Second order invariants and holography

0.19 MB·English

by  Luca Bonanno

Second order invariants and holography
Luca Bonanno¶♭, Gerardo Iannone♭ and Orlando Luongo‡†§
‡Dip. di Fisica, Universita` di Roma ”La Sapienza”, I-00185 Roma, Italy,
♭Institut fu¨r Theoretische Physik, J. W. Goethe - Univ. Max-von-Laue-Straße, 1 60438 Frankfurt am Main, Germany,
¶Dip. di Fisica, Universita` di Ferrara and INFN, via Saragat 1, 44100 Ferrara, Italy,
†Dip. di Scienze Fisiche, Universita` di Napoli ”Federico II”, Via Cinthia, Napoli, Italy,
§Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de M´exico, and
♭ Dip. di Fisica, Universita` di Salerno and INFN,I-84081, Salerno, Italy.
Motivated by recent works on the role of the Holographic principle in cosmology, we relate a
class of second order Ricci invariants to the IR cutoff characterizing the holographic Dark Energy
density. The choice of second order invariantsprovides an invariant way to account theproblem of
causality for the correct cosmological cutoff, since the presence of event horizons is not an a priori
1 assumption. Wefindthatthesemodelsworkfairlywell,byfittingtheobservationaldata,througha
1 combinedcosmological testwiththeuseofSNeIa,BAOandCMB.Thisclassofmodelsisalsoable
0 to overcome the fine-tuning and coincidence problems. Finally, to make a comparison with other
2 recent models, we adopt thestatistical tests AICand BIC.

n
a PACSnumbers: 98.80.-k,98.80.Jk,98.80.Es
J

0
I. INTRODUCTION where G is the gravitationalconstant,andR the radius
3 0
of the universe. If p and ρ are known (or equivalently

] H is known), the above coupled equations describe the
h EversincethediscoveryoftheDarkEnergy(DE)[1–3],
dynamics of the universe.
t the fundamentalproblemofunderstanding the natureof
- Inspiteofitssimplicityanditstriumphinfittingdata,
p the positive acceleration of the universe appears lacking
the ΛCDM model suffers from several shortcomings [9],
e ofsomeingredientsintheframeworkofGeneralRelativ-
h ase.g. the problemoffine tuning andthe problemofco-
ity (GR). Inaddition, observationsstronglysuggestthat
[ incidence. Theproblemoffinetuningisconnectedtothe
matter in the universe is dominated by a non-baryonic
large difference between the value of vacuum energy in
1 (cold) dark matter (DM) which seems to evolve sepa-
v rately from the unknown force driving the acceleration quantum field theory and the observed value of the cos-
8 [4, 5]. Despite neither DE nor DM have been directly mologicalconstant[10]. Theproblemofcoincidenceisre-
9 latedtotheincrediblysmalldifferencesbetweentheden-
detected, theyrepresentthe mostrelevantamountofen-
7 sitiesofDEandDM,whicharesupposedtoevolvediffer-
ergyandmatterintheuniverse,beingalmostthe70and
5
ently as the universe expands. Moreover,one can notice
. 25 percent of the total content, respectively. A theory
1 that at large enough energies (typically of the order of
propounded to explaintheir effects and to describe their
0 natures is strictly necessary in cosmology, although, up the Planck scale, Mpl ~c/G 1.22 1019GeV/c2),
1 ≡ ∼ ×
to now, it has not been still definitively formulated [6]. theΛCDMmodelissuppopsedtobenotreliableanymore;
1
: Probably, the most common paradigm trying to include inotherwords,itonlyprovidesalimiteddescription[11]
v both DE and DM, is the so-called ΛCDM model [7]. In of the early stages of the universe. As a consequence,
Xi this model, a cosmological constant term, Λ, is included the crucial role played by inflation still remains an effec-
within the Einstein equations and the total matter den- tiveapproachwithoutanybasisinafundamentaltheory.
r
a sity is assumed to be the sum of baryonic and DM den- Nevertheless, a Quantum Gravity (QG) theory seems to
sities, i.e. ρ=ρ +ρ . This model fits excellently and benecessarytoexplainthesestages[7],havingasalimit
b DM
with high-precision all observational data, meanwhile it theΛCDMmodel;thatisclearifonebelievesthatGRis
provides a remarkably small number of cosmologicalpa- a particular case of a more fundamental theory. This is
rameters [8]. The effects of Λ are hidden in the defini- theunderlyingphilosophyofwhatareusuallyreferredto
tions of the pressure, p, and of matter density, ρ, i.e. as modified theories of gravity, concerning the modified
p p+ Λc2, ρ ρ Λc2; thus, by assuming hereafter Einstein-Hilbert(EH)action[12,13]. Besidesfundamen-
→ 8πG → − 8πG talphysics motivations,cosmologistsacquireda hugein-
a (flat) Friedman-Robertson-Walker (FRW) metrics, i.e.
ds2 = cdt2 a(t)2 dr2/(1 kr2)+sin2θdφ2 +dθ2 , we terestinallofthese theories,thanks to the possibilityto
− − reach a unified scheme. Among the various alternatives,
caneasilywritedow(cid:2)nthecorrespondentFriedmanne(cid:3)qua-
crucial advances have been carried out in the studies of
tions:
Black Hole theory and String theory [14].
A very intriguing concept, within the framework of
H2 a˙ 2 = 8πGρ kc2 , (1) GR, is the so-called holographic principle (HP), which
≡(cid:18)a(cid:19) 3 − R2a2 provides some clues for solving the modern theoretical
0
a¨ 4πG and observational problems without directly modifying
= 3p/c2+ρ , the EH action. As it has been pointed out in QG, the
a − 3
(cid:0) (cid:1)
2
entropy of a system does not scale with the volume of invariants of curvature, derived from the Ricci scalar R,
that system, but with the area of its surface [15]. Start- the Weyl tensor C and the Riemann tensor R ,
µνξδ µνξδ
ingfromtheaboveconsideration,theHPpostulatesthat are based on such theoretical implicit assumptions and
the maximum entropy inside a region is not extensive, represent geometricalinvariants. Gao [23] suggested, for
butgrowsastheareaofthe surface. Therefore,the total the firsttime, thatthe DE densitycouldbe proportional
number of independent degrees of freedom should scale to the first order invariant R:1
with the surface area (in Planck units) as well. In par-
ρ R. (4)
ticular,theprincipleinvokesthatL3ρvac ≤LMP2l,where X ∝
MPl is the Planck mass and ρvac is the vacuum energy In this work we extend the analysis by Gao to the case
density of a system of size L. Actually, the HP repre- of second order (independent) invariants. Notice that
sents a new basic principle for both QG and GR, being a second order invariant is proportional to the inverse
supported by an effective quantum field theory. In order fourth power of an IR cutoff, then indicating with I the
i
to include this principle in a real cosmological scenario, i-th independent second order invariant, our assumption
oneneedstochoosethe correctcosmologicallengthscale should be formulated as follows
L, which is not a priori known.
ρ I . (5)
Since the cosmological system is the universe, the X ∝ | i|
p
DE should be associated with the scale density, namely The purposeofthe presentarticleis tostudy the models
ρDE ρX L. Writing the fraction of DE density in originating from these invariants and their implications
the fo≡rm ΩX∝≡ 3MρP2XlH2, we can imagine different holo- on the FRW spacetime. As shown below, these models
graphic approaches by choosing different IR cutoffs. For fairlyovercometheproblemofthelengthscale,providing
instance, for the case of the ΛCDM, if the Hubble pa- a good solution for the DE problem [7] and indicating a
rameter H is taken to be the characteristic scale of the good agreement with the theoretical predictions.
universe, then it is natural to postulate Λ−1/2 H; un- The paper is organized as follows: in Section II we
∝
fortunatelythis scenariofailsinreproducingthe positive develop the theory of the independent second order in-
accelerationofuniverse. Nevertheless,fromtheHPprin- variantsforaflatFRWuniverseandweanalyzethe out-
ciple, it would be possible to solve both the problems comesofthe assumption(5)incosmology. InSectionIII
of fine tuning and coincidence just by introducing the wefitourmodelswiththecosmologicaldatafromSuper-
correct length scale. In order to fix such a scale, many novae Ia (SNeIa), Baryonic Acoustic Oscillation (BAO)
approaches have been investigated in literature. In the andCosmologicalMicrowaveBackground(CMB).InSec-
so-calledHolographic Dark Energy model (HDE) [16, 17] tionIV,weprovideacomparisonwithotherrelevantcos-
thefutureeventhorizonoftheuniversecharacterizesthe mologicalmodelsandwemakeuseofmodelindependent
length L, i.e. statisticaltests,i.e. the AIC andBIC.FinallyinSection
V we present the conclusions.
∞ da′
L a . (2)
≡ Z Ha′2
a
II. SECOND ORDER HOLOGRAPHIC
An alternative is the so-called agegraphic model (ADE) INVARIANTS
[18–20], where the IR cutoff is the conformal age of the
universe, ThebasicpurposeofthisworkistoinvoketheHPand
torelatethesecondordercurvature(independent)invari-
da′
L η = . (3) ants to the DE density (as already discussed, the case of
≡ Z a′2H the first order curvature invariant has been extensively
treated by Gao [23–25]).
Unfortunately, the choice of a length scale is neither so
This can be easily performed by writing down the ex-
easynorarbitrary,sinceitleavesunclearsomeconceptual
plicit form of the second order geometrical invariants
problems. As pointed out by Cai [21], a drawback, con-
for a FRW metric and by using eq. (5). As pointed
cerning causality, appears in the above scenarios: how is
out in Ref.[27], among the 14 curvature scalar invariants
itpossiblethata localquantity,ase.g. the DE,couldbe
[26], the most interesting ones are the Kretschmann, the
explainedbyglobalconcepts,derivedfromthephysicsof
Chern-Pontryagin and Euler invariants. We can write,
space-time? Is there a mechanism allowinga localquan-
for every spacetime [28–32], their expressions as follows
titytobedeterminedbyaglobalone[22]? Acrucialissue
isthattheabovecosmologicallengthsareoriginatedfrom K =R Rαβγδ,
1 αβγδ
an expanding universe, which is an a priori assumption K =[∗R] Rαβγδ,
2 αβγδ
and notthe result ofa certainmodel, as it should be. In
order to solve this problem and inspired by the HP, we K3 =[∗R∗]αβγδRαβγδ, (6)
suggest to choose as length scales only those quantities
which are invariant under geometrical transformations,
avoiding the causality issue. This allows to solve both 1 UsuallythemodelisreferredtoastheRicciDarkEnergy(RDE)
the coincidence and fine-tuning problems, as well. The model.
3
where the starsindicate the correspondentdual counter- respectively. In order to fix the free parameters of our
parts. From the first Matt´e-decomposition of the Weyl two models, we need to find for which values of α, the
tensor, it is easy to get [33] conditions q < 0 and w 1 at redshift z = 0 are ful-
0
∼ −
filled. The corresponding intervals are α [0.042,0.052]
1 ∈
Rαβγδ =Cαβγδ+ 2 gαγRβδ− andα∈[0.039,0.047]formod1andmod3respectively. In
(cid:0) figures 1, 2, 3 and 4 we plot q(z) and w(z) as functions
−gβγRαδ−gαδRβγ +gβδRαγ of the redshift for both mod1 and mod3. As shown in
1 (cid:1) the pictures, if we set α=0.050for mod and α=0.046
1
(g g g g )R. (7)
−6 αγ βδ− αδ βγ formod3 wegettheexpectedacceleratedbehaviorofthe
universe, being q 0.6 and w 1. In particular, the
Therefore, K1, K2 and K3 can be expressed as follows results obtained w∼it−h the two m∼o−dels suggest that the
1 universe is accelerating since z 0.4.
K1 =CαβγδCαβγδ+2RαβRαβ R2 = ∼
− 3
1
=I +2R Rαβ R2, (8)
1 αβ − 3 0.4
K =[∗C] Cαβγδ =I 0.2
2 αβγδ 2 α=0.0502
2 0
K3 = CαβγδCαβγδ+2RαβRαβ R2 = -0.2
− − 3
2 q-0.4
= I +2R Rαβ R2.
1 αβ -0.6
− − 3
-0.8
From the above equations it is straightforward to infer
-1
the explicit expressions of the second order invariants in
-1.2
a FRW universe:
-1.4
I = 60 (H˙ +2H2)2+H4+ 2H2kc2 + k2c4 , -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0z.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4!
1 c4 (cid:26) R2a2 R4a4(cid:27)
0 0
I =0, FIG.1. Decelerationparameterqasafunctionoftheredshift
2
z for mod (in this case we use conventionally α = 0.050).
12 10kc2H2 5k2c4 1
I = 5(H˙ +2H2)2+5H4+ + Note that q is negative for z <0.4, providing an accelerated
3 −c4{ R2a2 R4a4 behavior of the universe since z =0.4; this is quite in agree-
0 0
kc2 ment with theprediction of ΛCDM which refers to z∼0.77.
+2(H˙ +2H2)H2+2(H˙ +2H2) . (9)
R2a2}
0
The HP postulate reads
0
3α α=0.0502
ρ = I , i=1,2,3; (10)
X i
8πG | |
p
where α is an dimensionless constant and should not be -0.5
confused with the tensorial index.
w
By combining together eqs. (10) and (1) and by using
the above expressions for I and I (the only two non-
1 3
-1
trivial invariants), we obtain two differential equations,
eachone providingthe temporalevolutionofthe Hubble
parameter. Afirstinterestingstepconsistsinsolvingthe
associated differential equations numerically for both I
1 -1.5
-1 0 1 2 3 4 5 6
and I , finding the correspondent acceleration parame- z
3
ters, by using the definition
FIG. 2. The barotropic factor w ≡ p as a function of the
ρ
H˙ redshift z for mod1 (in this case we use α = 0.050). Notice
q(t)= 1, (11) that w = −1 at z = 0, behaving as a cosmological constant
−H2 −
at low redshift.
and the effective barotropic parameters, expressed by .
1 d
w(t)= 1 lnρ, (12) Once determined q(z) and w(z) we can characterize
− − 3H dt
completelythekinematicsofthemodels. Inthefollowing
for both the models. section we will check the agreement of our models with
Fromnow on, we will refer to the cosmologicalmodels theobservations. Inparticular,wewillmakeuseofthree
arising from the invariants I and I as mod and mod , independent tests: SNeIa, BAO and CMB.
1 3 1 3
4
us rewrite the distance modulus
0.4
d
0.2 l
α=0.04621 µ=25+5log , (13)
0 10 Mpc
-0.2
where d (z) is the luminosity distance, defined by
L
q-0.4
-0.6 z dz′
d (z)=c(1+z) . (14)
-0.8 L Z H(z′)
0
-1
Sincethelikelihoodfunction isrelatedtothechi-square
-1.2
L
statistic, i.e. exp( χ2/2), we constrain the free
-1.4
L ∝ −
-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 parameters of a model, by minimizing the quantity
z
(µtheor µobs)2
FIG.3. Decelerationparameterqasafunctionoftheredshift χ2 = i − i . (15)
zformod3(inthiscaseweuseα=0.046). Againqisnegative SN Xi σi2
forz<0.4, providinganaccelerated behavioroftheuniverse
since z=0.4 until now. The second test that we perform is related to the ob-
servations of large scale galaxy clusterings, which pro-
vide the signatures of the BAO [36]. In particular we
0
use the measurement of the peak of luminous red galax-
α=0.04621
iesobservedinSloanDigitalSkySurvey(SDSS), usually
denoted by A and defined as follows
-0.5
2
w A= Ω H0 31 1 zBAO H0 dz 3 ,
m
H(z ) (cid:20)z Z H(z) (cid:21)
p (cid:2) BAO (cid:3) BAO 0
-1 (16)
with z = 0.35. In addition, the observed A is
BAO
estimated to be
-1.5-1 0 1 2 3 4 5 6 0.95 −0.35
z A =0.469 , (17)
obs
(cid:18)0.98(cid:19)
FIG. 4. Thebarotropic factor w as a function of theredshift
z formod3 (inthiscaseweuseα=0.046). Noticethatagain with an error σA = 0.017. In the case of the BAO mea-
w=−1 at z=0, behaving in this case too as a cosmological surement we should perform the following minimization:
constant term at low redshift.
A A 2
χ2 = − obs . (18)
BAO (cid:18) σ (cid:19)
A
III. COSMOLOGICAL CONSTRAINTS
Finally, concerning the CMB, we first analyze the so-
called CMB shift parameter
In this section we perform an experimental combined
zCMB H
procedure, by using a fairly typical combination of kine- R= Ω 0 dz. (19)
m
matical data. In particular we employ the three most p Z0 H(z)
common fitting procedures: SNeIa, BAO and CMB; the
Since this standardruler is fixed by the soundhorizon
first two concern low redshift data sets spanning from
atdecoupling(z =1091.36[37]),itgivesacomplemen-
z = 0 to z 2, while the third is a higher redshift test, dec
∼ tary bound to the SNeIa data and BAO as well. For the
since it is performed at z 1000.
∼ Rparameter,the observedvalue is Robs =1.726 0.018,
It is well established that standard candle data from as inferred from the WMAP 7 data [34]. Hence,±the cor-
SNeIa are indicators of distance, able to fit the corre- respondentCMBconstraintsaregivenbyminimizingthe
spondent luminosity distances for a particular class of chi square
models,byconsideringthedistancemodulusandthecor-
respondent redshift z. The problem of the systematics, R R 2
χ2 = − obs . (20)
which usually affects these measures, can be avoided or CMB (cid:18) σ (cid:19)
R
at least reduced, by the use of the most recent updated
Union 2 compilation [34], instead of other older samples Moreover,wenoticethat,differentlyfromSNeIa,BAO
[35]. Then,associatingtoeachSupernovamodulusµthe and CMB do not depend on H .
0
corresponding 1σ error, denoted by σ , we can perform Finally,bycombiningthe minimizationproceduresfor
µ
directly the experimental analysis. To this purpose, let SNeIa, CMB and BAO, we constrain the parameters of
5
Ωm(SN) Ωm(BAO) Ωm(CMB) oncegivenf(x)andg(xθ),thereexistsonlyasetofθmin
|
0.240±0.070 0.234±0.014 0.266±0.064 minimizing the difference between g(x,θ) and f(x) [48].
α(SN) α(BAO) α(CMB) However,withoutgoingintodetails,weonlynotethat
0.046±0.016 0.044±0.010 0.025±0.016 the AIC value for a single model is meaningless since
the exact model function f(x) is unknown. Therefore,
TABLE I. Summary of the results for mod1; in this case the quantities of interest are the differences ∆AIC
we have χ2 = 1.021, χ2 = 1.001, χ2 = 1.001, ≡
SNeIa BAO CMB AIC AIC , calculatedoverthe whole set ofmodels.
Ωm,(mean) =0.247±0.070, α(mean) =0.038±0.016. The g−enericmAinIC is given by
Ωm(SN) Ωm(BAO) Ωm(CMB) AIC= 2ln +2κ. (21)
max
0.260±0.090 0.312±0.080 0.343±0.060 − L
α(SN) α(BAO) α(CMB)
0.042±0.010 0.042±0.002 0.024±0.0015 A very similar criterionwas derivedby Schwarz[45] in a
Bayesiancontext (see [46] and references therein).
TABLEII.Summaryoftheresultsformod . Thereducedchi
3
squared are χ2 = 1.021, χ2 = 1.001, χ2 = 1.003; The BIC test provides
SNeIa BAO CMB
themeanvaluesΩ =0.305±0.090,α =0.036±
m,(mean) (mean)
0.015.
BIC= 2ln +klnN, (22)
max
− L
our two models. The numerical results are summarized
where, for both AIC and BIC, is the maximum
max
in the following tables: L
likelihood, k is the number of parameters, and N is the
The results show that the theoretical predictions dis-
number of data points used in the fit. If the errors are
cussed in Sec.II are in agreement with the experimental Gaussian, then χ2 = 2ln and the difference in
ones. The mean values of α in the tables are indeed BIC can be simplmifiiend to−∆BILCm=ax∆χ2 +∆klnN.
included in the intervals of values found in Sec.II. The min
CMBmeasurementsareusuallysmallerifcomparedwith We adopted both the AIC and BIC tests for different
the SNeIa and BAO ones. Notice finally that for the models.
SNeIa we had to fix a value for H (H = 2.33−18s−1),
0 0 AfirstnaturalextensionoftheΛCDMparadigmisthe
while for CMB and BAO it is not necessary.
wCDMmodel,alsocalledquintessencemodel;itprovides
an EoS of the form p = wρ with a negative barotropic
factor,whoseoriginisrelatedtothecouplingoftheRicci
IV. COMPARISON WITH OTHER MODELS
scalarwithanotevolvingscalarfieldφ. Sincethismodel
isstronglydependentonw,itiscalledwCDMinanalogy
In the Introduction we pointed out that the ΛCDM with the ΛCDM.
paradigmappearstobethe favoritefittingmodelamong
alargenumberofpossibilities. Thisshoulddependonits The theoretical explanation about the origin of the
smallnumberofparameters. Inparticular,foraflatcos- scalar field giving a negative EoS remains unsolved in
mology, the only parameter involved is the mass density the wCDM, then a varying quintessence has been pro-
Ω . posed: if w evolves with the redshift z, i.e. w = w[z],
m
Therefore,one canaskif there is a realnecessityto go the scalarfieldevolvesaswell. Sothe originofthe latter
beyond this approach, by considering other frameworks. should be found thermodynamically or in other ways.
Thisquestionsuggeststherequirementtofindatestable
One of the more recent and intriguing varying
to comparedifferent cosmologicalmodels, in orderto se-
quintessencemodelhasbeenproposedbythe Chevallier,
lectthe”best”one. Atthesametime,suchatestshould
Polarsky, Linder (CPL) [49]. This parametrization sug-
also be model independent. geststhatw[a]=w +w (1 a). Assuminga (1+z)−1,
0 1
Agoodchoiceisrepresentedbytheso-calledAkaikeIn- onegetsw[z]=w +w z −,whichforlowan≡dveryhigh
formationCriterium(AIC) andBIC[38,39]tests,which 0 11+z
redshifts becomes constant, i.e. w(z 0) = w and
are two of the most model-independent statistical meth- → 0
w[z ]=w +w .
odsforcomparingdifferentmodels. Moreover,sincetheir →∞ 0 1
firstusebyLiddle[40],thesetestsbecameastandarddi- WeperformtheAICandBICtestsforourtwomodels
agnostic tool [41–43] of regressionmodels [44–47]. (mod and mod ) and for the models discussed above.
1 3
The idea behind AIC is based on postulating two dis- Moreover, we also include the Ricci DE model (RDE),
tribution functions, namely f(x) and g(xθ): f(x) is as- studied by Gao [23]. The normalized Hubble rates, E
sumedtobetheexactone,whileg(xθ)ap|proximatesthe H ,forthesemodelsarewrittenbelow(noticethatther≡e
| H0
formerthroughasetofparametersdenotedbyθ. Hence, is not an analytic expression of E for the models mod
1
6
and mod ): it is interesting to notice that, to a slight variation of
3
w and w , it corresponds a large variation of AIC and
0 a
E = Ω (z)+1 Ω , BIC. This is in agreement with the fact that CPL is a
ΛCDM m m
−
p three parameters approach,then it appears disfavored if
compared with mod and mod .
1 3
E = Ω (z)+(1 Ω )(1+z)3(1+w),
wCDM m m
q −
(23)
V. CONCLUSION
E = Ω (z)+(1 Ω )f(z),
CPL m m
−
p
Inthis workwe have investigatedthe possibility to re-
ERDE =r2 2αΩm(z)+Ωf(1+z)4−α2 , lvaatreiatnhtesDIiE. Ttehrimshwasitbhetehnepseercfoonrmdeodrdinerainnadleopgeynwdietnhttihne-
− workofGao[23],whoproposedafirstorderinvariantap-
where f(z) = (1 + z)3(1+w0+w1)exp 3w1z and proach, namely the Ricci DE. The validity of our idea is
n−1+z o basedontheholographicprinciple,whichrequirestheex-
Ω (z) Ω (1+z)3.
m ≡ m istence of a cut-off scale, characterizing the dark energy.
The results of the tests are summarized in the table
Moreover, the latter must scale with the inverse square
below, in which we report, for each model, the number
of the cut-off (which means ρ √I ). After solving
and the names of the free parameters, the χ2 and the X ∝ i
numerically the Friedmann equations for the two non-
values of BIC and AIC:
trivial invariants in a flat FRW universe, we tested the
tworesultingmodels,namelymod andmod . Hence,we
1 3
Model N. Par. (k-1) Param. χ2 ∆BIC ∆AIC
min first analyzed the kinematics of the two models in terms
ΛCDM 1 Ωm 557.40 0 0 of the acceleration parameter q(z) and in terms of the
Quint. 2 Ωm,w 557.28 6.20 1.88 barotropic factor w(z), showing that they are compati-
CPL 3 Ωm,w0,wa 557.45 12.69 4.05 ble with the conditions q < 0 and w ∼ −1 at z = 0.
Then, we developed a test combining SNeIa, BAO and
RDE 2 Ωm,α 573.72 22.64 18.32
CMB in order to fix the cosmological parameters; again
mod1 2 Ωm,α 568.86 17.79 13.46 in this case we found a good agreement with the exper-
mod3 2 Ωm,α 568.91 17.84 13.51 imental results. It is also important to notice that the
two models, arising from different invariants, show an
incredibly similar behavior.
where we have used: Ω = 0.254 0.038 for ΛCDM; Arobustevidenceofthevalidityofthemodelsisgiven
m
±
Ω = 0.325 0.049 and w = 1.18 0.18 for wCDM; by the AIC and BIC tests, where we get results bet-
m
± − ±
Ω =0.246 0.034,w = 0.91 0.12andw = 0.32 ter than those obtained with the Ricci DE approach.
m 0 a
± − ± − ±
0.04forCPL[50,51];Ω=0.310 0.052andα=0.380 Therefore, our models are compatible with the holo-
± ±
0.049 for the Ricci DE. graphic principle and seem to behave better than the
The results have been obtained through a direct anal- first order invariant proposed by Gao. This encourages
ysisofeachmodel,followingthecombinedprocedureex- to study higher order invariants, what will be done in
plained in Section III. We can conclude that our models future works.
are disfavoredby the AIC and BIC analysis if compared
withΛCDM andwCDM. Itis clearthatΛCDMremains
ACKNOWLEDGEMENTS
the favorite model. On the other hand, the results ob-
tained for the Ricci DE are even worst. This seems to
suggest that using higher order invariants is better than It is a pleasure to thank Prof. Luca Amendola, Dr.
usinginvariantsoflowerorder,encouragingfurtherinves- Andrea Geralico, Prof. Salvatore Capozziello and Prof.
tigations in our direction. Concerning the CPL model, Hernando Quevedofor very fruitful discussions.
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