Galactic rotation curves in Einstein's conformal gravity

TL;DR

This paper demonstrates that Einstein's conformal gravity, with a specific rescaled metric, can accurately explain galactic rotation curves for a large galaxy sample without invoking dark matter, using only two universal parameters.

Contribution

It provides an exact solution within Einstein's conformal gravity that accounts for galactic rotation curves without dark matter, based on a Weyl conformally invariant metric.

Findings

Successfully fits rotation curves of 104 galaxies

Uses only two universal parameters derived from data

Eliminates the need for dark matter in galactic dynamics

Abstract

We show quantitatively that an exact solution of Einstein's conformal gravity can explain very well the galactic rotation curves for a sample of 104 galaxies without need for dark matter or other exotic modification of gravity. The metric is an overall rescaling of the Schwarzschild-de Sitter spacetime as required by Weyl conformal invariance, which has to be spontaneously broken, and the velocity of the stars depends only on two universal parameters determined on the base of the observational data.

Tables

Table 1. Table 1: Galactic parameters of our sample

		Distance	$L_B$	$R_0$(kpc)	$M_{HI}$	$M_{disk}$		Data sources
Galaxy	Type	(Mpc)	(${10}^{10}{L}_{⨀}$)	(kpc)	$\left({10}^{10}{M}_{⨀}\right)$	$\left({10}^{10}{M}_{⨀}\right)$	${\left(M/L\right)}_{stars}$	$L_B$	$R_0$	HI
M 33	HSB	0.9	0.85	2.5	0.11	1.13	1.33	mannheim2012fitting	kent1987surface	mannheim2012fitting
NGC 55	LSB	1.9	0.588	1.9	0.13	0.3	0.5	puche1991hi	puche1991hi	puche1991hi
DDO 64	LSB	6.8	0.015	1.3	0.02	0.04	2.87	smailagic2003uv	smailagic2003uv	smailagic2004lorentz
UGC 128	LSB	64.6	0.597	6.9	0.73	2.75	4.6	mannheim2012fitting	ansoldi2007non	ansoldi2007non
DDO 154	LSB	4.2	0.007	0.8	0.03	0.003	0.45	walter2008things	leroy2008star	walter2008things
DDO 168	LSB	4.5	0.032	1.2	0.03	0.06	2.03	mannheim2012fitting	mannheim2012fitting	mannheim2012fitting
DDO 170	LSB	16.6	0.023	1.9	0.09	0.05	1.97	lake1990distribution	lake1990distribution	lake1990distribution
UGC 191	LSB	15.9	0.129	1.7	0.26	0.49	3.81	van2000evolutionary	van2000evolutionary	van1997comparative
NGC 247	LSB	3.6	0.512	4.2	0.16	1.25	2.43	carignan1985surface	carignan1985surface	carignan1990hi
NGC 300	LSB	2.0	0.271	2.1	0.08	0.65	2.41	carignan1985surface	carignan1985surface	puche1990hi
UGC 477	LSB	35.8	0.871	3.5	1.02	1.00	1.14	fisher1981neutral	mannheim2012fitting	fisher1981neutral
F563-1	LSB	46.8	0.14	2.9	0.29	1.35	9.65	de1997dark	de1997dark	de1996h
F563-V2	LSB	57.8	0.266	2.0	0.2	0.6	2.26	de1997dark	de1997dark	de1996h
F568-2	LSB	80.0	0.351	4.2	0.3	1.2	3.43	de1997dark	de1997dark	de1996h
F571-8	LSB	50.3	0.191	5.4	0.16	4.48	23.49	de1996h	de1997dark	de1996h
F579-V1	LSB	86.9	0.557	5.2	0.21	3.33	5.98	de1996h	de1997dark	de1996h
F583-1	LSB	32.4	0.064	1.6	0.18	0.15	2.32	de1997dark	de1997dark	de1996h
F583-4	LSB	50.8	0.096	2.8	0.06	0.31	3.25	de1997dark	de1997dark	de1996h
F730-V1	LSB	148.3	0.756	5.8		5.95	7.87	mannheim2012fitting	mannheim2012fitting	…
NGC 925	LSB	8.7	1.444	3.9	0.41	1.372	0.95	walter2008things	leroy2008star	walter2008things
NGC 959	LSB	13.5	0.333	1.3	0.05	0.37	1.11	fisher1981neutral	esipov1991ubvr	fisher1981neutral
NGC 1003	LSB	11.8	1.48	1.9	0.63	0.66	0.45	mannheim2012fitting	broeils1992mass	sanders1996published
UGC 1230	LSB	54.1	0.366	4.7	0.65	0.67	1.82	de1997dark	van1993star	van1993star
UGC 1281	LSB	5.1	0.017	1.6	0.03	0.01	0.53	van2000evolutionary	de2002high	swaters2002westerbork
UGC 1551	LSB	35.6	0.78	4.2	0.44	0.16	0.2	swaters2002westerbork	de1996near	swaters2002westerbork
NGC 1560	LSB	3.7	0.053	1.6	0.12	0.17	3.16	mannheim2012fitting	mannheim2012fitting	mannheim2012fitting
UGC 2259	LSB	10.0	0.11	1.4	0.04	0.47	4.23	carignan1988hi	kent1987dark	carignan1988hi
NGC 2403	HSB	4.3	1.647	2.7	0.46	2.37	1.44	walter2008things	wevers1986palomar	walter2008things
IC 2574	LSB	4.5	0.345	4.2	0.19	0.098	0.28	walter2008things	pasquali2008large	walter2008things
NGC 2683	HSB	10.2	1.882	2.4	0.15	6.03	3.20	casertano1991declining	kent1985ccd	sanders1996published
NGC 2841	HSB	14.1	4.742	3.5	0.86	19.552	4.12	walter2008things	mannheim2012fitting	walter2008things
NGC 2903	HSB	9.4	4.088	3.0	0.49	7.155	1.75	walter2008things	wevers1986palomar	walter2008things
NGC 2976	LSB	3.6	0.201	1.2	0.01	0.322	1.6	walter2008things	simon2003high	walter2008things
NGC 3031	HSB	3.7	3.187	2.6	0.38	8.662	2.72	walter2008things	murphy2008connecting	walter2008things
NGC 3109	LSB	1.5	0.064	1.3	0.06	0.02	0.35	carignan1985light	carignan1985light	jobin1990dark
NGC 3198	HSB	14.1	3.241	4.0	1.06	3.644	1.12	walter2008things	wevers1986palomar	walter2008things
NGC 3521	HSB	12.2	4.769	3.3	1.03	9.245	1.94	walter2008things	leroy2008star	walter2008things
NGC 3621	HSB	7.4	2.048	2.9	0.89	2.891	1.41	walter2008things	de2008high	walter2008things
NGC 3627	HSB	10.2	3.70	3.1	0.10	6.622	1.79	walter2008things	leroy2008star	walter2008things
NGC 3716	HSB	17.4	3.34	3.2	0.60	3.82	1.15	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3769	HSB	15.5	0.684	1.5	0.41	1.36	1.99	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3877	HSB	15.5	1.948	2.4	0.11	3.44	1.76	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3893	HSB	18.1	2.928	2.4	0.59	5	1.71	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3917	LSB	16.9	1.334	2.8	0.17	2.23	1.67	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3949	HSB	18.4	2.327	1.7	0.35	2.37	1.02	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3953	HSB	18.7	4.236	3.9	0.31	9.79	2.31	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3972	HSB	18.6	0.978	2	0.13	1.49	1.53	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 3992	HSB	25.6	8.456	5.7	1.94	13.94	1.65	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4010	LSB	18.4	0.883	3.4	0.29	2.03	2.30	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4013	HSB	18.6	2.088	2.1	0.32	5.58	2.67	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4051	HSB	14.6	2.281	2.3	0.18	3.17	1.39	sanders1998rotation	tully1997ursa	sanders1998rotation
NGC 4085	HSB	19.0	1.212	1.6	0.15	1.34	1.11	sanders1998rotation	tully1996ursa	sanders1998rotation

Table 2. Table 2: Galactic parameters of our sample

		Distance	$L_B$	$R_0$(kpc)	$M_{HI}$	$M_{disk}$		Data sources
Galaxy	Type	(Mpc)	(${10}^{10}{L}_{⨀}$)	(kpc)	$\left({10}^{10}{M}_{⨀}\right)$	$\left({10}^{10}{M}_{⨀}\right)$	${\left(M/L\right)}_{stars}$	$L_B$	$R_0$	HI
NGC 4088	HSB	15.8	2.957	2.8	0.64	4.67	1.58	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4100	HSB	21.4	3.388	2.9	0.44	5.74	1.69	sanders1998rotation	tully1996ursa	sanders1998rotation
UGC 4115	LSB	5.5	0.004	0.3		0.01	0.97	de2001high	barazza2001structure	…
NGC 4138	LSB	15.6	0.827	1.2	0.11	2.97	3.59	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4157	HSB	18.7	2.901	2.6	0.88	5.83	2.01	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4183	HSB	16.7	1.042	2.9	0.30	1.43	1.38	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4217	HSB	19.6	3.031	3.1	0.30	5.53	1.83	sanders1998rotation	tully1996ursa	sanders1998rotation
UGC 4325	LSB	11.9	0.373	1.9	0.10	0.40	1.08	swaters2002westerbork	de2002high	swaters2002westerbork
NGC 4389	HSB	15.5	0.61	1.2	0.04	0.42	0.68	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 4395	LSB	4.1	0.374	2.7	0.13	0.83	2.21	swaters2002westerbork	de2002high	swaters2002westerbork
NGC 4826	HSB	10.2	3.7	3.1	0.10	6.622	1.79	walter2008things	leroy2008star	walter2008things
UGC 5005	LSB	51.4	0.2	4.6	0.28	1.02	5.11	de1997dark	van1993star	van1993star
NGC 5585	HSB	9.0	0.333	2.0	0.28	0.36	1.09	cote1991dark	cote1991dark	cote1991dark
UGC 5716	Sm	21.3	0.0588	1.14	0.1094	0.0588	1.0	lelli2016sparc	lelli2016sparc	lelli2016sparc
UGC 5750	LSB	56.1	0.472	3.3	0.10	0.10	0.21	de1997dark	van1993star	van1993star
UGC 5999	LSB	44.9	0.17	4.4	0.18	3.36	19.81	de1997dark	van1993star	van1993star
UGC 6399	LSB	18.7	0.291	2.4	0.07	0.59	2.04	sanders1998rotation	tully1996ursa	sanders1998rotation
UGC 6446	LSB	15.9	0.263	1.9	0.24	0.36	1.36	sanders1998rotation	tully1997ursa	sanders1998rotation
NGC 6503	HSB	5.5	0.417	1.6	0.14	1.53	3.66	mannheim2012fitting	wevers1986palomar	mannheim2012fitting
UGC 6614	LSB	86.2	2.109	8.2	2.07	9.70	4.60	de2001high	van1993star	van1993star
UGC 6667	LSB	19.8	0.422	3.1	0.10	0.71	1.67	sanders1998rotation	tully1996ursa	sanders1998rotation
UGC 6818	LSB	21.7	0.352	2.1	0.16	0.11	0.33	sanders1998rotation	tully1996ursa	sanders1998rotation
UGC 6917	LSB	18.9	0.563	2.9	0.22	1.24	2.20	sanders1998rotation	tully1996ursa	sanders1998rotation
UGC 6923	LSB	18.0	0.297	1.5	0.08	0.35	1.18	sanders1998rotation	tully1997ursa	sanders1998rotation
UGC 6930	LSB	17.0	0.601	2.2	0.29	1.02	1.69	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 6946	HSB	6.9	3.732	2.9	0.57	6.272	1.68	walter2008things	leroy2008star	walter2008things
UGC 6973	HSB	25.3	1.647	2.2	0.35	3.99	2.42	sanders1998rotation	tully1997ursa	sanders1998rotation
UGC 6983	LSB	20.2	0.577	2.9	0.37	1.28	2.22	sanders1998rotation	tully1996ursa	sanders1998rotation
UGC 7089	LSB	13.9	0.352	2.3	0.07	0.35	0.98	sanders1998rotation	tully1996ursa	sanders1998rotation
NGC 7137	LSB	25.0	0.959	1.7	0.10	0.27	0.28	bieging1977radio	mannheim2012fitting	bieging1977radio
NGC 7331	HSB	14.2	6.773	3.2	0.85	12.086	1.78	walter2008things	leroy2008star	walter2008things
NGC 7793	HSB	5.2	0.91	1.7	0.16	0.793	0.87	walter2008things	leroy2008star	walter2008things
UGC 11454	LSB	93.9	0.456	3.4		3.15	6.90	de2001high	mannheim2012fitting	…
UGC 11557	LSB	23.7	1.806	3.0	0.25	0.37	0.20	de2001high	swaters2002westerbork	swaters2002westerbork
UGC 11583	LSB	7.1	0.012	0.7		0.01	0.96	de2001high	mannheim2012making	…
UGC 11616	LSB	74.9	2.159	3.1		2.43	1.13	de2001high	mannheim2012making	…
UGC 11648	LSB	49.0	4.073	4.0		2.57	0.63	de2001high	mannheim2012making	…
UGC 11748	LSB	75.3	23.930	2.6		9.67	0.40	de2001high	mannheim2012making	…
UGC 11819	LSB	61.5	2.155	4.7		4.83	2.24	de2001high	mannheim2012making	…
UGC 11820	LSB	17.1	0.169	3.6	0.40	1.68	9.95	van1997comparative	mannheim2012fitting	van1997comparative
UGC 12632	Sm	9.77	0.1301	2.42	0.1744	0.1301	1.0	lelli2016sparc	lelli2016sparc	lelli2016sparc
ESO 079-G014	Sbc	28.7	5.1733	5.08	0.314	5.1733	1.0	lelli2016sparc	lelli2016sparc	lelli2016sparc
ESO 0840411	LSB	82.4	0.287	3.5		0.06	0.21	de2001high	mannheim2012fitting	…
ESO 1200211	LSB	15.2	0.028	2.0		0.01	0.20	de2001high	mannheim2012fitting	…
ESO 1870510	LSB	16.8	0.054	2.1		0.09	1.62	de2001high	bell2000star	…
ESO 2060140	LSB	59.6	0.735	5.1		3.51	4.78	de2001high	beijersbergen1999surface	…
ESO 3020120	LSB	70.9	0.717	3.4		0.77	1.07	de2001high	mannheim2012fitting	…
ESO 3050090	LSB	13.2	0.186	1.3		0.06	0.32	de2001high	mannheim2012fitting	…
ESO 4250180	LSB	88.3	2.6	7.3		4.79	1.84	de2001high	beijersbergen1999surface	…
ESO 4880490	LSB	28.7	0.139	1.6		0.43	3.07	de2001high	mannheim2012fitting	…
UGC A442	Sm	4.35	0.014	1.18	0.0263	0.014	1.0	lelli2016sparc	lelli2016sparc	lelli2016sparc
UGC A444	Im	0.98	0.0012	0.083	0.0067	0.0012	1.0	lelli2016sparc	lelli2016sparc	lelli2016sparc

Equations

$$S=\int d^{4}x\sqrt{-\hat{g}}\left({\varphi }^2\hat{R}+6{\hat{g}}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -2f{\varphi }^4\right),$$

$${\hat{g}}_{\mu \nu }^{\prime }={\Omega }^2{\hat{g}}_{\mu \nu },{\varphi }^{\prime }={\Omega }^{-1}\varphi ,$$

$$S_{\mathrm{EH}}=\frac{1}{16\pi G}\int d^{4}x\sqrt{-\hat{g}}\left(\hat{R}-2\Lambda \right),$$

$${\hat{g}}_{\mu \nu }^{\ast }=S(r){\hat{g}}_{\mu \nu }{\varphi }^{\ast }=S(r{)}^{-1/2}\varphi .$$

$$d{s}^2=Q^2(x)\left[-\left(1-\frac{2G{M}_{⨀}}{c^{2}x}-\frac{\Lambda }{3}{x}^2\right)c^{2}d{t}^2+\frac{d{x}^2}{1-\frac{2G{M}_{⨀}}{c^{2}x}-\frac{\Lambda }{3}{x}^2}+x^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\right],$$

$$Q(x)=\frac{1}{1-\frac{\gamma \ast }{2}x},$$

$$x=\frac{r}{1+{\gamma }^{\ast }\frac{r}{2}},$$

$$d{s}^2=-Q^2(r)\left(1-\frac{2G{M}_{⨀}Q(r)}{c^{2}r}-\frac{\Lambda }{3}\frac{r^2}{Q^2(r)}\right)c^{2}d{t}^2+\frac{d{r}^2}{Q^2(r)\left(1-\frac{2G{M}_{⨀}Q(r)}{c^{2}r}-\frac{\Lambda }{3}\frac{r^2}{Q^2(r)}\right)}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2),$$

$$Q(r)=1+\frac{\gamma \ast }{2}r(\text{mbox}noticethatx=r/Q).$$

$$V$$

$$V=-\frac{G{M}_{⨀}}{r}+\frac{{\gamma }^{\ast }{c}^2}{2}r.$$

$$\Sigma (R)={\Sigma }_0{e}^{-R/R_0},$$

$$v_{\mathrm{LOC}}^2(R)=\frac{N^{\ast }G{M}_{⨀}{R}^2}{2R_0^3}[I_0(x)K_0(x)-I_1(x)K_1(x)]+\frac{N^{\ast }{\gamma }^{\ast }{c}^2{R}^2}{2R_0}{I}_1(x)K_1(x),$$

$$r=\frac{\rho }{(1-{\gamma }_0\rho /4{)}^2},t=\int \frac{d\tau }{a(\tau )},$$

$$d{s}^2=-(1+{\gamma }_{0}r)c^{2}d{t}^2+\frac{d{r}^2}{1+{\gamma }_{0}r}+r^{2}d{\Omega }^{(2)}$$

$$d{s}^2$$

$$d{s}_{\mathrm{global}}^2$$

$$Q(R)$$

$$v_{\mathrm{GLOBAL}}^2=\frac{{\gamma }_0{c}^2}{2}R-\frac{\Lambda c^2}{3}{R}^2.$$

$$v_{\mathrm{TOT}}^2=v_{\mathrm{LOC}}^2+\frac{{\gamma }_0{c}^2}{2}R-\frac{\Lambda c^2}{3}{R}^2,$$

Full text

I Introduction

One of the greatest mysteries in cosmology in our days is the “dark matter’s” or “dark gravity’s” puzzle. Indeed, in order to take into account of all the observational evidences (galactic rotation curves, structure formation in the universe, CMB spectrum, bullet cluster) we need to somehow modify the right or the left side of the Einstein’s field equations. In this paper we do not pretend to provide a definitive answer to the long standing question of what dark matter is, but we want to make known an extremely interesting result based on some previous work by Mannheim. Therefore, we here concentrate on only one of the above listed issues, namely the galactic rotation curves.

In this paper we assume the “dark gravity” paradigma and we try to explain the anomalous behaviour of stars in galaxies in the contest of gravitational theories invariant under conformal transformations (see below). The analysis here reported is universal and apply to any conformally invariant theory that has the Schwarzschild-de Sitter metric as an exact solution. However, for the sake of simplicity we will focus on Einstein’s conformally gravity in presence of cosmological constant. The theory is specified by the following general covariant action functional,

[TABLE]

which is defined on a pseudo-Riemannian spacetime Manifold $\mathcal{M}$ equipped with a metric tensor field $\hat{g}_{\mu\nu}$, a scalar field $\phi$ (the dilaton), and it is also invariant under the following Weyl conformal transformation:

[TABLE]

where $\Omega(x)$ is a general local function. In (1) $f$ is a dimensionless constant that has to be selected extremely small in order to have a cosmological constant compatible with the observed value. The Einstein-Hilbert gravity is recovered whether the Weyl conformal invariance is broken spontaneously in exact analogy with the Higgs mechanism in the standard model of particle physics (for more details we refer the reader to bambi2017spacetime ; chakrabarty2018unattainable ). One possible vacuum of the theory (1) (exact solution of the equations of motion) is $\phi={\rm const.}=1/\sqrt{16\pi G}$, together with $R_{\mu\nu}\propto\hat{g}_{\mu\nu}$. Therefore, replacing $\phi=1/\sqrt{16\pi G}+\varphi$ in the action and using the conformal invariance to eliminate the gauge dependent degree of freedom $\varphi$, we finally get the Einstein-Hilbert action in presence of the cosmological constant,

[TABLE]

where $\Lambda$ is consistent with the observed value for a proper choice of the dimensionless parameter $f$ in the action (1). Ergo, Einstein’s gravity is simply the theory (1) in the spontaneously broken phase of Weyl conformal invariance bambi2017spacetime .

Besides the constant vacuum, if a metric $\hat{g}_{\mu\nu}$ is an exact solution of the equations of motion thus it is also the rescaled spacetime with a non trivial profile for the dilaton field, namely

[TABLE]

Therefore, in this paper we push further the relevance of conformal symmetry in the real world on the footprint of previous works that were mainly concerning the singularity issue bambi2017spacetime ; chakrabarty2018unattainable . Contrary to the latter papers, we here focus on a non asymptotically flat rescaling on the Schwarzschild metric (or Schwarzschild-de Sitter metric) in order to explain the data for the galactic rotation curves with a minimal and universal choice of only two free parameters that the reader will meet later on in the paper. Finally, in order to give physical meaning to the metric (4) the conformal symmetry has to broken spontaneously to a particular vacuum specified by the function $S(r)$.

II The galactic geometry

As explained in the introduction, given an exact solution of Einstein’s conformal gravity, any rescaled metric is an exact solution too, if the metric is accompanied by a non-trivial profile for the dilaton. Therefore, we here consider the following conformal rescaling of the Schwarzschild-de Sitter spacetime,

[TABLE]

where we identified $x$ with the radial coordinate. The reason of the particular rescaling $Q(r)$ will be clarify shortly. Now we perform a coordinate transformation in order to express the metric in the usual radial Schwarzschild coordinate, which identifies the physical radius of the two-sphere. Moreover, the transformation of coordinates should also be compatible with the usual relation $g_{00}=-1/g_{11}$. These requirements uniquely fix the rescaling factor. Indeed, in the new radial coordinate $r$, defined by

[TABLE]

the metric reads:

[TABLE]

It deserve to be notice that $x=r/Q$ and, therefore, the asymptotic contribution to the metric due to the cosmological constant is independent on the rescaling $Q(r)$. Moreover, it is interesting that any rescaling which differs from the one in (6) is not consistent with the two requirements above. Therefore, in the infinite class of exact solutions conformally equivalent to the Schwarzschild-de Sitter metric, there is only one geometry non-asymptotically flat consistent with $g_{00}=-1/g_{11}$ and two-dimensional transverse area $4\pi r^{2}$.

The local contribution to the stars’ velocity

Inside a galaxy the metric (8) represents the geometry around any star, and the coordinate $r$ has the usual interpretation of radial coordinate. Therefore, we can easily compute the Newtonian gravitational potential in which any other star moves in, i.e.

[TABLE]

At the galactic scale $\gamma^{*}r\ll 1$ mannheim2012fitting 111Notice that we can identify the radial coordinate $r$ with the radial physical length, namely $d\ell_{r}=\frac{dr}{1+\epsilon}\approx dr$, because $\epsilon$ is proportional to $GM/r$ or to integer powers of the product $\gamma^{*}\,r$ that are much smaller then $1$., hence a star of solar mass generates a Newtonian gravitational potential with a linear asymptotic correction in the galactic halo given by

[TABLE]

We have removed the contribution of the cosmological constant that we will reintroduce later.

The next step is to take into account the contribution to the gravitational potential on a probe solar mass star in the halo due to all the other stars in the galaxy sample. In other words (or in Newtonian terms) we have to consider the resulting force acting on the probe star and due to its interaction with all the other stars in the same galaxy.

For this purpose we have to use the distribution of the luminous matter in an thin disk galaxy whose surface brightness is consistent with the following profile,

[TABLE]

where $R$ is the distance from galactic center in cylindrical coordinates, which should not be confused with the radial coordinate $r$ between two stars in spherical coordinates, $R_{0}$ is the disk scale length of the galaxy, and $\Sigma_{0}$ is defined by means of the total luminosity of the galaxy $L=2\pi\Sigma_{0}^{2}$.

Therefore, we have to integrate equation (9) over the whole galaxy to finally get the rotation velocity for a star orbiting in the plane of the galactic disk. The result is the local galactic contribution to the velocity of a single star mannheim2006alternatives , namely

[TABLE]

where here $x\equiv R/2R_{0}$ (this definition should not be confuse with the radial coordinate in (6)) and $N^{*}$ is the total number of solar mass stars in the galaxy, while $I_{0}$, $I_{1}$ are the modified Bessel functions of first kind and $K_{0}$, $K_{1}$ are the modified Bessel functions of second kind.

III The Universe Geometry

In the previous section we derived the local geometry surrounding any star in the galaxy and integrated over all the stars in the same galaxy to finally get the force acting on a probe star. In this section we would like to investigate whether the other galaxies of the whole universe can significantly have influence on the motion of the stars in a sample galaxy.

As painted in Fig.1, the probe star, labeled as a point $P$ in a sample galaxy, feels the gravitational attraction of all the other stars in the same galaxy, but also the effect of all the galaxies in the Universe due to the non homogeneous nature of the latter. Indeed, the latter global effect would vanish in a perfectly homogeneous universe because, saying in Newtonian terms, the contributions to the total gravitational force coming from opposite directions erase each other. Furthermore, we also have to take into account the global contribution due to the cosmological constant that we will consider later. For the moment, the question we should find an answer is: what is the geometry due to all the other galaxies in the universe? First of all we observe that the metric (8) is not asymptotically flat and the linear contribution grows with the radial coordinate $r$, which makes a long distance interaction possible. Therefore, we can assume that the global contribution of the other galaxies in the Universe is well mimic by a metric very similar to (8). The last statement can be proved making a coordinate transformation that turns the metric (8) in a metric conformal to the Robertson-Walker (RW) spacetime mannheim2011impact . Indeed, under the coordinate transformation $(t,r,\theta,\phi)\,\,\rightarrow\,\,(\tau,\rho,\theta,\phi)$, i.e.

[TABLE]

it has been proved in mannheim2012fitting ; mannheim1989exact that the following metric (we here consider only the large distance leading contribution to (8), and, for the sake of simplicity we temporarily omit the cosmological constant that we will reintroduce later)

[TABLE]

turns into

[TABLE]

which is clearly conformal to a slightly inhomogeneous RW metric whether $\gamma_{0}$ is a very small parameter of inverse length dimension222In our case $\gamma_{0}\sim 10^{-30}{\rm cm}^{-1}$, hence for $\rho$ of the order of the observable Universe we get $\gamma_{0}^{2}\rho_{H}^{2}/16\sim(10^{-30}{\rm cm}^{-1}\times 10^{28}{\rm cm})^{2}/10\sim 10^{-5}$, which is negligible respect to $1$ in the metric (15).. Indeed, the metric (15) represents an homogeneous spacetime for $\gamma_{0}=0$.

In short, the RW metric (15) turns into (14) (or (8)) when expressed in terms of coordinates used by an observer at rest in the center of a chosen galaxy.

In force of the above global analysis we can now come back to the geometry (8). Since the RW metric (15) is isotropic and quasi-homogeneous, every point in the geometry can serve as the origin of the coordinates. Hence, we can place the origin of coordinates in the center of a given comoving galaxy, and we can identify $r$ ($r\sim\rho$) with $R$ introduced in the previous section. Indeed, the radial coordinate $r$ labels any point in the galaxy (respect to its center) and in particular the position of a probe star whose coordinate distance from the center of the galaxy was in the previous section indicated with $R$. Notice that for the global contribution we do not have to integrate over the all galaxies in the Universe, but we have to parametrize such effect with a constant $\gamma_{0}$.

Therefore, the geometry that provides a global contribution to the velocity is again (8) in which we have to replace the coordinate $r$ with $R$, namely

[TABLE]

where we assumed $\gamma_{0}^{2}\ll\Lambda$ in order to remove the contribution $\gamma_{0}^{2}R^{2}/4$ relative to $\Lambda R^{2}/3$. This assumption will be confirmed by our best fit that provides the value $\gamma_{0}\sim 10^{-30}{\rm cm}^{-1}$, while $\Lambda=1.1\times 10^{-56}{\rm cm}^{-2}$ (see next section). Using again $V=-(g_{00}(r)+c^{2})/2$ for the metric (8) we obtain the global contribution to the velocity of the stars, namely

[TABLE]

IV Fitting of the galactic rotation curves

Now collecting the local contribution (12) and global contribution (17) in the weak gravity limit mannheim1997galactic we end up with the following total contribution to the velocity of a probe star in a sample galaxy,

[TABLE]

The two universal parameters $\gamma^{*}$ and $\gamma_{0}$ will be determined in the this section on the base of the data in Table (1) and (2) for a sample of $104$ galaxies.

We acquired all the parameters for the luminosities ($L$), disk scale lengths ($R_{0}$), and HI gas mass ($M_{HI}$) from two sources: a previous work mannheim2012fitting and from the SPARC database lelli2016sparc . The parameters of $5$ galaxies (namely: ESO 079-G014, UGC 5716, UGC 12632, UGC A442, and UGC A444) have been extracted from SPARC. For the rotation velocities ($v$), $76$ out of $104$ galaxies can be found in SPARC database, the data for the other $28$ galaxies are here listed: ESO0840411mcgaugh2001high , ESO 1200211mcgaugh2001high , ESO 1870510mcgaugh2001high , ESO 2060140mcgaugh2001high , ESO 3020120mcgaugh2001high , ESO 3050090mcgaugh2001high , ESO 4250180mcgaugh2001high , ESO 4880490mcgaugh2001high , UGC 4115mcgaugh2001high , UGC 11454mcgaugh2001high , UGC 11583mcgaugh2001high , UGC 11616mcgaugh2001high , UGC 11648mcgaugh2001high , UGC 11748mcgaugh2001high , UGC 11819mcgaugh2001high , F 730-V1mcgaugh2001high , NGC 925de2008high , NGC 3031de2008high , NGC 3621de2008high , NGC 3627de2008high , NGC 4826de2008high , NGC 959de2008mass , NGC 7137de2008mass , UGC 1551de2008mass , NGC 4395de2006high , UGC 477de2006high , M 33warner1973high , NGC 1560broeils1992mass .

Our sample of galaxies cover high surface brightness (HSB) galaxies, which have large $N^{*}$ and $\Sigma_{0}$ ($\Sigma_{0}\propto N^{*}/R_{0}^{2}$), low surface brightness (LSB) galaxies with small $\Sigma_{0}$, and dwarf galaxies with small $N^{*}$.

From (18) it follows that the rotation velocity curve will fall off at large distance because of the presence of a negative quadratic term. Thus, in Fig.2-8, one can see that our formula (18) is in good agreement with the actual data.

In the fitting, we employ a disk model with exponential decay to describe HI gas distribution, and we choose the ratio of the gas scale length to the optical disk scale length to be four because the gas extend well beyond the optical disk. We have also multiplied the HI gas mass by $1.4$ to include the primordial Helium. As a result, we found good fits with two universal parameters fixed to the values: $\gamma^{*}=3.32\times 10^{-39}{\rm m}^{-1}$, $\gamma_{0}=3.3\times 10^{-28}{\rm m}^{-1}$.

Few further technical comments are needed. For the galaxy NGC 3109, we corrected the HI surface density to the value $1.67$ instead of $1.4$ as pointed in begeman1991extended . Following the argument in mannheim2012fitting we removed the first ten velocity-points for the galaxy NGC 4826 reported in de2008high . The galaxy NGC $2976$ was reported in simon2003high to have a double disk structure. However, considering that the rotation velocity data in de2008high is within a distance of $2.5$ kpc and most of the points are located at a distance less than $1.5$ kpc, we used the approximation of a single disk with a scale length equal to $1.2$ kpc. The plots in the Fig.s 2-8 show the fitting rotation velocity curves (in km ${\rm sec}^{-1}$) as functions of the radial distance (in kpc) from the galactic centers . Moreover, in all plots the observative data with errors (in km ${\rm sec}^{-1}$) are also displayed for the all sample of 104 galaxies

V Conclusions

In this paper we have selected a spherically symmetric solution of Einstein’s conformal gravity in presence of cosmological constant compatible with very minimal assumptions: (i) the metric is written in Schwarzschild coordinates in which the two dimensional transverse area equals $4\pi r^{2}$, (ii) the timelike and spacelike components of the metric satisfy the relation $g_{00}=-g_{11}$. This two requirements specify uniquely the metric in the infinite class of metrics conformally equivalent.

The resulting geometry captures the local and global structure of the universe up to two constant parameters that can be set based on observational data. The first of the two parameters ($\gamma^{*}$) is related to the only ambiguity present in the metric describing the local geometry, while the second parameter ($\gamma_{0}$) appears in a similar metric, but now describing the global geometry due to all the other galaxies in the Universe. Indeed, the metric proposed in this paper when expressed in comoving coordinates takes the form of a slightly non-homogeneous Robertson-Walker geometry.

From the geometry of the spacetime we have extracted the local and global contributions to the velocity of a probe star and fixed the two free parameters considering the actual rotations’ curves of a sample of $104$ galaxies. The results are shown in the Fig.s. 2-8. The plots shows an extraordinary agreement between the theoretical model and observational data, although for the larger galaxies the value of the cosmological constant is too small to make the model perfectly compatible with the data. Indeed, for example in UGC$128$, DDO$170$, NGC$1003$, UGC$1230$, NGC$3198$, UGC$6614$ the velocity at large distance is too large. On the other hand, for example the data relative to the galaxies F$563-1$, F$568-3$, NGC$2403$, NGC$2841$, NGC$3621$, NGC$7331$, UGC$11748$ are in perfect agreement with our model. However, our model is not completely realistic since we have modeled the surface brightness distribution and the intergalactic gas with simple exponentials profiles. Moreover, we did not take into account the thickness of the galactic disk and the uncertainty on the galactic inclination angle. These issues will be addressed in future works.

Nevertheless, we consider the outcome of our work a good reason to further reflect on the dark matter problem. Indeed, it is quite surprising that we can so well explain the galactic rotation curves in Einstein’s conformal gravity without need to add higher derivative terms or exotic matter in the Einstein’s equations. However, we would honestly emphasize that the missing matter (or missing gravity) seems need in many areas of cosmology and modern astrophysics that we here did not touch.