APPENDIX N

APPENDIX N: CMBA DETAILS

A group of \({\gamma _{cmb}}\) are represented by a transverse wave train crossing a canister. The train lies in a plane in polar coordinates that corresponds to \(\theta = \pi /2\) (equator). Eq. (4.20) leads one to the equation of motion for the transverse potential in the weak field limit: \[\begin{array}{l}{A_\theta }(\tau ,r,\phi ) = {r^{ – 1}}\sin \psi (\tau ,r,\phi ),\\\psi (\tau ,r,\phi ) = 2\pi \lambda {(\tau ,r,\phi )^{ – 1}}\xi (\tau ,r,\phi ),\\0 = {c^{ – 2}}{\partial _\tau }[(1 – V){\partial _\tau }\psi ] – {\partial _r}[(1 + U){\partial _r}\psi ] – {r^{ – 2}}{\partial ^2}\psi /\partial {\phi ^2},\end{array}\tag{N.1}\] Where \(\xi = 0\) at the center of the wave train (moving at velocity c). The gravitational potential \(U = V\), and use Eq. (F.3) to get \(V(r,\tau ) = 2{c^{ – 2}}{A_N}(r,\tau )\). The time dependence of Eq. (F.3) for this problem comes from substituting from Appendix Z the following in the summations of Eq. (F.3): \({Z_1} = – 958,{\rm{ }}{Z_2} = 958,{\rm{ }}{Z_3} = 0\) when \(\tau \) is before the early spring gravitational change, and \({Z_1} = – 657,{\rm{ }}{Z_2} = 958,{\rm{ }}{Z_3} = – 300\) after. The transition, Eq. (7.5), is completed during a tiny fraction of the millions of years for a \({\gamma _{cmb}}\) to travel within a canister, so the transition can be treated as instantaneous. Gravitational fields change everywhere in the universe at the same time.
One can neglect the small deflection of the wave train (and the train is very short compared to the size of the canister), so it is useful to use the approximate coordinates: \[\begin{array}{l}r = {({x^2} + {y_{IP}}^2)^{1/2}},\\{\partial _r} = x\,{r^{ – 1}}\,{\partial _x},\\{\partial _\varphi } = – {y_{IP}}\,{\partial _x},\\\psi = 2\pi \,\lambda {(\tau ,x)^{ – 1}}\,(x – c\tau ),\end{array}\tag{N.2}\] where \({y_{IP}}\) is a constant (the impact parameter) for a \({\gamma _{cmb}}\) moving from \(x = – \infty \) to \(x = \infty \). The center of the canister is at \(x = 0 = {y_{IP}}\). Next, use Eq. (N.2) to convert polar derivatives into x derivatives: \[\begin{array}{l}{\partial _r}\psi = x\,{r^{ – 1}}\,{\partial _x}\psi ,\\{\partial _\varphi }\psi = – {y_{IP}}\,{\partial _x}\psi ,\\{\partial _r}({\partial _r}\psi ) = {x^2}{r^{ – 2}}{\partial _x}({\partial _x}\psi ) + x{y_{IP}}^2{r^{ – 4}}({\partial _x}\psi ),\\{\partial _\varphi }({\partial _\varphi }\psi ) = {y_{IP}}^2{\partial _x}(\,{\partial _x}\psi ).\end{array}\tag{N.3}\] Next, evaluate the items of Eq. (N.3) in the limit where \(x – c\tau \to 0\) (the center of the wave train): \[\begin{array}{l}{\partial _\tau }\psi \to – 2\pi c{\lambda ^{ – 1}},\\{\partial _\tau }({\partial _\tau }\psi ) \to 4\pi c{\lambda ^{ – 2}}{\partial _\tau }\lambda ,\\{\partial _x}\psi \to 2\pi {\lambda ^{ – 1}},\\{\partial _x}({\partial _x}\psi ) \to – 4\pi {\lambda ^{ – 2}}{\partial _x}\lambda .\end{array}\tag{N.4}\] Finally, insert the items of Eq. (N.3) into the third element of Eq. (N.1) and then insert the limits of Eq. (N.4) into that equation to obtain the following, \[0 = 0.5({c^{ – 1}}{\partial _\tau }V – {\partial _x}V) + (1 – V){c^{ – 1}}{\lambda ^{ – 1}}{\partial _\tau }\lambda + (1 + V + {y_{IP}}^2{r^{ – 2}}){\lambda ^{ – 1}}{\partial _x}\lambda {\rm{ }}{\rm{.}}\tag{N.5}\] Eq. (N.5) is subject to some constraints. Let \({\tau _{GC}}\)be the universal time when local gravity is turned on in early spring—the same for every canister in the universe. \(V(\tau ,r)\) is a discontinuous function of r when \(\tau = {\tau _{GC}}\), and \(\lambda \) must be discontinuous as well (otherwise \[{\partial _\tau }V = 0\]). Furthermore, Eq. (N.5) is evaluated at the center of the (tiny) wave train, so \({\partial _\tau }\lambda = c{\partial _x}\lambda \) (except at the discontinuity).
It is necessary to divide the solution of Eq. (N.5) into two ranges which are connected by the discontinuity. The gravitational potential has two time-independent ranges, \({V_ < }(r)\) and \({V_ > }(r)\). \({V_ < }\) is calculated as described after Eq, (N.1) using the \({Z_3} = 0\) set, and \({V_ > }\) uses the \({Z_3} = – 300\) set. The discontinuity is at \({x_{GC}} = c{\tau _{GC}}\) or \({r_{GC}} = {({x_{GC}}^2 + {y_{IP}}^2)^{1/2}}\). \({V_ < }\) is used for \(x < {x_{GC}}\) and \({V_ > }\) after. The time derivative of V is infinite at the discontinuity as is the time derivative of \(\lambda \). These infinities must cancel, so one requires the time derivatives of Eq. (N.5) to vanish at the discontinuity, \[0 = 0.5{\partial _\tau }V + (1 – V){\partial _\tau }\ln \lambda {\rm{ (}}\tau = {\tau _{GC}}).\tag{N.6}\] Define \(\xi = {(1 + V)^{1/2}}\), then Eq. (N.6) is equivalent to \(\xi \lambda \) being continuous at the discontinuity. Thus, one finds the following condition across the discontinuity, \[{\lambda _ > }({x_{GC}}) = {\lambda _ < }({x_{GC}}){(1 + {V_ < }({r_{GC}}))^{1/2}}{(1 + {V_ > }({r_{GC}}))^{ – 1/2}}.\tag{N.7}\] The equations for \(\lambda \) are as follows, \[ \begin{aligned} \partial \ln \lambda_< / \partial x &= \left[ (\partial V_<(r) / \partial x)\{4 + 2y_{IP}^2 r^{-2}\}^{-1} \right], & x < x_{GC}, \\[6pt] \partial \ln \lambda_> / \partial x &= \left[ (\partial V_>(r) / \partial x)\{4 + 2y_{IP}^2 r^{-2}\}^{-1} \right], & x > x_{GC}, \\[6pt] &\quad [ \; ] \; \text{evaluated at } r = (x^2 + y_{IP}^2)^{1/2}. \end{aligned}\tag{N.8} \] The initial condition is \(x = – \infty \) and \({\lambda _ < } = \lambda ( - \infty )\). The integration continues until \(x = {x_{GC}}\) and \({\lambda _ < }({x_{GC}})\) is known. The next initial condition is \(x = {x_{GC}}\) and \({\lambda _ > } = {\lambda _ > }({x_{GC}})\)—known from Eq. (N.7). The integration continues until \(x = \infty \), and \(\lambda (\infty ) = {\lambda _ > }(\infty )\).
There are only two parameters that determine both \({V_ < }(r)\) and \({V_ > }(r)\), the total mass of the canister (solar masses) and the radius of the MDGO at the center of the canister (light years). There are only two additional parameters \({y_{IP}}\) (light years) and \({x_{GC}}\) (light years) required to determine the desired result \(T(\infty )/T( – \infty ) = \lambda ( – \infty )/\lambda (\infty )\).
There is an illustrative exact solution of Eq. (N.8) when \({y_{I{P_{}}}} = 0\) as follows: \[\begin{array}{l}T(\infty )/T( – \infty ) = {f_ > }/{f_ < },\\{f_ < } = {(1 + {V_ < }({x_{GC}}))^{1/2}}\exp ({V_ < }({x_{GC}})/4),\\{f_ > } = {(1 + {V_ > }({x_{GC}}))^{1/2}}\exp ({V_ > }({x_{GC}})/4).\end{array}\tag{N.9}\] Since the gravitational potentials are functions of r, it follows that the temperature ratio is a function of \({x_{GC}}^2\) (symmetric). The same holds for the solution of Eq. (N.8).
Numerical solutions of Eq. (N.8) are obtained through a routine program. The temperature ratio, \([T(\infty ) – T( – \infty )]/T( – \infty )\), is a function of two parameters for the canister gravitational-potential, and two photon-parameters relative to the location of the center of the canister.
The canister parameters are \({R_{MGO}}\) (light years) and \({M_{CAN}}\) (solar masses). The dimensionless potential, V, is calculated using Eq. (F.3), where \(V = {V_{00}} = 2{c^{ – 2}}{A_N}\), and \({R_E} = {R_{MGO}}\). For use of Eq. (F.2), \(M = {M_{CAN}}\). The photon parameters are \({y_{IP}}\) (light years) and \({x_{GC}}\) (light years). A photon that arrives at my telescope is assumed to travel along a straight line from a distant point much farther away than the center of the canister. \({y_{IP}}\) is the minimum distance of the photon from the center of the canister along that path (impact parameter). \({x_{GC}}\) is the distance of the photon along that path (relative to the point of closest approach to the canister center) when the gravitational change occurred. \({x_{GC}}\) is negative if the gravitational change occurred before the photon reached closest approach, else positive.
Figs. N1 and N2 illustrate some solutions for \({M_{CAN}} = {10^{14}}\) solar masses and \({R_{MGO}} = 2{\rm{x1}}{{\rm{0}}^6}\) light years—the same values used in Fig. 4 for deflections due to gravitational lensing. The temperature ratio is directly proportional to \({M_{CAN}}\).

Figure N.1. DeltaT/T for four values of Yip (0 mly red, 2 mly grey, 4 mly green, and 6 mly blue).

Figure N.2. DeltaT/T for four values of Xgc (0 mly red, 2 mly grey, 4 mly green, and 6 mly blue).

Assume that my telescope can accurately measure the temperature of infrared radiation (\({T_{OBS}}\)) coming from a sufficiently small area of the sky (excellent resolution and response). In that case, the temperature ratio is \(({T_{OBS}}/{T_{CMB}}) – 1\), where \({T_{CMB}} = 2.725\,\;K\). The slightly warm object of Fig. N2 is large (diameter about 3.4 arc minutes) compared to a galaxy at that distance. I believe that there would be about 10 of these objects per square degree. A carefully designed collection of baseline interferometers could possibly see these objects behind the foreground point sources. No existing observatory is sufficient. A sky map of these objects would lead to a more accurate value for the average density of matter throughout the universe.
The polarization of the photons (described above) is perpendicular to the plane of photon motion. There are also photons with polarization lying in the plane of motion. The model used above for canister gravitation includes the mass of the canister galaxies (as if they were smeared out in the central gas ball). Thus, the model-canister gravitational potential is a function of r, and there would be no net polarization for the whole object.
Galaxies in a canister follow random orbits. A more realistic model would include a gravitational perturbation that reflects the location of the galaxies at the time of interest, \[V \to V(r) + \delta V(r,\theta ,\phi ).\tag{N.10}\] The potential of Eq. (N.10) will give rise to a small net polarization for the whole object. A sky map of the object polarizations would not yield any significant information (due to randomness).
A foreground canister is transparent for photons from the object canisters of interest—no change of temperature. However, foreground canisters can deflect photons as per Fig. 4. A bundle of temperature-shifted photons from a small area of the object canister (a pixel) will be subjected to many small random deflections from foreground canisters on their journey to my telescope. The end result is that the size and temperature profile of an object-canister image should be weakly affected by the foreground canisters, but the position of the center of the image could be measurably displaced on the sky map. The apparent position of the center of a canister of interest undergoes a step (in a random walk) with each interception with a foreground canister. This randomness would not affect the average density of universal matter.