The first observations of an accelerated expansion of the universe were made in the late 1990ies by two independent groups, cf. [16] . The result is significant, because it implies that either there is an unknown form of energy (dark energy) involved or the influx of new matter causes the rate change. Actually, both approaches may be compatible, a virtual quantum ground state energy provides the source of a particle influx. The theoretical dark energy is an integrated rating of the instantaneous ground state values. It is important to determine an approximation of the acceleration, even if there does not exist a purely mathematical solution.

The Einstein Field Equation in its original form is

G μ ν = 8 π G T μ ν , (18)

where G_μν is the Einstein tensor and T_μν is the energy-momentum tensor. The Friedmann solution in Robertson-Walker metrics is, cf. [17] ,

( a ˙ a ) 2 = 8 π G ρ 3 − k c 2 ( a r c u r 0 ) 2 . (19)

Here a is the scale factor, such that r = ar₀, and r_cur is the curvature radius. The curvature factor k is −1, 0, +1 for negative, zero and positive curvature respectively.

We derivate Equation (19) with respect to time

2 a ˙ a ¨ = 8 π 3 ( G a 2 ρ ˙ + 2 G ρ a a ˙ + G ˙ ρ a 2 ) . (20)

We have

a ¨ a = 4 π G 3 ( a a ˙ ρ ˙ + 2 ρ + G ˙ G ρ a a ˙ ) . (21)

Equation (21) differs from the accustomed equation by accounting for the time derivative of G.

The time derivative of the density ρ ˙ contains the pressure P responsible for the expansion. The 1. Law of Thermodynamics states that dW + PdV = 0. We have

d W d t + P d V d t = 0. (22)

The rate of volume change is

d V d t = 32 π 3 d d t ( r 3 ) = 3 V a ˙ a . (23)

On the other hand the change of work is

d W d t = V ρ ˙ c 2 + ρ c 2 d V d t = V c 2 ( ρ ˙ + 3 ρ a ˙ a ) . (24)

Now, by combining Equations (23) and (24) with Equation (22) we have

ρ ˙ a a ˙ = − 3 ( ρ + P c 2 ) . (25)

We may now write the complete equation of acceleration

a ¨ a = − 4 π G 3 [ ρ ( 1 − G ˙ G a a ˙ ) + 3 P c 2 ] . (26)

It is easy to show that G ˙ G a a ˙ = − 1 . When G, ρ_mx and P_E from Equations (15),

(16) and (17) are substituted into Equation (26), the result is 0. This can be interpreted as the equation being correct, as well as the variable definitions, and that the need for the cosmological constant is necessary. However, we have solved the problem in a more plausible way, the difference between the density and the pressure (divided by c²) terms must be diminutive. Therefore we introduce a factor B + 1 before the pressure term. B is small, roughly 1/π², a more precise definition will be presented in Section 4.4. The new acceleration equation is

a ¨ a = c 2 2 r 2 B . (27)

We use the following relation as an Ansatz:

g = r 0 a ¨ = B c 2 2 r 0 a = B c 2 2 r . (28)

Here g has been used as the symbol for acceleration because a is reserved for the scale factor.

We conclude that r² in the denominator of Equation (27) should be multiplied with π² (cf. Figure 2) to indicate that a ¨ is an intrinsic 3d acceleration acting at each point of the space and giving rise to a Coriolis effect.

By integrating Equation (27) we obtain the rate of the universe expansion, i.e. the Hubble parameter. First

a ˙ d a ˙ = c 2 r 0 2 B d a a , (29)

and further

a ˙ = c r 0 B ln a a i = c r 0 B ln r r i . (30)

This is the Hubble parameter, a_i = r_i/r₀. As a first approximation we use B = 1/π² and obtain

h 0 = c π r 0 ln r 0 r i = 2.229 × 10 − 18   s − 1 ~ H 0 = 68.8 km s ⋅ Mpc . (31)

H₀ is close to the present best value of 67.7 km/sMpc.

The passage of time since the initial event is thought to be anchored to the curvature radius due to the relation l p / t p = c, where l p and t_P are the Planck length and the Planck time respectively, that is, length and time are synchronized even if the pull of gravity changes. Accordingly, we can treat universe size and time as synchronized variables, without considering the decreasing G.

The propagation of time is obtained by integration. We substitute u = ln(a/a_i) into Equation (30)

∫ a i e u d u u = ∫ c r 0 B d t . (32)

The integral on the left-hand side has an error function solution

∫ a i e u d u u = − a i i π ⋅ e r f i ( u ) . (33)

The real part solution of the right-hand side is obtained by using the Dawson integral function D + ( ln ( a / a i ) ) , to be found in Wolfram-Alpha.

The time as a function of the radius is obtained from

t ( r ) = D + 2 r c B . (34)

In our first approximation D + ( ln ( r 0 / r i ) ) = 0.05111 , B = 1/π² we get the age of the universe: t₀ = 4.51 × 10¹⁷ s ~ 14.3 Gyr, a slightly higher value than the official 13.8 Gyr.

Our attempt is to determine B as a function of time. It can be proved that lim ( L D + ( L ) ) = 1 / 2 for L → ∞ . We define a parameter

ν B = 1 4 L D + 2 , (35)

where L = ln(r/r_i) + 1/4. The addition of 1/4 stems from the need of considering the momentum change caused by the continuous influx of new matter. The momentum force is

F M = d p d t = M u ∂ V u ∂ t + V u ∂ M u ∂ t = M u ∂ V u ∂ r d r d t + V u ∂ M u ∂ r d r d t = 8 π b r B ( ln r r i + 1 4 ) = 8 π b r B L , (36)

where p = M_uV_u is the momentum and V_u the velocity of expansion.

It can be shown that

B = ( ν B π ) 2 . (37)

As r goes towards ∞, ν_B approaches 1 and B → 1/π². The universe is becoming flat and the frontier speed, see next section, is asymptotically approaching the speed of light.

Substituting L = 96.656 and D₊ = 0.0511248 into Equation (35) results in ν_B = 0.98957, which in turn substituted into Equation (37) results in B = 0.099219.

The new Hubble parameter estimate is h₀ = 2.2075 × 10⁻¹⁸ 1/s ~ H₀ = 68.12 km/(s·Mpc).

The corrected universe age is t₀ = 4.56 × 10¹⁷ s = 14.46 Gyr.

The cosmic age to proper distance (light cone) diagram is a central tool in judging the actual observable universe relative to the theoretical one. Because the expansion frontier is progressing at almost the speed of light, even near parts of our neighbourhood are disappearing.

We presume that the radius of the frontier of the expanding universe is r_F = r/π. From Equation (31) the velocity of the frontier expansion is

V F = c π B ln ( r r i ) = ν c , (38)

where the relative velocity is ν = V_F/c ≤ 1.

The calculation of the proper distance requires the geometric representation in Figure 3. Our task is to determine the radius x as a function of the cosmic age t_x. The perimeter of a circle of x represents the proper distance d_p at t = t_x.

We define a time factor

k x = 2 π D + B , (39)

such that t_x = k_xr_F/c, t₀ = k_0*r_F0/c and D₊ = Dawson integral function.

From Figure 3 we deduce

c 2 ( t 0 − t x ) 2 = r F 0 2 + r F 2 − 2 r F 0 r F 2 − x 2 , (40)

where from x is solved. Let κ = x/r_F0 be the normalized radius of x. From Equation (40) we then have

κ 2 = a 2 − [ 1 2 ( 1 − k x 0 2 ) + a 2 2 ( 1 − k x 2 ) + k x k x 0 a ] 2 . (41)

The proper distance is obtained from

d p = 2 κ r 0 . (42)

The relation between the cosmic age and the proper distance is shown in Figure 4. The computed curve (green) according to Equation (42) is compared to results originally published by T. H. Davis and C. H. Lineweaver, [18] , however updated with Planck 2013 satellite data, as published by N. Crichton in 2015, [19] . The shapes of the curves are almost identical, though the calculated curve shows a 5% higher maximum value.

In Figure 5, the projection of the universe is shown as the surface of a cone with logarithmic scaling. The radius of the observable universe is the half-perimeter of the end circle. The real observable universe is just a small part of the surface, the fast expansion prevents light from the neighbouring areas to reach the observer. In Figure 6, the time scale is linear, the radial scale logarithmic. The projection reaches from the time of the CMB to the present. Usually, the universe is described with similar images.

It was argued in Section 4.1 that the acceleration is an intrinsic 3d phenomenon that operates at any point in the universe. In rotating systems of fluids, particle clouds or star clusters, acceleration causes a Coriolis effect with a large impact on the dynamical patterns. This is best demonstrated by satellite images of low-pressure rotation on weather maps, or of hurricanes or typhoons, Figure 7.

Here we present a short analysis of the mathematical background of the Coriolis effect. In Figure 8 an observer is located at point O. The observer is studying the velocity relative to coordinate system A of an object at point P. The position vectors are defined as follows

X A = ∑ 1 3 x i e i , (43)

and

X O = X O A + X A , (44)

where e_i is the unit vector.

The observer measures the velocity

d X O d t = V O A + V A + ∑ 1 3 x i d e i d t = d X O A d t + ∑ 1 3 d x i d t e i + ∑ 1 3 x i d e i d t , (45)

where V_OA is the velocity difference between the systems, V_A is the velocity of P in A. The last term is the rate of expansion. The observer measures an acceleration g_obs of P according to

g o b s = d 2 X O d t 2 = g O A + d V A d t + ∑ 1 3 d x i d t d e i d t + ∑ 1 3 x i d 2 e i d t 2 . (46)

The time derivative dV_A/dt represents the “true” (in the reference frame of A) acceleration of the body at P. By derivation of the middle term in Equation (45) we obtain

g t r u = d V A d t = ∑ 1 3 d x i 2 d t 2 e i + ∑ 1 3 d x i d t d e i d t . (47)

The first term in the latter part represents the Newtonian acceleration g_N. The second term is the result of expansion.

Most galaxy systems are disk-like, therefore we assume the system A being a 2-dimensional cylindrical coordinate system with x₃ coordinates set to 0. The equations of the transformation between the Cartesian and the cylindrical system are

x 1 = r l cos θ , (48)

x 2 = r l sin θ , (49)

and

e x = cos θ e r − sin θ e θ , (50)

e y = sin θ e r + cos θ e θ , (51)

Here r l is the radial coordinate of the system A.

Next we substitute variables and unit vectors into Equations (46) and (47), and obtain

g o b s = d V O A d t = g O A + g t r u + ( r l d 2 e r d t 2 + d r l d t d e r d t ) e r                                       − ( d θ d t d r l d t + r l d 2 θ d t 2 + r l d θ d t d e θ d t ) e θ , (52)

and

g t r u = d V A d t = ( d 2 r l d t 2 + d r l d t d e r d t ) e r + ( d θ d t d r l d t + r l d 2 θ d t 2 + r l d θ d t d e θ d t ) e θ . (53)

As the tangential centripetal velocities are of special interest, we concentrate on the radial acceleration components g_obs and g_tru. We denote:

g N = d 2 r l / d t 2 , Newtonian gravitational acceleration in radial direction,

g e x p = r l ( d 2 e r / d t 2 ) , intrinsic expansion acceleration,

d r l / d t = V r l = g N / r l ,

d e r / d t = V e x p / r l = g e x p / r l = increase of local space.

We then define

g k i c = d r l d t ⋅ d e r d t = V r l V e x p r l = g N g e x p . (54)

The equations of the local radial accelerations are

g t r u = g N + g k i c = g N + g N g e x p , (55)

g o b s = g t r u + g k i c + g e x p . (56)

The velocity of an object orbiting in A obeys the law of centripetal force

V o r b i t = r l g , (57)

where g is either g_obs or g_tru.

Figure 9 shows true and observed accelerations from different published sources. The observation curves are compared with calculated functions based on Equations (55) and (56).

Results published by McGaugh et al., [20] [21] , indicate that for decreasing g_N the acceleration declines asymptotically towards ĝ = 0.92 × 10⁻¹¹ m/s². The number is close to the intrinsic acceleration g_exp = g₀ = 1.0533 × 10⁻¹¹ m/s² from Equation (28).

The MOND theory was developed by M. Milgrom early in the 1980ies, [22] . The MOND curve in Figure 9 follows almost exactly the theoretical Coriolis solution: g_MOND ≈ g_N + 2g_kic, without the additional g_exp term of Equation (56).

The comparison convincingly confirms that the Coriolis approach provides a credible explanation to the dynamic behavior of the galaxies.

In [6] the rotational velocities of the Milky Way were calculated according to the equations presented here. A spherical bulge and a thin disk, total masses m_MW = 2.3 × 10⁴¹ kg and m_bulge = 0.29 × 10⁴¹ kg, were used as a model of the galaxy. The acceleration as a function of the radius r_MW was obtained from gravitational potential computations. Figure 10 shows that the calculated V_obs ( = r l g o b s ) curve follows almost perfectly an average path of Lamost measurements, [25] .

The time-dependent Schrödinger equation is

∇ 2 ψ + 8 π 2 m e h 2 ( E i − U i ) ψ = 0 , (58)

where ψ is the quantum-mechanical wave function, E_i is the ground state energy, U i = 4 π b r i 2 is the potential energy. The spherically symmetric form of Equation (58) is

∂ 2 ψ ∂ r 2 + 2 r ∂ ψ ∂ r + 8 π 2 m e h 2 E i ψ − 32 π 3 m e b h 2 r 2 ψ = 0 , (59)

where r is the curvature radius and b the energy constant of Equation (14).

Let

a 1 4 4 = h 2 32 π 3 m e b , (60)

C = 8 π 2 m e h 2 E i . (61)

The Schrödinger equation takes the form

∂ 2 ψ ∂ r 2 + 2 r ∂ ψ ∂ r + C ψ − 4 a 1 4 r 2 ψ = 0. (62)

This Sturm-Liouville type differential equation has the solution

ψ ( r ) = 1 r e − ( r a 1 ) 2 { c 1 H n ( r a 1 ) + c 2 | F | ( a ; b ; x ) [ 1 4 ( 1 − C a 1 2 ) ; 1 2 ; ( r a 1 ) 2 ] } , (63)

where H_n is the Hermite polynomial function and | F | ( a ; b ; x ) is the Kummer confluent hypergeometric function. c₁ and c₂ are constants of integration.

The constant a₁ is in a key position

a 1 = h 2 π 1 2 π b m e . (64)

By substituting b from Equation (14) we have

1 2 π b m e = r i m e c , (65)

and by substituting G_i from Equation (7) we have

a 1 = h G i 2 π c 3 = l P i . (66)

By definition l P i is the Planck length of the virgin universe. In the CBU theory the Planck length is dependent on the radius r and time. The numerical value is l P i = 1.43516 × 10 − 14   m .

It can be shown that

( l P i 2 r i ) 2 = 2 π − 1 4 α f s , (67)

where α_fs is the fine structure constant, 1/137.036. The result is not a coincidence, but a purely physical relation based on known physical constants. The equation emphasizes the significance of the Planck scale and the curvature radius r_i of the virgin universe. It might be seen as a proof of the connection between gravity and quantum mechanics.

Equation (67) reflects the intuitive thought expressed by Richard Feynman in the 1950ies: “137 holds the answers to the Universe”, [26] .

Here is the confirmation.

Substituting G from Equation (15) into Equation (66) we obtain a general expression for the Planck length

l P ( r ) = l P r = 1 2 π h c b r . (68)

A simple check using r₀ = 4.20550 × 10²⁶ m shows that the Planck length according to Equation (68) results in 1.61625 × 10⁻³⁵ m, which exactly equals the current official value.

In analogy with the hydrogen atom the ground state energy E_GSi is postulated to be of the form

E G S i = h 2 8 m e l P i 2 = h c 4 π r i . (69)

The general expression of the instantaneous ground state energy takes the form

E G S ( r ) = E G S r = h 2 8 m e l P i l P r = π h c l P i 4 l P r r i = π 2 l P i 2 r i h c b r . (70)

The Planck energy W_P has an important bearing on the influx of new matter. By definition

W P i = h c 5 2 π G i = l P i r i W e . (71)

At the initial event a positron and an electron are exited, from Equation (71) we deduce that the particles originate from

2 W e = W P i l P i / 2 r i . (72)

The significance of l P i / 2 r i = 13.453499 in the CBU theory is obvious here, if W_Pi is considered the virtual dark energy of the virgin universe, then the real energy is obtained by dividing it with the ratio l P i / 2 r i .

The general expression for W_P(r) = W_Pr is

W P r = h c b r . (73)

For r = r₀, W_Pr is 1.9560815 × 10⁹ J, which is in full compliance with the official value.

A comparison of Equation (73) with Equation (70) shows the relation between the Planck and ground state energy

E G S r = π 2 l P i 2 r i W P r . (74)

The temporal change of the gravity strength and the momentum has an impact on the uncertainty window, i.e. ∆p∆x, which constitutes the classical Heisenberg formulation, cf. Ronald D. Adler [27] . We divide ∆x into a Heisenberg component ∆x_H = h/(4π∆p_H) and a gravity component

Δ x g = h 4 π Δ p g . (75)

The momentum uncertainty is

Δ p = d p d r l P r . (76)

We have

d p d r = 8 π b r c B L ( 1 + 1 4 L ) , (77)

where the term containing 1/4L is due to the fact that L is a ln(r) function.

Presuming that the location uncertainty is Δ x = l P r / 2 , we have

Δ p Δ x H ≤ 1 2 d p d r l P r 2 = h 4 π f r , (78)

where ∆p = f_r·∆p_H and

f r = 4 B L ( 1 + 1 4 L ) . (79)

Adler has derived an expression for the gravity component

Δ x g = π Δ p l P r 2 h = h 4 π Δ p ( 2 π l P r Δ p h ) 2 . (80)

After some algebraic manipulation we arrive at the final uncertainty equation

Δ p Δ x = Δ p ( Δ x H + Δ x g ) ≤ h 4 π ( f r + f r 2 ) = h 4 π F r , (81)

where F_r = f_r(1 + f_r) is the overall uncertainty factor and ∆p = F_r·∆p_g. When f_r is substituted into F_r, we obtain a proximity value to the real uncertainty factor: F r U = 4 B L ⋅ ( l P i / 2 r i ) , which applies to the universe BH. A similar analysis leads to F r B = B L ⋅ ( l P i / 2 r i ) for the galaxy BH.

We verify the validity of F_rU by substituting Equation (79) into F_r = f_r(1 + f_r), we have

F r U 4 B L = ( 1 + 1 / 4 L ) [ 1 + 4 B L ( 1 + 1 / 4 L ) ] = 13.4539. (82)

The result shows the resemblance with l P i / 2 r i = 13.4535 . It also proves that the reasoning of R.D. Adler is correct. We should consider the mathematical model as an approximation of the quantum process behind the injection of real matter particles.

A detailed analysis of the physics of galaxy BHs will be presented in Chapter 8.

The most compact version of the Field Equation is

G μ ν = 8 π G T μ ν , (96)

where G_μν is the Einstein tensor and T_μν is the energy-momentum tensor. Equation (96) has an analytical solution, provided T_μν is isotropic and homogeneous.

One objective of the study is to find a logical solution to the Field Equation without the cosmological constant Λ. We will show that incoming new matter and a variable G affect the energy density and the expansion pressure in a way making Λ unnecessary.

The Einstein tensor is divided into the Ricci curvature tensor R_μν and the Ricci scalar R according to

G μ ν = R μ ν − 1 2 R g μ ν . (97)

The components of the R_μν tensor are obtained by using the Christoffel symbols, the procedure is found in any textbook on General Relativity, cf. Carroll [17] . The Ricci tensor is

R μ ν = ( − 3 a ¨ a 0 0 0 0 a ¨ a + 2 ( a ˙ a ) 2 + 2 k c 2 a 2 r c u r 2 0 0 0 0 a ¨ a + 2 ( a ˙ a ) 2 + 2 k c 2 a 2 r c u r 2 0 0 0 0 a ¨ a + 2 ( a ˙ a ) 2 + 2 k c 2 a 2 r c u r 2 ) , (98)

where r_cur is the curvature radius and k the curvature parameter according to the Friedmann-Robertson-Walker metrics, k cf. Equation (19).

The Ricci scalar is

R = − 6 [ a ¨ a + ( a ˙ a ) 2 + k c 2 a 2 r c u r 2 ] . (99)

The metric tensor g_μν is

g μ ν = ( 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1 ) . (100)

Based on Equations (96) and (99) the unit of the energy-momentum tensor T_μν components are that of a mass density, kg/m³. The tensor is defined as follows

T μ ν = ( ρ e q 0 0 0 0 P e x p c 2 0 0 0 0 P e x p c 2 0 0 0 0 P e x p c 2 ) . (101)

Our task is to find the equivalent density ρ_eq and the pressure P_exp that accounts for both expansion and acceleration. From Equation (87) we have

P E M = − b 4 r ( 1 + 2 B L ) . (102)

The density ρ_eq affects the velocity of expansion, i.e. the Hubble parameter. It consists of two components

ρ e q = ρ m x + ρ E M , (103)

where ρ_EM is caused by the momentum change, ρ_mx is the matter + radiation density.

According to the 1^st Law of Thermodynamics we have

d W e x p d t + P E M d V d t = 0 , (104)

where W_exp = ρ_EMc²V, the time derivative of which is

d W e x p d t = ρ ˙ E M c 2 V + ρ E M c 2 d V d t . (105)

Here d V / d t = 3 V a ˙ a . We approximate that

ρ ˙ E M ≅ − ρ E M a ˙ a . (106)

From Equations (104) and (105) we now have

ρ E M = − 3 2 P E M c 2 . (107)

Substituting P_EM from Equation (102) we have

ρ E M = ρ m x ( 1 + 2 B L ) , (108)

and then Equation (103) becomes

ρ e q = ρ m x + ρ E M = 2 ρ m x ( 1 + B L ) . (109)

In order to get the total pressure P_exp responsible for both expansion and acceleration we need to include the gravitational parameter G into the 1^st Law of Thermodynamics

d ( G W e x p ) d t + G P e x p d V d t = 0 , (110)

where W_exp = ρ_eqc²V. After some manipulations, cf. Section 4.1, we have

G ˙ G ρ e q + ρ ˙ e q + 3 ρ e q a ˙ a = − 3 P e x p c 2 a ˙ a . (111)

From Equation (15) we conclude that G ˙ G = − a ˙ a . Further, we have that ρ ˙ e q = 2 ρ m x ( B − B L − 1 ) a ˙ a . Finally, the pressure is obtained from

P e x p c 2 = − 2 ρ m x 3 ( 1 + B + B L ) . (112)

Now the Einstein Field Equation can be completed in a form which eliminates the need for a cosmological constant Λ

R μ ν − 1 2 R g μ ν = 8 π G T μ ν . (113)

All components of the energy-momentum tensor contain 2ρ_mx. By substituting G from Equation (15) and ρ_mx from Equation (16) into Equation (113) we arrive at the following expression for the tensor

T μ ν = 3 ( c r 0 ) 2 ( 1 + B L a 2 0 0 0 0 − 1 3 ⋅ 1 + B + B L a 2 0 0 0 0 − 1 3 ⋅ 1 + B + B L a 2 0 0 0 0 − 1 3 ⋅ 1 + B + B L a 2 ) . (114)

where T_μν replaces 8πGT_μν.

Due to substitution G has vanished, the energy-momentum tensor is independent of the gravitational parameter. Making use of the definition of the Hubble parameter in Equation (30) we can write the tensor as follows

T μ ν = ( 3 ( h H 2 + c 2 a 2 r 0 2 ) 0 0 0 0 − ( h H 2 + ( 1 + B ) c 2 a 2 r 0 2 ) 0 0 0 0 − ( h H 2 + ( 1 + B ) c 2 a 2 r 0 2 ) 0 0 0 0 − ( h H 2 + ( 1 + B ) c 2 a 2 r 0 2 ) ) . (115)

The term 1 + B indicates the extra pressure required for acceleration.

The Field Equation written in its most compact form is now

G μ ν = R μ ν − 1 2 R g μ ν = T μ ν . (116)

The Friedmann-Robertson-Walker temporal equation, which also may be called the Hubble expansion equation, is

R 00 − 1 2 R g 00 = T 00 . (117a)

We have

− 3 a ¨ a + 3 a ¨ a + 3 ( a ˙ a ) 2 + 3 k c 2 a 2 r c u r 2 = 3 h H 2 + 3 c 2 a 2 r 0 2 . (117b)

Further,

( a ˙ a ) 2 + k c 2 a 2 r c u r 2 ≡ h H 2 + c 2 a 2 r 0 2 . (117c)

By definition a ˙ a = h H . k = 1 means that the universe is closed. ar_cur = ar₀

proves that the average curvature radius equals the radius r of the observable universe, a circumstance emphasized by Einstein in the cosmological constant paper [5] .

The acceleration equation is obtained from

R i j − 1 2 R g i j = T i j , (118)

where i and j = 1, 2, 3. After some algebraic manipulations and the substitution of Equation (117c) into the equation, we end up with

a ¨ = c 2 2 a r 0 2 B (119a)

or

g = r 0 a ¨ = c 2 2 a r 0 B . (119b)

For a = 1, B = 0.099219 we have g₀ = 1.0533 × 10⁻¹¹ m/s².

The equation corresponds to our Ansatz in Equation (28). The acceleration g is an inherent characteristic of the expanding universe. Over a comoving distance d in the vicinity of an observer, the scale factor a in Equation (119b) equals 1 − d/r₀ ≈ 1 – extremely small influence.

The inherent acceleration g provides a Coriolis effect, which explains the rotational behaviour of the galaxies and the celestial movements at large, and thereby eliminates the need for dark matter, cf. Eriksson [6] .

It is important to prove that the T_μν tensor fulfils the temporal energy conservation condition. The general expression using Christoffel symbols Γ is, cf. Carroll [17] ,

∇ μ T μ 0 = ∂ μ T μ 0 + Γ μ 0 μ T 0 0 − Γ μ 0 λ T μ λ = − ∂ 0 ρ e q − 3 a ˙ a ( ρ e q + P e x p c 2 ) = 0. (120a)

The time derivative is

∂ 0 ρ e q = ∂ ∂ t ( 1 + B L a 2 ) = − ( 2 ( 1 + B L ) a 2 − B a 2 ) ( a ˙ a ) . (120b)

We have

1 a 2 ( 2 + 2 B L − B − 3 − 3 B L + 1 + B + B L ) ( a ˙ a ) = 0. (120c)

The conservation condition is met. The result shows that the components ρ_eq and P_exp/c² are correctly derived.

The gravitational parameter G is extremely large in the initial phase, G_i ≈ 5 × 10³¹ m³/kg/s². The parameter decreases successively with the expansion. The photons in the cosmic microwave background (CMB) propagating faster than the expansion start from a deep gravitational well. We write the energy equation of the electromagnetic radiation as a gravitational gradient equation

h d f + g h f c 2 d r = 0 , (121)

where f is the frequency, g the gravitational acceleration along r. We obtain g from

g = G M r 2 = 2 c 2 r . (122)

Here we make use of M = 4πbr²/c² and G = c⁴/2πbr, cf. Equations (13) and (15). Having that f = c/λ, df = −cdλ/λ² and r = r₀a, where a is the scale factor, we write

∫ ​ d λ λ = 2 ∫ ​ d a a . (123)

Integration implies that λ is proportional to a². We deduce that

λ 0 − λ s λ s = ( 1 − a a ) 2 = z g . (124)

Here z_g is the gravitational redshift, λ_s and λ₀ are the wavelengths at the source and at the observer respectively.

As known, the cosmological redshift is

z c r = 1 a − 1. (125)

The combined redshift is z = z_g + z_cr. We have

λ 0 λ s = f s f 0 = z + 1 = 1 − a + a 2 a 2 . (126)

Let a_c stand for the scale factor at the CMB event. For vary small values of the scale factor the frequency of CMB photons decrease approximately according to

f 0 ≈ a c 2 f c . (127)

John Huchra at the Harvard-Smithsonian Center for Astrophysics has established the addition rule for combining multiple types of redshifts, [28] .

An interesting feature of Equation (126) is that the difference between z_cr and z changes very slowly for scale factors close to 1. The discrepancies of the current distance estimates in the vicinity of the Milky Way are still within error limits.

Data on proper distances (d_pΛ, d_pCBU), lookback times (t_Λz, t_Λ(z_cr) and t_CBU) are collected in Table 1. Λ refers to the standard ΛCDM model, cf. [29] and [30] , and CBU to calculated numbers of the current theory.

In Figure 11 the data of Table 1 show the correlation between the ΛCDM and CBU models. There is a good agreement between the lookback time graphs t_CBU and t_Λz. t_CBU is based on Equation (34), while t_Λz originates from the graph in [29] . z was determined from Equation (126) using the appropriate scale factor. d_pCMB was calculated according to a unique algorithm presented in Section 4.5, [6] . Even for the proper distances the similarity between the curves is good considering the different approaches.

The difference between the look-back times t_Λ(z_cr) and t_CBU, Figure 12, is interesting, because it indicates a deviation in the estimates of distant cosmic events. For instance, the most distant observed galaxy Glass-z13 has a redshift of 13, which would mean that a fully developed galaxy had evolved in about 0,35 Gyr after the Big Bang. This is an intriguing paradox; it is most uncertain that billions of stars would have developed in such a short time. According to CBU the lookback time would be 11 Gyr and, considering the age of the universe being 14.4 Gyr, the time for the galaxy to evolve would be a more plausible time of 3.4 Gyr.

We obtain the energy density w_BB of the black body radiation from the classical Stefani-Boltzmann equation

w B B = 8 π 5 k B 4 15 c 3 h 3 T 4 = α B T 4 . (128)

Here α_B = 4σ_SB/c = 7.565723 × 10⁻¹⁶ J/K⁴·m³ is the radiation density constant, σ_SB is the Stefani-Boltzmann constant, and k_B the Boltzmann constant.

The number density of the photons is obtained from, cf. Wikipedia: Photon Gas,

n p h = 16 π ζ ( 3 ) ( k B T h c ) 3 , (129)

where ζ(3) = 1.202056 is the Riemann zeta-function. By dividing Equation (128) with Equation (129) we obtain an expression for the photon energy

W p h = α B T 16 π ζ ( 3 ) ( h c k B ) 3 . (130)

If we assume that the photon energy at the CMB event equals one of the two photons caused by the annihilation, we have W_phc = W_e. The current photon energy is then

W p h 0 = a c 2 W e = α B T 0 16 π ζ ( 3 ) ( h c k B ) 3 . (131)

Accordingly, for T₀ = 2.72548 K we have

a C M B = a c = α B T 0 16 π ζ ( 3 ) ( c h k B ) 3 1 W e = 3.523 × 10 − 5 . (132)

Due to the square root the number is much larger than usually suggested (10⁻⁷ - 10⁻⁹).

The current photon energy is W_ph0 = 1.0164 × 10⁻²² J and the corresponding frequency f₀ = W_ph0/h = 153.4 GHz, slightly below the optimum frequency of the Planck black body spectrum of f_Imax = 160.23 GHz.

The black holes in the centre of the galaxies are seeds stemming from the CMB transition. During the transition, half of the energy forms the CMB radiation, while the other half is spread out in 10¹² - 10¹³ galaxy black holes. Right after the CMB event the total energy of the BHs is W B c t o t = 0.5 × 4 π r b r c 2 = 6.32 × 10 61   J . There are several indications that the radius r_B of an individual BH changes very slowly, which is also shown by the outcome of this study.

We use the inner photon radius R_B as the appropriate radius instead of the Schwarzschild radius r_s = 2·R_B. We have

R B = M B G c 2 , (141)

where M_B is the mass of the BH and G is the gravitation parameter of the universe at a given time. With G inversely proportional to the scale factor a and R_B constant M_B becomes inversely proportional to a, (M_B0 = M_Bc/a). Currently the energy content of all black holes is:

W B 0 t o t = W B c t o t / a c = 1.793 × 10 66   J or M B 0 t o t = 2.00 × 10 49   kg (10¹⁹ suns)

The estimates of the number of galaxies are so far very rough, a common number is 2 × 10¹¹ for the observable universe or 1.6 × 10¹² for the whole world, cf. Mario Livio, The Universe. Space. Tech., June 2022. The average BH mass would then be 1.25 × 10³⁷ kg or 6 × 10⁶ solar masses. These numbers are quite usual according to observations. The black hole of the Milky Way, Sagittarius* is estimated to have a mass of 8.3 × 10³⁶ kg.

When G from Equation (15) is substituted into Equation (141) we have

M B = 2 π b R B r c 2 . (142)

The mass of an individual BH increases at the same rate as the universe itself, i.e. inversely proportional to a. Like in the universe case there is an influx of electron-positron pairs.

The time derivative is

d M B d t = 2 π b R B B L c . (143)

The equation is needed later, when we want to determine the star formation change with time as a function of the galaxy’s stellar mass, cf. Figure 18.

The black hole has an inside gravitational parameter (on the surface), by applying Equation (6) we have

G B = c 4 2 π b R B . (144)

The parameter contains only constants and is accordingly a constant itself.

In the current study, the universe is considered to be a black hole without a singularity. The reason lies in the pressures that cause the expansion and their opposite action against gravitational contraction. The curvature of space governs electron-positron influx and thereby regulates the pace of expansion. In the galaxy BHs we can equally define a balance equation between expansion and contraction. Compared to the universe BH there is a much more moderate influx of electron-positron pairs (see next section) that causes an increase in the density without increasing the size.

The pressure driving the density increase is obtained from

d W d t + P d V d t + V d P d t = 0. (145)

The volume is constant along with R_B, so we are left with the equation

d W d t = − V d P d t . (146)

The expansion pressure is

P B e x p = − W B V B = − 3 b r 2 R B 2 , (147)

where V_B is the volume of the black hole.

The acceleration directed outwards is

g B e x p = 4 π G B P e x p 3 c 2 = − r c 2 R B 2 . (148)

Alternatively, the same result is obtained by considering that g = − ∇ ϕ , where ϕ = − G B M B / R B . For the symmetric spherical case we have

g B e x p = ∂ ∂ R B ( r c 2 R B ) = − r c 2 R B 2 . (149)

The contracting acceleration due to gravitation is obtained from the familiar equation

g B c o n t r = G B M B R B 2 = r c 2 R B 2 . (150)

We end up with the balance equation

g B e x p + g B c o n t r = 0. (151)

The black hole has two boundary spheres, the inner one or the photon sphere, r_B = 0.5·r_s and the outer one or the black hole event horizon at r_s. In between there is an energy gap. From these boundaries there is an influx of electron-positron pairs into the inside of the BH and proton-antiproton pairs into the space outside the BH, Figure 15. In the galaxy BH case the number of injected pairs, N_pair, is equal on the inside as on the outside, because R B / R B = 1 .

According to our earlier stated postulate all stellar matter of a galaxy originates from the central BH event horizon. As a result we obtain a clear-cut relation between the stellar mass, M_S, and the BH mass, M_B. The relation is

M S M B = 2 ν B i 2 m p N p a i r 2 m e N p a i r = ν B i 2 m p m e = 1786.4. (152)

This is a significant result. Some bold presumptions lead to a result that has a strong anchorage in current observations. The average example of Section 8.1, M_B = 1.25 × 10³⁷ kg, suggests a galaxy stellar content of 2.23 × 10⁴⁰ kg, which is a typical value.

During the last decade several papers have been published, wherein the mass ratio is discussed and different theories about the cause has been presented, cf. [31] [32] and [33] . The standard theory suggests that stellar matter feeds the black holes, not vice versa as the present theory (CBU) suggests.

An indication that the CBU result is the most plausible one, is shown in Figure 16. The line M B = [ ν B i 2 ( m p / m e ) ] − 1 ⋅ M S is shown in a chart created by Sandra Faber, [34] . There is a perfect fit with the “Small dense” galaxies. The line shows there is a linear correlation, which for the present theory means that M_B grows linearly and the assumption of R_B being constant is correct.

In Figure 17, a similar pattern as in Figure 16 is shown for galaxes of type “Active Galactic Nucleus”, [31] indicating that the ratio M S / M B = ν B i 2 ( m p / m e ) results in a perfect fit. In Figure 18 the change in stellar mass dM_S/dt is shown as a function of the mass. Here the observations are related to galaxies in a star forming phase, cf. [35] . The CBU theoretical time derivative hits into the middle of the statistical pattern.