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Inflation, bifurcations of nonlinear curvature Lagrangians and dark
energy
Eckehard W. Mielke
Departamento de F́ısica,
Universidad Autónoma Metropolitana–Iztapalapa,
Apartado Postal 55-534, C.P. 09340, México, D.F., MEXICO
ekke@xanum.uam.mx
Fjodor V. Kusmartsev
Department of Physics, Loughborough University, Loughborough,
Leicestershire LE11 3TU, United Kingdom
f.kusmartsev@lboro.ac.uk
Franz E. Schunck
Institut für Theoretische Physik, Universität zu Köln, 50923 Köln, Germany
fs@thp.uni-koeln.de
A possible equivalence of scalar dark matter, the inflaton, and modified gravity
is analyzed. After a conformal mapping, the dependence of the effective Lagrangian
on the curvature is not only singular but also bifurcates into several almost Ein-
steinian spaces, distinguished only by a different effective gravitational strength and
cosmological constant. A swallow tail catastrophe in the bifurcation set indicates the
possibility for the coexistence of different Einsteinian domains in our Universe. This
‘triple unification’ may shed new light on the nature and large scale distribution not
only of dark matter but also on ‘dark energy’, regarded as an effective cosmological
constant, and inflation.
1. Introduction
The origin of the principal constituents of the Universe in the form of dark matter
(DM) and dark energy (DE) (or rather dark tension due to its negative pressure),
remains a major puzzle of modern cosmology and particle physics. Ad hoc proposals
like modifying Newton’s law of gravity as in Milgrom’s MOND (Modified Newtonian
Dynamics) are difficult to reconcile with relativity, or need a phase coupling to
a complex scalar.3 The particle physics’ view is to leave gravity untouched but
postulate WIMPs (Weakly Interacting Massive Particles) like axions, dilatons or
neutralinos as dark matter candidates.66
The dominant non-visible “dark” fraction of the total energy of the Universe is
http://arxiv.org/abs/0704.1135v1
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2 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
known to exist from its gravitational effects. Since the dark matter part is distributed
over rather large distances, its interaction, including a possible self-interaction,60
must be weak. The main candidates for such weakly interacting particles are the
(universal) axion or the lightest supersymmetric particle, as the neutralino in most
models, cf. Ref.66
Recently, an observed excess of diffuse gamma rays has been attributed8 to the
annihilation of DM in our Galaxy. The flux of the presumed neutralino annihilation
allows a reconstruction of the distribution of DM in our Galaxy. Most probable is a
pseudo-isothermal profile but with a substructure of two doughnut-shape rings in
the galactic plane. It is believed that these transient substructures have their origin
in the hierarchical clustering of DM into galaxies. However, there are reservations
concerning the internal consistency in the interpretation of the observations: Given
the same normalization for the cross section of the DM particle, it appears unlikely1
that DM annihilation is the main source of the extragalactic gamma-ray background
(EGB). Moreover, then the large accompanied antiproton flux needs to meet several
constraints.5
On the other hand, heterotic string theory provides a very light universal axion
which may avoid12 the strong CP problem in quantum chromodynamics (QCD).50
Given the existence of such almost massless (pseudo-)scalars, it has been speculated
that a coherent non-topological soliton (NTS) type solution of a nonlinear Klein-
Gordon equation can account for the observed halo structure, simulating a Bose-
Einstein condensate of astronomical size;16,17,34 cf. Ref.53 In particular, a Φ6 toy
model43 yields exact Emden type solutions in flat spacetime, including a flattening37
of halos with ellipticity e < 1 as observed via microlensing.14,54
Both views are to some extend physically equivalent, i.e., scalar dark matter
minimally coupled to Einstein’s general relativity (GR) is equivalent4,27,49,55,59 to a
modified gravity in the relativistic framework of higher-order curvature Lagrangians.
Such effective Lagrangians may also arise from the low-energy limit of (super-)
strings. The model proposed by Carroll et al.6 also uses such an equivalence of a
nonlinear higher-order curvature Lagrangian, in order to explain the present cos-
mic acceleration, but takes resort to an 1/R type curvature Lagrangian, which is
unbounded for weak gravitational fields.
Quite generally, the dependence of the effective Lagrangian L on the scalar
curvature R is singular and cannot always be represented analytically in the (R,L)
plane. More stunning is our recent finding55 that, although nonlinear, a L = L(R)
type Lagrangian bifurcates in several branches of almost linear Lagrangians
2κeff
(R− 2Λeff) (1)
distinguished only by the effective gravitational constant κeff and a cosmological
constant Λeff .
This indicates a unifying picture of dark matter, ‘dark energy’53 with an equation
of state parameter wDE = p/ρ ≃ −1, and modified gravity which may account, on
different scales, for inflation,37,40 dark matter halos36,43 of galaxies or even dark
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 3
matter condensations, the so-called boson stars23,24,41,58 as candidates of MACHOs
(Massive Compact Halo Objects). The ‘landscape’ view25 of (super-)strings suggests
also such a triple unification based on coherent, possibly oscillating scalar field with
a mass larger than the Hubble scale at the present epoch, i.e. m > H∗ = 10
−23 eV.
2. Gravitationally coupled scalar fields
Conventionally, dark matter and inflation can be modeled by a real (or complex)
scalar field φ with self-interaction U(|φ|2) minimally coupled to gravity with the
Lagrangian density
LDM =
| g |
gµν(∂µφ
∗)(∂νφ)− 2U(|φ|2)
, (2)
where κ = 8πG is the gravitational constant in natural units, g the determinant of
the metric gµν , and R the scalar curvature of Riemannian spacetime with Tolman’s
sign conventions.63 A constant potential U0 = Λ/κ would simulate the cosmological
constant Λ.
In a wide range of inflationary models, the underlying dynamics is simply that
of a single scalar field — the inflaton — rolling in some underlying potential. This
scenario invented by Linde26 is generically referred to as chaotic inflation due to its
choice of initial conditions. Many superficially more complicated models can also
be rewritten in this framework.
3. General metric of a flat inflationary universe
A spatially flat (k = 0) Friedman-Robertson-Walker (FRW) Universe with metric
ds2 = dt2 − a2(t)
dr2 + r2
dθ2 + sin2 θdϕ2
is nowadays favored by observations.61 Its temporal evolution of the generic model
(2) is determined by the two autonomous first order equations
Ḣ = −3H2(1 + wDE) = κU(φ)− 3H2 =: V (H,φ) , (4)
φ̇ = ±
3H2 − κU(φ) = ±
V (H,φ) , (5)
where H = ȧ(t)/a(t) is the Hubble expansion rate.
The function V (H,φ) will turn to be the “height function” in Morse theory.46
Observe that V ≤ 0 in order to avoid scalar ghosts. For the FRW metric, the
Lagrangian density (2) reduces to
L = − 3
ȧ2a+
φ̇2 − U(φ)
a3 . (6)
Since the shift function is normalized to one for the FRW metric, the canonical
momenta are given by P = ∂L/∂ȧ = −6Ha2/κ and π = ∂L/∂φ̇ = a3φ̇. The
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4 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
Hamiltonian or “energy function” is given by
φ̇2 +
V (H,φ)
a3 ∼= 0 (7)
and, due to (5), vanishes for all solutions, which is a familiar constraint in GR.
Using the Hubble expansion rate H := ȧ(t)/a(t) as the new “inverse time”
coordinate
t = t(H) =
κŨ − 3H2
, (8)
we were able to find the general solution:56
a = a(H) = a0 exp
κŨ − 3H2
. (9)
ds2 =
κŨ − 3H2
)2 − a0
2 exp
κŨ − 3H2
dr2 + r2
dθ2 + sin2 θdϕ2
, (10)
φ = φ(H) = ∓
3H2 − κŨ
, (11)
where Ũ = Ũ(H) := U(φ(t(H))) is the reparametrized inflationary potential.
The singular case Ũ = 3H2/κ with wDE = −1 leads in (4) to the de Sitter
inflation with exponential expansion a(t) = a0 exp(
Λ/3 t), which for a later time
becomes physically unrealistic, since the inflationary expansion eventually needs to
merge into the usual Friedman cosmos. Therefore, in explicit models we use the
ansatz
Ũ(H) =
, (12)
for the potential, where g(H) = V (H,φ(t(h))) is a nonzero function which should
provide the graceful exit from the inflationary phase to the Friedman cosmos. In
order to have a positive acceleration during the inflationary phase, but on the other
hand a real scalar field in (11), its allowed range is −H2 < g < 0. This H–formalism
facilitates considerably the reconstruction of the inflaton potential.36,40,56 In second
order this is based on an Abel equation13 for the primordial density ǫ = −g/H2.
3.1. Classification by catastrophe theory
A general classification of all allowed inflationary potentials and scenarios has al-
ready been achieved by Kusmartsev et al.22 via the application of catastrophe theory
to the Hamilton–Jacobi type equations (4) and (5) regarded as an autonomous non-
linear system.
In phase space, the equilibrium states are given by the constraint {Ḣ, φ̇} = 0.
The critical or equilibrium points, respectively, of this system are determined by
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 5
V (Hc, φc) = 0. This constraint is globally fulfilled by κU(φ) = 3H
Λ, where the
Hubble expansion rate is constant, i.e. HΛ =:
Λ/3 corresponding to wDE = −1.
For φ̇ = 0 and Λ 6= 0, we recover the de Sitter inflation.
In general, the Jacobi matrix
−6H κU ′
±6H(−2κV )−1/2 ∓κU ′(−2κV )−1/2
, (13)
of the system (4) and (5) depends crucially on U ′ = dU/dφ. Since its determinant
vanishes, i.e. det J = 0, the system is always degenerate. For the analysis of stability,
it suffices therefore to consider only (4) and to reconstruct φ later. Since H and φ
are independent variables, we can introduce the non–Morse superpotentialW (H,φ),
defined via V := −∂W/∂H , and analyze the system with the aid of catastrophe
theory. From (4) we obtain
W (H,φ) = H3 − κU(φ)H + C(φ) , (14)
where C is an arbitrary function of φ. The function W is already in canonical
form in H–space, and belongs to a Whitney surface2,21 or to the Arnold singularity
class A2. The corresponding Whitney surface has only one control parameter, the
potential U . Thus, an evolution of critical points is determined via the values of the
potential U . Let us analyze the types of critical points at different fixed values of
the control parameter U .
If Uc := U(φc) < 0, the equation has no stable critical point due to the shape of
the Whitney surface. However if Uc > 0, there are two critical points: a stable one
at Hc =
Uc/3 and an unstable one at Hc = −
Uc/3.
The value Uc = 0 is the bifurcation point. Provided this is also an extremal of
V , we necessarily have ∂V/∂H |c= −6Hc = 0, ∂V/∂φ |c= κU ′c = 0, and φ̇c = 0.
Thus, also the critical points of the Klein–Gordon equation for the scalar field are
involved. Hence, the Hubble parameter has to vanish and φc is a double zero of the
potential U . The Hessian of (4) takes the form
Hess(V ) =
0 κU ′′
. (15)
The sub–determinants of the Hessian are ∆0 = ∂
2V/∂H2 = −6 < 0 and ∆1 =
detHess(V ) = (∂2V/∂H2) (∂2V/∂φ2)−(∂2V/(∂H∂φ))2 = −6κU ′′. For a maximum
of the potential U , i.e. U ′ = 0 and U ′′ < 0, the function V possesses a maximum; a
minimum of the potential U , however, corresponds to a saddle point of V .
3.2. Reheating
Since U(φc) can be associated with the latent heat of the Universe in this epoch, we
could demonstrate:22
The critical points of the non–Morse potential W (H,φ) determine the evolution
in the inflationary phase. Along the minima and maxima Hc = ±
U(φc)/3, the in-
flaton moves from the slow–roll to the hot regime. The saddle points of W , i.e. more
precisely, the minima of V , determine the onset of reheating.
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6 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
3.3. Scale-invariant spectrum as a limiting point
It is now possible to relate the “slow–roll” condition, for the velocity of the infla-
tionary phase, to the critical points resulting from catastrophe theory. For inflation
(with ä > 0) the two “slow–roll parameters” are, in first order approximation,
ǫ = −g/H2 and η = −dg/d(H2) , (16)
where g = V (H,φ) is the “graceful exit function”. In this reduced dynamics,
they are effectively determined by the first and second derivatives of the re-
duced non–Morse function W (H), i.e., more precisely by ǫ = (1/H2)(∂W/∂H)
and η = (1/2H)∂2W/(∂H)2. They will also determine the density fluctuations of
the early Universe.
The deviation ∆ := (n− 1)/2 from the scale-invariant Harrison-Zeldovich spec-
trum with n = 1 leads in the first order slow-roll approximation to the differential
equation
− ǫ = ∆ (17)
ǫ = −∆+AH2 , V = ∆H2 −AH4 (18)
as solutions for the density and the graceful exit function, respectively.
Then the non-Morse potential turns out to be
W (H,φ) = −
V dH = −∆
H5 + C(φ) (19)
which together with (14) provides us with the reparametrized potential
κŨ(φ) =
H2 − A
H4 , (20)
which for ∆ ≃ 0 is positive in the leading order.
In the next order slow-roll approximation arises a nonlinear Abel equation13
with a continuous spectrum for n < 1 and a discrete ‘blue‘ one for n > 1, see Ref.57
for more details. This division of the spectrum persists in the next to second order
slow-roll approximation.36 More important is our finding that in both higher order
approximations the Harrison-Zeldovich spectrum with n = 1 stays a limiting point,
which agrees quite well with recent constraints from the observations of WMAP.20
4. Higher–order curvature Lagrangians via field redefinition
Nonlinear modifications of the Einstein–Hilbert action are of interest, among oth-
ers, for the following reasons: First, some quadratic models can be renormalized
when quantized, cf. Ref.38 Second, specific nonlinear Lagrangians have the prop-
erty that the field equations for the metric remain second order as in GR; these are
the so–called Lovelock actions which arise from dimensional reduction of the Eu-
ler characteristics, cf. Ref.47 In Yang’s theory of gravity, this topological invariant
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 7
induces an effective cosmological constant for instanton solutions residing in Ein-
stein spaces.33,38 At times,64 this renormalizable model is referred to as Yang-Mielke
theory of gravity.
As an instructive example of higher-derivative theories of gravity,52 we consider
the Lagrangian density
L = L(R)
| g | =
| g |, (21)
where β is a dimensionless coupling constant. Through the conformal change30
gαβ → g̃αβ = Ωgαβ with Ω = 2κ
, (22)
of the metric, this Lagrangian can be mapped to the usual Hilbert–Einstein La-
grangian with a particular self–interacting scalar field. Indeed, Starobinsky62 con-
sidered earlier such models for inflation.
The scalar field, the inflaton, will arise via
lnΩ (23)
from the nonlinear parts of a higher–order Lagrangian L = L(R) in the scalar cur-
vature R. Recently, such modified gravity models are considered as alternatives7,48
for dark matter or even dark energy.
4.1. Reparametrized Lagrangian
Instead of studying the resulting complicated nonlinear field equations of higher–
order curvature Lagrangians, we follow the equivalence proof of Ref.27 for the con-
formal frame (22). Then, our inflaton Lagrangian density (2) acquires the form
L = 1
| g |
RΩ− 2κΩ2Û(Ω)
, (24)
where Û(Ω) := U(φ(Ω)) = U
3/2κ lnΩ
is the reparametrized potential. Thus
in our approach, the inflaton or dark matter scalar will not be regarded as an
independent field, but is induced via (23) by the non–Einsteinian pieces of the
general Lagrangian L = L(R). Solving for the potential via the method of Helmholz,
we obtain
Û(Ω) = H(R)/Ω2 =
/Ω2 . (25)
If we identify the conformal factor with the field momentum via Ω = 2κdL/dR,
the bracket in (25) can be regarded35 as a Legendre transformation L → H(R) =
RdL/dR − L from the original Lagrangian (2) to the general nonlinear curvature
scalar Lagrangian L = L(R). Then, the parametric reconstruction
R = 2κ exp
2κ/3φ
2U(φ) +
, (26)
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8 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
Fig. 1. The qualitative form of a Whitney surface with local extrema in the case of a nonlinear
curvature Lagrangian L(R) depending on φ as control parameter.
L = exp
2κ/3φ
U(φ) +
of the higher–order effective Lagrangian L(R) from the self–interacting inflaton
potential U(φ) arises.4 Here the scalar field plays merely the role of a control pa-
rameter. The form of the Whitney surface with its local valleys and mountains is
qualitatively shown in Fig. 1.
5. Bifurcations with effective ‘dark energy’
In order to model self-interacting dark matter, let us consider the potential
U = m2|φ|2
1− χ|φ|4
, (28)
where m is the mass of an ultra-light scalar and χ the coupling constant of the
nonlinear self-interaction. It provides us with a solvable non-topological soliton type
model of dark matter halos43,44 even with toroidal substructures45 and a reasonable
approximation of the rotation curves of dark matter dominated galaxies. Moreover,
the predicted scaling relation11 fits almost ideally astronomical observations. In
Ref.54 we predicted the effects of such scalar field halos for microlensing.
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 9
5.1. Free field
In the string landscape,25 a triple unification of inflation, DM and DE suggests
itself based on a simple quadratic potential (28) with χ = 0 corresponding to a free
massive field. Then, our scalar field toy model likewise incorporates ‘dark energy’
in a rather novel way: The exact parametric solution of the equivalent nonlinear
Lagrangian L(R) for the ‘free’ field reads
R = 6m2xex(1 + x), (29)
xe2x(2 + x), (30)
where x := lnΩ under the reality condition Ω > 0.
-1 -0.5 0 0.5 1 1.5 2
Fig. 2. The swallow tail behavior of the Lagrangian L(R) for a massive ‘free’ field (χ = 0 and
κ = m = 1).
The dependence of L(R) given in Fig. 2 is rather surprising and represents the
bifurcation set of the swallow tail catastrophe2,21,23 associated with some higher
dimensional grand manifolds. The scalar field φ, its mass m as well as χ play the
role of control parameters. According to the theory of singularities (more widely
known as catastrophe theory, cf. Arnol’d2), this bifurcation set indicates that the
Lagrangian manifolds are associated with two local “minima” and one “maximum”
(and saddle points at the meeting points of the grand manifold). Each of the “min-
ima” merges with the “maximum” at the cuspoidal points A and B and then dis-
appears. The minima (the semi-infinite segments A and B) are characterized by a
positive second derivative of the Hamiltonian H(Ω) with respect to the momentum
Ω, i.e. d2H/dΩ2 > 0. They correspond to an effective Lagrangian with vanishing
cosmological constant.
For the “maximum”, i.e. the segment AB, this derivative is negative and the ef-
fective Lagrangian has a modified gravitational constant and a positive cosmological
constant,
κeff = eκ , Λeff = 3m
2/(2e) , (31)
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10 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
respectively.
Fig. 3. Whitney surface for the nonlinear curvature Lagrangian L(R) corresponding to the model
(28) with χ = 0.3 and m = κ = 1 depending on φ as control parameter. (φ in units of
3/(2κ)
5.2. Self-interacting scalar
In the nonlinear case, the exact parametric solution L(R) reads
R = 6m2xex
1 + x−
, (32)
2 + x− 27χ
x4 − 9χ
. (33)
The resulting extremal curve of the Whitney surface is drawn in Fig. 3, and its
projections L = L(R), R = R(φ), and L = L(φ) in Fig. 4.
For χ 6= 0, the bifurcation diagram in Fig. 5 is of higher rank than for the free
field. Near the center, we find the butterfly part of the catastrophe. But, for each
non-vanishing χ, there exists a further cusp far away from the center, for a large
negative R value and L close to zero. From this cusp, the curve finally returns to
the origin, for L values very close to zero so that it cannot be seen in Fig. 5 or 6.
In an enlargement of the additional cusp in Fig. 7, one can see that L(R) actually
becomes negative and forms a cusp.
The effective strength of gravity and the value of cosmological constant
κeff = κe
−x , Λeff = 3m
, (34)
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 11
Fig. 4. Projections of the Whitney surface in Fig. 3 for the nonlinear curvature Lagrangian L(R)
corresponding to a nonlinear scalar (28) with χ = 0.3 and m = κ = 1.
0 5 10 15 20 25 30
Fig. 5. The wigwam singularity for the nonlinear curvature Lagrangian L(R) for a NTS model
with χ = 0, 0.1 (long dashes), 0.2 (short dashes), 0.3 (dashed) and κ = m = 1; here only the
butterfly part near the center is shown.
respectively, depend now on both, the mass m and, via x as a solution of R =
0, on the coupling constant χ of the scalar field, and need to be constrained by
cosmological data.
In quintessence models,65 e.g., the crossover scale of the scalar field is |φc| =
exp(−1/α)M2
/m, where α = 1/138 is about Sommerfeld’s fine structure constant
and MPl = 1/
κ the reduced Planck mass. Then the tiny observed cosmological
constant of the present epoch
ΛDE = m
2φ2c = exp(−2/α)M4Pl ≃ (10−3eV)4 (35)
is roughly reproduced for small χ.
It is important to stress that within the range limited by cuspoidal points, all
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12 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
-20 -10 0 10 20 30
Fig. 6. For the NTS model with χ = 0.3 and κ = m = 1, the wigwam catastrophe of the
Lagrangian L(R) can be decomposed into a butterfly plus additional elementary catastrophe.
Here, the butterfly catastrophe arises near the origin; at about (R,L) = (−250,−0.015) there is
a further cusp, cf. Fig. 7; the curve finally returns to the origin which cannot be seen due to our
scale.
-240 -220 -200 -180 -160
-0.04
-0.02
Fig. 7. Continuation of Fig. 6. Enlarging the additional cusp for the nontopological soliton field
with χ = 0.3 and κ = m = 1.
three states may coexist with each other. Thus our bifurcation set, cf. Ref.55 for
a generalization with χ 6= 0, indicates that each local patch of the Universe may
have a different strength and cosmological constant (‘dark energy’) controlled by
the mass m of the scalar field, but the effective Lagrangian has approximately the
same Einsteinian form (1). In “maximal” domains, inflation may be still going on,
thus realizing prospective ideas of Linde.26
5.3. Natural inflation from the axion?
In QCD, after integrating out the fermion fields, its generating functional including
a topological Pontrjagin term for the Yang-Mills gauge fields induces an effective
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 13
axion potential
U = Λ4QCD [1− cos(φ/fφ)] . (36)
This potential displays a periodicity with a period of 2πfφ, has a minimum at
φ = 0, as required, and leads to the induced axion mass of mφ = Λ
QCD/fφ. A
quintaxion may also be induced by spacetime torsion.39
Within natural inflation,51 such a potential has been proposed for an axion
coupling constant close to the Planck scale. For simplicity let us assume in the
following that fφ ≃
3/(2κ). Then, following the general prescription (26)-(27),
we find the reparametrized solution
R = 2κ exp (x) Λ4QCD [2− 2 cos (x) + sin (x)] , (37)
L = exp (2x) Λ4QCD [1− cos (x) + sin (x)] . (38)
Fig. 8 exhibits a swallow tail behavior similar as for a NTS potential with χ = 0.
-1 -0.5 0 0.5 1 1.5 2
-15000
-10000
-5000
Fig. 8. The axion gives rise again to a swallow tail catastrophe in the Lagrangian L(R), with the
choice of ΛQCD = 0.1, κ = 1, and fφ = 1.
6. Domain structure of dark matter and induced dark energy
Baryonic particles (such as protons and neutrons) account for visible matter in the
Universe, that is only a small fraction of the observed total matter. The major part
of it is a mysterious DM, which only interacts gravitationally. On the other hand
according to our model, the Universe splits into almost Einsteinian domains with
– depending on the physical scale and eras – a different gravitational strength and
cosmological constant, the latter being most likely a representation of DE. A natural
question is how these domains arise? To answer this, we have to look into the initial
inflationary stage of the Universe, long before decoupling of radiation and matter,
even before grand unified phase transitions where only an initial pre-field existed.
At the initial stage of the quantum era, the fundamental pre-field may have existed
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14 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
in a form of a scalar or vector gauge fields. Without lack of generality we can limit
ourselves to a scalar field φ.
During this quantum era, the scalar field was strongly fluctuating. Due to the
rapid inflation, the size of the Universe as well as the size of each individual fluc-
tuation have increased enormously while the amplitude remains almost the same.
During this fast spacelike expansion, just after the stretching, these space fluctu-
ations of φ were frozen while loosing their quantum nature due to the increasing
size. The decoupling of radiation and visible matter has imprinted fluctuations on
the electromagnetic spectrum which we observe now in the anisotropy microwave
background radiation. However, we cannot see such ripples of the visible matter
after the decoupling. Due to the gravitational interaction the process of large scale
formation has started just after the decoupling. And the state of nearly homoge-
neously distributed visible matter existing after the decoupling has eventually been
transformed into clusters of galaxies and star structures which we see nowadays. Ob-
viously, the decoupling of the dark part of matter from radiation and visible matter
may have happen even earlier, when the Universe was in the form of a quark-gluon
plasma. However, after decoupling the dark part of matter must be subjected to
the same gravitational forces and to a similar large scale formation. Since its total
mass is significantly larger than the visible mass, the formation of large structures
of DM may have a weak influence on the visible matter.
Dark matter will have an opposite effect by dictating where and how the visible
matter should be distributed, probably localized around clusters of DM. In the case
of scalar field DM, its peculiarities such as self-interaction may also have contributed
to the DE distribution. DM cannot be seen directly by traditional observations but
can be inferred from more sophisticated gravitational lensing,28 revealing that DM
exists in a form of a loose network of filaments, growing over time, which intersect in
massive structures at the locations of clusters of galaxies. This finding is consistent
with the conventional theory of large scale structure formation, where a smooth
distribution of DM collapses first into filaments and then into clusters, forming a
cosmic scaffold,28 then accumulating visible matter and later newly born stars.
The primary candidate for DM is a scalar field. If a scalar field exists, then it
may have different amplitudes or different mass densities in different space regions.
Such a distribution of φ has been created after the inflation of the Universe. It
is associated with different amplitudes of the frozen fluctuations. The process of
large scale structure formation is rather similar in each of these regions. Our re-
sults indicate that in each of these different regions the gravitational constant may
vary,55 depending on the amplitude of the scalar field of the given frozen space
fluctuation. In other words, in each of these regions defined by the original frozen
fluctuations there will arise different Newtonian potentials. This is consistent with
recent studies9 showing that, during expansion, a vector field having two different
initial amplitudes will give rise to different Newtonian potentials. This difference
in turn drives self-consistently enhanced growth of the density perturbations which
enables tiny perturbations to grow into the large structures we see today.
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 15
6.1. Cosmic domains
Our studies indicate that these different Newtonian potentials correspond to differ-
ent effective gravitational constants in various regions of the Universe. If this is true,
the data of recent observations28 should be reinvestigated. In regions with larger
gravitational constants one may see that the gravitational lensing will be stronger
and vice versa. We regard areas with different gravitational potentials as cosmic do-
mains characterized by their own gravitational and cosmological constants. There
occurs a coexistence of these domains (which primordially may be of topological
origin18,19). On the boundaries of the domains, changes in the gravitational and
cosmological constants are expected to be abrupt. In spite of the random or fluctu-
ational origin of these domains, the values of these gravitational and cosmological
constants take only a few universal numbers, corresponding to only a finite number
of types of domains.
For small curvature, each of these states can be described approximately by the
same effective Lagrangian (1) with different gravitational constant κeff and cosmo-
logical constant Λeff and emerge as a fixed point of the conformal transformation
from one side and universal classes of the smooth differential mappings from the
other side.21 These spaces are approximately Einstein spaces, but of different grav-
itational strength as well as with different cosmological constants. The distribution
of such cosmological constants in the Universe corresponds to DE induced by the
scalar field distribution. On a very large scale, φ is homogeneously distributed and
therefore we expect that DE will be distributed homogeneously, depending on the
details of inflationary cosmology.
6.2. Domains arising from free fields
If DM and DE were associated with a massive free scalar field gravitationally coupled
or/and associated with axions (see, Sec. 5.3) there will arise three types of domains:
1) The first type of domains, I, is described by Einstein‘s theory with conven-
tional gravitational constant and vanishing cosmological constant.
2) The second type of domains, II, is described by an Einsteinian model having a
very large gravitational constant and a vanishing cosmological constant. Such type
of domains may be gravitationally unstable, leading eventually to their contraction
and ultimately to a gravitational collapse and phenomena resembling supernova ex-
plosions. For very large masses, we may expect that these explosion will be stronger
than standard supernovae. Our approach may explain the fact why some supernovae
progenitors seem to have exceeded15 the Chandrasekhar limit.
3) The third type of domains, III, are described by an Einsteinian model with
large gravitational and positive cosmological constant ΛIIIeff given by (34). Then we
expect that such domains are still expanding according to effective Friedman equa-
tions and probably their sizes are also increasing. Such expansion may be ended
abruptly and such domains will be transformed into domains of type I.
One may speculate that the largest part of the Universe is occupied by domains
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16 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
of type I, corresponding to spaces with zero cosmological constant. The next largest
proportion of the Universe is expected to be occupied by domains of type III. Those
parts of the Universe occupied by domains of type II are decreasing with time. Each
disappearance of such domain would be accompanied by supernova type explosions.
On the average, the cosmological constant inferred from recent observations is
ΛDE =
VIIIΛ
VI + VII + VIII
, (39)
where the spacelike volumes VI, VII and VIII are associated with domains of type I,
II, and III, respectively.
In summary, our bifurcation set (see Fig. 1 and Fig. 2) indicates that the Universe
is locally described by Einstein’s GR and effectively split into domains. The splitting
originated from primordial quantum fluctuations which were frozen during inflation.
Each domain or local patch of the Universe may have different gravitational strength
and may have zero or negative pressure associated with DE. In domains of type II
associated with a positive cosmological constant the inflation may be still going
on. It will be stopped exactly at the bifurcation point A and this domain will be
transformed into a domain of type I, as in Fig. 2. For a massive free scalar field and
for axions the bifurcation set is the swallow tail catastrophe. There are two cusps
at the points A and B which are associated with the highest singularities of the
differential mappings. They are also related to the domain boundaries. In each of
these cuspoidal points A and B of the bifurcation diagram, the minimum of some
grand manifold merges with the maximum.22 Each local minimum (the segment A
and the semi-infinite segment B) is associated with two types of domains, I and
II, respectively. Each local maximum (saddle) of the grand manifold is associated
with an expanding domain of type III (the segment AB). The effective Einsteinian
gravity for this domain is stronger than that in the domain of type I, depending on
the mass of the scalar field. Domains with a positive effective cosmological constant
are still in an expanding inflationary phase.
The ‘strong’ gravity state, the domain II (the semi-infinite segment B) with a
negative or vanishing cosmological constant corresponds to deflation. Since the Uni-
verse is split into dynamical regions associated with different gravity, their bound-
aries change with time and present some kind of cosmological strings or membranes.
Changes of the boundary are associated with some sort of local phase transition re-
minding us of phase separations like those arising in first order phase transitions. At
some instant, all these domains are in quasi-equilibrium with each other although
their boundaries changes with time as the membranes are moving. At very long
time scales (much larger than the Planck time) when inflation in the domains of
type III will eventually be ceased, only stable phases, like the domains of type I
and some small concentration of domains of type II will remain. These domains
would produce explosions resembling supernovae with masses well beyond of the
Chandrasekhar limit, similar to those observed.15
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 17
6.3. Domains arising from self-interacting fields
In case of self-interacting scalar fields with χ 6= 0, the classification will be richer
(see Figures 4, 5 and 6), with five types of domains. The bifurcation diagram as-
sociated with such a wigwam catastrophe or A5 singularity according to Arnold’s
classification2 consists of four cusps or four A2 singularities, with a dramatic change
in the gravitational and cosmological constants.
There occurs a general decoupling of the wigwam catastrophe A5 into a butterfly
catastrophe A4, and the elementary cusp A2, depending on the values of R: If the
butterfly catastrophe arises near the origin or at small values of R, the additional
cuspoidal point is associated with a very large value of R ≃ −250. For some fixed
values of the coupling constant χ = 0.1, 0.2, 0.3, the butterfly part of bifurcation
diagram is presented in Figs. 5. In comparison with the free massive field, the in-
clusion of the self-interaction associated with the nonzero parameter χ gives rise to
two extra cusps leading to the appearance of two extra branches in the bifurcation
diagram, again associated with effective Einsteinian spaces. Although these almost
linear branches do exist at positive R, their major part is related to negative val-
ues of R. For one of these branches the cosmological constant vanishes while for
the other one the cosmological constant decreases when the value of χ increases,
and the part of the branch associated with R > 0 decreases. In general, the self-
interaction induced by the field φ increases the strength of gravity and decreases
the cosmological constant.
Let us describe in more detail the transitions in the bifurcations: Starting from
the first branch, denoted as I, through the first cuspoidal point we arrive at the
second branch II which is very similar to the case of a free massive scalar field,
although its length decreases significantly when χ increases and the cosmological
constant vanishes at any value of χ. This branch corresponds to the limit φ → 0
of small scalar fields, corresponding to the conformal factor Ω = 1. Therefore, on
this branch we recover the linear Hilbert-Einstein Lagrangian LHE = R/2κ, as ex-
pected. With decreasing Ω, we will come to the next cuspoidal point, where the new
branch III has a positive cosmological constant and gravity becomes stronger. With
increasing value of χ the length of this branch increases. Thus, there will be more
domains having positive cosmological constant and a gravitational strength higher
than the conventional. By decreasing Ω until reaching the next cuspoidal point of
the bifurcation diagram, we arrive at the branch IV. The spacetime associated with
this branch has a very strong gravity and negative cosmological constant. Similar
to the free massive case, this branch probably is gravitationally unstable. Finally,
for very small Ω, we will arrive at the branch V with R < 0, where gravity is the
strongest but without cosmological constant. Similar to the previous branch IV, the
domain V is unstable, as well. Thus in a Universe filled with self-interacting scalar
fields, we would expect that a rather strong phase separation into further domains
will arise.
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18 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
7. Discussion
For dilute DM the Lagrangian has the standard Hilbert-Einstein form LHE = R/2κ.
For dense DM, i.e. large values of the DM scalar, the resulting effective Lagrangian
is LHE = R/2κeff, only the slope is less steep, i.e. the effective gravitational coupling
κeff > κ is, by “renormalization”,
3,65 larger than Newton’s. This resembles some
aspects of MOND, but with the advantage that we still work in the standard general
relativistic framework. Depending on its sign, the cosmological constant Λeff in a
bifurcation approximated for small R by (1) can also model DE or an accelerating
phase of the present epoch of the Universe (like ‘anti-gravity’). There also may
arise “gravitational screening” for κeff smaller than Newton’s. MACHOs can now
be described by some specific nonlinear modifications of the Hilbert-Einstein action.
In the light of our finding we have to comment on recent observations,29 which
constrain the density of MACHO type objects in the Universe by measuring the
brightness of high redshift type Ia supernovae relative to low redshift samples. These
data favor DM made of microscopic particles (such as WIMPs) or scalar and vector
gauge fields over MACHOs with masses between 10−2 and 1010 solar masses. This
provides another evidence that our approach correctly describes the frozen space
fluctuations which lead to a formation of cosmic domains with a different density
of DM and DE in the Universe.
Acknowledgments
We would like to thank Burkhard Fuchs, Humberto Peralta, and Konstantin Zioutas for helpful
discussions. One of us (EWM) thanks Noelia, Markus Gérard Erik, and Miryam Sophie Naomi for
encouragement.
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http://arxiv.org/abs/astro-ph/9802258
http://arxiv.org/abs/astro-ph/0603449

  A possible equivalence of scalar dark matter, the inflaton, and modified
gravity is analyzed. After a conformal mapping, the dependence of the effective
Lagrangian on the curvature is not only singular but also bifurcates into
several almost Einsteinian spaces, distinguished only by a different effective
gravitational strength and cosmological constant. A swallow tail catastrophe in
the bifurcation set indicates the possibility for the coexistence of different
Einsteinian domains in our Universe. This `triple unification' may shed new
light on the nature and large scale distribution not only of dark matter but
also on `dark energy', regarded as an effective cosmological constant, and
inflation.

Introduction
The origin of the principal constituents of the Universe in the form of dark matter
(DM) and dark energy (DE) (or rather dark tension due to its negative pressure),
remains a major puzzle of modern cosmology and particle physics. Ad hoc proposals
like modifying Newton’s law of gravity as in Milgrom’s MOND (Modified Newtonian
Dynamics) are difficult to reconcile with relativity, or need a phase coupling to
a complex scalar.3 The particle physics’ view is to leave gravity untouched but
postulate WIMPs (Weakly Interacting Massive Particles) like axions, dilatons or
neutralinos as dark matter candidates.66
The dominant non-visible “dark” fraction of the total energy of the Universe is
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2 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
known to exist from its gravitational effects. Since the dark matter part is distributed
over rather large distances, its interaction, including a possible self-interaction,60
must be weak. The main candidates for such weakly interacting particles are the
(universal) axion or the lightest supersymmetric particle, as the neutralino in most
models, cf. Ref.66
Recently, an observed excess of diffuse gamma rays has been attributed8 to the
annihilation of DM in our Galaxy. The flux of the presumed neutralino annihilation
allows a reconstruction of the distribution of DM in our Galaxy. Most probable is a
pseudo-isothermal profile but with a substructure of two doughnut-shape rings in
the galactic plane. It is believed that these transient substructures have their origin
in the hierarchical clustering of DM into galaxies. However, there are reservations
concerning the internal consistency in the interpretation of the observations: Given
the same normalization for the cross section of the DM particle, it appears unlikely1
that DM annihilation is the main source of the extragalactic gamma-ray background
(EGB). Moreover, then the large accompanied antiproton flux needs to meet several
constraints.5
On the other hand, heterotic string theory provides a very light universal axion
which may avoid12 the strong CP problem in quantum chromodynamics (QCD).50
Given the existence of such almost massless (pseudo-)scalars, it has been speculated
that a coherent non-topological soliton (NTS) type solution of a nonlinear Klein-
Gordon equation can account for the observed halo structure, simulating a Bose-
Einstein condensate of astronomical size;16,17,34 cf. Ref.53 In particular, a Φ6 toy
model43 yields exact Emden type solutions in flat spacetime, including a flattening37
of halos with ellipticity e < 1 as observed via microlensing.14,54
Both views are to some extend physically equivalent, i.e., scalar dark matter
minimally coupled to Einstein’s general relativity (GR) is equivalent4,27,49,55,59 to a
modified gravity in the relativistic framework of higher-order curvature Lagrangians.
Such effective Lagrangians may also arise from the low-energy limit of (super-)
strings. The model proposed by Carroll et al.6 also uses such an equivalence of a
nonlinear higher-order curvature Lagrangian, in order to explain the present cos-
mic acceleration, but takes resort to an 1/R type curvature Lagrangian, which is
unbounded for weak gravitational fields.
Quite generally, the dependence of the effective Lagrangian L on the scalar
curvature R is singular and cannot always be represented analytically in the (R,L)
plane. More stunning is our recent finding55 that, although nonlinear, a L = L(R)
type Lagrangian bifurcates in several branches of almost linear Lagrangians
2κeff
(R− 2Λeff) (1)
distinguished only by the effective gravitational constant κeff and a cosmological
constant Λeff .
This indicates a unifying picture of dark matter, ‘dark energy’53 with an equation
of state parameter wDE = p/ρ ≃ −1, and modified gravity which may account, on
different scales, for inflation,37,40 dark matter halos36,43 of galaxies or even dark
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 3
matter condensations, the so-called boson stars23,24,41,58 as candidates of MACHOs
(Massive Compact Halo Objects). The ‘landscape’ view25 of (super-)strings suggests
also such a triple unification based on coherent, possibly oscillating scalar field with
a mass larger than the Hubble scale at the present epoch, i.e. m > H∗ = 10
−23 eV.
2. Gravitationally coupled scalar fields
Conventionally, dark matter and inflation can be modeled by a real (or complex)
scalar field φ with self-interaction U(|φ|2) minimally coupled to gravity with the
Lagrangian density
LDM =
| g |
gµν(∂µφ
∗)(∂νφ)− 2U(|φ|2)
, (2)
where κ = 8πG is the gravitational constant in natural units, g the determinant of
the metric gµν , and R the scalar curvature of Riemannian spacetime with Tolman’s
sign conventions.63 A constant potential U0 = Λ/κ would simulate the cosmological
constant Λ.
In a wide range of inflationary models, the underlying dynamics is simply that
of a single scalar field — the inflaton — rolling in some underlying potential. This
scenario invented by Linde26 is generically referred to as chaotic inflation due to its
choice of initial conditions. Many superficially more complicated models can also
be rewritten in this framework.
3. General metric of a flat inflationary universe
A spatially flat (k = 0) Friedman-Robertson-Walker (FRW) Universe with metric
ds2 = dt2 − a2(t)
dr2 + r2
dθ2 + sin2 θdϕ2
is nowadays favored by observations.61 Its temporal evolution of the generic model
(2) is determined by the two autonomous first order equations
Ḣ = −3H2(1 + wDE) = κU(φ)− 3H2 =: V (H,φ) , (4)
φ̇ = ±
3H2 − κU(φ) = ±
V (H,φ) , (5)
where H = ȧ(t)/a(t) is the Hubble expansion rate.
The function V (H,φ) will turn to be the “height function” in Morse theory.46
Observe that V ≤ 0 in order to avoid scalar ghosts. For the FRW metric, the
Lagrangian density (2) reduces to
L = − 3
ȧ2a+
φ̇2 − U(φ)
a3 . (6)
Since the shift function is normalized to one for the FRW metric, the canonical
momenta are given by P = ∂L/∂ȧ = −6Ha2/κ and π = ∂L/∂φ̇ = a3φ̇. The
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4 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
Hamiltonian or “energy function” is given by
φ̇2 +
V (H,φ)
a3 ∼= 0 (7)
and, due to (5), vanishes for all solutions, which is a familiar constraint in GR.
Using the Hubble expansion rate H := ȧ(t)/a(t) as the new “inverse time”
coordinate
t = t(H) =
κŨ − 3H2
, (8)
we were able to find the general solution:56
a = a(H) = a0 exp
κŨ − 3H2
. (9)
ds2 =
κŨ − 3H2
)2 − a0
2 exp
κŨ − 3H2
dr2 + r2
dθ2 + sin2 θdϕ2
, (10)
φ = φ(H) = ∓
3H2 − κŨ
, (11)
where Ũ = Ũ(H) := U(φ(t(H))) is the reparametrized inflationary potential.
The singular case Ũ = 3H2/κ with wDE = −1 leads in (4) to the de Sitter
inflation with exponential expansion a(t) = a0 exp(
Λ/3 t), which for a later time
becomes physically unrealistic, since the inflationary expansion eventually needs to
merge into the usual Friedman cosmos. Therefore, in explicit models we use the
ansatz
Ũ(H) =
, (12)
for the potential, where g(H) = V (H,φ(t(h))) is a nonzero function which should
provide the graceful exit from the inflationary phase to the Friedman cosmos. In
order to have a positive acceleration during the inflationary phase, but on the other
hand a real scalar field in (11), its allowed range is −H2 < g < 0. This H–formalism
facilitates considerably the reconstruction of the inflaton potential.36,40,56 In second
order this is based on an Abel equation13 for the primordial density ǫ = −g/H2.
3.1. Classification by catastrophe theory
A general classification of all allowed inflationary potentials and scenarios has al-
ready been achieved by Kusmartsev et al.22 via the application of catastrophe theory
to the Hamilton–Jacobi type equations (4) and (5) regarded as an autonomous non-
linear system.
In phase space, the equilibrium states are given by the constraint {Ḣ, φ̇} = 0.
The critical or equilibrium points, respectively, of this system are determined by
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 5
V (Hc, φc) = 0. This constraint is globally fulfilled by κU(φ) = 3H
Λ, where the
Hubble expansion rate is constant, i.e. HΛ =:
Λ/3 corresponding to wDE = −1.
For φ̇ = 0 and Λ 6= 0, we recover the de Sitter inflation.
In general, the Jacobi matrix
−6H κU ′
±6H(−2κV )−1/2 ∓κU ′(−2κV )−1/2
, (13)
of the system (4) and (5) depends crucially on U ′ = dU/dφ. Since its determinant
vanishes, i.e. det J = 0, the system is always degenerate. For the analysis of stability,
it suffices therefore to consider only (4) and to reconstruct φ later. Since H and φ
are independent variables, we can introduce the non–Morse superpotentialW (H,φ),
defined via V := −∂W/∂H , and analyze the system with the aid of catastrophe
theory. From (4) we obtain
W (H,φ) = H3 − κU(φ)H + C(φ) , (14)
where C is an arbitrary function of φ. The function W is already in canonical
form in H–space, and belongs to a Whitney surface2,21 or to the Arnold singularity
class A2. The corresponding Whitney surface has only one control parameter, the
potential U . Thus, an evolution of critical points is determined via the values of the
potential U . Let us analyze the types of critical points at different fixed values of
the control parameter U .
If Uc := U(φc) < 0, the equation has no stable critical point due to the shape of
the Whitney surface. However if Uc > 0, there are two critical points: a stable one
at Hc =
Uc/3 and an unstable one at Hc = −
Uc/3.
The value Uc = 0 is the bifurcation point. Provided this is also an extremal of
V , we necessarily have ∂V/∂H |c= −6Hc = 0, ∂V/∂φ |c= κU ′c = 0, and φ̇c = 0.
Thus, also the critical points of the Klein–Gordon equation for the scalar field are
involved. Hence, the Hubble parameter has to vanish and φc is a double zero of the
potential U . The Hessian of (4) takes the form
Hess(V ) =
0 κU ′′
. (15)
The sub–determinants of the Hessian are ∆0 = ∂
2V/∂H2 = −6 < 0 and ∆1 =
detHess(V ) = (∂2V/∂H2) (∂2V/∂φ2)−(∂2V/(∂H∂φ))2 = −6κU ′′. For a maximum
of the potential U , i.e. U ′ = 0 and U ′′ < 0, the function V possesses a maximum; a
minimum of the potential U , however, corresponds to a saddle point of V .
3.2. Reheating
Since U(φc) can be associated with the latent heat of the Universe in this epoch, we
could demonstrate:22
The critical points of the non–Morse potential W (H,φ) determine the evolution
in the inflationary phase. Along the minima and maxima Hc = ±
U(φc)/3, the in-
flaton moves from the slow–roll to the hot regime. The saddle points of W , i.e. more
precisely, the minima of V , determine the onset of reheating.
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6 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
3.3. Scale-invariant spectrum as a limiting point
It is now possible to relate the “slow–roll” condition, for the velocity of the infla-
tionary phase, to the critical points resulting from catastrophe theory. For inflation
(with ä > 0) the two “slow–roll parameters” are, in first order approximation,
ǫ = −g/H2 and η = −dg/d(H2) , (16)
where g = V (H,φ) is the “graceful exit function”. In this reduced dynamics,
they are effectively determined by the first and second derivatives of the re-
duced non–Morse function W (H), i.e., more precisely by ǫ = (1/H2)(∂W/∂H)
and η = (1/2H)∂2W/(∂H)2. They will also determine the density fluctuations of
the early Universe.
The deviation ∆ := (n− 1)/2 from the scale-invariant Harrison-Zeldovich spec-
trum with n = 1 leads in the first order slow-roll approximation to the differential
equation
− ǫ = ∆ (17)
ǫ = −∆+AH2 , V = ∆H2 −AH4 (18)
as solutions for the density and the graceful exit function, respectively.
Then the non-Morse potential turns out to be
W (H,φ) = −
V dH = −∆
H5 + C(φ) (19)
which together with (14) provides us with the reparametrized potential
κŨ(φ) =
H2 − A
H4 , (20)
which for ∆ ≃ 0 is positive in the leading order.
In the next order slow-roll approximation arises a nonlinear Abel equation13
with a continuous spectrum for n < 1 and a discrete ‘blue‘ one for n > 1, see Ref.57
for more details. This division of the spectrum persists in the next to second order
slow-roll approximation.36 More important is our finding that in both higher order
approximations the Harrison-Zeldovich spectrum with n = 1 stays a limiting point,
which agrees quite well with recent constraints from the observations of WMAP.20
4. Higher–order curvature Lagrangians via field redefinition
Nonlinear modifications of the Einstein–Hilbert action are of interest, among oth-
ers, for the following reasons: First, some quadratic models can be renormalized
when quantized, cf. Ref.38 Second, specific nonlinear Lagrangians have the prop-
erty that the field equations for the metric remain second order as in GR; these are
the so–called Lovelock actions which arise from dimensional reduction of the Eu-
ler characteristics, cf. Ref.47 In Yang’s theory of gravity, this topological invariant
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 7
induces an effective cosmological constant for instanton solutions residing in Ein-
stein spaces.33,38 At times,64 this renormalizable model is referred to as Yang-Mielke
theory of gravity.
As an instructive example of higher-derivative theories of gravity,52 we consider
the Lagrangian density
L = L(R)
| g | =
| g |, (21)
where β is a dimensionless coupling constant. Through the conformal change30
gαβ → g̃αβ = Ωgαβ with Ω = 2κ
, (22)
of the metric, this Lagrangian can be mapped to the usual Hilbert–Einstein La-
grangian with a particular self–interacting scalar field. Indeed, Starobinsky62 con-
sidered earlier such models for inflation.
The scalar field, the inflaton, will arise via
lnΩ (23)
from the nonlinear parts of a higher–order Lagrangian L = L(R) in the scalar cur-
vature R. Recently, such modified gravity models are considered as alternatives7,48
for dark matter or even dark energy.
4.1. Reparametrized Lagrangian
Instead of studying the resulting complicated nonlinear field equations of higher–
order curvature Lagrangians, we follow the equivalence proof of Ref.27 for the con-
formal frame (22). Then, our inflaton Lagrangian density (2) acquires the form
L = 1
| g |
RΩ− 2κΩ2Û(Ω)
, (24)
where Û(Ω) := U(φ(Ω)) = U
3/2κ lnΩ
is the reparametrized potential. Thus
in our approach, the inflaton or dark matter scalar will not be regarded as an
independent field, but is induced via (23) by the non–Einsteinian pieces of the
general Lagrangian L = L(R). Solving for the potential via the method of Helmholz,
we obtain
Û(Ω) = H(R)/Ω2 =
/Ω2 . (25)
If we identify the conformal factor with the field momentum via Ω = 2κdL/dR,
the bracket in (25) can be regarded35 as a Legendre transformation L → H(R) =
RdL/dR − L from the original Lagrangian (2) to the general nonlinear curvature
scalar Lagrangian L = L(R). Then, the parametric reconstruction
R = 2κ exp
2κ/3φ
2U(φ) +
, (26)
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8 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
Fig. 1. The qualitative form of a Whitney surface with local extrema in the case of a nonlinear
curvature Lagrangian L(R) depending on φ as control parameter.
L = exp
2κ/3φ
U(φ) +
of the higher–order effective Lagrangian L(R) from the self–interacting inflaton
potential U(φ) arises.4 Here the scalar field plays merely the role of a control pa-
rameter. The form of the Whitney surface with its local valleys and mountains is
qualitatively shown in Fig. 1.
5. Bifurcations with effective ‘dark energy’
In order to model self-interacting dark matter, let us consider the potential
U = m2|φ|2
1− χ|φ|4
, (28)
where m is the mass of an ultra-light scalar and χ the coupling constant of the
nonlinear self-interaction. It provides us with a solvable non-topological soliton type
model of dark matter halos43,44 even with toroidal substructures45 and a reasonable
approximation of the rotation curves of dark matter dominated galaxies. Moreover,
the predicted scaling relation11 fits almost ideally astronomical observations. In
Ref.54 we predicted the effects of such scalar field halos for microlensing.
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 9
5.1. Free field
In the string landscape,25 a triple unification of inflation, DM and DE suggests
itself based on a simple quadratic potential (28) with χ = 0 corresponding to a free
massive field. Then, our scalar field toy model likewise incorporates ‘dark energy’
in a rather novel way: The exact parametric solution of the equivalent nonlinear
Lagrangian L(R) for the ‘free’ field reads
R = 6m2xex(1 + x), (29)
xe2x(2 + x), (30)
where x := lnΩ under the reality condition Ω > 0.
-1 -0.5 0 0.5 1 1.5 2
Fig. 2. The swallow tail behavior of the Lagrangian L(R) for a massive ‘free’ field (χ = 0 and
κ = m = 1).
The dependence of L(R) given in Fig. 2 is rather surprising and represents the
bifurcation set of the swallow tail catastrophe2,21,23 associated with some higher
dimensional grand manifolds. The scalar field φ, its mass m as well as χ play the
role of control parameters. According to the theory of singularities (more widely
known as catastrophe theory, cf. Arnol’d2), this bifurcation set indicates that the
Lagrangian manifolds are associated with two local “minima” and one “maximum”
(and saddle points at the meeting points of the grand manifold). Each of the “min-
ima” merges with the “maximum” at the cuspoidal points A and B and then dis-
appears. The minima (the semi-infinite segments A and B) are characterized by a
positive second derivative of the Hamiltonian H(Ω) with respect to the momentum
Ω, i.e. d2H/dΩ2 > 0. They correspond to an effective Lagrangian with vanishing
cosmological constant.
For the “maximum”, i.e. the segment AB, this derivative is negative and the ef-
fective Lagrangian has a modified gravitational constant and a positive cosmological
constant,
κeff = eκ , Λeff = 3m
2/(2e) , (31)
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10 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
respectively.
Fig. 3. Whitney surface for the nonlinear curvature Lagrangian L(R) corresponding to the model
(28) with χ = 0.3 and m = κ = 1 depending on φ as control parameter. (φ in units of
3/(2κ)
5.2. Self-interacting scalar
In the nonlinear case, the exact parametric solution L(R) reads
R = 6m2xex
1 + x−
, (32)
2 + x− 27χ
x4 − 9χ
. (33)
The resulting extremal curve of the Whitney surface is drawn in Fig. 3, and its
projections L = L(R), R = R(φ), and L = L(φ) in Fig. 4.
For χ 6= 0, the bifurcation diagram in Fig. 5 is of higher rank than for the free
field. Near the center, we find the butterfly part of the catastrophe. But, for each
non-vanishing χ, there exists a further cusp far away from the center, for a large
negative R value and L close to zero. From this cusp, the curve finally returns to
the origin, for L values very close to zero so that it cannot be seen in Fig. 5 or 6.
In an enlargement of the additional cusp in Fig. 7, one can see that L(R) actually
becomes negative and forms a cusp.
The effective strength of gravity and the value of cosmological constant
κeff = κe
−x , Λeff = 3m
, (34)
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 11
Fig. 4. Projections of the Whitney surface in Fig. 3 for the nonlinear curvature Lagrangian L(R)
corresponding to a nonlinear scalar (28) with χ = 0.3 and m = κ = 1.
0 5 10 15 20 25 30
Fig. 5. The wigwam singularity for the nonlinear curvature Lagrangian L(R) for a NTS model
with χ = 0, 0.1 (long dashes), 0.2 (short dashes), 0.3 (dashed) and κ = m = 1; here only the
butterfly part near the center is shown.
respectively, depend now on both, the mass m and, via x as a solution of R =
0, on the coupling constant χ of the scalar field, and need to be constrained by
cosmological data.
In quintessence models,65 e.g., the crossover scale of the scalar field is |φc| =
exp(−1/α)M2
/m, where α = 1/138 is about Sommerfeld’s fine structure constant
and MPl = 1/
κ the reduced Planck mass. Then the tiny observed cosmological
constant of the present epoch
ΛDE = m
2φ2c = exp(−2/α)M4Pl ≃ (10−3eV)4 (35)
is roughly reproduced for small χ.
It is important to stress that within the range limited by cuspoidal points, all
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12 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
-20 -10 0 10 20 30
Fig. 6. For the NTS model with χ = 0.3 and κ = m = 1, the wigwam catastrophe of the
Lagrangian L(R) can be decomposed into a butterfly plus additional elementary catastrophe.
Here, the butterfly catastrophe arises near the origin; at about (R,L) = (−250,−0.015) there is
a further cusp, cf. Fig. 7; the curve finally returns to the origin which cannot be seen due to our
scale.
-240 -220 -200 -180 -160
-0.04
-0.02
Fig. 7. Continuation of Fig. 6. Enlarging the additional cusp for the nontopological soliton field
with χ = 0.3 and κ = m = 1.
three states may coexist with each other. Thus our bifurcation set, cf. Ref.55 for
a generalization with χ 6= 0, indicates that each local patch of the Universe may
have a different strength and cosmological constant (‘dark energy’) controlled by
the mass m of the scalar field, but the effective Lagrangian has approximately the
same Einsteinian form (1). In “maximal” domains, inflation may be still going on,
thus realizing prospective ideas of Linde.26
5.3. Natural inflation from the axion?
In QCD, after integrating out the fermion fields, its generating functional including
a topological Pontrjagin term for the Yang-Mills gauge fields induces an effective
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 13
axion potential
U = Λ4QCD [1− cos(φ/fφ)] . (36)
This potential displays a periodicity with a period of 2πfφ, has a minimum at
φ = 0, as required, and leads to the induced axion mass of mφ = Λ
QCD/fφ. A
quintaxion may also be induced by spacetime torsion.39
Within natural inflation,51 such a potential has been proposed for an axion
coupling constant close to the Planck scale. For simplicity let us assume in the
following that fφ ≃
3/(2κ). Then, following the general prescription (26)-(27),
we find the reparametrized solution
R = 2κ exp (x) Λ4QCD [2− 2 cos (x) + sin (x)] , (37)
L = exp (2x) Λ4QCD [1− cos (x) + sin (x)] . (38)
Fig. 8 exhibits a swallow tail behavior similar as for a NTS potential with χ = 0.
-1 -0.5 0 0.5 1 1.5 2
-15000
-10000
-5000
Fig. 8. The axion gives rise again to a swallow tail catastrophe in the Lagrangian L(R), with the
choice of ΛQCD = 0.1, κ = 1, and fφ = 1.
6. Domain structure of dark matter and induced dark energy
Baryonic particles (such as protons and neutrons) account for visible matter in the
Universe, that is only a small fraction of the observed total matter. The major part
of it is a mysterious DM, which only interacts gravitationally. On the other hand
according to our model, the Universe splits into almost Einsteinian domains with
– depending on the physical scale and eras – a different gravitational strength and
cosmological constant, the latter being most likely a representation of DE. A natural
question is how these domains arise? To answer this, we have to look into the initial
inflationary stage of the Universe, long before decoupling of radiation and matter,
even before grand unified phase transitions where only an initial pre-field existed.
At the initial stage of the quantum era, the fundamental pre-field may have existed
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14 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
in a form of a scalar or vector gauge fields. Without lack of generality we can limit
ourselves to a scalar field φ.
During this quantum era, the scalar field was strongly fluctuating. Due to the
rapid inflation, the size of the Universe as well as the size of each individual fluc-
tuation have increased enormously while the amplitude remains almost the same.
During this fast spacelike expansion, just after the stretching, these space fluctu-
ations of φ were frozen while loosing their quantum nature due to the increasing
size. The decoupling of radiation and visible matter has imprinted fluctuations on
the electromagnetic spectrum which we observe now in the anisotropy microwave
background radiation. However, we cannot see such ripples of the visible matter
after the decoupling. Due to the gravitational interaction the process of large scale
formation has started just after the decoupling. And the state of nearly homoge-
neously distributed visible matter existing after the decoupling has eventually been
transformed into clusters of galaxies and star structures which we see nowadays. Ob-
viously, the decoupling of the dark part of matter from radiation and visible matter
may have happen even earlier, when the Universe was in the form of a quark-gluon
plasma. However, after decoupling the dark part of matter must be subjected to
the same gravitational forces and to a similar large scale formation. Since its total
mass is significantly larger than the visible mass, the formation of large structures
of DM may have a weak influence on the visible matter.
Dark matter will have an opposite effect by dictating where and how the visible
matter should be distributed, probably localized around clusters of DM. In the case
of scalar field DM, its peculiarities such as self-interaction may also have contributed
to the DE distribution. DM cannot be seen directly by traditional observations but
can be inferred from more sophisticated gravitational lensing,28 revealing that DM
exists in a form of a loose network of filaments, growing over time, which intersect in
massive structures at the locations of clusters of galaxies. This finding is consistent
with the conventional theory of large scale structure formation, where a smooth
distribution of DM collapses first into filaments and then into clusters, forming a
cosmic scaffold,28 then accumulating visible matter and later newly born stars.
The primary candidate for DM is a scalar field. If a scalar field exists, then it
may have different amplitudes or different mass densities in different space regions.
Such a distribution of φ has been created after the inflation of the Universe. It
is associated with different amplitudes of the frozen fluctuations. The process of
large scale structure formation is rather similar in each of these regions. Our re-
sults indicate that in each of these different regions the gravitational constant may
vary,55 depending on the amplitude of the scalar field of the given frozen space
fluctuation. In other words, in each of these regions defined by the original frozen
fluctuations there will arise different Newtonian potentials. This is consistent with
recent studies9 showing that, during expansion, a vector field having two different
initial amplitudes will give rise to different Newtonian potentials. This difference
in turn drives self-consistently enhanced growth of the density perturbations which
enables tiny perturbations to grow into the large structures we see today.
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 15
6.1. Cosmic domains
Our studies indicate that these different Newtonian potentials correspond to differ-
ent effective gravitational constants in various regions of the Universe. If this is true,
the data of recent observations28 should be reinvestigated. In regions with larger
gravitational constants one may see that the gravitational lensing will be stronger
and vice versa. We regard areas with different gravitational potentials as cosmic do-
mains characterized by their own gravitational and cosmological constants. There
occurs a coexistence of these domains (which primordially may be of topological
origin18,19). On the boundaries of the domains, changes in the gravitational and
cosmological constants are expected to be abrupt. In spite of the random or fluctu-
ational origin of these domains, the values of these gravitational and cosmological
constants take only a few universal numbers, corresponding to only a finite number
of types of domains.
For small curvature, each of these states can be described approximately by the
same effective Lagrangian (1) with different gravitational constant κeff and cosmo-
logical constant Λeff and emerge as a fixed point of the conformal transformation
from one side and universal classes of the smooth differential mappings from the
other side.21 These spaces are approximately Einstein spaces, but of different grav-
itational strength as well as with different cosmological constants. The distribution
of such cosmological constants in the Universe corresponds to DE induced by the
scalar field distribution. On a very large scale, φ is homogeneously distributed and
therefore we expect that DE will be distributed homogeneously, depending on the
details of inflationary cosmology.
6.2. Domains arising from free fields
If DM and DE were associated with a massive free scalar field gravitationally coupled
or/and associated with axions (see, Sec. 5.3) there will arise three types of domains:
1) The first type of domains, I, is described by Einstein‘s theory with conven-
tional gravitational constant and vanishing cosmological constant.
2) The second type of domains, II, is described by an Einsteinian model having a
very large gravitational constant and a vanishing cosmological constant. Such type
of domains may be gravitationally unstable, leading eventually to their contraction
and ultimately to a gravitational collapse and phenomena resembling supernova ex-
plosions. For very large masses, we may expect that these explosion will be stronger
than standard supernovae. Our approach may explain the fact why some supernovae
progenitors seem to have exceeded15 the Chandrasekhar limit.
3) The third type of domains, III, are described by an Einsteinian model with
large gravitational and positive cosmological constant ΛIIIeff given by (34). Then we
expect that such domains are still expanding according to effective Friedman equa-
tions and probably their sizes are also increasing. Such expansion may be ended
abruptly and such domains will be transformed into domains of type I.
One may speculate that the largest part of the Universe is occupied by domains
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16 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
of type I, corresponding to spaces with zero cosmological constant. The next largest
proportion of the Universe is expected to be occupied by domains of type III. Those
parts of the Universe occupied by domains of type II are decreasing with time. Each
disappearance of such domain would be accompanied by supernova type explosions.
On the average, the cosmological constant inferred from recent observations is
ΛDE =
VIIIΛ
VI + VII + VIII
, (39)
where the spacelike volumes VI, VII and VIII are associated with domains of type I,
II, and III, respectively.
In summary, our bifurcation set (see Fig. 1 and Fig. 2) indicates that the Universe
is locally described by Einstein’s GR and effectively split into domains. The splitting
originated from primordial quantum fluctuations which were frozen during inflation.
Each domain or local patch of the Universe may have different gravitational strength
and may have zero or negative pressure associated with DE. In domains of type II
associated with a positive cosmological constant the inflation may be still going
on. It will be stopped exactly at the bifurcation point A and this domain will be
transformed into a domain of type I, as in Fig. 2. For a massive free scalar field and
for axions the bifurcation set is the swallow tail catastrophe. There are two cusps
at the points A and B which are associated with the highest singularities of the
differential mappings. They are also related to the domain boundaries. In each of
these cuspoidal points A and B of the bifurcation diagram, the minimum of some
grand manifold merges with the maximum.22 Each local minimum (the segment A
and the semi-infinite segment B) is associated with two types of domains, I and
II, respectively. Each local maximum (saddle) of the grand manifold is associated
with an expanding domain of type III (the segment AB). The effective Einsteinian
gravity for this domain is stronger than that in the domain of type I, depending on
the mass of the scalar field. Domains with a positive effective cosmological constant
are still in an expanding inflationary phase.
The ‘strong’ gravity state, the domain II (the semi-infinite segment B) with a
negative or vanishing cosmological constant corresponds to deflation. Since the Uni-
verse is split into dynamical regions associated with different gravity, their bound-
aries change with time and present some kind of cosmological strings or membranes.
Changes of the boundary are associated with some sort of local phase transition re-
minding us of phase separations like those arising in first order phase transitions. At
some instant, all these domains are in quasi-equilibrium with each other although
their boundaries changes with time as the membranes are moving. At very long
time scales (much larger than the Planck time) when inflation in the domains of
type III will eventually be ceased, only stable phases, like the domains of type I
and some small concentration of domains of type II will remain. These domains
would produce explosions resembling supernovae with masses well beyond of the
Chandrasekhar limit, similar to those observed.15
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Inflation, bifurcations of nonlinear curvature Lagrangians and dark energy 17
6.3. Domains arising from self-interacting fields
In case of self-interacting scalar fields with χ 6= 0, the classification will be richer
(see Figures 4, 5 and 6), with five types of domains. The bifurcation diagram as-
sociated with such a wigwam catastrophe or A5 singularity according to Arnold’s
classification2 consists of four cusps or four A2 singularities, with a dramatic change
in the gravitational and cosmological constants.
There occurs a general decoupling of the wigwam catastrophe A5 into a butterfly
catastrophe A4, and the elementary cusp A2, depending on the values of R: If the
butterfly catastrophe arises near the origin or at small values of R, the additional
cuspoidal point is associated with a very large value of R ≃ −250. For some fixed
values of the coupling constant χ = 0.1, 0.2, 0.3, the butterfly part of bifurcation
diagram is presented in Figs. 5. In comparison with the free massive field, the in-
clusion of the self-interaction associated with the nonzero parameter χ gives rise to
two extra cusps leading to the appearance of two extra branches in the bifurcation
diagram, again associated with effective Einsteinian spaces. Although these almost
linear branches do exist at positive R, their major part is related to negative val-
ues of R. For one of these branches the cosmological constant vanishes while for
the other one the cosmological constant decreases when the value of χ increases,
and the part of the branch associated with R > 0 decreases. In general, the self-
interaction induced by the field φ increases the strength of gravity and decreases
the cosmological constant.
Let us describe in more detail the transitions in the bifurcations: Starting from
the first branch, denoted as I, through the first cuspoidal point we arrive at the
second branch II which is very similar to the case of a free massive scalar field,
although its length decreases significantly when χ increases and the cosmological
constant vanishes at any value of χ. This branch corresponds to the limit φ → 0
of small scalar fields, corresponding to the conformal factor Ω = 1. Therefore, on
this branch we recover the linear Hilbert-Einstein Lagrangian LHE = R/2κ, as ex-
pected. With decreasing Ω, we will come to the next cuspoidal point, where the new
branch III has a positive cosmological constant and gravity becomes stronger. With
increasing value of χ the length of this branch increases. Thus, there will be more
domains having positive cosmological constant and a gravitational strength higher
than the conventional. By decreasing Ω until reaching the next cuspoidal point of
the bifurcation diagram, we arrive at the branch IV. The spacetime associated with
this branch has a very strong gravity and negative cosmological constant. Similar
to the free massive case, this branch probably is gravitationally unstable. Finally,
for very small Ω, we will arrive at the branch V with R < 0, where gravity is the
strongest but without cosmological constant. Similar to the previous branch IV, the
domain V is unstable, as well. Thus in a Universe filled with self-interacting scalar
fields, we would expect that a rather strong phase separation into further domains
will arise.
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18 E.W. Mielke, F.V. Kusmartsev, and F.E. Schunck
7. Discussion
For dilute DM the Lagrangian has the standard Hilbert-Einstein form LHE = R/2κ.
For dense DM, i.e. large values of the DM scalar, the resulting effective Lagrangian
is LHE = R/2κeff, only the slope is less steep, i.e. the effective gravitational coupling
κeff > κ is, by “renormalization”,
3,65 larger than Newton’s. This resembles some
aspects of MOND, but with the advantage that we still work in the standard general
relativistic framework. Depending on its sign, the cosmological constant Λeff in a
bifurcation approximated for small R by (1) can also model DE or an accelerating
phase of the present epoch of the Universe (like ‘anti-gravity’). There also may
arise “gravitational screening” for κeff smaller than Newton’s. MACHOs can now
be described by some specific nonlinear modifications of the Hilbert-Einstein action.
In the light of our finding we have to comment on recent observations,29 which
constrain the density of MACHO type objects in the Universe by measuring the
brightness of high redshift type Ia supernovae relative to low redshift samples. These
data favor DM made of microscopic particles (such as WIMPs) or scalar and vector
gauge fields over MACHOs with masses between 10−2 and 1010 solar masses. This
provides another evidence that our approach correctly describes the frozen space
fluctuations which lead to a formation of cosmic domains with a different density
of DM and DE in the Universe.
Acknowledgments
We would like to thank Burkhard Fuchs, Humberto Peralta, and Konstantin Zioutas for helpful
discussions. One of us (EWM) thanks Noelia, Markus Gérard Erik, and Miryam Sophie Naomi for
encouragement.
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