trend in the HO_2 emission could not be studied in this way.
Figure <ref> indicates that this trend is obviously generated
distinctly lower than 85 km. Nevertheless, if we assume that the
smoothing in the seasonal pattern in the WACCM data is similar for the
observations, it is likely that we would obtain a variability structure that
resembles the curve for the 1,510 nm feature plotted in
Fig. <ref>. Hence, the differences between modelled and
observed emission variations appear to be mostly caused by the WACCM-based
reproduction of temporal changes in the H density.

Table <ref> shows the mean centroid emission altitudes from the
corresponding climatologies of the different production processes of
HO_2 emission. Only the height of about 81 km for
Reaction <ref> is in the range from about 80 to 84 km for
the average profile derived from the -based analysis in
Sect. <ref>. The agreement looks even more promising if it is
considered that O, which is crucial for the upper limit of
84 km, is not involved in the chemical production of excited
HO_2 for this pathway. The simulated climatological variations show
increasing heights during the night and the highest values in austral summer
with maximum deviations from the mean lower than 2 km in most cases.
The emission of Reaction <ref> just reaches 80 km close to
sunrise, whereas the centroid heights are even below 70 km at the
beginning of the night due to the high concentrations of
O_2($\mathrm{a}^1\Delta_{\mathrm{g}}$) related to daytime O_3
photolysis in the lower mesosphere [Noll et al., 2016]. Reaction R14 shows a
similar variability pattern as Reaction <ref> but, with a mean value
of 86 km, the emission is too high. The reason is the impact of
O_3, which shows its density peak at a higher altitude than H in
the mesopause region in Fig. <ref>d. This plot also indicates that
the density distribution of all HO_2 peaks at 78 km, which is
lower than the emission maximum of excited HO_2 shown in
Fig. <ref>c.

Finally, Table <ref> shows that at least the results for
Reactions <ref> and <ref> appear to agree with a positive
but weak solar cycle effect that was derived for the 1,510 nm feature
in Sect. <ref>. Although the uncertainties are of the order of several
per cent, the almost doubled deviation from the measured value compared to the
other reactions, make Reaction R14 less likely with respect to the response to
solar activity.

Our investigation of the three proposed production mechanisms for excited
HO_2 shows promising results. It is likely that HO_2 is the
radiating molecule X that produces the 1,510 nm feature and the
strongly correlated X(NIR) continuum. Moreover, the recombination
reaction of H and O_2 with participation of an additional
collision partner is the most probable production process for excited
HO_2. All investigated properties such as total emission, nocturnal and
seasonal variations, emission heights, and solar cycle effect point to this
interpretation. It also helps that chemiluminescence by this mechanism was
already observed in the laboratory between about 800 and 1,550 nm
[Holstein et al., 1983], although an extension to higher wavelengths would be
important to test the presence of the 1,620 nm feature
(Fig. <ref>). Reaction <ref> that involves
O_2($\mathrm{a}^1\Delta_{\mathrm{g}}$) can only be a minor emission
source, otherwise the observed nocturnal trend would not fit. It is
also more challenging to produce emission below 1,270 nm for this
process. Finally, Reaction R14 that involves O_3 shows clear
discrepancies in the emission properties which can only be tolerated if the
contribution to the total emission is very small.

Our analysis of the nightglow (pseudo-)continuum with high-quality
data essentially reveals two contributions in the wavelength
range between 300 and 1,800 nm if remnants from different O_2
bands are excluded.

Our results of the correlation analysis of continuum structures and
non-negative matrix factorisation (NMF) of the continuum variability show that
the peak at 595 nm is well correlated with other features and the
underlying continuum in a wide wavelength range, especially between about 500
and 900 nm. The variations as mainly studied for the feature at
595 nm reveal a climatology with a mixture of semiannual and annual
oscillation with a main maximum in April/May and a main minimum in January
that confirms previous results based on a smaller sample. For the first time,
we estimated the effective solar cycle effect and found a weak positive
correlation. Using an approach for the estimate of effective emission heights
based on the analysis of a strong passing Q2DW that was initially developed
for OH lines, we obtained a range for the mean centroid height between
about 85 and 89 km.

In previous studies, the feature at 595 nm was identified as the main
peak of the FeO orange bands. Our simulations of the chemiluminescence
from the reaction of Fe and O_3 with WACCM can reproduce most of
the measured properties of the emission, which suggests that the NMF component
dominating the  VIS arm could have contributions from various
FeO bands. However, WACCM returns a maximum mean intensity of only
170 R, whereas the whole correlated spectrum could have
2.9 kR. We discovered that potential OFeOH emission (with
unknown spectral distribution) from the reaction between FeOH and
O_3 would have a very similar climatology according to our simulations.
Nevertheless, this reaction would only add up to 220 R. Therefore, a
major discrepancy remains. If there is another emitter, the basic precondition
for a good correlation with the 595 nm feature appears to be that the
variability is mainly determined by O_3.

The second continuum component dominates the  NIR arm. In
particular, a strong narrow peak at about 1,510 nm and a secondary
feature at about 1,620 nm were found, which indicates a complex band
system. Our best estimate of the average intensity of the entire system is
about 12 kR. The seasonal variations with maxima near the solstices
are actually in opposition to those of the 595 nm feature. There is
also a clear decrease of the intensity in the course of the night for the
entire year. The solar cycle effect is only weakly positive and the average
effective emission height appears to be most likely between about 80 and
84 km for the 1,510 nm feature.

The most promising candidate for the emitter is HO_2. Existing near-IR
spectra from the laboratory suggest that the 1,510 nm feature could be
the vibrational  transition of the electronic ground state
$^2\mathrm{A}^{\prime\prime}$. There would also be an explanation of the
enhanced emission near 1,270 nm, where only a part appears to be due
to residual O_2 emission. Other features from the experiments could not
be checked due to gaps in our continuum spectrum. We investigated different
potential production processes of excited HO_2 with WACCM. The
recombination reaction between H and O_2 under participation of
another collision partner showed the best performance. It is the main
production process of HO_2 in the mesosphere. With a modelled maximum
mean radiance of 82 kR, a moderate quantum yield of the reaction would
be sufficient to produce the continuum in the  NIR arm.
Moreover, this process indicates the best agreement with respect to the
climatological variations. Remaining discrepancies (especially a missing
secondary peak in austral winter) can be explained by deviations of the
modelled H densities from those of SABER-based retrievals. The
simulated weak solar cycle effect and the average centroid emission height of
about 81 km also show good agreement. Finally, the observed
chemiluminescence in the laboratory for this mechanism indicated emission down
to about 800 nm, which is consistent with the shape of the derived
continuum component. As wavelengths above about 1,550 nm were not
studied in the only known laboratory experiment, there is no evidence of the
existence of the 1,620 nm feature, so far. The other studied potential
emission processes appear to be much less efficient. Relatively weak or even
negligible emission rates are probably related to the direct radiative
recombination of H and O_2, the reaction of H and
O_3, and collisions of HO_2 with
O_2(a$^1\Delta_{\mathrm{g}}$). The latter would produce a steep decline
of the emission after dusk due to the decay of the population of excited
O_2 molecules produced by O_3 photolysis, which is not observed
in the  data. The reaction involving H and O_3
would generate a very different seasonal variability as observed.

The intriguing discoveries of this study will certainly stimulate further
investigations for a better understanding of the chemistry and dynamics in the
Earth’s mesopause region. The origin of the whole VIS-arm continuum still
needs to be solved. The study also revealed that the nighttime production of
O_2(a$^1\Delta_{\mathrm{g}}$) is not understood well, although these
excited molecules are essential for the strong emission at 1,270 nm.
These examples illustrate that there are still many things at these altitudes
that we do not know.

The basic  data for this project originate
from the ESO Science Archive Facility at
and are related to various observing programmes that were carried out
between October 2009 and September 2019. The raw spectra were processed
(using the corresponding calibration data) and then analysed. The NIR-arm
data were already used for the study of OH emission lines
[Noll et al., 2022, Noll et al., 2023]. Some results of these investigations, which are
available via the public repository Zenodo [Noll et al., 2022, Noll et al., 2023], were
also considered for this study. We performed dedicated WACCM6 runs with
modified chemistry for the years from 2003 to 2014. The crucial results are
stored at the University of Leeds. We also made use of TIMED/SABER data sets
that were already collected for previous studies for Cerro Paranal from the
SABER website at . These are the v2.0
products from 2002 to 2015 analysed by Noll et al., 2017 and the improved
O and H retrievals described by Mlynczak et al., 2018 for the
years 2002 to 2014 that were used by Noll et al., 2019. Results from the study
of v2.0 products from January and February 2017 by Noll et al., 2022 were also
considered. A comprehensive collection of data of our analysis (especially
with respect to the plotted data) is provided by Zenodo at
[Noll et al., 2023].

SN designed and organised the project, performed the
preparation and analysis of the  spectra, visualised the
results based on , WACCM, and SABER data, and is the main
author of the paper text. The co-authors contributed to the improvement of
the paper content. Moreover, JP designed the WACCM runs, investigated the
involved chemistry, and significantly influenced the scientific discussion.
WF performed the WACCM simulations and a preliminary analysis of the data.
KK significantly contributed to the discussion of the chemistry. In
particular, he first proposed HO_2 as possible emitter. WK carried
out the basic processing of the  spectra. CS contributed to
the discussion of the -based analysis. MB was involved in
the management of the project. SK managed the infrastructure for the
processing and storage of the  data and contributed to the
discussion of the measured continuum features.

Co-author John Plane is a member of the editorial board of
Atmospheric Chemistry and Physics. The authors do not have other competing
interests to declare.

Stefan Noll was financed by the project  of the German
Research Foundation (DFG). John Plane and Wuhu Feng were supported by grant
NE/T006749/1 from the UK Natural Environment Research Council. Konstantinos
Kalogerakis acknowledges financial support from the US National Aeronautics
and Space Administration (NASA grant 80NSSC23K0694) and the National Science
Foundation (NSF grant AGS-2009960). We thank Sabine Möhler from ESO for
her support with respect to the  calibration data.

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# Fully conservative $f(R,T)$ gravity and Solar System constraints

Nicolas R. Bertini<EMAIL_ADDRESS>Departamento de Ciências
Exatas e Aplicadas, Universidade Federal de Ouro Preto, Campus João Monlevade,
João Monlevade, MG, Brazil  Hermano Velten<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Ouro Preto, Campus Morro do
Cruzeiro, Ouro Preto, MG, Brazil

###### Abstract

The $f(R,T)$ gravity is a model whose action contains an arbitrary function of
the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$. We
consider the minimally coupled model $f(R,T)=\chi(R)+\varphi(T)$ and shown
that, for perfect fluids, the analysis of dynamical equations are sufficient
to determine how $\varphi$ depends on $T$, independently of the matter fields
equation of state and the geometry of space-time. Imposing the energy-momentum
tensor conservation we obtain that the trace dependent part $\varphi$ must be
linear in $T$, apart from the trivial case of a constant. However, the linear
dependence on $T$ is severely constrained using the full Will-Nordtvedt
version of the parameterized post-Newtonian (PPN) formalism. The result of the
PPN analysis is discussed and in addition it is shown that the diffeomorphism
invariance of the matter action imposes strong constraints on conservative
versions of $f(R,T)$ gravity.

##  I Introduction

General relativity is considered the correct description for the gravitational
interaction due to its huge success in explaining available astronomical data
at the solar system level and also for first predicting new gravitational
phenomena posteriorly observed. On the other hand, the existence of the dark
sector of the universe, consisting of dark matter and dark energy, is one of
the most intriguing scientific challenges of the last decades and remains
unsolved. By trusting that general relativity is the correct theory for
describing the gravitational interaction, specially above the galactic scales,
the standard cosmological model demands that about $25\%$ of current cosmic
energy budget should be composed by dark matter while other $70\%$ sums up the
dark energy sector. So far, though there exists a large number of
theoretically well motivated particles/fields candidates to compose the dark
sector, there is no direct detection of them. Such lack of detection has
motivated the search for a new theory in replacement of GR. Currently, this is
a vast research field [1, 2].

Either by modifying GR structures or by just adding new elements that yield to
extended GR versions, all existing proposals should be subjected to the fact
that there are no reasons to doubt GR validity at local scales i.e., the solar
system one. Apart from analytical and numerical results for the behavior of a
gravitational theory at the solar systems level, this analysis can also be
technically performed via the so called post-Newtonian formalism [3, 4, 5].

In this work we shall be concentrated on a widely known modified theory of
gravity in which the gravitational action of the total theory depends on the
trace of the energy momentum $T\equiv T^{\mu}_{\mu}$ in addition to the Ricci
scalar $R$ [6]. In short notation, starting with $f(R,T)$ gravity, GR is fully
recovered with $f(R,T)\equiv R$. In the literature there are many distinct
formulations for the functional dependence of $f(R,T)$. For example, the $T$
dependence has been written either in the polynomial, rational or the
exponential one [7, 8, 9, 10, 11, 12, 13, 14, 15].

When seeking to determine the functional form of $f(R,T)$, it is not strictly
required but highly desirable for the ensuing dynamics to exhibit conservative
behavior, i.e., preserve the vanishing of the four-divergence of the energy
momentum tensor, $\nabla^{\mu}T_{\mu\nu}=0$. Maintaining this property ensures
that the underlying physical system is conservative which is a fundamental
requirement for a wide range of physical phenomena but, specially for applying
the post-Newtonian formalism. Therefore, incorporating this criterion into the
search for the function $f(R,T)$ is a key factor. For example, cosmological
solutions have shown that energy-density conservation can be reinstated in a
manner similar to general relativity (GR) if $f(R,T)=f_{1}(R)+f_{2}(T)$, where
$f_{2}$ is defined as $A+BT^{1/2}$ and $A$ and $B$ are constants. [16]. The
issue of conservative $f(R,T)$ models have also been explored in the context
of spherically symmetric stellar configurations. In such analysis, it is
demanded that the effective energy momentum tensor should be conserved,
establishing a criteria for the allowed polytropic equation of state
parameters [17, 18]. Though the possibility that conservation can be violated,
in the sense that $\nabla^{\mu}T_{\mu\nu}\neq 0$, the entire mechanism behind
the post-Newtonian analysis demands conservation and has, as we will show
below, a strong constraining power on the functional form of $f(R,T)$. This is
the main result of this work.

At the level of cosmological solutions, the viability of $f(R,T)$ theories on
cosmological scales has been addressed in the literature. Though it has been
argued that the high redshift evolution of $f(R,T)$ cosmologies are
incompatible with standard cosmological model [19], a more complete recent
analysis using the age of the universe as a constraining cosmological
observable has proved that there is however a small free parameter space
$f(R,T)$ model that can lead to viable cosmological scenarios at the
background level [20].

Our goal in this work is to investigate the general feature of conservation in
$f(R,T)$ theories at the level of the field equations, i.e., before setting a
specific geometry. We show that the case $f_{2}\propto T^{1/2}$ represents a
particular “cosmological” case of the full conservative $f(R,T)$ theory. In
addition, by applying the parameterized post-Newtonian (PPN) formalism we
demonstrate that the existence of conservative $f(R,T)$ theories is
challenged. The only exception is the trivial cosmological constant like case.

This work is organized as follows. In the next section we review the field
equations for $f(R,T)$ gravity. In the third section we focus on discussing
the role played by a minimally coupled $f(R,T)=f_{1}(R)+f_{2}(T)$ choice. We
then work out the full conservative model at the field equations level and
present one of the main results of this work. In Sec. IV we briefly review
some essential PPN features and apply them to the case of minimally coupled
$f(R,T)$ gravity. In Sec. V the results from the previous section are
discussed and the observational bounds for the case developed in Sec. III are
determined. In Sec. VI we consider the action invariance under diffeomorphisms
and find the condition that must be satisfied for a general $f(R,T)$ theory to
be conservative. Our conclusions are in Sec. VII.

##  II The action and field equations

The $f(R,T)$ gravity was first proposed in [6], and characterized by an
arbitrary dependence of the gravitational action on the Ricci scalar and the
energy-momentum tensor trace $T=T^{\mu}_{\;\;\mu}$. The full action is

$\displaystyle S[g,\Psi]=\frac{1}{2\kappa^{2}}\int
f(R,T)\sqrt{-g}\,d^{4}x+S_{\mathrm{m}}[g,\Psi],$ (1)

where $\kappa^{2}=8\pi G$ being $G$ the gravitational coupling constant
($c=1$), $f(R,T)$ is an arbitrary function of the $R$ and $T$,
$S_{\mathrm{m}}$ is the action for the matter fields denoted collectively by
$\Psi$. The energy-momentum tensor is obtained from $S_{\mathrm{m}}$ by the
usual definition

$\displaystyle T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta
S_{\mathrm{m}}}{\delta g^{\mu\nu}}.$ (2)

The variation of action (1) with respect to the metric results in

$\displaystyle
f_{{}_{R}}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{{}_{R}}=\kappa^{2}T_{\mu\nu}-f_{{}_{T}}(T_{\mu\nu}+\Theta_{\mu\nu}),$
(3)

where $f\equiv f(R,T)$ and, for sake of simplicity, we defined

$\displaystyle f_{{}_{R}}\equiv\frac{\partial f(R,T)}{\partial R},\quad
f_{{}_{T}}\equiv\frac{\partial f(R,T)}{\partial
T}\quad\mathrm{and}\quad\Theta_{\mu\nu}\equiv g^{\alpha\beta}\frac{\delta
T_{\alpha\beta}}{\delta g^{\mu\nu}}.$ (4)

The equation for the energy-momentum tensor divergence, obtained directly from
(3), is

$\displaystyle\nabla^{\mu}T_{\mu\nu}=\frac{f_{{}_{T}}}{\kappa^{2}-f_{{}_{T}}}\left[-\frac{1}{2}\nabla_{\nu}T+(T_{\mu\nu}+\Theta_{\mu\nu})\nabla^{\mu}\ln
f_{{}_{T}}+\nabla^{\mu}\Theta_{\mu\nu}\right].$ (5)

Of course, $f_{T}=0$ restores the conservation. The energy momentum tensor for
perfect fluids reads

$\displaystyle T_{\mu\nu}=p(u_{\mu}u_{\nu}+g_{\mu\nu})+\varepsilon
u_{\mu}u_{\nu},$ (6)

where $p$ is the pressure, $\varepsilon$ is the energy density and
$u^{\alpha}$ is the four velocity of the fluid element. In this case

$\displaystyle\Theta_{\mu\nu}=-2T_{\mu\nu}+g_{\mu\nu}p,$ (7)

and $T$ depends on the energy density and pressure through

$\displaystyle T=-\varepsilon+3p.$ (8)

Furthermore, the matter field dynamics is related to $f$ by the equation

$\displaystyle\frac{1}{2\kappa^{2}}\int
d^{4}x^{\prime}\sqrt{-g^{\prime}}\frac{\delta
f^{\prime}}{\delta\Psi}=-\frac{\delta S_{\mathrm{m}}}{\delta\Psi},$ (9)

where $x^{\prime}$ denotes the integration variable. Therefore, it should be
clear that $p$ and $\varepsilon$ in above equations depend on $f$, by
construction.

##  III The case of a minimal coupling

This section outlines the fundamental characteristics of the gravitational
field equations when expressing the function $f(R,T)$ in the separable form of
$f_{1}(R)+f_{2}(T)$. Here, the matter sector, as given by the trace $T$, is
minimally coupled with the geometric quantities found in the gravitational
Lagrangian. It is noteworthy that in this case, the conservation of the energy
momentum tensor is essential for successful application of the PPN formalism,
and such conservation is indeed observed. However, it is worth mentioning that
nonminimally coupled scenarios are intrinsically nonconservative [20].

Imposing both the vanishing of the four-divergence of $T_{\mu\nu}$ in Eq. (5)
i.e., $\nabla^{\mu}T_{\mu\nu}=0$ and $f_{{}_{T}}-\kappa^{2}\neq 0$, and also
taking into account that
$\nabla^{\mu}\Theta_{\mu\nu}=\nabla^{\mu}(-2T_{\mu\nu}+g_{\mu\nu}p)=\nabla_{\nu}p$,
this yields to the constraint

$\displaystyle
f_{{}_{T}}\left[-\frac{1}{2}\nabla_{\nu}T+(g_{\mu\nu}p-T_{\mu\nu})\nabla^{\mu}\ln
f_{{}_{T}}+\nabla_{\nu}p\right]=0.$ (10)

Since $f_{{}_{T}}\nabla^{\mu}\ln f_{{}_{T}}=\nabla^{\mu}f_{{}_{T}}$ one has

$\displaystyle-\frac{1}{2}(\nabla_{\nu}T)f_{{}_{T}}+(g_{\mu\nu}p-T_{\mu\nu})\nabla^{\mu}f_{{}_{T}}+(\nabla_{\nu}p)f_{{}_{T}}=0.$
(11)

From this point one has to make a choice on the $f(R,T)$ function. Let us
consider the minimal coupling case

$\displaystyle f(R,T)=\chi(R)+\varphi(T).$ (12)

According to Eq. (9) it is technically possible to assume the dependence of
$p$ on $\varepsilon$ and $\varphi(T)$. For any equation of state of the type
$p=p(\varepsilon)$ admitting an inverse $\varepsilon=\varepsilon(p)$ one has
$T=-\varepsilon(p)+3p$, and therefore it is possible to write $p=p(T)$. This
is an important assumption made in our work.

Now, let $\zeta\equiv\zeta(R,T)$ be any function of $R$ and $T$. Then, without
loss of generality, the four-divergence of this quantity is directly computed
as
$\nabla_{\mu}\zeta=\nabla_{\mu}T\zeta_{{}_{T}}+\nabla_{\mu}R\zeta_{{}_{R}}$.
Therefore, it is trivial to write down the quantities
$\nabla_{\mu}f_{{}_{T}}=(\nabla_{\mu}T)f_{{}_{TT}}+(\nabla_{\mu}R)f_{{}_{RT}}=(\nabla_{\mu}T)\varphi_{{}_{TT}}$
and $\nabla_{\mu}p=(\nabla_{\mu}T)p_{{}_{T}}$. This allows one to rewrite (11)
into the form

$\displaystyle\left[(g_{\mu\nu}p-T_{\mu\nu})\varphi_{{}_{TT}}+\frac{1}{2}g_{\mu\nu}(2p_{{}_{T}}-1)\varphi_{{}_{T}}\right]\nabla^{\mu}T=0.$
(13)

Therefore, in general we have

$\displaystyle(p\delta^{\mu}_{\nu}-T^{\mu}_{\nu})\varphi_{{}_{TT}}+\frac{1}{2}(2p_{{}_{T}}-1)\varphi_{{}_{T}}\delta^{\mu}_{\nu}=0.$
(14)

The $\mu=\nu=0$ and $\mu=i,\;\;\nu=j$ components of above equation are (in the
comoving frame, $u^{i}=0$), respectively

$\displaystyle(p+\varepsilon)\varphi_{{}_{TT}}+\left(p_{{}_{T}}-\frac{1}{2}\right)\varphi_{{}_{T}}=0,$
(15)
$\displaystyle\left(p_{{}_{T}}-\frac{1}{2}\right)\varphi_{{}_{T}}\delta^{i}_{j}=0.$
(16)

In the cosmological context, considering the state equation $p=w\varepsilon$,
the solution obtained in the Refs. [16, 21] is consistent with that found
considering only the Eq. (15): $\varphi(T)=AT^{\frac{1+3w}{2(1+w)}}+B$, for
$w\neq 1/3$ and $A$ and $B$ being integration constants. As argued in [16]
this model should represent the only viable $f(R,T)$ theory, since it
constitutes the only case in which the standard conservation law is preserved.
However, the viability of this model in explaining cosmology background
observables is discussed in [19] and the findings of such reference represent
a challenge to this model as a viable modification of gravity. However the
approach in [16, 21] is such that Eq. (16) is not considered.

The solution of Eq. (15) using the condition (16), even if one does not know
the equation of state, is

$\displaystyle\varphi(T)=\sigma_{0}+\sigma_{1}T,\quad(\,p\neq-\varepsilon\,)$
(17)

being $\sigma_{0}$ end $\sigma_{1}$ constants. The only condition behind the
above result is $p\neq-\varepsilon$.

Concerning Eqs. (15) and (16) it is worth noting that, excluding the case
where $\varphi_{{}_{T}}=0$, the choice of the comoving frame implies a
relationship between $p$ and $T$ so that, given an equation of state
$p=p(\varepsilon)$, the fluid equation of state parameter can be determined.
For example, assuming $p=\omega\varepsilon$, Eq. (16) implies $\omega=1$, the
stiff matter model.

On the other hand, the solution (17) is found for any arbitrary reference
frame. To verify this one simply takes the trace of Eq. (14) and uses the
result to eliminate $(2p_{{}_{T}}-1)\varphi_{{}_{T}}$. This procedure results
in

$\left(T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T\right)\varphi_{{}_{TT}}=0\,.$ (18)

The above equation can be satisfied in two cases: (i) if $\varphi(T)$ is given
by (17); or (ii) if the term in parentheses is zero. The second case occurs,
in general, if $p=-\varepsilon$. This relation corresponds to the general
relativistic solution for a cosmological constant which is equivalent to the
case $\varphi=const$. However, the relation (17) is even more general by
including this case.

Since we have determined a functional form for $f(R,T)$ that leads to a
conservative energy-momentum tensor, in the next section we apply the full
Will-Nordevedt PPN formalism to this model and restrict the $\sigma$’s
parameters of the Eq. (17) in section V. For this we specify the functional
form

$\displaystyle f(R,T)=R+\varphi(T),$ (19)

which corresponds to $\chi(R)=R$ in Eq.(12), and

$\displaystyle\varphi(T)\equiv\varphi_{n}(T)=\sum^{n}_{i=0}\sigma_{i}T^{i}.$
(20)

The solution (17) is obtained for n=1. Substituting into the field equation
(3) we have

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{1}{2}g_{\mu\nu}\varphi+8\pi
GT_{\mu\nu}-\varphi_{{}_{T}}(T_{\mu\nu}+\Theta_{\mu\nu}).$ (21)

It is useful to write this expression in the form

$\displaystyle R_{\mu\nu}=-\frac{1}{2}g_{\mu\nu}\varphi+8\pi
G\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)-\varphi_{{}_{T}}\left(T_{\mu\nu}+\Theta_{\mu\nu}-\frac{1}{2}g_{\mu\nu}(T+\Theta)\right),$
(22)

where the trace was used to eliminate $R$.

##  IV Post-Newtonian expansion

In this section we apply the Will-Nordtvedt PPN formalism [3, 22] to $f(R,T)$
gravity in its minimally coupled form (19). In this formalism, a perfect fluid
is the source of the gravitational field. It describes the metric of a
gravitational theory in terms of ten observable PPN parameters in a theory-
independent way. The main small parameter of this formalism is the matter
velocity field $|\vec{v}|=v<1$. The metric is expanded about Minkowski
spacetime,

$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}\,,$ (23)

where $\eta_{\alpha\beta}$ is the Minkowski metric, which is of zeroth-order
on $v$, and $h_{\alpha\beta}\sim O(v^{2})$, at least. We use the signature
$(-,+,+,+)$.

Up to the first post-Newtonian order, the metric must be known as follows:
$g_{00}$ to order $v^{4}$, $g_{0i}$ to order $v^{3}$ and $g_{ij}$ to order
$v^{2}$ (Latin indices run from $1$ to $3$). Thus, up to the required order,
the Ricci tensor components can be expressed as

$\displaystyle R_{00}=$
$\displaystyle-\frac{1}{2}\nabla^{2}h_{00}-\frac{1}{2}\left(h^{k}_{~{}k,00}-2\,h^{k}_{~{}0,k0}\right)-\frac{1}{4}\,|\vec{\nabla}h_{00}|^{2}\
+$
$\displaystyle+\frac{1}{2}\,h_{00,l}\left(h^{lk}_{~{}~{},k}-\frac{1}{2}\,h^{k}_{~{}k,j}\delta^{j}_{l}\right)+\frac{1}{2}\,h^{kl}h_{00,lk}\,,$
(24) $\displaystyle R_{0i}=$
$\displaystyle-\frac{1}{2}\left(\nabla^{2}h_{0i}-h^{k}_{~{}0,ik}+h^{k}_{~{}k,0i}-h^{k}_{~{}i,k0}\right)\,,$
(25) $\displaystyle R_{ij}=$
$\displaystyle-\frac{1}{2}\\!\left(\nabla^{2}h_{ij}-\\!h_{00,ij}+\\!h^{k}_{~{}k,ij}-\\!h^{k}_{~{}i,kj}-\\!h^{k}_{~{}j,ki}\right).$
(26)

The comas refer to partial derivatives,
$\nabla^{2}\equiv\eta^{ij}\partial_{i}\partial_{j}$, and it is used that time
derivatives effectively yield to a higher order in the expansion. Thus, if a
quantity $X$ is of order $v^{n}$ then $X,_{k}\sim O(v^{n})$ and $X,_{0}\sim
O(v^{n+1})$.

Since the gravitational source is a perfect fluid, the energy-momentum tensor
is the one given in (6) but the energy density is decomposed into the mass
density $\rho$ and the specific energy density $\Pi$ in the form

$\varepsilon=\rho+\rho\Pi.$ (27)

The four velocity of the fluid element $u^{\mu}=(u^{0},v^{i})$, with

$u^{0}=\sqrt{\frac{1+v^{2}}{1-h_{00}}}\,,$ (28)

such that $u^{\mu}u_{\mu}=-1$. The density $\rho$, $\Pi$ and $p/\rho$ are of
order $v^{2}$ [3].

With the expressions above, we expand Eq. (22) to calculate the metric
components order by order on powers of $v$. As a first step, the zeroth order
equation in $v$ trivially leads to

$\sigma_{0}=0.$ (29)

This is expected since the constant $\sigma_{0}$ in Eq. (17) necessarily leads
to nonasymptotically flat spacetimes, like the cosmological constant in
general relativity, which is not considered in the standard PPN Solar System
analysis. This is also physically reasonable since, up to first post-Newtonian
order (1PN) and considering its value as inferred from the cosmological
observations, it has negligible impact on the Solar System dynamics [23]. The
next steps are described below.

  * •

$h_{00}$ up to order $v^{2}$ (Newtonian limit): Up to the required order one
finds

$R_{00}=-\frac{1}{2}\,\nabla^{2}h_{00},\;\;\varphi_{1}=\sigma_{1}T,\;\;T_{00}=-T=\rho\,\quad\mbox{and}\quad\Theta_{00}=-\Theta=2\rho.$
(30)

Inserting this into Eq. (22) we find that the terms containing $\sigma_{1}$
cancel out, therefore

$\nabla^{2}h_{00}=-8\pi G\rho\,.$ (31)

In order to be in agreement with the local Newtonian gravity it is demanded
then,

$h_{00}=2U,$ (32)

where $U$, the negative of the Newtonian potential111For conciseness, commonly
we will call $U$ the Newtonian potential, without writing “negative” in front
of it. We use $U$ since we are following the notation of Ref. [3] on the PPN
parameters and the potentials. [3], given by

$U(t,\vec{x})=G_{\mathrm{N}}\int\frac{\rho(t,\vec{x}\,^{\prime})}{|\vec{x}-\vec{x}\,^{\prime}|}d^{3}x^{\prime}\,,$
(33)

being $G_{\mathrm{N}}$ the Newtonian gravitational constant. Thus we must set

$G=G_{\mathrm{N}}=1.$ (34)

  * •

$h_{ij}$ up to order $v^{2}$: Imposing the three gauge conditions,

$h^{\mu}_{\;i,\mu}-\frac{1}{2}h^{\mu}_{\;\mu,i}=0,$ (35)

the spatial part of Eq. (22) reduces to

$-\frac{1}{2}\nabla^{2}h_{ij}=4\pi\rho\delta_{ij}+\sigma_{1}\rho\delta_{ij}.$
(36)

This equation is easily integrated, providing

$h_{ij}=2\left(1+\frac{\sigma_{1}}{4\pi}\right)U\delta_{ij}.$ (37)

  * •

$h_{0i}$ up to order $v^{3}$: With the fourth gauge condition

$h^{\mu}_{\;0,\mu}-\frac{1}{2}h^{\mu}_{\;\mu,0}=\frac{1}{2}h_{00,0}\,,$ (38)

the Eq. (22) becomes

$\displaystyle\nabla^{2}h_{0j}+U_{,0j}=16\pi\left(1+\frac{h_{1}}{8\pi}\right)\rho
v_{j}.$ (39)

The above equation can be integrated using the auxiliary potential
$\chi(t,\vec{x})$ [3], given by

$\chi(t,\vec{x})\equiv\int\rho(t,\vec{x}^{\prime})|x-\vec{x}^{\prime}|d^{3}x^{\prime}.$
(40)

From this definition one can write

$\nabla^{2}\chi=-2U\quad\mathrm{and}\quad\chi_{,0j}=V_{j}-W_{j},$ (41)

where

$V_{j}=\int\frac{\rho(x-\vec{x}^{\prime})v^{\prime}_{i}}{|x-\vec{x}^{\prime}|}d^{3}x^{\prime}\,,\quad\nabla^{2}V_{j}=-4\pi\rho
v_{j}\,.$ (42)

Therefore, Eq. (39) gives

$\displaystyle
h_{0j}=-\left(\frac{7}{2}+\frac{h_{1}}{2\pi}\right)V_{j}-\frac{1}{2}W_{j}\,.$
(43)

  * •

$h_{00}$ up to order $v^{4}$: To the required order

$\varphi_{2}=\sigma_{1}(-\rho+3p)+\sigma_{2}\rho^{2},$ (44)

and for the energy-momentum tensor, one finds

$T_{00}-\frac{1}{2}g_{00}T=\rho\left(v^{2}-U+\frac{\Pi}{2}+\frac{3p}{2\rho}\right).$
(45)

With the above relations the $0-0$ component of Eq. (22) gives

$\displaystyle R_{00}=-4\pi\rho
v^{2}\left(2+\frac{\sigma_{1}}{4\pi}\right)-4\pi\rho
U\left(-2+\frac{\sigma_{1}}{\pi}\right)-4\pi\rho\Pi-4\pi
p\left(3+\frac{\sigma_{1}}{2\pi}\right)+4\pi\rho^{2}\frac{\sigma_{2}}{8\pi}.$
(46)

Let us now find the $0-0$ component of the Ricci tensor. Considering the post-
Newtonian potentials introduced in [3]

$\displaystyle\nabla^{2}\phi_{1}=-4\pi\rho
v^{2},\quad\nabla^{2}\phi_{2}=-4\pi\rho U,$ (47)
$\displaystyle\nabla^{2}\phi_{3}=-4\pi\rho\Pi,\quad\nabla^{2}\phi_{4}=-4\pi
p,$ (48)

we take into account the gauge conditions (35) and (38), and using the
relation $|\vec{\nabla}U|=\nabla^{2}(U^{2}/2-\phi_{2})$, the Ricci tensor
component reads

$R_{00}=-\frac{1}{2}\nabla^{2}(h_{00}+2U^{2}-8\phi_{2}).$ (49)

With the above results, eq. (46) becomes

$\displaystyle
h_{00}=-2U^{2}+\left(4+\frac{\sigma_{1}}{2\pi}\right)\phi_{1}+\left(4+\frac{2\sigma_{1}}{\pi}\right)\phi_{2}+2\phi_{3}+\left(6+\frac{\sigma_{1}}{\pi}\right)\phi_{4}-\frac{\sigma_{2}}{4\pi}{\cal
T},$ (50)

where ${\cal T}$ is a new post-Newtonian potential defined as

$\displaystyle{\cal
T}(t,\vec{x})\equiv\int\frac{[\rho(t,\vec{x}^{\prime})]^{2}}{|x-\vec{x}^{\prime}|}d^{3}x^{\prime},\quad\nabla^{2}{\cal
T}=-4\pi\rho^{2}\,.$ (51)

With the above, we conclude the expansion of the separable $f(R,T)$ gravity as
a function of the PPN potentials. In the next section, we infer the values of
the PPN parameters and compare them with corresponding observational values.

##  V The PPN parameters in $f(R,T)$ gravity

The standard Will-Nordvedt PPN formalism [3, 22] does not include the last
term in Eq. (50). However, the realization
$\varphi_{2}(T)=\sigma_{0}+\sigma_{1}T+\sigma_{2}T^{2}$ does not keep null the
four-divergence of the field equations (22) due to the term $\propto T^{2}$.
Then, we assume

$\sigma_{2}=0.$ (52)

Before we proceed, a cautionary remark seems mandatory. The choice
$\sigma_{2}=0$ is not mandatory. For example, one can carry on the PPN
analysis keeping a nonvanishing $\sigma_{2}$ term. This is a similar to the
approach of Ref. [24] where the PPN parameters in the Palatini $f(R)$ is
performed. However, we have checked this would lead to a nontrivial
interpretation of the potentials and their connection with the PPN parameters.

With this, and the results obtained in the previous section, the metric up to
the first post-Newtonian order can be written as

$\displaystyle
g_{00}=2U-2U^{2}+\left(4+\frac{\sigma_{1}}{2\pi}\right)\phi_{1}+\left(4+\frac{2\sigma_{1}}{\pi}\right)\phi_{2}+2\phi_{3}+\left(6+\frac{\sigma_{1}}{\pi}\right)\phi_{4}\,,$
(53) $\displaystyle
g_{0j}=-\left(\frac{7}{2}+\frac{\sigma_{1}}{2\pi}\right)V_{j}-\frac{1}{2}W_{j}\,,$
(54) $\displaystyle
g_{ij}=\left[1+2\left(1+\frac{\sigma_{1}}{4\pi}\right)U\right]\delta_{ij}\,.$
(55)

To extract the PPN parameters from the above metric components we compare it
to the Will-Nordvedt general post-Newtonian metric [3]

$\displaystyle g_{00}=$ $\displaystyle-1+2U-2\beta
U^{2}+(2\gamma+2+\alpha_{3}+\zeta_{1}-2\xi)\phi_{1}+2(3\gamma-2\beta+1+\zeta_{2}+\xi)\phi_{2}$
$\displaystyle+2(1+\zeta_{3})\phi_{3}\
+2(3\gamma+3\zeta_{4}-2\xi)\phi_{4}-(\zeta_{1}-2\xi){\cal A}-2\xi\phi_{w},$
(56) $\displaystyle g_{0i}=$
$\displaystyle-\frac{1}{2}(4\gamma+3+\alpha_{1}-\alpha_{2}+\zeta_{1}-2\xi)V_{i}-\frac{1}{2}(1+\alpha_{2}-\zeta_{1}+2\xi)W_{i}\,,$
(57) $\displaystyle g_{ij}=$ $\displaystyle\ (1+2\gamma\,U)\,\delta_{ij}\,.$
(58)

From this comparison we obtain the following constraints

$\displaystyle\beta=1,\quad\gamma=1+\frac{\sigma_{1}}{4\pi},\quad\alpha_{1}=\alpha_{2}=\alpha_{3}=0,\quad\xi=0,$
(59)
$\displaystyle\zeta_{1}=0,\quad\zeta_{2}=\frac{\sigma_{1}}{4\pi},\quad\zeta_{3}=0,\quad\zeta_{4}=-\frac{\sigma_{1}}{2\pi}.$
(60)

The observational bounds for all PPN parameters are shown in Table 1. The
strongest limit for $\sigma_{1}$ comes from the parameter $\gamma$, which is
related to how much curvature is produced per unit rest mass, and states that

$|\sigma_{1}|<2.9\times 10^{-4}.$ (61)

Since all parameters $\alpha_{i}$ are null, the model does not predict
preferred-frame effects. On the other hand, nonvanishing values for the
parameters $\zeta_{2}$ and $\zeta_{4}$ indicate a violation of the total
momentum conservation. This classifies the model as semiconservative [22].

There are well known examples of theories that come from an action and have
$\alpha_{1}$ and $\alpha_{2}$ different from zero, but theories with an action
are not expected to yield nonzero values for any of the $\zeta$’s and
$\alpha_{3}$ if $\xi=0$ [25]. A further example of a theory with this
characteristic can be seen in [26].

Parameter | Limit | Parameter | Limit
---|---|---|---
$\gamma-1$ | $2.3\times 10^{-5}$ | $\xi$ | $4\times 10^{-9}$
$\beta-1$ | $8\times 10^{-5}$ | $\zeta_{1}$ | $2\times 10^{-2}$
$\alpha_{1}$ | $4\times 10^{-5}$ | $\zeta_{2}$ | $4\times 10^{-5}$
$\alpha_{2}$ | $2\times 10^{-9}$ | $\zeta_{3}$ | $1\times 10^{-8}$
$\alpha_{3}$ | $4\times 10^{-20}$ | $\zeta_{4}$ | —
Table 1:  Limits on the PPN parameters, considering only the strongest limits
for each parameter [22]. The $\zeta_{4}$ does not have a direct measurement.
These limits apply to the absolute value of each parameter.

##  VI Fully conservative model

The PPN analysis shows that, despite implying $\nabla^{\mu}T_{\mu\nu}=0$ at
the field equations level, the model with a linear dependence on $T$ is
severely constrained. One can further analyze this by considering the
invariance of the theory under diffeomorphisms.

First, we can consider that the energy-momentum tensor does not depend on
derivatives of the matter fields with respect to space-time, as is the case
with perfect fluids. Therefore ${\delta f^{\prime}}/{\delta\Psi}=(\partial
f^{\prime}/\partial\Psi^{\prime})\delta^{(4)}(x-x^{\prime})$, and the Eq. (9)
becomes

$\displaystyle\frac{\delta
S_{\mathrm{m}}}{\delta\Psi}=-\frac{1}{2\kappa^{2}}\frac{\partial
T}{\partial\Psi}f_{{}_{T}}\sqrt{-g}.$ (62)

Imposing that the theory is diffeomorphism-invariant [27], one obtains

$\displaystyle\int_{\cal
M}\left[(\nabla^{\mu}T_{\mu\nu})\xi^{\nu}\sqrt{-g}+\frac{\delta
S_{\mathrm{m}}}{\delta\Psi}\delta_{\xi}\Psi\right]d^{4}x=0,$ (63)

where $\delta_{\xi}\Psi=\xi^{\mu}\partial_{\mu}\Psi$ represents an
infinitesimal coordinate change given by the Lie derivative along the vector
$\xi^{\mu}$. Plug-in (62) and (63), we gets

$\displaystyle\nabla^{\nu}T_{\mu\nu}=\frac{1}{2\kappa^{2}}f_{{}_{T}}\nabla_{\mu}T.$
(64)

Therefore, any realization for $f(R,T)$ that leads to
$\nabla_{\mu}T^{\mu\nu}=0$ must be such that

$f_{{}_{T}}(R,T)=0.$ (65)

Then, $\varphi$ constant is the only solution that simultaneously satisfies
(15), (16) and (64) (with $\nabla_{\mu}T^{\mu\nu}=0$).

##  VII Final discussion

Modified gravity theories can display many distinct new geometrical structures
such as, e.g., violation of the usual energy-momentum conservation. This is
particularly observed in the cosmological context where the nonconservation
leads to a clear signature on the evolution of the pressureless (dark) matter
component, the main ingredient for structure formation. One of the most known
example of that is the class of $f(R,T)=f_{1}(R)+f_{2}(T)$ theories where $T$
is the trace of the energy momentum tensor. For this theory, assuming a power
law dependence $f_{2}(T)\propto T^{n}$, the standard cosmological conservation
law is recovered when $n=1/2$. We have shown however this is a sub case of a
more general notion of conservation in $f(R,T)$ gravity. Only in the expanding
cosmological background the solution $f_{2}(T)\propto T^{1/2}$ can be seen as
the full conservative one.

We then proceed to search for a general form of the $f(R,T)$ function that
maintains conservation at the field equations level. Our investigation
initially leads us to discover the function $f_{2}(T)=\sigma_{0}+\sigma_{1}T$.
Subsequently, we focus on evaluating this function using solar system data via
the Post-Newtonian formalism. As a result, the parameter $\sigma_{1}$ is found
to be considerably limited, as shown in (61). Therefore, we deduce that only
the inclusion of a trivial constant term in the gravitational Lagrangian is
compatible with both the complete conservation of the energy momentum tensor
and solar system tests. This outcome poses a significant challenge to the $T$
dependence of modified gravity theories in practice. Nonetheless, it cannot be
contended that this rules out this category of theories since it is possible
that the $T$ dependence works on cosmological scales while an auxiliary
screening mechanism comes into play to resolve the solar system tests.

It is also important to mention the discussion on possible approaches to
$f(R,T)$ gravity with minimal coupling that can be followed in [28, 29, 30].
One can modify the Lagrangian of $f(R,T)$ theories by including Lagrange
multipliers that lead to $\nabla_{\mu}T^{\mu\nu}=0$ at the field equations
level, as in [28]. On the other hand, one can also investigate which
particular realization of $f(R,T)$ agrees with $\nabla_{\mu}T^{\mu\nu}=0$ by
keeping the standard gravitational Lagrangian as in this reference and also as
argued by Harko and Moraes [29] (see also [17, 18]). In any case, the
fundamental thermodynamical variables of matter ($\rho$ and $p$) are
introduced in $S_{m}$ and are the same that also appear in the gravitational
part of the total Lagrangian, through $T$. Considering the model with minimal
coupling, $f(R,T)=f_{1}(R)+f_{2}(T)$, including $f_{2}(T)$ in the matter
sector is indeed a technically possible approach, but the new effective
thermodynamical variables appearing in the effective matter Lagrangia will
have a different meaning comparing to those included in $S_{m}$, and can be
distinguished from the fundamental variables. Imposing
$\nabla_{\mu}T^{\mu\nu}=0$ we conclude that the cosmological constant is the
favored result, and is part of $f_{2}(T)$. Including the cosmological constant
in the energy density or in the geometric part is a choice, but in any case
its value can be unequivocally determined by its physical effects. Therefore,
we emphasize that the search for the functional form of $f_{2}(T)$ and the
observational constraints is valid, contrarily to argued in Ref. [28, 30] and
that the result using the PPN analysis is consistent with the imposition set
by Eq. (64), for the case of minimal coupling.

###### Acknowledgements.

The authors thank FAPEMIG/FAPES/CNPq/CAPES for financial support. We thank
Junior Toniato for pointing out an important issue on the PPN analysis and
Davi Cabral Rodrigues for useful remarks.

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