Addressing the missing matter problem in galaxies through a new fundamental gravitational radius

TL;DR

This paper proposes a new gravitational radius derived from Noether symmetry in $f(R)$ gravity theories, which can explain galactic rotation curves and the baryonic Tully-Fisher relation without dark matter.

Contribution

It introduces a novel gravitational radius from $f(R)$ gravity symmetry, providing an alternative explanation for galactic dynamics.

Findings

Derivation of a new gravitational radius from Noether symmetry.

Explanation of galactic rotation curves without dark matter.

Reproduction of the baryonic Tully-Fisher relation.

Abstract

We demonstrate that the existence of a Noether symmetry in $f(R)$ theories of gravity gives rise to a further gravitational radius, besides the standard Schwarzschild one, determining the dynamics at galactic scales. By this feature, it is possible to explain the baryonic Tully-Fisher relation and the rotation curve of gas-rich galaxies without the dark matter hypothesis.

Tables

Table 1. Table 1: The data from the Baryonic Tully–Fisher relation of gas rich galaxies as a test of Λ Λ \Lambda CDM and MOND mcga11a : D 𝐷 D - distance of the galaxy, V c subscript 𝑉 𝑐 V_{c} - rotational velocity, M ∗ M* - mass of the stars, M ​ g 𝑀 𝑔 Mg - mass of the gas.

Galaxy	$D$	$V_c$	log$M\ast$	log$Mg$
	($Mpc$)	($km/s$)	($Msun$)	($Msun$)
DDO210	0.94	17 $\pm$ 4	5.88 $\pm$ 0.15	6.64 $\pm$ 0.20
CamB	3.34	20 $\pm$ 12	6.99 $\pm$ 0.15	7.33 $\pm$ 0.20
UGC8215	4.5	20 $\pm$ 6	6.81 $\pm$ 0.15	7.45 $\pm$ 0.20
DDO183	3.24	25 $\pm$ 3	7.24 $\pm$ 0.15	7.54 $\pm$ 0.20
UGC8833	3.2	27 $\pm$ 4	6.94 $\pm$ 0.15	7.30 $\pm$ 0.20
D564-8	6.5	29 $\pm$ 5	6.76 $\pm$ 0.20	7.32 $\pm$ 0.13
DDO181	3.1	30 $\pm$ 6	7.26 $\pm$ 0.15	7.56 $\pm$ 0.20
P51659	3.6	31 $\pm$ 4	6.67 $\pm$ 0.15	7.85 $\pm$ 0.20
KK9824	7.83	35 $\pm$ 6	7.72 $\pm$ 0.15	7.93 $\pm$ 0.20
UGCA92	3.01	37 $\pm$ 4	7.78 $\pm$ 0.15	8.32 $\pm$ 0.20
D512-2	14.1	37 $\pm$ 7	7.58 $\pm$ 0.20	7.96 $\pm$ 0.06
UGCA444	0.95	38 $\pm$ 5	7.34 $\pm$ 0.15	7.75 $\pm$ 0.20
KK98251	5.6	38 $\pm$ 5	7.34 $\pm$ 0.15	8.02 $\pm$ 0.20
UGC7242	5.4	40 $\pm$ 4	7.57 $\pm$ 0.15	7.78 $\pm$ 0.20
UGC6145	7.4	41 $\pm$ 4	7.20 $\pm$ 0.15	7.56 $\pm$ 0.20
NGC3741	3.0	44 $\pm$ 3	7.24 $\pm$ 0.15	8.45 $\pm$ 0.20
D500-3	18.5	45 $\pm$ 6	6.97 $\pm$ 0.20	7.94 $\pm$ 0.05
D631-7	5.5	53 $\pm$ 5	6.88 $\pm$ 0.20	8.29 $\pm$ 0.15
DDO168	4.3	54 $\pm$ 3	8.07 $\pm$ 0.15	8.74 $\pm$ 0.20
KKH11	3.0	56 $\pm$ 5	7.28 $\pm$ 0.15	7.85 $\pm$ 0.20
UGC8550	5.1	58 $\pm$ 3	8.25 $\pm$ 0.37	8.46 $\pm$ 0.39
D575-2	12.2	59 $\pm$ 7	7.63 $\pm$ 0.20	8.62 $\pm$ 0.07
UGC4115	7.5	59 $\pm$ 6	7.77 $\pm$ 0.15	8.58 $\pm$ 0.20
UGC3851	3.2	60 $\pm$ 5	8.45 $\pm$ 0.15	9.09 $\pm$ 0.20
UGC9211	12.6	64 $\pm$ 5	8.12 $\pm$ 0.39	9.21 $\pm$ 0.41
NGC3109	1.3	66 $\pm$ 3	7.41 $\pm$ 0.15	8.79 $\pm$ 0.20
UGC8055	17.4	66 $\pm$ 7	8.09 $\pm$ 0.15	9.02 $\pm$ 0.20
D500-2	17.9	68 $\pm$ 7	7.41 $\pm$ 0.20	9.06 $\pm$ 0.05
IC2574	4.0	68 $\pm$ 5	8.94 $\pm$ 0.15	9.20 $\pm$ 0.20
UGC6818	18.6	72 $\pm$ 6	9.22 $\pm$ 0.16	9.28 $\pm$ 0.20
UGC4499	13.0	74 $\pm$ 3	8.75 $\pm$ 0.27	9.32 $\pm$ 0.29
NGC1560	3.45	77 $\pm$ 3	8.70 $\pm$ 0.15	9.23 $\pm$ 0.20
UGC8490	4.65	78 $\pm$ 3	8.36 $\pm$ 0.15	8.96 $\pm$ 0.20
UGC5721	6.5	79 $\pm$ 3	8.17 $\pm$ 0.37	9.05 $\pm$ 0.39
F565-V2	48.	83 $\pm$ 8	8.30 $\pm$ 0.21	9.04 $\pm$ 0.24
F571-V1	79.	83 $\pm$ 5	9.00 $\pm$ 0.19	9.33 $\pm$ 0.22
IC2233	10.4	84 $\pm$ 5	8.96 $\pm$ 0.15	9.32 $\pm$ 0.20
NGC2915	3.78	84 $\pm$ 10	7.99 $\pm$ 0.15	8.78 $\pm$ 0.20
NGC5585	5.7	90 $\pm$ 3	8.98 $\pm$ 0.38	9.27 $\pm$ 0.40
UGC3711	7.9	95 $\pm$ 3	8.92 $\pm$ 0.15	9.01 $\pm$ 0.17
UGC6983	18.6	108 $\pm$ 3	9.53 $\pm$ 0.16	9.74 $\pm$ 0.20
F563-V2	61.	111 $\pm$ 5	9.41 $\pm$ 0.17	9.63 $\pm$ 0.21
F568-1	85.	118 $\pm$ 4	9.50 $\pm$ 0.18	9.87 $\pm$ 0.22
F568-3	77.	120 $\pm$ 6	9.62 $\pm$ 0.18	9.71 $\pm$ 0.22
F568-V1	80.	124 $\pm$ 5	9.38 $\pm$ 0.18	9.65 $\pm$ 0.22
NGC2403	3.18	134 $\pm$ 3	9.61 $\pm$ 0.15	9.77 $\pm$ 0.20
NGC3198	14.5	149 $\pm$ 3	10.12 $\pm$ 0.15	10.29 $\pm$ 0.20

Equations

$$A=\int d^{4}x\sqrt{-g}\left[f\left(R\right)+{\mathcal{L}}_m\right],$$

$$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R$$

$$-$$

$$f(R)=f_0{R}^n$$

$$d{s}^2=A(r)d{t}^2-B(r)d{r}^2-r^{2}d{\Omega }^2$$

$$f^{\prime }(R)\left(3\frac{R_{00}}{g_{00}}-R\right)+\frac{1}{2}f(R)-3\frac{f^{\prime }(R{)}_{;00}}{g_{00}}=0.$$

$$R_{00}(r)=\frac{2n-1}{6n}A(r)R(r)-\frac{n-1}{2B(r)}\frac{dA(r)}{dr}\frac{d\ln R(r)}{dr},$$

$$\square R^{n-1}(r)=\frac{2-n}{3n}{R}^n(r).$$

$$A(r)=\frac{1}{B(r)}=1+\frac{2\Phi (r)}{c^2}$$

$$\Phi (r)=-\frac{Gm}{2r}\left[1+{\left(\frac{r}{r_c}\right)}^{\beta }\right]$$

$$\beta =\frac{12n^2-7n-1-\sqrt{36n^4+12n^3-83n^2+50n+1}}{6n^2-4n+2}.$$

$$f(R)\propto R^{1+ϵ}.$$

$$R^{1+ϵ}$$

$$v_c^2(r)=\frac{GM}{2r}\left[1+(1-\beta ){\left(\frac{r}{r_c}\right)}^{\beta }\right].$$

$${\nabla }^2\Phi -\frac{R^{(2)}}{2}-f^{\prime \prime }(0){\nabla }^2{R}^{(2)}=\mathcal{X}\rho ,$$

$$-3f^{\prime \prime }(0){\nabla }^2{R}^{(2)}-R^{(2)}=\mathcal{X}\rho ,$$

$$f^{\prime }(\chi )\chi -2f(\chi )+3L_M^2\Delta f^{\prime }(\chi )=\frac{8\pi G{L}_M^2}{c^4}T,$$

$${\chi }^n\left(n-2\right)-3n{L}_M^2\frac{{\chi }^{(n-1)}}{r^2}\approx \frac{8\pi GM{L}_M^2}{c^2{r}^3}.$$

$$R{r}^2\le \frac{3n}{2-n}.$$

$$R_c>>r,$$

$$R^{(n-1)}\approx -\frac{8\pi GM}{3n{c}^{2}r{L}_M^{2\left(n-1\right)}}.$$

$$R=-\frac{2}{c^2}{\nabla }^2\Phi =\frac{2}{c^2}\nabla \cdot \mathbf{a}$$

$$a$$

$$a\approx -\frac{{\left(a_{0}GM\right)}^{1/2}}{r},$$

$$r_c=\sqrt{\frac{GM}{a_0}}.$$

$$r_f=\frac{\sqrt{a_{0}GM}}{a_f}=\frac{a_0}{a_f}{r}_c.$$

$$M=\frac{4a_0{v}_f^4}{G{a}_f^2{\left[1+(1-\beta ){\left(\frac{a_0}{a_f}\right)}^{\beta }\right]}^2}.$$

$$M=\frac{4a{v}_f^4}{G{a}_f^2{\left[1+(1-\beta ){\left(\frac{a}{a_f}\right)}^{\beta }\right]}^2}.$$

$$r_f\approx \frac{r_c}{2}.$$

$$\mathrm{d}{s}^2=A(r)c^2\mathrm{d}{t}^2-B(r)\mathrm{d}{r}^2-C(r)\mathrm{d}{\Omega }^2,$$

$$A=-\frac{c^3}{16\pi G{L}_M^2}\int \left[f(\chi )-\lambda (\chi -\bar{\chi })\right]\sqrt{-g}{\mathrm{d}}^{4}x.$$

Full text

Modified gravity; galactic dynamics; dark matter; Noether’s symmetries.

††preprint: APS/123-QED

I Introduction

The standard $\Lambda\mathrm{CDM}$ cosmological model predicts that about 70% of ”mass-energy budget” of the Universe is composed of dark energy ($\Omega_{\Lambda}\approx 0.69$), about 30% is the total mass of the Universe ($\Omega_{m}\approx 0.31$) which is dominated with more than 5/6 by dark matter ($\Omega_{c}\approx 0.26$), while ordinary baryonic matter constitutes only less than 1/6 ($\Omega_{b}\approx 0.05$) of the total mass (Planck Collaboration, Ade et al. (2016)). Flat rotation curves of spiral galaxies, together with discrepancy between the observed velocity dispersion and mass density of luminous matter in the clusters of galaxies, are usually assumed as the most significant observational evidences for the existence of dark mater (see e.g de Swart et al. (2017) and references therein). In order to account for the flat rotation curves of spiral galaxies, dark matter should dominate the total mass of the galaxy and should be concentrated in the outer baryonic regions of galactic disks, as well as in the surrounding haloes. Thus, according to $\Lambda\mathrm{CDM}$, the baryonic matter in spiral galaxies is mixed with and embedded in non-baryonic dark matter, causing rotation velocities of the visible matter within the disk that are constant or increasing with disk radius. However, very recently Genzel et al. (2017) studied the rotation curves in a sample of massive star-forming galaxies at the peak of cosmic galaxy formation 10 billion years ago, and found that in the case of 6 such galaxies at redshifts $z\sim 0.9-2.4$ the rotation velocities are not constant, but decrease with radius. Besides, the dark matter fractions near the half-light radius for all galaxies were modest or negligible. This indicated that a large fraction of the massive high-redshift galaxies were strongly baryon-dominated, with much less dark matter than in the local Universe, as well as that such baryon domination increased with redshift (Genzel et al. (2017)). These results are in discrepancy with $\Lambda\mathrm{CDM}$ predictions obtained by cosmological galaxy formation simulations, according to which dark matter should play a key role in formation of the galaxies and their clusters because, in a $\Lambda\mathrm{CDM}$ Universe, dark matter first clumps and forms haloes and then galaxies form at the centres of these dark matter haloes (see e.g. Springel et al. (2006)).

On the other hand, weak lensing observations of merging galaxy cluster 1E 0657-558 (the Bullet Cluster), were claimed to represent the first direct detection of dark matter (Clowe et al. (2006)). Such claim was based on the inferred spatial offset of the center of the total mass of this cluster from the center of its baryonic mass peaks, which allegedly could be explained only by cold dark matter hypothesis. However soon afterwards, Mahdavi et al. (2007) performed a weak lensing study of the cluster merger known as Abell 520 (the Train Wreck Cluster) and found a massive dark core, i.e. a mass peak without galaxies which coincided with the gas. This could not be satisfactorily explained within the current collisionless dark matter paradigm since dark matter, being non-interacting, is expected to always produce a peak offset in the maps obtained by weak lensing mass reconstruction. Therefore, in a sense, this was a counterexample to the Bullet Cluster and represented a challenge to cold dark matter hypothesis which is one of the cornerstones of $\Lambda\mathrm{CDM}$. There was an attempt to dispute the existence of this dark core (Clowe et al. (2012)), but more recent detailed observational studies of Abell 520 by Jee et al. (2012, 2014) confirmed its existence. Moreover, Lee & Komatsu (2010) demonstrated that the Bullet Cluster, initially considered as an evidence, actually represents a problem for $\Lambda\mathrm{CDM}$. Namely, high initial infall velocities of subclusters just before their collision, which are required to explain the observed physical and morphological properties of the Bullet Cluster, are incompatible with the much lower infall velocities predicted by $\Lambda\mathrm{CDM}$ (see Lee & Komatsu (2010) and references therein).

As a consequence, these results indicated that either some further constraints on dark matter interaction should be set, such as the non-gravitational forces acting on dark matter (see e.g. Harvey et al. (2015)), or an alternative explanation by modified gravity and without dark matter should be preferred.

Regarding the latter approach it was shown that MOND is able to successfully explain the dynamics of galaxies outside of galaxy clusters and a recently discovered tight relation between the radial acceleration inferred from their observed rotation curves and the acceleration due to the baryonic components of their disks (McGaugh et al. (2016)). In the case of the Bullet Cluster and other galaxy clusters, MOND can reduce the amount of unseen matter to a factor of two, leaving the possibility that it could be non-luminous ordinary baryonic matter, rather than cold dark matter (see e.g. Famaey & McGaugh (2012)). It was also shown that TeVeS, the relativistic version of MOND, is able to resemble the weak lensing density maps of the Bullet Cluster by a multicentred baryonic system (Angus et al. (2006)). Some other theories of modified gravity which give the weak field point particle gravitational potential with Yukawa correction term are also able to explain both flat rotation curves and the Bullet Cluster (see e.g. Brownstein & Moffat (2007)). Recently, Israel & Moffat (2016) showed that generalized Scalar-Tensor-Vector-Gravity (STVG) theory can also explain dynamics of both merging galaxy clusters, Abell 520 and the Bullet Cluster, without dark matter. This approach, however, requires modification of General Relativity and derivation of alternative field equations, as well as their cosmological reformulation.

In addition to the above arguments, it is well known that $f(R)$ theories of gravity gives rise, besides the standard Schwarzschild radius, to an additional ”characteristic length” salv11 ; annalen ; odintsov . Such a feature has a fundamental nature and is related to the further degrees of freedom emerging as soon as $f(R)\neq R$, where $R$ is the Ricci curvature scalar starobinsky . From the conformal transformation point of view, being $f(R)$ gravity equivalent to a scalar-tensor theory, this new gravitational length is related to the effective mass emerging from the potential describing the dynamics of the scalar field. The paradigm is that being General Relativity (GR) a second-order theory in metric description, one has only a characteristic gravitational radius (the Schwarzschild radius). $f(R)$ gravity, being a fourth-order theory, has two characteristic gravitational radii and, in general, any theory of gravity of order $2N$ has $N$ characteristic radii that emerge in the weak field limit schmidt .

These features can be of extreme interest in order to fix the size and the morphology of self-gravitating systems: in particular, the rotation curves of galaxies capo07 ; frig07 ; salucci . The further gravitational radius emerges in any $f(R)$ model and, in particular, is related to the conserved quantities derived from the Noether symmetries in any power-law function of $R$. As shown in arturo ; bern11 , for $f(R)\propto R^{n}$, where $n\in\mathbb{R}$, being $\mathbb{R}$ the set of real numbers, a Noether symmetry $\Sigma_{0}$ exists for any $n$. Specifically, $\displaystyle{\Sigma_{0}\equiv\frac{2GM}{c^{2}}}$, i.e. the Schwarzschild radius, is obtained for $n=1$, as it has to be in GR. For $n\neq 1$, a further characteristic length emerges besides the Schwarzschild radius (see arturo for details).

Here, we want to show that this new characteristic radius can account for the baryonic Tully–Fisher (BTF) relation of gas-rich galaxies. In this perspective, the dark matter hypothesis is not necessary for the stability and the dynamics of such systems. In other words, galactic self-gravitating systems could be explained, in principle, only by gravitational interaction and baryons.

The paper is organized as follows: in Sec. II we give the basic ingredients of $f(R)$ gravity; in the Sec. III, the observed BTF relation of gas-rich galaxies is derived in the framework of $f(R)$ gravity; we discussed the new gravitational radius depending on the mass of the system. Such an effective radius is straightforwardly related to the Modified Newtonian Dynamics (MOND) acceleration constant $a_{0}$ milg83 . Discussion and conclusions are drawn in Sec. IV. In Appendix A, we sketch the Noether Symmetry Approach by which it is possible to derive the further gravitational radius.

II Theory

Let us start from the action

[TABLE]

where $f(R)$ is a generic function of the Ricci curvature scalar and ${\cal L}_{m}$ is the standard matter Lagrangian. The field equations are

[TABLE]

in a Einstein-like form. Primes denote derivative with respect to $R$. The terms ${f^{\prime}(R)}_{;\mu\nu}$ and $\Box{f^{\prime}(R)}$ give rise to fourth order derivatives in $g_{\mu\nu}$. Let us consider the power - law case

[TABLE]

with $f_{0}$ a dimensional constant. Taking into account the gravitational field generated by a pointlike source and solving the field equations (2) in the vacuum case, we write the metric as :

[TABLE]

where $d\Omega^{2}=d\theta^{2}+\sin^{2}{\theta}d\varphi^{2}$ is the spherical line element (see arturo2 ; diego for a discussion). Combining the $00$ - vacuum component and the trace of the field equations (2) in absence of matter, we get the equation :

[TABLE]

Using Eq.(3), Eq.(5) reduces to :

[TABLE]

and the trace equation reads :

[TABLE]

For $n=1$, Eq.(7) reduces to $R=0$, which, inserted into Eq.(6), gives $R_{00}=0$ and then the standard Schwarzschild solution is recovered. Expressing $R_{00}$ and $R$ in terms of the metric (4), Eqs.(6) and (7) become a system of differential equations for $A(r)$ and $B(r)$. A physically motivated hypothesis is assuming

[TABLE]

with $\Phi(r)$ the gravitational potential generated by a pointlike mass $m$ at the distance $r$. A general solution is capo06 ; zakh06 :

[TABLE]

where $r_{c}$ is an arbitrary parameter, depending on the typical scale of the considered system and $\beta$ is a universal parameter:

[TABLE]

Clearly the gravitational potential deviates from the Newtonian one because of the presence of the second term on the right hand side. For $\beta=0$, it is $n=1$ and the Newtonian potential is fully recovered: the metric reduces to the Schwarzschild one and the weak field limit of GR is fully restored.

A comment is necessary at this point. The parameter $r_{c}$ is an integration constant depending on $\beta$. As shown in arturo , it is related to the fact that any $f(R)$ power-law model possesses a Noether symmetry. In other words, it is a fundamental radius because strictly depends on the existence of the Noether symmetry. In this perspective, it is a further gravitational radius besides the Schwarzschild one and depends on the mass and the size of the self gravitating system. Being this a central concept in this paper, details on the emergence of this further gravitational radius from Noether’s symmetries are given in Appendix A.

The other important point is related to the choice of the power-law action (3) that could appear non-natural in order to discuss deviations with respect to GR. Being $n$ any real number, it is always possible to recast the $f(R)$ power-law function as

[TABLE]

If we assume small deviation with respect to GR, that is $|\epsilon|\ll 1$, it is possible to re-write a first-order Taylor expansion as

[TABLE]

In this way, one can control the magnitude of the corrections with respect to the Einstein gravity. Lagrangian (12) has been investigated from Solar System up to cosmological scales. In particular, applications to gravitational waves maurofelix , binary star systems mnrasfelix , and neutron stars have been investigated farinelli . In the present paper, we consider the general case $n\in\mathbb{R}$ because, as shown in Appendix A, the Noether symmetry exists for any $n$ and large deviations from GR seem consistent with galactic dynamics as we will show below.

III Observational constraints for $r_{c}$ from BTF relation and circular velocity

Starting from the above solution (9), an excellent agreement between theoretical and observed rotation curves of low surface brightness galaxies has been obtained for $\beta$ = 0.817 (see capo07 ). A similar result has been achieved by Frigerio Martins and Salucci frig07 considering an enlarged sample of galaxies with respect to capo07 . Both these results can be framed into the BTF relation with the aim to show that the new fundamental gravitational radius $r_{c}$ can account for missing matter in galaxies.

Specifically, the empirical BTF relation is a universal relationship between the baryonic mass of a galaxy and its rotational velocity of the form $M_{b}\varpropto v_{c}^{4}$, which follows from the fact that luminosity $L$ traces baryonic mass $M_{b}$ through the mass-to-light ratio $\Upsilon$. The BTF relation can be recovered from $f(R)$ power-law gravity capo06 ; zakh06 ; zakh07 ; bork12 , but similar considerations can be developed for any $f(R)$ gravity model. As shown in capo07 , the BTF relation perfectly holds for low-surface brightness galaxies for $\beta$ = 0.817.

Before considering the general discussion of BTF relation, we have to say that the Gauss theorem does not hold for $f(R)$ gravity because they predict the scaling of gravitational force other than $1/r^{2}$, as it is in the Newtonian case capo07 . But nevertheless, the Bianchi identities hold so the fundamental conservation law of any theory of gravity is guaranteed. Furthermore, according to the results below, it seems that this violation, in the case of BTF relation, has no dramatic consequence. Therefore, we will study the simplest point-like case where the modified gravitational potential $\Phi(r)$ generated by a point-like mass $M$ at the distance $r$ is given by Eq. 9.

Circular velocity of a point mass, in the $R^{n}$ gravity potential (9), can be found in the standard way, that is $v^{2}_{c}(r)=r\dfrac{d\Phi}{dr}$, which gives binney :

[TABLE]

For a detailed explanation of the relation for circular velocity $v_{c}$, see capo07 . There, it is described the low energy limit of power-law $f(R)$ models. We emphasize that the circular velocity definition that we are using here can be assumed as the proper one. This point needs further explanation.

As shown in salv12 , considering the Newtonian limit of $f(R)$ gravity and discarding higher order terms than $\mathcal{O}(2)$, the field equations for a perfect-fluid energy-momentum tensor of dust ($p=0$) become:

[TABLE]

where $\rho$ is the mass density, $\mathcal{X}=\frac{8\pi G}{c^{4}}$ is the gravitational coupling, and $R^{(2)}$ is the Ricci scalar assumed up to the second order approximation. Due to the presence of $R^{(2)},$ these equations are of fourth-order. For $f^{\prime\prime}(0)\,=\,0$, the standard second-order Poisson equation. $\nabla^{2}\Phi\,=\,4\pi G\rho$, is recovered and Eq.(15) becomes the trace of standard GR in the weak field limit.

In general, the Poisson equation is different for $f(R)\neq R$, however, it is important to stress that the potential emerging as a solution, in our case that in Eq.(9), must satisfy the geodesic equation that has the same form both in GR and in $f(R)$ gravity. Furthermore, the gravitational field is conservative (and irrotational) in both cases, then the equation $\bm{a}=-\nabla{\Phi}$, where ${\bm{a}}$ is the acceleration and ${\Phi}$ is the potential, must hold for the standard Newtonian potential or other potentials derived in the Newtonian limit salv12 . This means that the circular velocity $v_{c}$, in the non-relativistic limit, can be obtained by $v^{2}_{c}(r)=r\dfrac{d\Phi}{dr}$. Eq. (13) holds for $f(R)\propto R^{n}$.

Finally, the derivation of Poisson equation, for general $f(R)$ gravity, is given by Zhao et al. zhao11 .

Let us now consider the fact that MOND can be recovered from $f(R)$ gravity in order to compare the predictions for BTF relation. In other words, it is possible to show that MOND is not only a phenomenological approach but it can be recovered as the weak field limit of Extended Theories of Gravity salv11 . Since this is a vital concept for the present paper, we will present a summary of the derivation reported in Ref. bern11 . We will proceed step by step to demonstrate this basic concept for our approach.

As reported in Appendix A, the Noether symmetries select a power-law for $f(R)$ gravity. This is the only general form of $f(R)$ function presenting symmetries. In particular, we assume $f(\chi)=\chi^{n}$, after introducing the dimensionless quantity $\chi:=L_{M}^{2}R$, where $R$ is the Ricci scalar, $L_{M}$ is the length fixed by the parameters of the theory, and $n$ any real number.

• -

The trace of field Eqs. (2) can be rewritten as

[TABLE]

with the trace of the energy-momentum $T:=T^{\alpha}_{\alpha}$. By substituting the power-law, it becomes:

[TABLE]

Here, we are assuming the weak field approximation with $d/d\chi\sim 1/\chi$, $\Delta\sim-1/r^{2}$, and matter density $\rho\sim M/r^{3}$. The second term in the l.h.s. of Eq.(17) is larger than the first if

[TABLE]

In this approximation, we can assume that the Ricci scalar corresponds to the Gaussian curvature and then $R\simeq R_{c}^{-2}$ where $R_{c}$ is the Gauss curvature radius. Immediately we have

[TABLE]

and then

[TABLE]

• -

At the second order, the Ricci scalar is

[TABLE]

that can be approximated as $R\approx-2\Phi/(c^{2}r^{2})\approx 2a/(c^{2}r)$, with $\Phi$ the gravitational potential and $a$ the acceleration. This gives:

[TABLE]

which converges to a MOND-like acceleration $a\propto 1/r$ if $n-2=-\left(n-1\right)$, that means $n=3/2$.

• -

With this value of $n$, we get the MOND relation

[TABLE]

reported in Ref. bern11 .

In other words, the weak field limit of $f(R)$ power-law gravity gives MOND as a particular case.

According to this derivation, the above characteristic length $r_{c}$ of $R^{n}$ gravity can be related to the MOND acceleration constant $a_{0}$ using the following expression bern11 :

[TABLE]

Assuming that rotation curve (13) is flat within the measurement uncertainties at some finite radius $r_{f}$, i.e. $v_{c}(r_{f})\approx v_{f}$, then $r_{f}$ could be also related to a certain MOND acceleration $a_{f}>a_{0}$ according to bern11 . This gives

[TABLE]

Hence, from (13), (23), and (24), the BTF relation of $R^{n}$ gravity expressed in terms of MOND accelerations is:

[TABLE]

We should point out that in the case of BTFR for spiral galaxies, McGaugh mcga11b has shown that instead of standard MOND acceleration constant $a_{0}$ one should use a slightly different, empirically calibrated constant $a$ (where $a_{0}=0.8a$), while the formula is unchanged. Therefore, for our calculations we use the following expression:

[TABLE]

As it can be seen from eq. (26), the slope of BTF relation in $R^{n}$ gravity is 4, which is the same as in the case of MOND: $M_{b}=Av_{f}^{4}$ where $A=47\pm 6\;\mathrm{M}_{\odot}\,\mathrm{km}^{-4}\,\mathrm{s}^{4}$. Fig. 1 shows the region in $(a_{f},\;\beta)$ parameter space where $41\leq A\leq 53\;\mathrm{M}_{\odot}\,\mathrm{km}^{-4}\,\mathrm{s}^{4}$ according to Eq. (26). As it can be seen from Fig. 1, $R^{n}$ gravity universal constant $\beta$ is correlated with MOND acceleration $a_{f}>a_{0}$.

In order to determine the best fit normalization, we adopted the following values of universal constants in these two theories: $\beta=0.817$, obtained by fitting the Type Ia Supernovae Hubble diagram capo07 and $a_{0}=1.24\times 10^{-13}$ km s*-2* mcga11a , and then we fitted eq. (26) to the observations of gas-rich galaxies given in Table 1 of mcga12 , considering MOND acceleration $a_{f}$ as a free parameter. The fit resulted with the following value of MOND acceleration: $a_{f}(\beta=0.817)=(2.56\pm 0.06)\times 10^{-13}$ km s*-2*, which yields the normalization of $51.4\;\mathrm{M}_{\odot}\,\mathrm{km}^{-4}\,\mathrm{s}^{4}$, being within the error interval of the corresponding MOND estimate. Except value $n$ = 3.5 ($\beta$ = 0.817) for $R^{n}$ gravity we also investigated values of $n$ = 1.5 and 2 corresponding $\beta$ are 0.518 and 0.667, respectively. In the same way like in the case of $\beta$ = 0.817, in order to determine the best fit normalization, we proceed with calculation. The fit resulted with the following value of MOND accelerations for $\beta$ = 0.518: $a_{f}(\beta=0.518)=(2.02\pm 0.05)\times 10^{-13}$ kms*-2*, and for $\beta$ = 0.667: $a_{f}(\beta=0.667)=(2.30\pm 0.06)\times 10^{-13}$ kms*-2*, respectively.

A graphical comparison between the best fit BTF relations in $R^{n}$ gravity, MOND and $\Lambda$CDM is presented in Fig. 2. The data (open circles) are for a sample of galaxies from mcga11a , and to make it more convenient for the readers, in Table 1 we present data for gas rich galaxies that we used. MOND and $\Lambda$CDM lines are obtained using the relations given in mcga11a . From Fig. 2, it can be seen that almost a perfect alignment is obtained between MOND and $R^{n}$ gravity predictions ($R^{n}$ gravity for values of $n$ = 1.5, 2 and 3.5 corresponding $\beta$ are 0.518, 0.667 and 0.817, respectively). On the contrary, $\Lambda$CDM fit is in unsatisfactory agreement with observations. We can see that the range for $\beta$ from 0.5 to 0.8 (corresponding to range of $n$ from 1.5 to 3.5) is in a good agreement with observations. For Fig. 2, we used data for a sample of galaxies which is used in mcga11a , which we designate with open circles. We want to stress that these values give two constraints on parameter $\beta$ obtained from fitting observations using $R^{n}$ gravity, from the Fundamental Plane (FP) bork16 and from MOND bern11 ; mendoza1 ; mendoza2 ; hidalgo ; sobouti .

By substituting the best fit value for $a_{f}$ in eq. (24), one obtains:

[TABLE]

The fact that the radii $r_{f}$ of all galaxies are constant when expressed in units of the characteristic length $r_{c}$ demonstrates the fundamental nature of $r_{c}$. Moreover, it is possible to make an analogy between this situation and, for example, the radius of innermost stable circular orbit around a Schwarzschild black hole which is always equal to 3 Schwarzschild radii, regardless of the black hole mass. Therefore, $r_{c}$ represents a new fundamental gravitational radius which plays an analogues role in the case of weak gravitational field at galactic scales, as the Schwarzschild radius in the case of strong gravitational field in the vicinity of compact massive objects.

IV Conclusions

In this paper we used power-law $f(R)$ gravity to demonstrate the existence of a new fundamental gravitational radius. This radius plays an analog role, in the case of weak gravitational field at galactic scales, like the Schwarzschild radius in the case of strong gravitational field in the vicinity of compact massive objects. Such a feature emerges from Noether’s symmetries that exist for any power-law $f(R)$ function. In particular, for $f(R)\propto R^{3/2}$, the MOND acceleration regime is recovered. Using this new gravitational radius, $f(R)$ theories of gravity are able to explain the baryonic Tully–Fisher relation of gas-rich galaxies in a natural way and without need for dark matter hypothesis.

We can conclude that the range for $\beta$ from 0.5 to 0.8 (corresponding to range of $n$ from 1.5 to 3.5) is in a good agreement with observations. This values represent two constraints on parameter $\beta$ obtained from fitting the FP and from MOND. In conclusion, we do not need any dark matter hypothesis in order to explain baryonic Tully-Fisher relation, and even more, $\Lambda$CDM fit is not in satisfactory agreement with observations.

We can conclude that $f(R)$ gravity can give a theoretical foundation both for rotation curve of galaxies salucci1 ; salucci2 and even for the empirical BTF relation. We have to stress that obtained value for parameter $\beta$ from BTF differs from parameter $\beta$ at Solar System scales or from S2 star orbit scales bork12 . The reason for this result is that gravity is not a scale-invariant interaction and then it differs at galactic scales with respect to local scales. The constraints working at Solar System level for GR could not be simply extrapolated at any astrophysical and cosmological scale. However further and detailed investigations are necessary to confirm this statement and to reproduce the whole dark sector phenomenology at all scales.

Acknowledgements.

S.C. acknowledges the support of INFN (iniziative specifiche TEONGRAV and QGSKY). P.J., V.B.J. and D.B. wish to acknowledge the support by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project 176003. The authors also acknowledge the support of the Bilateral Cooperation between Serbia and Italy 451-03-01231/2015-09/1 ”Testing Extended Theories of Gravity at different astrophysical scales”, and of the COST Actions MP1304 (NewCompStar) and CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology).

Appendix A The Noether Symmetry Approach

The fundamental feature on which are based the above results is the existence of a further gravitational radius related to a conserved quantity. In this Appendix, we briefly summarize the Noether Symmetry Approach that leads to the emergence of such a radius.

As discussed in details in arturo ; bern11 , the existence of a Noether symmetry selects a power law form for the $f(R)$ function.

Let us take into account an analytical $f(R)$ gravity. As discussed in Sec. III, we can transform $f(R)\rightarrow f(\chi)$ assuming $\chi:=L_{M}^{2}R$. With this transformation, we deal with homogeneous dimensional quantities bern11 .

A point–like $f(\chi)$ Lagrangian is obtained by imposing the spherical symmetry into the action (1). This means to reduce the infinite degrees of freedom of the field theory to a finite number. The approach consists in choosing suitable Lagrange multipliers into the action arturo ; defelice .

Let us assume a static spherically symmetric metric of the form

[TABLE]

where $\mathrm{d}\Omega^{2}:=\mathrm{d}\theta^{2}+\sin^{2}{\theta}\mathrm{d}\varphi^{2}$ is the angular component. A point–like Lagrangian $L$ for $f(\chi)$ is obtained by recasting action (1) as

[TABLE]

Here $\lambda$ is a Lagrange multiplier related to $\bar{\chi}=L_{M}^{2}\bar{R}$ expressed in terms of the metric (28). Due to the spherical symmetry, we can fix the functions $B(r)$ or $C(r)$. Both the assumptions are equivalent under a suitable change of coordinates arturo . With this consideration in mind, the Ricci scalar can be expressed by two of the three functions in (29), i.e.

[TABLE]

where prime is the derivative with respect to $r$. Varying with respect to $\chi$ gives the Lagrange multiplier

[TABLE]

and then substituting into (29) and discarding the boundary terms, the point–like Lagrangian reduces to

[TABLE]

which is canonical in the configuration space $\mathbb{Q}\equiv\{A,C,\chi\}$ and tangent space velocities $\mathbb{TQ}\equiv\{A^{\prime},A,C^{\prime},C,\chi^{\prime},\chi\}$. Assuming the regime given by (19) and the related weak field approximation, the last two terms of (32) are both much smaller than $L_{M}^{2}f_{\chi}$. This allows to rewrite Lagrangian (32) as

[TABLE]

If Noether symmetries exist, they are related to conserved quantities and cyclic variables that allow to reduce the dynamics arnold ; marmo ; morandi . Furthermore, the existence of a symmetry selects the functional form of $f(\chi)$ gravity.

In general, a symmetry, and then a conserved quantity, exists if the Lie derivative of the Lagrangian (33) along a vector field X vanishes:

[TABLE]

where $\,i=1,2,3,$ in the configuration space $\mathbb{Q}\equiv\{A,C,\chi\}$. Solving the vector equation (34) means to find out the functions $\alpha_{i}$ which constitute the components of the Noether vector

[TABLE]

defined on the tangent space $\mathbb{TQ}$.

Eq.(34) gives also the functional form of $f(\chi)$, see also prado ; rubano ; f(R)-cosmo ; defelice .

In the present case, a general form of Noether vector, related to the Killing equations of the model townsend , is

[TABLE]

where $k_{i}$ and $p_{i}$ are constants. The Lie condition (34) is satisfied for

[TABLE]

where $k$ is an integration constant and $n\in\mathbb{R}$. The related constant of motion $\Sigma_{0}$ is

[TABLE]

In the case of GR, i.e. for $n=1$, the Noether symmetry gives the conserved quantity $\Sigma_{0}=2r_{\text{g}}$, which is exactly the Schwarzschild radius. In the case of MOND, where equation (38) is valid for $n=3/2$, $C(r)=r^{2}$ and, at the lowest order of perturbation, $A(r)=1+2\Phi/c^{2}$, the constant of motion is given by

[TABLE]

Clearly this relation involves two characteristic lengths, a mass length-scale $l_{M}$, that is the above $r_{c}$, and the Schwarzschild radius $r_{\text{g}}$. In other words, the Noether symmetry gives a conserved quantity related to a further gravitational radius in the case $n\neq 1$.