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TYPE Original Research
PUBLISHED 06 February 2023
DOI 10.3389/fspas.2023.1071743
OPEN ACCESS
EDITED BY
Alvaro De La Cruz-Dombriz,
University of Cape Town, South Africa
REVIEWED BY
Christian Corda,
B. M. Birla Science Centre, India
Demosthenes Kazanas,
Goddard Space Flight Center (NASA),
United States
*CORRESPONDENCE
Václav Vavryčuk,
vv@ig.cas.cz
SPECIALTY SECTION
This article was submitted to Cosmology, a
section of the journal Frontiers in
Astronomy and Space Sciences
RECEIVED 16 October 2022
ACCEPTED 11 January 2023
PUBLISHED 06 February 2023
CITATION
Vavryčuk V (2023), Gravitational orbits in
the expanding Universe revisited.
Front. Astron. Space Sci. 10:1071743.
doi: 10.3389/fspas.2023.1071743
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comply with these terms.
Gravitational orbits in the
expanding Universe revisited
Václav Vavryčuk*
Institute of Geophysics, Czech Academy of Sciences, Prague, Czechia
Modied Newtonian equations for gravitational orbits in the expanding Universe
indicate that local gravitationally bounded systems like galaxies and planetary
systems are unaected by the expansion of the Universe. This result is derived
for the space expansion described by the standard FLRW metric. In this paper,
the modied Newtonian equations are derived for the space expansion described
by the conformal cosmology (CC) metric. In this metric, the comoving and
proper times are dierent similarly as the comoving and proper distances. As
shown by Vavryčuk (Front. Phys. 2022), this metric is advantageous, because it
properly predicts the cosmic time dilation, and ts the Type Ia supernova luminosity
observations with no need to introduce dark energy. Surprisingly, the solution of the
equations for gravitational orbits based on the CC metric behaves quite dierently
than that based on the FLRW metric. In contrast to the common opinion that
local systems resist the space expansion, they expand according to the Hubble
ow in the CC metric. The evolution of the local systems with cosmic time is
exemplied on numerical modelling of spiral galaxies. The size of the spiral galaxies
grows consistently with observations and a typical spiral pattern is well reproduced.
The theory predicts at rotation curves without an assumption of dark matter
surrounding the galaxy. The theory resolves challenges to the ΛCDM model such
as the problem of faint satellite galaxies, baryonic Tully-Fisher relation or the radial
acceleration relation. Furthermore, puzzles in the solar system are successfully
explained such as the Faint young Sun paradox or the Moon’s and Titan’s orbit
anomalies.
KEYWORDS
cosmological redshift, cosmic time dilation, conformal metric, dark energy, rotation curves,
dark matter, winding problem, galaxy expansion
1 Introduction
Observations of the cosmological redshi interpreted by Lemaître (1927) and Hubble
(1929) as an eect of the expansion of the Universe started a new era of cosmology and
opened space for applying theory of General Relativity (GR) to cosmological problems.
Subsequently, the Friedmann equations(Friedmann, 1922) became the basic equations
describing the expanding history of the Universe. Immediately, cosmology was faced with
the following fundamental questions: How does the global expansion of the Universe aect
local gravitational systems? How do local gravitational elds interact with the expansion and
where is a size threshold between systems aected by and resisting the global expansion? Do
galaxies and galaxy clusters expand or not? How does the global expansion aect our solar
system? ese theoretical problems paid attention of many cosmologists, because they have
essential consequences for understanding the evolution of the Universe and for interpreting
cosmological observations (McVittie, 1933;Einstein and Straus, 1945;Dicke and Peebles,
1964;Noerdlinger and Petrosian, 1971;Carrera and Giulini, 2010;Nandraetal., 2012).
Frontiers in Astronomy and Space Sciences 01 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
e simplest problem is to study the Newtonian equations of
motion for two-point particles placed in the expanding space and
mutually attracted by the gravitational force. If the local gravitational
eld is weak and velocities of particles are non-relativistic, the problem
can be solved by perturbations (McVittie, 1933;Noerdlinger and
Petrosian, 1971;Bolenetal., 2001;Faraoni and Jacques, 2007). In this
case, we assume that the metric of the space expansion is perturbed
by a weak gravitational eld. Assuming that the space expansion
is described by the Friedmann-Lemaitre-Robertson-Walker (FLRW)
metric, the metric tensor gμν of a gravitational eld produced by a
point mass Msituated in the expandingspace reads (Noerdlinger and
Petrosian, 1971, their Eq. 11).
ds2c212αdt2a2t12αdr2
1kr2r2dΩ2
dΩ2dθ2sin2θ dϕ2(1)
where tis time, cis the speed of light, atis the scale factor, kis the
Gaussian curvature of the space, ris the comoving distance, and θand
ϕare the spherical angles. Parameter
αGM
rc2α1(2)
is the Newtonian gravitational potential normalized to c2, and Gis the
gravitational constant. Assuming a massive non-relativistic particle
vcorbiting in the gravitational eld and using the geodesic
equation, we nally obtain the followingequation for the proper radius
Rof the orbit (Carrera and Giulini, 2010, their Eq. 12a, b).
RGM
R2L2
R3a
aR(3)
Lconst(4)
where LRVϕis the proper angular momentum, and Vϕis the proper
tangential velocity. For a more detailed derivation, see AppendixA.
Eqs.3,4are called the modied (or improved) Newtonian
equations and they dier from the standard Newtonian equations
describing the Kepler orbits by term a
aRin Eq.3related to the space
expansion. e analysis of the modied Newtonian equations applied
to the galaxy dynamics shows that the expansion term a
aRaects
the orbits within galaxies negligibly (Faraoni and Jacques, 2007). is
result led to the conclusion that galaxies and all smaller gravitational
systems are not aected by the space expansion and behave as in the
static Universe (Dicke and Peebles, 1964;Noerdlinger and Petrosian,
1971;Cooperstocketal., 1998;Faraoni and Jacques, 2007;Iorio,
2013).
e key in deriving the modied Newtonian equations is the
assumption that the space expansion is described by the FLRW
metric. However, this metric is not the only metric, which can
describe the evolution of the isotropic homogeneous Universe. A
potentially applicable metric is also the conformal cosmology (CC)
metric (Endean, 1994;Endean, 1997;Ibison, 2007;Kastrup, 2008;
Dabrowskietal., 2009;Grøn and Johannesen, 2011;Visser, 2015;
Haradaetal., 2018). is metric has the scale factor atnot only at
the space components but also at the time component of the metric
tensor gμν. Hence, the metric is characterized not only by the space
expansion but also by the time dilation during the cosmic evolution.
e CC metric has exceptional properties being intensively studied
in GR and its modications such as the Conformal Gravity theory
(Mannheim, 1990;Mannheim, 2006;Mannheim, 2012). Interestingly,
the CC metric is Lorentz invariant and leaves the Maxwell’s equations
unchanged from their form in the Minkowski spacetime (Infeld and
Schild, 1945;Infeld and Schild, 1946;Ibison, 2007).
Importantly, Vavryčuk (2022a) shows that the CC metric should
be preferable against the FLRW metric, because the FLRW metric is
actually inconsistent with observations of the cosmological redshi
and cosmic time dilation. He claims that the CC metric is necessary
for a proper description of the expansion of the Universe, because:
1) e time-time component of the metric tensor g00 must vary
with cosmic time similarly as the space-space components, and
2) the comoving and proper times must be dierent in analogy
to the comoving and proper distances. Consequently, time should
not be invariant as in the FLRW metric, but its rate must vary
during the evolution of the Universe. Without changing the time
rate, the frequency of photons propagating in the Universe cannot
be changed during the expansion and the photons cannot be
redshied.
e varying time rate during the evolution of the Universe is
also supported by observations of Type Ia supernovae (SNe Ia).
Since the SNe Ia display rather uniform light curves, they can serve
as the standard candles as well as the standard local clocks. e
spectral evolution of the light curves and stretching of time in the
observer frame was disclosed by many authors (Leibundgutetal.,
1996;Goldhaberetal., 1997;Phillipsetal., 1999;Goldhaberetal.,
2001). e stretching of light curves at high redshi is rmly
acknowledged and corrections for time dilation are now routinely
applied to the SNe Ia data (Leibundgut, 2001;Goobar and Leibundgut,
2011). e light curve stretching is commonly interpreted as the eect
of the cosmic time dilation even though the standard FLRW metric
does not allow it.
In addition, the CC model ts the SNe Ia luminosity observations
with no need to introduce dark energy and an accelerated expansion of
the Universe (Behnkeetal., 2002;Vavryčuk, 2022a). A possibility that
the varying time rate might solve the dark energy problem is reported
also by Visser (2015). Considering the varying time rate during the
cosmic evolution has also other important consequences. For example,
the cosmological and gravitational redshis are calculated from the
metric tensor by the same formula. is emphasizes a common
physical origin of both redshis. e gravitational redshi reects the
time distortion due to the presence of the local gravitational eld, while
the cosmological redshi is due to changes of the global gravitational
eld of the Universe.
Obviously, we can ask a question, whether does a local
gravitational eld in the CC metric behave dierently from that in
the FLRW metric or not. Hence, the aim of this paper is to study
local gravitational systems in the expanding space described by the CC
metric. e gravitational eld is assumed to be a perturbation of the
global gravitational eld of the Universe and velocities of particles are
non-relativistic. e improved Newtonian equations are derived using
the geodesic equation. It is shown that the local gravitational systems
behave quite dierently in the CC metric than in the FLRW metric.
In contrast to the common opinion that local systems resist the space
expansion, the results show that all local systems expand according to
the Hubble ow in the CC metric. e evolution of the local systems is
exemplied on numerical modelling of spiral galaxies. e presented
theory predicts at rotation curves and the observed morphology of
spirals. Also, other observations supporting the presented theory are
discussed.
Frontiers in Astronomy and Space Sciences 02 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
2 Theory
2.1 Expanding Universe described by the CC
metric
Let us assume an expanding Universe described by the CC metric
in the following form (Grøn and Johannesen, 2011;Vavryčuk, 2022a):
ds2a2tc2dt2dr2
1kr2r2dΩ2
dΩ2dθ2sin2θ dϕ2(5)
where atis the scale factor dening the cosmic expansion, tis the
comoving (contravariant) time, cis the speed of light, kis the Gaussian
curvature of the space, ris the comoving (contravariant) distance, and
θand ϕare the spherical angles.
To avoid confusions, we have to discuss the CC metric described
by Eq.5in more detail. is form of the metric is oen used in
cosmology but in a modied notation and in a dierent physical
context. Time tis called the “conformal time” and it is usually denoted
as η. However, the physical meanings of the comoving time tand the
conformal time ηare essentially dierent. Here, we consider time tin
Eq.5as the physical comoving cosmic time. Consequently, the time-
time component of the metric tensor g00 is time dependent. e rate
of time varies and Eq.5encompasses the space expansion as well as
the time dilation during the evolution of the Universe. By contrast,
the conformal time ηis commonly assumed to be a rescaled proper
time with no physical meaning and no eects on the metric. e
g00 component is still time independent as for the FLRW metric. In
this way, the metric with the conformal time ηdescribes the space
expansion with a uniform rate of time and no time dilation during
the evolution of the Universe.
When introducing the FLRW metric, it is oen argued that g00
can be assumed to be time invariant, because we have a freedom to
rescale time to keep g00 constant (Weinberg, 1972). From this point of
view, the FLRW and CC metrics would be equivalent (Visser, 2015).
However, this is not correct, because we cannot rescale arbitrarily
a cosmological coordinate system without physical consequences. A
trivial example is a relation between the space metrics for a static and
expanding Universe. Both metrics can be transformed each into the
other. However, this does not mean that the model of the static and
expanding Universe are physically equivalent. Similarly, the CC metric
and the FLRW metric are not physically equivalent, because the FLRW
metric is characterized by a uniform rate of time, but the CC metric
is characterized by a varying rate of time during the evolution of the
Universe.
2.2 Comoving and proper speeds in the CC
metric
Using Eq.5, the equation of the null geodesics, which describes
propagation of photons, ds20, reads
a2tc2dt2dl20
dl2dr2
1kr2r2dΩ2(6)
Consequently, we get for the comoving velocity vand the proper
velocity Vof photons (Vavryčuk, 2022a, his Eq. A6)
vdl
dt cVac(7)
e propagation velocity of massive particles is described by the
geodesic equation
d2xμ
ds2Γμ
αβ
dxα
ds
dxβ
ds 0(8)
Substituting the distance element ds by the time element dt, we obtain
d2xμ
dt2Γμ
αβ
dxα
dt
dxβ
dt Γ0
αβ
dxα
dt
dxβ
dt
dxμ
dt (9)
Considering the metric tensor gμν needed for calculating the
Christoel symbols Γμ
αβ in Eq.9dened by Eq.5, we get
ac2v ac2v av30(10)
hence v
v1v2c2 a
a(11)
Consequently, for a massive non-relativistic particle vcwe write
v
v a
a(12)
and the comoving velocity vand the proper velocity Vread
vv0a1Vav V0(13)
where subscript “0” refers to quantities at present.
Hence, the proper velocity of photons is not constant as in the
FLRW metric, but it increases in the CC metric with the expansion as
atc, where cis the speed of light for a1. By contrast, the proper
velocity of massive non-relativistic particles is not aected by the
Universe expansion. is is in contradiction with behaviour of non-
relativistic massive particles in the FLRW metric, where the comoving
velocity vdepends on aas a−2 and the proper velocity Vas a−1.
Note that the varying speed of light in Eq.7for the CC metric
is an inevitable consequence of the time dependence of g00. e
varying speed of light seems apparently against the basic principles of
theory of the Special and General Relativity, but according to Einstein
(1920): “e law of the constancy of the velocity of light in vacuo,
which constitutes one of the fundamental assumptions in the special
theory of relativity and to which we have already frequently referred,
cannot claim any unlimited validity”...“its results hold only so long
as we are able to disregards the inuences of gravitational elds on
the phenomena (e.g., of light)”. Since we do not study the speed of
light in free-falling inertial systems but in non-inertial systems, the
acceleration due to gravity is not cancelled with the gravitational eld.
Hence, the varying (coordinate-dependent) speed of light in the CC
metric is fully consistent with GR being analogous, for example, to the
varying (coordinate-dependent) speed of light known in the famous
Schwarzschild solution (Weinberg, 1972).
2.3 Conformal Friedmann equations
Assuming the FLRW metric described by Eq.A-1, the Friedmann
equation for the perfect isotropic uid reads (Peacock, 1999;Ryden,
2016)
a
a28πG
3ρkc2
a2(14)
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Vavryčuk 10.3389/fspas.2023.1071743
where a dadT is the derivative of the scale factor atwith respect
to the proper time T,Gis the gravitational constant, ρis the mean mass
density, and kis the spatial curvature of the Universe at present.
In order to express Eq.14 in the CC metric, we have to substitute
the proper time Tby the comoving time tand time derivative
adadT by adadt aa. Hence, the conformal Friedmann
equation reads
a
a28πG
3ρa2kc2(15)
where adenotes the derivative with respect to the comoving time t.
Considering the matter-dominated Universe, we get
8πG
3ρH2
0Ωma3(16)
Eq.15 is rewritten as (Vavryčuk, 2022a)
H2aH2
0Ωma1Ωk(17)
with the condition
ΩmΩk1(18)
where Ha aais the Hubble parameter, H0is the Hubble constant,
Ωmis the normalized matter density, and Ωkis the normalized space
curvature. As shown in Vavryčuk (2022a), Eq.17 describes the Type
Ia supernova (SNe Ia) dimming well with no need to introduce dark
energy, which is necessary in the standard ΛCDM model in order to
t the SNe Ia data.
Considering a11z, the comoving time tis expressed from
Eq.17 as a function of redshi as follows
dt 1
H01zΩm1zΩk12dz(19)
and the proper time Trelated to comoving tas dT atdt reads
dT 1
H01z2Ωm1zΩk12dz(20)
ese relations are needed for relating observations of redshi to
cosmic time.
2.4 Gravitational orbits in the expanding
Universe
Next, we study the inuence ofthe exp anding Universe on the local
gravity eld produced by a point mass. e gravity eld is assumed
to produce a small perturbation of the metric tensor gμν describing
the expanding space. So far, this problem has been studied under the
assumption that the space expansion is dened by the standard FLRW
metric (Carrera and Giulini, 2010), see AppendixA. Here, we derive
equations for the gravitational orbits for the space expansion dened
by the CC metric.
e homogeneous and isotropic expanding space characterized
by the CC metric will be disturbed by a spherically symmetric
gravitational eld produced by a point mass Msituated in the origin of
coordinates. Since we limit ourselves to the weak gravitational eects
of the point mass only, Eq.5is modied as follows
ds2a2tc212αdt212αdr2
1kr2r2dΩ2(21)
where
αGM
rc2α1 (22)
is the Newtonian gravitational potential normalized to c2,ris the
comoving distance of an orbiting particle from the point mass M, and
Gis the gravitational constant.
Let us assume a massive non-relativistic particle vcorbiting in
the gravitational eld in the plane dened by θ0. e metric tensor
gμν is dened by Eq.21. Calculating the Christoel symbols Γμ
αβ in
Eq.9, we get
rr
ϕ2αc2a
arα2
ϕ2r3r2
r1
c2a
arr2
ϕ2r20(23)
r
ϕ2r
ϕa
ar
ϕ2αr
ϕ1
c2a
a
ϕr2
ϕ2r20(24)
where dots over quantities mean derivatives with respect to the
comoving time t. Since |α|1 and c21, terms multiplied by α
or by 1c2in Eqs.23,24 can be neglected and we get the following
approximate equations
1
a
d
dt avrGM
r2vϕ2
rfgfc(25)
1
a
d
dt arvϕ0(26)
where vr ris the radial comoving velocity, vϕr
ϕis the tangential
comoving velocity, and fgand fcare the radial gravitational and
centrifugal forces in the comoving coordinate system. If we assume
that the orbit of the particle is stationary, the radial and centrifugal
forces are balanced fgfcand the RHS of Eq.25 equals zero.
Consequently, we get
avrVrconst(27)
arvϕrVϕconst(28)
where Vrand Vϕare the radial and tangential components of the
proper velocity V. For a circular orbit in the comoving coordinates,
Eqs.27,28 are further simplied as
Vr0Vϕconst(29)
rconstRaR0(30)
where subscript “0” refers to the quantity at present.
2.5 Physical consequences for the evolution
of local systems
Eq.30 is surprising and in contradiction to the common opinion
that the expansion of the Universe is without any appreciable eect on
local gravitational systems (Carrera and Giulini, 2010). is opinion
is based on equations for gravitational orbits in an expanding space
described by the FLRW metric (see AppendixA). e equations were
derived by many authors (Dicke and Peebles, 1964;Pachner, 1964;
Frontiers in Astronomy and Space Sciences 04 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
Callanetal., 1965;Cooperstocketal., 1998;Faraoni and Jacques,
2007;Sereno and Jetzer, 2007;Carrera and Giulini, 2010) and they
dier from the standard Newtonian equations for orbits in the static
Universe only slightly. e only dierence is that the equation for
the radial acceleration of an orbiting body in the gravitational eld
imbedded in the space with the FLRW metric contains a term a
aR,
which is related to the space expansion (see AppendixA, Eq. A-7).
It can be shown that this term might be appreciable in dynamics of
large-scale structures as galaxy clusters, but the orbits within galaxies
are aected negligibly. Consequently, all gravitationally bound systems
with size of galaxies or smaller should be unaected by the space
expansion.
By contrast, considering the CC metric for the expanding
Universe, the evolution of the local gravitational systems is essentially
dierent from that obtained for the FLRW metric. In particular, the
CC metric predicts:
1 An increase of the proper orbital radius Rorb with the expansion
irrespective of the size of the local system and an increase of the
proper orbital period Torb with the expansion. Consequently, the
size of galaxies (and all other local gravitational systems) must
growth in the expanding Universe. e rate of growth is 1z−1.
e orbit is stationary in comoving coordinates but non-stationary
in proper coordinates.
2 A constant proper rotation velocity Vϕof particles along the orbits
irrespective of the orbital radius during the space expansion. Hence,
the stationary circular orbits in the comoving coordinate system
become spirals in the proper coordinatesystem. Stars and gas within
spirals have at rotation curves. Since the at rotation curves are
a consequence of the expansion of the Universe, no dark matter is
needed to explain dynamics of galaxies.
3 Numerical modelling of the galaxy
dynamics
In this section, we will examine the evolution of local
gravitationally bounded systems in the expanding Universe by
numerical modelling. e modelling is simple and reects only
basic features of the galaxy evolution. It is focused on studying the
formation of spirals and their evolution in time. e gravitational eld
is axially symmetric and interactions between stars and gas inside
spirals are neglected. e aim is just to demonstrate the potential
of the derived Eqs29,30 of gravitational orbits in the CC metric
for the full realistic modelling of the galaxy evolution in future
studies.
3.1 Parameters for modelling
For modelling, we need to specify the expanding history of the
Universe described by the Hubble parameter Hz. Since we do not
use the FLRW metric, we cannot adopt the ΛCDM model. Instead,
the CC metric described by Eq.17 must be used. According to
Vavryčuk (2022a), we use parameters Ωm12, ΩΛ0 (no dark
energy), and Ωk02 (closed Universe). e Hubble constant is
H0698 km s−1 Mpc−1, obtained by Freedmanetal. (2019) from
observations of the SNe Ia data with a red giant calibration. Obviously,
Eqs.17,19 predict a quite dierent evolution of the Hubble parameter
and a time-redshi relation than the ΛCDM model. For example,
redshi z= 4 corresponds to the cosmic time of 12Gyr for the ΛCDM
model but 8.2Gyr for the CC model (see Figure1).
e galaxy is assumed to be formed by a bulge and disk with the
following exponential density proles (McGaugh, 2016)
ΣbRΣbRb0erRb0Rb(31)
ΣdRΣdRd0erRd0Rd(32)
where ΣbRand ΣdRare the surface densities of the bulge
and disk, respectively. e required parameters are for the bulge
ΣbRb036 M⊙pc−2,Rb025 kpc, Rb03 kpc, and for the disk
ΣdRd0992 M⊙pc−2,Rd080 kpc, Rd28 kpc. e total masses
of the bulge and disk are Mb= 8.5 × 109M⊙and Md= 8.5 × 1010M⊙,
respectively. e bulge is usually modelled as a prolate, triaxial bar
(Binneyetal., 1997;Bissantz and Gerhard, 2002), but we do not focus
on modelling a 3D geometry of the bulge and its evolution, so the
simplied approximation described by Eq.31 is satisfactory.
3.2 Bulge-bar region and spirals
e density proles and the Newtonian rotation curves predicted
for the used parameters are shown in Figure2. e total rotation
curve (Figure2B, black line) is separated into two domains. e
separation (critical) distance Rcis about 4kpc (Figure2B, dashed
vertical line) and corresponds to the maximum rotation velocity
associated with the disk density prole (Figure2B, red line). For
shorter distances (bulge-bar domain), the behaviour of the rotation
curve is complex being aected by both the bulge and disk. For larger
distances (spiral domain), the initial rotation curve is monotonously
decreasing aected mostly by the total mass in the bulge-bar area.
Obviously, the critical distance Rcmay vary for dierent galaxies being
dependent on their mass and density proles. As discussed below, the
rotation curve evolves in time and the nal rotation curve becomes at
in the spiral domain (Figure2B, green line).
Figure3 shows the bulge-bar and spiral regions for the spiral
galaxy UGC 6093 together with a scheme suggesting a possible
origin of spirals and their temporal evolution. e bulge-bar region is
characterized by a high concentration of stars and gas. Consequently,
gravitational forces maintain the shape of the galaxy inside this region,
irrespective of its rotation. Since the bar is continuously increasing
due to the space expansion, its ends cross the region boundary and
outow from the bulge-bar region into the spiral region. In this region,
the radial gravitational forces become dominant and stars and gas
move as bodies in a spherically symmetric gravitational eld. eir
motion will be described by Eqs.29,30: the rotation velocity will be
constant with time but the orbital radius will increase. Consequently,
the rotating bar will outstrip the material in the spiral region, which
will form a pattern of trailing spirals. e material will display at
rotation curves in all spirals, and the size of galaxies will increase with
time.
3.3 Scenarios of the galaxy evolution
e form of spirals depends on several factors: 1) e radius of
the bulge-bar area, 2) the age of the galaxy, 3) the mass of the galaxy,
density prole and its rotation velocity, and 4) the expansion history of
Frontiers in Astronomy and Space Sciences 05 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
FIGURE 1
The Hubble parameter (A) and the proper cosmic lookback time (B) as a function of redshift. The cosmological model is described by Eq.17 for the CC
metric (blue line) with Ωm12, and Ωk 02 (Vavryčuk, 2022a). The red line shows the standard ΛCDM model. The Hubble constant is
H0698 kms−1Mpc−1, obtained by Freedmanetal. (2019) from observations of the SNe Ia data with a red giant calibration. Note that panel (B) does not
imply that the age of the Universe is less than about 9 Gyr in the CC model. For redshifts z15, the cosmic evolution becomes more complicated due to
light-matter interactions, and Eq.17 must be modied, see Vavryčuk (2022b).
FIGURE 2
The density prole (A) and the initial and nal rotation velocity curves (B) for a simulated galaxy. The surface densities of the bulge (blue line in (A)) and disk
(red line in (A)) are calculated by Eqs31,32 with Σb(Rb0) = 3.6M⊙pc−2,Rb0= 2.5kpc, Rb= 0.3kpc for the bulge, and with Σd(Rd0) = 99.2M⊙pc−2,Rd0=
8.0kpc, Rd= 2.8kpc for the disk. The total masses of the bulge and disk are Mb= 8.5 × 109M⊙and Md= 8.5 × 1010M⊙, respectively. The total initial
rotation velocity (black line in (B)) is shown together with the contribution of the bulge (blue line in (B)) and disk (red line in (B)). The vertical dashed line in
(B) denes the boundary Rc= 4kpc between the bulge-bar regime and the trailing-spiral regime. The green line in (B) shows the nal rotation curve caused
by the space expansion (the stellar mass and gas with RRcare continuously moving out of the galaxy centre, but they keep their rotation velocity).
the Universe. erefore, we assume several alternative scenarios in the
numerical modelling, where we vary some of these parameters. e
radius of the bulge-bar area Rcwill be considered 3, 4, and 5kpc. e
age of a galaxy will be 8.0, 8.5, and 8.8Gyr. is age will correspond
to the redshi zof 3.2, 5.6, and 11.2. e masses of the bulge and disk
of a galaxy are Mb85109M⊙and Md851010 M⊙, but we will
also model a galaxy with a half of this mass and with a twice higher
mass.
In order to simulate a more realistic galaxy evolution, we assume
random conditions for the bulge-bar outow. e critical distance Rc
is not dened by a single value, but it varies according to the Gaussian
distribution with the standard deviation of 0.05kpc. Similarly, the
bulge-bar outow does not cross the domain boundary at two single
points dened at the two ends of the bar by angles ϕ0and 180.
Instead, angle ϕobeys the Gaussian distribution centred at 0and 180
with the standard deviation of 20.
3.4 Results
In modelling, we calculate orbits of stars and gas owed out from
the bulge-bar domain into the spiral domain and the evolution of the
orbits in time. Geometry of orbits is dened by Eqs.29,30, in which
we specify the rotation velocity at the boundary between the bulge-bar
and spiral regions and the expansion history dened by the Hubble
parameter Hz.Figure4 shows the evolution of a galaxy with age of
8.8Gyr. e galaxy started to evolve at redshi zof 11.2. e galaxy
was formed just by the bulge and bar with no spirals at the beginning
of the simulation. At this time, the radius of the galaxy was 4kpc
being the same as the radius of the bulge-bar area. e material, which
outowed from the bulge-bar domain due to the space expansion,
formed typical trailed spirals during the galaxy evolution (Figure4B).
Hence, the radius of the galaxy increased from 4kpc to 49kpc during
its life (Figure4A). e rotation velocity of the material in the spirals
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FIGURE 3
The spiral galaxy UGC 6093 observed by ESA/Hubble (A) and the
scheme of the spiral galaxy (B). The bulge and bar of the galaxy in plot
(B) is marked by the purple colour. The purple arrows in (B) show the
rotation direction of the bar. The black arrows in (B) show the direction
of the outow from the bulge-bar domain into the spiral domain. The
white dashed circle in (A) and the solid black circle in (B) mark the
boundary between the bulge-bar and spiral domains.
is uniform and attains a value of 217.7km/s. e classical problem in
the galaxy dynamics called the “winding paradox” of spirals (Ferreras,
2019) cannot appear, because the radius of spirals is continuously
increased and spiral arms are moving away from the bulge-bar
region.
Figure5 demonstrates a dependence of the spiral pattern on the
size of the bulge-bar area. e age of the galaxy is again 8.8 Gyr
and the evolution started at redshi zof 11.2. For a smaller radius
of the bulge-bar area (Figure5A,Rc= 3kpc), the rotation velocity
is slightly higher being 218.3km/s. As a result, the orbital period
is smaller and the recession of the spirals from the central part of
the galaxy is not so pronounced. By contrast, for a larger radius of
the bulge-bar area (Figure5C,Rc5 kpc), the rotation velocity is
slightly lower being 215.2km/s. e orbital period is higher and the
recession of the spirals from the central part of the galaxy is more
distinct.
Figure6 shows how is the spiral pattern aected by the age of
the galaxy. e gure shows three galaxies with the age of 8.0, 8.5,
and 8.8Gyr. e corresponding redshis are 3.2, 5.6, and 11.2. e
radius of the bulge-bar area is Rc4 kpc and the rotation velocity
is 217.7km/s for all three galaxies. As expected, the younger the
galaxy, the less evolved the spiral pattern. Consequently, the nal
radius of the galaxy increased from 4kpc to 17, 26, and 49kpc,
respectively.
Finally, Figure7 presents how is the spiral pattern aected by the
mass of a galaxy. Figure7B shows a galaxy with masses of the bulge
and disk of a galaxy are Mb85109M⊙and Md851010 M⊙
(rotation velocity at Rc4 kpc is 217.7km/s). ese values were used
in all previous simulations (Figures4–6). Figure7A shows a galaxy,
which has a twice higher mass. e corresponding rotation velocity
at Rc4 kpc is 307.9km/s. By contrast, Figure7C shows a galaxy,
which has a twice lower mass. e corresponding rotation velocity at
Rc4 kpc is 153.9km/s only. e age of the three galaxies is 8.8Gyr
and their evolution started at redshi z112. Since the galaxies have
the same age, their size increases in the same way: from 4kpc to 49kpc.
Still the spiral pattern is remarkably dierent for all three galaxies.
e massive galaxy rotates fast and the spirals are receding slowly
from the galaxy centre (Figure7A). e galaxy with the smallest mass
rotates slowly and the recession of spirals from the galaxy centre is high
(Figure7C).
4 Supporting observational evidence
e presented results are supported by many observations dicult
to explain under the standard cosmological model. is applies to
galaxy dynamics, morphology of spiral galaxies as well as dynamics
of the solar system. In the next, we review several puzzles in modern
cosmology resolved by the proposed theory.
4.1 Galaxy expansion
As shown in the previous sections, the size of galaxies should
increase with redshi as 1z−1 with no change of the galaxy
mass. Based on observations, it is accepted that the size of galaxies
evolves rapidly during the cosmic time (vanDokkumetal., 2008;
vanDokkumetal., 2010;Williamsetal., 2010), see Figure8. Using
observations from the Hubble Space Telescope (HST), galaxy sizes
dened by the eective radius, Re, have been extensively measured
with the Advanced Camera for Surveys (ACS) and the Wide Field
Camera 3/IR channel on board HST for massive galaxies at z3
(vanderWeletal., 2014) and z34 Lyman break galaxies (LBGs)
selected in the dropout technique (Trujilloetal., 2006;Dahlenetal.,
2007;McLureetal., 2013). e average size is reported to evolve
according to Re∼1z−B, with Branging most frequently between
0.8 and 1.2 (Bouwensetal., 2004;Oeschetal., 2010;Holwerdaetal.,
2015). For example, Shibuyaetal. (2015) studied the redshi evolution
of the galaxy eective radius Reobtained from the HST samples of
∼190,000 galaxies at z010, consisted of 176,152 photo-zgalaxies
at z06 from the 3D-HST + CANDELS catalogue and 10,454
Lyman break galaxies (LBGs) at z410 identied in the CANDELS,
HUDF 09/12, and HFF parallel elds. ey found that Revalues
at a given luminosity decrease toward high z, as Re∼1z−B, with
B110 006 for median, see Figure8A.
Since it is believed that the size of galaxies cannot be aected by
the expansion of the Universe, the observed expansion of galaxies is
explained by other mechanisms. e most popular theory suggests
the growth of galaxies being produced by galaxy mergers (Naabetal.,
2009;Kormendy and Ho, 2013;McLureetal., 2013;Conselice, 2014).
An important role in merging of galaxies play dark matter haloes
(Kaumannetal., 1993;Moetal., 1998). Two types of mergers are
distinguished: a major merger where the stellar masses of the galaxies
are comparable, and a minor merger where the stellar mass of one
galaxy is much lower.
However, the idea of the galaxy expansion due to galaxy mergers
is controversial for several reasons (Lerner, 2018). First, observations
indicate that the major and minor merger rates are much lower
to explain the galaxy expansion (Tayloretal., 2010;Manetal.,
2012;Manetal., 2016). For example, Mundyetal. (2017) report
approximately 0.5 major mergers at z35 representing an increase in
stellar mass of 20%–30% only when considering constant stellar mass
samples. As regards minor mergers, Newmanetal. (2012) studied
935 galaxies selected with 04z25 and concluded that minor
merging cannot account for a rapid growth of the size seen at
higher redshis. Manetal. (2016) studied massive galaxies using
the UltraVISTA/COSMOS catalogue, complemented with the deeper,
higher resolution 3DHST + CANDELS catalogue and estimated ∼1
major merger and ∼0.7 minor merger on average for a massive
M*1010.8 M⊙galaxy during z0125. e observed number of
major and minor mergers can increase the size of a massive quiescent
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FIGURE 4
The proper radius of a galaxy as a function of redshift (A), and the spiral arms formed during the galaxy evolution (B). The density of the material in the spiral
arms in (B) is colour coded. The age of the galaxy is 8.8Gyr and the maximum redshift is z112. The solid black circle in (B) marks the boundary between
the bulge-bar and spiral domains with radius Rc4kpc. The black dashed line in (B) denotes the central orbit with unperturbed parameters Rcand ϕ. The
other orbits forming the spirals are characterized by perturbed parameters Rcand ϕ, see the text. The red bulge and bar inside the black circle in (B)
illustrate schematically the orientation of the bar.
FIGURE 5
Geometry of spirals: dependence on the initial size of the bulge-bar region. The age of the galaxy is 8.8Gyr and the maximum redshift is z112. The
upper/lower plots show orbits with unperturbed/perturbed parameters Rcand ϕ. The critical radius Rcand the rotation velocity Vϕare 3kpc and 218.3km/s
in (A, B), 4kpc and 217.7km/s in (C, D), and 5kpc and 215.2km/s in (E, F). The masses of the bulge and disk are Mb= 8.5 × 109M⊙and Md= 8.5 × 1010M⊙,
respectively.
galaxy by a factor of two at most. Hence, additional mechanisms are
needed to fully explain the galaxy evolution. Second, mergers cannot
explain the growth of spiral galaxies, because mergers destroy disks as
shown by Bournaudetal. (2007). ird, the idea of mergers implies
an increase of stellar mass in galaxies over cosmic time. However,
observations show no or slight mass evolution in time (Bundyetal.,
2017;Kawinwanichakijetal., 2020).
4.2 Galaxy rotation curves
Another basic characteristics predicted by the presented theory are
at rotation curves of spiral galaxies. In Newton theory, a velocity of
stars in a rotating galaxy is controlled by gravitational and centrifugal
forces only. Assuming the Newton’s gravitation law, a balance between
the forces implies a decay of the orbital speed VRof a star with its
distance Rfrom the galaxy centre
V2RGMR
R(33)
where MRis the mass of the galaxy as a function of R. Hence,
the rotation curve VRdecays as R−1/2 provided the most of
mass is concentrated in the galaxy centre. e same applies to the
improved Newtonian equations in an expanding Universe described
by the standard FLRW metric. Obviously, the at rotation curves
predicted by the improved Newtonian equations in the Universe with
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FIGURE 6
Geometry of spirals: dependence on the galaxy age. The critical radius Rcand the rotation velocity Vϕare 4kpc and 217.7km/s. The upper/lower plots
show orbits with unperturbed/perturbed parameters Rcand ϕ. The galaxy age and the maximum redshift are 8.0Gyr and z320 in (A, B), 8.5Gyr and
z565 in (C, D), and 8.8Gyr and z1118 in (E, F). The masses of the bulge and disk are Mb= 8.5 × 109M⊙and Md= 8.5 × 1010M⊙, respectively.
FIGURE 7
Geometry of spirals: dependence on the galaxy mass. The critical radius Rc, the galaxy age and the maximum redshift are 4kpc, 8.8Gyr and z1118. The
upper/lower plots show orbits with unperturbed/perturbed parameters Rcand ϕ. The masses of the bulge and disk are Mb= 17 × 109M⊙and
Md17 1010M⊙in (A, B),Mb85109M⊙and Md= 8.5 × 1010M⊙in (C, D), and Mb= 4.25 × 109M⊙and Md= 4.25 × 1010M⊙in (E, F). The rotation
velocity Vϕis 307.9km/s in (A, B), 217.7km/s in (C, D), and 153.9km/s in (E, F).
the CC metric point to fundamental dierences between both the
metrics. Importantly, the at rotation curves of spiral galaxies are
observationally conrmed.
4.2.1 Observations of at rotation curves
Rubin and Ford (1970) discovered that the rotation curve of
the Andromeda Galaxy has a sharp maximum of V225 kms
at R400 pc, a deep minimum at R2 kpc, and it is nearly
at at R3 kpc with the maximum velocity of 270 10 kms.
Such behaviour was later conrmed also for other spiral galaxies
(Rubinetal., 1980;Rubinetal., 1985;vanAlbadaetal., 1985;
Begeman, 1989;Sanders, 1996;Swatersetal., 2000;Sofue and Rubin,
2001;deBlok and Bosma, 2002;deBloketal., 2008;McGaugh, 2019;
Tileyetal., 2019), for an example, see the rotation curve of the NGC
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FIGURE 8
Galaxy size evolution with redshift. (A) Median of the eective galaxy radius Ras a function of redshift for galaxies in the bin of LUV 031L*
z=3. The red
and cyan lled circles indicate radius Rfor star-forming galaxies measured in the optical (45008000Å) and UV (15003000Å) wavelength ranges,
respectively. The blue lled circles indicate radius Rfor the Lyman break galaxies measured in the UV wavelength range. For details, see Shibuyaetal. (2015,
their Figure8). (B) Median Petrosian radius of galaxies as a function of redshift for the mass-limited sample in the range of 109M⊙≤M* ≤ 1010.5M⊙. The
ratio of the surface brightness at radius Rto the mean surface brightness of a galaxy is η02. For details, see Whitneyetal. (2019, their Figure8). The
dashed lines in (A, B) show the size evolution predicted by the presented theory.
FIGURE 9
Rotation curve for the NGC 6503 Galaxy. The spiral domain covering
distances greater than 4kpc is indicated. In this domain, receding of
spirals from the galaxy centre causes the atness of the rotation curve.
For data, see Begeman (1987) and Lellietal. (2016a).
6503 Galaxy in Figure9. e measurements of rotation curves are
mostly based on (Sofue, 2017): observations of emission lines at
optical wavelengths such as Hαand [NII] lines, particularly, in HII
regions in galactic disks; at infrared wavelengths revealing kinematics
of dusty disks and nuclear regions of spiral galaxies with signicant
dust extinction; and at 21-cm HIline powerful to study kinematics of
entire spiral galaxy.
e rotation curves of spiral galaxies display a signicantsimil arity
irrespective of their morphology (Persicetal., 1996;Sofue, 2016;
Sofue, 2017). e dierences are mainly connected to the mass
and size of the galaxies. More massive and larger galaxies (Sa and
Sb) have high rotation velocity close to the nucleus, while smaller
galaxies (Sc) show slower rotation in the centre. e earlier-type (Sa
and Sb) galaxies display a at or slowly declining velocity at the
outermost part of the rotation curve, while the rotation velocity of the
later-type (Sc) galaxies monotonically increases. Similarly, dwarf
and LSB galaxies display monotonically increasing rotation velocity
until their galaxy edges (Swatersetal., 2000;deBloketal., 2001;
Swatersetal., 2003). In addition, Tully and Fisher (1977) revealed an
empirical statistical relation between the galaxy luminosity and the
maximum rotation velocity at a few galactic disk radii. e Tully-
Fisher relation is commonly used for estimating the luminosity of
distant galaxies (Jacobyetal., 1992;Mathewsonetal., 1992;Phillips,
1993) and for measuring the Hubble constant (Mouldetal., 2000;
Freedmanetal., 2001).
4.2.2 Dark matter
To explain the discrepancy between predicted and observed
rotation curves, several theories have been proposed. e most
straightforward way is to assume the presence of dark matter (DM)
with distribution calculated as (Schneider, 2015, his Eq.3.17)
Mdark RR
GV2
obs RV2R(34)
where VobsRis the observed rotation velocity and VRis calculated
according to Eq.33. e idea of dark matter origins from Zwicky
(Zwicky, 1937;Zwicky, 2009) who postulated “missing mass” to
account for the orbital velocities of galaxies in clusters. Originally,
the DM was assumed to be baryonic formed by gas, dust and
microscopic and macroscopic solid bodies including black-holes.
Later on, the baryonic origin of DM was questioned and rejected. Since
the analysis of galaxy rotation curves revealed that the mass of DM
(vanAlbadaetal., 1985;Dubinski and Carlberg, 1991;Navarroetal.,
1996;Persicetal., 1996;Navarroetal., 1997) is much higher than
estimates of dust and gas in galaxies (Calzettietal., 2000;Dunneetal.,
2000;Draine and Li, 2007;daCunhaetal., 2008;Sandstrometal.,
2013), DM was considered to be mostly of non-baryonic nature (White
and Rees, 1978;Davisetal., 1985;Whiteetal., 1987;Maddoxetal.,
1990;Mooreetal., 1999;Bergström, 2000;Bertone and Hooper, 2018).
To reconcile the theoretical and observed rotation curves, the DM
is signicant at large distances and forms a DM halo with the total
mass exceeding the stellar galaxy mass by about one order or more
(vanAlbadaetal., 1985;Dubinski and Carlberg, 1991;Navarroetal.,
1996;Persicetal., 1996).
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4.2.3 Conformal Gravity and MOND theory
e non-baryonic DM concept is not, however, unanimously
accepted. e non-baryonic DM is questioned, in particular, for its
exotic and mysterious nature and for diculties to be detected by other
methods independent of gravity. Also, signicant discrepancies with
predictions of the ΛCDM model on small scale are reported by many
authors (Kroupa, 2015;DelPopolo and LeDelliou, 2017). erefore,
several alternative theories have been proposed to explain the at
rotation curves (Mannheim, 2006). e most famous alternative
theories are the Conformal Gravity and Modied Newtonian
Dynamics theory (Mannheim, 2006).
e Conformal Gravity (CG) attempts to solve both the problems
of dark energy and dark matter. Similarly as the CC approach
presented in this paper, the CG theory emphasizes the importance of
conformal transformations for solving gravity problems (Mannheim,
2006). However, both the approaches are essentially dierent: the CC
metric is applied to the standard GR equations, while the CG theory
is based on modifying the GR equations, in order the equations to be
conformally invariant. e CG equations are based on minimizing the
Weyl action with the use of the Weyl conformal tensor rather than
on the Einstein-Hilbert action (Mannheim, 1990;Mannheim, 2006;
Mannheim, 2012). e CG theory avoids dark energy as well as ts
rotation curves without use of dark matter. Unlike the Newtonian
gravity, the CG is global theory, where both the local galactic and
the exterior gravitational elds are considered. e rotation velocity
is aected by a linear potential caused by the Hubble ow and by
a quadratic potential caused by inhomogeneities (Mannheim and
O’Brien, 2011;Mannheim, 2019). As reported by Mannheim (2019),
the CG theory has successfully tted rotation curves of 207 galaxies
including dwarf galaxies, see also Mannheim and O’Brien (2012) and
O’Brien and Mannheim (2012).
e Modied Newtonian Dynamics (MOND) theory was
presented by Milgrom (1983a),Milgrom (1983b),Bekenstein (2004),
Milgrom (2010), and Milgrom (2012), who proposed to modify the
Newton gravity law for very low accelerations. Below a0∼cH06, the
standard Newton gravity acceleration gNis substituted by a0gN. is
causes that the Newton’s law keeps valid for planetary and other small-
scale systems but it does not apply to galaxies and galaxy clusters. As a
consequence, rotation curves t well observations (Begemanetal.,
1991). Except for the at rotation curves, the MOND theory is
successful in accounting for some other phenomena listed below,
which are dicult to explain using the DM hypothesis (Sanders and
McGaugh, 2002;Bugg, 2015).
4.2.3.1 Faint satellite galaxies
e existence of the non-baryonic DM is questioned by a
detailed study of properties of faint satellite galaxies of the Milky
Way (MW), see Kroupaetal. (2010), which are distributed on a
planar structure. Similar alignments were observed also in isolated
dwarf galaxies in the local group (Pawlowski and Kroupa, 2013;
Pawlowski and McGaugh, 2014) as well as in more distant galaxies
(Galiannietal., 2010;Ducetal., 2014). is is a challenge for
cosmological simulations, because the DM sub-haloes are assumed to
be isotropically distributed.
4.2.3.2 Dual Dwarf Galaxy Theorem
e standard ΛCDM model predicts two types of dwarf galaxies:
Primordial DM dominated dwarf galaxies (Type A), and tidal and
ram-pressure dwarf galaxies (Type B). While Type A dwarfs should
surround the host galaxy spherically, the B dwarfs should be typically
correlated in phase space. However, only dwarf galaxies of Type B
are observed. is falsies the Dual Dwarf Galaxy eorem and the
presence of DM haloes (Kroupa, 2012).
4.2.3.3 Baryonic Tully-Fisher relation
Observation of the baryonic Tully-Fisher (BTF) relation, which
is a power-law relation between the rotation velocity of a galaxy
and its baryonic mass MS+G, calculated as the sum of stellar mass
MSand gas mass MG(Verheijen, 2001;Noordermeer and Verheijen,
2007;Zaritskyetal., 2014). is empirical relation is valid over several
orders of magnitude andw ith an extremely small scatter. In the ΛCDM
model, the rotation velocity should be primarily related to the total
virial mass, represented mostly by the dark matter halo, M∼V3, but
not to MS+G. Since the dark matter halo is largely independent of
baryonic processes, it is dicult to explain the observed extremely
low scatter of the BTF relation (Lellietal., 2016b;McGaughetal.,
2018;Lellietal., 2019). If the most mass of a galaxy is formed by
the baryonic dark matter located in the galaxy disk but not in the
halo, the close relation between the stellar, gas and dust masses is
expected.
4.2.3.4 Radial acceleration relation
A further close connection between mass of stars and gas MS+G
and the total mass Mof galaxies, was revealed by McGaughetal.
(2016), when they studied a relation between the acceleration gS+G
due to mass MS+G and the observed acceleration gobs due to total
mass M. e observed relation is fully empirical and points to a
strong coupling between the mass of the dark matter and mass
of stars and gas. Similarly as for the BTF relation, the observed
coupling between gS+G and gobs is dicult to explain in the ΛCDM
model.
4.2.3.5 Deciency of Conformal Gravity and MOND theory
Although, the MOND theory matches some observations quite
successfully, the theory was originally designed rather empirically to
t observations with no profound physical consistency. For example,
some deep reasoning, why value of a0is within an order of magnitude
of cH0, is missing.
A common deciency of MOND and CG seems violation of
the GR equations. By contrast, theory presented in this paper
explains satisfactorily the at rotation curves and the above mentioned
phenomena with no need to violate the GR theory. Nevertheless,
we have to admit that the denitive resolution, whether GR or its
modications such as MOND (Milgrom, 1983a;Bekenstein, 2004;
Milgrom, 2010;Milgrom, 2012) or Conformal Gravity (Mannheim,
2006;Mannheim and O’Brien, 2011;Mannheim and O’Brien, 2012;
Mannheim, 2019) describe the gravitational eld more appropriately,
will need other future tests, e.g., based on detection of gravitational
waves (Corda, 2009).
4.3 Morphology of spiral galaxies
Several theories have been proposed to explain the origin and
evolution of structure of spiral galaxies and to predict basic properties
of spiral arms (Toomre, 1977;Dobbs and Baba, 2014;Shu, 2016).
Lindblad (1962) was the rst, who assumed that the spiral structure
arose from interaction between the orbits and gravitational forces of
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the stars of the disk, and suggested to explain the spiral arms as density
waves. is idea was further elaborated by Lin and Shu (1964) and
Lin and Shu (1966) in their hypothesis of the quasi-stationary density
waves, in which the spirals are formed by standing waves in the disk.
ey assume that the spiral pattern rotates in a particular angular
frequency dierent from the rotation velocity of stars, which depends
on the star distance from the galaxy centre. e formation of the global
spiral pattern is considered as an instability of the stellar disk caused
by the self-gravity. e density-wave theory was further developed
and extended (Shu, 1970;Robertsetal., 1975;Sellwood and Carlberg,
1984;Elmegreenetal., 1999;Sellwood, 2011) and it is now the main
tool for studying the gravitational stability of disk galaxies. e results
are not decisive, but N-body simulations suggest that the spiral arms
are transient and recurrent rather than quasi-stationary (Babaetal.,
2009;Babaetal., 2013).
e density-wave theory faces, however, with several open
questions and limitations. First, the theory is based on the classical
Newtonian gravity, which neglects the space expansion. Second,
predictions of the theory are uncertain. Still it is not clearly resolved,
whether the spiral arms must be dynamic or whether the quasi-static
arms are a feasible solution. ird, the observationally documented
growth of the spiral galaxies with cosmic time is completely ignored
and unexplained in this theory.
All the mentioned diculties with modelling of spiral arms arise
from the fact that the theory starts with rejecting an idea of spirals
formed by stars and gas that remain xed in the spirals. e reason is
the so-called winding problem, when objects moving with the same
orbital speed in the disk cause a dierential rotation of material in
galaxies (Ferreras, 2019). Since the length of orbits is shorter near
the galaxy centre, the inner part of spirals winds up tighter than
its outer part. Hence, a typical spiral pattern disappears aer a few
rotations. is idea is, however, simplistic and incorrect, because the
GR eects in the galaxy evolution are ignored. If the space expansion
and time dilation are considered, the galaxy size is growing and galaxy
rotation speed varies with time. e spiral pattern is not destroyed,
because it is continuously expanding. Consequently, the winding
problem does not occur as demonstrated by numerical modelling in
Section3.
4.4 Solar system
Observations conrm that the global expansion aects also
the solar system. Here we mention some of prominent examples
(Andersonetal., 1998;Krizek, 2012;Iorio, 2015;Krizeketal., 2015;
Krizek and Somer, 2015).
4.4.1 Faint young Sun paradox
According to the Standard solar model (Bahcalletal., 2001), the
radius and luminosity of the Sun signicantly evolved during the
cosmic time. As the Sun is a star on the main sequence of the HR
diagram, the solar radius was 4Gyr ago about 89% of the solar radius at
present and the luminosity was about 73.8% of the present luminosity
(Bahcalletal., 2001, their tables1 and 2). For a constant distance
between the Sun and the Earth during this time span, such changes
would have dramatic consequences for life conditions on the Earth
(Ribasetal., 2010). e solar constant, i.e., the ux density at the
Earth’s mean orbital distance, is I136 kWm−2 at present (Kopp and
Lean, 2011), but 4Gyr ago it was I0100 kWm−2 only. Calculating
the equilibrium temperature as
Teq I1A
4σ14(35)
where A03 is the Earth’s albedo (Goodeetal., 2001) and σis
the Stefan-Boltzmann constant, we get Teq 2545 K and Teq
0
2360 K for the present time and for the past, respectively. Assuming
the same level of the greenhouse eect (325C), the global
average Earth’s temperature would be 47C in the past instead of
139C observed at present (https://www.climate.gov/news-features/
understanding-climate/climate-change-global-temperature). In fact,
because of the ice albedo, the temperature would be even lower in
the era of 4Gyr ago. For the Earth’s albedo of 0.5, we get Teq
0
2494 K and the global average Earth’s temperature would be 238C.
By contrast, no glaciation is indicated from geological observations
in the rst 2.7Gyr of the Earth’s evolution (Bertottietal., 2003)
and water-related sediments have been found 3.8Gyr ago (Windley,
1984). is severe discrepancy is known as the Faint young Sun
paradox.
e paradox is resolved, if the expansion of the solar system
is taken into account (Křížek, 2012;Křížek and Somer, 2015).
e age of 4Gyr corresponds to redshi z046 (see Figure1B)
and the orbital radius of the Earth was 1ztimes shorter
than at the present time. e ux corrected for a shorter orbital
radius is easily calculated as 100 1z2213 kWm−2 and the
corresponding Earth’s temperature is 445C, provided we assume
the same greenhouse eect as at present 325C. Hence, the
temperature conditions on the Earth were convenient for life over the
whole Earth’s history. Note that higher temperatures of oceans 70C
in the Precambrian era (3.5Gyr ago) are independently indicated by
observations of silicon and oxygen isotope data (Knauth, 2005;Robert
and Chaussidon, 2006). e recession velocity of the Earth from the
Sun comparable to the Hubble ow is indicated also from growth
patterns on fossil corals observed for the time span of the last 500Myr
(Zhangetal., 2010).
4.4.2 Lunar orbit anomaly
e Moon’s orbital distance is slowly increasing and the Earth’s
rotation rate is decreasing due to tidal forces transferring angular
momentum from the Earth to the Moon. In order to investigate the
Earth-Moon system, the Lunar Laser Ranging Experiment (LLR) from
Apollo 11, 14, 15 and Lunokhod missions was performed to measure
accurately the recession velocity of the Moon. e missions report
the Moon’s semimajor axis d384402 km, which increases at rate
(Dickeyetal., 1994) of 382 008cmyr. is value is anomalously
high and inconsistent with an expected lunar recession velocity due
to tidal forces, which should be lower by 3045(Křížek, 2009;
Riofrio, 2012;Křížek and Somer, 2022). e observed lunar recession
velocity would correspond to increasing Earth’s rotation period at
a rate of 23 ms per century, but only a rate of 18 ms per
century is observed (Stephensonetal., 2016). In addition, numerical
modelling of the orbital evolution under such tidal dissipation would
imply the age of the lunar orbit to be 15109years, instead of
4109years suggested by observations (Bills and Ray, 1999). is
discrepancy is known as the Lunar orbit anomaly and so far its origin is
unclear.
If the expansion of the solar system is considered, the
recession velocity of the Moon due to the expansion is 274 cmyr
assuming distance of the Moon d384402 km and the Hubble
Frontiers in Astronomy and Space Sciences 12 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
parameter H0698 kms−1Mpc−1 (Freedmanetal., 2019). Hence,
the lunar recession velocity due to tides is reduced to 108 cmyr
that is more realistic. If the tides were fully responsible for
the slowing Earth’s rotation at rate of 18 ms per century,
the lunar recession velocity would be ∼225 cmyr. However,
this is an upper limit only, because also other processes can
slow down the Earth’s rotation, such as impacts of massive
meteorites, large earthquakes, huge volcanic eruptions, and energy
dissipation in the Earth’s mantle and the outer Earth’s core due to
convection.
4.4.3 Other observations
e Hubble ow in the solar system is indicated by many other
observations (Křížek and Somer, 2015). Mars had to be much closer
to the Sun in the past; it is dusty and icy at the present, but
detailed images of the Martian surface reveal that it was formed by
rivers in the period of 34 Gyr ago (Carr, 1995;Davisetal., 2016;
Saleseetal., 2020). Measurements of the Titan’s orbital expansion rate
by the Cassini spacecra during ten close encounters of the Moon
between 2006 and 2016 revealed that Titan rapidly migrates away
from Saturn (Laineyetal., 2020). e Titan-Saturn mean distance
is d1221870 km and the Titan’s recession velocity is 113 cmyr.
e corresponding recession velocity due to the Hubble ow is
87 cmyr. If this velocity is subtracted, the anomaly disappears and
the resultant rate of 26 cmyr produced by tidal forces becomes
realistic. Also, the expansion of the solar system can explain the
formation of Neptune and Kuiper belt, the existence of fast satellites
of Mars, Jupiter, Uranus and Neptune that are below the stationary
orbit, or the large orbital momentum of the Moon (Křížek and Somer,
2015).
5 Discussion and conclusion
e presented theory and numerical modelling satisfactorily
explain several severe tensions between the standard ΛCDM model
and observations.
• e improved Newtonian equations derived for the CC metric
predict an increasing radius of gravitational orbits of local
systems. e rate of growth with cosmic time is 1z−1. is
applies to all local systems including galaxy clusters, galaxies,
and planetary systems. e gravitational orbits are stationary
in the comoving coordinates but non-stationary in the proper
coordinates.
• e existence of spirals in disk galaxies is a direct consequence of
the space expansion and time dilation. e spirals are formed by
the stellar mass and gas that remain xed in them. e stellar mass
and gas are continuously outowed from the bulge-bar region
into the spiral region. e spirals are detached from the bulge-bar
region due to the space expansion.
• Since the orbital velocity of particles is conserved and
the radius of orbits gradually increases during the
space expansion, the spiral galaxies display at rotation
curves.
e constant rotation velocity, the space expansion and time
dilation are the primary factors forming the morphology of spirals.
e time dilation is particularly important, because a slower rate of
time in the past signicantly helped to separate spirals from the bulge-
bar domain. As shown in the numerical modelling, the morphology
of spirals depends on the expansion history of the Universe, on the
galaxy mass, galaxy age and size of the bulge. Obviously, the presented
modelling is rather simple and denitely far from being complete. A
detailed parametric study based on observations of various types of
galaxies is necessary for drawing more specic conclusions about the
galaxy dynamics.
e previous theoretical attempts to correctly explain the galaxy
dynamics were unsuccessful for the following reasons: 1) e simplest
attempts ignored the space expansion and assumed galaxies in the
static Universe. Consequently, the GR eects related to the expansion
of the Universe were neglected and stars moved along stationary
orbits not evolving in time. 2) eories, which considered the space
expansion using the GR theory, applied an incorrect cosmological
model dened by the FLRW metric. is metric erroneously assumes
that time is invariant of the space expansion. is assumption has
fatal consequences for dynamics of local systems. An additional radial
acceleration originating in the space expansion is included in the
Newtonian equations for orbiting bodies, but the proper angular
momentum LRVϕkeeps constant. e constant Lcauses that
the eect of the Hubble ow on the orbiting bodies is eliminated
and the radius of the orbits is eectively insensitive to the space
expansion.
By contrast, the proper angular momentum LRVϕin the
Newtonian equations derived under the CC metric is not constant but
it increases with redshi. Consequently, the orbits are not stationary
any more, but their radius increases with cosmic time. Since the
velocity of orbiting massive particles does not depend on the space
expansion and the radius of orbits is continuously increasing, the
rotation curves are essentially at. Importantly, the rotation curves are
at without assuming non-baryonic dark matter haloes surrounding
galaxies. No dark matter is needed for explaining all basic properties
of the galactic dynamics. Applying the CC metric to interpretations of
the SNe Ia dimming reveals that also dark energy is unnecessary for
getting t with observations (Behnkeetal., 2002;Vavryčuk, 2022a).
Hence, dark energy and dark matter are false and superuous concepts
originating in a wrong description of the space expansion, when the
time dilation is ignored during the evolution of the Universe. Once a
correct metric is applied, the cosmological model is consistent with
observations with no need to introduce new unphysical concepts. A
controversy of dark matter and dark energy concepts is evidenced
also by many other observations (Kroupa, 2012;Weinbergetal., 2013;
Kroupa, 2015;Buchertetal., 2016;Bulletal., 2016;Koyama, 2016;
Bullock and Boylan-Kolchin, 2017;Ezquiaga and Zumalacárregui,
2017).
In addition, the presented theory resolves several other puzzles
and paradoxes in cosmology. It explains the origin of spirals in
a completely dierent way than proposed by the density-wave
hypothesis. e spirals are not an eect of standing waves in the
disk as so far believed (Toomre, 1977;Dobbs and Baba, 2014;Shu,
2016). Instead, they are objects formed by material remained xed
in spirals. Still, the winding problem is avoided. e theory also
removes tensions related to the observed galaxy growth explained
by major and/or minor mergers of galaxies. Since the hypothesis of
mergers (Naabetal., 2009;Kormendy and Ho, 2013;McLureetal.,
2013;Conselice, 2014) is refuted by observations of no evolution of
the galaxy mass (Bundyetal., 2017;Kawinwanichakijetal., 2020), the
problem of a galaxy growth is so far unsolved. e presented theory
Frontiers in Astronomy and Space Sciences 13 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
also resolves challenges to the ΛCDM model such as the problem
of faint satellite galaxies, the baryonic Tully-Fisher relation or the
radial acceleration relation. Furthermore, numerous puzzles in the
solar system are successfully explained such as the Faint young Sun
paradox, the lunar orbit anomaly, the presence of rivers on ancient
Mars, the Titan recession velocity anomaly, formation of the Kuiper
belt and others (Dumin, 2015;Křížeketal., 2015;Křížek and Somer,
2015).
Data availability statement
e original contributions presented in the study are included in
the article/supplementary material, further inquiries can be directed
to the corresponding author.
Author contributions
VV is responsible for the whole study presented in the paper.
Funding
e Institute of Geophysics of the Czech Academy of Sciences
supported this research.
Conict of interest
e author declares that the research was conducted in the absence
of any commercial or nancial relationships that could be construed
as a potential conict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their aliated organizations,
or those of the publisher, the editors and the reviewers. Any product
that may be evaluated in this article, or claim that may be made by
its manufacturer, is not guaranteed or endorsed by the publisher.
References
Anderson, J. D., Laing, P. A., Lau, E. L., Liu, A. S., Nieto, M. M., and Turyshev,
S. G. (1998). Indication, from Pioneer 10/11, Galileo, and Ulysses data, of an
apparent anomalous, weak, long-range acceleration. Phys. Rev. Lett. 81, 2858–2861.
doi:10.1103/PhysRevLett.81.2858
Baba, J., Asaki, Y., Makino,J., Miyoshi, M., Saitoh, T. R., and Wada,K. (2009). e origin
of large peculiar motions of star-forming regions and spiral structures of our galaxy. ApJ
706, 471–481. doi:10.1088/0004-637X/706/1/471
Baba, J., Saitoh, T. R., and Wada, K. (2013). Dynamics of non-steady spiral arms in disk
galaxies. ApJ 763, 46. doi:10.1088/0004-637X/763/1/46
Bahcall, J. N., Pinsonneault, M. H., and Basu, S. (2001). Solar models: Current epoch
and time dependences, neutrinos, and helioseismological properties. ApJ 555, 990–1012.
doi:10.1086/321493
Begeman, K. G., Broeils, A. H., and Sanders, R. H. (1991). Extended rotation
curves of spiral galaxies: Dark haloes and modied dynamics. MNRAS 249, 523–537.
doi:10.1093/mnras/249.3.523
Begeman, K. G. (1987). HI rotation curves of spiral galaxies. Ph.D. thesis. Groningen:
University of Groningen, Kapteyn Astronomical Institute.
Begeman, K. G. (1989). HI rotation curves of spiral galaxies. I. NGC 3198. Astron.
Astrophys. 223, 47–60.
Behnke, D., Blaschke, D. B.,Pervushin, V. N., and Proskurin, D. (2002). Description of
supernova data in conformal cosmology without cosmological constant. Phys. Lett. B 530,
20–26. doi:10.1016/S0370-2693(02)01341-2
Bekenstein, J. D. (2004). Relativistic gravitation theory for the modied Newtonian
dynamics paradigm. Phys. Rev. D. 70, 083509. doi:10.1103/PhysRevD.70.083509
Bergström, L. (2000). Non-baryonic dark matter: Observational evidence and
detection methods. Rep. Prog. Phys. 63, 793–841. doi:10.1088/0034-4885/63/5/2r3
Bertone, G., and Hooper,D. (2018). History of dark matter.Rev. Mod. Phys. 90, 045002.
doi:10.1103/RevModPhys.90.045002
Bertotti, B., Farinella, P., and Vokrouhlick, D. (2003). Physics of the solar system -
dynamics and evolution, space physics, and spacetime structure. Astrophysics and Space
Science Library.293. doi:10.1007/978-94-010-0233-2
Bills, B. G., and Ray,R . D. (1999). Lunar orbital evolution:A synthesis of recent results.
Geophys. Res. Lett. 26, 3045–3048. doi:10.1029/1999GL008348
Binney, J., Gerhard, O., and Spergel, D. (1997). e photometric structure of the inner
Galaxy. MNRAS 288, 365–374. doi:10.1093/mnras/288.2.365
Bissantz, N., and Gerhard, O. (2002). Spiral arms, bar shape and bulge microlensingin
the Milky Way. MNRAS 330, 591–608. doi:10.1046/j.1365-8711.2002.05116.x
Bolen, B., Bombelli, L., and Puzio, R. (2001). Expansion-induced contribution to the
precession of binary orbits. Class. Quantum Gravity 18, 1173–1178. doi:10.1088/0264-
9381/18/7/302
Bournaud, F., Jog, C. J., and Combes, F. (2007). Multiple minor mergers: Formation
of elliptical galaxies and constraints for the growth of spiral disks. Astron. Astrophys. 476,
1179–1190. doi:10.1051/0004-6361:20078010
Bouwens, R. J., Illingworth, G. D., Blakeslee, J. P., Broadhurst, T. J., and Franx, M.
(2004). Galaxy size evolution at high redshi and surface brightness selection eects:
Constraints from the Hubble ultra deep eld. ApJ 611, L1–L4. doi:10.1086/423786
Buchert, T., Coley, A. A., Kleinert, H., Roukema, B. F., and Wiltshire, D. L. (2016).
Observational challenges for the standard FLRW model. Int. J. Mod. Phys. D 25,
1630007–1630244. doi:10.1142/S021827181630007X
Bugg, D. V. (2015). Mond— a review. Can. J. Phys. 93, 119–125. doi:10.1139/cjp-2014-
0057
Bull, P., Akrami, Y., Adamek, J., Baker, T., Bellini, E., Beltrán Jiménez, J., et al. (2016).
Beyond ΛCDM: Problems, solutions, and the road ahead. Phys. Dark Universe 12, 56–99.
doi:10.1016/j.dark.2016.02.001
Bullock, J. S., and Boylan-Kolchin, M. (2017). Small-scale challenges to the ΛCDM
paradigm. Annu. Rev. Astronomy Astrophysics 55, 343–387. doi:10.1146/annurev-astro-
091916-055313
Bundy, K., Leauthaud, A., Saito, S., Maraston, C., Wake, D. A., and omas, D. (2017).
e Stripe 82 Massive Galaxy Project. III. A lack of growth among massive galaxies. ApJ
851, 34. doi:10.3847/1538-4357/aa9896
Callan, C., Dicke, R. H., and Peebles, P. J. E. (1965). Cosmology and Newtonian
mechanics. Am. J. Phys. 33, 105–108. doi:10.1119/1.1971256
Calzetti, D., Armus, L., Bohlin, R. C., Kinney, A. L., Koornneef, J., and Storchi-
Bergmann, T. (2000). e dust content and opacity of actively star-forming galaxies. ApJ
533, 682–695. doi:10.1086/308692
Carr, M. H. (1995). e Martian drainagesystem and the origin of valley networks and
fretted channels. J. Geophys. Res. 100, 7479–7508. doi:10.1029/95JE00260
Carrera, M., and Giulini, D.(2010). Inuence of global cosmological expansion on local
dynamics and kinematics. Rev. Mod. Phys. 82, 169–208. doi:10.1103/RevModPhys.82.169
Conselice, C. J. (2014). e evolution of galaxy structure over cosmic time. Annu. Rev.
Astron. Astrophys. 52, 291–337. doi:10.1146/annurev-astro-081913-040037
Cooperstock, F. I., Faraoni, V., and Vollick, D. N. (1998). e inuence of the
cosmological expansion on local systems. ApJ 503, 61–66. doi:10.1086/305956
Corda, C. (2009). Interferometric detection of gravitational waves: e denitive test
for General Relativity. Int.J. Mod. Phys. D 18, 2275–2282. doi:10.1142/S0218271809015904
Frontiers in Astronomy and Space Sciences 14 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
da Cunha, E., Charlot, S., and Elbaz, D. (2008). A simple model to interpret
the ultraviolet, optical and infrared emission from galaxies. MNRAS 388, 1595–1617.
doi:10.1111/j.1365-2966.2008.13535.x
Dabrowski, M. P., Garecki, J., and Blaschke, D. B. (2009). Conformal
transformations and conformal invariance in gravitation. Ann. Phys. 521, 13–32.
doi:10.1002/andp.20095210105
Dahlen, T., Mobasher, B., Dickinson, M., Ferguson, H. C., Giavalisco, M., Kretchmer,
C., et al. (2007). Evolution of the luminosity function, star formation rate, morphology,
and size of star-forminggalaxies selec ted atrest-frame 1500 and 2800 Å. ApJ 654, 172–185.
doi:10.1086/508854
Davis, J., Balme, M., Grindrod, P., Williams, R., and Gupta, S. (2016). Extensive
Noachian uvial systems in Arabia Terra: Implications for early Martian climate. Geology
44, 847–850. doi:10.1130/G38247.1
Davis, M., Efstathiou, G., Frenk, C. S., and White, S. D. M. (1985). e evolution of
large-scale structure in a universe dominated by cold dark matter. ApJ 292, 371–394.
doi:10.1086/163168
de Blok, W. J. G., Walter, F., Brinks, E., Trachternach, C., Oh, S. H., and Kennicutt, J.
R. C. (2008). High-resolution rotation curves and galaxy mass models from THINGS. AJ
136, 2648–2719. doi:10.1088/0004-6256/136/6/2648
de Blok, W. J. G., and Bosma, A. (2002). High-resolution rotation curves of low surface
brightness galaxies. Astron. Astrophys. 385, 816–846. doi:10.1051/0004-6361:20020080
de Blok, W. J. G., McGaugh, S. S., and Rubin, V. C. (2001). High-resolution
rotation curves of low surface brightness galaxies. II. Mass models. AJ 122, 2396–2427.
doi:10.1086/323450
Del Popolo, A., and Le Delliou, M. (2017). Small scale problems of the ΛCDM model:
A short review. Galaxies 5, 17. doi:10.3390/galaxies5010017
Dicke, R. H., and Peebles, P. J. (1964). Evolution of the solar system and the expansion
of the universe. Phys. Rev. Lett. 12, 435–437. doi:10.1103/PhysRevLett.12.435
Dickey, J. O., Bender, P. L., Faller, J. E., Newhall, X. X., Ricklefs, R. L., Ries, J. G., et al.
(1994). Lunar laser ranging: A continuing legacy of the Apollo program. Science 265,
482–490. doi:10.1126/science.265.5171.482
Dobbs, C., and Baba, J. (2014). Dawes review 4: Spiral structures in disc galaxies. Publ.
Astron. Soc. Aust., 31, e035. doi:10.1017/pasa.2014.31
Draine, B. T., and Li, A. (2007). Infrared emissionf rominterstellar dust. IV. e silicate-
graphite-PAH model in the Post-Spitzer era. ApJ 657, 810–837. doi:10.1086/511055
Dubinski, J., and Carlberg, R. G. (1991). e structure of cold dark matter halos. ApJ
378, 496. doi:10.1086/170451
Duc, P. A., Paudel, S., McDermid, R. M., Cuillandre, J. C., Serra, P., Bournaud, F., et al.
(2014). Identication of old tidal dwarfs near early-type galaxies from deep imaging and
H I observations. MNRAS 440, 1458–1469. doi:10.1093/mnras/stu330
Dumin, Y. V. (2015). e faint young Sunparadox in the context of modern cosmology.
Astron. Tsirkulyar 1623, 1–5.
Dunne, L., Eales, S., Edmunds, M., Ivison, R., Alexander, P., and Clements,
D. L. (2000). e SCUBA Local Universe Galaxy Survey - I. First measurements
of the submillimetre luminosity and dust mass functions. MNRAS 315, 115–139.
doi:10.1046/j.1365-8711.2000.03386.x
Einstein, A. (1920). “Relativity, the special and the general theory,” in Authorized
translation by. Editor W. L. Robert (Henry Holt and Company), 168.
Einstein, A., and Straus, E. G. (1945). e inuence of the expansion of space on
the gravitation elds surrounding the individual stars. Rev. Mod. Phys. 17, 120–124.
doi:10.1103/RevModPhys.17.120
Elmegreen, D. M., Chromey, F. R., Bissell, B. A., and Corrado, K. (1999). K-Band
observations of underlying symmetric structure in occulent galaxies. AJ 118, 2618–2624.
doi:10.1086/301127
Endean, G. (1997). Cosmology in conformally at spacetime. ApJ 479, 40–45.
doi:10.1086/303862
Endean, G. (1994). Redshi and the Hubbleconstant in conformally at spacetime. ApJ
434, 397. doi:10.1086/174741
Ezquiaga, J. M., and Zumalacárregui, M. (2017). Dark energy aer
GW170817: Dead ends and the road ahead. Phys. Rev. Lett. 119, 251304.
doi:10.1103/PhysRevLett.119.251304
Faraoni, V., and Jacques, A. (2007). Cosmological expansion and local physics. Phys.
Rev. D. 76, 063510. doi:10.1103/PhysRevD.76.063510
Ferreras, I. (2019). Fundamentals of galaxy dynamics. London: Formation and
Evolution (UCL Press).
Freedman, W. L., Madore,B. F., Gibson,B. K., Ferrarese, L., Kelson, D. D., Sakai,S., et al.
(2001). Final results from the Hubble space telescope key project to measure the Hubble
constant. ApJ 553, 47–72. doi:10.1086/320638
Freedman, W. L., Madore,B. F.,Hatt, D., Hoyt, T. J., Jang,I. S., Beaton, R. L., et al. (2019).
e Carnegie-Chicago Hubble program. VIII. An independent determination of the
Hubble constant based on the tip of the red giant branch. ApJ 882, 34. doi:10.3847/1538-
4357/ab2f73
Friedmann, A. (1922). Über die Krümmung des Raumes. Z. für Phys. 10, 377–386.
doi:10.1007/BF01332580
Galianni, P., Patat,F., Higdon, J.L., Mieske, S., and Kroupa, P. (2010). VLT observations
of NGC 1097’s “dog-leg” tidal stream. Dwarf spheroidals and tidal streams. Astron.
Astrophys. 521, A20. doi:10.1051/0004-6361/200913518
Goldhaber, G., Deustua, S., Gabi, S., Groom, D., Hook, I., Kim, A., et al. (1997).
“Observation of cosmological time dilation using Type Ia supernovae as clocks,” in
ermonuclear supernovae. Editors P. Ruiz-Lapuente, R. Canal, and J. Isern (Dordrecht:
Springer), 486, 777. doi:10.1007/978-94-011-5710-0_48
Goldhaber, G., Groom, D. E., Kim, A., Aldering, G., Astier, P., Conley, A., et al. (2001).
Timescale stretch parameterization of type Ia supernova B-band light curves. ApJ 558,
359–368. doi:10.1086/322460
Goobar, A., and Leibundgut, B. (2011). Supernova cosmology: Legacy and future.
Annu. Rev. Nucl. Part. Sci. 61, 251–279. doi:10.1146/annurev-nucl-102010-130434
Goode, P. R., Qiu, J., Yurchyshyn, V., Hickey, J., Chu, M. C., Kolbe, E., et al. (2001).
Earthshine observations of the Earth’s reectance. Geophys. Res. Lett. 28, 1671–1674.
doi:10.1029/2000GL012580
Grøn, Ø., and Johannesen, S. (2011). FRW universe models in conformally
at-spacetime coordinates I: General formalism. Eur. Phys. J. Plus 126, 28.
doi:10.1140/epjp/i2011-11028-6
Harada, T., Carr, B. J., and Igata, T. (2018). Complete conformal classication of the
Friedmann-Lemaître-Robertson-Walker solutions with a linear equation of state. Class.
Quantum Gravity 35, 105011. doi:10.1088/1361-6382/aab99f
Holwerda, B. W., Bouwens, R., Oesch, P., Smit, R., Illingworth, G., and Labbe, I. (2015).
e sizes of candidate galaxiesz∼9−10: conrmation of the brightCANDELS sample and
relation with luminosity and mass. ApJ 808, 6. doi:10.1088/0004-637X/808/1/6
Hubble,E. (1929). A relation between distance and radial velocity among extra-galactic
nebulae. Proc. Natl. Acad. Sci. 15, 168–173. doi:10.1073/pnas.15.3.168
Ibison, M. (2007). On the conformal forms of the Robertson-Walker metric. J. Math.
Phys. 48, 122501. doi:10.1063/1.2815811
Infeld, L., and Schild, A. (1945). A new approach to kinematic cosmology. Phys. Rev.
68, 250–272. doi:10.1103/PhysRev.68.250
Infeld, L., and Schild, A. E. (1946). A new approach tokinematic cosmology-(B). Phys.
Rev. 70, 410–425. doi:10.1103/PhysRev.70.410
Iorio, L. (2015). Gravitational anomalies in the solar system? Int. J. Mod. Phys. D 24,
1530015–1530343. doi:10.1142/S0218271815300153
Iorio, L. (2013). Local cosmological eects of the order of H in the orbital motion of a
binary system. MNRAS 429, 915–922. doi:10.1093/mnras/sts396
Jacoby, G. H., Branch, D., Ciardullo, R., Davies, R. L., Harris, W. E., Pierce, M. J., et al.
(1992). A critical review of selected techniques for measuring extragalactic distances. PASP
104, 599. doi:10.1086/133035
Kastrup, H. A. (2008). On the advancements of conformal transformations and their
associated symmetries in geometry and theoretical physics. Ann. Phys. 520, 631–690.
doi:10.1002/andp.200852009-1005
Kaumann, G., White, S. D. M., and Guiderdoni, B. (1993). e formation and
evolution of galaxies within merging dark matter haloes. MNRAS 264, 201–218.
doi:10.1093/mnras/264.1.201
Kawinwanichakij,L., Papovich, C., Ciardullo, R., Finkelstein,S. L., Stevans, M. L., Wold,
I. G. B., et al. (2020). On the (lack of) evolution of the stellar mass function of massive
galaxies from z = 1.5 to 0.4. ApJ 892, 7. doi:10.3847/1538-4357/ab75c4
Knauth, L. P. (2005). Temperature and salinity history of the precambrian ocean:
Implicationsfor the course of microbial evolution. Palaeogeogr. Palaeoclimatol. Palaeoecol.
219, 53–69. doi:10.1016/j.palaeo.2004.10.014
Kopp, G., and Lean, J. L. (2011). A new, lower value of total solar irradiance: Evidence
and climate signicance. Geophys. Res. Lett. 38, L01706. doi:10.1029/2010GL045777
Kormendy, J., and Ho, L. C. (2013). Coevolution (or not) of supermassive black holes
and host galaxies. Annu. Rev. Astron. Astrophys. 51, 511–653. doi:10.1146/annurev-astro-
082708-101811
Koyama, K. (2016). Cosmological tests of modied gravity. Rep. Prog.Phys. 79, 046902.
doi:10.1088/0034-4885/79/4/046902
Křížek, M. (2012). Dark energy and the anthropic principle. New Astron. 17, 1–7.
doi:10.1016/j.newast.2011.05.003
Křížek, M. (2009). Does a gravitational aberration contribute to the accelerated
expansion of the universe? Commun. Comput. Phys. 5, 1030–1044.
Křížek, M., Křížek, F., and Somer, L. (2015). Antigravity - its origin and manifestations.
Saarbruecken: Lambert Academic Publishing.
Křížek, M., and Somer, L. (2022). Anthropic principle and the Hubble-Lemaître
constant. Galaxies 10, 71. doi:10.3390/galaxies10030071
Křížek, M., and Somer, L. (2015). Manifestations of dark energy in the solar system.
Gravit. Cosmol. 21, 59–72. doi:10.1134/S0202289315010090
Kroupa, P., Famaey, B., de Boer, K. S., Dabringhausen, J., Pawlowski, M. S., Boily, C.
M., et al. (2010). Local-Group tests of dark-matter concordance cosmology. Towards a
new paradigm for structure formation. Astron. Astrophys. 523, A32. doi:10.1051/0004-
6361/201014892
Frontiers in Astronomy and Space Sciences 15 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
Kroupa, P. (2015). Galaxies as simple dynamical systems: Observational data disfavor
dark matter andstochastic star formation. Can. J. Phys. 93, 169–202. doi:10.1139/cjp-2014-
0179
Kroupa, P. (2012). e dark matter crisis: Falsication of the current standard model
of cosmology. Publ. Astron. Soc. Aust., 29, 395–433. doi:10.1071/AS12005
Lainey, V., Casajus, L. G., Fuller, J., Zannoni, M., Tortora, P., Cooper, N., et al. (2020).
Resonance locking in giant planets indicated by the rapid orbital expansion of Titan. Nat.
Astron. 4, 1053–1058. doi:10.1038/s41550-020-1120-5
Leibundgut, B. (2001). Cosmological implications from observations of type Ia
supernovae. Annu. Rev. Astron. Astrophys. 39, 67–98. doi:10.1146/annurev.astro.39.1.67
Leibundgut, B., Schommer, R., Phillips, M., Riess, A., Schmidt, B., Spyromilio, J., et al.
(1996). Time dilation in the light curve of the distant type IA supernova SN 1995K. ApJ
466, L21–L24. doi:10.1086/310164
Lelli, F., McGaugh, S. S., Schombert, J. M., Desmond, H., and Katz, H. (2019). e
baryonic Tully-Fisher relation for dierentvelocity denitions and implications for galaxy
angular momentum. MNRAS 484, 3267–3278. doi:10.1093/mnras/stz205
Lelli, F., McGaugh, S. S., and Schombert, J. M. (2016). Sparc: Mass models for
175 disk galaxies with Spitzer photometry and accurate rotation curves. AJ 152, 157.
doi:10.3847/0004-6256/152/6/157
Lelli, F.,McGaugh, S. S., and Schombert, J. M. (2016). e small scatter of the baryonic
Tully-Fisher relation. ApJ 816, L14. doi:10.3847/2041-8205/816/1/L14
Lemaître, G. (1927). Un Univers homogène de masse constante et de rayon croissant
rendant compte de la vitesse radiale des nébuleuses extra-galactiques. Ann. Société Sci.
Brux. 47, 49–59.
Lerner, E. J. (2018). Observations contradict galaxy size and surface brightness
predictions that areb ased on the expandinguniverse hypothesis. MNRAS 477, 3185–3196.
doi:10.1093/mnras/sty728
Lin, C. C., and Shu, F. H. (1964). On the spiral structure of disk galaxies. ApJ 140, 646.
doi:10.1086/147955
Lin, C. C., and Shu, F. H. (1966). On the spiral structure of disk galaxies, II. Outline of
a theory of density waves. Proc. Natl. Acad. Sci. 55, 229–234. doi:10.1073/pnas.55.2.229
Lindblad, B. (1962). “eories of spiral structure in galaxies,” in Problems of extra-
galactic research. Editor G. C. McVittie, 15, 146.
Maddox, S. J., Efstathiou, G., Sutherland, W. J., and Loveday, J. (1990). Galaxy
correlations on large scales. MNRAS 242, 43P–47P. doi:10.1093/mnras/242.1.43P
Man, A. W. S., To, S., Zirm, A. W., Wuyts, S., and van der Wel, A. (2012). e pair
fraction of massive galaxies at 0 ⩽z⩽3. ApJ 744, 85. doi:10.1088/0004-637X/744/2/85
Man, A. W. S., Zirm, A. W., and To, S. (2016). Resolving the discrepancy of galaxy
merger fraction measurements at z ∼0-3. ApJ 830, 89. doi:10.3847/0004-637X/830/2/89
Mannheim, P. D. (2006). Alternatives to dark matter and dark energy. Prog. Part. Nucl.
Phys. 56, 340–445. doi:10.1016/j.ppnp.2005.08.001
Mannheim, P. D. (1990). Conformal cosmology with no cosmological constant.
General Relativ. Gravit. 22, 289–298. doi:10.1007/BF00756278
Mannheim, P. D. (2019). Is dark matter fact or fantasy? — clues from the data. Int. J.
Mod. Phys. D 28, 1944022. doi:10.1142/S021827181944022X
Mannheim, P. D. (2012). Making the case for conformal gravity. Found. Phys. 42,
388–420. doi:10.1007/s10701-011-9608-6
Mannheim, P. D., and O’Brien, J. G. (2012). Fitting galactic rotation curves
with conformal gravity and a global quadratic potential. Phys. Rev. D. 85, 124020.
doi:10.1103/PhysRevD.85.124020
Mannheim, P. D., and O’Brien, J. G. (2011). Impact of a global quadratic potential on
galactic rotation curves. Phys. Rev.Lett. 106, 121101. doi:10.1103/PhysRevLett.106.121101
Mathewson, D. S., Ford, V. L., and Buchhorn, M. (1992). A southern sky survey of the
peculiar velocities of 1355 spiral galaxies. ApJS 81, 413. doi:10.1086/191700
McGaugh, S. S., Lelli, F., and Schombert, J. M. (2016). Radial acceleration
relation in rotationally supported galaxies. Phys. Rev. Lett. 117, 201101.
doi:10.1103/PhysRevLett.117.201101
McGaugh, S. S., Li, P., Lelli, F., and Schombert, J. M. (2018). Presenceof a fundamental
acceleration scale in galaxies. Nat. Astron. 2, 924. doi:10.1038/s41550-018-0615-9
McGaugh, S. S. (2019). e imprint of spiral arms on the galactic rotation curve. ApJ
885, 87. doi:10.3847/1538-4357/ab479b
McGaugh, S. S. (2016). e surface density proleof the galactic disk from the terminal
velocity curve. ApJ 816, 42. doi:10.3847/0004-637X/816/1/42
McLure, R. J., Pearce, H. J., Dunlop, J. S., Cirasuolo, M., Curtis-Lake, E., Bruce, V. A.,
et al. (2013). e sizes, masses and specic star formationrates of massive galaxies at 1.3 < z
< 1.5: Strong evidence in favour of evolution via minor mergers. MNRAS 428, 1088–1106.
doi:10.1093/mnras/sts092
McVittie, G. C. (1933). e mass-particle in an expanding universe. MNRAS 93,
325–339. doi:10.1093/mnras/93.5.325
Milgrom, M. (1983). A modication of the Newtonian dynamics - implications for
galaxies. ApJ 270, 371–383. doi:10.1086/161131
Milgrom, M. (1983). A modication of the Newtonian dynamics as a possible
alternative to the hidden mass hypothesis. ApJ 270, 365–370. doi:10.1086/161130
Milgrom, M. (2010). Quasi-linear formulation of MOND. MNRAS 403, 886–895.
doi:10.1111/j.1365-2966.2009.16184.x
Milgrom, M. (2012). Testing MOND over a wide acceleration rangein X-ray ellipticals.
Phys. Rev. Lett. 109, 131101. doi:10.1103/PhysRevLett.109.131101
Mo, H. J.,Mao, S., and White, S. D. M. (1998). e formation of galactic discs. MNRAS
295, 319–336. doi:10.1046/j.1365-8711.1998.01227.x
Moore, B., Ghigna, S., Governato, F., Lake, G., Quinn, T., Stadel, J., et al. (1999). Dark
matter substructure within galactic halos. ApJ 524, L19–L22. doi:10.1086/312287
Mould, J. R., Huchra, J. P., Freedman, W. L., Kennicutt J Robert, C., Ferrarese, L., Ford,
H. C., et al. (2000). e Hubble Space Telescope Key Project on the extragalactic distance
scale. XXVIII. Combining the constraints on the Hubble constant. ApJ 529, 786–794.
doi:10.1086/308304
Mundy, C. J., Conselice, C. J.,Duncan, K. J., Almaini, O., Häußler, B., and Hartley,W.G.
(2017). A consistent measure of the merger histories of massive galaxies using close-pair
statistics - I. Majormergers at z < 3.5. MNRAS 470, 3507–3531. doi:10.1093/mnras/stx1238
Naab, T., Johansson, P. H., and Ostriker, J. P. (2009). Minor mergers and the size
evolution of elliptical galaxies. ApJ 699, L178–L182. doi:10.1088/0004-637X/699/2/L178
Nandra, R., Lasenby,A. N., and Hobson, M. P. (2012). eeect of a massive object on
an expanding universe. MNRAS 422, 2931–2944. doi:10.1111/j.1365-2966.2012.20618.x
Navarro, J. F., Frenk, C. S., and White, S. D.M. (1997). A universal density prole from
hierarchical clustering. ApJ 490, 493–508. doi:10.1086/304888
Navarro,J. F., Frenk,C. S., and White, S. D. M. (1996). e structure of cold dark matter
halos. ApJ 462, 563. doi:10.1086/177173
Newman, A. B., Ellis, R. S., Bundy, K., and Treu,T. (2012). Can minor merging account
for the size growth of quiescent galaxies? Newresults from the CANDELS survey. ApJ 746,
162. doi:10.1088/0004-637X/746/2/162
Noerdlinger, P. D., and Petrosian, V. (1971). e eect of cosmological expansion on
self-gravitating ensembles of particles. ApJ 168, 1. doi:10.1086/151054
Noordermeer, E., and Verheijen, M. A. W. (2007). e high-mass end of the Tully-
Fisher relation. MNRAS 381, 1463–1472. doi:10.1111/j.1365-2966.2007.12369.x
O’Brien, J. G., and Mannheim, P. D. (2012). Fitting dwarf galaxy rotation curves with
conformal gravity. MNRAS 421, 1273–1282. doi:10.1111/j.1365-2966.2011.20386.x
Oesch, P. A., Bouwens, R. J., Carollo, C. M., Illingworth, G. D., Trenti, M., Stiavelli,
M., et al. (2010). Structure and morphologies of z ∼7-8 galaxies from ultra-deep
WFC3/IR imaging of the Hubble Ultra-Deep Field. ApJ 709, L21–L25. doi:10.1088/2041-
8205/709/1/L21
Pachner,J. (1964). Nonconservation of energy during cosmic evolution. Phys.Rev. Lett.
12, 117–118. doi:10.1103/PhysRevLett.12.117
Pawlowski, M. S., and Kroupa, P. (2013). e rotationally stabilized VPOS and
predicted proper motions of the Milky Way satellite galaxies. MNRAS 435, 2116–2131.
doi:10.1093/mnras/stt1429
Pawlowski, M. S., and McGaugh, S. S. (2014). Perseus I and the NGC 3109 association
in the context of the Local Group dwarf galaxy structures. MNRAS 440, 908–919.
doi:10.1093/mnras/stu321
Peacock, J. A. (1999). Cosmological physics. Cambridge: Cambridge University Press.
Persic, M., Salucci, P., and Stel, F. (1996). e universal rotation curve of spiral
galaxies — I. e dark matter connection. MNRAS 281, 27–47.
doi:10.1093/mnras/278.1.27
Phillips, M. M., Lira, P., Suntze, N. B., Schommer, R. A., Hamuy, M., and Maza,
J. (1999). e reddening-free decline rate versus luminosity relationship for type Ia
supernovae. AJ 118, 1766–1776. doi:10.1086/301032
Phillips, M. M. (1993). e absolute magnitudes of type IAsupernovae. ApJ 413, L105.
doi:10.1086/186970
Ribas, I. (2010). “e Sun and stars as the primary energy input in planetary
atmospheres,” in Solar and stellar variability: Impact on Earth and planets. Editors A. G.
Kosovichev, A. H. Andrei, and J. P. Rozelot, 264, 3–18. doi:10.1017/S1743921309992298
Riofrio, L. (2012). Calculation of lunar orbit anomaly. Planet. Sci. 1, 1.
doi:10.1186/2191-2521-1-1
Robert, F.,and Chaussidon, M. (2006). A palaeotemperature curve for the Precambrian
oceans based on silicon isotopes in cherts. Nature 443, 969–972. doi:10.1038/nature05239
Roberts, J. W. W., Roberts, M. S., and Shu, F. H. (1975). Density wave theory and the
classication of spiral galaxies. ApJ 196, 381–405. doi:10.1086/153421
Rubin, V. C., Burstein, D., Ford, J. W. K., and onnard, N. (1985). Rotation velocities
of 16 SA galaxies and a comparison of Sa, SB and SC rotation properties. ApJ 289, 81–104.
doi:10.1086/162866
Rubin, V. C., Ford, J. W. K., and onnard, N. (1980). Rotational properties of 21 SC
galaxies with a large range of luminosities and radii, from NGC 4605 (R=4 kpc) to UGC
2885 (R=122 kpc). ApJ 238, 471–487. doi:10.1086/158003
Rubin, V. C., and Ford, W. K. (1970). Rotation of the Andromeda nebula from a
spectroscopic survey of emission regions. ApJ 159, 379. doi:10.1086/150317
Ryden, B. (2016). Introduction to cosmology. Cambridge: Cambridge University Press.
Frontiers in Astronomy and Space Sciences 16 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
Salese, F., McMahon, W. J., Balme, M. R., Ansan, V., Davis, J. M., and Kleinhans, M. G.
(2020). Sustained uvial deposition recorded in Mars’ Noachian stratigraphic record. Nat.
Commun. 11, 2067. doi:10.1038/s41467-020-15622-0
Sanders, R. H., and McGaugh, S. S. (2002). Modied Newtonian dynamics
as an alternative to dark matter. Annu. Rev. Astron. Astrophys. 40, 263–317.
doi:10.1146/annurev.astro.40.060401.093923
Sanders, R. H. (1996). e published extended rotation curves of spiral galaxies:
Confrontation with modied dynamics. ApJ 473, 117–129. doi:10.1086/178131
Sandstrom, K. M., Leroy, A. K., Walter, F., Bolatto, A. D., Croxall, K. V., Draine, B. T.,
et al. (2013). e CO-to-H2conversion factor and dust-to-gas ratio on kiloparsec scales
in nearby galaxies. ApJ 777, 5. doi:10.1088/0004-637X/777/1/5
Schneider, P. (2015). Extragalactic Astronomy and cosmology: An introduction. New
York: Springer. doi:10.1007/978-3-642-54083-7
Sellwood, J. A., and Carlberg, R. G. (1984). Spiral instabilities provoked by accretion
and star formation. ApJ 282, 61–74. doi:10.1086/162176
Sellwood, J. A. (2011). e lifetimes of spiral patterns in disc galaxies. MNRAS 410,
1637–1646. doi:10.1111/j.1365-2966.2010.17545.x
Sereno, M., and Jetzer, P. (2007). Evolution of gravitational orbits in the expanding
universe. Phys. Rev. D. 75, 064031. doi:10.1103/PhysRevD.75.064031
Shibuya, T., Ouchi, M., and Harikane, Y. (2015). Morphologies of ∼190,000 galaxies at
z = 0-10 revealed with HST legacy data. I. Size evolution. ApJS 219, 15. doi:10.1088/0067-
0049/219/2/15
Shu, F. H. (1970). On the density-wave theory of galactic spirals. I. Spiral structure as
a normal mode of oscillation. ApJ 160, 89. doi:10.1086/150409
Shu, F. H. (2016). Six decades of spiral density wave theory. Annu. Rev. Astron.
Astrophys., 54, 667–724. doi:10.1146/annurev-astro-081915-023426
Sofue, Y. (2017). Rotation and mass in the Milky Way and spiral galaxies. PASJ 69, R1.
doi:10.1093/pasj/psw103
Sofue, Y. (2016). Rotation curve decomposition for size-mass relations of bulge,
disk, and dark halo components in spiral galaxies. PASJ 68, 2. doi:10.1093/pasj/
psv103
Sofue, Y., and Rubin, V. (2001). Rotation curves of spiral galaxies. Annu. Rev. Astron.
Astrophys. 39, 137–174. doi:10.1146/annurev.astro.39.1.137
Stephenson, F. R., Morrison, L. V., and Hohenkerk, C. Y. (2016). Measurement of
the Earth’s rotation: 720 BC to AD 2015. Proc. R. Soc. Lond. Ser. A 472, 20160404.
doi:10.1098/rspa.2016.0404
Swaters, R. A., Madore,B. F., and Trewhella, M. (2000). High-resolutionrotation curves
of low surface brightness galaxies. ApJ 531, L107–L110. doi:10.1086/312540
Swaters, R. A., Madore, B. F., van den Bosch, F. C., and Balcells, M. (2003). e
central mass distribution in dwarf and low surface brightness galaxies. ApJ 583, 732–751.
doi:10.1086/345426
Taylor, E. N., Franx, M., Glazebrook, K., Brinchmann, J., van der Wel, A., and van
Dokkum, P. G. (2010). On the dearth of compact, massive, red sequence galaxies in the
local universe. ApJ 720, 723–741. doi:10.1088/0004-637X/720/1/723
Tiley, A. L., Swinbank,A. M., Harrison, C. M., Smail, I., Turner, O. J., Schaller, M., et al.
(2019). e shapes of the rotation curves of star-forming galaxies over the last ≈ 10 Gyr.
MNRAS 485, 934–960. doi:10.1093/mnras/stz428
Toomre, A. (1977). eories of spiral structure. Annu. Rev. Astron. Astrophys. 15,
437–478. doi:10.1146/annurev.aa.15.090177.002253
Trujillo, I., Förster Schreiber, N. M., Rudnick, G., Barden, M., Franx, M., Rix, H. W.,
et al. (2006). e size evolution of galaxies since z∼3: CombiningSDSS, GEMS, and FIRES.
ApJ 650, 18–41. doi:10.1086/506464
Tully, R. B., and Fisher, J. R. (1977). A new method of determining distancesto galaxies.
Astron. Astrophys. 54, 661–673.
van Albada, T. S., Bahcall, J. N., Begeman, K., and Sancisi, R. (1985). Distribution of
dark matter in the spiral galaxy NGC 3198. ApJ 295, 305–313. doi:10.1086/163375
van der Wel, A., Franx, M., van Dokkum, P. G., Skelton, R. E., Momcheva,
I. G., Whitaker, K. E., et al. (2014). 3D-HST+CANDELS: e evolution of the
galaxy size-mass distribution since z = 3. ApJ 788, 28. doi:10.1088/0004-637X/
788/1/28
van Dokkum, P. G., Franx, M., Kriek, M., Holden, B., Illingworth, G. D., Magee, D.,
et al. (2008). Conrmation of the remarkable compactness of massive quiescent galaxies
at z∼2.3: Early-type galaxies did not form in a simplemonolithic collapse. ApJ 677, L5–L8.
doi:10.1086/587874
van Dokkum, P. G., Whitaker, K. E., Brammer, G., Franx,M., Kriek, M., L abbé, I., et al.
(2010). e growth of massive galaxies since z = 2. ApJ 709,1018–1041. doi:10.1088/0004-
637X/709/2/1018
Vavryčuk, V. (2022a). Cosmological redshi and cosmic time dilation in the FLRW
metric. Front. Phys. 10, 826188. doi:10.3389/fphy.2022.826188
Vavryčuk, V. (2022b). Considering light-matter interactions in Friedmann
equations based on the conformal FLRW metric. J. Adv. Res. doi:10.1016/j.jare.2022.
06.015
Verheijen, M. A. W. (2001). e Ursa Major Cluster of galaxies. V. H I rotation curve
shapes and the Tully-Fisher relations. ApJ 563, 694–715. doi:10.1086/323887
Visser, M. (2015). Conformally Friedmann-Lemaître-Robertson-Walker cosmologies.
Class. Quantum Gravity 32, 135007. doi:10.1088/0264-9381/32/13/135007
Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., Hirata, C., Riess, A. G., and
Rozo, E. (2013). Observational probes of cosmic acceleration. Phys. Rep. 530, 87–255.
doi:10.1016/j.physrep.2013.05.001
Weinberg, S. (1972). Gravitation and cosmology: Principles and applications of the
general theory of relativity. New York: John Wiley & Sons.
White, S. D. M., Frenk, C. S., Davis, M., and Efstathiou, G. (1987). Clusters, laments,
and voids in a universe dominated by cold dark matter. ApJ 313, 505. doi:10.1086/164990
White, S. D. M., and Rees, M. J. (1978). Core condensation in heavy halos:
A two-stage theory for galaxy formation and clustering. MNRAS 183, 341–358.
doi:10.1093/mnras/183.3.341
Whitney, A., Conselice, C. J., Bhatawdekar, R., and Duncan, K. (2019). Unbiased
dierential size evolution and the inside-out growth of galaxies in the deep CANDELS
GOODS elds at 1 ≤ z ≤ 7. ApJ 887, 113. doi:10.3847/1538-4357/ab53d4
Williams, R. J.,Quadri, R . F., Franx, M., vanD okkum,P., To, S., Kriek, M., et al. (2010).
e evolving relations between size, mass, surface density, and star formation in 3 × 104
galaxies since z = 2. ApJ 713, 738–750. doi:10.1088/0004-637X/713/2/738
Windley, B. F. (1984). e evolving continents/2nd revised and enlarged. edition/. New
York: Wiley.
Zaritsky, D., Courtois, H., Muñoz-Mateos, J. C., Sorce, J., Erroz-Ferrer, S., Comerón, S.,
et al. (2014). e baryonic Tully-Fisher relationship for S4G galaxies and the “condensed”
baryon fraction of galaxies. AJ 147, 134. doi:10.1088/0004-6256/147/6/134
Zhang, W., Li, Z., and Lei, Y. (2010). Experimental measurement of growth patterns
on fossil corals: Secular variation in ancient Earth-Sun distances. Chin. Sci. Bull. 55,
4010–4017. doi:10.1007/s11434-010-4197-x
Zwicky, F. (1937). On the masses of nebulae and of clusters of nebulae. ApJ 86, 217.
doi:10.1086/143864
Zwicky, F. (2009). Republication of: e redshi of extragalactic nebulae. General
Relativ. Gravit. 41, 207–224. doi:10.1007/s10714-008-0707-4
Frontiers in Astronomy and Space Sciences 17 frontiersin.org
Vavryčuk 10.3389/fspas.2023.1071743
Appendix A Gravitational orbits in the
expanding Universe described by the
FLRW metric
e metric tensor gμν of a gravitational eld produced by a point mass
Msituated in space obeying the FLRW metric reads (Noerdlinger and
Petrosian, 1971, their Eq. 11)
ds2c212αdT2a2T12αdr2
1kr2r2dΩ2(A-1)
where Tis the proper time and
αGM
rc2α1(A-2)
is the Newtonian gravitational potential normalized to c2, and Gis the
gravitational constant. Assuming a massive non-relativistic particle
vcorbiting in the gravitational eld in the plane dened by ϕ0
and calculating the Christoel symbols Γμ
αβ in the geodesic Eq.9, we
get the following approximate equations
rr
ϕ2GM
α2r32a
ar0(A-3)
r
ϕ2r
ϕa
ar
ϕ0(A-4)
where dots over quantities mean derivatives with respect to time T.
Inserting the proper distance RaTrinto Eq.A-3 we get
RGM
R2R
ϕ2a
aR(A-5)
Similarly, Eq.A-4 can be rewritten as
d
dT a2r2
ϕd
dT R2
ϕd
dT L0(A-6)
where LRVϕis the proper angular momentum, and Vϕis the proper
tangential velocity. Consequently, we can write (Carrera and Giulini,
2010, their Eq.12a,b)
RGM
R2L2
R3a
aR(A-7)
Lconst(A-8)
e equations are called the modied Newtonian equations and they
dier from the standard Newtonian equations describing the Kepler
orbits by term a
aRin Eq.A-7 related to the space expansion. e
analysis of the modied Newtonian equations applied to the galaxy
dynamics shows that assuming a constant L, the expansion term
a
aRaects the orbits within galaxies negligibly (Faraoni and Jacques,
2007).
Frontiers in Astronomy and Space Sciences 18 frontiersin.org
