#
An alternative to dark matter and dark energy: Scale-dependent gravity
in superfluid vacuum theory

Konstantin G. Zloshchastiev  https://orcid.org/0000-0002-9960-2874 Institute
of Systems Science, Durban University of Technology, P.O. Box 1334, Durban
4000, South Africa

(received: 17 June 2020 [RG], 29 August 2020 [MDPI], 26 November 2020 [arXiv])

###### Abstract

We derive an effective gravitational potential, induced by the quantum
wavefunction of a physical vacuum of a self-gravitating configuration, while
the vacuum itself is viewed as the superfluid described by the logarithmic
quantum wave equation. We determine that gravity has a multiple-scale pattern,
to such an extent that one can distinguish sub-Newtonian, Newtonian, galactic,
extragalactic and cosmological terms. The last of these dominates at the
largest length scale of the model, where superfluid vacuum induces an
asymptotically Friedmann-Lemaître-Robertson-Walker-type spacetime, which
provides an explanation for the accelerating expansion of the Universe. The
model describes different types of expansion mechanisms, which could explain
the discrepancy between measurements of the Hubble constant using different
methods. On a galactic scale, our model explains the non-Keplerian behaviour
of galactic rotation curves, and also why their profiles can vary depending on
the galaxy. It also makes a number of predictions about the behaviour of
gravity at larger galactic and extragalactic scales. We demonstrate how the
behaviour of rotation curves varies with distance from a gravitating center,
growing from an inner galactic scale towards a metagalactic scale: a squared
orbital velocity’s profile crosses over from Keplerian to flat, and then to
non-flat. The asymptotic non-flat regime is thus expected to be seen in the
outer regions of large spiral galaxies.

###### pacs:

04.60.Bc, 95.35.+d, 95.36.+x, 03.75.Kk

Keywords: quantum gravity; cosmology; superfluid vacuum; emergent spacetime;
dark matter; galactic rotation curve; quantum Bose liquid; logarithmic fluid;
logarithmic wave equation

††preprint: 17th Russian Gravitational Conference - International Conference
on Gravitation, Cosmology and Astrophysics (RUSGRAV-17) ††preprint: Universe
6, 180 (2020) [DOI: 10.3390/universe6100180]

##  I Introduction

Astronomical observations over many length scales support the existence of a
number of novel phenomena, which are usually attributed to dark matter (DM)
and dark energy (DE). Dark matter was introduced to explain a range of
observed phenomena at a galactic scale, such as flat rotation curves, while
dark energy is expected to account for cosmological-scale dynamics, such as
the accelerating expansion of the Universe. For instance, the $\Lambda$CDM
model, which is currently the most popular approach used in cosmology and
galaxy-scale astrophysics, makes use of both DE and cold DM concepts car01 .
In spite of being a generally successful framework purporting to explain the
large-scale structure of the Universe, it currently faces certain challenges
bb17 ; ty18 .

There is also growing consensus that a convincing theory of DM- and DE-
attributed phenomena cannot be a stand-alone model; but should, instead, be a
part of a fundamental theory involving all known interactions. In turn, we
contend that formulating this fundamental theory will be impossible without a
clear understanding of the dynamical structure of the physical vacuum, which
underlies all interactions that we know of. Moreover, this theory must operate
at a quantum level, which necessitates us rethinking of the concept of gravity
using basic notions of quantum mechanics.

One of the promising candidates for a theory of physical vacuum is superfluid
vacuum theory (SVT), a post-relativistic approach to high-energy physics and
gravity. Historically, it evolved from Dirac’s idea of viewing the physical
vacuum as a nontrivial quantum object, whose phase and derived velocity are
non-observable in a quantum-mechanical sense dir51 . The term ‘post-
relativistic’, in this context, means that SVT can generally be a non-
relativistic theory; which nevertheless contains relativity as a special case,
or limit, with respect to some dynamical value such as momentum (akin to
general relativity being a superset of the Newton’s theory of gravity).
Therefore, underlying three-dimensional space would not be physically
observable until an observer goes beyond the above-mentioned limit, as will be
discussed in more detail later in this article.

The dynamics and structure of superfluid vacuum are being studied, using
various approaches which agree upon the main paradigm (physical vacuum being a
background quantum liquid of a certain kind, and elementary particles being
excitations thereof), but differ in their physical details, such as an
underlying model of the liquid volbook ; huabook ; z11appb .

It is important to work with a precise definition of superfluid, to ensure
that we avoid the most common misconceptions which otherwise might arise when
one attempts to apply superfluid models to astrophysics and cosmology, some
details can be found in Appendix A. In fact, some superfluid-like models of
dark matter based on classical perfect fluids, scalar field theories or
scalar-tensor gravities, turned out to be vulnerable to experimental
verification lmo19 . Moreover, superfluids are often confused not only with
perfect fluids, but also with the concomitant phenomenon of Bose-Einstein
condensates (BEC), which is another kind of quantum matter occurring in low-
temperature condensed matter psbook04 . However, even though BEC’s do share
certain features with superfluids, this does not imply that they are
superfluidic in general.

In particular, quantum excitations in laboratory superfluids that we know of
have dispersion relations of a distinctive shape called the Landau “roton”
spectrum. Such a shape of the spectral curve is crucial, as it ensures the
suppression of dissipative fluctuations at a quantum level z12eb ; z20ijmpa ,
which results in inviscid flow kap38 ; am38 . If plotted as an excitation
energy versus momentum, the curve starts from the origin, climbs up to a local
maximum (called the maxon peak), then slightly descends to a local nontrivial
minimum (called the “roton” energy gap); then grows again, this time all the
way up, to the boundary of the theory’s applicability range. In fact it is not
the roton energy gap alone, but the energy barrier formed by the maxon peak
and roton minimum in momentum space, which ensures the above-mentioned
suppression of quantum fluctuations in quantum liquid and, ultimately, causes
its flow to become inviscid. In other words, it is the global characteristics
of the dispersion curve, not just the existence of a nontrivial local minimum
and related energy gap, which is important for superfluidity to occur.
Obviously, these are non-trivial properties, which cannot possibly occur in
all quantum liquids and condensates. Further details and aspects are discussed
in Appendix A.

This paper is organized as follows. Theory of physical vacuum based on the
logarithmic superfluid model is outlined in Section II, where we also
demonstrate how four-dimensional spacetime can emerge from the three-
dimensional dynamics of quantum liquid. In Section III, we derive the
gravitational potential, induced by the logarithmic superfluid vacuum in a
given state, using certain simplifying assumptions. Thereafter, in Section IV,
we give a brief physical interpretation of different parts of the derived
gravitational potential and estimate their characteristic length scales. In
Section V, profiles of induced matter density are derived and discussed for
the case of spherical symmetry. Galactic scale phenomena are discussed in
Section VI, where the phenomenon of galactic rotation curves is explained
without introducing any exotic matter ad hoc. In Section VII, we discuss the
various mechanisms of the accelerating expansion of the Universe, as well as
the cosmological singularity, “vacuum catastrophe” and cosmological
coincidence problems. Conclusions are drawn in Section VIII.

##  II Logarithmic Superfluid Vacuum

Superfluid vacuum theory assumes that the physical vacuum is described, when
disregarding quantum fluctuations, by the fluid condensate wavefunction
$\Psi(\boldsymbol{r},\,t)$, which is a three-dimensional Euclidean scalar. The
state itself is described by a ray in the corresponding Hilbert space,
therefore this wavefunction obeys a normalization condition

$\langle\Psi|\Psi\rangle=\int_{\cal V}\rho\,d{\cal V}={\cal M},$ (1)

where ${\cal M}$ and ${\cal V}$ are the total mass and volume of the fluid,
respectively, and $\rho=|\Psi|^{2}$ is the fluid mass density. The
wavefunction’s dynamics is governed by an equation of a $U(1)$-symmetric
Schrödinger form:

$\left[-i\hbar\,\partial_{t}-\frac{\hbar^{2}}{2m}\boldsymbol{\nabla}^{2}+V_{\text{ext}}(\boldsymbol{r},\,t)+F(|\Psi|^{2})\right]\Psi=0,$
(2)

where $m$ is the constituent particles’ mass,
$V_{\text{ext}}(\boldsymbol{r},t)$ is an external or trapping potential and
$F(\rho)$ is a duly chosen function, which effectively takes into account
many-body effects inside the fluid. This wave equation can be formally derived
as a minimizing condition of an action functional with the following
Lagrangian:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{i\hbar}{2}(\Psi\partial_{t}\Psi^{*}-\Psi^{*}\partial_{t}\Psi)+\frac{\hbar^{2}}{2m}|\boldsymbol{\nabla}\Psi|^{2}+V_{\text{ext}}(\boldsymbol{r},\,t)\,|\Psi|^{2}+{V}(|\Psi|^{2}),$
(3)

where ${V}(\rho)$ equals to a primitive of $F(\rho)$ up to an additive
constant: $F(\rho)={V}^{\prime}(\rho)$; throughout the paper the prime denotes
a derivative with respect to the function’s argument.

In this picture, massless excitations, such as photons, are analogous to
acoustic waves propagating with velocity
$c_{s}\propto\sqrt{|p^{\prime}(\rho)|}$, where fluid pressure $p=p(\rho)$ is
determined via the equation of state. For the system (2), both the equation of
state and speed of sound can be derived using the fluid-Schrödinger analogy,
which was established for a special case in Ref. dmf03 , and generalized for
an arbitrary $F(\rho)$ in works z11appb ; z19mat . In a leading-order
approximation with respect to the Planck constant, we obtain

$p=-\frac{1}{m}\int\\!\rho F^{\prime}(\rho)\,d\rho,\ \
c_{s}^{2}=\frac{1}{m}\rho|F^{\prime}(\rho)|,$ (4)

while higher-order corrections would induce Korteweg-type effects, thus
significantly complicating the subject matter z19mat .

Furthermore, it is natural to require that superfluid vacuum theory must
recover Einstein’s theory of relativity at a certain limit. One can show that
at a limit of low momenta of quantum excitations, often called a “phononic”
limit by analogy with laboratory quantum liquids, Lorentz symmetry does
emerge. This can be easily shown by virtue of the fluid/gravity analogy blv05
, which was subsequently used to formulate the BEC-spacetime correspondence
z11appb ; it can also be demonstrated by using dispersion relations z20ijmpa ;
z11pla , which are generally become deformed in theories with non-exact
Lorentz symmetry sch17 ; csa18 ; ol19 ; plj20 .

This correspondence states that Lorentz symmetry is approximate, while four-
dimensional spacetime is an induced phenomenon, determined by the dynamics of
quantum Bose liquid moving in Euclidean three-dimensional space. The latter is
only observable by a certain kind of observer, a F(ull)-observer. Other
observers, R(elativistic)-observers, perceive this superfluid as a non-
removable background, which can be modeled as a four-dimensional pseudo-
Riemannian manifold. What is the difference between these types of observers?

F-observers can perform measurements using objects of arbitrary momenta and
“see” the fundamental superfluid wavefunction’s evolution in three-dimensional
Euclidean space according to Eq. (2) or an analogue thereof. On the other
hand, R-observers are restricted to measuring only small-momentum small-
amplitude excitations of the background superfluid. This is somewhat analogous
to listening to acoustic waves (phonons) in the conventional Bose-Einstein
condensates, but being unaware of higher-energy particles such as photons or
neutrons.

According to BEC-spacetime correspondence, a R-observer “sees” himself located
inside four-dimensional curved spacetime with a pseudo-Riemannian metric. The
latter can be written in Cartesian coordinates as z11appb :

$g_{\mu\nu}\propto\frac{\rho}{c_{s}}\left(\begin{array}[]{ccc}-\left[c_{s}^{2}-\eta^{2}(\boldsymbol{\nabla}S)^{2}\right]&\vdots&-\eta\boldsymbol{\nabla}~{}S\\\
\cdots\cdots&\cdot&\cdots\\\
-\eta\boldsymbol{\nabla}S&\vdots&\textbf{I}\end{array}\right),$ (5)

where $\eta=\hbar/m$,
$S=S(\boldsymbol{r},\,t)=-i\ln{\left(\Psi(\boldsymbol{r},\,t)/|\Psi(\boldsymbol{r},\,t)|\right)}$
is a phase of the condensate wavefunction written in the Madelung
representation, $\Psi=\sqrt{\rho}\,\exp{(iS)}$, and I is a three-dimensional
unit matrix. To maintain the correct metric signature in Eq. (5), condition
$|c_{s}|>\eta\,|\boldsymbol{\nabla}S|$ must be imposed, which indicates that
$c_{s}$ is the maximum attainable velocity of test particles (i.e., small-
amplitude excitations of the condensate), moving along geodesics on this
induced spacetime. Therefore, $c_{s}$ is the velocity of those excitations of
vacuum, which describe massless particles in the low-momentum limit, whereas
massive test particles move along geodesics of a pseudo-Riemannian manifold
with metric Eq. (5). According to a R-observer, they are freely falling,
independently of their properties including their rest mass.

In this approach, we interpret Einstein field equations not as differential
equations for an unknown metric; but as a definition for an induced stress-
energy tensor, describing some effective matter to which test particles
couple. Therefore, this would be the gravitating matter observed by a
R-observer. We thus obtain

$\widetilde{T}_{\mu\nu}\equiv\kappa^{-1}\left[R_{\mu\nu}(g)-\frac{1}{2}g_{\mu\nu}R(g)\right],$
(6)

where $\kappa={8\pi G}/{c_{(0)}^{2}}$ is the Einstein’s gravitational
constant. An example of usage of this procedure will be considered in Section
VII.1. While Eq. (6) is in fact an assumption, it should hold not only under
the validity of conventional general relativity, but also in other Lorentz-
symmetric theories of gravity which are linear with respect to the Riemann
tensor, because the form of Einstein equations is quite universal (up to a
conformal transform). For other Lorentz-symmetric theories, whose field
equations cannot be transformed into this form, definition (6) can be adjusted
accordingly.

Furthermore, one can see from Eq. (4), that $c_{s}$ contains an unknown
function $F(\rho)$. To determine its form, let us recall that one of the
relativistic postulates implies that velocity $c_{s}$ should not depend on
density, at least in the classical limit. More specifically, at low momenta,
this velocity should tend to the value $c_{(0)}\approx c$, where
$c=2.9979\times 10^{10}$ $\text{cm}\,\text{s}^{-1}$ is called the speed of
light in vacuum, for historical reasons. Recalling Eq. (4), this requirement
can be written as a differential equation z11appb :

$\rho|F^{\prime}(\rho)|=mc_{s}^{2}\approx\text{const}(\rho),$ (7)

where $\text{const}(\rho)$ denotes a function which does not depend on
density. The solution of this differential equation is a logarithmic function:

$F(\rho)=-b\ln{(\rho/\bar{\rho})},$ (8)

where $b$ and $\bar{\rho}$ are generally real-valued functions of coordinates.
The wave Equation (2) thus narrows down to

$i\hbar\partial_{t}\Psi=\left[-\frac{\hbar^{2}}{2m}\boldsymbol{\nabla}^{2}+V_{\text{ext}}(\boldsymbol{r},\,t)-b\ln{(|\Psi|^{2}/\bar{\rho})}\right]\Psi,$
(9)

where $b$ is the nonlinear coupling; $b=b(\boldsymbol{r},\,t)$ in general.
Correspondingly, Equation (4) yields

$p=-(b/m)\rho,\ \ c_{s}=\sqrt{|b|/m},$ (10)

thus indicating that logarithmic Bose liquid behaves like barotropic perfect
fluid; but only when one neglects quantum corrections, and assumes classical
averaging. This reaffirms the statement made in the previous section about the
place of perfect-fluid models when it comes to gravitational phenomena. The
way gravity emerges in the superfluid vacuum picture is entirely different
from those models, as will be demonstrated shortly, after we have specified
our working model.

Some special cases of Eq. (9), for example when $b\to b_{0}=$ const, were
extensively studied in the past, although not for reasons related to quantum
liquids ros68 ; bb76 . There were also extensive mathematical studies of these
equations, to mention just some very recent literature ss18 ; af18 ; aff19 ;
wz19 ; bcs19 ; ct19 ; wtc19 ; lzh19 ; zh20 ; mm20 ; aj20 ; ss20 ; tb20je ;
tb20mj .

Interestingly, wave equations with logarithmic nonlinearity can be also
introduced into fundamental physics independently of relativistic arguments
z11appb ; z10gc ; szm16 . This nonlinearity readily occurs in the theory of
open quantum systems, quantum entropy and information yas78 ; bra91 ; as well
as in the theory of general condensate-like materials, for which
characteristic kinetic energies are significantly smaller than interparticle
potentials z18zna .

One example of such a material is helium II, the superfluid phase of helium-4.
For the latter, the logarithmic superfluid model is known to have been well
verified by experimental data z12eb ; sz19 . Among other things, the
logarithmic superfluid model does reproduce the sought-after Landau-type
spectrum of excitations, discussed in the previous section; detailed
derivations can be found in z12eb . One of underlying reasons for such
phenomenological success is that the ground-state wavefunction of free
(trapless) logarithmic liquid is not a de Broglie plane wave, but a spatial
Gaussian modulated by a de Broglie plane wave. This explains the liquid’s
inhomogenization followed by the formation of fluid elements or parcels; which
indicates that such models do describe fluids, rather than gaseous matter az11
; bo15 ; z17zna ; z18epl ; kdk19 ; z19ijmpb .

To summarize, a large number of arguments to date, both theoretical and
experimental, demonstrate the robustness of logarithmic models in the general
theory of superfluidity. In the next Section we shall demonstrate the
logarithmic superfluid model’s capabilities when assuming superfluidity of the
physical vacuum itself.

In what follows, we shall make use of a minimal inhomogeneous model for the
logarithmic superfluid which was proposed in Ref. z18zna , based on
statistical and thermodynamics arguments. In the F-observer’s picture, its
wave equation can be written as

$i\hbar\partial_{t}\Psi=\left[-\frac{\hbar^{2}}{2m}\boldsymbol{\nabla}^{2}+V_{\text{ext}}(\boldsymbol{r},t)-\left(b_{0}-\frac{q}{r^{2}}\right)\\!\ln{\\!\left(\frac{|\Psi|^{2}}{\bar{\rho}}\right)}\right]\\!\Psi,$
(11)

where $r=|\boldsymbol{r}|=\sqrt{\boldsymbol{r}\cdot\boldsymbol{r}}$ is a
radius-vector’s absolute value, and $b_{0}$ and $q$ are real-valued constants.
For definiteness, let us assume that $b_{0}>0$, because one can always change
the overall signs of the nonlinear term $F(\rho)$ and the corresponding field-
theoretical potential ${V}(\rho)$. As always, this wave equation must be
supplemented with a normalization condition (1), boundary and initial
conditions of a quantum-mechanical type; which ensure the fluid interpretation
of $\Psi$ ry99 .

One can show that nonlinear coupling $b=b(\boldsymbol{r})=b_{0}-q/r^{2}$ is a
linear function of the quantum temperature $T_{\Psi}$, which is defined as a
thermodynamic conjugate of quantum information entropy sometimes dubbed as the
Everett-Hirschman information entropy. The latter can be written as
$S_{\Psi}=-\langle\Psi|\ln{(|\Psi|^{2}/\bar{\rho})}|\Psi\rangle=-\int_{\cal
V}|\Psi|^{2}\ln{(|\Psi|^{2}/\bar{\rho})}\,d{\cal V}$, where a factor
$1/\bar{\rho}$ is introduced for the sake of correct dimensionality, and can
be absorbed into an additive constant due to the normalization condition (1).
Therefore, one can expect that the thermodynamical parameters

$b_{0}=b_{0}(T_{\Psi}),\ q=q(T_{\Psi})$ (12)

are constant at a fixed temperature $T_{\Psi}$. Thus, for a trapless version
of the model (11) we have four parameters, but only two of them, $m$ and
$\bar{\rho}$, are a priori fixed, whereas the other two, $b_{0}$ and $q$, can
vary depending on the environment.

##  III Induced Gravitational Potential

Invoking model (11), while neglecting quantum fluctuations, let us assume that
physical vacuum is a collective quantum state described by wavefunction
$\Psi=\Psi_{\text{vac}}(\boldsymbol{r},\,t)$, which forms a self-gravitating
configuration with a center at $\boldsymbol{r}=0$. Therefore, for this state,
the solution of Eq. (11) is equivalent to the solution of the linear
Schrödinger equation,

$i\hbar\partial_{t}\Psi=\left[-\frac{\hbar^{2}}{2m}\boldsymbol{\nabla}^{2}+V_{\text{eff}}(\boldsymbol{r},\,t)\right]\Psi,$
(13)

for a particle of mass $m$ driven by an effective potential

$\displaystyle V_{\text{eff}}(\boldsymbol{r},t)$ $\displaystyle=$
$\displaystyle
V_{\text{ext}}(\boldsymbol{r},t)-b\ln{\\!\left(\frac{|\Psi_{\text{vac}}(\boldsymbol{r},t)|^{2}}{\bar{\rho}}\right)}$
(14) $\displaystyle=$ $\displaystyle
V_{\text{ext}}(\boldsymbol{r},t)-\left(b_{0}-\frac{q}{r^{2}}\right)\\!\ln{\\!\left(\frac{|\Psi_{\text{vac}}(\boldsymbol{r},t)|^{2}}{\bar{\rho}}\right)}\\!,~{}~{}~{}~{}~{}$

when written in Cartesian coordinates z18zna . If working in curvilinear
coordinates, the last formula must be supplemented with terms which arise
after separating out the angular variables in the wave equation.

In the absence of quantum excitations and other interactions, it is natural to
associate this effective quantum-mechanical potential with the only non-
removable fundamental interaction that we know of: gravity. This
interpretation will be further justified in Section IV. Therefore, in
Cartesian coordinates one can write the induced gravitational potential as

$\Phi(\boldsymbol{r},t)=-\frac{1}{m}V_{\text{eff}}(\boldsymbol{r},t)=\frac{1}{m}\left(b_{0}-\frac{q}{r^{2}}\right)\ln{\\!\left(\frac{|\Psi_{\text{vac}}(\boldsymbol{r},t)|^{2}}{\bar{\rho}}\right)},$
(15)

where we assume that the background superfluid is trapless, i.e., we set
$V_{\text{ext}}=0$. It should also be remembered that in curvilinear
coordinates, this formula must be modified according to the remark after Eq.
(14); but for now we shall disregard any anisotropy and rotation.

It should be noticed that if one regards this potential as a multiplication
operator then its quantum-mechanical average would be related to the Everett-
Hirschman information entropy discussed in the previous section:
$\langle\Phi\rangle\sim
T_{\Psi}\left\langle\Psi\right|\,\ln{(|\Psi|^{2})}\left|\Psi\right\rangle\sim
T_{\Psi}S_{\Psi}$. This not only makes theories of entropic gravity (which are
essentially based on the ideas of Bekenstein, Hawking, Jacobson and others) a
subset of the logarithmic superfluid vacuum approach, but also endows them
with an underlying physical meaning and origin of the entropy implied.

We can see that the induced potential maintains its form as long as the
physical vacuum stays in the state $|\Psi_{\text{vac}}\rangle$. If the vacuum
were to transition into a different state, then it would change its
wavefunction; hence the induced gravitational potential would also change. We
expect that our vacuum is currently in a stable state, which is close to a
ground state or at least to a metastable state, with a sufficiently large
lifetime. It is thus natural to assume that the state
$|\Psi_{\text{vac}}\rangle$ is stationary and rotationally invariant.

As we established earlier, the wavefunction describing such a state should be
the solution of a quantum wave equation containing logarithmic nonlinearity.
In the case of trivial spatial topology and infinite extent, the amplitude of
such a solution is known to be the product of a Gaussian function, which was
mentioned in the previous section, and a conventional quantum-mechanical part,
which is a product of an exponential function, power function and a
polynomial. Thus we can write the amplitude’s general form as:

$|\Psi_{\text{vac}}|=\sqrt{\bar{\rho}}\,\left(\frac{r}{\bar{\ell}}\right)^{\chi_{0}/2}P(r)\,\exp{\\!\left(-\frac{a_{2}}{2\bar{\ell}^{2}}r^{2}+\frac{a_{1}}{2\bar{\ell}}r+\frac{a_{0}}{2}\right)},$
(16)

where $P(r)$ is a polynomial function, $\chi_{0}$ and $a$’s are constants, and
$\bar{\ell}=(m/\bar{\rho})^{1/3}$ is a classical characteristic length scale
of the logarithmic nonlinearity (alternatively, one can choose $\bar{\ell}$
being equal to the quantum characteristic length, $\hbar/\sqrt{mb_{0}}$, which
might be more useful for $\hbar$-expansion techniques). If quantum liquid
occupies an infinite spatial domain then the normalization condition (1)
requires

$a_{2}>0,$ (17)

which is also confirmed by analytical and numerical studies of differential
equations with logarithmic nonlinearity of various types bb76 ; ss18 ; af18 ;
bcs19 ; lzh19 ; z18zna ; ss20 .

Both the form of a function $P(r)$ and the values of $\chi_{0}$ and $a$’s must
be determined by a solution of an eigenvalue problem for the wave equation
under normalization and boundary conditions. At this stage, those conditions
are not yet precisely known; even if they were, we do not yet know which
quantum state our vacuum is currently in. Therefore, these constants’ values
remain theoretically unknown at this stage, yet can be determined empirically.

Furthermore, for the sake of simplicity, let us approximate the power-
polynomial term $(r/\bar{\ell})^{\chi_{0}/2}P(r)$, by the single power
function $(r/\bar{\ell})^{\chi/2}$, where the constant $\chi$ is the best
fitting parameter. Therefore, we can approximately rewrite Eq. (16) as

$|\Psi_{\text{vac}}|^{2}\approx\bar{\rho}\,\exp{\\!\left[-\frac{a_{2}}{\bar{\ell}^{2}}r^{2}+\frac{a_{1}}{\bar{\ell}}r+\chi\ln{\\!\left(\frac{r}{\bar{\ell}}\right)}+a_{0}\right]},$
(18)

which is more convenient for further analytical studies than the original
expression (16). From the empirical point of view, the function (18) can be
considered as a trial function, whose parameters can be fixed using
experimental data following the procedure we describe below.

For the trial solution (18), the normalization condition (1) immediately
imposes a constraint for one of its parameters:

$\text{exp}{(a_{0})}\approx\frac{{\cal M}a_{2}^{(\chi+3)/2}}{2\pi
m}\left[\,\breve{Z}\left(\frac{3}{2},\frac{1}{2}\right)+\frac{a_{1}}{\sqrt{a_{2}}}\,\breve{Z}\left(2,\frac{3}{2}\right)\right]^{-1},$
(19)

where we introduced an auxiliary function
$\,\breve{Z}\left(a,b\right)=\Gamma\left(a+\chi/2\right)\,_{1}F_{1}\\!\left(a+\chi/2,b;a_{1}^{2}/4a_{2}\right)$,
where $\Gamma(a)$ and $\,{}_{1}F_{1}\\!\left(a,b;z\right)$ are the gamma
function and Kummer confluent hypergeometric function, respectively. If values
of $a$’s and $\chi$ are determined, e.g., empirically, then this formula can
be used to estimate the ratio ${\cal M}/m$.

Furthermore, by substituting the trial solution (18) into the definition (15),
we derive the induced gravitational potential as a sum of seven terms:

$\displaystyle\Phi(r)$ $\displaystyle=$
$\displaystyle\Phi_{\text{smi}}(r)+\Phi_{\text{RN}}(r)+\Phi_{\text{N}}(r)$
(20)
$\displaystyle+\Phi_{\text{gal}}(r)+\Phi_{\text{mgl}}(r)+\Phi_{\text{dS}}(r)+\Phi_{0},$

where

$\displaystyle\Phi_{\text{smi}}(r)$ $\displaystyle=$
$\displaystyle-\frac{\chi\,q}{m}\frac{\ln{(r/\bar{\ell})}}{r^{2}}=-\zeta_{\chi
q}c_{b}^{2}\frac{L_{\text{smi}}^{2}\ln{(r/\bar{\ell})}}{r^{2}},~{}~{}~{}$ (21)
$\displaystyle\Phi_{\text{RN}}(r)$ $\displaystyle=$
$\displaystyle-\frac{a_{0}\,q}{m}\frac{1}{r^{2}}=-\zeta_{a_{0}q}c_{b}^{2}\frac{L_{\text{RN}}^{2}}{r^{2}}\,,$
(22) $\displaystyle\Phi_{\text{N}}(r)$ $\displaystyle=$
$\displaystyle-\frac{a_{1}q}{m\bar{\ell}}\frac{1}{r}=-\frac{GM}{r},$ (23)
$\displaystyle\Phi_{\text{gal}}(r)$ $\displaystyle=$
$\displaystyle\frac{\chi\,b_{0}}{m}\ln{(r/\bar{\ell})}=c_{b}^{2}\,\chi\ln{(r/\bar{\ell})},$
(24) $\displaystyle\Phi_{\text{mgl}}(r)$ $\displaystyle=$
$\displaystyle\frac{a_{1}b_{0}}{m\bar{\ell}}r=\zeta_{a_{1}}c_{b}^{2}\frac{r}{L_{\text{mgl}}},$
(25) $\displaystyle\Phi_{\text{dS}}(r)$ $\displaystyle=$
$\displaystyle-\frac{a_{2}b_{0}}{m\bar{\ell}^{2}}r^{2}=-c_{b}^{2}\frac{r^{2}}{L_{\text{dS}}^{2}},$
(26)

and

$\Phi_{0}=\frac{1}{m}\left(a_{0}b_{0}+\frac{a_{2}q}{\bar{\ell}^{2}}\right)=\frac{1}{m}\left(a_{0}b_{0}+\frac{q}{L_{\text{dS}}^{2}}\right)\,$
(27)

is the additive constant. Here, and throughout the paper, we denote the sign
functions by $\zeta$’s: $\zeta_{\alpha}=\,\text{sign}\left(\alpha\right)$, and
use the following notations:

$\displaystyle c_{b}=\sqrt{\frac{b_{0}}{m}},\ GM=\frac{a_{1}q}{m\bar{\ell}},\
L_{\text{smi}}=\sqrt{\frac{|\chi\,q|}{b_{0}}},$ $\displaystyle
L_{\text{RN}}=\sqrt{\frac{|a_{0}q|}{b_{0}}},\
L_{\text{mgl}}=\frac{\bar{\ell}}{|a_{1}|}=\frac{|q|}{mGM},$ (28)
$\displaystyle L_{\text{dS}}=\frac{\bar{\ell}}{\sqrt{a_{2}}},\
L_{\chi}=\frac{\chi\bar{\ell}}{a_{1}}=\frac{\chi q}{mGM},$

where $G$ is the Newton’s gravitational constant as per usual.

Furthermore, Lorentz symmetry emerges in the “phononic” low-momentum limit of
the theory, as discussed in the previous section. Therefore, a R-observer
would perceive the gravity induced by potential (20) as curved four-
dimensional spacetime, which is a local perturbation (not necessarily small)
of the background flow metric, such as the one derived in Section 5.3 of Ref.
z11appb , see Section VII.1 below. In a rotationally invariant case, the line
element of this spacetime can be written in the Newtonian gauge; if
$\Phi(r)/c_{(0)}^{2}\ll 1$, then it can be approximately rewritten in the form

$ds^{2}\approx-
c_{(0)}^{2}\\!\left[1+\frac{2\Phi(r)}{c_{(0)}^{2}}\right]dt^{2}+\frac{dr^{2}}{1+2\Phi(r)/c_{(0)}^{2}}+R^{2}(r)d\sigma^{2},$
(29)

where $R(r)=r\left[1+{\cal O}\left(\Phi(r)/c_{(0)}^{2}\right)\right]\approx
r$, $d\,\sigma^{2}=d\theta^{2}+\,\text{sin}^{2}\theta\,d\varphi^{2}$ is the
line element of a unit two-sphere, and a leading-order approximation with
respect to the Planck constant is implied, as usual. The mapping (29) is valid
for regions where the induced metric maintains a signature ‘$-+++$’, and its
matrix is non-singular. In other regions, such as close vicinities of
spacetime singularities or horizons, the relativistic approximation is likely
to fall outside its applicability range, thus it should be replaced with the
F-observer’s description of reality.

The main simplifying assumptions and approximations underlying the derivation
of our gravitational potential are summarized and enumerated in the Appendix
B.

##  IV Physical Interpretation

It should be noticed that if we did not have a logarithm in the original model
(11), then in Eqs. (14) and (15), then we would not have arrived at the
polynomial functions in Eqs. (20)–(26), which are easily recognizable. This
reaffirms our expectations that the underlying model can be successfully
confirmed by experiment; but first those functions must be endowed with
precise physical meaning.

In this Section, we shall assign a physical interpretation to each term of the
derived gravitational potential. For the sake of brevity, we shall be omitting
an additive constant $\Phi_{0}$, assuming that it is small compared to
$c_{(0)}^{2}$.

###  IV.1 Potential $\Phi_{\text{N}}$ and gravitational mass generation

We begin with term (23), which has the most obvious meaning. In a non-
relativistic picture, it represents Newton’s model of gravity. According to
the BEC-spacetime correspondence manifested through the mapping (29), a
R-observer can observe an effect of the potential $\Phi_{\text{N}}$ by
measuring probe particles moving along geodesics in the Schwarzschild
spacetime:

$ds^{2}_{(\text{N})}\approx-
c_{(0)}^{2}\left(1-\frac{r_{H}}{r}\right)dt^{2}+\frac{dr^{2}}{1-r_{H}/r}+r^{2}d\sigma^{2},$
(30)

where $r_{H}=2GM/c_{(0)}^{2}$ is the Schwarzschild radius. Therefore, in
absence of asymptotically non-vanishing terms, $M$ can be interpreted as the
gravitational mass of the configuration.

This mass can be expressed in terms of superfluid parameters as

$r_{H}=\frac{2a_{1}q}{mc_{(0)}^{2}\bar{\ell}},\ \
\,\text{sign}\left(a_{1}q\right)=\left\\{\begin{array}[]{rll}1&&\text{gravity},\\\
-1&&\text{anti-gravity},\end{array}\right.$ (31)

thus assigning physical meaning to a combination of parameters
$a_{1}q/m\bar{\ell}$. In particular, one can see that a sign of the product
$a_{1}q$ determines whether the $\Phi_{\text{N}}$ interaction is attractive
(gravity) or repulsive (anti-gravity). For most systems that we know of, anti-
gravitational effects have not yet been observed, therefore one can assume
that $M>0$ or

$a_{1}q>0$ (32)

from now on.

Nevertheless, it should be emphasized that the anti-gravity case is not a
priori forbidden in superfluid vacuum theory. Indeed, the spacetime
singularity occurs at $r=0$ in a relativistic picture only, which poses
certain issues for a R-observer, especially in the case of anti-gravity when a
singularity is not covered by an event horizon (the existence of naked
singularities is often doubted, on grounds of the cosmic censorship
hypothesis). However, a F-observer would see no singular behavior in either
case, because the wavefunction $\Psi_{\text{vac}}$ remains regular and
normalizable at each point of space and at any given time – as it should be in
a quantum-mechanical theory. This reaffirms the fact that spacetime
singularities are an artifact of incomplete information accessible to
observers operating with relativistic particles z11appb .

Thus, the mapping from Eqs. (23)–(30) can be used to reformulate black hole
phenomena in the language of continuum mechanics and the theory of
superfluidity; which can resolve certain long-standing problems occurring in
the relativistic theory of gravity. For instance, neglecting asymptotically
non-flat terms for simplicity, one can view Eqs. (23), (30) and (31) as the
gravitational mass generation mechanism: such mass is not a fundamental
notion, but a composite quantum phenomenon induced by the background
superfluid’s dynamics (through the elementary inertial mass $m$ and critical
density $\bar{\rho}$), its quantum temperature (through $q$), and an
exponential part of the condensate’s wavefunction (through $a_{1}$). Such a
mechanism can be thus considered as the quantum-mechanical version of the Mach
principle z11appb . For example, if either $a_{1}$ or $q$ vanish, then the
system would not possess any gravitational mass, but it still would be
gravitating in a non-Newtonian way, if other potentials from Eq. (20) are non-
zero.

###  IV.2 Potential $\Phi_{\text{RN}}$ and abelian charges

Equation (22) represents another potential which can be easily recognized.
According to the mapping (29), potential $\Phi_{\text{RN}}$ is observed by a
R-observer as Reissner-Nordström spacetime, when taken together with the
$\Phi_{\text{N}}$ potential:

$\displaystyle ds^{2}_{(\text{N}+\text{RN})}$ $\displaystyle\approx$
$\displaystyle-
c_{(0)}^{2}\left(1-\frac{r_{H}}{r}+\frac{r_{Q}^{2}}{r^{2}}\right)dt^{2}$ (33)
$\displaystyle+\frac{dr^{2}}{1-r_{H}/r+r_{Q}^{2}/r^{2}}+r^{2}d\sigma^{2},$

where $r_{Q}$ is a characteristic length scale:

$r_{Q}=\sqrt{2}L_{\text{RN}}\frac{c_{b}}{c_{(0)}}=\sqrt{\frac{2|a_{0}q|}{mc_{(0)}^{2}}},$
(34)

provided $a_{0}q<0$.

The Reissner-Nordström metric is known to be associated with the gravitational
field caused by a charge related to an abelian group. An example would be an
electric charge $Q$, which is related to the abelian group $U(1)$ of
electromagnetism. This charge can be thus revealed through the formula

$Q^{2}=\frac{r_{Q}^{2}c_{(0)}^{4}}{k_{e}G}\approx\frac{2|a_{0}q|c^{2}}{k_{e}Gm},$
(35)

where $k_{e}$ is the Coulomb constant. In other words, potential
$\Phi_{\text{RN}}$ describes, together with $\Phi_{\text{N}}$, the
gravitational field created by an object of a charge $Q$ and gravitational
mass $M$.

Thus, from a F-observer’s viewpoint, an electrical charge is not an elementary
notion, but a composite quantum phenomenon, induced by the background
superfluid’s dynamics (through the elementary inertial mass $m$), its quantum
temperature (through $q$), and overall constant coefficient of the
condensate’s wavefunction (through $a_{0}$).

It should be noted also that $\Phi_{\text{RN}}$ is a short-range potential,
therefore, it becomes substantial only at those microscopical length scales,
of an order $r_{QH}=r_{Q}^{2}/r_{H}=\bar{\ell}\,|a_{0}/a_{1}|$ or below. Since
$r_{Q}<r_{H}$ for most objects we know of, we have $0\leqslant r_{QH}<r_{H}$.
Thus, those scales would be causally inaccessible to a R-observer, but a
F-observer would have no problem accessing them, per usual.

###  IV.3 Potential $\Phi_{\text{smi}}$ and strong gravity

As the distance from a gravitating center decreases, it is term (21) which
eventually predominates. According to the mapping (29), this potential
$\Phi_{\text{smi}}$, when taken together with the $\Phi_{\text{N}}$ and
$\Phi_{\text{RN}}$ potentials, induces spacetime with the line element:

$\displaystyle ds^{2}_{(\text{N}+\text{RN}+\text{smi})}\\!$
$\displaystyle\approx$
$\displaystyle\\!-c_{(0)}^{2}\\!\left[1-\frac{r_{H}}{r}+\frac{r_{Q}^{2}}{r^{2}}-\zeta_{\chi
q}\frac{r_{W}^{2}}{r^{2}}\ln{\left(\frac{r}{\bar{\ell}}\right)}\right]\\!dt^{2}+\frac{dr^{2}}{1-r_{H}/r+r_{Q}^{2}/r^{2}-\zeta_{\chi
q}r_{W}^{2}\ln{\left(r/\bar{\ell}\right)}/r^{2}}+r^{2}d\sigma^{2},~{}~{}~{}~{}$
(36)

where $r_{W}$ is a characteristic length scale:

$r_{W}=\sqrt{2}L_{\text{smi}}\frac{c_{b}}{c_{(0)}}=\sqrt{\frac{2|\chi\,q|}{mc_{(0)}^{2}}}.$
(37)

The potential $\Phi_{\text{smi}}$ has a distinctive property: unlike other
sub-Newtonian potentials in Eq. (20), it can switch between repulsive and
attractive regimes, depending on whether the distance is larger or smaller
than $\bar{\ell}$.

The magnitudes of $\Phi_{\text{smi}}$ and $\Phi_{\text{RN}}$ become comparable
at two values of $r$, shown by the formula

$r_{WQ}^{(\pm)}=\bar{\ell}\exp{(\pm
r_{Q}^{2}/r_{W}^{2})}=\bar{\ell}\exp{(\pm|a_{0}/\chi|)},$ (38)

which indicates that $|\Phi_{\text{smi}}|$ overtakes $|\Phi_{\text{RN}}|$
either at $r<r_{WQ}^{(-)}$ or at $r>r_{WQ}^{(+)}$. If $|a_{0}/\chi|$ is large
then $r_{WQ}^{(+)}$ is exponentially large and $r_{WQ}^{(-)}$ is exponentially
small.

Magnitudes of $\Phi_{\text{smi}}$ and $\Phi_{\text{N}}$ become comparable at a
certain value of $r$, which is shown by the formula

$\displaystyle r_{WM}$ $\displaystyle=$ $\displaystyle-\ell_{W}\,{\cal
W}\\!\left(-\bar{\ell}/\ell_{W}\right)\leqslant\ell_{W},$ (39)

where $\ell_{W}\equiv{r_{W}^{2}}/{|r_{H}|}=|L_{\chi}|$, and by $\,{\cal
W}\\!\left(a\right)$ we denoted the Lambert function. The function
$|\Phi_{\text{smi}}(r)|$ becomes larger than $|\Phi_{\text{N}}(r)|$ at
$r<r_{WM}\leqslant|L_{\chi}|$.

Furthermore, considering $\Phi_{\text{smi}}$ as a perturbation of
$\Phi_{\text{N}}$, one can deduce that the effective gravitational coupling,

$G_{\text{eff}}\approx G\left[1+\zeta_{\chi
q}L_{\chi}\frac{\ln{(r/\bar{\ell})}}{r}\right],$ (40)

becomes larger as $r$ gets smaller. Here an approximation symbol reminds us
that we are working with wavefunction (18), and assume a leading-order
approximation with respect to the Planck constant.

One can see that gravity naturally becomes stronger at shorter scales, without
introducing any additional effects or matter, which suggests the way towards
resolving the hierarchy problem. Indeed, the latter states the large
discrepancy between magnitudes of forces of Standard Model interactions and
classical gravity, but in this case, the gravitational force grows stronger
than inverse square as $r$ decreases, as $\ln r/r^{3}$. Moreover, it is
possible to have even stronger short-length behaviour in our approach: if one
goes beyond a minimal model (11) and generalizes nonlinear coupling to a
series: $b(\boldsymbol{r})=b_{0}-q/r^{2}+q_{1}/r^{3}+...$, then this will
induce terms $\ln r/r^{4}$, $\ln r/r^{5}$, et cetera.

On the other hand, as $r$ grows, the function $G_{\text{eff}}$ converges to
$G$, especially if $|L_{\chi}|$ or $1/\bar{\ell}$ are sufficiently small;
which makes $G_{\text{eff}}$ approximately constant for a large range of
values of $r$.

###  IV.4 Potential $\Phi_{\text{dS}}$ and de Sitter spacetime

Let us turn our attention to terms which do not tend to zero at $r\to\infty$.
The physical meaning of one of these terms, given by Eq. (26), becomes clear
upon using the mapping (29):

$ds^{2}_{(\text{dS})}\approx-
c_{(0)}^{2}\left(1-\frac{r^{2}}{R_{\text{dS}}^{2}}\right)dt^{2}+\frac{dr^{2}}{1-r^{2}/R_{\text{dS}}^{2}}+r^{2}d\sigma^{2},$
(41)

where

$R_{\text{dS}}=\frac{1}{\sqrt{2}}\frac{c_{(0)}}{c_{b}}L_{\text{dS}}=\bar{\ell}\sqrt{\frac{mc_{(0)}^{2}}{2a_{2}b_{0}}}$
(42)

is a radius of de Sitter horizon.

This metric represents de Sitter spacetime (written in static coordinates),
which belongs to a class of Friedmann-Lemaître-Robertson-Walker (FLRW)
spacetimes. Indeed, by applying a coordinate transformation
$\tau-t=\tau_{\text{dS}}\ln{\\!\left(1-r^{2}/R_{\text{dS}}^{2}\right)}$,
$\varrho=\alpha_{0}^{-1}r\exp{\\!\left(-t/2\tau_{\text{dS}}\right)}\left(1-r^{2}/R_{\text{dS}}^{2}\right)^{-1/2}$,
one can rewrite Eq. (41) in isotropic coordinates

$ds^{2}_{(\text{dS})}\to-
c_{(0)}^{2}d\tau^{2}+\alpha_{0}^{2}\exp{\\!\left(\frac{\tau}{\tau_{\text{dS}}}\right)}\left(d\varrho^{2}+\varrho^{2}d\sigma^{2}\right),$
(43)

where $\tau_{\text{dS}}=R_{\text{dS}}/2c_{(0)}$ and $\alpha_{0}$ is an
integration constant.

The physical implications of the term $\Phi_{\text{dS}}$ will be further
examined in Section VII.

###  IV.5 Potentials $\Phi_{\text{mgl}}$ and $\Phi_{\text{gal}}$ and gravity
on astronomical scales

The remaining asymptotically non-vanishing potentials are given by Eqs. (24)
and (25). With respect to the dependence upon radial distance from the
gravitating center, they occupy an intermediate place between de Sitter term
(26) and Newtonian potential (23). It is thus natural to expect that these
terms are responsible for large scale dynamics – from galaxies (a kiloparsec
scale) to metagalactic objects, such as voids and superclusters (a megaparsec
scale).

According to the mapping (29), the terms $\Phi_{\text{gal}}$ and
$\Phi_{\text{mgl}}$ modify de Sitter metric (41): causing the resulting
spacetime to be asymptotically de Sitter only. The line element, which
corresponds to the terms $\Phi_{\text{dS}}$, $\Phi_{\text{mgl}}$ and
$\Phi_{\text{gal}}$ taken together, can be written in static coordinates as

$\displaystyle ds^{2}_{(\text{gal+mgl+dS})}$ $\displaystyle\approx$
$\displaystyle-
c_{(0)}^{2}\left[1+\beta_{\chi}^{2}\ln{\\!\left(\frac{r}{\bar{\ell}}\right)}+\frac{r}{R_{\mathfrak{M}}}-\frac{r^{2}}{R_{\text{dS}}^{2}}\right]dt^{2}+\frac{dr^{2}}{1+\beta_{\chi}^{2}\ln{\left(r/\bar{\ell}\right)}+r/R_{\mathfrak{M}}-r^{2}/R_{\text{dS}}^{2}}+r^{2}d\sigma^{2},$
(44)

where
$\beta_{\chi}=\sqrt{{2\chi\,b_{0}}/{mc_{(0)}^{2}}}=\sqrt{2\chi}{c_{b}}/{c_{(0)}}$,
and

$R_{\mathfrak{M}}=\frac{mc_{(0)}^{2}\bar{\ell}}{2a_{1}b_{0}}=\frac{\zeta_{a_{1}}}{2}\frac{c_{(0)}^{2}}{c_{b}^{2}}L_{\text{mgl}}$
(45)

is a characteristic length scale constant, which can be positive or negative
depending on the sign of $a_{1}$. Note that in Eq. (44), we included the
$\Phi_{\text{dS}}$-induced (de Sitter) term to remind us that at large $r$ we
still have a spacetime of a FLRW type.

Interestingly, linear terms in metrics occur also in an alternative theory of
gravity, Weyl gravity mk89 ; man93 . This coincidence can be explained by
conformal symmetry, which often emerges in logarithmic models at the
relativistic limit, see for example Section VII.1. However, Weyl gravity does
not produce a logarithmic term in metric, while in our theory it is induced by
$\Phi_{\text{gal}}$. This term is responsible for the flat rotation curves
phenomenon in galaxies, which will be discussed, along with other astronomical
implications of the terms $\Phi_{\text{mgl}}$ and $\Phi_{\text{gal}}$, in
further detail in Section VI.

It is also useful to know that the metric induced by the term (25) alone,

$ds^{2}_{(\text{mgl})}\approx-
c_{(0)}^{2}\left(1+\frac{r}{R_{\mathfrak{M}}}\right)dt^{2}+\frac{dr^{2}}{1+r/R_{\mathfrak{M}}}+r^{2}d\sigma^{2},$
(46)

transforms to the conformally FLRW-type metric

$\displaystyle ds^{2}_{(\text{mgl})}$ $\displaystyle\to$
$\displaystyle\frac{(1+\varrho/4R_{\mathfrak{M}})^{2}}{a_{\mathfrak{M}}^{2}(\tau)\,(1-\varrho/4R_{\mathfrak{M}})^{2}}\left[-c_{(0)}^{2}d\tau^{2}+\frac{a_{\mathfrak{M}}^{2}(\tau)}{(1-\varrho^{2}/16R_{\mathfrak{M}}^{2})^{2}}\left(d\varrho^{2}+\varrho^{2}d\sigma^{2}\right)\right],$
(47)

upon applying the coordinate transformation
$r=\varrho/(1-\varrho/4R_{\mathfrak{M}})^{2}$, $t=\int
d\tau/a_{\mathfrak{M}}(\tau)$. For R-observers, this represents a surrounding
homogeneous and isotropic spacetime with scale factor $a_{\mathfrak{M}}(\tau)$
and negative-definite spatial scalar curvature, $-1/(2R_{\mathfrak{M}})^{2}$.

Finally, one could mention that potentials of type (24) were studied, albeit
in the context of a linear Schrödinger equation, in Refs. zs18a ; zs18b .

##  V Density of Effective Gravitating Matter

In this Section, let us derive spherically-symmetric density profiles of
effective gravitating matter $\rho_{\Phi}$, which formally corresponds to the
superfluid-vacuum induced potential (20). Contrary to the directly observable
value of orbital velocity derived in Section VI, such density depends more
substantially on the choice of an observer. It can be defined either via the
Lorentz-covariant definition (6), or via the Poisson equation, which is a non-
relativistic version of Eq. (6). Correspondingly, one would obtain different
results, which will be discussed below.

In accordance with the last paragraph of Appendix B, this Section’s
computations cannot take into account any secondary induced matter, such as
the equilibrium configurations of mass-energy emerging as a result of
interaction between scalar and tensor modes of superfluid vacuum’s
excitations. One can show that such equilibria do exist, manifesting
themselves in a form of general relativistic nonsingular horizon-free stellar-
like objects or particle-like Q-balls, therefore, in reality such objects
would definitely contribute to the density associated with dark matter. In
other words, here we are restricting ourselves to background values of
density.

In this Section only, we shall be temporarily assuming that $M=M(r)$,
otherwise the corresponding contribution to the density profile would be
identically zero, and thus non-indicative.

###  V.1 Galilean symmetry

In this case, one defines an effective matter density by virtue of the Poisson
equation. Taking the whole potential (20) and assuming spherical symmetry, we
obtain

$\displaystyle\rho_{\Phi}(r)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4\pi
G}\left[\Phi^{\prime\prime}(r)+\frac{2}{r}\Phi^{\prime}(r)\right]=\rho_{\text{smi}}(r)+\rho_{\text{RN}}(r)+\rho_{\text{N}}(r)+\rho_{\text{gal}}(r)+\rho_{\text{mgl}}(r)+\rho_{\text{dS}}(r),$
(48)

where

$\displaystyle\rho_{\text{smi}}(r)$ $\displaystyle=$
$\displaystyle\frac{3\chi\,q}{4\pi
Gmr^{4}}\left[1-\frac{2}{3}\ln{\left(\frac{r}{\bar{\ell}}\right)}\right],$
(49) $\displaystyle\rho_{\text{RN}}(r)$ $\displaystyle=$
$\displaystyle-\frac{a_{0}q}{2\pi Gmr^{4}}=\frac{k_{e}Q^{2}}{4\pi r^{4}},$
(50) $\displaystyle\rho_{\text{N}}(r)$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi r}M^{\prime\prime}(r),$ (51)
$\displaystyle\rho_{\text{gal}}(r)$ $\displaystyle=$
$\displaystyle\frac{\chi\,b_{0}}{4\pi Gmr^{2}}=\frac{v_{\text{gal}}^{2}}{4\pi
Gr^{2}},$ (52) $\displaystyle\rho_{\text{mgl}}(r)$ $\displaystyle=$
$\displaystyle\frac{a_{1}\,b_{0}}{2\pi Gm\bar{\ell}r}=\frac{\chi\,b_{0}}{2\pi
GmL_{\chi}r},$ (53) $\displaystyle\rho_{\text{dS}}(r)$ $\displaystyle=$
$\displaystyle-\frac{3a_{2}\,b_{0}}{2\pi Gm\bar{\ell}^{2}}=-\frac{3b_{0}}{2\pi
GmL_{\text{dS}}^{2}}=\text{const},$ (54)

whereby the sum of the last three densities can be regarded as a density
corresponding to the astronomical-scale “dark matter” and “dark energy”, which
will be further justified in Sections VI and VII.

Furthermore, these formulae were derived on the assumption that the
gravitational coupling constant $G$ is the same for all length scales; which
is valid when any influence from the term (21) can be disregarded. However,
this term might cause an additional effect, discussed in Section IV.3: it
makes the gravitational coupling constant vary with distance. If this does
happen, then in Eqs. (48)–(54) one should replace $G$ with a running constant
given by Eq. (40).

###  V.2 Lorentz symmetry

In this case, one defines effective matter density by virtue of Einstein field
equations with the induced stress-energy tensor defined by Eq. (6). Using it
together with Eq. (20) and metric (29), we obtain

$\displaystyle{\tilde{\rho}}_{\Phi}(r)$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{4\pi
Gr}\left[\Phi^{\prime}(r)+\frac{1}{r}\Phi(r)\right]=\tilde{\rho}_{\text{smi}}(r)+\tilde{\rho}_{\text{RN}}(r)+\tilde{\rho}_{\text{N}}(r)+\tilde{\rho}_{\text{gal}}(r)+\tilde{\rho}_{\text{mgl}}(r)+\tilde{\rho}_{\text{dS}}(r)+\tilde{\rho}_{\Phi_{0}}(r),$
(55)

where

$\displaystyle\tilde{\rho}_{\text{smi}}(r)$ $\displaystyle=$
$\displaystyle\frac{\chi\,q}{4\pi
Gmr^{4}}\left[1-\ln{\left(\frac{r}{\bar{\ell}}\right)}\right],$ (56)
$\displaystyle\tilde{\rho}_{\text{RN}}(r)$ $\displaystyle=$
$\displaystyle-\frac{a_{0}q}{4\pi Gmr^{4}}=\frac{k_{e}Q^{2}}{8\pi r^{4}},$
(57) $\displaystyle\tilde{\rho}_{\text{N}}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi r^{2}}M^{\prime}(r),$ (58)
$\displaystyle\tilde{\rho}_{\text{gal}}(r)$ $\displaystyle=$
$\displaystyle-\frac{\chi\,b_{0}}{4\pi
Gmr^{2}}\left[1+\ln{\left(\frac{r}{\bar{\ell}}\right)}\right],$ (59)
$\displaystyle\tilde{\rho}_{\text{mgl}}(r)$ $\displaystyle=$
$\displaystyle-\frac{a_{1}\,b_{0}}{2\pi
Gm\bar{\ell}r}=-\frac{\chi\,b_{0}}{2\pi GmL_{\chi}r},$ (60)
$\displaystyle\tilde{\rho}_{\text{dS}}(r)$ $\displaystyle=$
$\displaystyle\frac{3a_{2}\,b_{0}}{4\pi Gm\bar{\ell}^{2}}=\frac{3b_{0}}{4\pi
GmL_{\text{dS}}^{2}}=\text{const},$ (61)
$\displaystyle\tilde{\rho}_{\Phi_{0}}(r)$ $\displaystyle=$
$\displaystyle-\frac{\Phi_{0}}{4\pi Gr^{2}}=-\frac{1}{4\pi
Gmr^{2}}\left(a_{0}b_{0}+\frac{q}{L_{\text{dS}}^{2}}\right)\\!,~{}~{}~{}$ (62)

whereby the sum of last four densities can be regarded corresponding to the
astronomical-scale “dark matter” and “dark energy”, which will be further
justified in Sections VI and VII. Density (62) is a somewhat surprising
contribution, because it corresponds to the constant term $\Phi_{0}$ in Eq.
(20), which is not supposed to affect trajectories; at least, in classical
mechanics. Unless its presence can be confirmed by observations, it must be
regarded as a gauge term, or as an artifact of approximations underlying Eq.
(29).

Furthermore, a running gravitational coupling constant can not be implemented
in the relativistic case as simply as in Section V.1. To preserve Lorentz
invariance, one has to associate this coupling with a four-dimensional scalar;
which automatically upgrades general relativity to a scalar-tensor gravity
with a non-minimally coupled scalar field. This theory cannot be written by
hand, but it must be derived in a way which is similar to that used in Section
VII.1.

Comparing results of Sections V.1 and V.2, one can conclude that the
definition of effective matter density is somewhat ambiguous: in particular,
it drastically depends on symmetry assumptions. Therefore, further
experimental studies should help to empirically establish which symmetry is
more appropriate to use when dealing with “dark” phenomena.

##  VI Galactic Rotation Curves

In this Section, we demonstrate how induced gravitational potential can
explain various phenomena, which are usually attributed to dark matter. Let us
focus on the terms (24) and (25), which were partially discussed in Section
IV.5. Because they become significant at a galactic scale and above (i.e., a
kiloparsec to megaparsec scale), it is natural to conform them to astronomical
observations; such as those of rotation curves in galaxies.

In a spherically symmetric case, velocity curves of stars orbiting with non-
relativistic velocities on a plane in a central gravitational potential
$\Phi(r)$ can be estimated using a simple formula
$v^{2}=Ra_{c}=R\,\Phi^{\prime}(R)$, where $v$ is the orbital velocity, $a_{c}$
is the centripetal acceleration, and $R$ is the orbit’s radius. The
cylindrically symmetric case can be considered by analogy, by assuming various
disk models too63 ; cas83 .

Considering the terms (24) and (25) in conjunction with the Newtonian term
(23), we thus obtain

$\displaystyle v(R)$ $\displaystyle=$
$\displaystyle\left\\{R\frac{d}{dR}\left[\Phi_{\text{N}}(R)+\Phi_{\text{gal}}(R)+\Phi_{\text{mgl}}(R)\right]\right\\}^{1/2}$
(63) $\displaystyle=$
$\displaystyle\sqrt{v_{\text{N}}^{2}+v_{\text{gal}}^{2}+\Phi_{\text{mgl}}(R)},$

where

$\displaystyle v_{\text{N}}^{2}$ $\displaystyle=$
$\displaystyle\frac{GM}{R}=\frac{a_{1}q}{m\bar{\ell}}\frac{1}{R}=-\Phi_{\text{N}}(R),$
(64) $\displaystyle v_{\text{gal}}^{2}$ $\displaystyle=$ $\displaystyle\chi
c_{b}^{2}=\frac{\chi b_{0}}{m}=\text{const},$ (65)

while the contribution from the term (26) is disregarded for now, due the
assumed smallness of the “local” cosmological constant $1/R_{\text{dS}}^{2}$;
and the contribution from the term (21) is disregarded due the assumed
smallness of the corresponding characteristic length, according to discussion
in Section IV.3.

In the case of a galaxy, the contribution from the Newtonian term
$\Phi_{\text{N}}$ rapidly decreases as $R$ grows. Correspondingly, the main
contribution would then come from the second term in a row,
$\Phi_{\text{gal}}$, and then from the third term, $\Phi_{\text{mgl}}$. From
Eq. (65) one can see that the contribution from $\Phi_{\text{gal}}$ is
constant, which explains the average flatness of galactic rotation curves.

Notice that the value of velocity $v_{\text{gal}}$ depends on one of the
wavefunction parameters $\chi$ and one of quantum temperature parameters
$b_{0}$. Both are not a priori fixed parameters of the model, cf. Eqs. (12)
and (18), but vary depending on the environment and conditions: background
superfluid gets affected by the gravitational potential it induces, because
this potential acts upon the surrounding conventional matter, thus creating
density inhomogeneity. Therefore, $\chi$ and $b_{0}$ should generally be
different for each galaxy; and hence should $v_{\text{gal}}$ be.

Similar to the case of $v_{\text{gal}}$, the parameter $a_{1}$ hence a value
$\Phi_{\text{mgl}}(R)$ at a fixed $R$ will also be dependent on the
gravitating object they refer to. This potential should usually be negligible
on the inner scale length of a galaxy, but as $R$ grows towards the
extragalactic length scale, $R\gtrsim 10\,\text{kpc}$, rotation curves should
start to deviate from flat:

$v(R)\approx v_{\text{gal}}\sqrt{1+\frac{R}{L_{\chi}}},$ (66)

which can be used for estimating the combination of superfluid vacuum
parameters $\chi\bar{\ell}/a_{1}$ empirically. In cases where the contribution
from other terms of the induced potential cannot be neglected, Equation (63)
must be generalized to include those too.

Possible galactic-scale regions, where this non-flat asymptotics should become
visible, depending on a value $L_{\chi}$, are the outer regions of large
spiral galaxies, such as M31 or M33 rvt80 ; cs00 ; ccf09 ; clw10 ; kcc17 ,
where $\Phi_{\text{mgl}}(r)$ can not only overtake $\Phi_{\text{N}}(r)$ but
also become comparable with $\Phi_{\text{gal}}(r)$.

##  VII Accelerating Expansion of the Universe

This phenomenon is usually explained by introducing exotic forms of
relativistic matter, such as dark energy; usually modeled by various long-
range scalar fields, which are assumed not to affect the numerous particle
physics experiments on the Earth. The superfluid vacuum approach offers a
simple framework, which can explain the Universe’s expansion as an observer-
dependent effect, without involving any matter other than the background
superfluid itself.

###  VII.1 Conformally flat spacetime and dilaton field

Following work z11appb , let us consider the most simple possible special
case: laminar flow of a logarithmic background condensate in a state
$|\Psi_{(0)}\rangle$, described by Eq. (9) at $b=\text{const}$, with a
constant velocity $\boldsymbol{u}^{(0)}$, if viewed as, from the F-observer’s
perspective, an embedding into underlying Euclidean space. On the other hand,
what does a R-observer see?

Due to a well-known separability property of the logarithmic Schrödinger
equation bb76 ; af18 ; bcs19 ; lzh19 ; z18zna , the phase of its simplest
ground-state solutions is a linear function of a radius-vector:

$i\ln{\left(\frac{\Psi_{(0)}(\boldsymbol{r},t)}{|\Psi_{(0)}(\boldsymbol{r},t)|}\right)}\propto\boldsymbol{u}^{(0)}\cdot\boldsymbol{r}+f(t),$
(67)

where $\boldsymbol{u}^{(0)}$ is a constant 3-vector, and $f(t)$ is an
arbitrary function of time. In this case, the fluid-Schrödinger analogy
confirms that the background condensate does flow with a constant velocity
$\boldsymbol{u}=-i\eta\boldsymbol{\nabla}\ln{\left(\Psi_{(0)}/|\Psi_{(0)}|\right)}\propto\boldsymbol{u}^{(0)}$.
Recalling Eq. (5), it means that the background geometry induced by such
solutions is conformally flat:

$ds^{2}_{(0)}\propto\frac{1}{\bar{\rho}}|\Psi_{(0)}(\boldsymbol{r},t)|^{2}\left[-c_{s}^{2}dt^{2}+\left(d\boldsymbol{r}-\boldsymbol{u}^{(0)}dt\right)^{2}\right],$
(68)

where $c_{s}$ is given by Eq. (10), in a leading-order approximation with
respect to the Planck constant.

Spacetime of a type (68) lies within a large class of manifolds with the
vanishing Weyl tensor – a type $\mathbf{O}$ in the Petrov classification. This
is the class that all FLRW spacetimes belong to, including those which
describe the Universe with accelerating expansion – simply written in
conformally-flat coordinates instead of comoving ones.

Using definition (6), we obtain an induced stress-energy tensor for our
system:

$\displaystyle\kappa\widetilde{T}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\tilde{D}\Bigl{[}\nabla_{\mu}\nabla_{\nu}{\phi}-\nabla_{\mu}{\phi}\nabla_{\nu}{\phi}$
(69) $\displaystyle-
g_{\mu\nu}\left(\nabla_{\lambda}\nabla^{\lambda}{\phi}+\tfrac{1}{2}(\tilde{D}-1)\nabla_{\lambda}{\phi}\nabla^{\lambda}{\phi}\right)\Bigr{]}\\!,~{}~{}~{}$

where $\tilde{D}=D-2=2$, $\nabla$ is a covariant derivative with respect to
metric $g$, and by ${\phi}$ we denote the induced scalar field:

${\phi}=\ln{\left(|\Psi_{(0)}(\boldsymbol{r},t)|^{2}/\bar{\rho}\right)},$ (70)

up to an additive constant.

This stress-energy tensor strongly resembles the one occurring in the theory
of gravity with a scalar field. One can verify that it can be indeed derived
from the following scalar-tensor gravity action functional

$\tilde{\cal S}[g,\,\not\\!{{\phi}}]\propto\int
d^{D}x\sqrt{-g}\,\text{e}^{\tilde{D}{\phi}}\left[R+\tilde{D}(\tilde{D}+1)(\nabla{\phi})^{2}\right],$
(71)

where the notation “$\not\\!{{\phi}}$” reminds us that the field ${\phi}$ is
fixed by the solution of the original quantum wave equation, cf. Eq. (70),
while the variation of action must be taken with respect to the metric only.
In other words, both metric $g$ and dilaton ${\phi}$ are induced by the
superfluid vacuum being in a state described by
$\Psi_{(0)}(\boldsymbol{r},t)$.

Thus, we have found yet another example of the differences between the
F-observer’s and R-observer’s pictures of reality. While the former sees a
background quantum fluid flowing with a constant velocity in three-dimensional
Euclidean space, the latter observes itself as being inside four-dimensional
spacetime governed by a Lorentz-covariant scalar-tensor gravity.

An action functional (71) therefore explains why covariant models involving
scalars provide a robust description of the large-scale evolution of the
Universe, agreeing with current observational data; yet no quanta of
relativistic dilaton have thus far been detected.

This correspondence also reveals the limitations of the relativistic
description itself: if the superfluid vacuum goes into a different quantum
state, then one gets a different expression for the induced metric, scalar,
stress-energy tensor and covariant action. In fact, for more complicated
superfluid flows, even the condition (67), leading to a conformal flatness,
can become relaxed to an asymptotic one. Therefore, depending on the physical
configuration (determined by external potential, if any, and boundary
conditions), nonlinear coupling behaviour and the quantum state the vacuum is
in, small fluctuations and probe particles would obey different covariant
actions. Consequently, a R-observer would have to tweak its field-theoretical
models by hand; with the unified picture being observable only at the level of
a F-observer.

###  VII.2 Cosmological constant and local expansion mechanism

In Section VII.1 we considered an example of the global superfluid flow which
would be seen by a relativistic observer as the accelerating expansion of the
observable Universe. What about locally induced gravity (20), can it cause
similar effects?

Let us consider once again the induced potentials of Sections IV.4 and IV.5.
The de Sitter term from Section IV.4 predominates if we consider the physical
vacuum alone, without any generated matter therein. This is perhaps only valid
for the early Universe, such as the one which existed during the inflationary
epoch. In the current epoch, our spacetime can only be de Sitter
asymptotically or even approximately; therefore the potential
$\Phi_{\text{dS}}$ must be negligible, unless one considers very large length
scales. For example, if a length scale $R_{\text{dS}}$, which generally
depends on a massive object defining the frame of reference of the function
(16), is comparable to a size of the observable Universe ($\sim$ 10 Gpc), then
$R_{\text{dS}}$ can be related to the cosmological constant $\Lambda$ as

$R_{\text{dS}}^{\text{(cos)}}=\sqrt{3/\Lambda},\ \ \Lambda\sim 10^{-56}\
\text{cm}^{-2},$ (72)

therefore, the term (26) becomes substantial only at a gigaparsec scale. This
relation yields an empirical constraint for a combination of characteristic
parameters of superfluid vacuum, average quantum temperature and
$\Psi_{\text{vac}}$:

$\frac{a_{2}^{\text{(cos)}}b_{0}^{\text{(cos)}}}{m\bar{\ell}^{2}}\approx\frac{1}{6}\Lambda
c^{2}\sim 10^{-36}\ \text{s}^{-2},$ (73)

see also the discussion around Eq. (12). All this essentially means that
$\Lambda$ is not a fundamental constant of Nature, but a combination of
various parameters of superfluid vacuum, including quantum temperature and
Gaussian width of the condensate function $\Psi_{\text{vac}}$. Therefore, its
smallness, sometimes referred as “vacuum catastrophe” car01 , can easily be
explained by the fact that: either average quantum temperature across the
Universe, or wavefunction’s width, or both, are sufficiently small compared to
$mc^{2}$ and $\bar{\ell}$, respectively; thus resulting in the overall
smallness of the ratio on the left-hand side of Eq. (73).

The line element, which results from taking $\Phi_{\text{dS}}$,
$\Phi_{\text{mgl}}$, $\Phi_{\text{gal}}$, $\Phi_{0}$ and $\Phi_{\text{N}}$
together, can be written in static coordinates as

$\displaystyle ds^{2}_{(\text{cos+N})}\\!$ $\displaystyle\approx$
$\displaystyle\\!-c_{(0)}^{2}\left[1+\delta_{0}-\frac{r_{H}}{r}+\beta_{\chi}^{2}\ln{\\!\left(\frac{r}{\bar{\ell}}\right)}+\frac{r}{R_{\mathfrak{M}}}-\frac{r^{2}}{R_{\text{dS}}^{2}}\right]dt^{2}$
(74)
$\displaystyle+\frac{dr^{2}}{1+\delta_{0}-r_{H}/r+\beta_{\chi}^{2}\ln{\left(r/\bar{\ell}\right)}+r/R_{\mathfrak{M}}-r^{2}/R_{\text{dS}}^{2}}+r^{2}d\sigma^{2},$

where $\delta_{0}=2\Phi_{0}/c_{(0)}^{2}$. In this metric, the Schwarzschild
term ensures that spacetime singularity at $r=0$ is “dressed” by the black
hole horizon, while at large $r$ we still have spacetime of a FLRW type. While
from the viewpoint of a F-observer, no spacetime singularity would pose a
problem, because quantum wavefunction remains regular and normalizable at any
non-negative value of $r$. This reaffirms our earlier statement that
cosmological singularity is an artifact of the low-momentum approximation of
superfluid vacuum z11appb ; z20rss .

According to Eq. (47), the linear potential term (25), when taken alone,
induces the universal acceleration $c_{(0)}^{2}/2R_{\mathfrak{M}}$, occurring
due to the spatial curvature, when seen by a R-observer in its own local
static coordinate system. This contributes to the Hubble expansion induced by
the quadratic term (26). Thus, in the relativistic picture, the non-small
fluctuation of superfluid vacuum produces an effect at the center of a
gravitating configuration; and therefore contributes to the explicit
rotational motions of the stars inside this configuration, which can be seen
as a consequence of curved spacetime.

###  VII.3 Expansion mechanisms and cosmological coincidences

Comparing Sections VII.1 and VII.2, one can see that they describe different
expansion mechanisms. The mechanism of Section VII.1 occurs due to the global
flow of background superfluid, assumed to be laminar, which is “seen” by a
R-observer as a FLRW-type spacetime; the resolution of various cosmological
problems related to this mechanism was discussed in Section 5 of Ref. z11appb
. On the other hand, in Section VII.2, expansion is explained as a cumulative
effect from terms in metric (74), which do not vanish at spatial infinity,
induced by a “local” wavefunction associated with a gravitating configuration
or body. This wavefunction can be regarded as a fluctuation (not necessarily
small) of the global wavefunction from Section VII.1.

The interplay between these mechanisms depends on the length scales of the
quadratic and linear terms, $R_{\text{dS}}$ and $R_{\mathfrak{M}}$. Unless a
cumulative expansion effect from asymptotically non-vanishing terms taken
together is, by some extraordinary coincidence, exactly equal to the expansion
due to the global flow mechanism, its rate must be different from that of the
global flow-induced expansion.

The occurrence of an additional expansion mechanism, at the scale of a
supercluster, such as our Virgo or Laniakea, could explain the remarkable
discrepancy between measurements of the Hubble constant using different
methods from those based on the whole Universe expansion, such as cosmic
microwave background radiation (CMB). Among non-CMB methods one could mention
Cepheid calibration, time-delay cosmography, and geometric distance
measurements to megamaser-hosting galaxies fmg01 ; cfs19 ; pbr20 . From a
theoretical point of view, a scenario with different expansion rates seems
slightly more plausible, because it does not require an explanation why
accelerations from different mechanisms should be exactly equal to each other
(this coincidence should not be confused with the conventional cosmological
coincidence which we will discuss next).

Furthermore, our approach offers a simple explanation of the cosmological
coincidence problem itself car01 , in both the simplified and quantitative
versions thereof.

The simplified formulation of the cosmological coincidence states that if dark
matter and dark energy were different kinds of matter, then during the
Universe’s evolution they should have evolved independently of each other;
therefore their distributions would be uncorrelated by now - which does not
seem to be the case. Superfluid vacuum theory trivially resolves this paradox:
because “dark matter” and “dark energy” are actually induced phenomena and
manifestations of the same object, superfluid vacuum, they cannot be
independent from each other.

The quantitative formulation of the coincidence problem is an explanation
requirement for why the ratio of DM- and DE-associated densities is of order
one, despite the reasons given in the simplified formulation. In our approach,
we can regard all terms of the induced potential, which do not vanish at
spatial infinity, as being associated with “dark” effects; but inside this
group we cannot unambiguously separate DM-attributed effects from DE ones. For
example, potential (25) is intermediate between de Sitter and logarithmic,
thus it affects both galactic rotation curves and Hubble expansion, cf.
Sections VI and VII.2.

If, for simplicity, we consider only the local expansion mechanism of Section
VII.2 and omit the contribution from $\Phi_{0}$, then the cosmological
coincidence can be reformulated as the following condition:

$\frac{\widetilde{\Omega}_{\text{gal}}+\widetilde{\Omega}_{\text{mgl}}}{\widetilde{\Omega}_{\text{mgl}}+\widetilde{\Omega}_{\text{dS}}}={\cal
O}(1),$ (75)

where $\widetilde{\Omega}_{\text{}}$’s are average values corresponding to
densities from Section V.2. The numerator of this ratio represents average
density of effective “dark matter”, while the denominator represents effective
“dark energy” density; or at least the predominating proportions thereof.

In general, this condition simply imposes yet another constraint for the
parameters of the theory. It is trivially satisfied if the involved parts of
the gravitational potential, hence the associated densities, are of the same
order of magnitude, if averaged on a large scale. Moreover, even if
$\widetilde{\Omega}_{\text{gal}}$ or $\widetilde{\Omega}_{\text{dS}}$ are much
smaller than the remaining involved densities, but the value
$\widetilde{\Omega}_{\text{mgl}}$ is substantial, then the relation (75) still
holds, due to the presence of $\widetilde{\Omega}_{\text{mgl}}$ in both parts
of the ratio.

##  VIII Conclusions

Working within the framework of the post-relativistic theory of physical
vacuum, based on the logarithmic superfluid model, we derived induced
gravitational potential, corresponding to a generic quantum wavefunction of
the vacuum. This mechanism is radically different from the one used in models
of relativistic classical fluids and fields, which are based on modifying the
stress-energy tensor in Einstein field equations.

The form of such a wavefunction is motivated by ground-state solutions of
quantum wave equations of a logarithmic type. Such equations find fruitful
applications in the theory of strongly-interacting quantum fluids, and have
been successfully applied to laboratory superfluids z12eb ; z19ijmpb ; sz19 .
We note that, in principle, one is not precluded from adding other types of
nonlinearity, such as polynomial ones, into the condensate wave equations, but
the role of logarithmic nonlinearity is crucial

Thus, we used a logarithmic superfluid model with variable nonlinear coupling,
because it accounts for an effect of the environment in a more realistic way
than the logarithmic model with a constant coupling. As a result, for the
trapless version of our model, we have four parameters, but only two of them
are a priori fixed, whereas the other two can vary, depending on the quantum
thermodynamic properties of the environment under consideration. Additionally,
a number of parameters come from the wavefunction solution itself. Those are
not independent parameters of the theory, but functions thereof. Because we do
not yet know the exact form of the superfluid wavefunction, see remarks at the
end of Section III, we leave those parameters to be empirically estimated, or
bound, at the stage of current knowledge.

It turns out that gravitational interaction has a multiple-scale structure in
our theory: induced potential is dominated by different terms at each length
scale; such that one can distinguish sub-Newtonian, Newtonian (inverse-law),
galactic (logarithmic-law), metagalactic (linear-law), and cosmological
(square-law) parts. A relativistic observer, who operates with low-momentum
small-amplitude fluctuations of superfluid vacuum, observes this induced
potential by measuring the trajectories of probe particles moving along
geodesics in induced four-dimensional pseudo-Riemannian spacetime. The metric
of the latter is determined by virtue of the BEC-spacetime correspondence and
fluid-Schrödinger analogy, applied jointly.

The sub-Newtonian part of the induced gravitational potential is defined as
one which grows faster than the inverse law, as distance tends towards zero.
It can be naturally divided into the following two parts. One part has an
inverse square law behaviour, and thus can be associated with the
gravitational field caused by a $U(1)$ gauge charge, such as an electric
charge. On a relativistic level, it is described by Reissner-Nordström
spacetime. The other part has ‘inverse square times logarithm’ law behaviour,
which might become substantial at both ultra-short and macroscopic distances,
depending on the values of the corresponding parameters. If it “survives” at
macroscopic distances, then it upgrades Newton’s gravitational constant to a
function of length, such that gravity has both strong and weak regimes.

With the potential or spacetime metric in hand, one can, in principle, assign
effective fictitious matter density to our potential, which corresponds to
“dark matter” and “dark energy”. This can be done in two ways: either by
Einstein field equations in a relativistic case, or the Poisson equation in a
non-relativistic one. It should be noted that the resulting density in each of
the cases can be modified, depending on whether the gravitational constant is
considered to be running or not. This will require more verification from
future experimental and theoretical studies.

Furthermore, on a galactic scale and above, the potential is dominated by non-
Newtonian terms, which do not vanish at spatial infinity. This explains the
non-Keplerian behaviour of rotation curves in galaxies, which is often
attributed to dark matter. Our model, not only explains the average flatness
of galactic rotation curves, but also makes a number of new predictions. One
of them is the approximately linear law behavior of gravitational potential on
a metagalactic scale, which is an intermediate scale between galactic
distances and the size of the observable universe. This should partially
affect galactic rotation curves too: as the distance from the gravitating
center grows further towards the metagalactic length scale, a squared
velocity’s profile asymptotically changes from being flat towards linear, cf.
Eq. (66).

On the other hand, at the largest length scale, the induced potential displays
square law behaviour. If the quadratic term is negative-definite, then the
corresponding metric describes (asymptotically) de Sitter space, merely
written in static coordinates. Taken together with the contribution from the
linear potential term, this explains the accelerating expansion of the
corresponding spacetime region, which is usually associated with dark energy.

Such expansion could supplement the “global” one, caused by laminar flow of
background logarithmic superfluid absent any other matter, which induces a
FLRW-type spacetime. The occurrence of more than one type of expansion
mechanism, could be responsible for the discrepancy between measurements of
the Hubble constant using different methods.

The relevant problems, such as smallness of cosmological constants and
cosmological coincidence, were also discussed.

To conclude, we used the BEC-spacetime correspondence and fluid-Schrödinger
analogy to argue that the description of reality and fundamental symmetry
crucially depend on the choice of an observer. We demonstrated that both dark
matter and dark energy are related phenomena, and different manifestations of
the same object, superfluid vacuum, which acts by inducing both gravitational
potential and spacetime.

Abbreviations
The following abbreviations are used in this manuscript:

BEC | Bose-Einstein condensate
---|---
CDM | Cold dark matter
CMB | Cosmic microwave background
DE | Dark energy
DM | Dark matter
dS | de Sitter
FLRW | Friedmann-Lemaître-Robertson-Walker
F-observer | Full observer
RN | Reissner-Nordström
R-observer | Relativistic observer
SVT | Superfluid vacuum theory

###### Acknowledgements.

This research is supported by Department of Higher Education and Training of
South Africa and in part by National Research Foundation of South Africa.
Proofreading of the manuscript by P. Stannard is greatly appreciated.

##  Appendix A Superfluid or not

When it comes to astrophysics and cosmology, including dark matter and dark
energy applications, the notion of superfluid is probably the most
misunderstood and misused term of all. In addition to having a catchy, but not
exactly informative, prefix, the first thing which causes misunderstanding of
the term is inviscid flow. This, the most striking feature of superfluids, is
often regarded as the definition thereof. As a result, superfluids are often
confused with perfect fluids, which have no viscosity by definition, and can
be easily implemented into Einstein field equations by virtue of a stress-
energy tensor borrowed from classical fluid mechanics.

However, perfect fluids, being classical hydrodynamical objects by
construction, can not properly reflect the essentially quantum nature of the
superfluidic matter that we know of. Therefore, perfect or non-quantum
inviscid fluids can be used as a crude approximation, at best, of superfluids.

Additionally, superfluids are often also confused with the concomitant
phenomenon of Bose-Einstein condensation (BEC), which is another kind of
quantum matter occurring in low-temperature condensed matter, such as cold
cesium atoms in a trap. However, in the superfluid helium phase, for example,
the BEC’s content comprises only about ten per cent. Therefore, this
condensate alone cannot fully account for dissipation-free flow and the other
distinctive features of superfluids. Even though Bose-Einstein condensates do
share certain features with superfluids, this does not imply that they are
superfluidic in general.

In particular, quantum excitations in laboratory superfluids are known to have
dispersion relations of a distinctive shape called the Landau “roton”
spectrum. This shape of the spectral curve is crucial, as it ensures the
suppression of dissipative fluctuations at a quantum level z12eb . If plotted
as an excitation energy versus momentum, the curve starts from the origin,
climbs up to a local maximum (called the maxon peak), then slightly descends
to a local nontrivial minimum (called the “roton” energy gap); and then grows
again, this time all the way up, to the boundary of the theory’s applicability
range, cf. a solid curve in the Figure 1a from Ref. z20ijmpa . In fact it is
not the roton energy gap alone, but the energy barrier formed by the maxon
peak and roton minimum in momentum space, which ensures the above-mentioned
suppression of quantum fluctuations in quantum liquid and, ultimately, causes
its flow to become inviscid. In other words, it is the global characteristics
of the curve, not just the existence of a nontrivial local minimum and related
energy gap, which are required for superfluidity to occur. Obviously, Landau’s
shape is a non-trivial property which cannot possibly occur in all quantum
liquids.

The final reason for the misuse of the term ‘superfluid’ is the extensive, but
not always careful, utilization of relativistic scalar field models in
astrophysics and cosmology. Historically, the four-dimensional scalar field
came about as a bold extrapolation of a non-relativistic wavefunction into the
realm of Lorentz-symmetric theories. However, when it comes to quantum liquids
and condensates, there is a fundamental difference between relativistic scalar
field and condensate wavefunction, which makes a correspondence between them
far from isomorphic.

The fluid condensate wavefunction obeys both a normalization condition (1) and
a wave Equation (2), therefore, it is a three-dimensional Euclidean scalar
related to a ray in the associated Hilbert space. One can see that Eqs.
(1)–(3) are essentially non-relativistic and three-dimensional. While one can
still make the last two relativistic, by replacing derivative parts with the
Lorentz-covariant analogues thereof, Eq. (1) strongly violates the Lorentz
invariance. This condition requires the foliation of a spacetime manifold into
$(3+1)$-dimensional spacelike hypersurfaces, and makes mass-energy a three-
dimensional scalar, not a time component of a four-dimensional vector.

As a result, relativistic scalar fields models offer a useful, but approximate
description of superfluidic phenomena (which is valid for small wavefunction
amplitudes and low momenta of excitations, running ahead), while the rigorous
extrapolation of BEC and superfluid notions into the relativistic domain
requires special treatment. Therefore, it is not surprising that some
“superfluid” models of dark matter; which are based on classical perfect fluid
models, scalar field theories or scalar-tensor theories of gravity, turn out
to be vulnerable to experimental verification lmo19 .

##  Appendix B Assumptions and approximations

Let us summarize and enumerate the main simplifying assumptions and
approximations underlying the derivation of our gravitational potential. We
used them to keep our calculations as analytical and non-perturbative as
possible, which is crucial for an essentially nonlinear theory, such as ours.

First, it should be emphasized that superfluid vacuum theory, even when
narrowed down to its logarithmic version, is a framework which potentially
contains a set of models. Our chosen underlying quantum superfluid model,
defined by Eq. (11), is a minimal one. It can easily be expanded by adding
polynomial terms $|\Psi|^{k}$ to the logarithmic nonlinearity in wave
equations, to make it describe the phenomena in a more precise way. The reason
for this is that the logarithmic nonlinearity is a leading-order approximation
for the condensate-like matter, as discussed in Ref. z18zna ; but non-
logarithmic terms can also come into play: one example is to be found in Ref.
sz19 . The nonlinear coupling function $b=b(\boldsymbol{r},t)$ can also be
made more detailed, to account for the thermodynamic environment in a more
realistic way.

Second, for a chosen model, the signs of coupling parameters can be changed,
which is often equivalent to changing the topological sector a model belongs
to. Because we do not precisely know the topological structure of the physical
vacuum we live in, it essentially doubles the number of candidate models to be
tested empirically. For example, changing the sign of the field-theoretical
potential; or, alternatively, the overall sign of the nonlinear coupling $b$;
switches between the topologically trivial, which is considered in this paper,
and the topologically non-trivial sectors. In a logarithmic liquid model with
constant nonlinear coupling, switching to the topologically non-trivial sector
of the theory means that wavefunction changes from a droplet-like non-
topological soliton type to a bubble-like topological soliton type, as
discussed in Refs. z18epl ; z19ijmpb . Consequently, the forms of a trial
wavefunction and induced potential in Section III would also change.

Third, we have avoided the problem of ambiguity in choosing trapping potential
and boundary conditions by making the former identically zero, and the latter
of a conventional quantum-mechanical type on an infinite spatial domain of
trivial topology $\mathbb{R}^{3}$. These are simplifying assumptions which are
yet to be proven to work, otherwise they must be replaced with something more
sophisticated.

Fourth, even if we have chosen our model correctly, there still remains the
ambiguity of how to determine the state of the vacuum—is it in a ground state,
an excited but metastable state, a pure or mixed state, a superposition of
states, or even in a quantum transition between states? Any new empirical
information about this could drastically change the assumptions underlying the
computations in Section III.

Fifth, the technical approximation which led us from Eqs. (16)–(18) might
oversimplify the picture. It is suitable for the purposes of this study, but
should be modified in more precise considerations.

Finally, throughout the paper, we consider the physical vacuum alone: assuming
that its small excitations, which would be observed by a R-observer as
relativistic matter with deformed dispersion relations z20ijmpa ; z11pla ;
sch17 ; csa18 ; ol19 ; plj20 , do not back-react, for example, via interaction
between scalar and tensor modes. This is obviously an over-simplified picture
of reality, which is sufficient for our current purpose, but unlikely to be
valid in general.

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