Special cases of the Multi-Measure Model – understanding the prolonged inflation

Denitsa Staicova1, ∗
1
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria
The multi-measure model (MMM), in which one modifies the action to include both the Rieman-
nian measure and a non-Riemannian one, has proven to be able to produce viable Universe evolution
scenarios. In this article we consider two special cases of the multi-measure model, in which we re-
move some of the non-linearity of the Lagrangian or we decouple the two kinetic terms entirely. We
show numerically that one can still get the needed evolutionary stages, but furthermore, one can
obtain a sufficient number of e-folds of the early inflation in both cases. Furthermore, we connect
the model with hyperinflationary models and investigate how the different epochs are born from the
interplay between the two scalar fields. We also demonstrate that there is a dynamically induces
slow-roll epoch, which is prolonged by the complicated movement of the two scalars in the field
space. Finally, we show that while the adiabatic speed of sound can become imaginary, the phase
speed of sound is equal to one.
arXiv:2011.02967v1 [gr-qc] 5 Nov 2020

INTRODUCTION

The main challenge in front of modern cosmology is not so much to construct an inflationary model as there
are already many of them ([1, 2]), but to produce a model which satisfies all the observational requirements, while
attempting to solve the known problems. In order for a theory to be considered viable, it needs to reproduce the main
known cosmological periods – early inflation, matter domination and late-time exponential expansion. It also needs
to be able to produce a powerful enough inflationary stage and to not contradict the predictions of the observational
data. The latter means, it needs to reduce to the very extensively tested Λ − CDM model at post-inflationary times.
The multi-measure model has been created by Guendelman, Nissimov, and Pacheva [3–13]. Its main advantage is
that in the one scalar (darkon) case, it possesses a dynamically generated cosmological constant. This means that it
allows for spontaneous symmetry breaking, starting from Weyl (local conformal) invariant theory. This is important,
because the Planck results predict almost scale invariant spectrum of primordial fluctuations in the CMB [14]. It does
not change the speed of light and thus it remains within the constraints set by LIGO [15, 16]. Finally, the models
based on the non-Riemannian measure have been studied extensively in various physical situations and have shown
interesting results, for some recent examples, see [17–23].
The multi-measure model employs a number of scalar fields coupled to more than one independent volume form.
In all the models, one of the volume forms is the standard Riemannian volume form, and the other volume forms
can be defined trough the derivatives of auxiliary third rank anti-symmetric gauge field(s) (exact four-form). These
new, auxiliary fields add only gauge degrees of freedom, but they dynamically generate a cosmological constant as a
consequence of the equations of motion. Furthermore, they lead to a perfect fluid energy-momentum tensor, describing
the dark energy and the dark matter sector simultaneously.
In our previous work [24–26], we studied the cosmological aspect of a two measure model with two scalar fields
– the darkon and the inflaton. We demonstrated that it can describe well phenomenologically the evolution of the
Universe, but we also noted some weakness. Notably, the model could not produce the needed number of e-folds.
This seemed like a numerical problem, but due to the large number of parameters, we were not able to prove it. In
this article, we continue our work on the model by considering two special cases of the model – first we remove the
non-linearity in the Lagrangian and second, we remove the dark charge. This simplifies the problem and allows us to
better study the parameter-space of the problem. We see that we are able to obtain the the needed number of e-folds
(> 60) and that they depend strongly on the initial size of the universe. Also, we show numerically that there is a
dynamically induced slow-roll period produced by the complicated movement of the two scalar fields which leads to
the early inflation. Finally we investigate the speed of sound in the two cases and we show that as expected, there is
a difference between the adiabatic and the effective speed of sound.

OVERVIEW OF THE MULTI-MEASURES MODEL

The multi-measure model has been described in details in a number of articles ([10, 11] and also [24–26]). Here we
will discuss only the relevant to the special cases parts of the theory. We will start with the full effective Lagrangian
of the theory in Einstein frame. For the formulations in Jordan frame where one introduces the Riemannian and
2

the non-Riemannian measures, and details about the derivation of the equations of motion, see [26] and references
therein. The effective Lagrangian depends on two scalar fields: the inflaton φ and the darkon u and has the following
form:

L(ef f ) = X̃ − Ỹ (V (φ) + M1 − χ2 b0 e−αφ X̃) + Ỹ 2 (χ2 (U (φ) + M2 ) − 2M0 ). (1)

Here, X̃ = − 21 g̃ µν ∂µ φ∂ν φ and Ỹ = − 12 g̃ µν ∂µ ũ∂ν ũ, are the respective kinetic terms for two scalar fields in the
Weyl-rescaled metric g̃ µν . The potential terms V (φ) = f1 e−αφ , U (φ) = f2 e−2αφ enter the Lagrangians of the scalar
fields in the original Jordan frame. M0 , M1 , M2 are the integration constants coming from the equations of motion
√
(EOM) and χ2 = Φ(B)/ −g is the ratio between the non-Riemannian and the Riemannian volume forms. Note that
this is the only left-over from the non-Riemannian mesures in the action.
This effective Lagrangian is non-linear, it has a non-canonical kinetic terms of both scalar fields and thus can be
classified as a generalized k-essence type. We also have a coupling parameter b0 between the two kinetic terms X and
Y. √
The action of the model is the standard general relativity (GR) action S (ef f ) = d4 x −g̃(R̃ + L(ef f ) ) satisfying
R

a perfect fluid energy-momentum tensor.

In the Friedman–Lemaitre–Robertson–Walker space-time metric, the effective equations of motion are:
v 3 + 3av + 2b = 0 (2)
r
ρ
ȧ(t) − a(t) = 0 (3)
6
d  χ2 
a(t)3 φ̇(1 + b0 e−αφ v 2 ) +
dt 2
2
φ̇ 1 v4
a(t)3 (α χ2 b0 e−αφ v 2 + Vφ v 2 − χ2 Uφ ) = 0 (4)
4 2 4
The dot over the fields indicates the time derivative and the subscript φ – the derivative with respect to the field
φ. The algebraic Equation (2) comes from the conservation of the dark charge where v = u̇ and the parameters are:

1 V (φ) + M1 − 12 χ2 b0 e−αφ φ̇2 pu

a=− ,b = −
3 χ2 (U (φ) + M2 ) − 2M0 2a(t)3 (χ2 (U (φ) + M2 ) − 2M0 )

with pu – an integration constant. Equation (3) is the first Friedman equation where a(t) is the metric scaling function,
and the energy density is:

1 2 3 v2 3pu v
ρ= φ̇ (1 + χ2 b0 e−αφ v 2 ) + (V + M1 ) + .
2 4 4 4a(t)3

The second Friedman equation is:

1
ä(t) = − (ρ + 3p)a(t), (5)
12
where the pressure of the perfect fluid is: p = 21 φ̇2 (1 + 14 χ2 b0 e−αφ v 2 ) − 41 v 2 (V + M1 ) + pu v/(4a(t)3 ).

SPECIAL CASES

First we will consider the case b0 = 0. This means that we are removing the coupling between the two kinetic terms
in the effective Lagrangian. Namely, if we omit the φ-dependence from the potentials, one gets:

L(ef f ) = X̃ − Ỹ (V + M1 ) + Ỹ 2 (χ2 (U + M2 ) − 2M0 ). (6)

This is still a non-linear Lagrangian of the k-essence type with inflaton equation as follows:

ȧ(t)
φ̈ + 3φ̇ − f1 αe−αφ(t) v(t)2 /2 + χ2 f2 αe−2αφ(t) v(t)4 /4 = 0 (7)
a(t)
 3

and the velocity of the darkon scalar field v becomes:

  q  31
2Uef f
v(t) = 2
VM
pu /(a(t)3 + − 16
81 U ef f M n + p 2 /a(t)6
u +
  q − 31
4Uef f 2Uef f
3VM 2
VM
pu /(a(t)3 + − 16 U
81 ef f M n + p 2 /a(t)6
u . (8)

We recall that the effective potential is defined as:

(f1 e−αφ + M1 )2
Uef f (φ) = . (9)
4χ2 (f2 e−2αφ + M2 ) − 8M0

and VM = f1 e−αφ(t) + M1 .
From Eq. 8 we can easily see that there is an initial singularity in our equations connected with the term 1/a(t)3 .
In this case, the energy density becomes:

1 2 v2 pu
ρ= φ̇ + VM + 3v .
2 4 4a(t)3

The asymptotic for v(t), is such that for t → 0

  31  − 31
2Uef f 2pu 4Uef f 2Uef f 2pu
v(t) = 2 (a(t)3 + 2 (a(t)3 . (10)
VM 3VM VM
Here the two terms have equal real parts
 but with opposite imaginary parts so that v(t) remains real. Thus, one
1
2Uef f 2pu 3
can assume that in this limit, v(t) ≈ 2< V 2 a(t)3
.
M

If we use that value to find an approximation for the inflaton equation around the singularity at t = 0, we find:

φ̈ + 3φ̇H + W = 0 (11)
√ 4/3 √ 2/3
Uef f pu pu Uef f ȧ(t)
where W = 2χ2 f2 αe−2αφ(t) 2 VM2 a(t)3 − f1 αeαφ(t) 2 a(t) 3 VM2 and H = a(t) is the Hubble constant and
0
the prime denotes derivative with respect to φ. This term is qualitatively different from Uef f due to the critical
dependence on a(t). The density ρ and the pressure p also depend strongly on a(t). This approximation is applicable
only very close to t = 0, until about t ∼ 10−3 .
Far away from the singularity, the velocity of the darkon field becomes:

r s
Uef f (f1 e−αφ(t) + M 1)
v(t) = 2 = (12)
VM χ2 (f2 e−2αφ(t) + M2 ) − 2M0 )

The approximation of the darkon velocity is excellent fit for the actual velocity for t > 10−3 . Accounting for the
much simpler form of v(t), the inflaton equation becomes:
0
φ̈ + 3φ̇H + Uef f = 0. (13)
This is the standard inflaton equation of a single scalar field rolling down a potential. In this case the density and
the pressure become: ρ = φ̇2 /2 + Uef f and p = φ̇2 /2 − Uef f , thus simplifying dramatically the Friedman equation.
These are the equations for the second special case we consider in this article, namely b0 = 0, pu = 0. For it, the two
scalar fields are related only trough the algebraic Equation 8. Also, for these much simpler EOM, one can see that
the equation of state (EOS) w = p/ρ still satisfies the observational requirements (wa(t)→0 → − 1/3, wa(t)→∞ → − −1)
analytically.
Numerical methods
Our numerical methods have already been described extensively in previous articles. For more details see [26]. Here,
again we work with the Fehlberg fourth - fifth order Runge–Kutta method with degree four interpolation implemented
in Maple and we choose for our cosmological constant Λasymp = 1.025. We perform our calculations in units in which
4

c = 1, G = 1/16π, and tu = 1, where c is the speed of light, G is Newton’s constant, and tu is the present day age of
the Universe. Furthermore, we always require that our potential is step-like with left plateau higher than the right
f2 M2
one (i.e. we require f12 >> M12 ). As noted in previous works, one may easily center the effective potential around
φ = 0, but this do not change qualitatively the observed results.
The case b0 = 0.
We will work with the following parameters:
M0 = −.03, M1 = 0.8, M2 = 0.01, α = 2.4, pu = 10−85 , χ2 = 1, f1 = 5.7, f2 = 10−6 .
For them, one can use two initial conditions:
A. a(0) = 10−31 , φ(0) = −1.8, φ̇(0) = 0
B. a(0) = 10−30 , φ(0) = −3.8, φ̇(0) = 0

FIG. 1. From left to right: the equation of state w = p(t)/ρ(t), the inflaton field φ(t), and the effective potential Uef f . The
dashed line corresponds to the case A, the solid – to the case B. The cross and the diamond denote the start of the integration
in the two cases, the circle - the final value for φ(t).

The plots of the relevant quantities for those two sub-cases are shown on Fig. 1. From the evolution of the EOS, one
can see that in both case, we have a universe with 3 stages – early inflation (w → −1), matter-domination (w < −1/3)
and late-time inflation (w → −1). The two sub-cases match very closely in all time, except for the initial moments,
when the first solution (A.) posses an ultra-relativistic stage (w → 1/3), marked on the plot with dashed line. The
evolution of the scalar field φ(t) in both cases is very similar, except for the first few time-steps of the integration.
To understand better the movement of the inflaton, on the last plot we show the effective potential (same for both
cases), with the starting points of our integration in the two cases marked with a cross for case A and with a diamond
for case B. One can see that while case A starts much lower on the effective potential than case B, the inflaton field
“climbs up the slope”, i.e. it goes backwards instead of forward, with case A much more pronounced (climbing to to
φ = 4). In both cases, the inflaton stays on the slope of the potential – it doesn’t reach the plateau characterized by
0
Uef f (φ0 ) → 0 – but it climbs to a much flatter part of the potential. This corresponds to what we have previously
established in the general case, that the plateau is not accessible for the inflaton scalar field.
A phenomenon similar to “climbing up the slope” has already been observed in other inflationary theories. It has
been proposed in [27] (generalised to more than 2 fields in [28]) in a two-scalar fields model with a field space of a
hyperbolic plane. For them, the second scalar field contributes to the so-called angular momentum. Instead of rolling
down the potential, the scalar field would orbit the bottom of the potential until it has lost all its angular potential.
This would lead to a prolonged inflation. According to the article, its perturbations are adiabatic and approximately
scale invariant. While in our model, the inflaton doesn’t orbit the bottom, but the top of the potential, it’s still
interesting to investigate the parallels between the two theories.
The Lagrangian of the hyperinflation model (assuming FRWL metric) in our notations is:

Lhyp = X̃ + Ỹ f (φ) − Vef f (φ).

The EOM for ψ will lead to a conserved quantity J(0) = a(t)3 Jhyp (t), where Jhyp = f (φ)v is the angular momentum.
Inflation will happen until Jhyp (t) 6= 0. In some regimes any angular perturbation may grow exponentially.
The comparison with the MMM can be done easily for b0 = 0 when we put it in the form

LM M M = X̃ + Ỹ (−(V + M1 ) + Ỹ (χ2 (U + M2 ) − 2M0 )).

 5

FIG. 2. The evolution of the two fields and their time derivatives. On the third plot we can see the so-called hyper-inflationary
angular momentum. The color legend on the three plots is Case A: green dashed line, Case B - red solid line and pu = 0 the
black dotted line,

Then the angular momentum will be: J = −v (V + M1 ) + v 3 (χ2 (U (φ) + M2 ) − 2M0 ). Note that this corresponds
exactly to the conserved current of the dark fluid im MMM:

∂L(ef f )
 
p
∂µ −g̃g̃ µν ∂ν ũ = 0. (14)
∂ Ỹ
To study the relationship between the two types of theories, we plot on Fig. 2 the scalar fields and their derivatives
along with the angular momentum. Here, the value for u(t) is obtained after point-wise numerical integration using
the modified Simpson’s rule applied to 10 000 points. One can see that the angular momentum starts very high for
the cases when pu 6= 0, while it starts from approximately 0 for pu = 0. In the three cases, the angular momentum
starts oscillating around the zero while the inflation lasts and when inflation ends, it settle to zero. Thus the angular
momentum indeed traces the early inflation but it doesn’t give information about the other stages trough which the
evolution passes. What we can see is that the velocities of the two fields seem to exchange energy, except for the
beginning of the evolution. Both of them demonstrate loops – there is the late-time loop on the right (phi ˙ > 0)
˙
common for both cases and the early-time one on the left (phi < 0) which is much larger in the case A (the cut by
the axes blue line), than it is in the case B (shown on the zoomed in plot with red).
The inflation in the two cases happens for φIA ∈ (−3.84, −2.21) and φIB ∈ (−3.26, −2.29) which on the plots are the
regions where u is steeply rising while φ is almost constant. This correlates with the idea in hyperinflation theories
that the scalar field orbits the potential. On the other hand, the matter domination happens while the darkon field
is almost constant.
We can conclude that the just like in the hyperinflation case, the observed epochs are born from the interplay
between the two scalar fields and the exchange of energy between them. Inflation occurs while the inflaton remains
approximately constant, i.e. φ̇ ≈ 0, φ̈ ≈ 0, even though we start on the steep slope of the effective potential and not
on its plateaus. Thus we have slow-roll regime dynamically generated by the exchange of energy between the two
scalar fields which sends the inflaton back on the more slowly varying upper part of the potential. The inflation ends
when the angular momentum falls to almost zero and the system can no longer keep the inflaton on the flatter part
of the potential and it starts rolling down the steep slope.
We turn our attention to the number of e-folds, which can be calculated as N = ln(af /ai ), where ai and af are the
beginning and the end of inflation, i.e we have removed the ultra-relativistic stage. The number of e-folds in case A
is N = 68, versus N = 63 in case B. In both cases, it is enough to put the model in the error-bounds of observational
expectations N > 65. This is much higher than what we obtained in the general case (b0 6= 0), when the maximal
value which we got was N = 22. This effect seems to be mostly numerical as we will discuss later.
The case b0 = 0, pu = 0. The second special case which we consider is the one with b0 = 0 and pu = 0. This
case is particularly interesting because it removes the initial singularity from our equations and it allows us to study
the Universe evolution when we do not have a Big Bang. In this case, the velocity of the darkon field is Eq. 12 and
the inflaton equation is Eq. 13. Basically in this case, we have removed the evolution before t = 10−3 and we deal
with much simplified equations of motion. The numerical evoluion of the parameters follows very closely the ones
shown on Fig. 1 for case B differing only that now the EOS starts from w(0) = −1, while the inflaton field starts
form its minimum shown on Fig. 1 (φ0 = −4.045) and it increases monotonously afterwards. For this reasons, we
6

omit showing it on Fig. 1, and we add it only to the zoomed in figure of the dependence u(φ) on Fig. 2, where it’s
shown with a black dash-dotted line. As for the second plot (u̇(t)(φ̇(t)) the pu = 0 case has only the loop on the left,
coinciding exactly with that on the plot. For third plot, the difference is negligible, again closely resembling case B,
only this time J decreases monotonously, i.e. there’s no inflexion point at the beginning. The most notable difference
in this case is the lack of the “climbing up the slope” in the movement of the inflaton – the scalar field just rolls down
the slope as expected. This means that it is the dark charge what generates the "climbing up the slope" phenomenon.
Note that while, we have removed the singularity from v(t), there is still the much weaker dependence on a(0) coming
from the inflaton equation itself. Because of this, different a(0) will still change our solutions.

Study of the parameters

a b c

FIG. 3. On the panels we compare 3 values of the initial condition f a(0) = 10−10 , 10−20 , 10−30 denoted with asterisks, diagonal
crosses and diamonds accordingly. We plot: a) the dependence of the number of e-folds from the starting point on the slope
φ0 , b) f1 (φ0 ) c) f1 (pu ) for the 3 different cases.

In our previous works, we made the claim that the number of e-folds depend on the starting position of the
integration and how close a(0) is to 0. To study this phenomenon we wrote a code which automatically searches
for solutions of the equations of motion, fulfilling our normalizations. This allowed us to study a much wider set of
parameters by varying a(0), φ(0), f1 , pu to get the normalization a(1) = 1 and ä(0.71) = 0. The results are shown on
Fig. 3, where we plot 3 sets of points corresponding to 3 different initial conditions: a(0) = {10−10 , 10−20 , 10−30 }.
The number of e-fold (Fig. 3 a))clearly grows with the decrease of the initial value of a(0). This is apparently due
to the initial singularity, but notably, not the one in v(t), but the one in the inflaton equation itself. One can see
this by noting the position of the black diamonds which correspond to the limit case pu = 0. This seems to confirm
that indeed the problem of the too weak inflation numerically is due to the initial condition for a(0) and that close
enough to the singularity of the equations, one can get arbitrarily large number of e-folds. This seems as a numerical
instability, but one must note that we do not know the initial conditions of our universe and thus we know only the
minimal number of e-folds needed to produce our universe. Note that all our solutions are normalized to a(1) = 1,
meaning the Universe itself doesn’t grow bigger. Also to be noted, the duration of the inflation does not grow for
larger number of e-folds.

On Fig. 3 b) we have shown the relation f1 (φ0 ) and on Fig. 3 c) the dependence f1 (pu ). Surprisingly in the
latter case, the values of different initial conditions a(0), φ0 fall on the same curve, which show that this relation is
independent from the initial conditions. We recall that both f1 and pu are physical quantities, one of them comes
from the potential term of one of the inflaton Lagrangians while pu is the so called conserved dark charge.

The speed of sound

Finally, we would like to discuss the speed of sound in this two special cases. We have plotted the so called adiabatic
sound speed, cs 2 a = ρ̇ṗ on Fig. 4 for a) the pu 6= 0 cases from above, and on b) the pu = 0 cases for two different
starting points a(0) = 10−5 and a(0) = 10−45 .
 7

FIG. 4. On the panels are the speed of sound of the two main case a) b0 = 0, pu 6= 0, a(0) = 10−30 (case A) denoted with
solid yellow line , case (B) denoted with blue dash-dot line b) b0 = 0, pu = 0, a(0) = 10−5 denoted with yellow, solid line,
b0 = 0, pu = 0, a(0) = 10−45 – with blue, dashed line

Despite the different initial conditions and parameters in the 4 cases, it doesn’t seem possible to avoid the negative
region. Even in the case (B) where there is a positive initial speed of sound it still reverts to −1 during inflation. This
is well-known property of perfect fluid dark energy models and it is considered related to the difference between the
adiabatic speed of sound defined above (generated by pressure perturbations) and the actual speed of propagation of
the perturbation, the phase speed (generated by entropic perturbations).
The phase speed in the special case b0 = 0, pu = 0, on-shell, coincides with that of a standard one scalar field
FX
theory L = F (X, φ). For it: c2phase = p,X /ρ,X = FX +2XF XX
= 1 [29] (where we have taken into account that
p = F (X, φ)). A generalization for a wide range of multifield theories of the form L = F (X IJ , φK ) can be found in
[30, 31]. According to [31], the perturbations along the field-space trajectory move with the single-field cs , while the
orthogonal ones move with the speed of light. Calculating the phase speed of sound in theories with non-canonical
kinetic terms of two scalars is not trivial. If we take [31] as a reference, and we apply it to a Lagrangian depending
on 3 scalar fields - φ, v, v 2 , we are able to reconstruct the inflaton equation. From there we obtain c2s,phase = 1. This
seems correct since this case differs from the simpler pu = 0 case only in first initial moments.
There are a number of theories in which the adiabatic speed of sound shows non-standard behavior but the effective
speed of sound gives a scale on which perturbations may be dampened, the so called effective sound horizon (see
[32, 33]). For example, in quintessence, the speed of sound is imaginary, while in k − essence theories, c2s > 1 and
thus perturbations can travel faster than light. In [34], quintessence with non-minimal derivative coupling to gravity
has been shown to suffer from both superluminal perturbations and Laplacian (gradient) instability – c2s < 0. In
[35] the authors have studied variable dark energy speed of sound and have found that cDE s 6= 1 when the model has
non-canonical kinetic term. In a study [36, 37] about inspired by the sidetracked inflation, but generalized to models
allowing an effective single field theory, the imaginary sound speed leads to exponentially increasing and decreasing
ones (instead of positive and negative modes). The exponentially growing fluctuation becomes constant after the
sound Hubble crossing, so they are named transient tachionic instability. An interesting study [38] shows that to fit
WMAP data, c2s < 0.04. Similarly, in [39], the authors predict that dark energy clustering is more efficient when
cs → 0. In our case, we can see that while the adiabatic speed of light indeed is variable and imaginary during
inflation, the phase speed doesn’t imply instabilities. We leave the complete investigation of the perturbation in the
special cases and in the full case for future works.

CONCLUSIONS

In this article, we studied two special cases of the multi-measure model of Guendelman–Nissimov–Pacheva with
which one can decouple the two scalar fields in the Lagrangian. Again we see solutions which can reproduce the known
stages of the Universe without any further constraints. We show numerically that one is able to get the necessary
number of e-folds and thus it is possible to obtain a strong-enough early inflation. We study the parameter-space
more extensively than in previous works and see the dependence of some of the parameters on each other. Also we
show numerically that while the adiabatic speed of sound becomes imaginary during inflation, the phase speed is
equal to one. Our most interesting result is the connection of our model with the hyperinflationary models in which
8

the ‘centrifugal force’ of a field orbiting the hyperbolic plane of the two fields leads to prolonged inflation. We have
studied how this movement of the scalar fields with respect to each other is related to the equation of state of the
universe and to the so-called angular momentum of the model. We have seen that during the early inflation epoch
we have a dynamically induced slow-roll period in which the effective potential varies more slowly thus allowing the
application of slow-roll approximations.
Acknowledgments The work is supported by the Bulgarian National Science Fund for support via research grants
DN 08-17, DN-18/17, KP-06-N 8/11. We have received partial support from European COST actions CA15117 and
CA18108. It is a pleasure to thank Emil Nissimov, Svetlana Pacheva, Michail Stoilov and David Benisty for the
discussions.

∗
dstaicova@inrne.bas.bg
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