Crossing SNe Ia and BAO observational constraints with local ones in hybrid metric-Palatini gravity

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by  Iker Leanizbarrutia

SNe Ia and BAO observational constraints on hybrid metric-Palatini gravity
Iker Leanizbarrutia,1, Francisco S.N. Lobo,2, and Diego Sa´ez-G´omez3,
∗ † ‡
1Fisika Teorikoaren eta Zientziaren Historia Saila, Zientzia eta Teknologia Fakultatea,
Euskal Herriko Unibertsitatea, 644 Posta Kutxatila, 48080 Bilbao, Spain
2Instituto de Astrof´ısica e Ciˆencias do Espa¸co, Faculdade de Ciˆencias da Universidade de Lisboa,
Edif´ıcio C8, Campo Grande, P-1749-016 Lisbon, Portugal
3Institut de Ci`encies de l’Espai, ICE/CSIC-IEEC, Campus UAB,
Carrer de Can Magrans s/n, 08193 Bellaterra (Barcelona), Spain
(Dated: February 1, 2017)
Inthispaper,weconsidertherecentlyproposedhybridmetric-Palatinigravitationaltheory,which
consists of adding to the Einstein-Hilbert Lagrangian an f(R) term constructed `a la Palatini. Us-
ing the respective dynamically equivalent scalar-tensor representation, we explore the cosmological
7 evolutionofaspecificmodel,givenbyf(R)∝R2,andobtainconstraintsonthefreeparametersby
1 usingdifferentsourcesofcosmologicaldata. Theviabilityofthemodelisanalysedbycombiningthe
0
conditions imposed by the Supernovae Ia and Baryonic Acoustic Oscillations data and the results
2
are compared with thelocal constraints.

n
a PACSnumbers: 04.50.Kd,98.80.-k,95.36.+x
J

1
3 I. INTRODUCTION `a la Palatini [13]. Using the respective dynamically
equivalentscalar-tensorrepresentation,itwasshownthat

] the theory can pass the Solar System observational con-
c Thediscoveryofthelate-timeacceleratedexpansionof
straints even if the scalarfield is very light. This implies
q the Universe [1, 2] has motivated a tremendous amount
- the existence of a long-range scalar field, which is able
r of research on dark energy models [3] and modified the- to modify the cosmological and galactic dynamics, but
g ories of gravity [4–9] as a possible source for this cosmic
[ leaves the Solar System unaffected. This has motivated
speed-up. In the latter framework, it is assumed that further research in this promising model. In fact, it was
1 atlargescalesEinstein’sGeneralRelativity(GR) breaks
shownthatthetheorycanalsobeformulatedintermsof
v down,andoneneedstointroducenewdegreesoffreedom
thequantityX κ2T+R,whereT andRarethetraces
0 to the gravitationalsector. In this context, f(R) gravity ≡
ofthe energy-momentumandRiccitensors,respectively,
8
has been widely exploredto addressthis problem, where
9 andthe variableX representsthe deviationwithrespect
essentially two approacheshave been analysedin the lit-
8 tothefieldequationtraceofGR.Thecosmologicalappli-
erature. Thefirstapproachconsistsonvaryingtheaction
0 cationsofthishybridmetric-Palatinigravitationaltheory
1. with respect to the metric [4], and the second approach, wereexplored[14],andcriteriatoobtaincosmicaccelera-
denoted by the Palatini formalism, consists on treating
0 tion were discussed. Severalclasses of dynamical cosmo-
the metric and the connection as separate variables [10],
7 logicalsolutions,dependingonthefunctionalformofthe
andtheseareusedinvaryingtheactiontoobtainthere-
1 effective scalar field potential, describing both accelerat-
: spectivefieldequations. Inf(R)gravity,bothformalisms
v ing and decelerating Universes were explicitly obtained
are related to two different theories, which easily be ver-
i andadynamicalsystemwasexplored,whichwasfurther
X ified by the respective scalar-tensor representations. In
explored in [15]. The problem of dark matter was also
r fact,onecanshowthatthef(R)metricformalismcorre- addressedinthecontextofthehybridtheory[16–18]. We
a sponds to a Brans-Dicketheory (in the presence of a po-
refer the reader to Ref. [19] for a recent review.
tential)withaBrans-Dickeparametergivenbyw =0
BD
and the Palatini formalism to a Brans-Dicke parameter The cosmologicalperturbation equations were derived
with w = 3/2. Moreover, within the metric ap- and applied to uncover the nature of the propagating
BD
−
proach,specificviablemodelshavebeenproposed,which scalardegreeoffreedomandthe signaturesthesemodels
are capable of keeping the GR results at local scales [11] predict in the large-scalestructure. The evolution of the
and provide good fits when compared with cosmological linear perturbations in the hybrid theory was analysed
data [12]. in [20], where the full set of linearized evolution equa-
tions, for the perturbed potentials in the Newtonianand
Now, recently, a novel approach to f(R) gravity has
synchronous gauges, were derived in the Jordan frame.
been proposed that consists of adding to the metric
It was concluded, that for the specific model used, that
Einstein-Hilbert Lagrangian an f( ) term constructed
R the main deviations from GR arise in the distant past,
with an oscillatory signature in the ratio between the
Newtonianpotentials. Furthermore,twoparticularmod-
els of the hybrid metric-Palatini theory were introduced
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] [21], and their background evolution was explored. It
‡Electronicaddress: [email protected] was shown explicitly that one recovers GR with an ef-
2
fective CosmologicalConstant at late times. This is due varying the action (2.1) with respect to the independent
to the fact that the PalatiniRicci scalar evolvestowards connection Γ˜λ yields
µν
andasymptoticallysettlesattheminimumofitseffective
potential during cosmologicalevolution. 1 1
˜ √ g δλf gβγ δβf gλγ δγf gβλ =0.
In this work, we consider the respective dynami- ∇λ − α R − 2 α R − 2 α R
(cid:20) (cid:18) (cid:19)(cid:21)
callyequivalentscalar-tensorrepresentationofthehybrid (2.4)
metric-Palatinitheory and explore the cosmologicalevo- It is straightforward to show that the connection Γ˜λ
lution ofa specific model, givenby f( ) 2. Further- iscompatiblewiththemetricf g ,whichisbasicallyµaν
R ∝R µν
more, we obtain constraints on the free parameters by conformaltransformationoftheRmetricg . Hence,both
µν
usingdifferentsourcesofcosmologicaldata. Morespecif- Riccitensorsarerelatedbyaconformalfactor,suchthat:
ically, the viability of the model is analysed by combin-
ing the conditions imposed by Supernovae Ia and Bary- 3f ,µf ,ν f ;µν 1gµν(cid:3)f
onic Acoustic Oscillations (BAO) data and the results Rµν =Rµν + 2 Rf2R − Rf − 2 f R . (2.5)
are compared with the local constraints. R R R
Thispaperisoutlinedinthefollowingmanner: InSec- In addition, by taking the trace of the field equations
tion II, we present the general formalism of the hybrid (2.3), the curvature can be expressed in terms of the
R
metric-Palatini theory, in particular, the action and the curvature R and the trace of the energy-momentumten-
field equationsinthe curvatureapproachandthe scalar- sor [14]:
tensor representation. In Section III, we consider the
f( ) 2f( )=κ2T +R=X , (2.6)
weakfieldlimitofthehybridgravity,andpresentthecon- R R− R
dition to pass the local tests. In Section IV, we consider
which can be solved algebraically to obtain = (X).
a particularmodel which is then confrontedwith cosmo- R R
Note that the trace of the Einstein equations gives
logical observations. More specifically, the cosmological κ2T + R = 0, such that the new variable X may be
evolutionofthemodelisanalysedandconstraintsonthe
usedto measuredeviationsfromGR.Moreover,the field
freeparametersareobtainedbyusingdifferentsourcesof
equations can be expressed in terms of the metric g
µν
data, and the viability of the model is analysed by com-
and X, such that for a particular cosmological solution
bining the cosmologicalandlocalconstraints. We finally
the corresponding action can be easily reconstructed by
presentanddiscussourresultsinSectionVandconclude
inverting the equation (2.6).
in Section VI.
In the next section,the scalar-tensorrepresentationof
thetheoryisexplored,whichyieldsinterestingproperties
that are shown in the subsequent sections and allows us
II. HYBRID METRIC-PALATINI GRAVITY
tostudythecosmologicalbehaviourofaviablemodelby
using such a framework.
A. Action and field equations
In the present manuscript, we consider a class of hy- B. Scalar-tensor representation
bridmetric-PalatinigravitygivenbytheHilbert-Einstein
term and an arbitrary function of the curvature scalar, AsinthecaseofthemetricandPalatinif(R)gravities,
whichisconstructed`alaPalatini. Thiscanbe expressed theaction(2.1)canbeexpressedintermsofascalarfield,
by the following Lagrangian[13]: which simplifies the analysis, such as the study of the
Newtonian law corrections or the study of cosmological
1
= d4x√ g[R+f( )]+ d4x√ g (g ,ψ ), solutions within hybrid f( ) gravity, as shown in the
S 2κ2 − R − Lm µν i R
Z Z next sections. In such a case, the action is given by [13]
(2.1)
whereκ2 =8πG,RistheRicciscalardefinedintermsof 1
the Levi-Civita connection of the metric g while = d4x√ g[R+φ V(φ)]
gµν is the Palatinicurvature, where thµeνRicci teRnso≡r S 2κ2 Z − R−
µν
is dRefined in terms of an independent connection Γ˜αµν as + d4x√−gLm(gµν,ψi) , (2.7)
Z
Rµν =Γ˜λµν,λ−Γ˜λµλ,ν +Γ˜λλσΓ˜σµν −Γ˜λµσΓ˜σλν . (2.2) Then, by varying the action with respect to the scalar
field φ, it yields,
Byvaryingthe action(2.1)withrespectto themetric,
the following field equation is obtained: ∂V
=0 φ=φ( ) . (2.8)
R− ∂φ ⇒ R
1 1
R g R+f f( )g =κ2T , (2.3)
µν µν µν µν µν
− 2 RR − 2 R Hence,theoriginalaction(2.1)isrecoveredprovidedthat
Eq. (2.8) is invertible,
where f = ∂f( )/∂ and the energy-momentum ten-
sor is deRfined asRusuaRl by Tµν = √−−2gδ(√δ−gµgνLm). Then, f(R)=φ(R)R−V (φ(R)) , (2.9)
3
where the scalar field and the potential are given by III. NEWTON LAW CORRECTIONS IN R+f(R)
GRAVITY
φ=f , V(φ)= f f( ), (2.10)
R R R− R
In order to study the correctionsinduced by the extra
respectively.
scalar field at local scales such as the Earth or the Solar
Besides Eq. (2.8), the field equations are obtained by
System,weproceedtoanalysethecorrectiontotheweak
varying the action (2.7) with respect to the metric and
fieldlimit,similarlyasperformedinRef.[22]forthemet-
the independent connection, leading to:
ric/Palatini case. Then, one considers the metric to be
quasi-Minkowski at local scales where the cosmological
1 1
Rµν gµνR+φ µν [φgµν V(φ)]=κ2Tµν , (2.11) evolutionhasnegligibleeffects andthe scalarfieldisalso
−2 R −2 −
approximated to its asymptotic value, which is assumed
and constant around the present time. Hence, the weak field
limit may be expressed as follows:
˜ √ gφgµν =0, (2.12)
∇ − g η +h (x) , h 1 ,
µν µν µν µν
respectively. These(cid:0)equations(cid:1)are equivalent to those φ ≈ φ +φ˜(x) , φ˜ | 1|.≪
given in Eqs. (2.3) and (2.4) as can easily be shown by ≈ 0 | |≪
using Eq. (2.10). As stated above, the solution for Eq.
As shown in [13] for the hybrid action (2.1), the field
(2.12) is given by the Levi-Civita connection of the met- equations (2.16) and (2.17) for h and φ (x) at linear
µν 1
ric φg , that is conformally related to g , such that
µν µν order yield:
the Ricci tensor R is conformally transformed (2.5),
µν
and we have 2 1 V + 2φ˜
2h = T η T + 0 ∇ η ,
µν µν µν µν
=R + 3 ∂ φ∂ φ 1 φ+ 1(cid:3)φ , ∇ −1+φ0 (cid:18) − 2 (cid:19) 2(1+φ0)
Rµν µν 2φ2 µ ν − φ ∇µ∇ν 2 (3.1)
(cid:18) (cid:19)(2.13) and
κ2φ
R=gµνRµν =R+ 2φ32∂µφ∂µφ− φ3(cid:3)φ . (2.14) (∇2−m2φ)φ= 3 0ρm . (3.2)
Here 2 is the 3D Laplacian of flat space and
Herethelasttermintheexpressionof isatotalderiva-
∇
R
tive, such that the action (2.7) is given by:
1
m2 = [2V V φ(1+φ)V ] , (3.3)
1 3 φ 3 − φ− φφ φ=φ0
S = 2κ2 d4x√−g (1+φ)R+ 2φ∂µφ∂νφ−V(φ) is the effective mass of the scalar field.(cid:12)(cid:12) The general so-
Z (cid:20) (cid:21)
lution assuming spherical symmetry and far from the
+ d4x√ g (g ,ψ ) . (2.15)
− Lm µν i sources was found in Ref. [13], and yields:
Z
Thisisthe actionofanon-minimallycoupledscalarfield 2G M V r2
eff 0
h (r) = + , (3.4)
withanon-canonicalkinetictermwhichmimicssomehow 00 r 1+φ 6
0
the Brans-Dicke theory except for the self-interacting
2γG M V r2
term of the scalar field and the coupling to the curva- h (r) = eff 0 , (3.5)
ij
r − 1+φ 6
ture. The field equations become (cid:18) 0 (cid:19)
R 1g R= 1 κ2T + φ g (cid:3)φ whereM = d3xρisthemassofthesource,Geff istheef-
µν µν µν µ ν µν fective Newtonian constant and γ is the post-Newtonian
− 2 (1+φ) ∇ ∇ −
R
h parameter, both given by:
3 3 1
∂ φ∂ φ+ g ∂ φ∂λφ g V , (2.16)
µ ν µν λ µν
− 2φ 4φ − 2 G φ
i Geff = 1 0e−mφr ,
1+φ − 3
0 (cid:18) (cid:19)
(cid:3)φ+ 1 ∂ φ∂µφ+ φ[2V (1+φ)V ]= κ2φT,(2.17) γ = 1+(φ0/3)e−mφr . (3.6)
2φ µ 3 − φ 3 1 (φ0/3)e−mφr
−
respectively. In comparison to the pure Palatini case, In order to avoidlarge corrections at scales of the Earth
the scalar field is dynamical when assuming the hybrid or the Solar System, the effective Newtonian constant
action (2.1), as shown by Eq. (2.17). has to be approximately G and the post-Newtonian pa-
In the next section, we review the weak field limit of rameter γ 1. Hence, contrary to the case of met-
≈
R+f( ) gravity and consider a particular model which ric f(R) gravity where a large mass of the scalar field
R
is then confronted with cosmological observations. m is required, which scales with the curvature through
φ
4
the chameleon mechanism in order to satisfy the obser- Inaddition,thecontinuityequationforthematterper-
vational constraints [23], in the hybrid gravity case de- fect fluid is given by
scribedbytheaction(2.1),justasmallvalueof φ 1
is required. Nevertheless, a positive mass m2 >|00is|≪also ρ˙m+3H(ρm+pm)=0 . (4.5)
φ
needed in order to avoid instabilities. Here we are focusing on the late-time epochs, so that
Let us now introduce the model that is constrained we can neglect the radiation contribution and assume a
in the next section in order to test the local constraints pressureless fluid for the barionic and dark matter con-
provided by (3.6), tent w = p /ρ = 0. Then, the continuity equation
m m m
(4.5)canbe easilysolvedintermsofthescalefactorand
2
f( )= R V , (3.7) the redshift, z:
0
R 4V −
1
a 3
where {V0,V1} are constants. By using the scalar-tensor ρm =ρm0 a0 =ρm0(1+z)3 , (4.6)
representation (2.7) or equivalently (2.15), the corre- (cid:16) (cid:17)
where1+z =(a /a)isused. Sinceouraimistoconstrain
spondingself-interactingtermforthescalarfieldisgiven 0
the model (3.7) by using specific observational data, the
by
FLRW equations should be expressed in terms of an ob-
V(φ)=V0+V1φ2 . (3.8) servableastheredshiftinsteadofthecosmictime,which
leads to:
Hence,inordertoavoiddeviationsfromNewton’slawat
local scales (3.6), one requires to have φ0 1 while the H2(1+φ) = Ω H2(1+z)3+ V
scalar mass should be positive: ≪ m 0 6
φ
m2φ =2(V0−2V1φ)>0, ⇒ φ<V0/V1. (3.9) + φ′H(1+z) H −(1+z)H4φ′ ,(4.7)
(cid:18) (cid:19)
Thus, the model (3.7) is consistent at local scales as far
and
as the above conditions hold. In the next section, the
cosmologicalevolutionofthemodel(3.7)isanalysedand 2HH′(1+z)(1+φ)=3ΩmH02(1+z)3
constraints onthe free parametersare obtained by using 3 φ2
different sources of data. The viability of such models is +2H2(1+z)φ′ (1+z)2H2 ′
− 2 φ
analysed by combining the cosmological and local con-
+(1+z)2HH φ +(1+z)2H2φ , (4.8)
straints. ′ ′ ′′
where Ω =κ2ρ /(3H2) and H is the Hubble param-
m m0 0 0
eter evaluated today. Hence, we can solve the equations
IV. FLRW COSMOLOGIES IN HYBRID
(4.7) and (4.8) to draw the cosmological evolution pro-
GRAVITY
vided by the model (3.8). To do so we identify the free
parameters of the model as V ,V ,φ ,φ ,H , where
0 1 0 1 0
Let us now study the cosmological evolution of the φ and φ are the scalar field{and its first deri}vative at
0 1
above model. Here we assume a flat Friedmann- z =0 respectively. However,by the sources of data used
Lemaˆıtre-Robertson-Walker(FLRW) metric: in this analysis, the Hubble parameter is dropped out
whileV canalsoberemovedbyusingthe equation(4.7)
3 0
ds2 = dt2+a(t)2 dxi 2, (4.1) as a constraint equation to satisfy the flatness condition
− evaluated at z =0:
i=1
X
where a(t) is the scale factor. In this work, we consider 1=Ωm+Ωφ . (4.9)
a perfect fluid, T diag( ρ ,p ,p ,p ), where ρ
µν ≡ − m m m m m Here Ωφ accomplishes for all the extra terms in (4.7).
andp aretheenergydensityandpressureforthematter
m Then,weobtainthefollowingexpressionrelatingthefree
content of the universe. Hence, by the field equations
parameters:
(2.16) and (2.17), the FLRW equations become [14, 19]:
φ
3H2 = 1 κ2ρm+ V 3φ˙ H + φ˙ , (4.2) V0 =6(cid:20)1+φ0−Ωm−φ1(cid:18)1− 4φ10(cid:19)(cid:21)−V1φ20 . (4.10)
1+φ" 2 − 4φ!# Hence, the only free parameters that remain are
1 3φ˙2 Ωm,V1,φ0,φ1 , which are constrained by using Super-
2H˙ = κ2(ρm+pm)+Hφ˙ + φ¨ ,(4.3) n{ovae Ia and B}aryonic Acoustic Oscillations (BAO) and
1+φ"− 2 φ − #
the results compared with the local conditions obtained
whereH =a˙/aistheHubbleparameter,whilethescalar in the section above.
field equation yields: Inthefollowingalltheobservationaldatausedtocon-
strain the model is explained. The analytical form of
φ¨+3Hφ˙ φ˙2+φ 2V (1+φ)dV = κ2φ(ρ 3p ). the χ2 expression used for each dataset is shown, which
m m
−2φ 3 − dφ − 3 − willbethenminimizedtoperformthestatisticalanalysis
(cid:20) (cid:21)
(4.4) from the Bayesian approach.
5
A. SNe Ia data where the angular-diameter distance is
c z dz
The Union2.1 compilation [24] dataset was chosen for D (z)= ′ . (4.17)
A
(1+z) H(z )
our SNe Ia test, which comprises of 580 Type Ia Super- Z0 ′
novaedistributedintheredshiftrange0.015<z <1.414.
The defined BAO observables are measured in three dif-
Besides the distance modulus µ(z ) for each SN, the full
i ferent overlapping redshift slices in the WiggleZ survey,
statistical plus systematics covariance matrix of the sur-
where the effective redshift in each bin are (z , z , z )=
vey is also given. The definition of distance modulus is 1 2 3
(0.44, 0.60, 0.73). The values that observables take in
each redshift bin are
µ(z)=5log d (z)+µ , (4.11)
10 L 0
F (z ) = 0.482
where the dimensionless luminosity distance d is given 1 1
L
by F (z ) = 0.650
2 2
F (z ) = 0.865,
z dz 3 3
′
d (z)=(1+z) , (4.12)
L Z0 E(z′) and the covariance matrix 103CBAO in this case is
and µ is a nuisance parameter which include all the in-
0 2.401 1.350 0.0
formationrelatedwithconstants,likethevalueofHubble
1.350 2.809 1.934 . (4.18)
constant H and the SNeIa absolute magnitude. E(a) is  
0 0.0 1.934 5.329
givenbythe adimensionalHubble functionofthe model,
 
once itis numericallysolvedfromEq.(4.7)-(4.8). Avec- Thus,thelaststeptocomputetheχ2 contributionofthe
torcanbedefinedwiththedifferencebetweenmodeland
BAO data is
observed magnitudes,
µ(z ) µ (z ) χ2BAO =(Xobs−Xmod)TC−B1AO(Xobs−Xmod), (4.19)
1 obs 1
XSN = −...  , (4.13) where Xobs = (F1(z1),F2(z2),F3(z3)) and Xmod is the
µ(zn) µobs(zn) datavectorcreatedusingthemodelwhichisbeingtested.
−
 
and we build the SNe contribution to the χ2 using the
covariance matrix C from [24], V. RESULTS AND DISCUSSION
χ2SN =XTSN ·C−1·XSN. (4.14) To perform the statistical Bayesian analysis, the min-
imisation of the χ2 is done by using the Markov Chain
However, this contribution to the χ2 from the SNe
Monte Carlo (MCMC) Method [27–29]. The model con-
data contains the nuisance parameter µ . An analyt-
0
tains four free parameters Ω ,V ,φ ,φ , where φ(z =
ical marginalization can be done over the µ0 nuisance { m 1 0 1}
0) = φ and φ(z = 0) = φ are the initial conditions
parameter [25], changing the χ2 contribution of the SNe 0 ′ 1
for the scalar field. The priors used in the MCMC com-
Ia data, after marginalizing,to the form of
putation are all broad constraints: for the matter we set
d b2 0 < Ωm < 1, and for the initial condition parameters
χ2SN =a+log2π − d , (4.15) −10< φ0 <10 and −10< φ1 < 10, while the quadratic
term of the potential V /H2 is left free. The parameter
1 0
where V isfixedbytheconstraintequation(4.10). Inaddition,
0
note that the equations (4.2) and (4.3) are not well de-
a ≡ XTSN ·C−1·XSN fined at φ = 0, since the equivalence among the hybrid
b XT C 1 1 model (2.1) and the scalar-tensor representation (2.15)
≡ SN · − ·
d 1T C 1 1, is not longer valid at φ = 0, where the conformal factor
−
≡ · · thatrelatesR and diverges. Atsuchpoint, the theory
with 1 standing for the identity matrix and T for trans- (2.1) reduces to GRRwith a cosmologicalconstant as can
pose. be seen by the action (2.7), i.e., the theory reduces to
ΛCDM in that case, a limit that may be achieved in the
model dealt in this paper (3.7) when 0.
R→
B. BAO data TheconvergenceoftheMCMC’sresultsweretestedus-
ingthemethodof[30],wherealmosteveryparameterhas
From WiggleZ Dark Energy Survey data [26], we have correctly converged. The single parameter that did not
usedtheAlcock-PaczynskidistortionparameterF(z)for fully converge was V1/H02, being its convergence at the
the BAO test. This observable is defined as edge of acceptable. Nevertheless, all chains have shown
the same minimum in the parameter space, even for the
F(z) (1+z)D (z)H(z)/c, (4.16) V /H2 parameter. Thus, these results of the MCMC’s
≡ A 1 0
6
TABLEI:ConstraintsforthePalatiniHybridmodeltogetherwiththeΛCDMmodel. Boldparametersareused
in MCMC analysis, plain ones are derived.
Model V1/H20 φ0 φ1 Ω0m V0/H02 χ2 χ2red
ΛCDM - - - 0.289+0.041 - 552.523 0.951
−0.037
Hybrid grav. −15+27 −0.12±0.62 0.02±0.91 0.43+0.20 3.87+0.68 552.126 0.955
−10 −0.26 −0.57
are summarised in Table I. As shown, the minimum χ2 cluding a non-metric curvature term. Then, by using an
achieved by the model is 552.126, being the reduced χ2 auxiliaryscalarfield,onecaneasilyshowthatsuchathe-
0.955. IncomparisontoΛCDMmodel,bycontrast,when ory of gravity is equivalent to a Brans-Dicke-like action,
using the same datasets, this results in χ2 = 552.523, such that the analysis turns out easier. Nevertheless,
min
ortakingintoaccountthedegreeoffreedomofthemodel, suchanequivalencebreaksdowninthelimitwhereGRis
χ2 = 0.951, with Ω = 0.289+0.041. Since the ΛCDM recovered. Bystudyingtheweakfieldlimitofthetheory,
red 0.037
modelcontainsjustonefreepar−ameter,thereducedχ2 the effective Newton constant and the post-Newtonian
red
becomes smaller. However, the differences on the value parameters are obtained, which depend not only on the
of χ2 among the models is not significant enough to mass of the scalar field (as occurs in chameleon fields)
red
rule out this model of hybrid gravity. In addition, as but on the background value of the scalar field itself, as
pointed above, the hybrid model (3.7) contains ΛCDM shown in Ref. [13], such that a small enough value of
as a possible limit. As shown in Table I and Fig. 2, the the scalar field may be enough to avoid violations at lo-
case φ = 0 is within the 1σ region, as expected. More- cal scales, in contrast to the large mass required in the
0
over, the range values for Ω becomes very large as the chameleonmechanism. Then,asimplemodelisproposed
m
auxiliary scalar field may behave as a pressureless fluid where the constraints on the free parameters of the the-
for large redshifts. ory are obtained.
On the other hand, combining the results of the fits
summarised in Table I and the local constraint given by Then, the cosmologicalevolutionofthe model is stud-
the conditions (3.9) to avoid Newtonian corrections, we ied by assuming a flat FLRW universe, where the aux-
may conclude whether hybrid gravity may be kept as a iliary scalar field is responsible for the late-time cosmic
viable candidate for dark energy. Note that the initial acceleration. Using Supernovae Ia data and Baryonic
value for the scalar field should be φ0 1, such that Acoustic Oscillations data, the free parameters of the
≪
we restrict the analysis to those values of the 1σ region modelarefittedbyusinganMCMCanalysis. Thevalues
in Fig. 1 which accomplishes such constraint. Then, in ofχ2 andχ2 areobtainedandcomparedtothe ΛCDM
red
Fig. 2 the evolution of the scalar field is depicted. In model. Thevaluesofχ2 donotdiffersignificantly,such
red
the right panel, the initial condition for φ0 is fixed while that hybrid gravity remains as a possible candidate for
the scalar potential varies according to the values of the dark energy. Nevertheless, the errors on the free param-
1σ regionfor V1 andV0, while the dashedlines represent eters turn out much larger, particularly on the value of
theconstraint(3.9)onthemassofthe scalarfield,which the matter density Ω , since the number of parameters
m
varies according to V1 and V0. Here we have assumed is larger than in the ΛCDM model and the scalar field
the mean value in Table I for φ0. In the right panel, the may mimic a pressureless fluid at large redshifts. Fi-
scalarpotentialisfixedatV1 = 15andtheinitialvalue nally,we comparethe cosmologicalconstraintsto the lo-
−
ofthescalarfieldvaries. Asshownintherightpanel,the cal constraint, where we found that the initial condition
scalarfield tends to zeroatlargeredshifts, convergingto on the scalar field determines whether the cosmological
the ΛCDM model in the past, but only some values of constraints and the local ones are both satisfied. More-
the scalar potential satisfy the constraint (3.9). Hence, over, the quadratic term of the scalar potential is well
whether the quadratic term of the potential V1 < 0, the constrained, concluding that it would likely be greater
initial value of the scalar field should be φ0 > V0/2V1 , than zero.
| |
and close to zero, while the initial conditions can be set
more freely when V >0.
1 Hence, we can conclude that modifications of GR in
thewayofbreakingthemetricityconditionarenotruled
out, but they lead to larger errors than the concordance
VI. CONCLUSIONS
model of cosmology. Future analysis of the growth of
large scale structure and the anisotropies of the CMB
Inthispaperwehavefocusedontheanalysisoftheso- may provide better constraints on this type of models.
calledhybridgravity,whereGRisslightlymodifiedbyin-
7
Acknowledgments edgesfinancialsupportfromtheUniversityoftheBasque
CountryUPV/EHUPhDgrant750/2014. ILalsothanks
FSNL acknowledgesfinancialsupportofthe Funda¸ca˜o the hospitality of the Instituto de Astrof´ısica e Ciˆencias
paraaCiˆenciaeTecnologiathroughanInvestigadorFCT do Espa¸co, Faculdade de Ciˆencias da Universidade de
Researchcontract,withreferenceIF/00859/2012,funded Lisboa where part of this work was carried out. This
byFCT/MCTES(Portugal). DSGisfundedbytheJuan article is basedupon workfromCOST ActionCA15117,
delaCiervaprogram(Spain)No.IJCI-2014-21733andby supported by COST (European Cooperation in Science
MINECO (Spain), project FIS2013-44881. IL acknowl- and Technology).
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8
40
0
2
0
H
/
V1 −40
−80
2
1
φ0 0
−1
−2
2
1
φ 0
−2
5
2 4
0
H
/
0
V
3
2
0.2 0.4 0.6 0.8 −80 −40 0 40 −2 −1 0 1 2 −2 0 2 2 3 4 5
Ω V /H 2 φ φ V /H 2
m 1 0 0 1 0 0
FIG. 1: Contours for 1σ and 2σ with also posterior probabilities of thePalatini Hybrid model.
9
0.2
0.05
0.1 0.00
-0.05
HLΦz 0.0 HLΦz -0.10
-0.15
-0.1 -0.20
-0.25
-0.30
-0.2
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
z z
FIG. 2: Evolution of the scalar field φ = φ(z). Left panel: the blue region represents the 1σ region for the values of the
potential V1/H02=−15+−2170 from table I, theinner red line refers to themean value. The dashed lines represent the constraint
φ < V0/2V1 from (3.9) for the mean value (red) and the upper and down limits (blue). Right panel: evolution of the scalar
field for different initial conditions φ0, thered line gives the mean value in table I and the dashed line providesthe newtonian
constraint 3.9.