150– | $\infty$ |  |  $2.55$$\pm$ | $0.10$ |  $2.23$$\pm$ | ${}^{1.01}_{0.82}\,(^{0.78}_{0.61})$ |  $2.18$$\pm$ | ${}^{1.00}_{0.82}\,(^{0.78}_{0.61})$

An important point for the assessment of the STXS stage-1.2 results of the CB-
and NN-analyses, and their constraining power relates to the separation of
individual STXS bins. The CB-analysis primarily targets the separation of
signal from background, while the distinction of STXS bins is left to the
selection based event categorization. In contrast to this strategy the
multiclassification ansatz of the NN-analysis targets the best possible
separation of individual STXS bins and backgrounds in the 14D input space to
the NNs. In this way, the NN-analysis generally achieves smaller migration
effects, increased separation of individual STXS bins, and reduced
vulnerability to variations of individual features within systematic
uncertainties, compared to the CB-analysis.

Both, the CB- and NN-analyses observe the same trends across STXS stage-1.2
bins. In particular, both analyses observe no signal in the 0-jet STXS bins,
while their sensitivity to observe a signal larger than zero for a signal as
expected from the SM is at the 2–3 s.d. level. In all other STXS bins, a
signal compatible with the expectation of the SM is observed.

The linear correlation matrices of the STXS stage-1.2 measurements are shown
in Fig. 13. For the NN-analysis the observed correlations are generally below
0.4 with trends of a more global anticorrelation of $\Pg\Pg\PH$
$N_{\text{jet}}=1$ and $N_{\text{jet}}\geq 2$ STXS bins with the $\PQq\PQq\PH$
STXS bins, reflecting the difficulty of experimentally distinguishing
processes in these topological regions, especially for $\pt^{\PH}>200\GeV$. As
expected, no correlation of the $\Pg\Pg\PH$ and $\PQq\PQq\PH$ with the
$\PV\PH$ STXS bins is observed. For the CB-analysis, global and local trends
are again similar, which confirms that both analyses capture the same features
in data. Single correlation coefficients are generally significantly higher
for the CB-analysis, demonstrating the NN-analysis’s ability to separate the
individual STXS bins.

Figure 13:  Correlation matrices of the POIs of the measured STXS stage-1.2
signal strengths for the combination of the (upper) CB-, resp. (lower) NN-
analysis with the $\PV\PH$-analysis.

The same results, as shown in Figs. 10 and 12, may be obtained as measurements
of the product of cross sections and branching fraction to leptons maximizing
the same likelihood function, where those theoretical uncertainties related to
pure normalization changes of the signal templates for individual processes or
within individual STXS stage-1.2 bins are dropped from the uncertainty model.
A summary of these measurements is shown in Fig. 14. The upper left panel of
the figure refers to the inclusive and STXS stage-0 results, the lower left
panel to the STXS stage-1.2 measurements in the $\PV\PH$, and the right panel
to the STXS stage-1.2 measurements in the $\Pg\Pg\PH$ and $\PQq\PQq\PH$ bins.
The measured cross sections span several orders of magnitude and apart from
that follow the same trends, as discussed in the previous paragraphs.

Figure 14:  Cross section measurements in the (upper left) stage-0 bins, and
in the stage-1.2 bins related to the (lower left) $\PV\PH$, (upper right)
$\PQq\PQq\PH$, and (lower right) $\Pg\Pg\PH$ processes. The combination of the
CB- and $\PV\PH$-analyses is labeled by CB, the combination of the NN- and
$\PV\PH$-analyses is labeled by NN. Central values and combined statistical
and systematic uncertainties are given for each measurement.

Figure 15 shows the most probable value, as well as the 68 and 95% confidence
interval contours in the plane of multiplicative modifiers of the coupling
strengths to vector bosons and fermions, $\kappa_{\PV}$–$\kappa_{\mathrm{F}}$.
This figure has been extensively referred to in the past [109]. For this
figure, $\PH\to\PW\PW$ decays have been treated as signal, giving increased
sensitivity to the $\Pe\PGm$ final state in the CB- and NN-analyses, and the
$\PV\PH$-analysis. Also here the measurement, as obtained from the NN-analysis
is compatible with the expectation from the SM, within the 95% confidence
interval. The measured value of $\kappa_{\PV}$ turns out to be close to 1,
while the value of $\kappa_{\mathrm{F}}$ is roughly 15% lower than the SM
expectation. These findings coincide with the previously discussed findings
for $\mu_{\Pg\Pg\PH}$ and $\mu_{\PQq\PQq\PH}$, shown in Fig. 10. The
measurements from the CB- and NN-analyses are consistent with each other. The
measurement of the CB-analysis is less precise but closer to the SM
expectation. These findings again coincide with the $p$-values under the
assumption of the SM hypothesis, previously reported for the corresponding
inclusive measurements. A complete set of the reported measurements is
available in the HepData database [110].

Figure 15:  Most probable values, 68, and 95% confidence interval contours
obtained from a negative log-likelihood scan, for a model treating the
couplings to vector bosons ($\kappa_{\PV}$) and fermions
($\kappa_{\mathrm{F}}$) as POIs. The combination of the CB- (NN-) with
$\PV\PH$-analyses is shown in light red (dark blue). For the likelihood
evaluation all nuisance parameters are profiled in each point in the plane.
The $\PH\to\PW\PW$ decay is treated as signal.

##  0.12 Summary

Measurements of Higgs boson production, where the Higgs boson decays into a
pair of leptons, have been presented. The analyzed data are selected from
proton-proton collisions at a center-of-mass energy of 13collected by the CMS
experiment at the LHC from 2016–2018, corresponding to an integrated
luminosity of 138. Results are presented in the form of signal strengths
relative to the standard model predictions and products of cross sections and
branching fraction to leptons, in up to 16 different kinematic regions,
following the simplified template cross section scheme of the LHC Higgs
Working Group. For the simultaneous measurements of a neural network based
analysis and an analysis targeting vector boson associated Higgs boson
production signal strengths are found to be $0.82\pm 0.11$ for inclusive Higgs
boson production, $0.67\pm 0.19$ ($0.81\pm 0.17$) for the production mainly
via gluon fusion (vector boson fusion), and $1.79\pm 0.45$ for vector boson
associated Higgs boson production. The latter result significantly improves an
earlier measurement performed by the CMS Collaboration on a smaller data set.

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# Unified description of galactic dynamics and the cosmological constant

Mariano Cadoni<EMAIL_ADDRESS>Andrea P. Sanna<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria,
09042, Monserrato, Italy  Istituto Nazionale di Fisica Nucleare (INFN),
Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy

(August 27, 2024)

###### Abstract

We explore the phenomenology of a two-fluid cosmological model, where the
field equations of general relativity (GR) are sourced by baryonic and cold
dark matter. We find that the model allows for a unified description of small
and large scale, late-time cosmological dynamics. Specifically, in the static
regime we recover the flattening of galactic rotation curves by requiring the
matter density profile to scale as $1/r^{2}$. The same behavior describes
matter inhomogeneities distribution at small cosmological scales. This traces
galactic dynamics back to structure formation. At large cosmological scales,
we focus on back reaction effects of the spacetime geometry to the presence of
matter inhomogeneities. We find that a cosmological constant with the observed
order of magnitude, emerges by averaging the back reaction term on spatial
scales of order $100\ \text{Mpc}$ and it is related in a natural way to matter
distribution. This provides a resolution to both the cosmological constant and
the coincidence problems and shows the existence of an intriguing link between
the small and large scale behavior in cosmology.

The $\Lambda$-Cold Dark Matter (CDM) model represents our current best
understanding of the observed properties of the universe, by assuming that
only $\sim 5\%$ of its energy content is constituted by baryonic matter, while
the remaining $\sim 95\%$ is exotic. Specifically, $\sim 30\%$ is associated
to non-baryonic CDM, $\sim 65\%$ to dark energy in the form of a cosmological
constant (CC) $\Lambda$ Aghanim:2018eyx . The former allows for a simple
explanation of a wide variety of observations, ranging from the flattening of
rotation curves in disk galaxies and the internal dynamics of galaxy clusters
to cosmological structure formation and evolution, the abundances of light
elements and the power spectrum of the cosmic microwave background radiation
Aghanim:2018eyx ; Bertone:2004pz ; Cuoco:2003cu . The CC accounts instead for
the observed accelerated expansion of the universe Riess:1998cb ;
Perlmutter:1998np .
Despite these successes, several questions remain open. The most pressing ones
are perhaps the understanding of the nature of the dark components, the
explanation of the origin of galactic and cluster dynamics in terms of their
formation and explaining why the CC has the observed value and why its energy
density is so closed to that of matter in the present epoch (the cosmological
constant and the coincidence problems)Peebles:2002gy .
In this letter, we build on the results of Ref. Cadoni:2020jxe , which are
sufficiently general to be applied to various scenarios where cosmology is
sourced by a two-fluid system. Working in the standard $\Lambda$CDM framework
and using baryonic matter and CDM (whose existence is here assumed) as sources
of the gravitational field, we tackle some of the aforementioned problems of
late-time cosmology. Specifically, our model reproduces, in the static regime,
the flattening of galactic rotation curves, whereas at small cosmological
scales explains local inhomogeneities and structure formation. This last
result, in particular, traces the origin of galactic dynamics back to
structure formation. At large cosmological scales, back reaction effects of
the geometry to the presence of matter inhomogeneities are investigated. It is
shown that, when averaged on spatial scales of order $100\ \text{Mpc}$, they
reproduce an effective cosmological constant, whose order of magnitude agrees
with observations (this solves the CC problem). The origin of $\Lambda$ is
thus linked to matter distribution, which solves the coincidence problem.

The model.—Our model of late-time cosmology is GR sourced by a two-fluid
system, consisting of baryonic and cold dark matter. We adopt the standard
description and we model them as two pressureless perfect fluids, interacting
only gravitationally one with each other, with densities $\rho_{B}$ and
$\rho_{DM}$ and 4-velocities $U_{\mu}$ and $W_{\mu}$ respectively. The stress-
energy tensor is then :

$T_{\mu\nu}=\rho_{B}U_{\mu}U_{\nu}+\rho_{DM}W_{\mu}W_{\nu}.$ (1)

It is known that Eq. (1) can be recast as the stress-energy tensor of an
anisotropic fluid by an appropriate rotation of $U_{\mu}$ and $W_{\mu}$
Bayin:1985cd . In the current case, this transformation yields

$T_{\mu\nu}=\rho u_{\mu}u_{\nu}+p_{\parallel}w_{\mu}w_{\nu},$ (2)

describing an anisotropic fluid with zero tangential pressure $p_{\perp}$.
$\rho\simeq\rho_{B}+\rho_{DM}$ (for $W^{\mu}U_{\mu}\simeq 1$),
$u_{\mu}u^{\mu}=-w_{\nu}w^{\nu}=-1$ and ${p_{\parallel}}$ can be interpreted
as an effective radial pressure stabilizing the dark matter halo. This is
conceptually equivalent to the description of a fluid of collisionless
particles, where an effective pressure term can be associated to the stress
tensor modeling the anisotropy of the velocity distributions Jeans22 ;
Binney:1982jf ; Herrera:1997plx . This is particularly suited for dark matter,
which is believed to be made of collisionless particles. The two-fluid
approach has the advantage to provide such a description in a natural and
straightforward way.

To describe both the galactic and cosmological regime, we use the following
general, spherically symmetric, spacetime metric (we use units with the speed
of light $c=1$)

$ds^{2}=a^{2}(t)\left[-e^{\alpha(t,r)}dt^{2}+e^{\beta(t,r)}dr^{2}+r^{2}d\Omega^{2}\right],$
(3)

where $\alpha(t,r)$ and $\beta(t,r)$ are metric functions, $a$ is the
cosmological scale factor and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}$.
The resulting independent Einstein field and conservation equations are (we
use ${}^{\prime}=\partial_{r},\,\dot{\,}=\partial_{t}$):

$3\frac{\dot{a}^{2}}{a^{2}}e^{-\alpha}+\frac{e^{-\beta}}{r^{2}}\left(-1+e^{\beta}+r\beta^{\prime}\right)+\frac{\dot{a}}{a}\dot{\beta}e^{-\alpha}=8\pi
Ga^{2}\rho;$ (4)

$\frac{\dot{a}}{a}\alpha^{\prime}+\frac{\dot{\beta}}{r}=0;$ (5)

$\frac{\left(1+r\alpha^{\prime}\right)-e^{\beta}}{e^{\beta}a^{2}r^{2}}-e^{-\alpha}\left(2\frac{\ddot{a}}{a^{3}}-\frac{\dot{a}}{a^{3}}\dot{\alpha}-\frac{\dot{a}^{2}}{a^{4}}\right)=8\pi
G{p_{\parallel}}$ (6)

$\dot{\rho}+\frac{\dot{a}}{a}\left(3\rho+{p_{\parallel}}\right)+\frac{\dot{\beta}}{2}\left(\rho+{p_{\parallel}}\right)=0;$
(7)

${p_{\parallel}}^{\prime}+\frac{\alpha^{\prime}}{2}\left(\rho+{p_{\parallel}}\right)+\frac{2}{r}{p_{\parallel}}=0.$
(8)

In particular, in the static case, Eq. (8) is the generalization of the
Newtonian hydrostatic equilibrium equation

$p^{\prime}=-\frac{\partial_{r}e^{\alpha}}{2}\rho=-\Phi^{\prime}\rho=-\frac{Gm(r)\rho}{r^{2}},$
(9)

where $\Phi(r)$ is the gravitational potential. In fact, using the weak field
result $e^{\alpha}\simeq 1+2\Phi(r)$, Eq. (8) gives the Tolmann-Oppenheimer-
Volkoff-like equation:

$\partial_{r}\left(r^{2}{p_{\parallel}}\right)=-r^{2}\partial_{r}\Phi\left(\rho+{p_{\parallel}}\right).$
(10)

Galactic regime.— The galactic regime is obtained by taking the static limit
of Eqs. (4)-(8), which implies the scale factor being constant (we set $a=1$),
and the metric functions depending on $r$ only. Integration of Eq. (4) in this
case gives

$e^{-\beta}=1-\frac{2GM(r)}{r},$ (11)

where $M(r)\equiv 4\pi\int\rho(r)r^{2}dr$ is the Misner-Sharp (MS) mass of the
system. The remaining equations are solved together with a given profile for
the matter density $\rho=\rho(r)$. We are looking for solutions reproducing
the flattening of rotation curves at galactic scales. This can be achieved by
choosing the following matter density profile

$\rho=\frac{\sigma}{r^{2}},$ (12)

with $\sigma$ a constant. This gives the MS mass $M(r)=4\pi\sigma r$. In fact,
virializing the galactic motion, we get the velocities $v^{2}=8\pi G\sigma$.
We skip a constant term in $M(r)$, which represents the contribution of the
mass contained in the central regions of the galaxy. Here, we are considering
only galactic scales $\gg\text{kpc}$, where, according to observations (see
e.g. Refs. Rubin:1980zd ; Bosma:1981zz ), rotation curves starts flattening.
With the density profile (12), the field equations can be integrated to give

$\displaystyle\alpha=2\ln\left[\mathcal{C}\ln\left(\frac{r}{L}\right)\right];$
(13a) $\displaystyle{p_{\parallel}}=-\frac{\sigma}{r^{2}}+\frac{1-8\pi
G\sigma}{4\pi G}\frac{1}{r^{2}\ln\left(r/L\right)},$ (13b)

with $\mathcal{C}$ and $L$ integration constants. These are galactic
parameters and could be determined, together with $\sigma$, by combining
rotation curves and gravitational lensing (see, e.g. Ref. Faber:2005xc ).
We note that

${p_{\parallel}}=-\rho=-\frac{\sigma}{r^{2}},\quad\alpha=0$ (14)

also solves the field equations. This solution, giving an equation of state
(EOS) ${p_{\parallel}}=-\rho$ and a negative pressure, dominates for
$r\to\infty$, i.e in the transition to the cosmological regime. Conversely,
the second, positive, term in Eq. (13b) dominates at smaller (galactic)
scales. Physically, this means that, in this regime, the hydrostatic
equilibrium of dark matter halos is obtained by contrasting the gravitational
pull with a positive radial pressure. On the other hand, at large distances,
in the transition to the cosmological regime, the hydrostatic equilibrium is
not reached in the usual intuitive way. It is a local equilibrium in which
both sides of Eq. (10) separately vanish. This is possible only if the EOS is
${p_{\parallel}}=-\rho$ and the pressure is negative. As it is already evident
from the form of the EOS, this is strongly related to the generation of the
cosmological constant in the cosmological regime (see below).
The existence of solution (14) is a peculiar feature of fluids with
anisotropies. Static isotropic fluids do not allow for solutions with
$\alpha^{\prime}=0$ and $p,\,\rho$ satisfying $p=-\rho$.

Our static solution, describing hydrostatic equilibrium of the DM medium,
models a spacetime with a conical singularity (the relevance of this kind of
solution for DM has been already noted in Muckprivate ). In the simplest case,
given by Eq. (14), the metric is:

$ds=-dt^{2}+\frac{dr^{2}}{(1-8\pi G\sigma)}+r^{2}d\Omega^{2}.$ (15)

The solution is physically acceptable in the weak field limit when the the
deficit angle is very small:

$\sigma\ll\frac{1}{8\pi G}.$ (16)

In this limit, the spacetime can be well approximated by flat Minkowski space.
The flattening of galactic rotation curves is observed when the acceleration
drops below $a_{0}\sim\ell^{-1}$ (with $\ell$ the size of the cosmological
horizon). This implies that condition (16) is satisfied for $r\ll l$, which
covers not only galactic, but also larger scales where our universe appears
inhomogeneous, with the density of inhomogeneities scaling as $1/r^{2}$
Cadoni:2020jxe . The behavior $\rho\sim 1/r^{2}$ is thus responsible not only
for the flattening of galactic rotation curves, but also well describes
structure distribution at small cosmological scales Cadoni:2020izk . This
means, physically, that the dynamical properties of galaxies are inherited
from structure formation.
The condition $r\ll l$ holds true also at scales $\mathcal{R}\sim 100\
\text{Mpc}$, where our universe begins to appear homogeneous and isotropic. It
breaks down for $r\sim\ell$, i.e. in the cosmological regime where the static
approximation is no longer valid and we have to consider the full form of the
metric (3).

Cosmological regime.—When $a\neq 1$ and the assumption of staticity of the
metric functions is dropped, the system of equations (4)-(8) describes the
cosmological regime of our model. The matter density determines the metric
function $\beta(t,r)$, whereas the back reaction of the metric to the presence
of matter is codified in the function $\alpha(t,r)\neq\text{constant}$
Cadoni:2020jxe . In the decoupling limit, when the back reaction can be
neglected, the cosmological dynamics is described by the standard Friedmann-
Lemaitre-Robertson-Walker (FLRW) cosmology and decouples completely from
inhomogeneities Cadoni:2020izk .

In the cosmological regime, exact solutions of the field equations (4)-(8) can
be found using a method similar to that used in Ref. Cadoni:2020jxe . One
first integrates Eq. (5), defines the rescaled quantities $\hat{\rho}\equiv
3e^{\alpha}(3-r\alpha^{\prime})^{-1}\rho,\,\hat{p}_{\parallel}\equiv
e^{\alpha}{p_{\parallel}}$ and then uses an ansatz to separate the standard
FLRW dynamics from that of inhomogenities and the back reaction:

$\displaystyle a^{2}\hat{\rho}(t,r)\equiv
a^{2}{\rho}^{(1)}(t)+\frac{3e^{\alpha}}{3-r\alpha^{\prime}}\left(\rho^{(2)}(r)+\rho^{(3)}(t,r)\right),$
$\displaystyle a^{2}\hat{p}_{\parallel}(t,r)\equiv
a^{2}{p}_{\parallel}^{(1)}(t)+e^{\alpha}\left({p}_{\parallel}^{(2)}(r)+{p}_{\parallel}^{(3)}(t,r)\right).$
(17)

The system (4)-(8) splits in the usual FLRW equations for $a$, sourced by
${\rho}^{(1)}$ and ${p}_{\parallel}^{(1)}$, together with the solutions for
$\beta$ and $\alpha$ (see Ref. Cadoni:2020jxe  for the calculation details) .
The solutions for $\beta$ is also here given by Eq. (11), with the MS mass
being $M(t,r)\equiv M^{(1)}(t)+M^{(2)}(r)+M^{(3)}(t,r)$, where $M^{(1)}$ is an
integration function, while $M^{(2,3)}$ are the MS masses associated to
$\rho^{(2,3)}$ respectively. Finally, the solution for $\alpha$ turns out to
be

$\displaystyle\alpha(t,r)=\mathcal{A}(t)+2G\
\frac{a}{\dot{a}}\int\frac{\dot{M}}{r^{2}}\left(1-\frac{2GM}{r}\right)dr,$
(18)

with $\mathcal{A}(t)$ integration function. We note that this solution, and in
particular the $(r,t)$-dependent terms, in general, removes the conical
singularity of the static solution.

The next step is to describe cosmology near the transition scale $\mathcal{R}$
to homogeneity and isotropy. In this situation, we cannot simply use the
general solution written above, since it describes also inhomogeneities and
their interaction with the cosmological dynamics encoded in the scale factor
$a$. Also the decoupling limit does not seem appropriate because it completely
neglects the back reaction. The simplest way to circumvent this problem is
first to split $\alpha$ into functions depending only on $r$ and $t$, i.e.
$\alpha\equiv\alpha_{r}(r)+\alpha_{t}(t)$. Then, we expand the solutions near
the decoupling limit, i.e. $r\alpha_{r}^{\prime}=0$, and near the present
epoch of our universe, i.e. $a^{2}=1$. Finally, we perform the spatial average
of the resulting $r$-dependent quantities. Keeping only the leading terms in
the expansions, Eqs. (17) give ($\alpha_{t}$ can be absorbed by a rescaling of
$t$)

$\displaystyle a^{2}\rho(t)=\frac{3}{8\pi
G}\left(\frac{\dot{a}}{a}\right)^{2}+a^{2}\langle\rho^{(3)}\rangle_{r};$ (19)
$\displaystyle a^{2}p(t)=\frac{1}{8\pi
G}\left[\left(\frac{\dot{a}}{a}\right)^{2}-2\frac{\ddot{a}}{a}\right]-a^{2}\Bigg{\langle}\frac{M^{(3)}}{4\pi
r^{3}}\Bigg{\rangle}_{r},$ (20)

where $\rho^{(3)}$, for consistency reasons, is function of $r$ only and the
spatial averaging is performed on spatial scales of order $100\ \text{Mpc}$
(see Cadoni:2020jxe  for further details). When $\rho^{(3)}\sim 1/r^{2}$,
these equations describe standard FLRW cosmology with the averaged back
reaction term $\langle\rho^{(3)}\rangle_{r}$ playing the role of a
cosmological constant,

$\displaystyle\Lambda\sim-8\pi G\langle\rho^{(3)}\rangle_{r},$ (21)

corresponding to a perfect fluid with equation of state $p=-\rho$. It is
important to stress that spatial averaging at length scales of order $100\
\text{Mpc}$ solves also the conical singularity issue. In fact, the weak field
condition (16) is satisfied and the $t=\text{constant}$ sections of our metric
are regular, flat Minkowski spacetime. Notice that the emergence of a
cosmological constant at large scales could have been guessed directly from
the existence of the static solution (14), which dominates at large $r$.

Let us now evaluate the order of magnitude of the cosmological constant. In
Ref. Cadoni:2020jxe  it is argued that $\rho^{(3)}\sim\rho^{(2)}$, since
$\rho^{(3)}$ is the energy density of the back reaction of the geometry to the
presence of matter inhomogeneities, described by $\rho^{(2)}$. As we have
previously seen, modelling both dark matter at galactic scales and the
structure formation at small cosmological scales Cadoni:2020izk ;
Cadoni:2020jxe  is consistent with $\rho^{(2)}$ scaling as $1/r^{2}$. We have
therefore $\rho^{(3)}\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}-\frac{1}{r^{2}}$, where the minus sign is due
to the fact that $\rho^{(3)}$ should give rise to an attractive force and must
behave as an inverse power of $r$, hence $\Lambda\sim 8\pi
G\langle\rho^{(2)}\rangle_{r}$. The spatial average can be easily computed in
terms of the total mass $M$ of dark matter inside a sphere of radius $R$ of
order $100\ \text{Mpc}$: $\langle\rho^{(2)}\rangle_{r}=3M/4\pi R$. We have
$M\sim 10^{18}\ M_{\odot}$ and $\Omega_{\Lambda 0}\sim 1$, giving the correct
order of magnitude of the observed cosmological constant Aghanim:2018eyx . The
fact that the energy densities associated to matter and $\Lambda$ are of the
same order of magnitude at the present epoch does not appear here as a
coincidence. This solves the coincidence problem of the standard $\Lambda$CDM
cosmological scenario.

Conclusions.—In the present work, we have presented and explored the
phenomenology of a cosmological model sourced by baryonic and cold dark matter
in late-time cosmology. Our model allows for a unified description of galactic
dynamics and cosmological dynamics as well as structure formation. The
flattening of the rotation curves of galaxies, the formation and distribution
of structures at small cosmological scales and the cosmological constant all
have the same origin in the distribution of matter inhomogeneities and in back
reaction of the geometry to the presence of the latter. The observed galactic
dynamics and structure formation and distribution are correctly reproduced by
assuming the presence of an anisotropic component of the pressure and of an
$1/r^{2}$ scaling of matter density. The observed order of magnitude of the
cosmological constant is explained as the average of the back reaction of the
geometry at scales where our universe starts appearing homogeneous and
isotropic. The results of our work show the existence of an intriguing link
between the small and large scale behavior in cosmology. Hints on this
direction come also from the presence of the fundamental acceleration scale
$a_{0}$ in the baryonic Tully-Fisher relation, which is of the order of
magnitude of the present Hubble acceleration McGaugh:2004aw .

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