A model for a non-minimally coupled scalar field interacting with dark matter

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by  J. B. Binder

A model for a non-minimally coupled scalar field interacting with dark matter
J. B. Binder and G. M. Kremer
Departamento de F´ısica, Universidade Federal do Paran´a
Caixa Postal 19044, 81531-990 Curitiba, Brazil
In this work we investigate the evolution of a Universe consisted of a scalar field, a dark matter
field and non-interacting baryonic matter and radiation. The scalar field, which plays the role of
darkenergy,isnon-minimallycoupledtospace-timecurvature,anddrivestheUniversetoapresent
acceleratedexpansion. Thenon-relativisticdarkmatterfieldinteractsdirectlywiththedarkenergy
and has a pressure which follows from a thermodynamic theory. We show that this model can
reproducethe expected behavior of thedensity parameters, deceleration parameter and luminosity
distance.
6 I. INTRODUCTION and radiation. Furthermore, we shall consider that the
0 scalar field, which plays the role of dark energy, is non-
0 minimally coupled to the scalar curvature [5] and that
Recently, well known astronomical observations with
2 the dark matter interacts with the scalar field according
super-novae of type Ia suggested that our Universe is

n presently submitted to an accelerated expansion [1, 2] to a model proposed by Wetterich [16, 17].
a For an isotropic, homogeneous and spatially flat Uni-
butthenatureoftheresponsibleentity,theso-calleddark
J verse described by the Robertson-Walker metric
energy, still remains unknown. The simplest theoretical
5
explanation for the acceleration consists in introducing
2 ds2 =dt2−a(t)2 dx2+dy2+dz2 , (1)
a cosmological constant and investigating the so-called

1 ΛCDM model (see [3]), which fits the present data very witha(t)denotingthecosm(cid:0)icscalefactor,the(cid:1)Friedmann
v wellbuthassomeimportantunsolvedproblems. Another
and acceleration equations read
6 possibilityistointroduceaminimallycoupledscalarfield
0 φ(t),whichhasbeenextensivelystudiedbythescientific ρ a¨ ρ+3p
11 community and could be invalidated if future observa- H2 = 3(1−ξφ2), a =−6(1−ξφ2), (2)
tions imply that the ratio between the pressure and the
0
6 energy density of the scalar field is restricted to values respectively. The quantities ρ = ρb + ρr + ρdm + ρφ
0 pφ/ρφ <−1 (see [4]). and p = pb +pr +pdm +pφ denote the energy density
/ In this work we consider a scalar field non-minimally and the pressure of the sourcesof the gravitationalfield,
c
q coupled to space-time curvature, which was investigated respectively,andtheyaregivenintermsoftherespective
- in [5] and widely studied recently, among others, in the quantitiesforitsconstituents. Moreover,thedotdenotes
gr works[6,7,8,9,10,11,12]. Also,asideseveralmodelsfor the differentiation with respect to time, H = a˙/a is the
: the dark sector interaction [13, 14, 15], we consider here Hubble parameter, and ξ the coupling constant between
v
a direct coupling between dark matter and dark energy, the scalar field φ and the curvature scalar.
i
X following a model proposed in [16] and studied more re- The modified Klein-Gordon equation for the scalar
r centlyin[17]and[18]. Wealsouseathermodynamicthe- field, whichtakesintoaccountthe non-minimalcoupling
a ory[19]inordertorelatetheeffectsofthenon-minimally and the interaction with the dark matter, reads
coupling to the dark matter pressure [12]. All compo-
nentswillbedescribedbyasetoffieldequations,andthe φ¨+3Hφ˙+ dV + ξφ (ρ−3p)−βρ =0, (3)
resulting observables – namely, the density parameters, dφ 1−ξφ2 dm
thedeceleratingparameterandtheluminositydistance–
whichareobtainedassolutionsofthefieldequations,will where β is a constant that couples the fields of dark en-
be compared to the available data set in order to draw ergy and dark matter. The energy density and the pres-
the conclusions about the viability of this model. We sure of the dark energy are given by
show that physically acceptable solutions are obtained
1
and that there exist some freedom parameters that will ρ = φ˙2+V +6ξHφφ˙, (4)
φ
2
be important to fit the data fromincoming experiments.
Units have been chosen so that 8πG = c = ~ = k = 1, p = 1φ˙2−V −2ξ φφ¨+φ˙2+2Hφφ˙ , (5)
φ
whereas the metric tensor has signature (+,−,−,−). 2
(cid:16) (cid:17)
respectively, where V(φ) denotes the dark energy poten-
tialdensity. From(3)–(5)itfollowsthe evolutionequa-
II. FIELD EQUATIONS
tion for the energy density of dark energy, namely
We shall model the Universe as a mixture of a scalar 2ξφφ˙ρ
ρ˙ +3H(ρ +p )=− +βρ φ˙. (6)
field, a dark matter field and non-interacting baryons φ φ φ 1−ξφ2 dm
2
Thefirsttermontheright-handsideof(6)istherespon- 1.0 Ω +Ω
dm b
sible for the energy transfer from the dark energy to the 1.4 0.8
gravitationalfieldwhereasthesecondreferstoanenergy 1.3 0.6 ξξ == 00 ββ == 0-0.5
1.2
tpraaWrntisecfelserhsatsololtathhsesautdmaprbek=tmh0aattattnehdretfibheaaldrty.tohnesbaarreotnroonp-irceelaqtuivaitsitoinc Ωeters () 0011....8901 000...024 Ωr ξξ == +-00..000033 ββ == --00..55
m 1000 2000 3000 4000 5000 6000
of state for the radiation field pr = ρr/3 holds. Hence, para 00..67 Ωφ
tinhteereavcotliuntgiobnareyqounastiaonnds froarditahteioennberegcyomdeensities of non- ensity 00..45
d
0.3 Ω
0.2 dm
ρ˙ +3Hρ =0, ρ˙ +4Hρ =0, (7)
b b r r 0.1
Ω
0.0 b
respectively, and we get from (7) the well-known rela- 0.0 0.5 1.0 1.5 2.0 2.5
tionships: ρ ∝1/a3 and ρ ∝1/a4. redshift (z)
b r
Now by using equations (2) together with (6) and (7)
FIG. 1: Evolution of density parameters vs. red-shift.
it follows the evolution equation for the energy density
of the dark matter field:
ρ˙dm+3H(ρdm+pdm)=−βρdmφ˙. (8) 0.8 ξξ == 00 ββ == 0-0.5
ξ = +0.003 β = -0.5
ξ = -0.003 β = -0.5
Note that the term on the right-hand side of (8) rep- 0.6
resents the energy transfer from the dark energy to the er (q) 0.4
dark matter. et
m
ara 0.2
p
n
o
III. APPLICATION erati 0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ecel -0.2 redshift (z)
d
Two different situations for the dark matter could be -0.4
analyzed,namely, the one where it is consideredas pres-
-0.6
sureless and the other with a non-vanishing pressure.
Here we shall restrict ourselves to the latter case. In
FIG. 2: Deceleration parameter vs. red-shift.
order to determine the expression for the dark matter
pressure,werecallthatinthepresenceofmattercreation
the first law of thermodynamics for adiabatic (dQ = 0)
open systems reads [19] 1/a−1 by using (4), (5), (11) together with ρ ∝ 1/a3
b
and ρ ∝ 1/a4 and the usual exponential potential V =
dQ=d(ρV)+pdV − ρdm+pdmd(ndmV)=0, (9) V0e−αrφ, where α and V0 are constants. Furthermore,
n
dm theinitialconditionsfortheenergydensitieswerechosen
from the present known values given in the literature
where V denotes the volume and it was supposed that
[20] for the density parameters Ω (z) = ρ (z)/ρ(z), i.e.,
only dark matter creation is allowed. By considering i i
n ∝ ρ and V ∝ a3 it follows from (9) together Ωb(0) = 0.04995, Ωr(0) = 5×10−5 and Ωdm(0) = 0.23.
dm dm
with (2) and (6) – (8): The constant V0 was determined from the present value
forthedarkenergydensityparameter,i.e.,Ω (0)=0.72.
φ
2ξφφ˙ρ ρ +p InFig. 1wehaveplottedthered-shiftevolutionofthe
1−ξφ2 =− dmn dm(n˙dm+3Hndm). (10) density parameterswhere itwasconsidereda fixedvalue
dm
for the exponent α, namely α=0.1, and different values
Nowweuseagain(8)andobtainthefollowingexpression for the coupling constants β and ξ. From this figure
for the pressure of the dark matter field we infer that all kinds of coupling delay the growth of
darkmatterenergydensityandthedecayofdarkenergy
2 density with respect to the red-shift, by a comparison
p 1 βφ˙ 1 βφ˙ 2ξφφ˙ρ
dm =− − ± − + . with the uncoupled case, i.e., β = ξ = 0. Also, positive
ρ 2 6H v 2 6H 3Hρ (1−ξφ2)
dm u ! dm values ofξ imply thatthe radiation-matter(darkmatter
u
t (11) plus baryons) equality takes place at higher red-shifts,
We note that only the plus sign must be chosen in the whereas negative values of ξ at lower ones.
aboveequationin orderto obtaina positive dark matter The deceleration parameter q =−a¨a/a˙2 as a function
pressure. Moreover, we observe that for this case the of the red-shift is plotted in Fig. 2. According to cur-
dark matter pressure vanishes when β =ξ =0. rentexperimentalvalues [21] the decelerated-accelerated
We have solved numerically the system of differential transitionoccursatz =0.46±0.13,whereasthepresent
T
equations (3) and (8) as functions of the red-shift z = value of the deceleration parameter for the ΛCDM case
3
β = 0 ξ = 0 means that, by adjusting the coupling constants in this
46 ββ == --00..55 ξξ == 0+0.003 model,itispossibletohaveconcordancewiththe exper-
µmagnitudes 0 444345 βe x=p -e0r.im5 e ξn t=a l- 0d.a0t0a3 3490 iampIpennatrFaeinlgtd.am3taawg.enhitauvdeepmlotatneddtthheeadbiffseorluentecemµa0gbniettuwdeeenMthoef
difference between 34449012 3333345678 arbo5efylpsotorgRyuepdisreeecsnes+Itaae2.tst5hT,aeawhl.feehuxenp[e2rcxee1trpi]diormefnoseirsonsifot1tanh8tlh5efveoldaurrlameuµtdeai0n-ssorphtesoaiaifitktdnyestwndsµhifos0ertfora=emsnuctmpehteheg−rei-cvniMweroncovlarie=nkes
33 L L
38 0.00 0.05 0.10 0.15 0.20 Mpc: d = (1+z)cH−1 zdz/H(z). We infer from the
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 L 0 0
redshift (z) figure that allcases fit well the experimentaldata at low
R
red-shifts, but when ξ becomes more negative, the curve
FIG. 3: Differenceµ0 vs. red-shift. goes away from the expected experimental values.
Thepressureofthedarkmatterasfunctionofthered-
shiftisplottedinFig. 4. Weinferfromthisfigurethatfor
0.4 ξ = 0 β = -0.5
ξ = +0.003 β = -0.5 a fixed value of the coupling constant β, negative values
ξ = -0.003 β = -0.5 forξ leadtohigherpressureofthedarkmatterfield,and
by increasing ξ to positive values the pressure decreases.
0.3
In all cases the dark matter pressure tend to a constant
ρp/dmdm0.2 value for higher red-shifts.
0.1 IV. CONCLUSIONS
The model analyzed here includes two kinds of cou-
0.0
0 2 4 6 8 10 plings, namely, the coupling of the scalar field with the
redshift (z)
curvature scalar and the coupling of the dark matter
field with the scalarfield. Like the model investigatedin
FIG. 4: Dark matter pressure vs. red-shift.
the work [12], it has a strong dependence on the initial
conditions, since we have solved the system of coupled
differential equations by imposing that the slope of the
is q0 ≈ −0.55. We note from this figure that the scalar field has small positive values at very high red-
decelerated-accelerated transition shifts from z ≈ 0.7 shifts, which was a necessary condition to have positive
T
for the uncoupled case (β = ξ = 0) to z ≈ 0.2 for values for the dark energy density. However, as we have
T
β = −0.5 and ξ = 0. Moreover, by keeping a fixed shown, there exist some freedom in this model because
value for β, positive values for ξ cause later transitions the two coupling constants can be adjusted in order to
wherenegativevaluesforξimplyearliertransitions. This fit present and future experimental data.
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