Effective dark energy through spin-gravity coupling

2025-04-24 0 0 537.46KB 7 页 10玖币
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Effective dark energy through spin-gravity coupling
Giovanni Otaloraa,1, Emmanuel N. Saridakisb,c,d,2
aDepartamento de F´ısica, Facultad de Ciencias, Universidad de Tarapac´a, Casilla 7-D, Arica, Chile
bNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece
cCAS Key Laboratory for Research in Galaxies and Cosmology, University of Science and Technology of China, Hefei, Anhui 230026,
China
dDepartamento de Matem´aticas, Universidad Cat´olica del Norte, Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile
Abstract
We investigate cosmological scenarios with spin-gravity coupling. In particular, due to the spin of the baryonic and
dark matter particles and its coupling to gravity, they probe an effective spin-dependent metric, which can be calculated
semi-classically in the Mathisson-Papapetrou-Tulczyjew-Dixon formalism. Hence, the usual field equations give rise to
modified Friedmann equations, in which the extra terms can be identified as an effective dark-energy sector. Additionally,
we obtain an effective interaction between the matter and dark-energy sectors. In the case where the spin-gravity
coupling switches off, we recover standard ΛCDM cosmology. We perform a dynamical system analysis and we find a
matter-dominated point that can describe the matter era, and a stable late-time solution corresponding to acceleration
and dark-energy domination. For small values of the spin coupling parameter, deviations from ΛCDM concordance
scenario are small, however for larger values they can be brought to the desired amount, leading to different dark-energy
equation-of-state parameter behavior, as well as to different transition redshift from acceleration to deceleration. Finally,
we confront the model predictions with Hubble function data.
1. Introduction
Modified gravity is one of the two ways one may follow
in order to explain universe acceleration [1, 2] (the other
one being the introduction of the dark-energy sector [3, 4]),
while it has been also proved to be efficient in alleviating
the two famous tensions of ΛCDM cosmology, namely the
H0and the σ8ones [5]. However, a crucial additional ad-
vantage of modified gravity is that it may have improved
renormalizability and thus is closer to a quantum descrip-
tion [6].
One possibility is the addition of quantum corrections
to the action [7–9]. Other approaches to quantum gravity
consider that the standard energy-momentum dispersion
relation is deformed near the Planck scale, since this may
arise from string field theory [10], loop quantum gravity
[11], and non-commutative geometry [12]. Furthermore,
in the context of “doubly general relativity” [13–16] the
modification of the dispersion relation leads to an effective
spacetime metric which depends on the energy and mo-
mentum of the probe particle. Since spacetime is repre-
sented by a one-parameter family of metrics, parametrized
by the energy of the probe particle, this semiclassical ap-
proach is called Gravity’s Rainbow [13]. Applications of
Gravity’s Rainbow to dark energy and inflation can be
found in [17–23].
1giovanni.otalora@academicos.uta.cl
2msaridak@noa.gr
However, interestingly enough, an effective spacetime
metric can alternatively arise in the context of a semiclas-
sical description of the spinning particle in an arbitrary
gravitational field [24–27], without the need of a modified
dispersion relation. For instance, in [27] (see also Refs.
[28, 29]) the authors have constructed a Lagrangian for-
mulation for the Mathisson-Papapetrou-Tulczyjew-Dixon
(MPTD) equations [30–33], in the context of a semiclassi-
cal vector model for the spin space. Hence, in the minimal-
coupling prescription of gravity, a spinning particle effec-
tively probes a different geometry, lying within the general
class of Riemann-Cartan geometry, which is determined by
an effective metric g(ef f)
µν that depends on its spin.
In particular, in vector models of spin the basic variables
are the non-Grassmann vector ωµand its conjugated mo-
mentum πµ. The spin-tensor is constructed using these
variables as Sµν = 2 (ωµπν−ωνπµ). Then, one starts
from the free theory in flat space, for which there is a La-
grangian formulation without auxiliary variables [27, 29],
and the minimal coupling to gravity is achieved by covari-
antization of this Lagrangian [27–29]. After constructing
the Hamiltonian formulation, one can eliminate the mo-
menta from the Hamilton equations by using the mass-
shell condition. Thus, a closed system for the equations
of motion (the Lagrangian form of MPTD equations) is
obtained, with the emergence of the effective metric geff
µν
[27].
The effective metric, produced along the world-line of
the particle through interaction of the spin with gravity,
Preprint submitted to Elsevier May 23, 2023
arXiv:2210.06598v2 [gr-qc] 22 May 2023
is given by [27]
g(eff)
µν =gµν +1
8m2Sσ
µθσν +Sσ
νθσµ
+1
8m22
SσαθσµSτ
αθτ ν ,(1)
where mis the mass of the particle. In the above expres-
sion, the spin-tensor of the particle is defined as Sµν =
2 (ωµπν−ωνπµ) = Si0=Di, Sij = 2ijkSk, where Diis
the dipole electric moment and Siis the three-dimensional
spin-vector [34, 35], and it satisfies the relation Sµν Sµν ≡
8σ=const. with σthe absolute spin which is a constant
of motion. Finally, the tensor
θµν ≡Rαβµν Sαβ ,(2)
where Rαβµν is the Riemman tensor related to the phys-
ical metric gµν , quantifies the coupling between spin and
gravity.
In the present manuscript we desire to investigate the
implications of this effective spin-dependent metric in the
context of cosmology, by deriving the corresponding mod-
ified Friedmann equations. In particular, we identify the
extra spin-gravity coupling terms as an effective dark-
energy sector. We mention here that since the spin-gravity
coupling above is based on the coupling of Riemann tensor
to spin, one expects that is would be larger in the early
Universe, or around black holes. Nevertheless, since the
coupling is present, even at late-times it can play a role if
it generates a collective effect from all spining dark-matter
particles of the Universe. In some sense the situation is
similar to modified gravity, where the modification at late-
times (where curvature is small) is extremely small, and
thus it is impossible to be observed in Solar System exper-
iments or in scales below galaxy clusters, however, in the
whole Universe collectively, it can lead to deviations from
ΛCDM paradigm that can improve cosmological behavior.
Finally, note that concerning dark matter there are many
theories which suggest that it could correspond to massive
higher spin particles, typically found in string theory, and
this could lead to enhanced spin-gravity coupling effects
[6].
The plan of the paper is the following: In Section 2
we present the construction at hand, extracting the mod-
ified Friedmann equations and the effective dark energy
sector. In Section 3 we investigate the resulting cosmo-
logical behavior, performing a dynamical system analysis
and elaborating the model numerically. Finally, in Sec-
tion 4 we summarize the obtained results. Throughout
the manuscript, we adopt natural units c=~= 1 and we
use the metric signature (−,+,+,+).
2. Modified Friedmann equations and effective
dark energy
In this section we apply the above formulation in a
cosmological framework, namely we consider the back-
ground metric gµν to be a flat Friedmann-Robertson-
Walker (FRW) one, with form
ds2=−dt2+a(t)2δij dxidxj,(3)
with a(t) the scale factor. Concerning the matter sec-
tor, we consider baryonic and dark matter particles corre-
sponding to the standard perfect-fluid energy-momentum
tensor
Tµν = [(ρm+pm)UµUν+gµν pm],(4)
with ρmand pmthe energy density and pressure re-
spectively, while the four-velocity of the fluid is Uµ=
(−1,0,0,0) such that UµUµ=−1. Finally, concerning
the field equations we consider the ones of standard gen-
eral relativity, namely
Gµν ≡Rµν −1
2gµν R+ Λgµν =κ2Tµν ,(5)
where Gµν is the Einstein tensor, κ2= 8πG is the gravi-
tational constant, and Λ is the cosmological constant.
As we mentioned in the Introduction, due to the spin-
gravity coupling the matter particles feel the effective met-
ric g(eff)
µν of (1). In order to calculate it one starts with the
calculation of the averaged effective metric hg(ef f )
µν i[26].
The volume element contains a large number of particles
with a randomly oriented spin distribution. Several differ-
ent authors have studied the effects at cosmological scales
of matter distributions that locally contain a large number
of randomly oriented spin particles, and they have found
that the microscopic gravitational field equations can as-
sume a pseudo-Einsteinian form that includes spin correc-
tions terms [24, 26]. Definitely, one needs an averaging
procedure for these fluctuating terms in the microscopic
domain. This is similar to what is done when obtaining
the macroscopic Maxwell equations.
The above averaging procedure typically leads to zero
spin average and zero spin gradient, however to non-zero
average for the spin-squared terms arising in the field
equations of the theories with spin-gravity couplings [24].
Therefore, the averaged effective metric is obtained by
substituting (3) into (1) and then averaging over all pos-
sible directions of the three-dimensional spin-vector ~
S(t)
and the dipole electric moment ~
D(t), namely h~
S·~
Si=
~
S2,h~
D·~
Di=~
D2and h~
Si=h~
Di= 0. In the flat
FRW metric, under the assumption that the absolute spin
Sµν Sµν ≡8σ=const., we have the relations ~
S(t)2/m2=
3α/(4a(t)4) and ~
D(t)2/m2= 6β/a(t)2, where αand βare
constants with dimensions of mass−2, and thus we find
8σ=hSµν Sµν i= 6m2(α−2β) [26]. Hence, we finally re-
sult to the averaged effective metric Gµν ≡ hg(ef f )
µν igiven
by
ds2= (−1 + F1)dt2+a(t)2(1 + F2)δij dxidxj,(6)
where
F1= 3β˙
H+H2
1 +
(5α−19β)˙
H+H2
20
,(7)
2
摘要：

E ectivedarkenergythroughspin-gravitycouplingGiovanniOtaloraa,1,EmmanuelN.Saridakisb,c,d,2aDepartamentodeFsica,FacultaddeCiencias,UniversidaddeTarapaca,Casilla7-D,Arica,ChilebNationalObservatoryofAthens,LofosNymfon,11852Athens,GreececCASKeyLaboratoryforResearchinGalaxiesandCosmology,UniversityofS...

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