# On the Superstrings-Induced Four-Dimensional Gravity and Its Applications to Cosmology

Authors: Masao Iihoshi; Sergei V. Ketov
Journal: Advances in High Energy Physics
 (2008)

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## Abstract

We review the status of the fourth-order (quartic in the spacetime curvature) terms induced by superstrings/M-theory (compactified on a warped
torus) in the leading order with respect to the Regge slope parameter, and
study their (nonperturbative) impact on the evolution of the Hubble scale
in the context of the four-dimensional FRW cosmology. After taking into
account the quantum ambiguities in the definition of the off-shell superstring
effective action, we propose the generalized Friedmann equations, find the
existence of their (de Sitter) exact inflationary solutions without a spacetime
singularity, and constrain the ambiguities by demanding stability and
the scale factor duality invariance of our solutions. The most naive (Bel-Robinson tensor squared) quartic terms are ruled out, thus giving the evidence
for the necessity of extra quartic (Ricci tensor-dependent) terms in
the off-shell gravitational effective action for superstrings. Our methods are
generalizable to the higher orders in the spacetime curvature.


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## Body

## 1. Introduction

The  homogeneity and isotropy of our Universe, as well
as the observed spectrum of density perturbations, are explained by
inflationary cosmology [1, 2]. Inflation is usually realized by introducing a scalar
field (inflaton) and choosing an appropriate scalar potential. When using
Einstein equations, it gives rise to the massive violation of the strong energy
condition and the exotic matter with large negative pressure. Despite the
apparent simplicity of such inflationary scenarios, the origin of their key
ingredients, such as the inflaton and its scalar potential, remains obscure.Theory of superstrings is the leading candidate for a
unified theory of Nature, and it is also the only known consistent theory of quantum
gravity. It is therefore natural to use superstrings or M-theory for the
construction of specific mechanisms of inflation. Recently, many brane
inflation scenarios were proposed (see, e.g., [3] for a review), together with
their embeddings into the (warped) compactified superstring models, in a good
package with the phenomenological constraints coming from particle physics
(see, e.g., [4]).
However, it did not contribute to revealing the origin of the key ingredients
of inflation. It also greatly increased the number of possibilities up to 10500 (known as the string landscape),
hampering-specific theoretical predictions in the search for the signatures of
strings and branes in the Universe.The inflaton driven by a scalar potential and
its engineering by strings and branes are by no means required. Another possible
approach can be based on a modification of the gravitational part of Einstein
equations by terms of the higher order in the spacetime curvature [5]. It does not require an
inflaton or an exotic matter, while the specific higher-curvature terms are
well known to be present in the effective action of superstrings [6, 7].The perturbative strings are defined on-shell (in the
form of quantum amplitudes), while they give rise to the infinitely many
higher-curvature corrections to the Einstein equations, to all orders in the
Regge slope parameterα′ and the string coupling gs.
The finite form of all those corrections is unknown and beyond our control. However,
it still makes sense to consider the leading corrections to the Einstein
equations, coming from strings and branes. Of course, any result to be obtained
from the merely leading quantum corrections cannot be conclusive. Nevertheless,
they may offer both qualitative and technical insights into the early Universe
cosmology, within the well-defined and highly restrictive framework. In this paper,
we adopt the approach based on the Einstein equations modified by the leading
superstring-generated gravitational terms which are quartic in the spacetime curvature. We
treat the quartic curvature terms on
equal footing with the Einstein term, that is, nonperturbatively.We consider only geometrical (i.e., pure gravity)
terms in the low-energy M-theory effective action in four space-time
dimensions. We assume that the quantumgs-corrections can be suppressed against the
leading α′-corrections, whereas all the moduli,
including a dilaton and an axion, are somehow stabilized (e.g., by fluxes,
after the warped compactification to four dimensions and spontaneous
supersymmetry breaking).Our paper is organized as follows. In Section2, we
review our starting point: M-theory in 11 spacetime dimensions with the leading
quantum corrections, and the dimensional reduction to four spacetime
dimensions. In Section 3, we discuss the problem of the off-shell extension of
the gravitational part of the four-dimensional effective action for
superstrings. In Section 4, we review the physical significance of the on-shell
quartic curvature terms. In Section 5, we prove that it is impossible to
eliminate the fourth-order time derivatives in the four-dimensional equations
of motion with a generic metric. The structure of equations of motions for the
special (FRW) metrics is revealed in Section 6, which contains our main new
results. The exact (de Sitter) solutions, stability, and duality constraints
are also discussed in Section 6. Our conclusion is Section 7. In Appendix A, we
give our notation and compute some relevant identities. The two-component
spinor formalism (for completeness) is summarized in Appendix B.


## 2. M-Theory and Modified Einstein Equations

There are five perturbatively consistent superstring
models in ten spacetime dimensions (see, e.g., the book [6, 7]). All those models are
related by duality transformations. In this paper, we are going to consider
only the gravitational sector of the heterotic and type-II strings. In
addition, there exists a parent theory behind all those superstring models, it
is called M-theory, and it is eleven-dimensional [6, 7]. Not so much is known about
the nonperturbative M-theory. Nevertheless, there are the well-established
facts that (i) the M-theory low-energy effective action is given by the
11-dimensional supergravity [8], and (ii) the leading quantum gravitational
corrections to the 11-dimensional supergravity from M-theory in the bosonic
sector are quartic in the curvature [9–12] (see, e.g., [13] for some recent progress). Our purpose in this section
is to emphasize what is not known.All the bosonic terms of the M-theory corrected
11-dimensional action read as follows [9–12]:(2.1)S11=−12κ112∫d11x−g[R−12⋅4!F2−16⋅3!⋅(4!)2ε11CFF]−T2(2π)4⋅32⋅213∫d11x−g(J−12E8)+T2∫C⋀X8,where κ11 is the 11-dimensional gravitational constant, T2 is the M2-brane tension given by(2.2)T2=(2π2κ112)1/3,where C is a 3-form gauge field
of the 11-dimensional supergravity [8], and F=dC is its four-form field strength, R is the gravitational scalar curvature, ε11 stands for the 11-dimensional Levi-Civita
symbol in the Chern-Simons-like coupling, while (J,E8,X8) are certain quartic polynomials in the 11-dimensional
curvature. The J is given by(2.3)J=3⋅28(RmijnRpijqRmrspRqrsn+12RmnijRpqijRmrspRqrsn)+𝒪(Rmn),where the E8 is the 11-dimensional extension of the
eight-dimensional Euler density(2.4)E8=13!εabcm1n1⋯m4n4εabcm′1n′1⋯m′4n′4Rm′1n′1m1n1⋯Rm′4n′4m4n4,and the X8 is the eight-form(2.5)X8=1192⋅(2π2)4[trR4−14(trR2)2],where the traces are taken with
respect to (implicit) Lorentz indices in eleven space-time dimensions. The
(world) vector indices are also suppressed in (2.1).TheJ-contribution (2.3) is defined modulo Ricci-dependent terms by its derivation
[9–12]. The basic reason is the on-shell nature of the perturbative
superstrings [6, 7],
whose quantum on-shell amplitudes determine the gravitational effective action
modulo field redefinitions. Via the Einstein-Hilbert term, the metric field
redefinitions contribute to the next (quartic) curvature terms with at least
one factor of Ricci curvature. Therefore, some additional physical requirements
are needed in order to fix those Ricci-dependent terms in the off-shell
M-theory effective action.To match the constraints imposed by particle physics,
M-theory is supposed to be compactified to one of the superstring models in ten
dimensions, and then down to four spacetime dimensions, for example, on a
Calabi-Yau complex three-fold [6, 7]. Alternatively, M-theory may be directly compactified
down to four real dimensions on a 7-dimensional special (G2) holonomy manifold [14]. The bosonic fields of the
action (2.1) are just an eleven-dimensional metric and a 3-form (there is no
dilaton in eleven dimensions). In other words, the 11-dimensional action (2.1)
is the most general starting point to discuss the M-theory/superstrings
compactification.In the presence of fluxes, we should consider thewarped compactification, whose metric is
of the form [15](2.6)ds112=e2A(y)dsFRW2+e−2A(y)ds72,where dsFRW2 is the FRW metric in (uncompactified)
four-dimensional spacetime (see (5.1) below), ds72 is a metric in compactified seven dimensions
with the coordinates ya, a=4,5,6,7,8,9,10,
and A(y) is called a warp factor.Since we are interested in the gravitational sector of
the four-dimensional type-II superstrings, an explicit form of the 7-metricds72 is not needed. In the case of heterotic
strings, one has to include the “anomalous” term quadratic in the
curvature (see below). We put all the four-dimensional scalars (like a dilaton,
an axion, and moduli) into the matter stress-energy tensor (in Einstein frame),
and assume that they are somehow stabilized to certain fixed values. In
addition, we do not consider any M-theory/superstrings solitons such as M- or
D-branes. After dimensional reduction, the only gravitational terms coming from
type-II superstrings in four dimensions are given by(2.7)S4=−12κ2∫d4x−g(R+βJR),where we have introduced the
Einstein coupling κ in four dimensions, and the four-dimensional
counterpart JR of J in (2.3), i,j=0,1,2,3,(2.8)JR=RmijnRpijqRmrspRqrsn+12RmnijRpqijRmrspRqrsn+𝒪(Rmn).The relation between the coupling constantsκ11 and κ is given by(2.9)κ2=e5AMKK7κ112,where we have introduced the Kaluza-Klein (KK) compactification scale MKK−7=Vol7≡∫d7yg7 and the average warp factor A (with an integer weight p)(2.10)epA=1Vol7∫d7yg7epA(y).We also find(2.11)β=13(κ2223/2π5e14AMKK7)2/3of mass dimension −6.
For instance, when substituting the Planck scale κ≈10−33cm and MKK−1≈10−15cm, and ignoring the warp factor, A=0,
we get the incredibly small (and, in fact, unacceptable—see Section 6) value(2.12)β≈10−118cm6.As regards the four-dimensional heterotic strings, the
action (2.7) is to be supplemented by the term [16, 17](2.13)SH=−12κ2∫d4x−g(18JH),where(2.14)JH=RijklRijkl+𝒪(Rmn)again modulo Ricci-dependent
terms.The gravitational action is to be added to a matter
action, which lead to themodified Einstein equations of motion (in the type-II case, for definiteness)(2.15)Rij−12gijR+β1−gδδgij(−gJR)=κ2Tij,where Tij stands for the energy-momentum tensor of all
the matter fields (including dilaton and axion).Due to the ambiguities in the definition of theJR-polynomial, it is also possible to replace it
by(2.16)JC=CmijnCpijqCmrspCqrsn+12CmnijCpqijCmrspCqrsn+𝒪(Rmn),where we have introduced the
Weyl tensor in four dimensions [18], which is the traceless part of the curvature tensor—see Appendices A and B.


## 3. Going Off-Shell with the Curvature Terms

There are about102 Ricci-dependent terms in the most general
off-shell gravitational effective action that is quartic in the curvature. It also means
about 100 new coefficients, which makes the fixing of
the off-shell action to be extremely difficult. The quartic curvature terms are
thus different from the quadratic curvature terms, present in the on-shell heterotic string effective action
(2.13), whose off-shell extension is very simple (see below). It is, therefore,
desirable to formulate some necessary conditions that any off-shell extension
has to satisfy.(i) The first condition is, of course, the vanishing
of all extra terms (i.e., beyond those in (2.8)) in the Ricci-flat case
[19, 20]. The perturbative
superstring effective action is usually deducted from the superstring
amplitudes, whose on-shell condition is just the Ricci-flatness. In the
alternative method, known as the nonlinear sigma-model beta-function approach,
the Ricci-dependent ambiguities in the effective equations of motion
(associated with the vanishing sigma-model beta-functions) arise via the
dependence of the renormalization group beta-functions of the nonlinear
sigma-model upon the renormalization prescription, starting from two loops
(see, e.g., [21] for
details).(ii) Supersymmetry requires all quantum bosonic
corrections to be extendable to locally supersymmetric invariants. It can be
made manifest in four spacetime dimensions, where the off-shell superspace
formalism ofN=1 supergravity is available [22]. The Weyl tensor, Ricci
tensor and scalar curvature belong to three different N=1 superfields called Wαβγ, Gαα˙ and R,
respectively, while the first superfield is
chiral.  (We use the two-component spinor notation [22], α,β,…=1,2—see Appendix
B.)  In
particular, the Weyl tensor Cαβγδ appears in the first order of the N=1 superspace chiral anticommuting coordinates θα as(3.1)Wαβγ(x,θ)=Wαβγ(x)+θδCαβγδ(x)+⋯so that the JH terms (with all curvatures being replaced by
Weyl tensors) are easily supersymmetrizable in superspace as(3.2)∫d2θℰ−1Wαβγ2.The JC terms in (2.16) are also extendable to the
manifest superinvariant(3.3)∫d4θE−1Wαβγ2W¯α˙β˙γ˙2,where we have introduced the
supervielbein densities, ℰ and E,
in the chiral and central superspaces, respectively (see [22] for details).Those invariants were extensively studied in the past
because they naturally appear as the possible counterterms (with divergent
coefficients) in quantum four-dimensional supergravity (see, e.g., [23, 24]). In superstring theory,
one gets the same structures,
though with finite coefficients
(see, e.g., [25, 26]). Thus, in four dimensions, the structure of the on-shell superstrings quartic curvature
terms is fixed by local N=1 supersymmetry alone, up to normalization.(iii) The absence of the higher-order time derivatives
is usually desirable to prevent possible unphysical solutions to the equations
of motion, as well as to preserve the perturbative unitarity, but it is by no
means necessary. As is well known, the standard Friedmann equation of general
relativity is an evolution equation, that is, it contains only the first-order
time derivatives of the scale factor [1, 2, 27].
It happens due to the cancellation of terms with the second-order time
derivatives in the mixed 00-component of Einstein tensor—see, for
example, the appendix of [28] for details. It can also be seen as the consequence
of the fact that the second-order dynamical (Raychaudhuri) equation for the
scale factor in general relativity can be integrated once, by the use of the
continuity equation (3.5), thus leading to the evolution (Friedmann) equation
[1, 2]. As regards the
quadratic curvature terms present in the heterotic case, their unique off-shell
extension is given by the Gauss-Bonnet-type combination [29, 30](3.4)JH→G=RijklRijkl−4RijRij+R2.In the expansion around
Minkowski space, gij(x)=ηij+hij(x),
the fourth-order derivatives (at the leading order in 𝒪(h2)) coming from the first term in (3.4) cancel
against those in the second and third terms [31]. As a result, the off-shell extension (3.4) appears
to be ghost-free in all dimensions. As regards four space-time dimensions, the terms
(3.4) can be rewritten as the four-dimensional Euler density
(A.7). Therefore,
being a total derivative, (3.4) does not contribute to the four-dimensional
effective action. (Of course, adding Euler
densities to the Einstein-Hilbert term matters in higher (than four) dimensions [32, 33], or with the dynamical dilaton and axion fields
[34–36].) The higher-time derivatives are apparent in the
gravitational equations of motion with the quartic curvature terms (see also
[37]). It is natural
to exploit the freedom of the metric field redefinitions in order to get rid of
those terms. However, in Section 5 we prove that it is impossible to eliminate
the fourth-order time derivatives in the quartic curvature terms via a metric
field redefinition. It may still be possible for some special (like FRW)
metrics, after imposing the string duality requirement (Section 6).(iv) The matter equations of motion in general
relativity imply the covariant conservation law of the matter energy-momentum
tensor(3.5)(Tij);j=0.By the well-known identity (Rij−(1/2)gijR);j=0,
(2.15) and (3.5) imply(3.6)[1−gδδgij(−gJ)];j=0.For instance, when J=G as in (3.4), (3.6) reads(3.7)−12(RijklRijkl−4RijRij+R2);m+2(RmjklRnjkl);n−4(RminjRij);n−4(RmiRin);n+2(RRmn);n=0.By the use of Bianchi identities
for the curvature tensor, we find by an explicit calculation that the left-hand
side of (3.7) identically vanishes. We believe that
(3.6) should be identically satisfied by any off-shell gravitational correction J because, otherwise, the consistency of the
gravitational equations of motion may be violated.Given the quartic curvature terms (2.8), the modified
Einstein equations of motion (2.15) are(3.8)κ2Tij=Rij−12gijR+β[−12gijJR−Rmhk(iRj)rtm(RkqsrRtqsh+RksqtRhrqs)−Rkqs(iRj)rmt(RhsqtRkrmk−RthsqRhrmk)+(RitrjRksqtRhsqr)(;k;h)+(RisqtRrktmRjsqk)(;r;m)−(Rhrs(iRj)mnrRhmnk+Rsht(iRj)mnlRkmnh)(;k;s)].(v) We may also add thecausality constraint as our next
condition: the group velocity of ultraviolet perturbations on a gravitational
background with the higher-curvature terms included must not exceed the speed
of light. As was demonstrated in [38, 39], the causality condition merely affects the sign
factors of the full curvature terms in the action, namely, the signs in front
of (RmnpqRmnpq)2 and (Rmnpq*Rmnpq)2 should be positive. It must be automatically
satisfied by the perturbative superstring quartic corrections (2.8) due to the
known unitarity of superstring theory, and it is the case indeed—see the
identity (A.20—just because β>0.Of course, our list is not complete, and it could be
easily extended by more conditions, for example, by requiring the consistency
with black hole physics, gravitational waves, nucleosynthesis, and so forth.
For example, in Section6, we impose the scale factor duality as yet another constraint.


## 4. On-Shell Structure and Physical Meaning of the Quartic Curvature Terms

The detailed structure and physical meaning of the
quartic curvature terms in (2.8) and (2.16) are easily revealed via their
connection to the four-dimensional Bel-Robinson (BR) tensor [40–43]. The latter is well known
in general relativity [44–47]. We review here the main properties of the BR tensor,
and calculate the coefficients in the important identities—see
(4.4) and
(4.5) in the section below.  (Those coefficients
were left undetermined in [45–47]).The BR tensor is defined by(4.1)TRiklm=RipqlRkpqm+*Ripql*Rkpqm,whose structure is quite similar
to that of the Maxwell stress-energy tensor(4.2)TijMaxwell=FikFjk+*Fik*Fjk,Fij=∂iAj−∂jAi
(see also Appendix A for
more details).The
Weyl cousinTCijlm of the BR tensor is obtained by replacing all
curvatures by Weyl tensors in (4.1)—see
(A.10). The Weyl BR tensor can be
factorized in the two-component formalism (see Appendix B)(4.3)(TC)αβγδα˙β˙γ˙δ˙=CαβγδC¯α˙β˙γ˙δ˙.In this section, we consider all the quartic terms
on-shell, that is,modulo Ricci
tensor-dependent terms. Therefore, we are not going to distinguish between TR and TC here. The Ricci tensor-dependent additions will
be discussed in Sections 5 and 6.The significance of the BR tensor to the quartic
curvature terms is already obvious from superspace (see Section3), where the
locally N=1 supersymmetric extension of the quartic Weyl terms (2.16) is given
by (3.3) whose bosonic part is the BR tensor squared due to (4.3). As regards a
straightforward proof, see Appendix A and our derivation of
(A.20) there, which
imply(4.4)Tijkl2=8JR=14(RijklRijkl)2+14(*RijklRijkl)2.In addition, when using another
identity
(A.17), (4.4) yields(4.5)Tijkl2=8JR=−14(*Rijkl2)2+14(*RijklRijkl)2=14(P42−E42)=14(P4+E4)(P4−E4),where we have introduced the
Euler and Pontryagin topological densities in four dimensions—see
(A.7) and
(A.8), respectively.In addition [40–43, 45–47], the on-shell BR tensor is fully symmetric with respect to its vector
indices, it is traceless,(4.6)Tijkl=T(ijkl),Tikli=0.(ii) It is covariantly conserved (though the BR tensor is not a
physical current),(4.7)∇iTijkl=0,and it has positive “energy”
density(4.8)T0000>0.Equation (4.6) is most easily
seen in the two-component formalism (see Appendix B),
(4.7) is the consequence
of Bianchi identities [48], whereas
(4.8) just follows from the definition
(4.1).The BR tensor is related to the gravitational
energy-momentumpseudo tensors
[45–47]. It can be most
clearly seen in Riemann normal
coordinates (RNCs) at any given point in spacetime. The RNCs are defined by the relations(4.9)gij=ηij,gij,k=0,gij,mn=−13(Rimjn+Rinjm),so that the derivatives of
Christoffel symbols read as follows:(4.10)Γjk,li=−13(Rijkl+Rikjl).Raising and lowering of vector
indices in RNCs are performed with Minkowski metric ηij and its inverse ηij,
whereas all traces in the last two equations (4.9) and
(4.10) vanish(4.11)ηijgij,mn=ηijΓij,lk=Γij,ki=Γjk,ii=0.Moreover, there exists the remarkable noncovariant
relation (valid only in RNC) [45–47](4.12)Tijkl=∂k∂l(tijLL+12tijE),where the symmetric Landau-Lifshitz (LL) gravitational
pseudotensor [27](4.13)(tLL)ij=−ηipηjqΓpmkΓqkm+ΓmniΓpqjηmpηnq−(ΓnpmΓmqjηinηpq+ΓnpmΓmqiηjnηpq)+hijΓnpmΓmqnηpq,and the nonsymmetric Einstein (E) gravitational pseudotensor
[49, 50](4.14)(tE)ji=(−2ΓmpiΓjqm+δjiΓpmnΓqnm)ηpqhave been introduced in RNC, in
terms of Christoffel symbols.


## 5. Off-Shell Quartic Curvatures in Cosmology

The main cosmological principle of aspatially homogeneous and isotropic (1+3)-dimensional universe (at large scales) gives
rise to the standard Friedman-Robertson-Walker (FRW) metrics of
the form [49, 50](5.1)dsFRW2=dt2−a2(t)[dr21−kr2+r2dΩ2],where the function a(t) is known as the scale factor in “cosmic”
coordinates (t,r,θ,ϕ);
we use c=1 and dΩ2=dθ2+sin2θdϕ2,
while k is the FRW topology index taking values (−1,0,+1).
Accordingly, the FRW metric (5.1) admits a 6-dimensional isometry group G that is either SO(1,3), E(3), or SO(4),
acting on the orbits G/SO(3),
with the spatial 3-dimensional sections H3, E3,
or S3,
respectively. By the coordinate change, dt=a(t)dη,
the FRW metric (5.1) can be rewritten to the form(5.2)ds2=a2(η)[dη2−dr21−kr2−r2dΩ2],which is manifestly
(4-dimensional) conformally flat in the case of k=0.
Therefore, the 4-dimensional Weyl tensor of the FRW metric obviously vanishes
in the “flat” case of k=0.
It is well known that the FRW Weyl tensor vanishes in the other two cases, k=−1 and k=+1,
too [28, 51]. Thus we have(5.3)CijklFRW=0.Inflation in an early universe is defined as the epoch
during which the scale factor is accelerating [1, 2](5.4)a¨(t)>0,orequivalentlyddt(H−1a)<0,where the dots denote time
derivatives, and H=a˙/a is Hubble “constant.” The amount of
inflation is given by a number of e-foldings [1, 2]:(5.5)N=lna(tend)a(tstart)=∫tstarttendHdt,which should be around 70 [1, 2].Though the leading purely geometrical (perturbative)
correction in the heterotic string case is given by the Gauss-Bonnet
combination (3.4), and thus it does not contribute to the equations of motion
in four space-time dimensions, the situation changes when the dynamical moduli (axion and dilaton) are
included. The effective string theory couplings are moduli dependent, which
gives rise to a nontrivial coupling with the moduli in front of the
Gauss-Bonnet term, so that the latter is not a total derivative any more. At
the level of the one-loop corrected heterotic superstring effective action in
four dimensions, the cosmological solutions were studied in [34–36]. As regards the realization
of inflation in M-theory, see, for example, [52].In the case of type-II superstrings (after stabilizing
the moduli) we are left with the quartic curvature terms in the
four-dimensional effective action (Section2). Let us address the issue of the
higher-time derivatives in the general setting. It is quite natural to use the
freedom of the metric field redefinitions in string theory in order to try to
get rid of the higher-time derivatives in the effective action. The successful
example is provided by the Gauss-Bonnet gravity (Section 3) that we are now
going to follow. Let us consider a weak gravitational field(5.6)gij(x)=ηij+hij(x)in the harmonic gauge(5.7)(hij),j=12∂ih,h=ηijhij.The linearized curvatures are
given by(5.8)Rijkl=12[hil,jk−hjl,ik−hik,jl+hjk,il],whereas the Ricci tensor and the
scalar curvature in the gauge (5.7) read(5.9)Rij=−12□hij,R=−12□h,□≡∂i∂i.(We assign the lower caselatin letters to spacetime indices, i,j,k,…=0,1,2,3,
and the lower case middle greek letters to spatial indices, μ,ν,…=1,2,3.)As is
clear from the structure of those equations, it is possible to form the Ricci
terms after integration by parts in the quadratic
curvature action. As a result, there is a cancellation of all terms with the
fourth-order time derivatives in the leading order𝒪(h2) of the Gauss-Bonnet action (3.4) in all
spacetime dimensions, as was first observed in [31].Unfortunately, we find that it does not work for the
quartic curvature terms, even in four spacetime dimensions, as we now going to
argue.When using the linearized curvature (5.8), the quartic
terms (2.8) in four spacetime dimensions have the structure(5.10)25JR=AikjlAiljk+2AikjlBiljk+BikjlBiljk+Aikjl{Cilkj+Clijk}+Bikjl{Cilkj+Clijk}+2Cikjl{Cilkj+Clijk+Ckjil+Cjkli}−Cikjl{Ciljk+Clikj+Cjkil+Ckjli},where we have introduced the
notation ∂ij2=∂i∂j and(5.11)Aikjl=∂mn2hik(∂2mnhjl+∂2jlhmn−∂2jmhln−∂2lnhjm),Bikjl=∂2ikhmn(∂2mnhjl+∂2jlhmn−∂2jmhln−∂2lnhjm),Cikjl=∂m2ihnk(∂2mnhjl+∂2jlhmn−∂2jmhln−∂2lnhjm),while all the index contractions
above are performed with Minkowski metric.Equation (5.10) is not very illuminating, but it is
enough to observe that the dangerous terms (∂002hμν)4 and (∂0∂λhμν)(∂002hμν)3 do contribute, and thus lead to the terms with
the fourth- and third-order time derivatives in the equations of motion, when
all hμν are supposed to be independent. The last
possibility is to convert those terms into some Ricci tensor-dependent
contributions. However, in the harmonic gauge (5.7), getting the Ricci tensor
requires the two spacetime derivatives to be contracted into the wave operator,
as in (5.9), in each dangerous term, which is impossible for the quartic
curvature terms, unlike their quadratic counterpart because any integration by
parts in the quartic terms does not end up with a wave operator in each term.
The equations of motion in the case of (BR)2-gravity with the FRW metric are explicitly
computed in Section 6, as an example.Having failed to remove the higher-time derivatives
for a generic metric, one can try to get rid of them for a special class of
metrics, namely, the FRW metrics of our interest. The simplest example arises
when all the Riemann curvatures in the quartic curvature terms are replaced by
the Weyl tensors, as in (2.16). It also amounts to adding certain quartic
curvature terms with at least one Ricci factor to the effective action (2.7).
This proposal is based on the reasonable assumption [53] coming from the AdS/CFT
correspondence that the AdS7×S4 and AdS4×S7 spaces seem to be the exact solutions to the
(eleven-dimensional) M-theory equations of motion. Of course, such assumption
is just the sufficient condition, not the necessary one, because there may be
many more solutions. The substitution Rijkl→Cijkl leads to the contributions with three Weyl
tensors (from the quartic terms) in the equations of motion, which implies no perturbative superstring corrections to
the FRW metrics at all because of (5.3).In Section6, we find that the scale factor duality
requirements allow a family of the generalized Friedmann equations coming from
the most general quartic curvature terms, with just a few real parameters.


## 6. Exact Solutions, Stability, and Duality

Our motivation in this paper is based on the
observation that the standard model (SM) of elementary particles does not have
an inflaton.  (The proposal [54] to identify the inflaton
with the SM Higgs boson requires its nonminimal coupling to gravity, which does
not fit to string theory.)
In addition, M-theory/superstrings have plenty of inflaton
candidates but any inflationary mechanism based on a scalar field is highly
model dependent. When one wants the universal geometrical mechanism of inflation
based on gravity only, it should occur due to some Planck scale physics to be
described by the higher-curvature terms (cf. [5]).On the experimental side, it is known that the vacuum
energy densityρinf during inflation is bounded from above by a
(non)observation of tensor fluctuations of the cosmic microwave background
(CMB) radiation [55]:(6.1)ρinf≤(10−3MPl)4.It severely constrains but does
not exclude the possibility of the geometrical inflation originating from the
purely gravitational sector of string theory because the factor of 10−3 above may be just due to some numerical
coefficients (cf. Section 2).In this section, we consider the structure of our
generalized Friedmann equation withgeneric quartic curvature terms. We get
the conditions of stability of our inflationary solutions, and solve the
duality invariance constraints coming from string theory [56, 57].Due to a single-arbitrary functiona(t) in the FRW Ansatz (5.1), it is enough to take
only one gravitational equation of motion in (2.15) without matter, namely, its
mixed 00-component. As is well known [1, 2], the spatial
(3-dimensional)
curvature can be ignored in a very early universe, so we choose the manifestly
conformally-flat FRW metric (5.1) with k=0 in our Ansatz. It leads to a purely
gravitational equation of motion having the form(6.2)3H2≡3(a˙a)2=βP8(a˙a,a¨a,a⃛a,a⃜a),where P8 is a polynomial with respect to its arguments:(6.3)P8=∑n1+2n2+3n3+4n4=8,n1,n2,n3,n4≥0cn1n2n3n4(a˙a)n1(a¨a)n2(a⃛a)n3(a⃜a)n4.Here, the sum goes over the integer partitions (n1,2n2,3n3,4n4) of 8,
the dots stand for the derivatives with respect to time t,
and cn1n2n3n4 are some real coefficients. The highest
derivative enters linearly at most, n4=0,1.The FRW Ansatz withk=0 gives the following nonvanishing curvatures:(6.4)R0μ0ν=δμνa¨a,Rμνλρ=(δλμδνρ−δρμδνλ)(a˙)2,Rνμ=−δνμ[a¨a+2(a˙a)2],where μ,ν,λ,ρ=1,2,3.
For example, in the case of the (BR)2 gravity (3.8), after a straightforward (though
quite tedious) calculation of the mixed 00-equation without matter and with the
curvatures (6.4), we find(6.5)3H2+β[9(a¨a)4−36H2(a¨a)3+84H4(a¨a)2−36H(a¨a)2(a⃛a)+63H8−72H3(a¨a)(a⃛a)+48H6(a¨a)−24H5(a⃛a)]=0.It is remarkable that the
fourth-order time derivatives (present in various terms of (3.8)) cancel,
whereas the square of the third-order time derivative of the scale factor, a⃛2,
does not appear at all in this equation.
(Taking Weyl tensors instead of Riemann curvatures leads
to the vanishing coefficients.)Our generalized Friedmann equation (6.2) applies to any combination of the quartic curvature
terms in the action, including the Ricci-dependent terms. The coefficients cn1n2n3n4 in (6.3) can be thought of as linear
combinations of the coefficients in the most general quartic curvature action.
The polynomial (6.3) merely has 12 undetermined coefficients, that is
considerably less than a 100 of the coefficients in the most general
quartic curvature action.The structure of (6.2) and (6.3) admits the existence
of rather generic exact inflationary solutions without a spacetime singularity.
Indeed, when using the most naive (de Sitter) Ansatz for the scale factor,(6.6)a(t)=a0eBtwith some real positive
constants a0 and B,
and substituting (6.6) into (6.2), we get 3B2=(#)βB8,
whose coefficient (#) is just a sum of all c-coefficients in (6.3). Assuming the (#) to be positive, we find an exact solution:(6.7)B=(3#β)1/6.This solution in nonperturbative
in β,
that is, it is impossible to get it when considering the quartic curvature
terms as a perturbation. Of course, the assumption that we are dealing with the
leading correction implies Bt≪1.
Because of (2.11) and
(6.7), it leads to the natural hierarchy(6.8)κMKK≪1orlPl≪lKK,where we have introduced the
four-dimensional Planck scale lPl=κ and the compactification scale lKK=MKK−1.Theeffective Hubble scale B of (6.7) should be lower than the effective (with warping) KK scale MKKeff.=eAMKK in order to validate our four-dimensional
description of gravity, that is, the ignorance of all KK modes:(6.9)B<MKKeff..It rules out the naive KK
reduction (with A=0) but still allows the warped compactification
(2.6), when the average warp factor is tuned(6.10)eA<(κMKK)2/5(9/#)3/10223/10π∼𝒪(10−3),where we have used (2.11) and
have estimated (#) by order 10.The exact solution (6.6) is nonsingular, while it
describes an inflationary isotropic and homogeneous early universe.
(The
exact de Sitter solutions in the special case (2.8) were also found in
[58–61].)  Given the expanding universe, the curvatures
decrease, so that the higher-curvature terms cease to be the dominant
contributions against the matter terms we ignored in the equations of motion.
The matter terms may provide a mechanism for ending the geometrical inflation
and reheating (i.e., a graceful exit to the standard cosmology).To be truly inflationary solutions, (6.6) and (6.7) should
correspond to the stable fixed points (or attractors) [1, 2]. The stability conditions
are easily derived along the standard lines (see, e.g., [62, 63]). When using the
parameterization(6.11)a(t)=eλ(t),we easily find(6.12)a˙a=λ˙,a¨a=λ¨+(λ˙)2,a⃛a=λ⃛+3λ¨λ˙+(λ˙)3,a⃜a=λ⃜+4λ⃛λ˙+6λ¨(λ˙)2+3(λ¨)2+(λ˙)4.Equation (6.3) now takes the form(6.13)P8=∑n1+2n2+3n3+4n4=8,n1,n2,n3,n4≥0dn1n2n3n4(λ˙)n1(λ¨)n2(λ⃛)n3(λ⃜)n4,where the d-coefficients are linear combinations of the c-coefficients (easy to find). Equations (6.6)
and (6.7) are also simplified(6.14)λ(t)=Bt+λ0,wherea0=eλ0,d8000=#.The solution (6.14) can be
considered as the fixed point of the equations of motion (6.2) in a generic
case(6.15)3y12=βP8(y1,y2,y3,y˙3)≡βP8,0(y1,y2,y3)+βP4(y1,y2,y3)y˙3,where we have introduced the
notation(6.16)y1=λ˙,y2=λ¨,y3=λ⃛.Equation (6.15) can be brought
into an autonomous form(6.17)y˙1=y2,y˙2=y3,y˙3=3y12−βP8,0(y1,y2,y3)βP4(y1,y2,y3)≡f(y1,y2,y3),that is quite suitable for the
stability analysis against small perturbations about the fixed points, ya=yafixed+δya,
where a=1,2,3.
We find(6.18)δy˙1=δy2,δy˙2=δy3,δy˙3=∂f∂y1|δy1+∂f∂y2|δy2+∂f∂y3|δy3,where all the partial
derivatives are taken at the fixed point (denoted by |). The fixed points are stable when all the eigenvalues
of the matrix(6.19)M^=(010001∂f∂y1|∂f∂y2|∂f∂y3|)in (6.18) are negative or have negative real parts
[62, 63]. Then the fixed point is a
stable attractor.To the end of this section, we would like to
investigate how the symmetries of string theory are going to affect the
coefficients of our generalized Friedmann equation. Here, we apply the scale
factor duality [56, 57] by
requiring our equation (6.2) to be invariant under the duality transformation(6.20)a(t)⟷1a(t)≡b(t).This duality is a cosmological
version of the genuine stringy T-duality (which is the symmetry of the
nonperturbative string spectrum), in the case of time-dependent backgrounds.
The scale factor duality is merely the symmetry of the (perturbative) equations
of motion of the background fields. It is used, for example, in the so-called
prebig bang scenario [64] in order to avoid the cosmological singularity.In theλ-parameterization (6.11), the duality
transformation (6.20) takes the very simple form(6.21)λ(t)⟷−λ(t).The equations of motion in the
form (6.15) are manifestly invariant under λ(t)→λ(t)+λ0,
where λ0 is an arbitrary constant.It is straightforward to calculate how the right-hand
side of (6.2) transforms under the duality
(6.20) by differentiating
(6.20). We
find(6.22)a˙a=−b˙b,a¨a=−b¨b+2(b˙b)2,a⃛a=−b⃛b+6(b˙b)(b¨b)−6(b˙b)3,a⃜a=−b⃜b+6(b¨b)2+8(b˙b)(b⃛b)−36(b˙b)2(b¨b)+24(b˙b)4.To see how the duality affects the polynomialP8,
we consider the case with the third-order time derivatives, motivated by
(6.5).
We introduce the notation(6.23)a˙a=x,a¨a=y,a⃛a=z,so that the duality invariance
condition reads(6.24)P8(−x,2x2−y,6xy−6x3−z)=P8(x,y,z).The structure of the polynomial P8 in
(6.3), as the sum over partitions of 8,
restricts a solution to (6.24) to be most quadratic in z:(6.25)P8(x,y,z)=a2(x,y)z2+b5(x,y)z+c8(x,y),whose coefficients are
polynomials in (x,y),
of the order being given by their subscripts, that is,(6.26)a2(x,y)=a0x2+a1y,b5(x,y)=b0x5+b1x3y+b2xy2,c8(x,y)=c4y4+c3y3x2+c2y2x4+c1yx6+c0x8.After a substitution of (6.25)
and (6.14) into
(6.24), we get an over determined system of linear equations
on the coefficients. Nevertheless, we find that there is a consistent general
solution(6.27)P8(x,y,z)=a0x2z2+(b0x5−3a0xy2)z+c4y4+(9a0−4c4)y3x2+c2y23x4+(8c4−18a0−3b0−2c2)yx6+c0x8parameterized by merely five real
coefficients (a0,b0,c4,c2,c0).
Requiring the existence of the exact solution (6.6), that is, the positivity of (#) in (6.7) yields(6.28)5c4+c0>11a0+2b0+c2.As regards the(BR)2 gravity representing the “minimal”
candidate for the off-shell superstring effective action, we checked that
neither the duality invariant structure (6.15) nor the inequality
(6.28) are
satisfied by the coefficients present in
(6.5). We interpret it as the clear
indications that some additional Ricci-dependent terms have to be added to the (BR)2 terms or, equivalently, the (BR)2 gravity is ruled out as the off-shell
effective action for superstrings.Finally, we would like to mention about some possible
simplifications and generalizations.The last equation
(6.4) apparently implies that the
Ricci-dependent terms in P8 should have the factor of (y+2x2).
Hence it may be possible to completely eliminate both the fourth- and
third-order time derivatives in our generalized Friedmann equations, though we
are not sure that this choice is fully consistent. However, if so, instead of
(6.24), we would get another duality condition(6.29)P8(−x,2x2−y)=P8(x,y),whose most general solution is
simpler:(6.30)P8(x,y)=c0x8+c5y(y−2x2)[y(y−2x2)−4x6]+c6x4y(y−2x2)with merely three, yet to be
determined coefficients (c0,c5,c6).We would like to emphasize that our results above can
be generalized to any finite order with respect to the spacetime curvatures in
the off-shell superstring effective action because it amounts to increasing the
order of the polynomialP.
The list (6.10) can be continued to any higher order in the derivatives. We can
now speculate about the form of the generalized Friedmann equation to all orders in the curvature. It depends
upon whether (i) there will be some finite maximal order of the time
derivatives there, or (ii) the time derivatives of arbitrarily high order
appear (we do not know about it). Given the case (i), we just drop the
requirement that the right-hand side of our cosmological equation
(6.2) is a
polynomial, and take a duality-invariant functionP instead. In the case (ii), we should replace
the function by a functional,
thus getting a nonlocal equation having the form(6.31)H2=a˙2a2=βP[a(t)],whose functional P is subject to the nontrivial duality
constraint(6.32)P[a(t)]=P[1a(t)].Imposingsimultaneously both conditions of
stability and duality invariance leads to severe constraints on the c-coefficients. Hence it also severely restricts
the quantum ambiguities in the superstring-generated quartic curvature gravity.
Finding their solutions seems to be a nontrivial mathematical problem, we would
like to investigate it elsewhere [65].


## 7. Conclusion

The higher-curvature terms in the gravitational action
defy the famous Hawking-Penrose theorem [66–68] about the existence of a spacetime singularity in any
exact solution to the Einstein equations. As we demonstrated in this paper, the
initial cosmological singularity can be easily avoided by considering the
superstring-motivated higher-curvature terms on equal footing (i.e.,
nonperturbatively) with the Einstein-Hilbert term.Our results predict the possible existence of the very
short de Sitter phase driven by the quartic curvature terms, in the early
inflationary epoch.Though we showed the natural existence of inflationary
(de Sitter) exact solutions without a spacetime singularity under rather
generic conditions on the coefficients in the higher-derivative terms, it is
not enough for robust physical applications. As a matter of fact, we assumed
the dominance of the higher-curvature gravitational terms over all matter
contributions in the very early Universe at the Planck scale. However, given
the expansion of the Universe under the geometrical inflation, the spacetime
curvatures should decrease, so that the matter terms can no longer be ignored.
The latter may effectively replace the geometrical inflation by another
matter-dominated mechanism, thus allowing the inflation to continue
substantially below the Planck scale.In addition, the number of e-foldings (5.5) is just
about one in our scenario based on the quartic curvature terms, which makes it
difficult to compete with the conventional inflation mechanisms [1, 2]. An investigation of the
possible “graceful exit” strategies, toward a matter-driven inflation is,
however, beyond the scope of the given paper.The quartic curvature terms are also relevant to the
Brandenberger-Vafa cosmological scenario of string gas cosmology [69]—see, for example,
[63] for a recent
investigation of the higher-curvature corrections there.
(The higher-curvature terms were considered only perturbatively in
[63].)The higher-time derivatives in the equations of motion
may be unavoidable when using the higher-curvature terms, but we do not see
that they constitute a trouble.Gravity with the quartic curvature terms is a good
playground for going beyond the Einstein equations. Our analysis may be part of
a more general approach based on superstrings, including dynamical moduli and
extra dimensions.



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