Dark matter from dark energy-baryonic matter couplings

0.39 MB·English

by  Alejandro Aviles

Dark matter from dark energy-baryonic matter couplings
Alejandro Avil´es1,2,∗ and Jorge L. Cervantes-Cota2,3,†
1Instituto de Ciencias Nucleares, UNAM, M´exico
2Depto. de F´ısica, Instituto Nacional de Investigaciones Nucleares, M´exico
3Berkeley Center for Cosmological Physics, University of California, Berkeley, California, USA
(Dated:)
We present a scenario in which a scalar field dark energy is coupled to the trace of the energy
momentum tensor of the baryonic matter fields. In the slow-roll regime, this interaction could
give rise to the cosmological features of dark matter. We work out the cosmological background
solutions and fit the parameters of the model using the Union 2 supernovae data set. Then, we
1 develop cosmological perturbations up to linear order, and we find that the perturbed variables
1 havean acceptable behavior, in particular the density contrast of baryonic matter grows similar to
0 that in theΛCDM model for a suitable choice of thestrength parameter of thecoupling.
2
PACSnumbers: 98.80.-k,98.80.Cq,95.35.+d
n
a
J
I. INTRODUCTION easiestway-mathematicallyandconceptually-toaccom-
5
modate these ideas into the observed history of the Uni-

] Currently, the most successful model of cosmology verse is decomposing Tµdνark in two species, dark energy
O and dark matter,
that we have is ΛCDM. It is constructed in order to
C match a wide variety of modern cosmological observa-
Tdark =TDE +TDM. (2)
. tions that have stunned the physicist community in the µν µν µν
h
p last decades. Among them, there are precision mea- But this decomposition is not unique. In fact, the dark
- surements of anisotropies in the cosmologicalmicrowave sector could be composed by a large zoo of particles and
o
background radiation [1, 2], baryon acoustic oscillation
r complicated interactions between them. Or, it could be
t [3, 4], and Type Ia supernovae [5–7]. All these observa- even just one unknown field. To accomplish this last
s
tionspointoutthatourUniverseatpresentisdominated
a possibilitywenotethatintheΛCDMmodeltheequation
[ by a cosmological constant -dark energy- and there are of state parameter of the total dark sector is given by

about 5 times of some unknown nonbaryonic, dark mat-
2
ter, over the baryonic matter which is well understood
v
3 by the standard model of particles. w Σρiwi 1 , (3)
0 Nonetheless, some objections to the ΛCDM model ex- eff ≡ Σρi ≃−1+ 0.315a−3
2 ist, both theoretical [8, 9] and observational (see for ex-
3 ample [10]). Thus, alternative proposals have appeared where i index the dark matter component and the cos-
. mologicalconstant,andinthelastequalitywehaveused
2 in the literature giving rise to the idea that dark energy
(0) (0)
1 varies with time. Scalar fields have attracted special at- Ω /Ω 0.315 [2]. In order to mimic this model
DM DE ≈
0 tention, mainly quintessence [11–13], among some other with just one dark field, we must have wdark weff.
≃
1 alternatives [9]. Anyfluidwithequationofstateparameterequaltoweff
v: Recently there has been a lot of interest in studying willproduce the same expansionhistoryof the Universe.
i couplings in the dark sector species [14–23]. This is in Ontheotherhand,somestringtheory-inspiredmodels
X
part motivated by the fact that until today we can only ofdarkenergysharethepeculiaritythatscalarfields,like
ar extract information of these components through gravi- the dilaton, couple directly to matter with gravitational
tationalinteraction,afeaturethathasbeendubbedDark strength. Tohavecosmologicalinfluenceatpresent,these
Degeneracy[24–28]. Specifically,wecandefinetheenergy fieldsmustbenearlymassless,leadingtolong-rangefifth-
contentofthe darksector,and infactthe darksectorit- forcesandtolargeviolationsoftheequivalenceprinciple.
self, using the Einstein field equations Some mechanisms have been proposed in order to avoid
this unacceptable behavior, as the Damour-Poliakov ef-
8πGTdark =G 8πGTobs, (1) fect [29, 30], in which the interaction is dynamically
µν µν − µν driven to zero by the expansion of the Universe, and as
whereG comesfromtheobservedgeometryoftheUni- thechameleonmechanism[31,32],wherethemassofthe
µν
verse and Tobs from its observedenergy content. In this scalar field has an environment density dependence, be-
µν
sense the darksector reflects ourlack ofknowledge. The comingveryhuge inoverallhighdensity regions,suchas
those in which Einstein principle equivalence and fifth-
force search experiments are performed.
In this work we follow the both lines of thought out-
∗Electronicaddress: [email protected] lined above. To this end we consider an interaction La-
†Electronicaddress: [email protected] grangian between a nearly massless scalar and the trace
2
of the energy momentum tensor of the ordinary matter
fields given by
1
T = ρ. (7)
−1 A(φ)
−
=√ gA(φ)T. (4)
int Then, the total action can be written as
L −
This type of coupling has been investigated in a cos-
mological context in [33–35]. This interaction has some S = d4x√ g R 1φ,αφ V(φ) eα(φ)ρ ,
,α
attractiveproperties,thefielddoesnotcoupletotheelec- − 16πG − 2 − −
Z (cid:20) (cid:21)
tromagnetic field and in this sense is dark; also, it does (8)
not couple to relativistic matter and then do not affects where we have defined the function eα(φ) through
the success of the early Universe cosmology, although it
could couple to the inflaton field. As we shall see, the
1
coupling could also give mass to the field. eα(φ) = . (9)
1 A(φ)
Atamorefundamentallevelwecanconsider,inafirst −
approximation,afermionicmatterfreefieldswithenergy Thesetheoriesreducetothosewithaninteractionterm
mT(ofm) =entuimψ¯γtµen∂soψr=Tµ(fν)m=−ψ¯ψiψ¯,γwµh∂eνrψe ianntdhetrlaacsetgeiqvueanlibtyy fife(φld)sLamre,ppreorpfeocstedfluinid[s2,9a]l,tihnotuhgehctahseeiinntwerhaiccthiotnhevamnaisthteers
µ ψ
− −
we have used the Dirac equation. The coupling then be- forrelativisticfluids(seeAppendix). The fieldequations
comesaYukawa-likeoneandgivesmassto the fermions. derived from action 8 are
IfwechosecorrectlythefunctionA(φ)wecanalsointer-
pret this result as that the interactionhas given mass to
G =8πG(T(φ)+eα(φ)T(m)), (10)
the field φ. µν µν µν
This paper is organized as follows: in Section II, we
where the energy momentum tensors of the fields are
present the details of the general theory; in Section III,
we derive the background cosmology equations and we 1
T(φ) =φ φ g φ,αφ g V(φ), (11)
choose a specific model that mimics the ΛCDM model, µν ,µ ,ν − 2 µν ,α− µν
thenwenumericallyobtainthecosmologicalsolutions;in
and
Section IV, we work the theory of linear perturbations;
finally, in Section V, we present our conclusions. T(m) =ρu u . (12)
µν µ ν
For the scalar field the evolution equation is
II. THE GENERAL THEORY
(cid:3)φ V′(φ)=α′(φ)eα(φ)ρ, (13)
−
The action we consider is
where prime means derivative with respect to φ. This
equation shows that, as in the chameleon models, the
evolution of the scalar field is governed by the effec-
R 1
S = d4x√ g φ,αφ V(φ) tive potential V (φ) = V(φ) + eα(φ)ρ, and that the
,α eff
− 16πG − 2 −
Z (cid:20) (cid:21) mass of slow oscillations about the minimum is given by
+Sint+Sm, (5) m2 = Ve′f′f(φmin), where φmin is the value of the scalar
field that minimizes the effective potential. By using the
wherethecouplingofthescalarfieldwiththetraceofthe Bianchiidentities the conservationequationfor the mat-
energymomentumtensorofthe ordinarymatter fields is ter fields becomes
givenby(4). Thereisadifficultyherebecausethematter
fields not only appears in the matter action, but also in
the interactionaction. Thus, we have to define the trace µT(m) = α′(φ)ρ(g +u u )∂µφ. (14)
∇ µν − µν µ ν
of the energy momentum tensor of baryonic matter as
From this equation we obtain the continuity and
geodesic equations
2 δ(S +S )
T = int m gµν. (6)
−√ g δgµν
− µ(ρu )=0, (15)
µ
∇
To solve this redundant definition we will be specific
and
andwewillworkwithaperfectfluidofdustinthematter
seencetrogry,dthenesLitaygorfatnhgeiaflnuiids iLnmits=re−stρf√ra−mge,[w36h]e.reThρeitsrathcee uµ∇µuν =−α′(φ)(gµν +uµuν)∂µφ. (16)
ofthe energymomentumtensorbecomes (seeAppendix, The geodesic equation (16) shows that only the chan-
or for an alternative derivation see [35]) ges of the scalar field perpendicular to the four velocity
3
of a particle affect its motion. In particular, if the vec- where
tor field uµ defines a global time, only the spatial gradi-
ent of the field influences the particle’s motions. This is φ˙2/2 V(φ)
the case of the homogeneous and isotropic cosmology,in w = − . (24)
which there is a family of preferred comoving observers dark φ˙2/2+V(φ)+(eα(φ) 1)ρ0a−3
−
with the expansion. If the field is also homogeneous and Ifthe fieldisinslow-rollregimeweneglecttheφ˙2 con-
isotropic, these observers are affected by the field only
tribution and the dark equation of state parameter be-
through gravity.
comes
In the nonrelativistic limit with static fields, the
geodesic equation (16) for a point particle reduces to
1
w = . (25)
dark −1+ eα(φ)−1ρ a−3
d2X~ V(φ) 0
dt2 =−∇ΦN −α′∇φ−α′V~(V~ ·∇)φ, (17) Then, the following cosmological scenario is plausible
withoutdarkmatter. Duringtheradiationera,thetrace
whereX~ isthespatialpositionoftheparticle,V~ itsthree of the energy momentum tensor of the baryonic matter
velocity and Φ = 1h the Newtonian potential. The is equal to zero and the background cosmology is as the
last term can bNe dr−op2pe0d0 out because it is of the order standardbigbang. Then,intheearlytimesofmatterera
(v/c)2, but we prefer to leave it there because it is a theenergydensityρishigh,andthustheinteractionterm
dissipative term that arises due to the interaction with dominates in the last expression, giving rise to wdark
the scalar field and it shows explicitly how the energy 0. As the time goes on, ρ is redshifted as a−3, th≈e
∼
transfer occurs. Equation (17) also shows that the force potential becomes important and eventually dominates
exertedbythescalarfielduponatestparticleofmassm over the interaction term, and in this limit wdark 1.
→−
is given by To obtain an acceptable late time cosmology we have
to impose a set of constrictions in the free functions of
the theory. First, to obtain equation (3) we impose the
F~ = m (α(φ)). (18) constriction
φ
− ∇
eα(φ) 1 Ω(0)
III. BACKGROUND COSMOLOGY C (φ) − ρ DM. (26)
1 ≡ V(φ) 0 ≃ Ω(0)
DE
Considering a spatially flat FRW metric, we write the Second,toaccountforthecorrectamountofdark mat-
background evolution equations as ter, here given by the interaction term, we impose the
constriction
8πG 1
H2 = φ˙2+V(φ)+(eα(φ) 1)ρ+ρ , (19)
3 2 − Ω(0)
φ¨+(cid:16)3Hφ˙+V′(φ)+α′(φ)eα(φ)ρ=0,(cid:17) (20) C2(φ)≡eα(φ)−1≃ ΩD(0M) . (27)
b
ρ˙+3Hρ=0. (21)
C andC aregeneralfunctionsofthescalarfieldthat
1 2
we expect to have a very little variation in the slow-roll
The last equation is a consequence of (14) (see also the
regime. Also, to obtain the observed accelerated expan-
discussion after equation (16)), and it can be integrated
to give ρ = ρ a−3, where ρ is the value of the energy sion of the Universe, we must have
0 0
densityofbaryonicmattertodayandwehavenormalized 8πGV(φ )=Λ, (28)
0
the value of the scale factor today to a =1. This set of
0
equations is a general feature of interactions of the type where φ0 is the value of the scalar field today and Λ is
f(φ) = f(φ)ρ (see for instance [15, 18, 20]). The the measured cosmological constant. Note that this last
m
conseLrvation−equation suggests to interpret the interac- equation is not independent from the two constrictions.
tion as a pure dark sector feature. Then we define Theinteractionoverfermionsimposeotherconstraints
because,asweshowinthe Appendix,itseffectistoshift
the fermion masses from m to m(φ) = eα(φ)m. Fol-
ρ =ρ +(eα(φ) 1)ρ, (22) lowing [38] the abundance of light elements constrains
dark φ
−
the variation of protons’ and neutrons’ masses to be
where ρ = φ˙2/2+V(φ). The total energy density is at most of about 10% from nucleosynthesis until to-
φ
then ρ =ρ +ρ, and the Friedman equation, 3H2 = day. In ourmodel this is translatedinto ∆m(φ)/m(φ)=
T dark
8πG(ρdark+ρ). The conservation equation for ρdark is (eα(φn) −eα(φ0))/eα(φ0) < 0.1, where φn is the value of
the scalar field at nucleosynthesis and φ is the value of
0
the scalar field today. According to this the scalar field
ρ˙ +3H(1+w )ρ =0, (23) has to be in slow-roll at least since that epoch.
dark dark dark
4
Fixing the free functions Numerical Analysis
Inthechameleonmodelthepotentialisoftherunaway With the selection of the free functions given in last
formandthecouplingwithmatterisanexponentialone. section,thecosmologicalbackgroundevolutionequations
The effectiveequationofstate parameterbecomesofthe can be written as
formw = (1+φ(exp(βφ) 1)a−3)−1 andthe argu-
dark
− −
ments shown above are applicable as long as the field is
8πG 1 1 1 ρ ρ
in the slow-rollregime. Neverthelesswe will follow a dif- H2 = φ˙2+ m2φ2+ ǫφ2 0 + 0 , (36)
3 2 2 φ 2 a3 a3
ferentapproachandwiththeonlyaimtomimic asfaras
(cid:16) (cid:17)
possible the ΛCDM model we choose the free functions and
of the theory as
ρ
φ¨+3Hφ˙ +m2φ+ǫ 0φ=0. (37)
φ a3
1
eα(φ) =1+ ǫφ2, (29) Todothe numericalanalysis,itisconvenienttodefine
2
the dimensionless strength parameter of the interaction
and
β ǫM2. (38)
≡ p
1
V(φ)= 2m2φφ2, (30) UsingH0 =70.4Km/s/Mpc,Λ/3H02=ΩΛ =0.73and
equation (35), the requiredvalue of the scalar field mass
where ǫ is a constant with inverse squared mass units. as a function of β and C (φ ) is
2 0
Note that with this selection the minimum of the effec-
tive potential is always at φ=0, and the evolution field
β
equation for the scalar field (13) becomes mφ =3.6 10−4 Mpc−1. (39)
× sC2(φ0)
Weevolvetheequations(36)and(37)fordifferentval-
(cid:3)φ (m2 +ǫρ)φ=0, (31)
− φ uesofβ,usingthe numericalconstrictions(33)and(34).
and it is explicit that the field has an effective environ-
1.0
ment dependent mass
m2 =m2 +ǫρ, (32)
eff φ 0.5
and that the force mediated by the field has a range ark
1/meff. Although this is a feature on chameleon fields, Ωd 0.0
the evolution equation here is linear and then we do not
expect to obtain the thin-shell suppression [31, 32], also
-0.5
a feature in chameleon fields.
The two constriction equations (26) and (27) become
-1.0
0.0 0.5 1.0 1.5 2.0
a
(0)
ǫρ Ω
C = 0 DM 0.315 (33)
1 m2φ ≃ Ω(D0E) ≃ FteIrGo.f1t:hSehdoawrkstsheceteovro,lwutdiaornk.oTfhtheedeaqsuhaedtiolnineofissttahteerpeasrualmt ien-
theΛCDMmodel. Thethickoscillating curvecorrespondsto
and
β = 1; the short-dashed, to β = 0.2; and the thick nonoscil-
(0) lating to β =0.04.
1 Ω
C (φ)= ǫφ2 DM 5. (34)
2 2 ≃ Ω(0) ≃ Figure 1 shows the results for w . The equation of
b dark
state that is oscillating comes from a field that is oscil-
The equation that relates the scalar field mass with Λ lating about the minimum at φ = 0 and then it is not
becomes in the slow-roll regime. The nonoscillating curves come
from fields that are in the slow-roll regime. Note that
2
m2φ ≃2Λ Mφp , (35) tchoemceusrfvroemwiathfieβld=in0s.0lo4w,-wrohlilcrhegciomrree.sponds to ǫ ≈ G,
(cid:18) 0 (cid:19)
We plot the scale factor a(t) as a function of time.
where the reduced Planck mass is given by Mp = The results are shown in figure 2. We also show the
(8πG)−1/2. results for the ΛCDM model, and the initial conditions
5
1.0
Β=0.01
0.8
1.5
W0.6
0.4
1.0
a 0.2
0.0
0.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
N
0.0
0.00 0.02 0.04 0.06 0.08 0.10
m t
Φ FIG. 4: Evolution of the different density parameters as a
Β=0.04 function of N = loga, we use β = 0.01. The upper line is
Ωdark; the dotted line is Ωb; the dashed line, Ωint; and the
1.5 thick line, Ω . The vertical line denotes present time.
φ
1.0
a
loga. These are given by
0.5
8πG 1 1 1 ρ
Ω = φ˙2+ m2φ2+ ǫφ2 0 ,
0.0 dark 3H2 2 2 φ 2 a3
0.00 0.05 0.10 0.15 (cid:18) (cid:19)
m t
Φ 8πGρ0
Ω = ,
b 3H2a3
FIG. 2: Evolution of the scale factor as a function of cosmic 8πG1 ρ
time. The dashed lines are the results for theΛCDM model. Ωint = 3H22ǫφ2a03,
The vertical lines denote the present time. Note that the
8πG 1 1
temporal scale is different for each case. Ω = φ˙2+ m2φ2 .
φ 3H2 2 2 φ
(cid:18) (cid:19)
Note that Ω =Ω +Ω and Ω +Ω =1.
dark int φ dark b
5.0
4.5 Fit to supernovae data
2
C
4.0
Figure 5shows the best fit to the Union2 data set[7],
3.5 a recentsample of 557supernovae. We use β =0.01and
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 the minimum value of the χ2 function is given by C =
a 1
0.305 and C (φ ) = 4.66, where φ is the cosmological
2 0 0
value of the scalar field today. The module distance µ
FIG. 3: Evolution of C2(φ) as a function of the scale factor and the luminosity distance d are related by
using: β=0.01, thick line; and β=0.04, dashed line. L
d
L
µ=5log +25. (40)
10 Mpc
(cid:18) (cid:19)
Figure 6 showsthe contour plots at1σ and2σ for this
are such that at some early time both models have the
same fit.
same amount of dark matter, here given by the interac-
In the background cosmological solutions the model
tion. Note that the scale factor grows more slowly than
looks very similar as the ΛCDM one as far as β < 0.04.
in the ΛCDM model. This is because the emulated dark
In the next section we consider the linear pertur∼bation
mattercomponentdoesnotredshiftwitha−3,butrather
theory of the cosmologicalsolutions.
with φ2a−3. The age of the Universe is, in the case of
β =0.01,about3%largerthanintheΛCDMmodel,and
about 10% for the case β =0.04.
IV. LINEAR PERTURBATIONS
Figure 3 is a plot of the function C (φ) as a function
2
of the scale factor for the strength parameters β = 0.01 Inthe lastsectionwehaveshownhow the background
andβ =0.04,withinitialvalue C (φ )=5. Note thatin dynamicscanbemadeverysimilartotheΛCDMmodel.
2 i
bothcases,afteraninitialtransientperiod,thefunctions Thisismainlybecausethebackgroundfluidsareaffected
become less steep. bythescalarfieldonlygravitationally,seeequation(21).
This comes ultimately from the fact that the right-hand
Figure 4 shows the evolution of the different density side of the conservation equation (14) projects out tem-
parameters as a function of the number of e-folds, N = poral variations of φ, with respect to the observer uµ.
6
Union2data Β=0.01
44 200
42
150
Μ 40 ∆
100
38
50
36
34 0
0.00 0.02 0.04 0.06 0.08 0.10
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0
z
-200
-400
FIG. 5: Supernovae Union 2 data and the prediction for the Φ
model (solid line). We have used β = 0.01, C1 = 0.305 and m -600
C2(φ0)=4.66. (cid:144)
Θ -800
-1000
Union2 -1200
0.40 0.00 0.02 0.04 0.06 0.08 0.10
0.002
0.35 0.001
p
M
0.000
1 (cid:144)
C0.30 Φ
∆ -0.001
0.25 -0.002
0.00 0.02 0.04 0.06 0.08 0.10
0.20
0.0002
4.2 4.4 4.6 4.8 5.0 5.2
C
2 0.0001
FIG.6: Contourplotsat1σand2σ. ThecentralpointisC1 = 2cs 0.0000
0.305 and C2(φ0) = 4.66, which corresponds to a minimum -0.0001
valueχ2d.o.f. =0.98.
-0.0002
0.00 0.02 0.04 0.06 0.08 0.10
-0.002
In the background cosmology only temporal variations
ofthescalarfieldareconsidered,andsoitdoesnotexert -0.003
any net force over the baryons. This is not true in the
inhomogeneouscosmologywhere spatial gradientsof the Y-0.004
fields have to be taken into account. Accordingly one
-0.005
expects that the evolution of cosmologicalperturbations
of baryonic fields could differ from the obtained in the -0.006
ΛCDM model. In this section we show that the model
hasanacceptablegrowthofbaryonicmatterdensityper- -0.007
0.00 0.02 0.04 0.06 0.08 0.10
turbations. m t
Φ
We consider scalar perturbations in the longitudinal
gauge. The metric is given by
FIG.7: Evolution oftheinitial perturbationsforacomoving
wavenumber k = 0.05Mpc−1, using β = 0.01. For subhori-
zon scales k ≫ H, the relation δ(k2)/δ(k1) = k22/k12 for the
ds2 = (1+2Ψ)dt2+a2(t)(1 2Φ)δ dxidxj. (41) density contrast holds approximately. The dashed lines are
ij
− − theresults in theΛCDM model and the vertical lines denote
present time. The background dynamics are the same as in
figure2.
7
Β=0.04 The fields perturbations are given by
ρ (t,~x) = ρ(t)(1+δ(t,~x)), (42)
b
φ(t,~x) = φ (t)+δφ(t,~x), (43)
0
150 uµ = uµ+uµ, (44)
0 p
∆100 where ρ(t) and uµ are the background energy density
0
andvelocityofthebaryonicmatterfluid,andφ (t)isthe
0
background value of the scalar field. As usual we define
50
the variable θ of the baryonic fluid as the divergence of
thepeculiarvelocityofthefluidinmomentumspace(see
0
[37] for details),
0.00 0.05 0.10 0.15
0
θ =ik vi, (45)
i
-100
where k is the i component of the comoving wave num-
mΦ-200 ber vectior ~k of a perturbation, related to the physical
(cid:144) -300 wavelength by λ=2πa/k.
Θ Inthisanalysisweneglectanisotropicstressesandthen
-400
thescalarpotentialsareequal,Ψ=Φ. Theperturbation
-500 equations in Fourier space are given by
0.00 0.05 0.10 0.15
k2
0.0010 Ψ+3H(Ψ˙ +HΨ)=4πG φ˙2Ψ φ˙ δ˙φ m2φ δφ
a2 0 − 0 − φ 0
0.0005
Mp 0.0000 −ρ 1+ 21ǫφ02 δ+ 1h+ǫφ10ǫφ2δφ , (46)
(cid:144) -0.0005 (cid:18) (cid:19)(cid:16) 2 0 (cid:17)i
Φ
-0.0010
∆
2k2 k2 1 θ
-0.0015 (Ψ˙ +HΨ)=8πG φ˙ δφ+ρ 1+ ǫφ2 ,
a2 a2 0 2 0 a
-0.0020 (cid:20) (cid:18) (cid:19) (cid:21)
(47)
0.00 0.05 0.10 0.15
0.0003
Ψ¨ +4HΨ˙+(2H˙ +3H2)Ψ=
0.0002
0.0001 4πG[−φ˙02Ψ+φ˙0δ˙φ−m2φφδφ], (48)
2cs 0.0000
and the hydrodynamical perturbation equations
-0.0001
-0.0002
1
-0.0003 δ˙ 3Ψ˙ + θ =0, (49)
− a
0.00 0.05 0.10 0.15
-0.002
1 ǫφ φ˙ 1 k2 ǫφ
θ˙+ 2H+ 0 0 θ = Ψ+ 0 δφ ,
-0.003 a 1+ 21ǫφ02!a a2 (cid:18) 1+ 12ǫφ02 (cid:19)
Y (50)
-0.004
and
-0.005 k2
δ¨φ + 3Hδ˙φ+ +m2 +ǫρ δφ
a2 φ
-0.006 (cid:20) (cid:21)
0.00 0.05 m 0.1t0 0.15 = −2m2φφ0Ψ−ǫρ0φ0(δ+2Ψ)+4φ˙0Ψ˙. (51)
Φ
These six equations are not all independent, we use
(47), (49), (50) and (51) to numerically evolve a set of
FIG.8: Thesameasfigure7butwithβ=0.04, whichcorre- initial perturbations. The background dynamics are the
sponds to ǫ≈G. same as in figure 2. In fact, for the case β = 0.01, they
yield the best fit to supernovae data in the last section.
8
Theresultsofthenumericalanalysisareshowninfigures V. CONCLUSIONS
7 and 8. The dashed lines are the results in the ΛCDM
model. Itispossiblethatfutureobservationsandexperiments
revealthatthereexistsomeinteractionsbetweenthedark
Aninterestingparameteristhesquaredoftheeffective
sector and the known fields of the standard model; in-
soundspeedofthe perturbationsofthe scalarfield, c2 =
s deed, this is the hope for terrestrial experiments in de-
δp /δρ ,
φ φ
tecting dark matter. The dark degeneracy allows us to
study awide varietyofmodels thatmimic the ΛCDMat
today’sobservationaccuracybutwitharicherdynamics.
φ˙ δ˙φ φ˙2Ψ m2φ δφ
c2 = 0 − 0 − φ 0 . In this paper we have presented a plausible mecha-
s φ˙ δ˙φ φ˙2Ψ+m2φ δφ+ 1ǫφ2(δ+ ǫφ0 δφ) nism inwhich aninteractionbetweena quintessence-like
0 − 0 φ 0 2 0 1+21ǫφ02 field and baryonic matter gives rise to effects similar to
(52)
those of dark matter. The interaction is given through
For an uncoupled scalar field dark energy component, the trace of the energy momentum tensor of the matter
as quintessence, the sound speed is exactly one in the fields,andinthecasethatwehaveconsidered,dustmat-
slow-rollapproximation,preventingscalarfieldstructure ter, it is the same as an interaction of the type f(φ) .
L
to form. In our model, the scalar field acts both as dark Thedifferencesaremanifestifweinsteadusearadiation
energy and, by the coupling to baryons, as dark matter. field, e.g. photons, in this case the interaction vanishes
As a dark energy component, scalar field perturbations -neglecting the trace anomaly [40]- and in this sense is
tend to be erased,but on the other hand the interaction dark. Because of this coupling the radiation era is the
drivestozerothespeedofsoundsquared,duetothelast same as in the standard big bang and the differences
term in the denominator of equation (52), allowing the appear once matter dominates. We have described the
scalar field perturbations to grow. As a result of this background cosmology and by fitting the parameters of
interplay scalar field perturbations are not completeley the model we can mimic as far as we want the late time
damped,insteadinitialscalarfieldinhomogeneitiescould ΛCDM background solutions, from decoupling until to-
beamplifiedoratleastfrozen. Thisbehaviorisshownin day. Specifically we have shown the differences for the
figures 7 and 8, where the scalar field perturbations do cases of β = 0.4 (ǫ G) and β = 0.01. For this last
≈
not decay to zero as the time goes on, but they tend to value of β, we made fits to the Union 2 supernovae data
a negative value about which they oscillate. setandfoundthatthebestfitisobtainedbychoosingthe
other two parameters of the model equal to C = 0.305
1
The scalar field perturbations become a source of the
and C (φ )=4.66.
2 0
gravitational potential (see equation 46) that acts upon
We have worked out the first order perturbation the-
the baryonic matter through equation (49). This is a
ory and shown that these models are capable to form
feature of dark matter. However, note that the sign of
linear structure without dark matter. Initial perturba-
the average of the scalar field perturbations is negative
tionsofafreescalarfielddarkenergyareerasedbecause
and,fromthe Poissonequation(46),they actas arepul-
its sound speed is equal to one, and by the fact that
sive gravitationalsource. Nevertheless, the gravitational
dark energy accelerates the Universe, it tends to freeze
potential, read from figures 7 and 8, is similar to the
any matter perturbation. One of the effects of the inter-
one in the ΛCDM model. This is because the baryonic
action, when the field is slow-rolling, is to decrease the
matter energy density appears multiplied by the factor
sound speed. In our model this induces the scalar field
(1 + ǫφ2/2) in equation (46) and it contributes as an
perturbationstooscillateaboutanonzeronegativevalue
attractive gravitational source. On the other hand, the
andyieldarepulsivegravitationalforceoverthebaryonic
scalefactorgrowsatslowerratethanintheΛCDMmodel
matterperturbations. Despitethiseffect,weshowedthat
andthis yields anincrease inthe gravitationalperturba-
the baryonic matter density contrast could grow as fast
tions through the second term of the left-hand side. of
as in the ΛCDM model. This is because the interaction
equation (46). As a net result of these effects the gravi-
enhances the gravitationalstrength ofthe baryons. This
tational potential is slightly stronger (Ψ more negative),
lattereffectincreasesasβ does,butatthesametimethe
and therefore the density contrast grows faster, than in
increase of β can break the slow-roll. Thus, suitable β
the ΛCDM model. These can be seen in figures 7 and 8. are found for β <0.04.
∼
Asitwaspointedoutin[39]coupleddarkenergymod-
els could suffer from unwanted severe instabilities char-
acterized by a negative speed of sound squared. This is VI. APPENDIX
notthecaseinourmodelinwhichthenumericalanalysis
showsthatthescalarfieldperturbationsdonotgrowbut In this Appendix, we show how the interaction affects
insteadoscillatearoundanonzerovalue. Amoredetailed different types of matter, namely a fermion field and a
analysis [21] has shown that these instabilities are sup- perfect fluid.
pressed when the scalar field is in the slow-roll regime, For a fermion Dirac field we focus on the microscopic
as in our case. interactionandweconsiderafixedMinkowskispacetime.
9
TheLagrangianis = + + =( ψ¯(iηµνγ ∂ where we have used the identity δρ = 1(ρ+p)(u u +
m)ψ+A(φ)T)√ ηL+ Lψ,anLditnhtetLraφceis−givenbyµequνa−- g )δgµν (see for example [36]). Takin2g the tracµe,νwe
φ µν
− L
tion (6), which now reads obtain the differential equation
2 δ(S +S )
T = ψ int ηµν. (53)
−√ η δηµν
dT
−
3A(φ)(1+w)ρ +(1 4A(φ))T+(1 3w)ρ=0, (57)
Forafreefermionfield,oneobtains T(0) = iψ¯γ ∂ ψ = dρ − −
µ µ
mψ¯ψ,wherewehaveusedtheDiracequati−on. Now,for
t−hetotalLagrangianwemaketheansatz T =meα(φ)ψ¯ψ, with general solution
and inserting it into equation (53), we obtain that
eα(φ) =(1 A(φ))−1. So, the Lagrangiannow reads
−
(1 3w)
= ψ¯(iγµ∂ eα(φ)m)ψ+ . (54) T = − ρ+Cρ−(1−4A)/3A(1+w), (58)
L − µ− Lφ −1 (1 3w)A(φ)
− −
The effect of the interactionis to shift the mass of the
fermions from m to eα(φ)m. whereC isanintegrationconstantthatwehavetochoose
Now, let us consider in the matter sector a perfect equaltozeroifwewantforaradiationfluidT =0. Then
fluid with an equation of state parameter w constant. we obtain the next equation for the trace
The matter Lagrangianis = ρ√ g, where ρ is the
m
L − −
energy density of the fluid in its rest frame [36]. We will
assume that the trace of the energy momentum tensor 1 3w
T = − ρ. (59)
depends only on ρ, T = T(ρ). It also depends on the
−1 (1 3w)A(φ)
metric, but only through ρ-. Then δT = (dT/dρ)δρ. − −
Let us consider the action
For dust, w = 0 and this equation reduces to (7). For
a relativistic fluid w = 1/3 and the trace vanishes. This
result agrees with the one obtained through an iterative
S +S = d4x√ g(A(φ)T(m) ρ), (55)
int m − − procedure in [34, 35].
Z
Acknowledgments
performing variations with respect to the metric, equa-
tion (6) gives
J.L.C.C. thanks the Berkeley Center for Cosmologi-
cal Physics for hospitality, and gratefully acknowledges
T = ρ(1+w)u u +wρg +A(φ)Tg support from a UC MEXUS-CONACYT Grant, and a
µν µ ν µν µν
CONACYT Grant No. 84133-F. A.A. acknowledges a
dT
A(φ) ρ(1+w)(u u +g ), (56) CONACYT Grant No. 215819.
µ ν µν
− dρ
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