Observational tests of Galileon gravity with growth rate

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by  Koichi Hirano

Observational tests of Galileon gravity with growth rate
∗
Koichi Hirano
Department of Physics, Ichinoseki National College of Technology, Ichinoseki 021-8511, Japan
Zen Komiya
Department of Physics, Tokyo University of Science, Tokyo 162-8601, Japan
(Dated: January 25, 2011)
We compare observational data of growth rate with the prediction by Galileon theory. For the
same value of the energy density parameter Ωm,0, the growth rate in Galileon models is enhanced
compared with the ΛCDM case, due to the enhancement of Newtonfs constant. The smaller Ωm,0
1
1 is, the more suppressed growth rate is. Hence the best fit value of Ωm,0 in the Galileon model is
0.16 from only the growth rate data, which is considerably smaller than such value obtained from
0
observations of supernovae Ia, the cosmic microwave background and baryon acoustic oscillations.
2
This result seems to be qualitatively thesame in most of thegeneralized Galileon models. Wealso

n findtheupperlimitoftheBrans–Dickeparametertobeω<−40,fromthegrowthratedata. More
a and bettergrowth rate data are required to distinguish between dark energy and modified gravity.
J

4
2 I. INTRODUCTION in the 4-dimensional effective theory of the DGP model.
The self-interaction term (cid:3)φ( φ)2 induces decoupling

] ∇
of the Galileon field φ from gravity at small scales by
O Cosmologicalobservations,suchastype Ia supernovae
the so-called Vainshtein mechanism [26]. This allows
(SNIa) [1, 2] and the cosmic microwave background
C
the Galileontheorytorecovergeneralrelativityatscales
(CMB) anisotropies [3], indicate that the universe is un-
.
h dergoing an accelerated phase of expansion. This late- around the high density region, which is consistent with
p time accelerationis one of the biggest mysteries in mod- solar-systemexperiments.
o- ern cosmology. The conventional explanation is that it Galileontheory hasbeen covariantizedandconsidered
r is caused by the cosmological constant or dark energy in curved backgrounds [27, 28]. It has been shown that
st [4–7]. This would meanthat the universeis mostly filled Galileon symmetry cannot be preserved once the theory
a with an unknown energy-momentum component. The is covariantized;however,it is possible to keep the equa-
[ cosmologicalconstantis the standardcandidate for dark tion of motion as a second-order differential equation,
2 energy. To explain the current acceleration of the uni- that is, free from ghostlike instabilities. Galileon grav-
v verse,the cosmologicalconstantmusthaveanincredibly ity caninduce self-acceleratedexpansionofthe late-time
1 small value. However, its value cannot be explained by universe. Hence, inflation models inspired by Galileon
5 current particle physics and it is affected by fine-tuning theory have been proposed [29–31]. In Ref. [32], the pa-
4
problems and the coincidence problem. rametersofthegeneralizedGalileoncosmologywerecon-
5
. An alternative explanation for the current accelerated strained by observational data of supernovae Ia (SN Ia),
2 expansion of the universe is to extend general relativ- the cosmic microwave background (CMB) and baryon
1 ity to a more general theory of gravity at long dis- acousticoscillations(BAO).Theevolutionofmatterden-
0
tances. Several modified gravity approaches have been sityperturbationsinGalileonmodelshasalsobeenstud-
1
: proposed including f(R) gravity [8], scalar-tensor the- ied [19–21, 33, 34].
v ories [9–11], and the Dvali–Gabadazde–Porrati (DGP) Inthisletter,wecompareobservationaldataofgrowth
i
X braneworld model [12–14]. The DGP model, however, is ratewiththepredictionsofGalileontheory. Eventhough
plagued by the ghost problem [15] and is incompatible the background expansion history in modified gravity is
r
a with cosmologicalobservations [16]. almost identical to that of the standard ΛCDM model
As an alternative to general relativity, Galileon grav- or dark energy models, the evolution of matter density
ity models have recently been proposed [17–25]. These perturbations in modified gravity is different from that
modelsareconstructedbyintroducingascalarfieldwith of the ΛCDM model or dark energy models. Thus it is
a self-interaction whose Lagrangian is invariant under importanttostudythegrowthhistoryofperturbationsin
Galileonsymmetry∂ φ ∂ φ+b ,whichmaintainsthe order to distinguish modified gravity from models based
µ µ µ
equation of motion as a→second-order differential equa- on the cosmological constant or dark energy. Hence, we
tion. This prevents the theory from exhibiting a new computethegrowthrateofmatterdensityperturbations
degree of freedom, and perturbation of the theory does inGalileoncosmologyandcompareitwithobservational
not lead to ghost or instability problems. The simplest data.
term for the self-interaction is (cid:3)φ( φ)2, which appears This letter is organizedasfollows. Inthe next section,
∇
we describe the background evolution and the evolution
oflinearperturbationsinGalileoncosmology. InSec. III
we study the observationaltests ofGalileongravitywith
∗ [email protected] growth rate. Finally, a summary is given in Sec. IV.
2
II. GALILEON GRAVITY MODEL where ( φ)2 = gµν φ φ, (cid:3)φ = gµν φ, L is
µ ν µ ν m
∇ ∇ ∇ ∇ ∇
the matter Lagrangian. ω is the Brans–Dicke parameter
A. BACKGROUND EVOLUTION and f(φ) is a function of the Galileon field φ. Variation
with respect to the metric gives the Einstein equations,
and variation with respect to the Galileon field φ leads
The action we consider is given by [19, 20]
to the equation of motion. For Friedmann–Robertson–
ω Walker spacetime, the Einstein equations give
S = d4x√ g φR ( φ)2+f(φ)(cid:3)φ( φ)2+L ,
m
− − φ ∇ ∇
Z (cid:20) (cid:21)
(1)
ρ ω d
3H2 = 3HP + P2+φ2f(φ) 3H 1P P3, (2)
2φ − 2 − 2
(cid:18) (cid:19)
p ω d +2
3H2 2H˙ = +P˙ +P2+2HP + P2 φ2f(φ) P˙ + 1 P2 P2 (3)
− − 2φ 2 − 2
(cid:18) (cid:19)
and the equation of motion for the Galileon field gives
6 2H2+H˙ ω 2P˙ +P2+6HP φ2f(φ) 6 2HP˙ +H˙P +2HP2+3H2P P 4d P2P˙ d 2+3d +d P4 =0,
1 1 1 2
− − − −
(cid:16) (cid:17) (cid:16) (cid:17) h (cid:16) (cid:17) (cid:0) (cid:1) (4i)
where an overdot represents differentiation with respect B. DENSITY PERTURBATIONS
to cosmic time t, H =a˙/a is the Hubble expansion rate,
P φ˙/φ, and dn = dnlnf(φ)/dlnφn. ρ is the energy The evolution equation for the cold dark matter over-
≡
density and p is the pressure. In this letter, we consider density δ in linear theory is governedby
a spatially flat Universe (k =0) only.
δ¨+2Hδ˙ 4πG ρδ 0, (8)
TheFriedmannequation(2)canbewrittenintheform − eff ≃
where G represents the effective Newtonfs constant in
eff
1
3H2 = (ρ+ρ ), (5) Galileon gravity, which is obtained as
M2 φ
pl 1 (1+f(φ)φ˙2)2
G = 1+ , (9)
eff
where the effective dark energy density ρφ is defined as 16πφ" F #
where
ω d
ρ =2φ 3HP + P2+φ2f(φ) 3H 1P P3
φ − 2 − 2 φ¨ φ˙2 φ˙ φ˙4
(cid:20) (cid:18) (cid:19) (cid:21) F =3+2ω+φ2f(φ) 4 2 +8H φ2f(φ) .
+3H2 Mp2l−2φ . (6) " φ − φ2 φ − φ4#
(10)
(cid:0) (cid:1)
In the numerical analysis in this letter, since we are TheeffectiveNewtonfsconstantG isclosetoGatearly
eff
interested in the basic parametersof Galileon theory, we timesbutincreasesatlatetimes. (TheeffectiveNewtonfs
consider a specific model with constantmaydepend onthe scalek,but we arenotcon-
cerned about this point as we do not consider the power
r2 spectrum of δ in this letter.)
f(φ)= c , (7)
φ2 We setthe initialconditionδ aatearlytimes. Solv-
≈
ing the evolution equation numerically we obtain the
growth factor δ/a in Galileon gravity as shown in Fig.
where r is the crossover scale [18]. The energy density
c
parameter of matter is defined as Ω =ρ/3M2H2. 1 and Fig. 4. The linear growth rate is written as
m pl
At early times, we set φ 1/16πG, and the Einstein dlnδ
equations (2) and (3) reduc≃e to the usual forms: 3H2 = f = dlna. (11)
8πGρand3H2+2H˙ = 8πGp,thatis,generalrelativity
− The growth rate can be parameterized by the growth
isrecoveredbelowacertainscale. Atlatetimes,inorder
index γ defined by
to describe the cosmicaccelerationtoday,the value ofr
c
must be fine-tuned. f =Ωγ . (12)
m
3
1.2 1.2
1.1 1.1
1 1
a a
δ/ δ/
0.9 ω=–10000 0.9 Ω =0.10
ω=–1000 Ωm,0=0.30
ω=–100 Ωm,0=0.50
0.8 ω=–10 0.8 Ωm,0=0.70
m,0
ΛCDM ΛCDM
0.7 0.7
0 1 2 3 4 5 0 1 2 3 4 5
z z
FIG.1. Growthfunctionδ/ainGalileongravityasafunction FIG. 4. Growth function δ/a in Galileon gravity as a func-
of redshift z forvarious valuesof theBrans–Dicke parameter tion of redshift z for various values of today’s energy den-
ω. The parameters are given by Ωm,0 =0.30. sity parameter of matter Ωm,0. The parameters are given by
ω=−10000.
1.8 1.8
1.7 1.7
1.6 1.6
1.5 1.5
1.4 1.4
1.3 1.3
1.2 1.2
f 1.1 f 1.1
1 1
0000....6789 ωω=ω=–Λω=–1C=1–0D01–00010M0000 0000....6789 ΩΩΩΩmmmmΛ,,,,0000====C0000D....1357M0000
0.5 Observations 0.5 Observations
0.4 0.4
0 1 2 3 4 5 0 1 2 3 4 5
z z
FIG. 2. Growth rate f in Galileon gravity as a function of FIG.5. Growthratef inGalileongravityasafunctionofred-
redshiftz forvariousvaluesoftheBrans–Dickeparameterω. shift z forvariousvaluesof today’senergy densityparameter
The parameters are given by Ωm,0 =0.30. of matter Ωm,0. The parameters are given by ω=−10000.
1 1
Ω =0.10
0.9 0.9 Ωm,0=0.30
0.8 0.8 ΩΩmm,,00==00..5700
0.7 0.7 mΛ,0CDM
0.6 0.6
γ 0.5 ω=–10000 γ 0.5
0.4 ω=–1000 0.4
ω=–100
0.3 ω=–10 0.3
ΛCDM
0.2 0.2
0.1 0.1
0 0
0 1 2 3 4 5 0 1 2 3 4 5
z z
FIG. 3. Growth index γ in Galileon gravity as a function of FIG. 6. Growth index γ in Galileon gravity as a function of
redshiftz forvariousvaluesoftheBrans–Dickeparameterω. redshiftzforvariousvaluesoftoday’senergydensityparame-
The parameters are given by Ωm,0 =0.30. terofmatterΩm,0. Theparametersaregivenbyω=−10000.
4
We plot the growth rate f in Galileon gravity in Fig.
2 and Fig. 5, and the growthindex γ inGalileongravity
Ω =0.16+0.18. (95% C.L.) (14)
in Fig. 3 and Fig. 6. ω is the (constant) Brans–Dicke m,0 −0.11
parameter and Ω is the energy density parameter of
m,0
matter at the present day. We set Ω = 0.30 for the
m,0
Galileon model in Fig. 1, 2, 3 and ω = 10000 for the
GalileonmodelinFig. 4,5,6. Ω forthe−ΛCDMmodel 1
m,0
is 0.30 in these Figs.
For the same value of Ω , the growth rate f in 0.8
m,0
Gcaaslei,ledounemtoodtheelseinsheannhcaenmceendtcoofmNpeawrteodnwfsitchontshteanΛt.CTDhMe bility 0.6
a
smaller Ωm,0 is, the more suppressed the growth rate is. ob
We describe the observational data of Fig. 2 and Fig. 5 r 0.4
P
in the next section.
0.2
III. OBSERVATIONAL TESTS
0
0 0.1 0.2 0.3 0.4 0.5
Ω
A. Observational data
m,0
In Table I we list the growth rate data used in our
FIG. 7. 1D probability distribution of the energy density
analysis: the linear growth rate f ≡ dlnδ/dlna from parameter of matter Ωm,0 for the Galileon model from the
a galaxy power spectrum at low redshifts [35–40] and a growth rate data.
Lyman-α growth factor measurement obtained with the
Lyman-α power spectrum at z = 3 [41]. In Fig. 8, we plot the probability distribution of the
Brans–Dicke parameter ω for the Galileon model from
the growth rate data. We obtained the constraint as
TABLEI.Currentlyavailabledataforlineargrowthratesfobs follows:
usedinouranalysis. z isredshiftandσ isthe1σ uncertainty
of the growth rate data.
ω < 40. (95% C.L.) (15)
z fobs σ Ref. −
0.15 0.51 0.11 [35, 36]
0.35 0.70 0.18 [37]
0.55 0.75 0.18 [38]
0.77 0.91 0.36 [39] 1
1.40 0.90 0.24 [40]
3.00 1.46 0.29 [41]
0.8
y
The corresponding χ2 is given by: bilit 0.6
a
b
6 (f(z ) f (z ))2 o
χ2 = i − obs i . (13) Pr 0.4
σ(z )2
i
i=1
X
0.2
To determine the best value and the allowed region of
the parameters, we minimize the χ2 and use the maxi-
0
mum likelihood method. 100 101 102 103 104 105 106
| ω |
B. Numerical results
FIG. 8. 1D probability distribution of absolute value of the
Brans–Dickeparameter|ω|(notethatω<0)fortheGalileon
We now presentour main results for the observational model from thegrowth rate data.
tests with growth rate.
In Fig. 7, we plot the probability distribution of Ω InFig. 9,weplottheprobabilitycontoursinthe(Ω ,
m,0 m,0
for the Galileon model from the growth rate data. The ω )-planefortheGalileonmodel. Thedotted(blue)and
| |
bestfitvalueisΩ =0.16. Thisisconsiderablysmaller solid (red) contours show the 1σ (68%) and 2σ (95%)
m,0
than such value obtained from observations of SNIa and confidencelimits,respectively,fromthegrowthratedata.
CMB and BAO [32, 33]. We obtained the constraint as (ω is the absolute value of the Brans–Dicke parameter;
| |
follows: ω <0.)
5
106 formationcriterion(AIC)[42]andtheBayesianinforma-
tion criterion (BIC) [43], for the Galileon model and the
105 1σ (68%) 2σ (95%) ΛCDM model, from growth rate data. The χ2 value for
the Galileon model is smaller than that for the ΛCDM
104 model. However, the values of AIC and BIC for the
ΛCDM model are smaller than for the Galileon model,
ω | 103 because there is one more free parameterin the Galileon
| model than in the ΛCDM model. In the ΛCDM model,
102 the value of Ωm,0 from only the growthrate data is con-
sistent with value obtained from observations of SNIa
101 and CMB and BAO.
In Fig. 10, we plot the growth rate f in the best fit
100
modelsoftheGalileontheoryandΛCDM.Moreandbet-
0 0.1 0.2 0.3 0.4 0.5
Ω ter growth rate data are required to distinguish between
m,0
dark energy and modified gravity.
TABLEII.Resultsofobservationaltestsfromthegrowthrate
FIG. 9. Probability contours in the (Ωm,0, |ω|)-plane for the data.
Galileon model. The dotted (blue) and solid (red) contours
show the 1σ (68%) and 2σ (95%) confidence limits, respec- Model Best fit parameters χ2 ∆AIC ∆BIC
tively,from thegrowth rate data. ΛCDM Ωm,0 =0.25 3.138 0.00 0.00
Galileon Ωm,0 =0.17, ω=−2.0×106 2.943 1.81 1.60
1.8
1.7
1.6 IV. SUMMARY
1.5
1.4
1.3 For the same value of Ω , the growth rate f in
m,0
1.2
Galileon models is enhanced compared with the ΛCDM
f 1.1
case,duetotheenhancementofNewtonfsconstant. The
1
smaller Ω is, the more suppressed growth rate is.
0.9 m,0
0.8 Hence the best fit value of Ωm,0 in the Galileon model
0.7 from only the growth rate data is Ωm,0=0.16. This is
Galileon
0.6 ΛCDM considerably smaller than such value obtained from ob-
0.5 Observations servations of SNIa and CMB and BAO [32, 33]. This
0.4
result seems to be qualitatively the same in most of the
0 0.5 1 1.5 2 2.5 3 3.5
generalized Galileon models. On the other hand, in the
z
ΛCDM model, the value of Ω from only the growth
m,0
ratedataisconsistentwithvalueobtainedfromobserva-
FIG. 10. Growth rate f in best fit models of Galileon theory
tions of SNIa and CMB and BAO. We also find the up-
and ΛCDM with parameters in Table II.
per limit of the Brans–Dicke parameter to be ω < 40,
−
fromthe growthrate data. More and better growthrate
In Table II, we list the best fit parameters, the χ2 dataarerequiredtodistinguishbetweendarkenergyand
values (Eq. (13)), and the differences of the Akaike in- modified gravity.
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