This is a report on the application of the graviton redshift theory (GRST) [1], which is the theory that gravitons lose energy thru gravitational redshift while traveling in the accelerating system of a gravitational field. Accessing the Spitzer Photometry and Accurate Rotation Curves (SPARC) database, we give the results of fits to all 175 galaxies using a fixed set of parameters. What we are given from the SPARC database are the observed galaxy rotation radial positions and velocities. Also given are the velocity profiles representing the mass-to-light ratios for the rotation disk and bulge of the galaxy and the velocity profile for the Hydrogen (H-I) and Helium (He) gas in the galaxy. We are given other observables such as the distance to the galaxy and some photometry which we do not use here. We assume that there are only baryons in the galaxy which emit light from the stars and radio emissions from the gases. We assume the existence of gravitons which transfer the force of the gravitational field due to the mass of the baryons. Given an estimate of the total galaxy baryonic mass and assuming that the disk and bulge mass-to-light ratios are constant and proportional, we can determine a mass-to-light ratio ϒ ∗ for the stars. What is seen in galaxy observations is that the actual rotational velocities are very often greater than the Newtonian velocities due to only the baryons especially in the outer galaxy positions, for which non-baryonic dark matter has been hypothesized. Thus, we seek a model to account for the observed rotation velocities of any spiral or dwarf galaxy based on only baryons and the gravitons hypothesized to be associated with them. We propose that the effect traditionally attributed to unknown particles of dark matter can be fulfilled by the gravitational redshift of the gravitons.

First off, we need to determine a value for the total baryonic mass for a SPARC galaxy [2]. This is just a pure guess but we can base it on the baryonic Tully-Fisher relation (BTFR) [3][4] since this has been used to fit many galaxies and especially for the SPARC galaxies which we will access exclusively in this report. So, we say that the total baryonic mass in solar units M ⊙ for a galaxy is given by

M b g a l / M ⊙ = A g a l 50 ( v f ) 4 for some dimensionless multiple A g a l with v f the

final velocity observed for the galaxy. Next, since we assume that gravitons exist and that they travel at constant speed c in the galaxy, the same as the speed of light, then the gravitons will experience a lessening of their energy due to the gravitational redshift phenomenon, which is essentially a gravitational time speed up effect at lesser gravitational field strengths farther away from the center of the galaxy mass. This energy loss can be accounted as a mass loss using Einstein’s relation δ { mass } = δ { energy } / c 2 . We provide a specific form of how this energy couples into the gravitational field to deliver the observed results of the rotations of stars and gas, which is expressed in the form,

where K ( r ; k , β ) is a coupling function which has two constant coupling parameters k and β , m is the orbiting test mass, δ v = a ( t ) δ t is the change in velocity of an observer in freefall along the way in the gravitational field with respect to an observer in an inertial system at a large distance from the galaxy, δ t = δ r / c is the short travel time of the gravitons at speed c over distance δ r . The acceleration a at a point r in the field is given by a ( r ( t ) ) = − G M b ( r ) / r 2 and the integral is the sum of the work done by the system as the gravitons travel from the inner galaxy to the orbiting mass. We assume that K ( r ; k , β ) takes the form,

where k and β are coupling constants, M b ( r ) is the baryon mass at radial position r and M b ( r f ) = M b g a l is the total baryon mass of the galaxy at the final galaxy radial position r f .

Just applying Newton’s laws using the baryonic mass to obtain the rotational velocities falls short of the observed rotational velocities in most cases. To fix this, since we think that the gravitational redshift energy loss of the gravitons can provide the necessary missing mass, then, using (1) and (2), our function for the rotational velocity of an orbiting mass takes the form,

From the second equation of (3), we can obtain a value for the coupling constant k at the final position r f since the factor ( r f / r ) β = 1 for r = r f , this yields,

With the value obtained for k then the constant β can be determined by minimizing the error of the fit of the predicted rotation curve for v 2 ( r ; k , β ) given by (3) to the observed rotation curve v o b s 2 ( r ) . This is expressed by the general form,

for a sequence of β x . Subsequently, we will explain the algorithm to setup the lists of parameter values for A g a l , k and β . Additionally, there are also the mass-to-light ratio parameters ϒ d s k and ϒ b u l obtained from the mass and velocity profiles for the stars in the disk and bulge, respectively, of the galaxy data.

The values for A g a l are taken from a list of N A elements defined by,

This gives the 152 list values of { 0.01 , 0.275 , 0.54 , ⋯ , 39.48 , 39.745 , 40.01 } . To obtain the galaxy mass for a particular instance j of the fitting process with normalization multiplier A g a l , j , and the final observed galaxy velocity v f , the total baryonic mass M b g a l , j for the instance is given by,

The values for coupling constant k are taken from a list of N k elements defined by,

This gives the 62 list values for k of { 0 , 0.08 , 0.16 , ⋯ , 4.72 , 4.8 , 4.88 } .

The values for β are taken from a list of N β elements defined by,

This gives the 62 list values for β of { − 6.0 , − 5.803 , − 5.607 , ⋯ , 5.607 , 5.803 , 6.0 } . A constant mass-to-light ratio ϒ d s k is assumed for a galaxy disk and it is assumed that the galaxy bulge, if it exists, has a mass-to-light ratio proportional to that of the disk, namely ϒ b u l = ( b u l f a c ) ϒ d s k . Then, since the mass-to-light ratio is assumed constant, we can evaluate it at the final position r f for each instance j of the baryonic mass M b g a l , j obtained from (7), which is expressed by,

where absolute values |   | are needed because v g a s can sometimes be negative (Ref. [2], p. 5). For the fits we set b u l f a c = 1.4 for all galaxies [2].

The algorithm for the fits performed in this case consists of selecting a possible total baryonic mass M b , j from (7), determine the mass-to-light ratios ϒ d s k j and ϒ b u l j using (10). From the photometric data which has been reduced to the equivalent velocities for the galaxy bulge, disk and gas mass content, we calculate instance j of the baryonic mass function M b , j ( r ) within radial distance r from the galaxy center as due to a spherically symmetric distribution using the Newtonian relation,

where r = r k , k = 0 , 1 , ⋯ , N o b s − 1 , where N o b s is the number of radial distances observed, and the absolute values of the velocities are needed because they can sometimes be negative (Ref. [2], p. 5). The baryon mass profile velocities for the disk, bulge and gas are taken from Table2.mrt of the SPARC database.

With the total baryon mass for the instance j being M b , j ( r f ) = M b g a l , j at the last galaxy radial position r f , since for r = r f the value of ( r f / r ) β = 1 , a value for the constant coupling scale k is determined at the final radial position. By determining for each instance of the theoretical circular rotation velocity v j , i ( r ) for instance j computed at the final radial position r = r f with each of the possible instances of k i from the list (12) and determining the error with the observed final velocity v o b s ( r f ) , given by,

where i = 0 , 1 , ⋯ , N k − 1 . After determining each error k e r r , j , i for the instance of mass M b , j , the value of k j for that instance is determined by the minimum error,

Now having the mass distribution instances M b , j ( r ) and the coupling constant scale value k j for each instance, the best coupling coefficient β j can be determined by the minimum error between the observed galaxy rotation velocity v o b s ( r ) and the theoretical velocity v j , n ( r ) . The theoretical velocity has the form,

The error between the observed and theoretical rotation velocities is expressed,

The best value for the exponent β j is determined by the minimum error,

Finally, the errors between the observed and theoretical velocities are determined using the instance values for the triplets ( M b , j , k j , β j ) and then the minimum of the errors gives the best triplet. To do this we use the observed v o b s ( r ) and the theoretical v j , j ( r ) using (14), where the error is expressed by,

The best galaxy value for the triplet ( M b g a l , k , β ) is determined by the minimum error of E r r j in (17), yielding instance j = j m i n for the triplet,

We use the velocities from the SPARC database [2], derived from near-infrared surface photometry at 3.6 μm. To simplify the analysis, we set the mass-to-light ratios for the disk and bulge to be constant and proportional [2] where ϒ b u l = ( b u l f a c ) ϒ d s k with b u l f a c = 1.4 . From the photometric data which has been reduced to the equivalent velocities for the galaxy bulge, disk and gas mass content, we calculate the baryonic mass function M b ( r ) within radial distance r from the galaxy center as due to a spherically symmetric distribution using the Newtonian relation,

where r = r n , n = 0 , 1 , ⋯ , N − 1 , N > 1 , N the number of radial distances observed, and the absolute values of the velocities are needed because they can sometimes be negative (Ref. [2], p. 5). The rotation curve observed velocities and the mass profile velocities for the disk, bulge and gas are taken from Table2.mrt of the SPARC database.

Using the SPARC results at radial distance r for the gas, disk and bulge velocities for the galaxy, where the mass internal to r is given by M b ( r ) of (19), we obtain the equivalent graviton energy loss using (3), where the graviton mass function M g ( r ) is given by,

where n = 0 , 1 , ⋯ , N − 1 . The predicted velocity (3) using (19) and (20) is expressed in the form,

where n = 0 , 1 , ⋯ , N − 1 , N > 1 . For clarity, in (20) the logical expression term ( j > 0 ) equals 1 if j is greater than 0, and zero otherwise. Table 1 shows data used for all the 175 SPARC galaxy fits.

Upon the hypothesis that gravitons exist then they would be bosons of zero mass thus traveling in vacuum at the same speed as photons. Then, as for photons traveling against a gravitational field, the same kind of energy loss due to gravitational redshift would occur for gravitons. However, since gravitons and the gravitational field are one and the same then it is reasonable to hypothesize that the gravitational field will decrease in energy. It is this decrease in the field that the coupling factor K ( r ; k , β ) of (2) seeks to address. It is not unreasonable to assume that a more physical form for the coupling factor can be derived from first principles. What the GRST offers is a connection of dark matter in the universe to baryons via gravitons.

The basic strategy of this study was to use a fixed set of parameters { ( A g a l , j , k j , β j ) } as input to the GRST equation of motion (3) to fit any galaxy of the SPARC data base. The parameter set was determined after a single rough trial run through the 175 galaxies, where some fit failures were addressed by adjusting the input parameter set. The second run through all the SPARC galaxies is what this report describes. No further adjustments were made to make the fits. Any future pass through the data will be based on the fit parameters found already for each galaxy, searching in a vicinity thereof for a better fit to the galaxy rotation curve. Ultimately the quality of the fits depends on the accuracy of the stellar photometry and gas observations, which makes the GRST algorithm a possible tool for use in astrophysical observations since it can indicate where measurements may be excessive or lacking.

The author declares that the data supporting the findings of this study are available within the paper, its supplementary information files, and from the SPARC website http://astroweb.cwru.edu/SPARC/. Plots of all 175 fits to the SPARC galaxies are available in a compressed folder at https://doi.org/10.13140/RG.2.2.17064.64008.