4.0: Cosmology

         This page gives material relating to cosmology. References to Appendices A to D of this Page are listed below. Appendix A describes the Metric Equation. Appendix B gives data concerning the Density of Matter in the Universe Appendices C and D are obtained from Appendices C and D of Ref. [4.1], which present analyses relating to the Yilmaz Cosmology Model. This material is modified to change the Hubble constant from 25 to 20 km/sec per million light years. Appendix E gives data concerning the density of a neutron star. Appendix F gives a revised (2007) analysis of Cosmic Background Radiation.

References for Appendices A to D:

[4.1]  Adrian Bjornson, A Universe that We Can Believe, Addison Press, Woburn, MA, 2000, ISBN 09703231-0-7.

[4.2]  Adrian Bjornson, The Scientific Story of Creation, Addison Press, Woburn, MA, 2000, ISBN 09703231-1-5.

[4.3]  Website. This internet website, www.olduniverse. com 

[4.4]   Huseyin Yilmaz, “New Approach to General Relativity”, Physical Review, vol. 111, No. 5, Sept. 1, 1958, pp 1417-1426.  

[4.5]  J. V. Narlikar, Introduction to Cosmology, 1993, 2nd Ed., Cambridge U. Press, Cambridge, England, ISBN 0-521-42352-X. 

References 4.1 to 4.3 are referred to as Believe [4.1], Story [4.2], and Website [4.3].

Appendix A: The Metric Equation

      This appendix explains how relativistic effects are calculated from the metric equation, which is based on the metric tensor. The metric equation is derived as follows. Consider two points, (1), (2) separated by distances Dx, Dy, Dz in the x, y, z directions. The Pythagorean theorem shows that the distance Dd between the two points is

      (Dd)2  =  (Dx)2 + (Dy)2 + (Dz)2                                         (A-1)

Assume that a light pulse is emitted at point (1) and travels to point (2) in time Dt. The distance Dd between the points is equal to the speed of light c multiplied by the time interval Dt. Hence the following relation holds:

      0  =  (cDt)2 - (Dd)2  =  (cDt)2 - (Dx)2 - (Dy)2 - (Dz)2           (A-2)

To handle more general experiments, the difference between (cDt)2 and (Dd)2 is defined as (Ds)2. This yields the metric equation, which is  

      (Ds)2  =  (cDt)2 - (Dd)2  =  (cDt)2 - (Dx)2 - (Dy)2 - (Dz)2      (A-3)  

A relativistic event is characterized by the time of the event and the three-dimensional spatial coordinates where the event occurs. The generation of the light pulse at point (1) is event (1), and the reception of the light pulse at point (2) is event (2). In this case the quantity (Ds)2 is zero. 

      Assume that a missile is fired from point (1) to point (2). Event (1) is the firing of the missile at point (1), and event (2) is the missile impact at point (2). Since the missile speed is less than the speed of light, the time difference Dt between the events is greater than in the first experiment, and so (Ds)2 is greater than zero. A case where (Ds)2 is greater than zero is called time-like. In a time-like case, one event can influence the other.

      Now assume that point (2) has an early-warning system, which detects the flash of the missile firing at point (1). In response to this flash, point (2) fires a missile in return. Event (1) is the firing of the missile from point (1) and event (2) is the firing of the second missile from point (2). The difference between these two events is still time-like. The quantity (Ds)2 is greater than zero, and so one event can (and did) influence the other. 

      Now assume that two warring parties have a truce that ends at noon. Not trusting the other party, each one decides to launch a preemptive strike exactly at noon. Event (1) is when point (1) fires its missile and event (2) is when point (2) fires its missile. In this case (Ds)2 is negative, and the condition is called space-like. In a space-like case where (Ds)2 is negative, there is not enough time between the two events for a light pulse to travel between them, and so one event cannot influence the other.

      Let us apply the metric equation for (Ds)2 to the space travel experiment, discussed in Story [4.2], Chapter 7, where two observers traveling at different speeds are measuring the speed of light. The two observers measure different values for the time and spatial variables involved in the experiment. However if (Ds)2 is zero for one observer, it is zero for the other, because the speed of light is the same for both observers. The two observers always measure the same value for (Ds)2, and so (Ds)2 is called invariant. By recognizing that (Ds)2 is the same for all observers one can derive the equations of Special Relativity.

Effect of Gravity on the Metric Equation  

      In a gravitational field the speed of light is not constant, and so the simple metric equation does not apply and must be generalized. The metric equation is first expressed in terms of the general coordinates x0, x1, x2, x3, where x0 is normalized time (equal to ct and denoted t), and x1, x2, x3, are the spatial coordinates, x, y, z. The metric equation becomes

      (Ds)2  =  (Dt)2 - (Dx)2 - (Dy)2 - (Dz)2 

                =  (Dx0)2 - (Dx1)2 - (Dx2)2 - (Dx3)2               (A-4)

In a gravitational field, the terms of the metric equation are modified by the metric tensor elements g00, g11, g22, g33 to give:

      (Ds)2  =  g00(Dx0)2 + g11(Dx1)2 + g22(Dx2)2 + g33(Dx3)2                  

    =  g00(Dt)2 + g11(Dx)2 + g22(Dy)2 + g33(Dz)2                  (A-5)

This assumes that the metric tensor is diagonal. For a nondiagonal metric tensor with 16 elements, the metric equation has 16 terms, which includes cross-product terms, such as  g13Dx1Dx3. However, our discussion is limited to diagonal metric tensors.

      Compare the general metric formula of Eq. A-5 with Eq. A-4, which applies in the absence of a gravitational field. This shows that with no gravitational field, g00 is equal to +1 and g11, g22, g33 are equal to -1. Within our solar system the gravitational field is weak, and so the values of the metric tensor elements are very close to these ideal values.

Calculating Relativistic Effects from the Metric Equation

      With the Yilmaz theory, the metric tensor elements g11, g22, g33 in rectangular coordinates are equal. Consequently the speed of light and the contraction of distance is the same for all directions. This allows us to simplify the metric equation by considering a single spatial dimension, which we call x. The metric equation reduces to

      (Ds)2  =  g00(Dt)2 + g11(Dx)2                              (A-6)                     

      For the Schwartzschild solution, the speed of light and the contraction of distance are not the same in all directions. This confusing issue is discussed in Appendix G of Believe [4.1] and in the Website [4.3], Addendum (Page 5), Chapter 2. At this point we ignore it. For the Schwartzschild solution, we only consider motion in the radial direction.

      If Ds in the metric equation is set to zero, a light pulse can just travel between the two events. Hence this condition yields the speed of light.

      If Dx is set to zero, (Ds)2 is positive because g00 is positive. Hence the interval between the events is time-like. In this case the value for Ds is called the proper time between the two events, which we denote Dtp. Proper time is the time difference read by a clock that is moved at constant velocity between the two events.

      If Dt is set to zero, (Ds)2 is negative because g11 is negative. Hence the interval between the events is space-like. In this case the value for Ö[-(Ds)2] is called the proper distance between the two events, which we denote Dxp. Proper distance is the distance read on a ruler stretched between the two events, where the ruler is at rest in the coordinates for which the two events are simultaneous.

      Applying these principles to the metric equation (Eq. A-6) gives the results in Table A-1. In the first case, Ds is set to zero, and the ratio Dx/Dt is calculated, which (as we will see) gives the apparent relative speed of light. In the second case, Dx is set to zero, and the value for Ds is the proper time, which is denoted Dtp. In the third case, Dt is set to zero, and the value for Ö[-(Ds)2] is the proper distance, which is denoted Dxp. The formulas that are calculated from the metric equation for the variables shown in the second column are given in the third column.

Table A-1: Derivation of relativistic parameters from  metric equation

condition        variable           formula         meaning

Ds = 0        Dx/Dt                  Ö[-g00/g11]       speed of light
Dx = 0        Dtp = Ds           Ö[g00] Dt        proper time
Dt = 0        Dxp = Ö[-(Ds)2]  Ö[-g11] Dx      proper distance

Table A-2: Relativistic effects obtained from metric equation.

meaning                  physical            equation                formula
                                 ratio                ratio

speed of light              cap/c              Dx/Dt              Ö[-g00/g11]
spatial contraction    Dxap/Dxp        Dx/Ö[-(Ds)2]      1/Ö[-g11]
 clock period              Dtap/Dtp         Dt/Ds              1/Ö[g00]
wavelength ratio        l'/l                  Dt/Ds              1/Ö[g00]

      The variables Dt and Dx in Table A-1 are usually called coordinate time and coordinate distance. These are time and distance intervals that are measured at infinity relative to a star, and so are the values that appear to exist to an observer on earth. We use the term "apparent" to represent the term that is usually called "coordinate" in the relativity literature. The variables Dt and Dx in Table A-1 are called apparent time and apparent distance intervals, and are denoted Dtap and Dxap.

      Since Dt is equal to cDt, the ratio Dx/Dt is equal to (1/c)(Dx/Dt). The velocity Dx/Dt for this case is the apparent speed of light, which we denote cap. Hence the ratio Dx/Dt is equal to cap/c, which is the relative value of the apparent speed of light.

      The above principles are applied to Table A-1 to give the ratios shown in Table A-2. The second column shows the variables in terms of physically meaningful terms, and the third column shows the values specified in the metric equation. The fourth column gives the formulas for these ratios that are calculated from the metric equation. A gravitational field causes the speed of light to decrease, it causes a dimension to contract, and it causes a time interval to expand. Because of the expansion of time interval, a clock runs slower.

      Since a gravitational field causes a clock to run slower, an excited atom on the surface of a star oscillates at a lower frequency, and so its spectrum has a longer wavelength. The wavelength ratio l'/l is equal to Dt/Ds, where l is the normal wavelength and l' is the wavelength that is observed. This wavelength ratio is shown in the last row of Table A-2.       

      Table A-3 shows the g00 and g11 elements for the Einstein and Yilmaz theories given in Story [4.2], Appendix E.. The values for the Einstein Schwartzschild solution are obtained from Table E-3 and those for the Yilmaz solution are obtained from Eqs. E-19, E-20.   

Table A-3: Metric tensor elements for Einstein Schwartzschild solution and Yilmaz solution of the gravitational effects of a star

                       Einstein               Yilmaz

         g00            1 - 2(m/r)               exp[-2m/r]
          g11          -1/[1 - 2(m/r)]         - exp[2m/r]  

      These g00 and g11 values are applied to the formulas of Table A-2 to obtain the specific formulas in Table A-4 for the Einstein Schwartzschild solution and for the Yilmaz solution of a single star. The clock rate in Table A-4 is equal to the reciprocal of Dtap/Dtp in Table A-2, which is the clock period. These specific formulas are plotted in Story [4.2], Chapter 10, Figs. 10-1 to 10-3.

Table A-4: Relativistic effects caused by a gravitational field for Einstein and Yilmaz theories

Characteristic              General            Einstein              Yilmaz

Speed of light              Ö[-g00 /g11]      [1 - 2(m/r)]          exp[-2m/r]
Clock rate                       Ö[g00]        Ö[1 - 2(m/r)]          exp[-m/r]
Spatial contraction          1/Ö[-g11]      Ö[1 - 2(m/r)]          exp[-m/r] 
Wavelength ratio (l'/l)      1/Ö[g00]      1/Ö[1 - 2(m/r)]      exp[m/r]  

   To generalize the Yilmaz solution, the m/r ratio is replaced by the relativistic gravitational potential f. The resultant values of g00 and g11 for the general Yilmaz solution are shown in the second column of Table A-5. For the Yilmaz cosmology model, 2f is equal to (r/r0)2, as will be shown in Eq. C-4 of Appendix C.. The third column of Table A-5 gives the metric tensor elements for the Yilmaz cosmology model.

Table A-5: Metric tensor elements g00 and g11 for the general Yilmaz solution and for the Yilmaz cosmology model  

element     General Yilmaz      Yilmaz Cosmology Model       

g00                    e-2f                         exp[-(r/r0)2]      
g11                    - e2f                        - exp[(r/r0)2]       

      The metric tensor values of Table A-5 are applied to Table A-2 to obtain the values of the relativistic effects for the Yilmaz cosmology model shown in Table A-6.  

Table A-6: Effects of gravitational field on speed of light, spatial contraction, and  clock rate reduction, for the Yilmaz cosmology model and the general Yilmaz solution

meaning           variables          formula             Yilmaz            cosmology model         

speed of light     cap/c         Ö[-g00/g11]             e-2f              exp[-(r/r0)2]
contraction      Drap/Dr        1/Ö[-g11]                e-f              exp[-(r/r0)2/2]    
clock rate         Dt/Dtap        Ö[g00]                   e-f              exp[-(r/r0)2/2]    

Appendix B: Density of Matter in the Universe

B.1 Luminosity Density of the Universe  

      Narlikar [4.5] (p. 304) gives in his Eq. 9.19 the following measured luminosity density of the universe, which was derived from the Revised Shapley-Ames Catalog: 2.18x108 Lsunh0/Mpc3, where Lsun is the luminosity of the sun and Mpc means million parsecs, which is equal to 3.26 MLyr (3.26 million light years). Our assumed Hubble constant, 20 km/sec per MLyr, is equal to 65 km/sec per Mpc, and so the normalized Hubble constant h0 is 0.65. Hence the luminosity density can be expressed as

      Luminosity density  =  4.09x106 Lsun/MLyr3                    (B-1)

B.2 Dark Matter

      Measurements of the motions of galaxies and clusters of galaxies show that there must be much more dark matter (which we cannot see) than there is luminous matter (which we can see). Otherwise these groups of stars would fly apart. Narlikar [22] (p. 310) gives in his Table 9.1 the data shown in the first data column of Table B-1. These values are expressed in terms of the normalized Hubble constant h0. The last column gives the values for the ratio h that correspond to our normalized Hubble constant h0, which is 0.65. The parameter h in Table D-1 is the ratio of total mass to luminous mass.

      In Table B-1, items (1) to (4) apply to the rotational motions of galaxies. These data suggest that the total mass associated with the rotation of a single galaxy is about 8 times the luminous mass.

      Items (5) to (8) involve motions of groups of galaxies. The mass ratio is much greater for a galaxy cluster than for a single galaxy. The rotation of a single galaxy involves matter in the vicinity of the galaxy, whereas the rotation of a cluster involves the total matter of the cluster.

      The average distance between galaxies is about 10 million light years (10 MLyr), and so we can allocate to each galaxy a volume of (10 MLyr)3, which is 1000 MLyr3. This volume is about 5 million times greater than the volume of the galaxy itself. Even though the density of dark matter is much smaller in the intergalactic space between galaxies, than in the vicinity of a galaxy, the total intergalactic matter is much greater than the dark matter close to the galaxy.

Table B-1: Average ratio (h) of total mass per luminous mass  

Object                                      h/h0                              h

(1) Our Galaxy (inner part)      6 ± 2                3.9 ± 1.3
(2) Our Galaxy (outer part)      40 ± 30             26 ±  20
(3) Spiral galaxies                    9 ± 1                5.9 ± 0.7
(4) Elliptical galaxies               10 ± 2               6.5 ± 1.3
(5) Galaxy pairs                      80 ± 20             52 ±  13
(6) Local Group                     160 ± 80             104 ± 52
(7) Statistics of clustering       500 ± 200           325 ± 130
(8) Abell clusters                   500 ± 200           325 ± 130

      Item (5) involves a pair of galaxies, and item (6) for our local group involves a very small galaxy cluster. In contrast, items (7), (8) involve large clusters of galaxies, and so should give a better indication of the effects of intergalactic mass. Items (7), (8) show that the mass of intergalactic dark matter is 325 times the luminous mass of the galaxy itself. Therefore intergalactic dark matter (which we cannot see) is about 325 times greater than the luminous matter (which we can see).

      Stellar mass is approximately proportional to luminosity for a large collection of stars. Hence if the sun luminosity Lsun is replaced by the sun mass Msun, Eq. B-1 gives the approximate mass density of luminous matter. Multiplying this by 325 to account for dark matter gives the following for the total mass density of the universe

      Mass density  =  1.33x109 Msun/MLyr3                  (B-2)

Replace Msun by the sun mass, 1.99x1033 gram. Since one light year (Lyr) is 9.46x1012 km, MLyr is 9.46x1021 meters. This gives the following for the total mass density of the universe:

      Mass density =  3.125x10-24 grams/meter3                (B-3)

The mass of a hydrogen atom is 1.67x10-24 gram, and so this mass density is equivalent to 1.87 hydrogen atoms per cubic meter. This shows that our best estimate of the average mass density of the universe is about 2 hydrogen atoms per cubic meter.

B.3 Predicted Density of Matter

      The Yilmaz cosmology model predicts an average density of matter in the universe of (3/8pGT02), where T0 is the apparent universe age and G is Newton's gravitational constant (6.674x10-8 cm3/gm-sec2). There are 31.558 million seconds per year, and so the apparent universe age  T0, which is 15 billion years, is 4.734x1017 seconds. The above formula gives an average density of matter of 7.98x10-30 gm/cm3 or 7.98x10-24 grams per cubic meter. A hydrogen atom has a mass of 1.67x10-24 gm, and so this density is equivalent to 4.78 hydrogen atoms per cubic meter.

      Thus the Yilmaz theory predicts an average density of matter in the universe equivalent to 4.8 hydrogen atoms per cubic meter. This predicted density is remarkably consistent with the 2 hydrogen atoms per cubic meter density derived from astronomical measurements.

      Big Bang theories define a critical mass density for the universe. If the density of matter is less than critical, the universe should expand forever; and if the density is greater than critical the universe should eventually collapse. The critical mass density for the Big Bang theory has the same value (3/8pGT02) that is required by the Yilmaz theory, and so is also equivalent to 4.8 hydrogen atoms per cubic meter.

B.4 The Source of Dark Matter

      What is the source of dark matter? Big Bang theorists are searching hard for dark matter, because their theories have difficulty explaining the early evolution of the universe unless the density of matter is close to the critical density. They have not found sufficient dark matter in the universe to achieve critical mass density.

      A fundamental mistake has been made in the Big Bang search for dark matter. Astronomers have used the radiation from quasars to measure the density of intergalactic gas. Since they assume that quasars are at enormous distances, they have concluded that the density of intergalactic gas must be extremely small. However, quasars are very much closer, and so these estimates of intergalactic gas derived from quasar radiation are not meaningful. Besides, most of the gas in space is probably molecular hydrogen (H2), rather than atomic hydrogen (H), and molecular hydrogen is very difficult to detect.

      Dark matter probably consists primarily of hydrogen atoms in the enormous spaces between galaxies. A frantic search for missing dark matter is being performed by some astronomers. This is a consequence of the lack of open debate in astronomy today. If astronomers listened to Halton Arp, they would recognize that quasars are probably close, and so they would not be using quasar light spectra to determine the density of intergalactic hydrogen.

B.5 Rate of Creation of Matter

      Narlikar [4.5] (p. 240, Eq. 8-4) shows that the rate of mass creation to compensate for the Hubble expansion is (3r/T0). Setting r equal to 4.8 hydrogen atoms per cubic meter shows that 0.96 (approximately 1.0) hydrogen atom is created per cubic meter every billion years, or one hydrogen atom is created every year within a cubic kilometer.

      Our earth, with a radius of 6378 km, has a volume of 1.09x1012 cubic kilometers, and so 1.09x1012 hydrogen atoms would be created per year within a volume the size of the earth, which is 34,500 hydrogen atoms per second. The equivalent energy (Mc2) of the hydrogen atom (1.67x10-27 kilogram) is 1.50x10-10 watt-second. Hence 34,500 hydrogen atoms per second is equivalent to 5.18x10-6 watt (5.18 microwatt). This shows that the predicted rate of creation of matter is equivalent to the continual conversion of 5 microwatts of energy into matter within a volume the size of the earth.    

B.6 Total Mass in the Universe

      If the universe has critical mass density, the total mass in the Big Bang universe is the density (3/8pGT02) multiplied by the volume of the observable universe, (4/3)pr03, where r0 is the radius of the observable universe. This gives a total mass of (r03/2GT02). The ratio (r0/T0) is equal to the speed of light c, and so the total universe mass becomes (c2r0/2G). The normalized relativistic mass of the sun, denoted (msun), is defined as (MsunG/c2). Hence the total mass of the observable Big Bang universe for critical mass density is equal to

      Mass of observable universe  =  (c2r0/2G)  =  Msun(r0/2msun)       (B-4

The normalized mass of the sun msun is 1.475 km. We assume that r0 is 15 billion light years, or 142x1021 km. Substituting these values into the above equation gives 48x1021 Msun for the universe mass.

      For critical mass density, the observable Big Bang universe has 48x1021 times the sun mass. This is the predicted mass of the Yilmaz cosmology model for a sphere with a radius of 15 billion light years.

      The volume of the observable Big Bang universe is as follows, where the radius r0 is 15,000 MLyr (15,000 million light years):

      Universe volume  =  (4/3)pr03  =  14.1x1012 MLyr3                    (B-5)

Multiply the Mass density of Eq. B-2 by this Universe volume to obtain the measured mass of the observable Big Bang Universe:

      Mass of observable universe  =   18.8x1021 Msun            (B-6)  

This gives a measured mass of the observable Big Bang universe equivalent to 18.8x1021 suns. This is about 40 percent of the theoretical mass obtained from Eq. B-4, which is very good agreement.

B.7 Size of the Initial Big Bang Universe

      A star with our sun's mass but the density of water would have a radius of 780,000 km, obtained by multiplying 700,000 km (sun radius) by the cube root of 1.4 (sun density). Multiply 780,000 km by the cube root of 18.8x1021 (from Eq. B-6) to obtain the radius for the observable universe if it were squeezed into a single body with the density of water:

  Universe radius  =  20.7x1012 km  =  2.19 Lyr   (water density)   (B-7

One light year (Lyr) is 9.46x1012 km. The density of a neutron star is 2.0x1014 times the density of water. The cube root of this ratio is 58,480. Hence the radius of Eq. B-7 is divided by 58,480 to obtain the following initial radius of the observable universe for neutron-star density:

  Universe radius  =  354 million km   (neutron-star density)               (B-8)   

This radius is 1.55 times the radius (228 km) of the orbit of Mars. From astronomical data, the observable Big Bang universe would have had an initial radius 1.55 times the radius of the Mars orbit if it began with the density of a neutron star, the maximum possible density of matter

Appendix C: Derivation of Yilmaz Cosmology Model

C.1 Basic Derivation

      The Yilmaz cosmology model is a simple application of the Yilmaz gravitational theory to cosmology. It assumes a constant average density of matter that extends to infinity and does not change with time. The model predicts that the universe should expand locally according to the Hubble law. In order for the mass density of the universe to remain constant with the Hubble expansion, the model accepts the steady-state universe postulate that matter is being created to offset the expansion.

      In the original 1958 paper on his theory, Yilmaz gave the metric equation for his cosmology model, and showed that the model predicts a local expansion that agrees with the Hubble law. After that, Yilmaz has ignored applications of his theory to cosmology, and has not considered his cosmology model since the paper was published over 40 years ago.

       The first step in applying the Yilmaz theory is to calculate the gravitational potential. The Website [4.3] Addendum (Page 5) gives in Chapter 3 the following formula for the gravitational potential for a medium of constant mass density r at an observation point (p2) located at a distance r from a specified central point (p1):

      f  =  - (2pr'/3)r2                                                 (C-1)

The mass density r' is given in normalized relativistic mass units. Let us consider point (p1) to be a distant galaxy, which is being observed on the earth at point (p2). This gives the gravitational potential on earth relative to the distant galaxy. To obtain the gravitational potential of the galaxy, relative to the earth, we take the negative of Eq. C-1. Let us also convert the mass density to conventional mass units by multiplying r' by G/c2. Since we are interested in 2f, we multiply the equation by 2. Equation C-1 becomes

      2f  =  (4prG/3c2)r2        (uniform average mass density)                  (C-2)

This analysis assumes a uniform average density of matter. For his cosmology model, Yilmaz derived in Ref. [4.4] the following value for 2f, which assumed a statistical distribution of point masses:

      2f  =  (8prG/3c2)r2        (statistical mass distribution)           (C-3)

This is twice the value given in Eq. C-2. This Yilmaz value was calculated by applying the formula for gravitational potential in a statistical analysis. Unfortunately, the original statistical analysis has been lost. Although this derivation is not available at this time, we assume that it can be recreated later.

      A statistical distribution of point masses should be more realistic than a uniform distribution of matter. Therefore we use the Yilmaz formula for 2f given in Eq. C-3. The quantity 2f can be expressed in a convenient form by defining the parameter r0 as follows:

      2f  =  (r/r0)2                                                    (C-4)

Comparing this with Eq. C-3 gives the following expression for r0:

      r02  =  r2/2f  =  3c2/8prG                                    (C-5)

      Story [4.3] shows in Appendix E that  the metric tensor element g00 of the Yilmaz theory is equal to e-2f, and g11 is equal to -e2f. Hence, the values of these elements for the Yilmaz cosmology model are as shown in Table C-1. The symbol exp(x) is used to represent the exponential relation ex when the expression for x is complicated.

Table C-1: Primary metric tensor elements for Yilmaz cosmology model

element         Yilmaz formula         cosmology model            

  g00                           e-2f                   exp[-(r/r0)2]
  g11                         - e2f                    - exp[(r/r0)2]     

Table C-2: Parameters derived from Yilmaz cosmology model  

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