC−DM model. On the cosmology through the crossing of the phantom divide.
other hand, the sign changes also for E =E which cor- For sake of simplicity, we will match with observations
s
responds to a finite redshift z =z . The evolutionof the onlythenon-phantomsolutionsincewewanttotestthe
s
Hubble parameter is achieved by inserting the previous viabilityofgeneralizedEoSbut,inprinciple,themethod
expressions for dρ /dz and (1 +w )ρ in Eq.(4), we to constrain method which we will discuss below can be
X X X
finally get: extended also to phantom cosmology.
dE 3Ω (1+z)2
M C. Generality of generalized EoS of the universe
=
dz 2E
3 E2 ΩM(1+z)3 [Cs(1 Es/E)]1/2 TheaboveIM andQEmodelsareparticularexamples
− − (24)
± 2E(1+z) ofgeneralizedEoSwhichcanbeusedtofitobservational
(cid:2) (cid:3)
6
data. A general approach to our generalized EoS is pos- Since
sible froma theoreticalpoint of view. We will show that
a¨ αβ
anyobservationalcosmologymayemergefromsuchEoS. =
By using a single function f(t), we now assume the fol- a κ2 1 β ln γ+ t2 2 γ+ t2 2
lowing EoS: − 2 κ2 κ2
(cid:16) β (cid:0) (cid:1)t2(cid:17) (cid:0) (cid:1)t2
1 ln γ+ γ
× − 2 κ2 − κ2
2 ρ (cid:26)(cid:18) (cid:18) (cid:19)(cid:19)(cid:18) (cid:19)
p= ρ f′ f−1 κ . (27) β(1+α)t2
− − κ2 3 + , (35)
(cid:18) (cid:18) r (cid:19)(cid:19) κ2
(cid:27)
Then it is straightforwardto show that a solution of the
ift>0,therearetwosolutionsofa¨=0,onecorresponds
Friedmann equations (1), (2), and the continuity equa-
to late time and another corresponds to early time. The
tion (3) is given by
latetimesolutiont=t ofa¨=0isobtainedbyneglecting
l
γ. One obtains
3 3 2
H =f(t) , ρ= f(t)2 , p= f(t)2 f′(t) .
κ2 κ2 − κ2
(28) t=t κe1/β−α−1 . (36)
l
Then any cosmology given by H = f(t) can be realized ∼
by the EoS (27). Ontheotherhand,theearlytimesolutioncouldbefound
As first example, if f(t) and therefore H is given by by neglecting β, which is 10−2 , to be
O
(cid:0) (cid:1)
2 1 1 t=t κ√γ . (37)
f(t)=H = + , (29) e ∼
3(w +1) t t t
m (cid:18) s− (cid:19) Thentheuniverseundergoesacceleratedexpansionwhen
the EoS has the following form: 0<t<t and t <t<t . Here t is Rip time:
e l s s
p= ρ (wm+1)ρ t2 8 3 . (30) ts =κ −γ+e2/β ∼κe1/β . (38)
− ∓ ts s s− 3(wm+1)κrρ t may be identifiedpwith the time when the inflation
e
ended. Oneisabletodefinethenumberofthee-foldings
Since
N as
e
2 1 1
H˙ = + , (31) N =ln(a(t )/a(0)) . (39)
3(wm+1) −t2 (ts t)2! e e
−
Hence, it follows
the EoS parameter w defined by
eff
1 β ln(2γ)
p 2H˙ N = αln − 2 . (40)
weff = ρ =−1− 3H2 , (32) e − 1− β2 ln(γ) !
goestow > 1whent 0andgoesto 2 w < 1 Thus, we demonstrated that generalized EoS of the uni-
m m
at large times−. The cro→ssing w = 1−oc−curs wh−en verse,inthesamewayasscalarmodelofRef.[31,32]may
eff
H˙ =0, that is, t=t /2. Note that − presentthe naturalunificationofthe earlytime inflation
s
and late time acceleration. Working in the same direc-
tion, with more observational constraints, one can sug-
a¨ 16t
= s gest even more realistic generalized EoS of the universe,
a 27(wm+1)3(ts t)2t2 describing the cosmologicalevolution in great detail.
−
(3w +1)t However, some comments are necessary at this point.
m s
t . (33) As for the specific models analyzed above, to test mod-
× − 4
(cid:26) (cid:27) els against observations is convenient to translate all in
Hence, if w > 1/3, the deceleration of the universe terms of z, i.e.
m
−
turns to the acceleration at t=t (3w +1)t /4.
a m s
≡
As second example, we now consider
z˙
H(z)=f(z,z˙)= , (41)
−z+1
αβt
f(t)=H(t)= . (34) and then E(z). This allows to select interesting ranges
κ2 1− β2 ln γ+ κt22 γ+ κt22 of z to compare withobservations, for example 100 <
(cid:16) (cid:0) (cid:1)(cid:17)(cid:0) (cid:1)
7
z <1000 for very far universe (essentially CMBR data), We thus set h = 0.664 in order to be consistent with
10<z <100 (structure formation), 0<z <10 (present their work, but we have checked that varying h in the
universe probed by standard candles, lookback time, 68% CL quoted above does not alter the main results.
etc.). Then different datasets, coming from different ob- Furthermore, the value we are using is consistent also
servational campaigns, have to be consistently matched with H = 72 8 km s−1 Mpc−1 given by the HST Key
0
±
with the same cosmological solution ranging from infla- project [39] based on the local distance ladder and with
tion to present accelerated era. This program could be the estimates coming from the time delay in multiply
hard to be realized in details because of the difficulties imaged quasars [40] and the Sunyaev-Zel’dovich effect
to join together a reasonable patchwork of data coming in X-ray emitting clusters [41].
from different epochs, but it is possible in principle. ThesecondterminEq.(43)makesitpossibletoextend
the redshift range over which y(z) is probed resorting to
thedistancetothelastscatteringsurface. Actually,what
III. CONSTRAINING THE MODELS canbe determinedfromthe CMBRanisotropyspectrum
is the so called shift parameter defined as [42, 43]:
Having in mind the theoretical considerations of last
subsection,letusnowdevelopamethodtoconstraindark
energymodelswithgeneralizedEoSagainstobservations. ΩMy(zls) (45)
R≡
In particular, we will use data coming from SNeIa and
p
where z is the redshift of the last scattering surface
radio-galaxies for the above IM and QE models which ls
which can be approximated as [44],:
reproduce the present day cosmic acceleration by means
of inhomogeneous corrections in the dark energy EoS.
z =1048 1+0.00124ω−0.738 (1+g ωg2) (46)
ls b 1 M
A. The method with ω = Ω h(cid:0)2 (with i = b,M f(cid:1)or baryons and total
i i
matter respectively) and (g ,g ) given in Ref.[44]. The
1 2
In order to constrain the EoS characterizing parame-
parameterω iswellconstrainedbythebaryogenesiscal-
b
ters, we maximize the following likelihood function:
culations contrasted to the observed abundances of pri-
mordial elements. Using this method, Kirkman et al.
χ2(p) [45] have determined:
exp (42)
L∝ − 2
(cid:20) (cid:21) ω =0.0214 0.0020 .
b
±
where p denotes the set of model parameters and the
pseudo-χ2 merit function is defined as: Neglecting the small error, we thus set ωb = 0.0214 and
use this value to determine z . It is worth noting, how-
ls
ever, that the exact value of z has a negligible impact
ls
N yth(z ,p) yobs 2 on the results and setting zls = 1100 does not change
χ2(p) = i − i constraints on the other model parameters.
σ
i=1(cid:20) i (cid:21) Finally,thethirdterminthedefinitionofχ2takesinto
X
(p) 1.716 2 (p) 0.469 2 account the recent measurements of the acoustic peak in
+ R − + A − .(43) the large scale correlation function at 100 h−1 Mpc sep-
0.062 0.017
(cid:20) (cid:21) (cid:20) (cid:21) aration detected by Eisenstein et al. [46] using a sample
LetusdiscussbrieflythedifferenttermsenteringEq.(43). of 46748 luminous red galaxies (LRG) selected from the
Inthefirstone,weconsiderthedimensionlesscoordinate SDSS Main Sample [47]. Actually, rather than the posi-
distance y to an object at redshift z defined as: tion of acoustic peak itself, a closely related quantity is
better constrained from these data defined as [46]:
z dz′
y(z)=Z0 E(z′) (44) = √ΩM zLRG y2(zLRG) 1/3 (47)
A z E(z )
andrelatedtotheusualluminositydistanceD asD = LRG (cid:20) LRG (cid:21)
L L
(1+z)r(z). Daly & Djorgovki[37]havecompileda sam- with z =0.35 the effective redshift of the LRG sam-
LRG
plecomprisingdataony(z)forthe157SNeIaintheRiess ple. As it is clear, the parameter depends not only on
A
et al. [5] Gold dataset and 20 radio-galaxies from [38], the dimensionless coordinate distance (and thus on the
summarized in Tables1 and 2 of [37]. As a preliminary integratedexpansionrate),butalsoonΩ andE(z)ex-
M
step,theyhavefittedthelinearHubblelawtoalargeset plicitly which removes some of the degeneracies intrinsic
of low redshift (z <0.1) SNeIa thus obtaining: in distance fitting methods. Therefore, it is particularly
interesting to include as a further constraint on the
A
h=0.664 0.008 . model parameters using its measured value [46]:
±
8
Par bf 1σ CR 2σ CR
1.75
wDE −1.03 (−1.18,−0.91) (−1.39,−0.82) 1.5
Ω˜M 0.31 (0.27,0.36) (0.23,0.41)
ΩM 0.29 (0.27,0.31) (0.25,0.33) 1.25
weff 0.018 (−0.022,0.081) (−0.068,0.142) 1
M y
wX(z=0) −1.13 (−1.39,−0.92) (−1.71,−0.76) 0.75
q0 −0.54 (−0.71,−0.41) (−0.94,−0.31)
zT 0.61 (0.51,0.71) (0.41,0.83) 0.5
t0 (Gyr) 14.08 (13.45,14.71) (12.90,15.44) 0.25
0
0 0.5 1 1.5 2
TABLE I: Summary of the results of the likelihood analysis z
for the IM model. The maximum likelihood value (bf) and
the1and2σ confidencerange(CR)forthemodelparameters FIG. 1: The best fit curve superimposed on the observed
(wDE,Ω˜M,ΩM) and some derived quantities are reported. dimensionless coordinate distance y(z) for IM model.
not) Hubble parameter. This example stresses the need
for going beyond the SNeIa Hubble diagram in order to
=0.469 0.017 .
A ± not only lessen the impact of eventualsystematic errors,
Note that, although similar to the usual χ2 introduced but also directly break possible theoretical degeneracies
in statistics, the reduced χ2 (i.e., the ratio between the among model parameters. TableI reports a summary of
χ2 and the number of degrees of freedom) is not forced the results giving best fit values at 1 and 2σ confidence
to be 1 for the best fit model because of the presence of ranges for some interesting derived quantities. Since the
the priors on and andsince the uncertainties σi are uncertainties on (wDE,Ω˜M,ΩM) are not Gaussian dis-
R A
notGaussiandistributed,buttakecareofbothstatistical tributed, we do not apply a naive propagation of errors
errors and systematic uncertainties. With the definition to determine the constraints on each derived quantity.
(42)ofthelikelihoodfunction,thebestfitmodelparame- We thus estimate the 1 and 2σ confidence ranges on the
tersarethosethatmaximize (p). However,toconstrain derived quantities by randomly generating 20000 points
L
agivenparameterp ,oneresortstothemarginalizedlike- in the space of model parametersusing the marginalized
i
lihood function defined as: likelihood functions of each parameter and then deriv-
ingthelikelihoodfunctionofthecorrespondingquantity.
This procedure gives a conservative estimate of the un-
Lpi(pi)∝ dp1... dpi−1 dpi+1... dpnL(p) (48) cperrotcaeidnutireesmwahyicahlsios beneouusgehdfiofranouarnaailmytsi.caNloexteprtehsastiotnhiiss
Z Z Z Z
not available.
that is normalized at unity at maximum. Denoting with
χ2 thevalueoftheχ2 forthebestfitmodel,the1and2σ The best fit model parameters gives a theoretical y(z)
0
confidence regions are determined by imposing ∆χ2 = diagram which is in very good agreement with the ob-
χ2 χ2 =1 and ∆χ2 =4 respectively. served data as convincingly shown in Fig.1. It is worth
− 0 noting that the effective dark energy EoS w is quite
DE
close to the ΛCDM value which is not surprising since
B. Results for the IM model this latter model is known to fit very well a larger set
of observations (including also CMBR anisotropy spec-
trum and matter power spectrum [7, 8]). For the same
Before discussing the results of the likelihood analy-
reason, it is also not unexpected that Ω turns out to
sis,itisworthstressingthatthemethoddescribedabove M
be perfectly consistent with estimates coming from, e.g.,
makes it possible to obtain constraints on Ω directly
M
and not only on Ω˜ . Actually, since we have used, WMAP CMBR spectrum [4] and clustering data [3]. As
M a result, the best fit Ω˜ is only slightly larger than Ω
in our analysis, only the dimensionless coordinate dis- M M
tance, the likelihood function should depend only on thus suggesting that weff is vanishingly small. As re-
M
the parameters entering E(z) Lwhich determines the di- ported in TableI, the best fit weff is indeed very small
M
mensionless coordinate distance through Eq.(44). As a and weff = 0 is well within the 1σ confidence range.
consequence, only Ω˜M and wDE could be constrained BasedMon Eq.(11), one may argue that wH is practically
without any possibility to infer ΩM. The use of the pri- null and hence conclude that there is no need of any H2
ors on the shift and acoustic peak parameters is correctionterminthe darkenergyEoS.However,sucha
R A
the way to break the degeneracy between ΩM and the resultisbiasedbyatheoreticalprejudice. Actually,ifwe
ratio weff/w because both and depends explic- assumethatEq.(10)is correct,thenthe presentdayEoS
M DE R A
itly onΩ andnotimplicitly throughthe (integratedor turns out to be in the phantom region w < 1 and,
M X
−
9
Par bf 1σ CR 2σ CR
1.75
q0 −0.54 (−0.66,−0.48) (−0.78,−0.42)
1.5
Es 0.941 (0.937,0.945) (0.924,0.988)
ΩM 0.28 (0.26,0.29) (0.24,0.31) 1.25
wX(z=0) −0.93 (−1.0,−0.71) (−1.0,−0.50) y 1
zT 0.70 (0.62,0.76) (0.48,0.82) 0.75
t0 (Gyr) 14.21 (13.89,14.56) (13.37,14.85)
0.5
0.25
TABLE II: Summary of theresults of the likelihood analysis
for the QE model. The maximum likelihood value (bf) and 0
the1and2σ confidencerange(CR)forthemodelparameters 0 0.5 1 1.5 2
z
(q0,Es,ΩM) and some derived quantitiesare reported.
FIG. 2: The best fit curve superimposed on the observed
dimensionless coordinate distance y(z) for theQE model.
using Eq.(16), the true barotropicfactor w is estimated
f
(at 2σ CR) as:
expansion. This opens the way to determine the transi-
1.35 w 0.75
f
− ≤ ≤− tion redshift z defined as the solution of the equation
T
which is still more in the domain of phantom fields. A q(z )=0. For the IM model, it is possible to derive an
T
possiblewaytorecoverw > 1withoutalteringnoneof analytical expression:
f
−
the fit results is to resort to Eq.(18) rather than Eq.(9).
In such an approach, while the present day value of the
1
dark energy EoS wX(z = 0) is still given by the value z = 1−2q0+3wDE 3wDE 1 . (49)
reported in TableI, the constraints above on wf are no T (cid:20)(1−2q0)(1+wDE)(cid:21) −
more valid and should be replaced by:
The constraints we get are summarized in TableI. As a
1.35 w 1 wdHκ2 −1 0.75 . generalremark,wenote thatthe bestfitvaluezT =0.61
− ≤ f − 2 ≤− is once again close to that for the concordance ΛCDM
(cid:18) (cid:19)
modelforthe samereasonsexplainedaboveforq . Most
0
It is then possible to fit the same dataset with the same of the z estimates available in literature are model de-
T
accuracy by choosing a whatever value of wf > 1 pro- pendent since they have been obtained as a byproduct
−
vided that wdH is then set in such a way that the above of fitting an assumed model (typically the ΛCDM one)
constraint is not violated. to the data. A remarkable exception is the reported
Having discussed the constraints on the dark energy z =0.46 0.13foundbyRiessetal. [5]fittingtheansatz
T
EoS,it is interesting to compare some model predictions q(z) = q ±+(dq/dz) z to the Gold SNeIa sample. Al-
0 z=0
with the current estimates in literature. As a first test, though our best fit estimate is formally excluded at 1σ
we consider the present day value q0 of the decelera- level by the result of Riess et al., there is a wide overlap
tion parameter. Since, as a general rule, q0 = 1/2 + between our estimated1σ CR and that ofRiess et al. so
(3/2)ΩXwx(z = 0) and we get that both ΩX = 1 ΩM that we may safely conclude that the two results agree
−
and wx(z = 0) are consistent with the ΛCDM values, it each other.
is not surprising that the best fit q = 0.54 is quite
0 As a final check, we consider the present age of the
−
similar to the concordance model prediction q 0.55
0 universe that may be evaluated as:
≃ −
[4,5,7,8]. Ontheotherhand,q mayalsobedetermined
0
inamodelindependentwaybyusingacosmographicap-
∞
proach, i.e. expanding the scale factor a(t) in a Taylor dz
t =t (50)
series and fitting to distance related data. Using a fifth 0 H (1+z)E(z)
Z0
order expansion and the Gold SNeIa sample, John [48]
hasobtainedq = 0.90 0.65whichisasolargerangeto with t = 1/H = 9.778h−1 Gyr the Hubble time. Our
0 H 0
− ±
be virtually in agreement with almost everything. How- best fitestimate turns outto be t =14.08Gyr with the
0
ever,wenotethatourbestfitvalueissignificantlylarger 2σCRextendingfrom12.90upto15.44Gyr. Thisresult
thanhis estimate. Althoughthis couldbe aproblem, we is in satisfactory agreement with previous model depen-
donotconsideritparticularlyworrisomegiventheuncer- dentestimatessuchast =13.24+0.89GyrfromTegmark
0 +0.41
taintiesrelatedtotheimpactoftruncatingtheexpansion et al. [7] and t = 13.6 0.19 Gyr given by Seljak et al.
0
±
of a(t) to a finite order. [8]. Aging of globular clusters [49] and nucleochronol-
According to the most recent estimates, the cosmic ogy [50] give model independent (but affected by larger
speed up has started quite recently so that it is possi- errors) estimates of t still in good agreement with our
0
ble to detect the first signature of the past decelerated one. Note also that, the largervalue of the best fit t we
0
10
10
8
0.1
6
L 4 L
H% H% 0.001
DH 2 DH
0
0.00001
-2
-4
0.001 0.0050.01 0.05 0.1 0.5 1 0.001 0.0050.01 0.05 0.1 0.5 1
a a
FIG. 3: The percentage difference ∆H with respect to the FIG. 4: The percentage difference ∆H with respect to the
ΛCDM scenario for the IM model. We set the parameters ΛCDM scenario for the QE model. We set the parameters
ΩM (for both the IM and the ΛCDM model) and Ω˜M to ΩM (forboth theQE andtheΛCDM model) andq0 totheir
their best fit values in TableI, and choose three values for best fit values in TableII, and choose three values for Es,
wDE, namely wDE = −1.39 (short dashed), −1.03 (solid) namely Es = 0.924 (short dashed), 0.941 (solid) and 0.988
and −0.82 (long dashed). (long dashed).
obtainisalsoaconsequenceofourassumptionh=0.664 namical point of view) to the concordance ΛCDM sce-
that is smaller than h = 0.71 used by Tegmark et al. nario. As a further support to this picture, let us note
andSeljak etal. asa resultoftheirmoreelaboratedand that wX(z = 0) 1 as for the cosmological constant.
≃ −
comprehensive data analysis. Note that, because ofour choice to be in a non-phantom
regime,w isforcedtobelargerthan 1sothatwemay
X
−
give only upper limits on this quantity. If we had cho-
sen the phantom regime, and the possibility to cross the
C. Results for QE model
phantom divide, we should have given also lower limits
to w .
X
The results of the likelihood analysis obtained in this The fact that the model is indeed close to the usual
case are summarized in TableII where constraints on ΛCDMcanbealsounderstoodlookingattheconstraints
the model parameters (q0,Es,ΩM) and on some derived onEs. Eq.(23)shows thatthe ΛCDM modelis obtained
quantities aregiven,while Fig.2 showsthe best fitcurve in the limit E = 1 so that, in a certain sense, this pa-
s
superimposed to the data. As it is clear, the agreement rametermeasureshowfarthe darkenergyfluidisfroma
isquitegoodand,toalargeextent,theQE modelworks simple cosmological constant. Indeed, the best fit value
as well as the IM one in fitting the used dataset. E =0.941isquitecloseto1sothatonecouldarguethat
s
Although the dark energy EoS is significantly differ- only minor deviations from the standard ΛCDM model
ent, it is interesting to note that most of the QE model are detected.
predictedquantitiesareinverygoodagreementwiththe Asafinalremark,wenotethatboththetransitionred-
same quantities for the IM scenario. It is worthwhile to shift z (that, in this case, has to be evaluated numer-
T
spendsomewordsonthepresentdayvalueofthedeceler- ically) and the age of the universe t are in agreement
0
ationparameter. While fortheIM modelq0 isaderived with what is expected for the ΛCDM model (once the
quantity, hereq0 is used to assignthe model characteris- value of h is raised to the commonly adopted h = 0.70)
ticsandisthereforeobtaineddirectlyfromthelikelihood so that it is not surprising that we find good agreement
procedure. Notwithstanding the different role played in with the values reported in literature and quoted above
the two cases, the best fit value is identical, while the when discussing the results for the IM model.
1 and 2σ CR overlap very well4. This result may be
qualitatively explained noting that, in both cases, set-
ting the model parameters to their best fit values leads IV. HIGH REDSHIFT BEHAVIOUR
to a cosmological model that is quite close (from a dy-
The likelihood analysis presented above has demon-
strated that both the IM and the QE model are able
to fit the data on the dimensionless coordinate distance
4 It is not surprising that the 1 and 2σ CR on q0 in TableI are to SNeIA and radio-galaxies. As a further test, we have
slightlylargerthanthoseinTableII.Actually,fortheIM model,
also imposed priors on the shift and acoustic peak pa-
the constraints on q0 are determined by the uncertainties on
rameters in order to better constrain the models. Nev-
theothermodelparameters,while,inthepresentcase,theyare
obtainedfromthefittingproceduredirectly. ertheless, the bulk of the data we have considered only