11institutetext: Physics Department, Ben-Gurion University of the Negev, Beer-
Sheva 84105, Israel 22institutetext: Frankfurt Institute for Advanced Studies
(FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany
33institutetext: Bahamas Advanced Study Institute and Conferences, 4A Ocean
Heights, Hill View Circle, Stella Maris, Long Island, The Bahamas
44institutetext: Institute for Nuclear Research and Nuclear Energy, Bulgarian
Academy of Sciences, Sofia, Bulgaria

# Non-Canonical Volume-Form Formulation of Modified Gravity Theories and
Cosmology

David Bensity 11 Eduardo I. Guendelman 112233 Alexander Kaganovich 11 Emil
Nissimov 44 Svetlana Pacheva _Present address:_ Insert the address here if
needed44

###### Abstract

A concise description is presented of the basic features of the formalism of
non-canonical spacetime volume-forms and its application in modified gravity
theories and cosmology. The well known unimodular gravity theory appears as a
very special case. Concerning the hot issues facing cosmology now, we
specifically briefly outline the construction of: (a) unified description of
dark energy and dark matter as manifestations of a single material entity – a
special scalar field “darkon”; (b) quintessential models of universe evolution
with a gravity-“inflaton”-assisted dynamical Higgs mechanism – dynamical
suppression/generation of spontaneous electroweak gauge symmetry breaking in
the “early”/“late” universe; (c) unification of dark energy and dark matter
with diffusive interaction among them; (d) mechanism for suppression of 5-th
force without fine-tuning.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Non-Riemannian Volume-Form Formalism

A broad class of actively developed modified/extended gravitational theories
is based on employing alternative non-Riemannian spacetime volume-forms
(metric-independent generally covariant volume elements) in the pertinent
Lagrangian actions instead of, or alongside with, the canonical Riemannian
volume element given by the square-root of the determinant of the Riemannian
metric. This method was originally proposed in Guendelman:1999qt ;
Guendelman:1999tb  and for a concise geometric formulation using differential
forms combined with canonical Hamiltonian formalism for systems with
constraints (gauge symmetries), see Guendelman:2014lea ; Guendelman:2015qva
(an earlier geometric formulation with a “quartet” of scalar fields appeared
in Gronwald:1997ei ).

Volume-forms are fairly basic objects in differential geometry – they exist on
arbitrary differentiable manifolds and define covariant (under general
coordinate reparametrizations) integration measures. It is important to stress
that the existence of volume-forms is completely independent of the presence
or absence of additional geometric structures on the manifold – Volume forms
are defined spivak  by nonsingular maximal rank differential forms $\omega$:

$\displaystyle\int_{{\mathcal{M}}}\omega\bigl{(}\ldots\bigr{)}=\int_{{\mathcal{M}}}dx^{D}\,\Omega\bigl{(}\ldots\bigr{)}\;\;,$
$\displaystyle\omega=\frac{1}{D!}\omega_{\mu_{1}\ldots\mu_{D}}dx^{\mu_{1}}\wedge\ldots\wedge
dx^{\mu_{D}}\;,$ (1)
$\displaystyle\omega_{\mu_{1}\ldots\mu_{D}}=-\varepsilon_{\mu_{1}\ldots\mu_{D}}\Omega\;,$

(our conventions for the alternating symbols
$\varepsilon^{\mu_{1},\ldots,\mu_{D}}$ and
$\varepsilon_{\mu_{1},\ldots,\mu_{D}}$ are: $\varepsilon^{01\ldots D-1}=1$ and
$\varepsilon_{01\ldots D-1}=-1$). The volume element density $\Omega$
transforms as scalar density under general coordinate reparametrizations.

In standard generally-covariant theories (with action $S=\int
d^{D}\\!x\sqrt{-g}{\mathcal{L}}$) the Riemannian spacetime volume-form is
defined through the “D-bein” (frame-bundle) canonical one-forms
$e^{A}=e^{A}_{\mu}dx^{\mu}$ ($A=0,\ldots,D-1$):

$\displaystyle\omega=e^{0}\wedge\ldots\wedge
e^{D-1}=\det\|e^{A}_{\mu}\|\,dx^{\mu_{1}}\wedge\ldots\wedge
dx^{\mu_{D}}\longrightarrow\quad\Omega=\det\|e^{A}_{\mu}\|\,d^{D}x=\sqrt{-\det\|g_{\mu\nu}\|}\,d^{D}x\;.$
(2)

Instead of, or alongside with, $\sqrt{-g}$ we can employ one or several
different alternative non-Riemannian volume elements as in (1) given by non-
singular exact $D$-forms $\omega^{(j)}=dB^{(j)}$ where:

$\displaystyle
B^{(j)}=\frac{1}{(D-1)!}B^{(j)}_{\mu_{1}\ldots\mu_{D-1}}dx^{\mu_{1}}\wedge\ldots\wedge
dx^{\mu_{-1}}\longrightarrow\quad\Omega^{(j)}\equiv\Phi(B^{(j)})=\frac{1}{(D-1)!}\varepsilon^{\mu_{1}\ldots\mu_{D}}\,\partial_{\mu_{1}}B^{(j)}_{\mu_{2}\ldots\mu_{D}}\;.$
(3)

In other words, the non-Riemannian volume elements are defined in terms of the
dual field-strengths of auxiliary rank $D-1$ tensor gauge fields
$B^{(j)}_{\mu_{1}\ldots\mu_{D-1}}$.

Let us again strongly emphasize that the term “non-Riemannian” concerns only
the nature of the non-canonical volume elements, which exist on the spacetime
manifold with a standard Riemannian geometric structure, torsionless affine
connection $\Gamma^{\lambda}_{\mu\nu}$ either independent of $g_{\mu\nu}$
(first-order metric-affine / Einstein-Palatini formalism) or as a Levi-Civita
connection w.r.t. $g_{\mu\nu}$ (second-order purely metric / Einstein-Hilbert
formalism).

The generic form of modified gravity actions involving (one or more) non-
Riemannian volume-elements, called for short actions, read (henceforth $D=4$,
and we will use units with $16\pi G_{Newton}=1$):

$\displaystyle S=\int
d^{4}x\,\Phi(B^{(1)})\bigl{(}R+{\mathcal{L}}^{(1)}\bigr{)}+\int
d^{4}x\,\sum_{j\geq 2}\Phi(B^{(j)})\,{\mathcal{L}}^{(j)}+\int
d^{4}x\,\sqrt{-g}{\mathcal{L}}^{(0)}\;,$

where $R$ is the scalar curvature. The equations of motion of (1) w.r.t. the
auxiliary tensor gauge fields $B^{(j)}_{\mu\nu\kappa}$ according to (3) imply:

$\displaystyle\partial_{\mu}\bigl{(}R+{\mathcal{L}}^{(1)}\bigr{)}=0\;\;,\;\;\partial_{\mu}{\mathcal{L}}^{(j)}=0\;\;(j\geq
2)\;,\,\longrightarrow\;\;R+{\mathcal{L}}^{(1)}=M_{1}\quad,\quad{\mathcal{L}}^{(j)}=M_{j}\;,$
(4)

where all $M_{j}$ ($j\geq 1$) are free integration constants not present in
the original NRVF gravity action (1).

A characteristic feature of the NRVF gravitational theories (1) is that when
starting in the first-order (Palatini) formalism all non-Riemannian volume-
elements $\Phi(B^{(j)})$ yield almost pure-gauge degrees of freedom,
additional physical (field-propagating) gravitational degrees of freedom
except for few discrete degrees of freedom with conserved canonical momenta
appearing as the arbitrary integration constants $M_{j}$ in (4). The reason is
that the NRVF gravity action (1) in Palatini formalism is linear w.r.t. the
velocities of some of the components of the auxiliary gauge fields
$B^{(j)}_{\mu\nu\kappa}$ defining the non-Riemannian volume-element densities,
and does not depend on the velocities of the rest of auxiliary gauge field
components. The (almost) pure-gauge nature of the latter is explicitly shown
in Guendelman:2015qva ; Guendelman:2016lea  (appendices A) employing the
standard canonical Hamiltonian treatment of systems with gauge symmetries,
i.e., systems with first-class Hamiltonian constraints a’la Dirac henneaux-
teitelboim ; rothe .

However, in the second-order formalism (where $\Gamma_{\mu\nu}^{\lambda}$ is
the usual Levi-Civita connection w.r.t. $g_{\mu\nu}$) the first non-Riemannian
volume form $\Phi(B^{(1)})$ in (1) is not any more pure-gauge. The reason is
that the scalar curvature $R$ (in the metric formalism) contains second-order
(time) derivatives (the latter amount to a total derivative in the ordinary
case $S=\int d^{4}x\sqrt{-g}R+\ldots$). Now defining
$\chi_{1}\equiv\Phi(B^{(1)})/\sqrt{-g}$, the latter field becomes physical
degree of freedom as seen from the equations of motion of (1) w.r.t.
$g^{\mu\nu}$:

$R_{\mu\nu}+\frac{1}{\chi_{1}}\bigl{(}g_{\mu\nu}\Box\chi_{1}-\nabla_{\mu}\nabla_{\nu}\chi_{1}\bigr{)}+\ldots=0\;.$
(5)

As a final introductory remark let us note that the well-known covariant
formulation of unimodular gravity Henneaux:1989zc  can be viewed as a simple
particular case within the general class (1) of modified gravity actions based
on the non-Riemannian volume-form formalism. Indeed, the original action of
unimodular gravity Henneaux:1989zc  reads:

$\mathcal{S}=\int d^{4}x\sqrt{-g}\left(R+2\Lambda+\mathcal{L}_{m}\right)-\int
d^{4}x\,\Phi\,2\Lambda$ (6)

with $\Lambda$ being a dyamical field, and $\Phi\equiv\partial_{\mu}F^{\mu}$
where the vector density $F^{\mu}$ can be written as Hodge-dual
$F^{\mu}\equiv\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}B_{\nu\kappa\lambda}$
w.r.t. rank 3 auxiliary gauge field $B_{\nu\kappa\lambda}$ (cf. (3) for
$D=4$). Variation w.r.t. $F^{\mu}$ implies $\Lambda=const$, whereas variation
w.r.t. $\Lambda$ yields $\Phi=\sqrt{-g}$, in what follows, for general NRVF
gravity models (1) the field ratio $\chi_{1}$ is either a non-trivial
algebraic function of the matter fields in ${\mathcal{L}}^{(j)}$ within the
first-order (Palatini) formalism (cf. Eq.(20) below), or it becomes a new
dynamical scalar field within the second-order (metric) formalism (cf.
Eq.(5)).

##  2 Simple Model of Unified Dark Energy and Dark matter

A simple NRVF gravity model providing a unified description of dark energy and
dark matter defined by an action, particular representative of the class (1),
was proposed in Guendelman:2014bva ; Guendelman:2015jii :

$S=\int
d^{4}x\,\Bigl{[}\sqrt{-g}\bigl{(}R+X-V_{1}(\phi)\bigr{)}+\Phi(B)\bigl{(}X-V_{2}(\phi)\bigr{)}\Bigr{]}\;,$
(7)

or equivalently:

$\displaystyle S=\int
d^{4}x\,\sqrt{-g}\bigl{(}R-U(\phi)\bigr{)}+\bigl{(}\sqrt{-g}+\Phi(B)\bigr{)}\bigl{(}X-V(\phi)\bigr{)}$

using the notations: $V\equiv V_{2},\;\;U\equiv
V_{1}-V_{2},\,X\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$,
and $\Phi(B)\equiv
1/3!\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}B_{\nu\kappa\lambda}$ (cf.
(3)). Variation of the action (LABEL:eq:NRVF-1) w.r.t. auxiliary gauge field
$B_{\nu\kappa\lambda}$ yields (cf. the general Eqs.(4)):

$X-V(\phi)=-2M_{0},$ (9)

where $M_{0}$ is free integration constant. The variation of (LABEL:eq:NRVF-1)
w.r.t. scalar field $\phi$ can be written in the following suggestive form:

$\displaystyle\nabla_{\mu}J^{\mu}=-\sqrt{2X}U^{\prime}(\phi)\;,$ (10)
$\displaystyle
J_{\mu}\equiv-\bigl{(}1+\chi\bigr{)}\sqrt{2X}\partial_{\mu}\phi\;\;,\;\;\chi\equiv\frac{\Phi(B)}{\sqrt{-g}}\;.$
(11)

The dynamics of $\phi$ is entirely determined by the dynamical constraint (9),
completely independent of the potential $U(\phi)$. On the other hand, the
$\phi$-equation of motion written in the form (10) is in fact an equation
determining the dynamics of $\chi$. The energy-momentum tensor $T_{\mu\nu}$ in
the Einstein equations can be written in a relativistic hydrodynamical form
as:

$T_{\mu\nu}=\rho_{0}u_{\mu}u_{\nu}+g_{\mu\nu}{\widetilde{p}}\;\;,\quad
J_{\mu}=\rho_{0}u_{\mu}$ (12)

where $u_{\mu}$ is a fluid velocity unit vector:

$u_{\mu}\equiv-\frac{\partial_{\mu}\phi}{\sqrt{2X}}\quad({\rm
note}\;u^{\mu}u_{\mu}=-1\;)\;,$ (13)

and the energy density ${\widetilde{\rho}}$ and pressure ${\widetilde{p}}$ are
given as:

${\widetilde{\rho}}=\rho_{0}+2M_{0}+U(\phi),\quad{\widetilde{p}}=-2M_{0}-U(\phi)$
(14)

with $\rho_{0}\equiv(1+\chi)2X={\widetilde{\rho}}+{\widetilde{p}}$. Energy-
momentum conservation $\nabla^{\nu}T_{\mu\nu}=0$ implies:

$\nabla^{\mu}\bigl{(}\rho_{0}u_{\mu}\bigr{)}=-\sqrt{2X}\,U^{\prime}(\phi),\quad
u_{\nu}\nabla^{\nu}u_{\mu}=0\;,$ (15)

the last Eq.(15) meaning that the matter fluid flows along geodesics. In
Eqs.(12), (14) the quantity $\rho_{\rm DE}\equiv
2M_{0}+U(\phi)=-{\widetilde{p}}$ has the interpretation as dark energy
density, whereas $\rho_{0}$ is the dark matter energy density. For
$U(\phi)={\rm const}$ or $U(\phi)=0$ the model (LABEL:eq:NRVF-1) possesses a
non-trivial hidden nonlinear Noether symmetry under:

$\displaystyle\delta_{\epsilon}\phi=\epsilon\sqrt{X},\quad\delta_{\epsilon}g_{\mu\nu}=0\;,\quad\delta_{\epsilon}{\mathcal{B}}^{\mu}=-\epsilon\frac{1}{2\sqrt{X}}\phi^{,\mu}\bigl{(}\Phi(B)+\sqrt{-g}\bigr{)}\;,$
(16)

where
${\mathcal{B}}^{\mu}\equiv\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}B_{\nu\kappa\lambda}$,
with a Noether conserved current $J^{\mu}=\rho_{0}u_{\mu}$ according to (11):
$\nabla_{\mu}J^{\mu}=0$. Specifically, for Friedmann-Lemaitre-Robertson-Walker
metric with Friedmann scale factor $a(t)$ Eq.(11) with $U(\phi)=0$ implies:
$\rho_{0}=c_{0}/a^{3}$, $c_{0}$ being a free integration constant.

Thus, according to (12), (14) the model provides an exact description of
$\Lambda$CDM model, and for a non-trivial potential $U(\phi)$, breaking the
hidden Noether symmetry (16), we have interacting dark energy and dark matter.

The above interpretation justifies the alias “darkon” for the scalar field
$\phi$. Let us specifically emphasize that both dark energy and dark matter
components of the energy density (14) have been dynamically generated thanks
to the non-Riemannian volume element construction – both due to the appearance
of the free integration constant $M_{0}$ and of the hidden nonlinear Noether
symmetry (16) (“darkon” symmetry). In Ref.Benisty:2020nql  the correspondence
between $\Lambda$CDM model and the “darkon” Noether symmetry was exhibited up
to linear order w.r.t. gravity-matter perturbations and the implications of
the “darkon” symmetry breaking for possible explanation of the cosmic tensions
was briefly discussed. Staicova:2016pfd  confront some potential with the late
accelerated expansion data.

##  3 Quintessential Inflationary Model with Dynamical Higgs Effect in Metric-
Affine Formulation

The starting point is the following specific NRVF gravity action from the
class (1) involving coupling to a scalar “inflaton” $\varphi$ and to the
bosonic sector of the standard electroweak particle model where, following the
remarkable Bekenstein’s idea from 1986 Bekenstein:1986hi  about gravity-
assisted dynamical spontaneous symmetry breakdown, the Higgs-like $SU(2)\times
U(1)$ iso-doublet scalar $\sigma_{a}$ enters with a standard positive mass-
squared and without self-interaction in sharp distinction w.r.t. standard
particle model. The pertinent NRVF action reads explicitly Guendelman:2016lea
; Brink:2020fie ; Benisty:2020vvm :

$\displaystyle S=\int
d^{4}x\,\Phi_{1}(A)\Bigl{[}R(g,\Gamma)-2\Lambda_{0}\frac{\Phi_{1}(A)}{\sqrt{-g}}+L^{(1)}(\varphi,\sigma)\Bigr{]}+\int
d^{4}x\,\Phi_{2}(B)\Bigl{[}f_{2}e^{2\alpha\varphi}+L_{\rm EW-
gauge}-\frac{\Phi_{0}(C)}{\sqrt{-g}}\Bigr{]}\;,$ (17)

with notations:

  * •

$\Phi_{1}(A)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\,\partial_{\mu}A_{\nu\kappa\lambda}$
and similarly for $\Phi_{2}(B)$, $\Phi_{0}(C)$ according to (3);

  * •

The scalar curvature $R(g,\Gamma)=g^{\mu\nu}R_{\mu\nu}(\Gamma)$ is given in
terms of the Ricci tensor $R_{\mu\nu}(\Gamma)$ in the first-order (Palatini)
formalism;

  * •

The matter Lagrangian reads:

$\displaystyle
L^{(1)}(\varphi,\sigma)=X_{\varphi}+f_{1}e^{\alpha\varphi}+X_{\sigma}-m_{0}^{2}\sigma^{*}_{a}\sigma_{a}e^{\alpha\varphi},\quad
X_{\varphi}\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi\;\;,\;\;X_{\sigma}\equiv-g^{\mu\nu}\nabla_{\mu}\sigma^{*}_{a}\nabla_{\nu}\sigma_{a}\;;$

  * •

$L_{\rm EW-gauge}$ denotes the Lagrangian of the $SU(2)\times U(1)$ gauge
fields.

  * •

$\Lambda_{0}$ is small dimensionful constant which will be identified in the
sequel with the “late” universe cosmological constant in the dark energy
dominated accelerated expansion’s epoch.

The equations of motion w.r.t. auxiliary tensor gauge fiels in $\Phi_{1}(A)$,
$\Phi_{2}(B)$ and $\Phi_{1}(C)$ yield (cf. (4)):

$\displaystyle
g^{\mu\nu}\Bigl{(}R_{\mu\nu}(\Gamma)-\frac{1}{2}\partial_{\mu}\varphi\partial_{\nu}\varphi-\nabla_{\mu}\sigma^{*}_{a}\nabla_{\nu}\sigma_{a}\Bigr{)}-4\Lambda_{0}\frac{\Phi_{1}(A)}{\sqrt{-g}}+\bigl{(}f_{1}-m^{2}_{0}\,\sigma^{*}_{a}\sigma_{a}\bigr{)}e^{\alpha\varphi}=M_{1},$
(18) $\displaystyle f_{2}e^{-2\alpha\phi}+L_{\rm EW-
gauge}-\frac{\Phi_{0}(C)}{\sqrt{-g}}=-M_{2},\,\,\frac{\Phi_{2}(B)}{\sqrt{-g}}=\chi_{2}$
(19)

where $M_{1,2},\chi_{2}$ are integration constants. The $g^{\mu\nu}$-equations
of motion together with (18)-(19) imply that the ratio
$\chi_{1}\equiv\frac{\Phi(A)}{\sqrt{-g}}$ is an algebraic function of the
matter fields:

$\chi_{1}(\varphi,\sigma)\equiv\frac{\Phi_{1}(A)}{\sqrt{-g}}=\frac{2\chi_{2}\bigl{(}f_{2}e^{2\alpha\varphi}+M_{2}\bigr{)}}{M_{1}+\bigl{(}m^{2}_{0}\,\sigma^{*}_{a}\sigma_{a}-f_{1}\bigr{)}e^{\alpha\varphi}}.$
(20)

The equation of motion w.r.t. $\Gamma^{\lambda}_{\mu\nu}$, following analogous
derivation in Guendelman:1999qt , yields a solution for
$\Gamma^{\mu}_{\nu\lambda}$ as a Levi-Civita connection w.r.t. to a Weyl-
conformally rescaled metric:

${\bar{g}}_{\mu\nu}=\chi_{1}(\varphi,\sigma)\,g_{\mu\nu}$ (21)

with $\chi_{1}(\varphi,\sigma)$ as in (20). The conformal transformation
$g_{\mu\nu}\to{\bar{g}}_{\mu\nu}$ via (21) on the NRVF action (17) converts
the latter into the physical Einstein-frame action (objects in the Einstein-
frame are indicated by a bar):

$\displaystyle S_{\rm EF}=\int
d^{4}x\,\sqrt{-{\bar{g}}}\Bigl{[}R({\bar{g}})-\frac{1}{2}{\bar{g}}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-{\bar{g}}^{\mu\nu}\nabla_{\mu}\sigma^{*}_{a}\nabla_{\nu}\sigma_{a}-U_{\rm
eff}(\varphi,\sigma)+L_{\rm EW-gauge}({\bar{g}})\Bigr{]}\;.$ (22)

Here the interesting object is the effective Einstein-frame scalar potential:

$U_{\rm
eff}(\varphi,\sigma)=\frac{\Bigl{[}M_{1}+e^{\alpha\varphi}\bigl{(}m^{2}_{0}\,\sigma^{*}_{a}\sigma_{a}-f_{1}\bigr{)}\Bigr{]}^{2}}{4\chi_{2}\bigl{(}f_{2}e^{2\alpha\varphi}+M_{2}\bigr{)}}+2\Lambda_{0}\;,$
(23)

which is entirely dynamically generated due to the appearance of the free
integration constants $M_{1,2}$ and $\chi_{2}$ (18)-(19). $U_{\rm
eff}(\varphi,\sigma)$ exhibits a number of remarkable features:

  * •

$U_{\rm eff}(\varphi,\sigma)$ possesses two (infinitely) large flat regions as
a function of $\varphi$ at $\sigma_{a}={\rm fixed}$.

  * •

The first one – the (-) flat “inflaton” region for large negative values of
$\varphi$ (and $\sigma_{a}$ – finite) corresponds to the “slow-roll”
inflationary evolution of the “early” universe driven by $\varphi$ where:

$U_{\rm eff}\bigl{(}\phi,\sigma\bigr{)}\simeq
U_{(-)}=\frac{M_{1}^{2}}{4\chi_{2}\,M_{2}}+2\Lambda_{0}\;,$ (24)

independent of the finite value of $\sigma_{a}$, which is energy scale of the
inflationary epoch. Thus, in the “early” universe there is no spontaneous
breaking of electroweak $SU(2)\times U(1)$ symmetry. Moreover, $\sigma_{a}$
does not participate in the “slow-roll” inflationary evolution, so $\sigma$
stays constant there equal to the “false”vacuum value $\sigma=0$
Benisty:2020vvm .

  * •

The second flat region is the (+) flat “inflaton” region for large positive
values of $\varphi$ (and $\sigma_{a}$ – finite) which corresponds to the
evolution of the post-inflationary (“late”) universe. Here:

$U_{\rm eff}\bigl{(}\varphi,\sigma\bigr{)}\simeq
U_{(+)}(\sigma)=\frac{\Bigl{(}m^{2}_{0}\,\sigma^{*}_{a}\sigma_{a}-f_{1}\Bigr{)}^{2}}{4\chi_{2}f_{2}}+2\Lambda_{0}$
(25)

Figure 1: Qualitative shape of the two-dimensional plot for the effective
scalar potential $U_{eff}$.

becomes a dynamically induced $SU(2)\times U(1)$ spontaneous symmetry breaking
Higgs-like potential with a Higgs “vacuum” at $|\sigma_{\rm
vac}|=\frac{1}{m_{0}}\sqrt{f_{1}}$.

  * •

Relations (24)-(25) allow the following natural identification of the scales
of the parameters: $\Lambda_{0}\sim 10^{-122}M^{4}_{Pl}$ (current epoch
observable cosmological constant); $f_{1}\sim f_{2}\sim M^{4}_{EW}$ and
$m_{0}\sim M_{EW}$ ($M_{EW}$ being the electroweak mass scale); $M_{1}\sim
M_{2}\sim 10^{-8}M_{Pl}^{4}$ corresponding to the “early” universe’s energy
scale of inflation being of order $10^{-2}M_{Pl}$.

Concerning confrontation with the observational data, the viability of the
present model (in a slightly simplified form without the Higgs scalar, which
as already mentioned does not influence the slow-roll inflationary dynamics)
has been analyzed and confirmed numerically in Ref.Guendelman:2014bva . The
results for the tensor-to-scalar ratio $r\simeq 10^{-3}$ and for the scalar
spectral index $n_{s}\simeq 0.96$ which are in a good agreement with the
latest Further detailed numerical studies on the NRVF models have been
presented in Refs. Staicova:2016pfd ; Staicova:2018bdy ; Staicova:2019ksr .

Let us also note that Ref.Guendelman:2014bva  (for an earlier version, see
Guendelman:2002js ) exhibits an explicit realization of the cosmological
“seesaw” mechnaism through the NRVF formulation, as well as it yields an
additional “emergent universe” cosmological solution without a “Big-Bang”
initial singularity. For a brief illustration of the latter effects let us
consider the “inflaton-only” NRVF action studied in Guendelman:2014bva  (for
simplicity we skip the $R^{2}$ term):

$\displaystyle S=\int
d^{4}x\,\Phi_{1}(A)\Bigl{[}R+X_{\varphi}-f_{1}e^{-\alpha\varphi}\Bigr{]}+\int
d^{4}x\,\Phi_{2}(B)\Bigl{[}-b\,e^{-\alpha\varphi}\,X_{\varphi}+f_{2}e^{-2\alpha\varphi}-\frac{\Phi_{0}(C)}{\sqrt{-g}}\Bigr{]}\;,$
(26)

where $b$ is an additional dimensionless parameter.

The “inflaton” potential in the Einstein frame (analog of (23)) is:

$U_{\rm
eff}(\varphi)=\frac{1}{4\chi_{2}}\bigl{(}f_{1}e^{-\alpha\varphi}+M_{1}\bigr{)}^{2}\bigl{(}f_{2}e^{-2\alpha\varphi}+M_{2}\bigr{)}^{-1}$
(27)

so that on the (-) and (+) “inflaton” flat regions $U_{\rm eff}(\varphi)$
reduces to: $U_{(-)}\simeq\frac{f_{1}^{2}}{4\chi_{2}\,f_{2}}$ and
$U_{(+)}\simeq\frac{M_{1}^{2}}{4\chi_{2}\,M_{2}}$, accordingly. Therefore,
choosing $f_{1}\sim f_{2}\sim 10^{-8}M^{4}_{Pl}$ conforming to the
inflationary scale, and taking $M_{1}\sim M^{4}_{EW}$ and $M_{2}\sim
M^{4}_{Pl}$ we achieve $U_{(+)}\sim 10^{-122}M^{4}_{Pl}$ vastly smaller than
$U_{(-)}$. If we take $\alpha\to-\alpha$ in (26) the roles of $f_{1,2}$ and
$M_{1,2}$ are interchanged.

Similar “seesaw” effect is found in Refs.Benisty:2019vej ; Benisty:2020xqm
where the scalar potential is extracted from the slow-roll parameters. 111The
paper Benisty:2020xqm  was awarded second prize in the 2020 Essay Competition
of the Gravity Research Foundation.. Furthermore, the NRVF model (26) yields
in EInstein-frame “emergent universe” solution for the range of the
$b$-parameter: $-4(2-\sqrt{3})<b\frac{f_{1}}{f_{2}}<-1$.

Figure 2: Qualitative shape of the one-dimensional plot for the effective
scalar potential $U_{eff}$.

##  4 Dynamical Generation of Inflation in Metric Formulation

Let us now consider a substantionally truncated version of the model (17)
without any matter fields, involving few non-Riemannian volume elements
Benisty:2019tno :

$S=\int
d^{4}x\Bigl{\\{}\Phi_{1}(A)\Bigl{[}R(g)-2\Lambda_{0}\frac{\Phi_{1}(A)}{\sqrt{-g}}\Bigr{]}+\Phi_{2}(B)\frac{\Phi_{0}(C)}{\sqrt{-g}}\Bigr{\\}}\;,$
(28)

where now unlike (17) $R(g)\equiv
g^{\mu\nu}R_{\mu\nu}\bigl{(}\Gamma(g)\bigr{)}$ is the scalar curvature in the
second order (metric) formalism ($\Gamma^{\lambda}_{\mu\nu}(g)$ being the
Levi-Civita connection w.r.t. $g_{\mu\nu}$).

The equations of motion w.r.t. auxiliary tensor gauge fields
$A_{\mu\nu\lambda}$, $\Phi_{2}(B)$ and $\Phi_{1}(C)$ are special cases of the
dynamical constraint Eqs.(18)-(19) with all matter field terms being zero,
which again introduce the three free integration constants
$M_{1,2},\,\chi_{2}$.

Passage to the physical Einstein frame is again realized via the conformal
transformation (21), however this time we have to use the well-known formulas
for conformal transformations within the metric formalism (Dabrowski:2008kx ;
bars indicate magnitudes in the ${\bar{g}}_{\mu\nu}$-frame):

$\displaystyle
R_{\mu\nu}(g)=R_{\mu\nu}(\bar{g})-3\frac{{\bar{g}}_{\mu\nu}}{\chi_{1}}{\bar{g}}^{\kappa\lambda}\partial_{\kappa}\chi_{1}^{1/2}\partial_{\lambda}\chi_{1}^{1/2}+\chi_{1}^{-1/2}\bigl{(}{\bar{\nabla}}_{\mu}{\bar{\nabla}}_{\nu}\chi_{1}^{1/2}+{\bar{g}}_{\mu\nu}{\bar{\Box}}\chi_{1}^{1/2}\bigr{)}\;,$
(29)

with $\chi_{1}\equiv\frac{\Phi_{1}(A)}{\sqrt{-g}}$. Redefining $\chi_{1}$ as
$\chi_{1}=\exp{\bigl{(}u/\sqrt{3}\bigr{)}}$ allows to write the Einstein-frame
NRVF action in the form:

$\displaystyle S_{\rm EF}=\int
d^{4}x\sqrt{-{\bar{g}}}\Bigl{[}R({\bar{g}})-\frac{1}{2}{\bar{g}}^{\mu\nu}\partial_{\mu}u\partial_{\nu}u-U_{\rm
eff}(u)\Bigr{]}\;,$ (30) $\displaystyle U_{\rm
eff}(u)=2\Lambda_{0}-M_{1}\exp{\bigl{(}-\frac{u}{\sqrt{3}}\bigr{)}}+\chi_{2}M_{2}\exp{\bigl{(}-2\frac{u}{\sqrt{3}}\bigr{)}}\;.$
(31)

Thus, from the original pure-gravity NRVF action (28) we derived a physical
Einstein-frame action (30)-(31) containing a dynamically created scalar field
$u$ with a non-trivial effective scalar potential $U_{\rm eff}(u)$ (31)
entirely dynamically generated by the initial non-Riemannian volume elements
in (28) because of the appearance of the free integration constants
$M_{1,2},\,\chi_{2}$ in their respective equations of motion. There are two
main features of the effective potential (31) which are relevant for
cosmological applications with the dynamically created field $u$ as an
“inflaton”.

  * •

$U_{\rm eff}(u)$ (31) possesses one flat region for large positive values of
$u$ where $U_{\rm eff}(u)\simeq 2\Lambda_{0}$, which corresponds to “early”
universe’ inflationary evolution with energy scale $2\Lambda_{0}$.

  * •

$U_{\rm eff}(u)$ (31) has a stable minimum for a small finite value $u=u_{*}$
where $e^{-u_{*}/\sqrt{3}}=M_{1}/(2\chi_{2}M_{2})$.

  * •

The region around the stable minimum at $u=u_{*}$ correspond to “late”
universe’ evolution where the minimum value of the potential:

$U_{\rm eff}(u_{*})=2\Lambda_{0}-\frac{M_{1}^{2}}{4\chi_{2}M_{2}}\equiv
2\Lambda_{\rm DE}$ (32)

is the dark energy density value.

Remark. The effective potential $U_{\rm eff}$ (31) generalizes the well-known
Starobinsky inflationary potential Starobinsky:1979ty  ((23) reduces to
Starobinsky potential upon taking the following special values for the
parameters: $\Lambda_{0}=\frac{1}{4}M_{1}=\frac{1}{2}\chi_{2}M_{2}$).

In Ref.Benisty:2019tno  a thorough analysis has been performed of the slow-
roll inflationary dynamics driven by the dynamically created “inflaton” $u$
with its dynamically generated effective potential (31), including explicit
calculation of the standard slow-roll parameters $\epsilon$ and $\eta$, as
well we have obtained explicit expressions for the tensor-to-scalar ratio $r$
and the scalar spectral index $n_{s}$ of density perturbations as functions of
the number of e-folds ${\mathcal{N}}=\log a$ ($a$ being the Friedmann scale
factor):

$\displaystyle
r\simeq\frac{12}{\Bigl{[}{\mathcal{N}}+\frac{\sqrt{3}}{4}u_{i}({\mathcal{N}})+c_{0}\Bigr{]}^{2}},\quad
n_{s}\simeq 1-\frac{r}{4}-\sqrt{\frac{r}{3}},$ (33)

with
$c_{0}\equiv\frac{\sqrt{3}}{2}-\frac{3}{4}\log\Bigl{(}2\bigl{(}1+2/\sqrt{3}\bigr{)}\Bigr{)}$.
$u_{i}({\mathcal{N}})$ is the value of the “unflaton” at the start of
inflation as function of ${\mathcal{N}}$.

For a plausible assumption about the scales of $M_{1,2},\,\chi_{2}$ and taking
${\mathcal{N}}=60$ $e$-folds till end of inflation the observables are
predicted to be: $n_{s}\approx 0.969\;\;,\;\;r\approx 0.0026$, which conform
to the PLANCK constraints Akrami:2018odb  ($0.95<n_{s}<0.97\;\;,\;\;r<0.064$).

##  5  Dynamical Spacetime Formulation

Let us now observe that the non-Riemannian volume element $\Omega=\Phi(B)$ (3)
on a Riemannian manifold can be rewritten using Hodge duality (here $D=4$) in
terms of a vector field
$\chi^{\mu}=\frac{1}{3!}\frac{1}{\sqrt{-g}}\varepsilon^{\mu\nu\kappa\lambda}B_{\nu\kappa\lambda}$
so that $\Omega$ becomes
$\Omega(\chi)=\partial_{\mu}\bigl{(}\sqrt{-g}\chi^{\mu}\bigr{)}$, i.e. it is a
non-canonical volume element different from $\sqrt{-g}$, but still involving
the metric. It can be represented alternatively through a Lagrangian
multiplier action term yielding covariant conservation of a specific energy-
momentum tensor of the form ${\mathcal{T}}^{\mu\nu}=g^{\mu\nu}{\mathcal{L}}$:

$\mathcal{S}_{(\chi)}=\int
d^{4}x\sqrt{-g}\,\chi_{\mu;\nu}{\mathcal{T}}^{\mu\nu}=\int
d^{4}x\partial_{\mu}\bigl{(}\sqrt{-g}\chi^{\mu}\bigr{)}\bigl{(}-{\mathcal{L}}\bigr{)}\;,$
(34)

where
$\chi_{\mu;\nu}=\partial_{\nu}\chi_{\mu}-\Gamma_{\mu\nu}^{\lambda}\chi_{\lambda}$.
The vector field $\chi_{\mu}$ is called “dynamical space time vector”, because
of the energy density of ${\mathcal{T}}^{00}$ is a canonically conjugated
momentum w.r.t. $\chi_{0}$, which is what we expected from a dynamical time.

In what follows we will briefly consider a new class of gravity-matter
theories based on the ordinary Riemannian volume element $\sqrt{-g}$ but
involving action terms of the form (34) where now ${\mathcal{T}}^{\mu\nu}$ is
of more general form than ${\mathcal{T}}^{\mu\nu}=g^{\mu\nu}{\mathcal{L}}$.
This new formalism is called “dynamical spacetime formalism” Guendelman:2009ck
; Benisty:2016ybt  due to the above remark on $\chi_{0}$.

Ref.Benisty:2018qed  describes a unification between dark energy and dark
matter by introducing a quintessential scalar field in addition to the
dynamical time action. The total Lagrangian reads:

$\mathcal{L}=\frac{1}{2}R+\chi_{\mu;\nu}{\mathcal{T}}^{\mu\nu}-\frac{1}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}-V(\phi),$
(35)

with energy-momentum tensor
${\mathcal{T}}^{\mu\nu}=-\frac{1}{2}\phi^{,\mu}\phi^{,\nu}$. From the
variation of the Lagrangian term $\chi_{\mu;\nu}{\mathcal{T}}^{\mu\nu}$ with
respect to the vector field $\chi_{\mu}$, the covariant conservation of the
energy-momentum tensor $\nabla_{\mu}{\mathcal{T}}^{\mu\nu}=0$ is implemented.
The latter within the FLRW framework forces the kinetic term of the scalar
field to behave as a dark matter component:

$\nabla_{\mu}{\mathcal{T}}^{\mu\nu}=0\quad\Rightarrow\quad\dot{\phi}^{2}=\frac{2\Omega_{m0}}{a^{3}}.$
(36)

where $\Omega_{m0}$ is an integration constant. The variation with respect to
the scalar field $\phi$ yields a current:

$-V^{\prime}(\phi)=\nabla_{\mu}j^{\mu},\quad
j^{\mu}=\frac{1}{2}\phi_{,\nu}(\chi^{\mu;\nu}+\chi^{\nu;\mu})+\phi^{,\mu}$
(37)

For constant potential $V(\phi)=\Omega_{\Lambda}=const$ the current is
covariantly conserved. In the FLRW setting, where the dynamical time ansatz
introduces only a time component $\chi_{\mu}=(\chi_{0},0,0,0)$, the variation
(37) gives:

$\dot{\chi}_{0}-1=\xi\,a^{-3/2},$ (38)

where $\xi$ is an integration constant. Accordingly, the FLRW energy density
and pressure read:

$\rho=\left(\dot{\chi}_{0}-\frac{1}{2}\right)\dot{\phi}^{2}+V,\quad
p=\frac{1}{2}\dot{\phi}^{2}(\dot{\chi}_{0}-1)-V.$ (39)

Plugging the relations (36,38) into the density and the pressure terms (39)
yields the following simple form of the latter:

$\rho=\Omega_{\Lambda}+\frac{\xi\Omega_{m0}}{a^{9/2}}+\frac{\Omega_{m0}}{a^{3}},\quad
p=-\Omega_{\Lambda}+\frac{\xi\Omega_{m0}}{2\,a^{9/2}}.$ (40)

In (40) there are 3 components for the "dark fluid": dark energy with
$\omega_{\Lambda}=-1$, dark matter with $\omega_{m}=0$ and an additional
equation of state $\omega_{\xi}=1/2$. For non-vanishing and negative $\xi$ the
additional part introduces a minimal scale parameter, which avoids
singularities. If the dynamical time is equivalent to the cosmic time
$\chi_{0}=t$, we obtain $\xi=0$ from Eq.(38), whereupon the density and the
pressure terms (40) coincide with those from the $\Lambda$CDM model precisely.
The additional part (for $\xi\neq 0$) fits to the late time accelerated
expansion data Anagnostopoulos:2019myt .

Figure 3: The posterior distribution for the simplest case of DST cosmology in
the parameter space $\xi$ vs. $H_{0}$.

In order to constraint our model, we deploy the following data sets: Cosmic
Chronometers (CC) exploit the evolution of differential ages of passive
galaxies at different redshifts to directly constrain the Hubble parameter
Jimenez:2001gg . We use uncorrelated 30 CC measurements of $H(z)$ discussed in
Moresco:2012by ; Moresco:2012jh ; Moresco:2015cya ; Moresco:2016mzx . As
Standard Candles (SC) we use uncorrelated measurements of the Pantheon Type Ia
supernova Scolnic:2017caz  that were collected in Anagnostopoulos:2020ctz . In
addition, we use the uncorrelated data points from different Baryon Acoustic
Oscillations (BAO) collected in Benisty:2020otr  from Percival:2009xn ;
Beutler:2011hx ; Busca:2012bu ; Anderson:2012sa ; Seo:2012xy ; Ross:2014qpa ;
Tojeiro:2014eea ; Bautista:2017wwp ; deCarvalho:2017xye ; Ata:2017dya ;
Abbott:2017wcz ; Molavi:2019mlh . Studies of the BAO features in the
transverse direction provide a measurement of $D_{H}(z)/r_{d}=c/H(z)r_{d}$,
with the comoving angular diameter distance defined in Hogg:2020ktc ;
Martinelli:2020hud . In our database we use the parameters $D_{A}=D_{M}/(1+z)$
and

$D_{V}(z)\equiv[zD_{H}(z)D_{M}^{2}(z)]^{1/3}.$ (41)

which is a combination of the BAO peak coordinates. $r_{d}$ is the sound
horizon at the drag epoch. Finally, for very precise "line-of-sight" (or
"radial") observations, BAO can also measure directly the Hubble parameter
Benitez:2008fs .

The posterior distribution yields: $\Omega_{m}=0.2811\pm 0.0743$,
$\Omega_{\Lambda}=0.7187\pm 0.07351$, $\xi=(-2.776\pm 3.023)\cdot 10^{5}$ and
with the Hubble parameter $H_{0}=69.72\pm 1.239Mpc/(km/sec)$. Benisty:2018gzx
shows that with higher dimensions, the solution derived from the Lagrangian
(35) describes inflation, where the total volume oscillates and the original
scale parameter exponentially grows.

The dynamical spacetime Lagrangian can be generalized to yield a diffusive
energy-momentum tensor. Ref. Calogero:2013zba  shows that the diffusion
equation has the form:

$\nabla_{\mu}{\mathcal{T}}^{\mu\nu}=3\sigma j^{\nu},\quad j^{\mu}_{;\mu}=0,$
(42)

where $\sigma$ is the diffusion coefficient and $j^{\mu}$ is a current source.
The covariant conservation of the current source indicates the conservation of
the number of the particles. By introducing the vector field $\chi_{\mu}$ in a
different part of the Lagrangian:

$\mathcal{L}_{(\chi,A)}=\chi_{\mu;\nu}{\mathcal{T}}^{\mu\nu}+\frac{\sigma}{2}(\chi_{\mu}+\partial_{\mu}A)^{2},$
(43)

the energy-momentum tensor ${\mathcal{T}}^{\mu\nu}$ gets a diffusive source.
From a variation with respect to the dynamical space time vector field
$\chi_{\mu}$ we obtain:

$\nabla_{\nu}{\mathcal{T}}^{\mu\nu}=\sigma(\chi^{\mu}+\partial^{\mu}A)=f^{\mu},$
(44)

a current source $f^{\mu}=\sigma(\chi^{\mu}+\partial^{\mu}A)$ for the energy-
momentum tensor. From the variation with respect to the new scalar $A$, a
covariant conservation of the current emerges $f^{\mu}_{;\mu}=0$. The latter
relations correspond to the diffusion equation (42). Refs.Benisty:2017eqh ;
Benisty:2017rbw ; Benisty:2017lmt ; Bahamonde:2018uao  study the cosmological
solution using the energy-momentum tensor
${\mathcal{T}}^{\mu\nu}=-\frac{1}{2}g^{\mu\nu}\phi^{,\lambda}\phi_{\lambda}$.
The total Lagrangian reads:

$\mathcal{L}=\frac{1}{2}R-\frac{1}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}-V(\phi)\\\
+\chi_{\mu;\nu}{\mathcal{T}}^{\mu\nu}+\frac{\sigma}{2}(\chi_{\mu}+\partial_{\mu}A)^{2}.$
(45)

The FLRW solution unifies the dark energy and the dark matter originating from
one scalar field with possible diffusion interaction. Ref.Benisty:2018oyy
investigates more general energy-momentum tensor combinations and shows that
asymptotically all of the combinations yield $\Lambda$CDM model as a stable
fixed point. Banerjee:2019kgu  shows that the DST theories and Diffusive
extensions can describe a Lagrangian formulation for Running Vacuum Models.

##  6 Scale Invariance, Fifth Force in Fermionic and Dust Matter

In this class of theories the fifth force problem can be solved, this can be
checked most simply in a theory with two volume elements (integration measure
densities) Guendelman:1999qt ; Guendelman:1999tb , where at least one of them
was a non-canonical one and short-termed “two-measure theory” (TMT). The
result is expected to be generic however. This model has a number of
remarkable properties if fermions are included in a self-consistent way
Guendelman:1999tb . In this case, the constraint that arises in the TMT models
in the Palatini formalism can be represented as an equation for
$\chi\equiv\Phi/\sqrt{-g}$, in which the left side has an order of the vacuum
energy density, and the right side (in the case of non-relativistic fermions)
is proportional to the fermion density. Moreover, it turns out that even cold
fermions have a (non-canonical) pressure $P_{f}^{noncan}$ and the
corresponding contribution to the energy-momentum tensor has the structure of
a cosmological constant term which is proportional to the fermion density. The
remarkable fact is that the right hand side of the constraint coincide with
$P_{f}^{noncan}$. This allows us to construct a cosmological model
Guendelman:2012vc  of the late universe in which dark energy is generated by a
gas of non-relativistic neutrinos without the need to introduce into the model
a specially designed scalar field.

In models with a scalar field, the requirement of scale invariance of the
initial actionGuendelman:1999qt  plays a very constructive role. It allows to
construct a modelGuendelman:2006af  where without fine tuning we have
realized: absence of initial singularity of the curvature; k-essence;
inflation with graceful exit to zero cosmological constant. Of particular
interest are scale invariant models in which both fermions and a dilaton
scalar field $\phi$ are present. Then it turns out that the Yukawa coupling of
fermions to $\phi$ is proportional to $P_{f}^{noncan}$. As a result, it
follows from the constraint, that in all cases when fermions are in states
which constitute a regular barionic matter, the Yukawa coupling of fermions to
dilaton has an order of ratio of the vacuum energy density to the fermion
energy density Guendelman:2006ji . Thus, the theory provides a solution of the
5-th force problem without any fine tuning or a special design of the model.
Besides, in the described states, the regular Enstein’s equations are
reproduced. In the opposite case, when fermions are very deluted, e.g. in the
model of the late Universe filled with a cold neutrino gas, the neutrino dark
energy appears in such a way that the dilaton $\phi$ dynamics is closely
correlated with that of the neutrino gas Guendelman:2006ji .

Scale invariant model containing a dilaton $\phi$ and dust (as a model of
matter)Guendelman:2007ph  possesses similar features. Dilaton to matter
coupling "constant" $f$ appears to be dependent of the matter density. To see
this more explicitly, let us consider the action containing a non Riemannian
measure $\Phi$ which is a total divergence and is invariant under the global
scale transformations:

$g_{\mu\nu}\rightarrow
e^{\theta}g_{\mu\nu},\quad\Gamma^{\mu}_{\alpha\beta}\rightarrow\Gamma^{\mu}_{\alpha\beta},\quad\phi\rightarrow\phi-\frac{M_{p}}{\alpha}\theta,\quad\Phi\rightarrow
e^{2\theta}\Phi$ (46)

where $\theta=const$. It is convenient to represent the action in the
following form:

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{g}+S_{\phi}+S_{m}$ (47)
$\displaystyle S_{g}$ $\displaystyle=$
$\displaystyle-\frac{1}{\kappa}\int(\Phi+b_{g}\sqrt{-g})R(\Gamma,g)e^{\alpha\phi/M_{p}}d^{4}x\,;$
$\displaystyle S_{\phi}$ $\displaystyle=$ $\displaystyle\int
e^{\alpha\phi/M_{p}}\left[(\Phi+b_{\phi}\sqrt{-g})\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-\left(\Phi
V_{1}+\sqrt{-g}V_{2}\right)e^{\alpha\phi/M_{p}}\right]d^{4}x\,;$
$\displaystyle S_{m}$ $\displaystyle=$
$\displaystyle\int(\Phi+b_{m}\sqrt{-g})L_{m}d^{4}x\,,$

where the Lagrangian for the matter, as collection of particles, which
provides the scale invariance of $S_{m}$ reads

$L_{m}=-m\sum_{i}\int
e^{\frac{1}{2}\alpha\phi/M_{p}}\sqrt{g_{\alpha\beta}\frac{dx_{i}^{\alpha}}{d\lambda}\frac{dx_{i}^{\beta}}{d\lambda}}\,\frac{\delta^{4}(x-x_{i}(\lambda))}{\sqrt{-g}}d\lambda$
(48)

where $\lambda$ is an arbitrary parameter. For simplicity we consider the
collection of the particles with the same mass parameter $m$. We assume in
addition that $x_{i}(\lambda)$ do not participate in the scale transformations
(46). We will assume that $d\vec{x}_{i}/d\lambda\equiv 0$ for all particles.
It is convenient to proceed in the frame where $g_{0l}=0$, $l=1,2,3$. Then the
particle density is defined by

$n(\vec{x})=\sum_{i}\frac{1}{\sqrt{-g_{(3)}}}\delta^{(3)}(\vec{x}-\vec{x}_{i}(\lambda))$
(49)

where $g_{(3)}=\det(g_{kl})$ and

$S_{m}=-m\int
d^{4}x(\Phi+b_{m}\sqrt{-g})\,n(\vec{x})\,e^{\frac{1}{2}\alpha\phi/M_{p}}$ (50)

It turns out that when working with the new metric ($\phi$ remains the same)

$\tilde{g}_{\mu\nu}=e^{\alpha\phi/M_{p}}(\zeta+b_{g})g_{\mu\nu},$ (51)

which we call the Einstein frame, the connection becomes Riemannian. Notice
that $\tilde{g}_{\mu\nu}$ is invariant under the scale transformations (46).
The transformation (51) causes the transformation of the particle density

$\tilde{n}(\vec{x})=(\zeta+b_{g})^{-3/2}\,e^{-\frac{3}{2}\alpha\phi/M_{p}}\,n(\vec{x})$
(52)

After the change of variables to the Einstein frame (51) and some simple
algebra, the gravitational equations take the standard GR form

$G_{\mu\nu}(\tilde{g}_{\alpha\beta})=\frac{\kappa}{2}T_{\mu\nu}^{eff}$ (53)

where $G_{\mu\nu}(\tilde{g}_{\alpha\beta})$ is the Einstein tensor in the
Riemannian space-time with the metric $\tilde{g}_{\mu\nu}$. The components of
the effective energy-momentum tensor are as follows:

$\displaystyle T_{00}^{eff}$ $\displaystyle=$
$\displaystyle\frac{\zeta+b_{\phi}}{\zeta+b_{g}}\left(\dot{\phi}^{2}-\tilde{g}_{00}X\right)$
$\displaystyle+$
$\displaystyle\tilde{g}_{00}\left[V_{eff}(\phi;\zeta,M)-\frac{\delta\cdot
b_{g}}{\zeta+b_{g}}X+\frac{3\zeta+b_{m}+2b_{g}}{2\sqrt{\zeta+b_{g}}}\,m\,\tilde{n}\right]$
$\displaystyle T_{ij}^{eff}$ $\displaystyle=$
$\displaystyle\frac{\zeta+b_{\phi}}{\zeta+b_{g}}\left(\phi_{,k}\phi_{,l}-\tilde{g}_{kl}X\right)$
$\displaystyle+$
$\displaystyle\tilde{g}_{kl}\left[V_{eff}(\phi;\zeta,M)-\frac{\delta\cdot
b_{g}}{\zeta+b_{g}}X+\frac{\zeta-
b_{m}+2b_{g}}{2\sqrt{\zeta+b_{g}}}\,m\,\tilde{n}\right]$

Here the following notations have been used:

$X\equiv\frac{1}{2}\tilde{g}^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}\qquad
and\qquad\delta=\frac{b_{g}-b_{\phi}}{b_{g}}$ (56)

and the function $V_{eff}(\phi;\zeta)$ is defined by

$V_{eff}(\phi;\zeta)=\frac{b_{g}\left[M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}\right]-V_{2}}{(\zeta+b_{g})^{2}}$
(57)

The dilaton $\phi$ field equation in the Einstein frame is as follows

$\displaystyle\frac{1}{\sqrt{-\tilde{g}}}\partial_{\mu}\left[\frac{\zeta+b_{\phi}}{\zeta+b_{g}}\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\phi\right]-\frac{\alpha}{M_{p}}\,\frac{(\zeta+b_{g})M^{4}e^{-2\alpha\phi/M_{p}}-(\zeta-
b_{g})V_{1}-2V_{2}-\delta b_{g}(\zeta+b_{g})X}{(\zeta+b_{g})^{2}}$
$\displaystyle=\frac{\alpha}{M_{p}}\,\frac{\zeta-
b_{m}+2b_{g}}{2\sqrt{\zeta+b_{g}}}\,m\,\tilde{n}$ (58)

In the above equations, the scalar field $\zeta$ is determined as a function
$\zeta(\phi,X,\tilde{n})$ by means of the following constraint:

$\displaystyle\frac{(b_{g}-\zeta)\left(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}\right)-2V_{2}}{(\zeta+b_{g})^{2}}-\frac{\delta\cdot
b_{g}X}{\zeta+b_{g}}=\frac{\zeta-
b_{m}+2b_{g}}{2\sqrt{\zeta+b_{g}}}\,m\,\tilde{n}$ (59)

One should now pay attention to the interesting result that the explicit
$\tilde{n}$ dependence involving the same form of $\zeta$ dependence

$\frac{\zeta-b_{m}+2b_{g}}{2\sqrt{\zeta+b_{g}}}\,m\,\tilde{n}$ (60)

appears simultaneously222Note that analogous result has been observed earlier
in the model Guendelman:2012vc ; Guendelman:2006af  where fermionic matter has
been studied instead of the macroscopic (dust) matter in the present model. in
the dust contribution to the pressure (through the last term in Eq. (6)), in
the effective dilaton to dust coupling (in the r.h.s. of Eq. (58)) and in the
r.h.s. of the constraint (59).

Let us analyze consequences of this wonderful coincidence in the case when the
matter energy density (modeled by dust) is much larger than the dilaton
contribution to the dark energy density in the space region occupied by this
matter. Evidently this is the condition under which all tests of Einstein’s
GR, including the question of the fifth force, are fulfilled. if the dust is
in the normal conditions there is a possibility to provide the desirable
feature of the dust in GR: it must be pressureless. This is realized provided
that in normal conditions (n.c.) the following equality holds with extremely
high accuracy:

$\zeta^{(n.c.)}\approx b_{m}-2b_{g}$ (61)

Remind that we have assumed $b_{m}>b_{g}$. Then $\zeta^{(n.c.)}+b_{g}>0$, and
the transformation (51) and the subsequent equations in the Einstein frame are
well defined. Inserting (61) in the last term of Eq. (6) we obtain the
effective dust energy density in normal conditions

$\rho_{m}^{(n.c.)}=2\sqrt{b_{m}-b_{g}}\,m\tilde{n}$ (62)

When we get only a slight deviation of from $\zeta$ from $b_{m}-2b_{g}$, when
the matter energy density is many orders of magnitude larger than the dilaton
contribution to the dark energy density, we obtain an effective 5th force
coupling $f$. For this look at the $\phi$-equation in the form (58) and
estimate the Yukawa type coupling constant in the r.h.s. of this equation. In
fact, using the constraint (59) and representing the particle density in the
form $\tilde{n}\approx N/\upsilon$ where $N$ is the number of particles in a
volume $\upsilon$, one can make the following estimation for the effective
dilaton to matter coupling "constant" $f$ defined by the Yukawa type
interaction term $f\tilde{n}\phi$ (if we were to invent an effective action
whose variation with respect to $\phi$ would result in Eq. (58)):

$f\equiv\alpha\frac{m}{M_{p}}\,\frac{\zeta-
b_{m}+2b_{g}}{2\sqrt{\zeta+b_{g}}}\approx\alpha\frac{m}{M_{p}}\,\frac{\zeta-
b_{m}+2b_{g}}{2\sqrt{b_{m}-b_{g}}}\sim\frac{\alpha}{M_{p}}\,\frac{\rho_{vac}}{\tilde{n}}\approx\alpha\frac{\rho_{vac}\upsilon}{NM_{p}}$
(63)

becomes less than the ratio of the "mass of the vacuum" in the volume occupied
by the matter to the Planck mass. The model yields this kind of "Archimedes
law" without any especial (intended for this) choice of the underlying action
and without fine tuning of the parameters. The model not only explains why all
attempts to discover a scalar force correction to Newtonian gravity were
unsuccessful so far but also predicts that in the near future there is no
chance to detect such corrections in the astronomical measurements as well as
in the specially designed fifth force experiments on intermediate, short (like
millimeter) and even ultrashort (a few nanometer) ranges. This prediction is
alternative to predictions of other known models.

Finally, we want to point out fundamental differences of our solution of the
fifth force force problem to the Chameleon approach. The important point to
make is that we are talking of totally different mechanisms, in the Chameleon
model, the proposed quintessential scalar, the Chameleon field has a mass in
vacuum which is very small, of the order of the Hubble parameter for example
(or in any case very very small). The Chameleon scalar however becomes massive
in presence of dense matter, in compact objects, like Earth, a typical number
for this mass has been cited, $m^{-1}\sim 60_{mm}$ Khoury:2003aq . This is why
a quanta of this scalar field can penetrate only into a thin shell of the body
in the depth about 60micrometer, and the fifth force acts only on the thin
shell. This is a way the Chameleon model is argued to explain the smallness of
the fifth force. In our case there is no mass generation whatsoever since for
our dilaton field, what happens here is the vanishing of the effective
coupling constant between the dilaton field and the dense matter, while the
dilaton keeps is mass zero or very close to zero. The elimination of
interaction between our dilaton field and dense matter is total and absolute,
in comparison, a Chameleon wave can suffer a total reflection from a dense
matter region, in such a situation it will not be a total elimination of the
fifth force, but it may be hard indeed to prepare such an experiment. The
elimination of the fifth force in the Chameleon model is argued to exist
because in a spherically symmetric static configuration of a macroscopic
object only a very small shell of the object can be a source of the Chameleon
scalar, while in our case there would be no source for the scalar, not even
the edge or surface of the dense object or at any place of the dense object.
Higher-order theories of gravity also have been also studied in connection of
fifth force suppression and have been shown to produce an explicit realization
of the Chameleon scenario from first principles Capozziello:2007eu ;
Capozziello:2012ie .

## Acknowledgements

We all are grateful for support by COST Action CA-15117 (CANTATA), COST Action
CA-16104 and COST Action CA-18108. D.B. thanks Ben-Gurion University of the
Negev and Frankfurt Institute for Advanced Studies for generous support. E.N.
and S.P. are partially supported by Bulgarian National Science Fund Grant DN
18/1.

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While addition, subtraction and comparison of large integers can be performed
in linear time (compared to the size of the input), this is no longer the case
for multiplication. In fact, the construction of efficient multiplication
algorithms for large integers is an important problem in the theory of
computation. In practice, multiplication is usually realized by the Toom–Cook
algorithm (see [19, 3]), which for medium-sized inputs is one of the fastest
known algorithm. Given
$\mathtt{x}_{1},\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Z}}$ it computes an
expression $\mathtt{x}_{1}\odot\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Z}}$
representing $\langle\mathtt{x}_{1}\rangle\cdot\langle\mathtt{x}_{2}\rangle$
in time $O(T_{1}(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|))$, where
$T_{1}(N)=N^{\theta}$ with $\theta=\log 5/\log 3\approx 1.465$.

However, for theoretical purposes (or very large inputs), there are
multiplication algorithms which are even faster: Schönhage and Strassen gave
an algorithm which given
$\mathtt{x}_{1},\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Z}}$ computes an
expression $\mathtt{x}_{1}\odot\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Z}}$
representing $\langle\mathtt{x}_{1}\rangle\cdot\langle\mathtt{x}_{2}\rangle$
in time $O(T_{2}(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|))$, where
$T_{2}(N):=N\log(N)\log\log(N)$ and conjectured that the optimal time
complexity of such an algorithm should be
$O(T_{3}(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|))$, where $T_{3}(N):=N\log(N)$.
An algorithm with this time complexity was provided very recently by Harvey
and van der Hoeven (see [11]), but it is still unknown whether its time
complexity is optimal. Moreover, no superlinear lower bound is currently
known.

######  Convention A.3.

From now on we fix a function $T$ with the following properties:

  1. (T1)

$T$ is superadditive, i.e. $T(N_{1}+N_{2})\geq T(N_{1})+T(N_{2})$.

  2. (T2)

$T$ is asymptotically at least linear, i.e. $N=O(T(N))$.

  3. (T3)

For each $C>1$ one has $T(N)=O(T(CN))$.

  4. (T4)

Multiplication of integers of size $N_{1}$ and $N_{2}$ can be performed in
time $O(T(N_{1}+N_{2}))$.

It is easy to check that the functions $T_{1}$, $T_{2}$ and $T_{3}$ above all
satisfy conditions (T1)–(T3); they also satisfy (T4) by the work of Toom–Cook,
Schönhage–Strassen and Harvey–van der Hoeven respectively.

###  A.2 A complexity convexity lemma

In the previous subsection we have seen that the time complexity of integer
multiplication is governed by a function $T$ which is both superadditive and
asymptotically at least linear. These properties of $T$ have the following
crucial consequence:

######  Lemma A.4.

Let $g$ be a real-valued function and let $T$ be a real-valued function
satisfying Properties (T1) and (T2) of Convention A.3. If $g(x)=O(T(x))$, then

$g(x_{1})+\dots+g(x_{k})=O(T(x_{1}+\dots+x_{k})),$

where the implied constants are independent of $k$.

More explicitly, the conclusion of the lemma says that

$\limsup\left\\{\frac{g(x_{1})+\dots+g(x_{k})}{T(x_{1}+\dots+x_{k})}\mid
k\in\mathbb{N},x_{1},\dots,x_{k}\in\mathbb{N}\right\\}<\infty.$

###### Proof.

Let $x:=x_{1}+\dots+x_{k}$. Since $x_{j}\in\mathbb{N}$ we have $x_{j}\geq 1$
for all $j=1,\dots,k$ and hence $k\leq x$. Since $g(x)=O(T(x))$ there exist
constants $C_{0},C_{1},C_{2}\in\mathbb{N}$ such that

$g(x)\leq\left\\{\begin{array}[]{ll}C_{1}T(x),&\text{if }x>C_{0},\\\
C_{2},&\text{if }x\leq C_{0}.\end{array}\right.$

Up to reordering the $x_{j}$ we may assume that $x_{1},\dots,x_{r}>C_{0}$ and
$x_{r+1},\dots,x_{k}\leq C_{0}$. Then using superadditivity of $T$ we obtain

$\displaystyle g(x_{1})+\dots+g(x_{k})$ $\displaystyle\leq$ $\displaystyle
C_{1}T(x_{1})+\dots+C_{1}T(x_{r})+C_{2}+\dots+C_{2}$ $\displaystyle=$
$\displaystyle C_{1}(T(x_{1})+\dots+T(x_{r}))+C_{2}(k-r)$ $\displaystyle\leq$
$\displaystyle C_{1}T(x_{1}+\dots+x_{r})+C_{2}k$ $\displaystyle\leq$
$\displaystyle C_{1}T(x)+C_{2}x.$

Since $T$ is asymptotically at least linear, the lemma follows. ∎

The relevance of this property for our runtime analysis is explained in Remark
4.1.

###  A.3 Encoding arithmetic of rational numbers

We now encode the field $\mathbb{Q}$ of rational numbers. We will store such
numbers as mixed fractions, and it will be crucial for many of our algorithms
to allow _unreduced mixed fractions_ to appear in the computations. In
particular, this is necessary in order to implement addition of rational
numbers efficiently (since reducing fractions takes even more time than
multiplication). Working with unreduced mixed fractions will have the side-
effect that infinitely many different words will represent the same rational
number, but this will not cause any problems.

To define our encoding we set
$\Sigma_{\mathbb{Q}}:=\\{\mathtt{0},\mathtt{1},\mathtt{+},\mathtt{-},\mathtt{/}\\}$
and define

$\mathtt{Num}_{\mathfrak{\mathbb{Q}}}:=\\{\varepsilon\\}\cup\\{\mathtt{s}\mathtt{K}/\mathtt{m}/\mathtt{n}\mid\mathtt{s}\in\\{\mathtt{+},\mathtt{-}\\},\;\mathtt{K,m,n}\in\mathtt{Num}_{\mathbb{N}_{0}},\;\langle\mathtt{m}\rangle<\langle\mathtt{n}\rangle\\}.$

We then define an encoding

$\mathtt{Num}_{\mathbb{Q}}\to\mathbb{Q},\quad\mathtt{x}\mapsto\langle\mathtt{x}\rangle$

as follows: The emptyword is interpreted as $0$, and if
$K,m,n\in\mathtt{Num}_{\mathbb{N}_{0}}$, then we set

$\langle\mathtt{s}\mathtt{K}/\mathtt{m}/\mathtt{n}\rangle:=\langle\mathtt{s}\mathtt{K}\rangle+\frac{\langle\mathtt{m}\rangle}{\langle\mathtt{n}\rangle}.$

Since we do not require $\mathtt{m}$ and $\mathtt{n}$ to be relatively prime,
the map $\mathtt{x}\to\mathtt{\langle x\rangle}$ is not injective. For
example, $\mathtt{-11/10/101}$ and $\mathtt{-11/100/1010}$ both represent
$-3\frac{2}{5}=-3\frac{4}{10}$. If
$\mathtt{s}\mathtt{K}/\mathtt{m}/\mathtt{n}\in\mathtt{Num}_{\mathbb{Q}}\setminus\\{\varepsilon\\}$,
then the amount of memory needed to store
$\mathtt{s}\mathtt{K}/\mathtt{m}/\mathtt{n}$ is given by

$|\mathtt{s}\mathtt{K}/\mathtt{m}/\mathtt{n}|_{\Sigma_{\mathbb{Q}}}=\lceil\log_{2}(|\langle\mathtt{K}\rangle|+1)\rceil+\lceil\log_{2}(|\langle\mathtt{m}\rangle|+1)\rceil+\lceil\log_{2}(|\langle\mathtt{n}\rangle|+1)\rceil+3,$

whereas $|\varepsilon|_{\Sigma_{\mathbb{Q}}}=0$. However, it will be
computationally convenient to work with a slightly different notion of size:
We define the _size_ of an element of $\mathtt{Num}_{\mathfrak{\mathbb{Q}}}$
by $\|\varepsilon\|:=0$ and

$\|\mathtt{s}\mathtt{K}/\mathtt{m}/\mathtt{n}\|:=\lceil\log_{2}(|\langle\mathtt{K}\rangle|+1)\rceil+2\lceil\log_{2}(|\langle\mathtt{n}\rangle|+1)\rceil+3,$
(A.1)

Since $0\leq m<n$ we then have

$\|x\|<|x|_{\Sigma_{\mathbb{Q}}}<2\|x\|,$ (A.2)

hence we do not lose anything by working with the size $\|\cdot\|$. The main
advantage of our notion of size is that it behaves better with respect to
arithmetic operations, as the following lemma shows.

######  Lemma A.5.

Let $T$ be a function as in Convention A.3. Then for all
$\mathtt{x}_{1},\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Q}}$ there exist
elements
$\mathtt{x}_{3}=:\mathtt{x}_{1}\oplus\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Q}}$
and
$\mathtt{x}_{4}=:\mathtt{x}_{1}\ominus\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Q}}$
of size
$\|\mathtt{x}_{3}\|,\|\mathtt{x}_{4}\|\leq\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|$
which can be computed in time $O(T(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|))$
and satisfy
$\langle\mathtt{x}_{3}\rangle=\langle\mathtt{x}_{1}\rangle+\langle\mathtt{x}_{2}\rangle$
and
$\langle\mathtt{x}_{4}\rangle=\langle\mathtt{x}_{1}\rangle-\langle\mathtt{x}_{2}\rangle$.

###### Proof.

The case of subtraction follows immediately from the case of addition, so we
will focus on the latter. The cases where either $\mathtt{x}_{1}$ or
$\mathtt{x}_{2}$ are empty are obvious.

Thus assume $\mathtt{x}_{1}=\mathtt{s_{1}K_{1}/m_{1}/n_{1}}$ and
$\mathtt{x}_{2}=\mathtt{s_{2}K_{2}/m_{2}/n_{2}}$ and let
$K_{1},m_{1},n_{1},K_{2},m_{2},n_{2}$ denote the respective interpretations of
$\mathtt{K_{1},m_{1},n_{1},K_{2},m_{2},n_{2}}$.

First, consider the case when both signs $\mathtt{s_{1}}$ and $\mathtt{s_{2}}$
are positive. The sum of the fractional parts of $\mathtt{x}_{1}$ and
$\mathtt{x}_{2}$ is

$\frac{m_{1}}{n_{1}}+\frac{m_{2}}{n_{2}}=\frac{p}{q},\quad\text{where
}{p:=m_{1}n_{2}+m_{2}n_{1}}\text{ and }{q:=n_{1}n_{2}}.$

We thus define

$\mathtt{p}:=(\mathtt{m}_{1}\otimes\mathtt{n}_{2})\oplus(\mathtt{m}_{2}\otimes\mathtt{n}_{1})\text{
and }\mathtt{q}:=\mathtt{n}_{1}\otimes\mathtt{n}_{2}.$

It is clear from the formulas that $\mathtt{p}$ and $\mathtt{q}$ both have
size $O(||\mathtt{x}_{1}||+||\mathtt{x}_{2}||)$ and in view of Properties (T1)
and (T3) of $T$ we can compute them in time
$O(T(||\mathtt{x}_{1}||+||\mathtt{x}_{2}||))$ using $3$ multiplication and $2$
addition routines.

Once $\mathtt{p}$ and $\mathtt{q}$ have been computed, we can compute the
integer part of the fraction $p/q$ in linear time
$O(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|)$. Indeed, since $0\leq m_{1}<n_{1}$
and $0\leq m_{2}<n_{2}$, we have $m_{1}n_{2}<n_{1}n_{2}$ and
$m_{2}n_{1}<n_{1}n_{2}$, hence $p<2q$. Then $K^{\prime}=\lfloor p/q\rfloor$
can be equal to $0$ or $1$, so to find $K^{\prime}$ we need only to check the
condition $p<q$. If it is true, then $K^{\prime}=0$, otherwise $K^{\prime}=1$,
and this can be checked in time $O(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|)$. If
$K^{\prime}=1$ we can also compute
$\mathtt{p}^{\prime}=\mathtt{p}\ominus\mathtt{q}$ in linear time. Note that in
this case $p/q=K^{\prime}\frac{p^{\prime}}{q}$, where
$p^{\prime}:=\langle\mathtt{p}^{\prime}\rangle$.

Finally, define $\mathtt{K}:=\mathtt{K_{1}}\oplus\mathtt{K}_{2}$ if
$K^{\prime}=0$ and
$\mathtt{K}:=\mathtt{K}_{1}\oplus\mathtt{K}_{2}\oplus(\mathtt{+1})$ if
$K^{\prime}=1$. Again, this can be computed in linear time. If $K^{\prime}=0$
then we define $\mathtt{x}_{3}:=\mathtt{+K}/\mathtt{p}/{\mathtt{q}}$. If
$K^{\prime}=1$, then we define
$\mathtt{x}_{3}:=\mathtt{+K}/\mathtt{p^{\prime}}/{\mathtt{q}}$. In either
case, $\mathtt{x}_{3}$ represents
$\langle\mathtt{x}_{1}\rangle+\langle\mathtt{x}_{2}\rangle$ and has been
computed in time $O(T(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|))$.

It remains to prove the inequality
$||\mathtt{x}_{3}||\leq||\mathtt{x}_{1}||+||\mathtt{x}_{2}||$. For any
$n_{1},n_{2}\geq 1$ we have

$\lceil\log_{2}(n_{1}n_{2}+1)\rceil\leq\log_{2}((n_{1}+1)(n_{2}+1))+1\leq\log_{2}(n_{1}+1)+\log_{2}(n_{2}+1)+1.$

Hence the denominator $n_{1}n_{2}$ can be stored as a word of length

$\lceil\log_{2}(n_{1}+1)\rceil+\lceil\log_{2}(n_{2}+1)\rceil+1.$ (A.3)

Since the nominator of the fractional part (which is $p$ or $p^{\prime}$) is
strictly smaller than the denominator $n_{1}n_{2}$, it can be stored as a word
of smaller length.

Similarly, for any $K_{1},K_{2}\geq 1$ we have

$\lceil\log_{2}(K_{1}+K_{2}+2)\rceil\leq\log_{2}((K_{1}+1)(K_{2}+1))+1\leq\log_{2}(K_{1}+1)+\log_{2}(K_{2}+1)+1.$

In the cases $0\leq K_{1},K_{2}\leq 1$ this inequality is also true and is
easily verified. This assures that $\mathtt{K}$ can be stored using
$\lceil\log_{2}(K_{1}+1)\rceil+\lceil\log_{2}(K_{2}+1)\rceil+1$ symbols.
Adding to this $1$ symbol for the sign, $2$ symbols for the separators and two
values of (A.3) which we use to save the fractional part, we see that
$\mathtt{x}_{3}$ less than $N$ symbols altogether, where

$N=\lceil\log_{2}(K_{1}+1)\rceil+\lceil\log_{2}(K_{2}+1)\rceil+2(\lceil\log_{2}(n_{1}+1)\rceil+\lceil\log_{2}(n_{2}+1)\rceil)+6=||\mathtt{x}_{1}||+||\mathtt{x}_{2}||.$

This finishes the proof in the case
$\mathtt{s}_{1}=\mathtt{s}_{2}=\mathtt{+}$.

If the signs $s_{1}$ and $s_{2}$ are both negative, we make the same
computations, but define instead
$\mathtt{x}_{3}:=\mathtt{-K}/\mathtt{p}/{\mathtt{q}}$ (or
$\mathtt{x}_{3}:=\mathtt{-K}/\mathtt{p^{\prime}}/{\mathtt{q}}$ if
$K^{\prime}=1$). Clearly this does not influence the result.

If the signs $s_{1}$ and $s_{2}$ are different then the computations are
similar, except that some sums are replaced by differences and we need a few
more comparisons to produce the integer part and the sign. The time complexity
is still $O(T(\|\mathtt{x}_{1}\|+\|\mathtt{x}_{2}\|))$, and the size
estimations are exactly the same since the denominator of the fractional part
is again equal to $n_{1}n_{2}$ and the absolute value of the integral part
does not exceed $|K_{1}|+|K_{2}|+1$. ∎

Concerning comparison of rational numbers we have the following statement:

######  Lemma A.6.

Let $\mathtt{x}_{1},\mathtt{x}_{2}\in\mathtt{Num}_{\mathbb{Q}}$. Then we can
decide whether $\mathtt{x}_{1}\equiv\mathtt{x}_{2}$ or not in time
$O(T(||\mathtt{x}_{1}||+||\mathtt{x}_{2}||))$.

###### Proof.

With the same notation as in the proof of Lemma A.5, the lemma follows from
the straight-forward check of one of the equalities
$K_{1}n_{1}+m_{1}=K_{2}n_{2}+m_{2}$ or $K_{1}n_{1}+m_{1}=-(K_{2}n_{2}+m_{2})$.
∎

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Authors’ addresses:
Tobias Hartnick
Institut für Algebra und Geometrie, KIT,
Englerstr. 2, 76131 Karlsruhe, Germany
<EMAIL_ADDRESS>

Alexey Talambutsa
Steklov Mathematical Institute of Russian Academy of Sciences,
Gubkina Str. 8, 119991, Moscow, Russia
<EMAIL_ADDRESS>