Cosmological implications of Bumblebee vector models

2.1 MB·English

by  Diogo Capelo

Cosmological implications of bumblebee vector models
Diogo Capelo∗
Departamento de Física, Instituto Superior Técnico,
Universidade de Lisboa,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Jorge Páramos†
Centro de Física do Porto and Departamento de Física e Astronomia,
Faculdade de Ciências da Universidade do Porto,
Rua do Campo Alegre 687, 4169-007 Porto, Portugal
(Dated: May 20, 2015)
The bumblebee model of spontaneous Lorentz symmetry breaking is explored in a cosmological
context, considering a single nonzero time component for the vector field. The relevant dynamic
5 equationsfortheevolutionoftheUniversearederivedandtheirpropertiesandphysicalsignificance
1 studied. We conclude that a late-time de Sitter expansion of the Universe can be replicated, and
0 attempt to constrain the parameter of the potential driving the spontaneous symmetry breaking.
2

y PACSnumbers: 04.20.Fy,04.50.Kd,98.80.Jk
a
M
I. INTRODUCTION theRiccitensor, asdepictedin theactionfunctional(1).

9 The bumblebee model incorporates a mechanism of
1 spontaneous breaking of the Lorentz symmetry, which
Several outstanding questions in cosmology, such as
is compatible with general Riemann-Cartan geometries
the current accelerated expansion of the Universe [1] or
] [9, 12, 19–22]. This mechanism is inspired by the Higgs
c the presence and origin of dark matter [2], have motived
q mechanism of the Standard model of fundamental par-
many efforts to expand upon the conventional formalism
- ticles and interactions (itself an allusion to the Landau
r of general relativity (GR), as the latter cannot provide a
g direct explanation of such phenomena. A great number theory of phase transitions that first originated in Con-
[ densed Matter Physics), and serves to both safeguard
ofattemptshavebeenmadetophenomenologicallystudy
Lorentz symmetry as a symmetry of the action and reg-
2 putativemodificationsofGR,rangingfromtheinclusion
ulate the way in which that symmetry is broken.
v of nontrivial curvature terms in the Einstein-Hilbert ac-
5 tion[3,4]totheadditionofsuitablescalar[5]andvector The coupling of curvature to the vector field and the
8 presence of a potential leads to a higher complexity of
fields [6–9], among others [10].
6 theequationofmotionthatcannotbeextrapolatedfrom
Amongstthelatter,thebumblebeemodelwasinitially
7 approaches that eschew the use of a potential [23, 24] or
0 putforwardedin1989[11]asatoymodelforspontaneous
uncouple the vector field from the curvature [25].
. Lorentz symmetry breaking using a vector field; being
1 The aim of this work is to further assess the impact of
an extension of GR, it takes the usual Einstein-Hilbert
0 the bumblebee model and explore the effects of a spon-
5 actionandexpandsitwithavectorfieldandapotential.
taneous LSB in a cosmological context — namely its po-
1 Itprovidedasimpleandmoretractablescenariothanthe
tential as a candidate for dark energy.
: moregeneralStandardModelextension,aframeworkfor
v This work is organized as follows: first, the model
studying Lorentz-breaking terms added to the Standard
i
X model of interactions and general relativity [12]; several and its governing equations are described; the impact
of the latter on cosmology is then presented, prompting
r aspects of the impact of the SME on the gravitational
a the study of the ensuing dynamical system; finally, the
sector have been addressed e.g. in Refs. [13–18].
obtainedresultsarediscussedandconclusionsaredrawn.
The bumblebee model can be considered as a particu-
larcaseofvector-tensormodels[6,7], whichalsoinclude
the more well-known Einstein-aether models [8]; in the
II. THE BUMBLEBEE MODEL FOR
latter, the potential of the vector field resorts to a La-
SPONTANEOUS LORENTZ SYMMETRY
grange multiplier that constrains its norm, and exhibits
BREAKING
couplings between the vector field and the Ricci tensor
and scalar. Distinctively, the bumblebee model assumes
As mentioned in the previous section, the bumblebee
only that the potential has a nonvanishing vacuum ex-
model extends the standard formalism of general rela-
pectation value (VEV), but restricts the coupling to just
tivity by allowing a LSB; this is dynamically driven by
a suitable potential exhibiting a nonvanishing VEV, so
that the bumblebee vector field B acquires a specific
µ
∗Electronicaddress: [email protected] four-dimensional orientation.
†Electronicaddress: [email protected] The action functional should capture the most rele-
2
vant features of the bumblebee model, namely the cou- Variation of Eq. (1) with respect to the bumblebee
plingbetweenthebumblebeefieldandgeometryandthe field yields its equation of motion,
presence of a general potential. As discussed in detail (cid:18) (cid:19)
ξ
in Ref. [7], the simultaneous inclusion of kinetic terms ∇ Bµν =2 V(cid:48)Bν − B Rµν . (3)
for the vector field of the forms F Fµν (with F the µ 2κ µ
µν µν
field-strength) and (∇µBµ)2 leads to the appearance of If the lhs of the equation vanishes, the above results in
ghost degrees of freedom, which vanish if only one of the a simple algebraic relation between the bumblebee, its
latter is considered. potential and the geometry of spacetime.
Furthermore, an extensive discussion of the symme-
tries of a model with couplings of the form BµBνR
µν
and B BµR was presented in Ref. [14]: in particular, if III. COSMOLOGY
µ
the potential is quadratic and includes a Lagrange mul-
tiplier λ, V(λ,Bµ) = λ(BµBµ ± b2), so that variation In most cosmological studies, the Friedmann-
with respect to the former fixes BµBµ = ±b2, then the Robertson-Walker metric (FRW) metric is assumed, re-
coupling term BµBµR can be gauged away by rescaling flecting the cosmological principle which posits an ho-
G. mogeneous and isotropic Universe. However, if Lorentz
This said, the fixed points obtained in this study indi- symmetryisspontaneouslybroken,itispossiblethatthe
catethatthebumblebeefielddoesnotrestatitsnonvan- bumblebee field acquires a nonvanishing spatial orienta-
ishingVEV,forageneralpotential, sothattheinclusion tion,whichwoulddynamicallybreaktheaforementioned
of both types of couplings should have physical conse- isotropy and require a more evolved geometry. This pos-
quences. However,forsimplicitywedropthecouplingto sibility (elaborated in the final section) shall not be pur-
the Ricci scalar and thus consider the model posited in sued in this study: instead, one considers the ansatz for
Ref. [9],thusobtainingamorecomputationallytractable the bumblebee field,
problem which still retains the main features mentioned
(cid:16) (cid:17)
above; a discussion of the effect of an additional cou- B = B(t),(cid:126)0 , (4)
µ
plingwiththeRicciscalarcanbefoundintheAppendix,
where it is shown that — although the strength of the thusupholdingthevalidityofthecosmologicalprinciple,
additionalcouplingcouldleadtonewfixedpoints—the i.e. maintainingtheassumptionoflarge-scalehomogene-
overall dynamical structure of the field equations is not ityandisotropyoftheUniverse. Onethusadoptstheflat
substantially modified. FRW metric, as given by the line element,
Given the above, we consider the action functional ds2 =−dt2+a(t)2(cid:2)dr2+r2dθ2+r2sin2(θ)dφ2(cid:3) , (5)
(cid:90) √ (cid:20) 1
S = −g (R+ξBµBνR )− where a(t) is the scale factor.
2κ µν
From Eq. 4, one sees that the field-strength tensor
(1)
(cid:21)
− 1BµνB −V (cid:0)BµB ±b2(cid:1)+L d4x , vanishes, Bµν =0, and the only nontrivial component of
4 µν µ M the bumblebee Eq. (3) is
(cid:18) (cid:19)
whereκ=8πG,ξisacouplingconstant(withdimensions 3ξ a¨
V(cid:48)− B =0 , (6)
[ξ]=M−2), Bµ is the bumblebee field (with [Bµ]=M), 2κa
B ≡ ∂ B −∂ B is the field-strength tensor, b2 ≡
µν µ ν ν µ which, for a nonvanishing bumblebee field, establishes a
b bµ = (cid:104)B Bµ(cid:105) (cid:54)= 0 is the expectation value for the
µ µ 0 relation between the dynamics of the potential and the
contracted bumblebee vector, V is a potential exhibiting
scale factor. This is probably the reason that motivated
aminimumatB Bν±b2 =0,andL istheLagrangian
µ M the authors of Ref. [7] to describe bumblebee models
density for the matter fields.
with only a Maxwell kinetic term as “non-dynamical”;
By varying Eq. (1) with respect to the metric, the
we maintain, however, that the nonminimal coupling of
modified Einstein equations are obtained,
curvature in our case is enough to drive a meaningful
(cid:20) (cid:18) (cid:19) (cid:21)
1 process, as shall be studied below.
G = κ 2V(cid:48)B B −B Bα− V + B Bαβ g
µν µ ν µα ν 4 αβ µν Considering the matter energy-momentum tensor for
(cid:20) pressureless dust,
1
+ξ BαBβR g −B BαR −B BαR
2 αβ µν µ αν ν αµ T =ρu u , (7)
µν µ ν
1 1
+2∇α∇µ(BαBν)+ 2∇α∇ν(BαBµ) vwehloerceityρ(iws itthheuene=rg(y1,d(cid:126)0e)nsdiutye otof mthaettneorr,muaµliizsattihoenfcoounr--
1 1 (cid:21) µ
−2∇α∇β(BαBβ)gµν − 2 (BµBν) +Tµν ,(2) dgeittihoenruwµituhµE=q.−(14)),, tahnedtu−sitngcotmhpeoanbeonvteoefxtphreemssioodnifiteod-
where V(cid:48) denotes the derivative of the potential V with field Eqs. (2) becomes
rteenspsoecrtotfomitastaterrg.umentandTµν istheenergy-momentum H2(1−ξB2)= 1κ(ρ+V)+ξHBB˙ , (8)
3
3
while the diagonal i−i components read the action functional (1), one sees that it collapses to
the Einstein-Hilbert form with a Cosmological Constant
(cid:18) (cid:19)
a¨
H2+2 (1−ξB2)= (9) Λ = κV0. Indeed, Eq. (15) can be directly integrated,
a yielding a de Sitter solution
(cid:16) (cid:17)
κV +ξ 4HBB˙ +B˙2+BB¨ , a(t)=a eH0(t−t0) . (17)
0
where H ≡a˙/a is the Hubble parameter. with H2 ≡ κV /3 = Λ/3; Eq. (14) then requires that
0 0
Using the Bianchi identities, one also obtains the fol- ρ=0, as expected.
lowing modified equation of conservation of energy, As the next section will show, the existence of a phase
... of exponential acceleration of the Universe is not re-
ξ a¨ ξ a
ρ˙ =−3Hρ−3 B(HB+B˙)+3 B2 , (10) strained to the uncoupled scenario ξ =0, but for a non-
κa κ a vanishing coupling as well.
showingthatthereisanenergyexchangebetweenmatter
and the bumblebee field.
B. Generality of a de Sitter solution
Due to the complexity of the equations derived in this
section, a simple, closed-form solution cannot be ob-
tained by purely analytical means. Given this difficulty, Following the preceding discussion, we consider a so-
it becomes more enlightening to study instead the full lution of the form Eq. (17) (with unknown H0), so that
dynamical picture presented by the Friedmann and Ray- the bumblebee field equation (6) becomes
chaudhuri equations under the constraints imposed by
3ξ
the bumblebee field equation of motion. V(cid:48)B2 = H 2B2 . (18)
2κ 0
For a nonvanishing bumblebee field, any potential V
A. Absence of coupling between the bumblebee which varies nonlinearly yields a quantity V(cid:48) that de-
field and the Ricci tensor pend on B, and the above implicitly establishes that the
bumblebeefieldmustbeconstantthroughV(cid:48)(B2±b2)=
Atthispoint,itisusefulnonoticetherelevanceofthe (3ξ/2κ)H2 (if B = 0 this relation does not hold, but
0
coupling strength ξ: if this is “switched off” by setting the bumblebee remains constant nevertheless). Using
ξ =0, the modified Einstein Eqs. (2) become B =B =const on Eqs. (8) and (10) yields
0
G = T + (11) κV
µν µν H2 = 0 , ρ˙ =−3H ρ=0 . (19)
(cid:20) (cid:18) 1 (cid:19) (cid:21) 0 3(1−ξB2) 0
κ 2V(cid:48)B B −B Bα− V + B Bαβ g , 0
µ ν µα ν 4 αβ µν
Asintheprevioussection,thiscorrespondstoaUniverse
without matter and dominated by the bumblebee field
while the bumblebee equation of motion (3) reads
alone, which acts as a form of dark energy.
∇ Bµν =2V(cid:48)Bν . (12) GivenaparticularpotentialV,Eq. (18)andtheabove
µ
fully determine both H and the nonvanishing value of
0
Althoughsimplified,theaboveequationsinprinciplestill B . Anticipating the next study, one considers a power-
0
allowsfordynamicalbehavior. However, theassumption law potential of the form
Eq. (4) leads to a vanishing field force Bµν =0, so that
nV
the bumblebee Eq. (6) collapses to V(B2±b2)=M4−2n(B2±b2)n →V(cid:48) = , (20)
B2±b2
V(cid:48)B =0 , (13)
where M has dimensions of mass. Using Eqs. (19) and
forcing the bumblebee field to either vanish or rest at (18) for B (cid:54)=0, one obtains
one of the extrema of its potential, and keeping it from
2n∓ξb2
evolving with time. B2 = → (21)
Thus, setting ξ = 0 and B˙ = B¨ = 0 in Eqs. (8-10) 0 ξ(1+2n)
yields the simplified set, (cid:18) 2n 1±ξb2(cid:19)n
V = M4 →
1 0 1+2n ξM2
H2 = κ(ρ+V ) , (14)
3 0 2nκ(cid:18) 2n 1±ξb2(cid:19)n−1
H2 = M2 .
H2+2a¨ = κV , (15) 0 3 ξ 1+2n ξM2
a 0
ρ˙ = −3Hρ . (16) Ifthebumblebeefieldvanishes,thenonetriviallyobtains
H2 =κV /3.
0 0
Thus, thebumblebeefieldcontributessolelythroughthe Noticethatintheaboveonedoesnotsimplyfindthat
value of its constant potential V(B2) ≡ V : checking B2 = ∓b2 and V(B2 ± b2) = V(0) = 0, as could be
0
4
naively expected if one assumed that the cosmological We notice that the dimensionless variable x is directly
2
dynamics should evolve the bumblebee fielduntilit rests proportional to B2, so that no expansion is performed
at its nonvanishing VEV <B2 >=b2: indeed, following around the nonvanishing VEV b2 (cid:54)= 0 arising after the
the discussion of the previous section, the dynamical ef- LSB; conversely, this definition appears to indicate that
fect of the coupling term ξBµBνR found in the action the vector field is expanded around its symmetric phase
µν
functional(1)onthefieldequationscanbeinterpretedas B2 = 0. However, since the set of fixed points does
a friction term that arrests the evolution of the bumble- not change with a shift in the definition of x , there is
2
beefield,withthedissipatedenergyactingtocounteract no distinguishable physics in adopting either form (this
thegravitationalattractionofmatteranddriveanaccel- wascheckeddirectlybyinsteadusingthedefinitionx =
2
erated expansion of the Universe. ξ(B2−b2)).
One can now look back into the on-shell form of the Furthermore, the calculations below reveal that no
action(1),which—withtheprescription(4),B(t)=B fixed point arises with B2 = b2 (i.e. x = ξb2): the
0 2
and the de Sitter solution (19) — is reduced to bumblebeefieldneverrollstoitsnonvanishingVEV,due
to the effect of the coupling with the Hubble parameter
(cid:90) 1 (cid:20) (cid:18) ξB2(cid:19) (cid:21)√ (and its derivative) in the equations of motion.
S = R−6 1− 0 H 2 −gd4x ,(22)
2κ 2 0 The modified Friedmann equation (8) reduces to
which, as noted in the preceding section, corresponds 1=x1+x2+x3+ΩM , (27)
to the Einstein-Hilbert action with an added constant
while the bumblebee equation of motion becomes
term which acts as a cosmological constant Λ: if the
bumblebee field vanishes, the latter is given by its usual (2αx +q)x =0 . (28)
1 2
definition Λ = 3H2, by comparison with the usual La-
0
grangian density L = (R − 2Λ)/(2κ); if one considers The two equations above are algebraic constraints that
B (cid:54)= 0, then it also includes a contribution from the mustbeobeyedbyanysolutiontothedynamicalsystem
0
coupling between the bumblebee field and the Ricci ten- (25).
sor, Λ∗ ≡ 3H2(1−ξB2/2) (see Ref. [26] for a study of The form of Eq. (28) indicates two possible branches,
0 0
the contribution to a phase of accelerated expansion of corresponding to whether x is vanishing or not. If the
2
theUniversearisingfromanonminimalcouplingbetween latteristrue,thisequationcollapsestoq =−2αx ,which
1
matter and the Ricci scalar). can then be used to further simplify the dynamical sys-
tem(25). Theoppositecasex =0ismoretroublesome,
2
as it implies that the dynamical system is no longer au-
IV. ANALYSIS OF THE DYNAMICAL SYSTEM tonomous[inthesensethatx(cid:48) =f(x )alone]: onewould
i j
have to promote q and Ω to dynamical variables and
m
InordertoexplorethepossiblesolutionstoEqs. (8-10) consider the two additional differential equations
and confirm the results obtained in the previous section,
Ω(cid:48) = (2q−1)Ω +βx −2αx (x +x ) , (29)
one begins by defining the following dimensionless vari- M M 2 1 2 3
ables(seeRefs. [27]forasimilartreatmentinthecontext q(cid:48) = −β+q+2q2 .
of quintessence),
Furthermore, it can be shown that the determination of
the nature of any fixed points would be problematic, as
κV ξBB˙
x = , x =ξB2 , x = , (23) the linearization of the Jacobian matrix of the system
1 3H2 2 3 H
around x =0 diverges.
2
This caveat, however, may be circumvented if one no-
together with the usual relative matter density and de-
ticesthatx =0isnotdynamicallyrelevantifx(cid:48) (cid:54)=0,as
celeration parameter 2 2
the variable just rolls out of its vanishing value and the
κρ a¨a relation q =−2αx becomes immediately valid, allowing
1
Ω = , q =− . (24)
M 3H2 a˙2 the aforementioned substitution into Eqs. (25).
Furthermore, since one is not interested in modeling
Using Eqs. (6), (8) and (10), one obtains the spontaneous LSB, it is natural to assume that the
bumblebee field has had time to roll from its previous
x(cid:48) = 2(1+αx +q)x , (25)
1 3 1 vanishing VEV (before the LSB) towards the broken
x(cid:48) = 2x , phase <B2 >=b2.
2 3
x(cid:48)3 = (1−2q)(1−x2)−3x1+x3(q−3) , x(cid:48)T=hu0s,,wohniechisislepfthyosnilcyalwlyithhartdhetopainthteorlpogreicta,lascaistewxo2ul=d
2
where the prime denotes differentiation with respect to implythatthespontaneousLSBhasnodynamicaleffect
the number of e-folds N ≡loga and one defines on the Bumbleebee field, which forever rests at its pre-
LSB vanishing VEV.
...
V(cid:48)(x ) a Fortunately, it can be solved analytically; since the
α(x )≡ 2 , β ≡ . (26)
2 V(x ) aH3 physical implication of this is that the bumblebee field
2
5
onlyimprintsaneffectonthedynamicsthroughthecon- The presence of this multiplicative term on two of the
stant value of the potential V , one expects to recover three dynamical equations (33) renders its behavior very
0
the usual picture found in GR for a Universe composed important in understanding the evolution of the overall
of matter and a cosmological constant, where the former system. Since Eq. (27) implies that, for the matter-
becomesmoreandmoredilutedandtheexpansiongrows dominated Universe Ω = 1 → x + x + x = 0, if
m 1 2 3
exponentially at late times. x > ξb2 > 0, x will be positive by definition, as seen
2 1
Indeed, taking Eqs. (25) and (29) and replacing x = in Eq. (35). In that case, x will have to be negative,
2 3
x(cid:48) =0 immediately yields x =0, so that causingx todecreaseuntilitcrossesthevalueofξb2 and
2 3 2
α diverges (and with it, the trajectories in phase space).
x(cid:48)1 = 2(1+q)x1 , (30) Therefore, one must enforce x2i < ξb2, where the for-
3x = 1−2q , mer is the initial value of the dimensionless variable x :
1 2
Ω(cid:48) = (2q−1)Ω . notice that, since one is studying the cosmological dy-
M M
namics after the spontaneous LSB has occurred and the
Notice that the last equation is equivalent to the first, potential V has settled into the form (35), this initial
as can be seen from the Friedmann equation (27), which value is endowed with physical meaning, as it is related
reads Ω =1−x →Ω(cid:48) =−x(cid:48). One obtains the single to the value acquired by the bumblebee field after the
m 1 m 1
differential equation LSB (which is not modeled directly).
The inequality x < ξb2 will imply that α is always
2i
x(cid:48)1 =3(1−x1)x1 , (31) negative, in order to avoid the divergence of the system:
as the value of x gets closer to ξb2, α becomes larger
2
with solution
and(aswillbeseennumerically),theevolutioninphase-
1 space becomes more abrupt; thus, it is expected that
x (N) = → (32)
1 1+Ce−3N some distance must be allowed between the initial val-
C ues of x2 and ξb2 for sufficiently smooth trajectories in
Ωm(N) = 1−x1 = C+e3N , phase space. Additionally, since the value of ξb2 is not
expectedtobeverylarge(sinceξ istheconstantrespon-
1−3x 3 C
q(N) = 1 =−1+ , sible for the Lorentz-violating curvature coupling and no
2 2C+e3N
evidence of such effects have yet been observed), the im-
where C is an integration constant. As expected, one posed restraint on the initial values x2 < ξb2 becomes
finds that Ω becomes vanishingly small as the Uni- very demanding.
m
verse expands and approaches a de Sitter phase q =−1. Under this potential, the system exhibits the fixed
One concludes that, even if the bumblebee field becomes points shown on I: in particular, fixed points xB and
locked in the symmetric phase < B2 >= 0 so that xC correspondtothedeSittersolutionsfoundinsection
x = x(cid:48) = 0, this does not give rise to any unphysi- IIIB—theformercorrespondstoeitheravanishingcou-
2 2
cal dynamics, and can proceed towards the study of the pling ξ or a bumblebee field frozen at its unbroken VEV
more relevant scenario x (cid:54)=0. B2 =0,andisthusanunnaturalcandidateforadeSitter
2
As discussed previously, a nonvanishing bumblebee attractor;conversely,xC displaysanadditionalcontribu-
field implies that q =−2αx ; replacing this into the dy- tionduetothecouplingbetweenthebumblebeefieldand
1
namical system (25) yields the closed set the Ricci tensor.
x(cid:48) = 2(1+αx −2αx )x , (33) (x1,x2,x3,Ωm,q)
x1(cid:48)2 = 2x3 , 3 1 1 xA (cid:0)0,0,0,1,21(cid:1)
x (1,0,0,0,−1)
x(cid:48)3 = (1+4αx1)(1−x2)−3x1−x3(3+2αx1) , xB (cid:16)1+ξb2,2n−ξb2,0,0,−1(cid:17)
C 1+2n 1+2n
and the relation x (0,1,0,0,0)
D
x +x +x +Ω =1 . (34)
1 2 3 M TABLE I: Fixed points of the model, Eq. (33).
Following the results obtained in the previous section,
one adopts a power-law potential of the form (20), here ThefixedpointxA correspondstoamatterdominated
rewritten as Universe before the bumblebee-induced dark energy be-
comes relevant. This transition requires that this fixed
V(x )=M4(cid:0)x −ξb2(cid:1)n , (35) point is repulsive, with x and/or x attractive.
2 ∗ 2 B C
The fourth fixed point, x , is unphysical, as it cor-
D
wherethesignisfixedandξb2 isallowedtohavenegative responds to a Universe without matter and only a con-
values. It is worthy of note that, for this potential, the stant bumblebee field, which either expands or contracts
variable α becomes: linearly or is static (so that q = 0); although it is not
n mandatory for this fixed point to be repulsive (as it suf-
α= . (36)
x −ξb2 fices that our Universe undergoes a trajectory in phase
2
6
space sufficiently far from it), it is a desirable feature. Denoting the three roots of each polynomial r , r ,
B,1 B,2
The analysis of the stability of the system can be r and r , r , r respectively, the eigenvalues for
B,3 C,1 C,2 C,3
achieved from studying the eigenvalues of the Jacobian D and D become:
B C
matricespertainingtothelinearisedsystemineachfixed
(cid:32) (cid:33)
point, which are: 4n r r r
λ = −1, B,1 , B,2 , B,3 , (45)
B ξb2 (ξb2)2 (ξb2)2 (ξb2)2
 
2 0 −3− 4n 4n
ξb2 ξb2
0 0 −1 0  and
D =  , (37)
A 0 2 −3 0  (cid:18) r r r (cid:19)
λ = −3, C,1 , C,2 , C,3 . (46)
0 0 0 −1 C n(ξb2−1) n(ξb2−1) n(ξb2−1)
The constraints on n and ξb2 imposed by the stability of
2+ 8n 0 −3− 4n 0  the fixed points xB and xC are shown in Fig. 1.
ξb2 ξb2
 4n 0 −1+ 4n(ξb2−1) 0 
DB = ξb2 ξb2  , (38)
 −2n 2 −3+ 2n 2n 
 ξb2 ξb2 ξb2 
0 0 0 −1+ 4n
ξb2
 
2 0 −1 0
1 0 −3− 1 0 
D =n n  , (39)
C 1 2 −4 −2n2+3n−2nξb2
 nξb2−n 
0 0 0 −3
 
2 0 −3 0
0 0 −1 0 
D =  , (40)
D  
0 2 −3 0
 
0 0 0 −1
The eigenvalues λ for matrices D and D do not de-
i A D
pend on n or ξb2:
FIG. 1: Parameter constraints assuming x and x are
B C
unstable (white), only x is stable (light grey) and both
C
λ = (−2,−1,−1,2) , (41)
A x and x are stable (dark grey). The points used in
B C
λD = (−2,2,−1,−1) , (42) the analysis of phase-space trajectories are highlighted.
and,sinceatleastoneoftheircomponentshasapositive Itcanbeseenimmediatelythatonlytheregion−1/2<
real part, the two associated fixed points are unstable. n<0 warrants the instability of all of the system’s fixed
For the remaining two fixed points, the issue of stability points, regardless of the value of ξb2. Additionally, if n
willvary,dependingonthevaluestakenbynorξb2. The and ξb2 have the same sign, then only one stable fixed
specific form of the dependence is quite complex, but it point is to be found (x ); this fact being especially rel-
C
can be understood in a fairly simple way by defining the evant since the simpler and more readily interpretable
third degree polynomials (of variable λ): cases are those for which n is positive (and small).
If one allows ξb2 to assume negative values while n re-
pB(λ) = λ3+ξb2(cid:0)ξb2−10n(cid:1)λ2+ (43) mains positive, then the two possible cases of only xC
2(cid:0)ξb2(cid:1)2(cid:104)4n+4n2−17nξb2−2(cid:0)ξb2(cid:1)2(cid:105)λ− being stable or both xB and xC being stable are sep-
arated by a line that can be roughly described by the
4(cid:0)ξb2(cid:1)3(cid:104)8n2−2nξb2−16n2ξb2+(cid:0)ξb2(cid:1)3(cid:105) , equation ξb2 ≈ −4.76n, with the former lying below the
lineandlatter,above. Whatthismeansisthattoensure
the stability of x , n must increase with ξb2. A similar
B
behavior can be noted if, on the other hand, n assumes
p (λ) = λ3+12n3(cid:0)ξb2−1(cid:1)3+ (44) negative values (lower than −1/2) and ξb2 is positive,
C
2n(cid:0)ξb2−1(cid:1)(cid:0)3λ+ξb2−1(cid:1)λ+ although in this case the equation describing the divid-
ing line appears shifted (closer to ξb2 ≈ −4.76n−2.38)
3n2(cid:0)ξb2−1(cid:1)2(cid:0)5λ+2ξb2−2(cid:1) . and the area in which both fixed points are stable only
7
asymptotically approaches the axis at ξb2 = 0. Again,
this is worthy of note since, as mentioned before, ξb2 is
expected to be small, ξb2 (cid:28)1.
V. TRAJECTORY AND EVOLUTION
ANALYSIS
Followingthepreviousresults,onenowstudiesthetra-
jectoriesofthevariablesinthephase-space,especiallyx ,
2
Ω and q, i.e. the bumblebee field, matter density and
M
deceleration parameter, respectively - which better pro-
vide a physical picture of the conditions of the Universe
at the given time. One thus numerically integrates the FIG. 3: Projection of the phase-space into the x -q
2
system (33), with initial values corresponding to the de- plane for n=4 (two darker plots) and n=6 (two
parture from the matter dominated era, ΩM ≈ 1 and lighter plots). Fixed points are marked as dots in dark
q ≈ 0.5 (the evolution is not affected by choosing ini- grey (n=4) and light grey (n=6). For both cases
tial conditions sufficiently close to these values). A note ξb2 =10−2 and x =10−3 (darker),10−12 (lighter).
2i
shouldbemadeconcerningtheinitialvalueofx ,orx .
2 2i
The default would be to start the numerical simulation
at x = 0, but since the bumblebee field is in that posi- beforethatovershootandcosmographicsurveysindicate
2
tion only before symmetry is broken, it is reasonable to thatq hasbeensteadilydeclining[28],itisreasonableto
expect a deviation from the origin, albeit small. assume that the lowest value of q lies still in the future.
Furthermore, because of the discussion surrounding
Eq. (36), it can be seen that the restrictions imposed
A. Example of system convergence on the initial values of x drive this overshoot deeply
2
into negative values, q (cid:28) −1: mathematically, this is-
Thecasewheren=4isdepictedinFigs. 2-4,showing sue could be settled by allowing x2 <0, which allows for
that the system collapses entirely to the single attractor, muchsmoothertrajectories(ifstillovershootingq =−1),
as determined by the constraints displayed in Fig. (1). butresultsonanaddedstrainontherestrictionEq. (27),
causing Ω to dip into unphysical (negative) values.
M
FIG. 2: Projection of the phase-space into the Ω -q
M
plane for n=4 (two darker plots) and n=6 (two FIG. 4: Projection of the phase-space into the x -Ω
2 M
lighter plots). Fixed points are marked as dots in dark plane for n=4 (two darker plots) and n=6 (two
grey (n=4) and light grey (n=6). For both cases lighter plots). Fixed points are marked as dots in dark
ξb2 =10−2 and x =10−3 (darker),10−12 (lighter). grey (n=4) and light grey (n=6). For both cases
2i
ξb2 =10−2 and x =10−3 (darker),10−12 (lighter).
2i
These trajectories describe a Universe that transitions
from an initial stage of matter domination (Ω ≈1,q ≈ Fig. 4 shows that the trajectories decrease mono-
M
0.5)toadarkenergydominatedera(Ω ≈0,q ≈−1). It tonically with a physically meaningful, positive matter-
M
is interesting to note that the behavior indicates that — density profile. Two linear phases are discernible: a very
instead of evolving smoothly and monotonically towards steepphasecorrespondingtotheinitialplungeofthede-
adeSitterphase—thesystemovershootsthatmarkand celeration parameter and consequent overshoot q < 1,
crossesbelowq <−1,beforeasymptoticallyapproaching andalesspronounceddecreaseduringthecompensation
q = −1. Since the current value of Ω ≈ 0.3 is well of that overshoot and evolution towards the final state
M
8
x =(1+ξb2)/(1+2n). the trajectories lead the variables into physically mean-
2
ingless values. Even if convergent trajectories for this
caseindeedexist,findingthemconstituteslargelyaprob-
B. Example of system divergence lem of fine-tuning.
The second case, for n = 2, is depicted in Figs. 5, 6
and 7, and exhibits a divergence of the trajectories. In
this case, the system’s attractor is insufficient to coun-
teract the repulsive effect of the repulsor x close to the
A
system’s initial values at the start of the simulation.
FIG. 7: Projection of the phase-space into the x -Ω
2 M
plane for n=4 (two darker plots) and n=6 (two
lighter plots). Fixed points are marked as dots in dark
grey (n=4) and light grey (n=6). For both cases
ξb2 =10−2 and x =10−3 (darker),10−12 (lighter).
2i
FIG. 5: Projection of the phase-space into the Ω -q
M
plane for n=4 (two darker plots) and n=6 (two
lighter plots). Fixed points are marked as dots in dark C. Overshoot variation
grey (n=4) and light grey (n=6). For both cases
ξb2 =10−2 and x2i =10−3 (darker),10−12 (lighter). As mentioned before, the analysed stable trajectories
presentabehaviorforthedecelerationparameterq over-
shootingitsfinalconvergencevalueattherespectivefixed
point for the chosen potential. The magnitude of this
effect, though it may appear to depend on the initial
value taken for x , indeed flattens out after an initial
2
phase (for approximately −5.5 < x < −2) and re-
2i
sponding linearly, eventually stabilising at a final value
of q ≈−15.78, for n=4, as can be seen in Fig. 8.
Min
FIG. 6: Projection of the phase-space into the x -q
2
plane for n=4 (two darker plots) and n=6 (two
lighter plots). Fixed points are marked as dots in dark
grey (n=4) and light grey (n=6). For both cases
ξb2 =10−2 and x =10−3 (darker),10−12 (lighter).
2i
Inthiscasetheanalysisismuchmorestraightforward: FIG. 8: Variation of the minimum of q with the initial
the fixed point x is so repulsive that it causes the tra- value of x for n=4.
A 2
jectoriestosteerclearoftheclosevicinityofthebasinof
convergence of the attractor, and therefore no physically This result seems to suggest two zones of effective
significant convergence towards the latter is attained, as system evolution: the first one [−5.5 < log (x ) <
10 2i
9
log (ξb2)] more dynamical and vulnerable to the initial be strongly repelled initially, thus diverging from the de
10
values, and the second one [log (x )<−5.5] more sta- Sitter attractor. This can be explained by the fact that,
10 2i
ble and robust to variations of the initial conditions. for odd n, the variable x will be initially negative if
1
As for the other studied case, n=6, presented in Fig. x < ξb2 and positive if x > ξb2. But the discussion
2 2
9, it depicts a similar behavior, with the differences be- concerning Eq. 36 guarantees the system will diverge
ing a faster stabilization [a linear response is seen for in the latter case, and in the former, the negativity of
log (x ) < −4.2, leveling out at q ≈ −20.47], com- x disturbs the balance of the Friedmann Equation (27),
10 2i Min 1
pared to when n=4. disallowing convergence.
The above does not guarantee that the system does
not allow converging trajectories, only that they are not
dense in the (continuous) set of all possible trajectories
with physically significant starting points, and even if
one was to be found, there would be no basis to argue
the great fine-tuning needed for its manifestation.
Additionally, the analysis of convergent trajectories in
phase-space predict that q always overshoots the final
value of −1, but since the present-day Universe is still
on the descending slope of the trajectory, prior to that
overshoot, such behavior constitutes a prediction of the
model,ratherthananexperimentalconstraint(albeitone
not to be verified in the foreseeable future).
FIG. 9: Variation of the minimum of q with the initial Expectationsconcerningξb2,x2andconvergenceofthe
value of x for n=6. system may, however, be used in restricting the window
2
of accessible values to the dynamical system’s variables:
the major factors in that process can be categorized into
three primary groups: (1) regularity of the solutions, (2)
VI. CONCLUSIONS AND OUTLOOK eliminationofunphysicalbehaviorand(3)mathematical
stability of dynamical system.
Inthiswork,onehasconsideredthebumblebeemodel,
a vectorial extension of general relativity under a spon- 1. n<−1/2 or n>0: This bound is required from a
taneous symmetry breaking mechanism. Since the study stability analysis of the fixed points of the dynam-
is made considering that the bumblebee vector is only ical system, as there are no converging trajectories
nontrivial in the time component, which depends only for values of the exponent n outside the indicated
ontime,noadditionalbreakingofLorentzsymmetryoc- window. However,itcanbearguedthatthereisno
curs, aside from theonederivedfrom adopting the FRW guarantee that the evolution of the Universe must
metric, which is equivalent to a foliation of space-time lead to stable convergence — or that, even in the
into spatial pictures labelled by a cosmological time. presenceofsuitableattractors,aregionofconverg-
It was found that the model predicts four points of ing trajectories can be found for a specific choice
equilibrium, in agreement with the preliminary study of of model parameters n and ξb2 without fine-tuning
theequations; twoareunstableregardlessofthevalueof the initial conditions.
the model’s parameters, corresponding to the static and
matter-dominatedcases. TheremainingtwoyieldaUni- 2. x <ξb2: AfterthespontaneousLSBhasoccurred
2i
verseundergoinganacceleratedexpansion,whichcanbe and the potential has settled into its form (35),
ascribed to the arise of a quantity akin to a cosmological this upper bound for the value of the bumblebee
constant: this can be caused by the effects of the po- fieldmustbeobeyed,otherwisethesystemdiverges
tentialalone(forpracticallyallvaluesoftheparameters, through the action of α.
with the exception of −1 < n< 0), making the identifi-
2
cationΛ≡3H 2,ortotheeffectsofaconcertedevolution 3. x (cid:28) ξb2: The stronger restriction, not contem-
0 2i
between the potential and the bumblebee vector’s non- plated in the Friedmann Equation, is related with
zero component, resulting in Λ∗ ≡ −3H 2(1+V )/2, the behavior of the deceleration parameter q, and
0 C
where V =(1±ξb2)/(1+2n) is the value of the poten- ascertainsthattheminimumofthesolutionforthis
C
tial on the fixed point x . variable(theso-calledovershoot)showsatendency
C
It was noted that, when considering a power-law type to explode if the absolute initial values of x and
2
of potential, even within the parametrical constraints ξb2 are too close. Once more, this is due to the di-
that guarantee the existence of one attractor, a conver- mensionless quantity α diverging and dominating
gent scenario isn’t always found: in particular, this was the dynamic of the system, making impossible the
not attained in the case of odd powers, which, for all smoothtransitionrequiredinorderfortherenotto
tested values, appear to cause the system’s variables to be an overshoot.
10
Regarding the obtained interval for the exponent n mechanism[30],albeitwiththeaddedcomplexityofcon-
where at least one of the de Sitter fixed points is an at- sidering a vector field instead of a scalar. By the same
tractor(firstpointofthelistabove),thecomparisonwith token, one remarks that the interface between regions
a more complex, nonpolynomial potential could provide with different spatial LSB could give rise to topologi-
interesting hints: indeed, if the latter can be expressed cal defects, which would physically correspond to areas
analytically as a series of powers of B2, the two should where Lorentz symmetry holds.
match in the relevant regime for B2 where the latter can However mesmerizing, the above scenario clearly re-
be truncated. quires heavy-duty numerical computations, and is thus
Such a potential naturally arises from the asymptoti- incompatible with the stated purpose of this work: to
cally free solutions of a quantized bumblebee model [15], clearly identify the possibility of driving an accelerated
being expressed in terms of the Kummer (confluent hy- expansion of the Universe via a nonminimally coupled
pergeometric) function M(κ−2,2,z)−1. However, the bumblebee vector field.
stability analysis found there did not assess the dynam-
ical (cosmological) evolution of the bumblebee field, but
concentratedontheexistenceofminimaofthepotential, Acknowledgments
so that the former could rest on a nonvanishing VEV —
with excitations around the latter giving rise to Nambu- The authors thank O. Bertolami for fruitful discus-
Goldstone modes associated with the ensuing LSB. sion, and the referees for their useful remarks. D.C.
Our scenario differs significantly: it is classical in na- was partially supported by the research grant Ref.
ture,hasonenonvanishingminimadefinedbythepower- BIC.UTL/Santander.05/12;J.P.waspartiallysupported
lawformofthepotential, andthefixedpointsshowthat by Fundação para a Ciência e a Tecnologia under the
the bumblebee field does not evolve towards this nonva- project PTDC/FIS/111362/2009.
nishing VEV; furthermore, the results obtained in this
study rely extensively on the presence of a coupling with
geometry, which is absent from Ref. [15]. As such, no Additional bumblebee-curvature coupling
direct comparison is feasible.
As a final remark, we note that the fairly rich struc- As discussed in section III, the action functional (1)
ture of the model prompts more focused studies: going consideredinthisstudyfollowsRef. [9],withonlyacou-
beyondfurtherinquiriesintocosmologicaldynamics[e.g. pling between the bumblebee field and the Ricci tensor
by selecting a different potential V(B2±b2) or including considered: wearguethatthiscapturestheessentialfea-
a coupling between the bumblebee field and the Ricci turesofthemodel,namelytheensuingdeSitterattractor
scalar], one anticipates that interesting results should xC depending on the coupling strength ξ.
arisefromaspatiallyorientedbumblebeethatbreaksthe If one additionally considers the coupling term in the
homogeneity and isotropy postulated by the cosmologi- action functional
cal principle: this could bear a possible relation with the (cid:90) χ √
putative “axis of evil” reported in the cosmological back- S = 2κBµBµR −gd4x , (47)
ground radiation [29].
A different approach could also consider several local where χ is a coupling strength (with dimensions [χ] =
instances of spontaneous Lorentz symmetry breaking, in [ξ]=M−2), the modified bumblebee Eq. (3) becomes
analogy with Condensed Matter Physics [30]: one can (cid:18) (cid:19)
ξ χ
argue that this is compatible with the above-mentioned ∇ Bµν =2 V(cid:48)Bν − B Rµν − BνR , (48)
µ 2κ µ 2κ
possibilityofaspontaneousLSBinaspatialdirection,so
that the bumblebee field acquires a particular (random)
and the r.h.s. of the modified Einstein field equations
orientation only within a region with a well defined co-
acquires the extra terms
herence length, beyondwhich anotherorientationis ran-
domly adopted during the LSB. If the size of the observ- −χ[B BαG +RB B + (49)
α µν µ ν
able Universe is much larger than the coherence length, g (B Bα)−∇ ∇ (B Bα)] .
µν α µ ν α
it is physically plausible that the large scale superposi-
tion of several regions with different spatial orientations Introducing the FRW metric (5), Eq. (48) becomes
for the bumblebee field gives rise to a globally homoge-
(cid:20) (cid:21)
3(2χ+ξ)a¨ 3χ
neous and isotropic Universe. The collision of different V(cid:48)− − H2 B =0 , (50)
“bubbles” could also produce interesting physics and hy- 2κ a κ
pothetically lead to observable relics of the LSB.
so that, considering the additional contribution to the
The in-depth study of such a mechanism would most
derivative of the potential the curvature appearing on
probablyrequiremodelingtheactualLSB(i.e. theevolu- the r.h.s. of the field equations, ∆V(cid:48) ≡ (χ/2κ)R, the
tionoftheshapeofthepotential,sothattheVEVofthe
scalar curvature transforms as
bumblebee field becomes nonvanishing) and encompass
similar considerations to the well-known Kibble-Zurek R→(1−ξB2)R−3χ (B Bα) , (51)
α