Prospects of Non-Riemannian Cosmology

0.2 MB·English

by  Dirk Puetzfeld

Texas Symposium on Rel. Astrophysics, Stanford University, 13. Dec - 17 Dec 2004 1
Prospects of Non-Riemannian Cosmology
DirkPuetzfeld
DepartmentofPhysicsandAstronomy,IowaStateUniversity,Ames,IA50011,USA∗
Inthisworkweprovidethemotivationforconsideringnon-Riemannianmodelsincosmology. Non-Riemannian
extensions of general relativity theory have been studied for a long time. In such theories the spacetime
continuumisnolongerdescribedbythemetricalonebutendowedwithadditionalgeometricquantities. These
newquantities canbecoupledtotheintrinsicpropertiesofmatterinaverynaturalwayandthereforeprovide
a richer gravitational theory, which might be necessary in view of the recent cosmological evidence for dark
matter and dark energy. In this work we mainly focus on the concepts in metric-affine gravity and point out
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theirpossiblesignificanceintheprocessofcosmologicalmodelbuilding.
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1. Introduction 2. Theoretical landscape
n
a
J Withthe rightamountofcrudenessonecouldsum-
2 marize the reasons to consider a drastic step, like the
1 change of the gravity theory which underlies cosmol-
ogy, as follows:

1
v Cosmology, especially its observational sector, is • Largeamountsofdarkmatter/energynecessary
1 currently a thriving field of physics. On the theoreti- to fit current observations within the CCM.
3
cal side opinions have convergedto what is nowadays
2
dubbed cosmological concordance model (CCM). But, • No direct observation of a dark matter particle
1
despite ofallthe successesof this modelin describing in the laboratory.
0
5 differentcosmologicalobservations,weshouldnotfool
• No theoretical explanation for the smallness of
0 ourselves to believe that the grand picture of cosmol-
the dark energy component when compared to
/ ogystands onafirmbasis. Thereasonforthis issim-
h quantum field theory.
ple: Interpretationofthedatawithintheconcordance
p
- modelleads inevitably to the introductionofthe con- • NoreasontobelievethatGRisvalidintheearly
o cepts of dark matter and dark energy1. We surely
r universe, i.e. at high energies.
t couldlivewith suchconcepts bystating thatthey de-
s
a pend on some peculiar details which yet have to be • NotestofNewtonian/generalrelativisticgravity
: added to the description of our universe. Unfortu- on cosmological scales.
v
nately,darkmatterandenergymakeupthecomplete
i
X energybudgetwithinoursimplepictureandtherefore In the following we have a glance at some of the
r cannotbetreatedassomeminordetailswhichremains proposed remedies for this situation.
a to be workedout. This is clearlyanembarrassingsit-
uation which needs to be addressed by cosmologists.
2.1. Alternatives
Inthefollowingwewillhaveaglanceatthetheoret-
AlthoughweknowforsurethatGRhastobemod-
icallandscape ofcosmologyand pay specialattention
ified in order to make it compatible with quantum
to the non-Riemannian approach. Non-Riemannian
theory [12], we do not have any final form of this
extensions of our current gravity theory, i.e. General
new gravitational theory. Additionally, we do not
Relativity (GR), represent a well motivated frame-
knowhowpossiblelow-energymodifications,andthus
work and have been discussed extensively in the lit-
modificationsthatmayplayanimportantroleforthe
erature. In this short review we only address ques-
aforementioned observational problems in cosmology,
tions which are related to the possible cosmologi-
caused by such a new theory will look like.
cal significance of such an approach. Readers who
In table I we provided a very rough overview over
want to learn more about the fundamentals of non-
current theoretical approaches to extend/replace our
Riemanniangravitytheories,thegaugetheoreticalap-
current gravity theory and thereby also our cosmo-
proach to gravity, and metric-affine gravity (MAG)
logical model. The separation into different model
should consult the excellent reviews [3, 6, 7, 9, 10].
classes is sometimes not unique. For example, one
couldalsocountthenon-symmetricgravitymodelsas
non-Riemannian models and all of the models listed
∗Electronicaddress: [email protected] in table I could in principle also have a non-trivial
1See[4]foraninventoryofcosmological parameters. topology.
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2 Texas Symposium on Rel. Astrophysics, Stanford University, 13. Dec - 17 Dec 2004
Table I Some examples for different classes of models
recently used to explain observations of cosmological
significance.
Model type Description
Scalar-tensor theories Modified Lagrangian, addi-
tional scalar field (maybe a
leftover from some higher the-
ory) non-minimally coupled to
theRicci scalar.
Higherdimensions Our universe represents only a
4-d brane in a 5-d bulk, gravity
assumed to be the only inter-
action which propagates in the
bulk.
Figure 1: Classification of different spacetime types
f(R,Rαβ,...) models Modified gravitational La- according to thenon-Riemannian scheme. By switching
grangian in terms of the off torsion and nonmetricity we arrive at theusual
curvature. Riemannian spacetime as encountered in GR.
Topological models Universe assumed to have non-
trivial topology, i.e. impose
some additional global proper- be viewed as the natural prolongation of the line of
ties of spacetime which GR, as
reasoning which led to the formulation of the so far
a local theory, makes no state-
most successful gravity theory, namely GR.
mentsabout.
Non-symmetricgravity Theories in which the metric
gαβ is nolonger symmetric. 2.3. Metric-affine gravity
Tensor-vector-scalarthe- Additional vector field intro-
ory duced by hand into the defini-
In metric-affine gravity the spacetime continuum
tionofthemetric,extendedLa-
which contains matter carries both stresses σαβ (or
grangianwhichcontainsthead-
momentum currents) and hyperstresses ∆αβγ (or hy-
ditional vectorial quantity and
permomentum currents). The geometry of spacetime
an extra scalar field.
isdescribedbymeansofametricg andanindepen-
Non-Riemannian models Spacetime no longer Rieman- αβ
dent affine connection Γγ . The metric is still sym-
nian,newfieldstrengthstorsion αβ
Tαβγ and nonmetricity Qαβγ metric,i.e.gαβ =gβα,buttheconnectionisno-longer
couple to intrinsic properties of given by the metric compatible connection α 2
nβγo
mattersuch as thespin.
known from GR and may be asymmetric Γγ 6=Γγ .
αβ βα
IfwedefineCartan’storsiontensorTα :=Γα and
βγ [βγ]
the nonmetricity tensor Q := −∇ g then, cf.
αβγ α βγ
2.2. Non-Riemannian gravity [22],the affineconnectionmightbe splitupasfollows
One ofthe generalframeworksfor non-Riemannian Γαβγ =(cid:8)αβγ(cid:9) + Tβγα−Tγαβ +Tαβγ
gravity theories is metric-affine gravity (MAG) as re- 1
+ (Q α+Q α −Qα ). (1)
viewedbyHehletal.in[10]. Inthefollowingwefocus 2 βγ γ β βγ
on the new geometrical notions in metric-affine grav-
ity and try to explain their possible impact on cos- Furthermore, one assumes that the momentum cur-
mology. The reasons to pick MAG as starting point rent Σαβ of matter essentially couples to the metric
arevaried: (i)Amongthedifferentalternativegravity gαβ whereas the hypermomentum current ∆αβγ cou-
theories MAG represents a very well motivated and ples to the affine connection Γα of the spacetime.
βγ
naturalgeneralization,c.f. the introductionof[10] for From the last assumption and the splitting of the
a list of arguments, and (ii) there exists a general connection as given in eq. (1) it becomes clear that
Lagrangian formulation of MAG according to which MAG incorporates several other alternative gravita-
many other non-Riemanniantheories may be system- tional theories, as well as GR itself. For example it
atically classified. (iii) There exist several exact (also is well known from Einstein-Cartan (EC) theory that
non-cosmological) solutions for MAG which rank it
among the best studied alternative gravity theories
duringthelastyears. (iv)Theideatocoupleintrinsic
features of matter to new geometrical quantities can 2(cid:8)αβγ(cid:9):= 12gαµ(cid:0)∂βgγµ+∂γgβµ−∂µgβγ(cid:1).
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thetorsionofspacetimecouplestotheintrinsicspinof
particles. In figure 1 we sketchedhow differentspace-
times,andtherebydifferentalternativetheorieswhich
make use of these richer spacetime concepts, maybe
classified with respect to the torsion and nonmetric-
ity.
The gravitational field Lagrangian of MAG is ex-
pected to be of the form L = L(g,∂g,Γ,∂Γ) and a
matterLagrangianminimallycoupledtothenewgeo-
metricalfieldsL =L (ψ,∇ψ,g). Thegravitational
m m
fieldequationsaregivenbythevariationalderivatives Figure 2: In more complex fluid models matter may be
with respect to the metric δL/δgαβ ∼ σαβ and the represented by a medium with dislocations or finely
connection δL/δΓα ∼ ∆ βγ. We only mention here dispersed voids. Non-Riemannian models allow for a
βγ α
that a very general suggestion for the dynamics of natural coupling of thenew geometrical quantities like
this theory, which makes use of a slightly different torsion and nonmetricity to such kind of fluid properties.
but equivalent notation than the one used here, has
been made in [11].
Since the field equations of the FLRW model are
extremely simple and the main evidence (see [2, 23]
for the latest SNIa samples) for the new dark energy
3. Cosmology
component comes from cosmological tests which are
related to the expansion history of the universe, it is
In[17]weprovidedabriefchronologicalguidetothe
naturalto study the impactofchangesin this history
literature on non-Riemannian cosmological models.
andtheirpossibleorigin. Therehavebeenseveralsug-
Therein the developments in cosmology were traced
gestionsformodificationsduringthelastyearscoming
back to the early seventies and were given in table
fromdifferentdirections,cf.tableI. Non-Riemannian
form. Most of the early non-Riemannian cosmologi-
models,especiallyMAG,provideaverygoodstarting
cal models were basedon Einstein-Cartantheory. In-
pointforthestudyofchangesofthecosmologicalfield
vestigations mainly revolved around the construction
equations which are justified by a grander theory, in
of exact solutions and the question of whether or not
the case of MAG by the general Lagrangianprovided
an initial singularity can be avoided in such models.
in [11]. Let us now come to the rhs, i.e. the matter
In the 1980s more general types of Lagrangians were
side, of the field equations.
considered. The inclusion of quadratic terms in the
Lagrangian,leading to dynamical degreesof freedom,
was mainly motivated by the framework of Poincar´e 3.1. Continua with microstructure
gauge theory (PGT) and led to new classes of ex-
act solutions. The story continued with the advent A very promising approach to construct cosmolog-
of the inflationary model, which led to investigations ical models in a non-Riemannian setup is related to
which tried to mimic or justify this new idea within the availabilityofmore sophisticatedfluid models. In
different non-Riemannian scenarios. Till the end of thecaseofmetric-affinegravityObukhov&Tresguer-
the 1990s most of the works in non-Riemannian cos- res [15, 16] devised a fluid model termed the hyper-
mology (NRC) were focused on the description of the fluid 4. This kind of fluid can be used as a natural
early stages of the universe. This bias can mainly be sourceforthehypermomentumcurrentwhichappears
ascribed to the estimates for the new spin-spin con- ontherhsofthefieldequationsofMAG.Thenewde-
tact interaction encountered in Einstein-Cartan the- greesof freedomin sucha fluid model can be coupled
ory. This interaction shows up at extremely high en- to the new geometrical properties, i.e. torsion and
ergydensities3 andmightthereforeplayonlyacrucial nonmetricity. In the hyperfluid picture, which can
roleintheearlyuniverse. Thisfocushaschangeddur- viewed as a generalization of early spin-fluid models
ingrecentyears,mostlyduetothe persistingneedfor [13, 14, 19, 20, 21, 24], the motion of the fluid is de-
the large amount of dark matter and more recently scribed by usual four-velocity and a triad attached
also dark energy. The requirement of dark energy at to each fluid element, which can undergo arbitrary
late stages of the cosmic evolution might be taken as deformations during the motion of the fluid. This
an indicator for the presence of new physics possibly is analogous to the description of continua with mi-
due to some non-Riemannian relics in cosmology. crostructure [5] in the theory of elasticity. In figure
2 we sketched two examples of such media, namely
3In [8, 9] it was estimated that this may be the case at
approximately1047g/cm3. 4Seealso[1].
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Figure 3: How to build and compare cosmological models?
one withdislocations andanotherone withfinely dis- at the same time explain the effects caused by dark
persed voids. The hyperfluid and special cases of it matter/energy in a purely gravitational way. Most
wereusedinseveralcosmologicalmodelsinthepast5, of the models proposed so far are either not worked
asystematictreatmentforafairlygeneralLagrangian out to a sufficient level of detail, fail one or more of
of MAG will be published in [18]. thecosmologicaltests,orarenotdistinguishablefrom
the CCM with the currently available data. But, and
this cannot be stressed enough, we are clearly only
4. How to test and compare? beginning to explore the different possibilities of non-
Riemannian models. This is especially true for the
As we sketched in figure 3, the list of different cos- cosmologicalsectorofmetric-affinegravitywhichcur-
mological tests is quite long, and still growing. Usu- rently only covers a very small region of the theoreti-
allyonestartswithatheoreticalmodelfromtheupper cally permissible parameter space.
portion of the figure and then compares it to some of
the observations in the lower portion of the figure in
ordertofalsifyit. Inviewofthesometimesverydiffer-
enttheoreticalapproachesthiscanbecomequitecum-
Acknowledgments
bersome, i.e. one has to spend a lot of time to work
out the single tests in a scenario which significantly
deviates from the cosmological concordance model. The author wants to thank F.W. Hehl for constant
Thereforeoneofthemostpressingtasksincosmology adviceandsupport. ThefinancialsupportbyM.Pohl
is the development of a post-Newtonian framework and SLAC is gratefully acknowledged. Additionally,
which allows us to compare different theoretical ap- the author wants to thank S. LeBohec for the hospi-
proaches in a systematic and somewhat standardized tality during his stay at University of Utah.
way, and hopefully allows for the a fast backreaction
of the cosmologicaltests on the theoretical model. In
figure 3 we denoted such a framework, which has yet
to be developed, by the connecting middle part.
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Texas Symposium on Rel. Astrophysics, Stanford University, 13. Dec - 17 Dec 2004 5
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