(* Copyright 2017 Daniel Strano *)

(* Strano 2017 shows radiating vacuum solutions for Schwarzschild and Reissner-Nordstroem black holes, and implies radiating vacuum solutions for Kerr-Newmann and its Lorentz-boosted extension. *)

(* Naive application of Fermi's golden rule implies a mass flux (per time) of natural 1/2 out of any black hole. However,single gravitons would have to potentially carry half the mass of a body like Sgr A* by blind application of the perturbation theory. Gravitons with energy equal to or greater than the Planck energy would have de Broglie wavelengths which fit inside their Schwarzschild radii, with exactly extremal amounts of momentum,implying naked singularities. (While this implies that gravitons with Planck energy or greater are not allowed in this case, it invokes Penrose’s cosmic censorship hypothesis without pointing to physical principles that enforce that hypothesis.) However, a more realistic approximate model is that gravitons of the Planck energy or heavier, heavy enough to be black holes themselves, are too heavy to escape black holes, capping the average emissive power at approximately Ep/(2*rs), with “Ep” being the Planck energy.  Then, we can talk about graviton temperature in an observable universe. *)

(* Define unit conversions up front: *)

mpToKg=217647051*10^(-16)

timePToSeconds=53911613*10^(-51)

tempPToK=141680833*10^24

lpToMeters=161622938*10^(-43)

epToJoules=1956*10^6

SolarToKg=198855*10^25

mpToSolar =mpToKg/SolarToKg

yearToSeconds=31557600

lyToMeters=9460730472580800

JoulesPerKg=89875517873681764

eVtoJoules=1602176620898*10^(-31)

protonMassKg=16726218*10^(-34)

higgsMassEV=12509*10^7

metersPerParsec=30856775814913700

hubbleP=719*10^2/(10^6*metersPerParsec )*timePToSeconds

(* For Stefan-Boltzmann law: *)

sigmaPlanck=8*Pi^5/(15*(2*Pi)^3)

(* For Wien's displacement law: *)

wiensMK=2987772917*10^(-12)

speedOfLightMS=299792458

(* Define area function for simplicity of expression: *)

area[rs_]:=4*Pi*rs^2

(* The scalar monopole waves should have a background temperature. If these waves are physical, a cosmological remnant temperature would be left at the beginning of the Grand Unified Epoch, after gravity freezes out from the grand unified (strong, electro-, weak) force. It would decrease in density with change in scale factor as does the cosmic microwave background. However, black holes would tend to convert baryonic matter to this form of ("dark") radiation, and baryonic particles would also gain ("dark") mass when interacting with this background temperature, since these are mass monopole waves. *)

(* Suppose that dark energy is actually the background temperature of mass monopole interaction waves.*)

(* Approximate dark energy density, in Planck masses per Planck volumes: *)

Clear[deDensity]

deDensity=7*10^(-27)/mpToKg*lpToMeters^3

deDensity*1.0

(* If this is a monopole interaction temperature, this is the temperature in Planck units: *)

deTemp=(deDensity/(sigmaPlanck /2))^(1/4)

deTemp*1.0

deTemp*tempPToK*1.0

(* Peak frequency of thermal spectrum in Hz: *)

speedOfLightMS/(wiensMK/(deTemp*tempPToK*1.0))

(* Spectral radiance at this temperature is neglible at LIGO background measurement from 41.5Hz to 169.25Hz. (Estimate by highest spectral power in range over whole range): *)

16*Pi^2*(169.25*timePToSeconds)^3/(Exp[2*Pi*100*timePToSeconds/deTemp]-1)*((169.25-41.5)*timePToSeconds)*1.0

6.182342343007825`*^-153

6.182342343007825`*^-153/(3*hubbleP^2/(8*Pi))

(* (For above, see Abbott,B.,et al."Searching for a stochastic background of gravitational waves with the laser interferometer gravitational-wave observatory." The Astrophysical Journal 659.2 (2007):918.) *)

(* To give the appearance of a cosmological constant, the temperature would have to be buffered by energy emission from black holes. *)

(* Approximate energy density change in one Planck time, due to Hubble expansion, in Planck power per Planck volume: *)

(1-1/(1+hubbleP)^4)*(7*10^(-27)/mpToKg*lpToMeters^3)*1.0

(* If we estimate, what is the power output of astronomical black holes? Estimates put the number of black holes in Milky Way as between 10^7 and 10^9, based on the masses of stars in our galaxy, compared to approximately 10^11 stars in our galaxy. We can estimate there to bo very roughly 10^16 to to 10^17 stars in the observable universe, based on number of galaxies counted and number of stars in our galaxy. How many black holes does this imply "on the back of an envelop"? *)

10^17/(10^11)*10^9*1.0

10^16/(10^11)*10^7*1.0

(* We estimate there to be very roughly between 10^12 and 10^15 astronomical black holes in the observable universe. *)

(* The rate of loss of mass from a black hole to monopole waves, in Planck units, is approximately 1/m. This means, at time t after formation, the mass of a perfectly quiescent black hole, neglecting Hawking radiation, goes like m[t_]:=Sqrt[m0^2-2*t]. It can be checked that that only a neglible fraction of the mass of 3 solar mass black hole is lost in 14 billion years, i.e. the temperature of an astronomical black hole can be assumed almost exactly constant between its formation and the modern epoch:*)

Sqrt[(3/mpToSolar)^2-2*((14*10^9)*(3.154*10^7)/timePToSeconds)]*mpToSolar

(* However, even though black hole temperature change is negligible, there might be significant average output. Averaged over the entire observable universe, what upper and lower bounds do we estimate on the astronomical black hole scalar monopole interaction wave output? *)

(mpToSolar/3*10^15)/(4/3*Pi*(465*10^8*lyToMeters/lpToMeters)^3)*1.0

(mpToSolar/15*10^12)/(4/3*Pi*(465*10^8*lyToMeters/lpToMeters)^3)*1.0

(* Neglible output compared to change due to expansion. *)

(* How many single Planck mass black holes would it take to match the energy density change? *)

Solve[(1*n)/(4*10^80/lpToMeters^3)*1.0==(1-1/(1+hubbleP)^4)*(7*10^(-27)/mpToKg*lpToMeters^3)*1.0,n]

mpToKg*64.64365541529443`

(* Of course, 64 approximately point-like sources wouldn't provide a spatially homogeneous effect. Further, if Planck mass black holes were present at any given time, they'd only persist for about a Planck time. This would require about 10^45 Planck mass black holes evaporating per second within the observable universe. *)

(* How many kilograms is this over the age of the universe? *)

64.64365541529443`/timePToSeconds*13.82*10^6*3.154*10^7*mpToKg

(* It's about an order of magnitude lower than the observed total baryonic mass. This is a very tight bound for us, because astronomy tells us that primordial black holes do not account for this large of a mass fraction, in the modern day observable universe. However, our model implies being at the critical point of a phase change, with primordial black holes being like melting ice in a freezing temperature glass of water in a warmer room. There need not be as much "ice in the glass" today as when it started "melting." As the local density of primordial black holes go down, the local temperature decreases, and the net energy output of the black holes increases somewhat. There comes a point, though, where there is too little "ice" (too few small black holes) to maintain a relatively steady and uniform temperature. *)

(* If a baryon interacts with the scalar monopole wave background, what are the chances of gaining enough mass to become a black hole? (Maxwell-Boltzmann distribution): *)

(* Assume about 10^80 baryons in the observable universe: *)

Integrate[1/Sqrt[Pi*deTemp]*Exp[-x^2/deTemp]*10^80,{x,1,Infinity}]

Integrate[1/Sqrt[Pi*deTemp]*Exp[-x^2/deTemp]*10^80,{x,1,Infinity}]*1.0

(* Negligible chance of a baryon gaining enough energy to become a black hole. *)

(* What would be the mass tipping point for a black hole to be in equilibrium with this supposed background, in solar masses? *)

(m/.Solve[deDensity==1/(area[2*m]*4*m),m])*mpToSolar*1.0

(* How about for just the cosmic microwave background? *)

cmbPlanck=2.72548/tempPToK

(m/.Solve[sigmaPlanck/2*(cmbPlanck^4)==1/(area[2*m]*4*m),m])*mpToSolar*1.0

(* Based on the Maxwell-Boltzmann distribution, if dark energy is the monopole interaction temperature, roughly how much "dark matter" do we expect due to baryonic rest mass excitation? *)

(* Coupling constant actually depends on level of excitation. Is this significant? *)

(* Assume that dark matter is at the same temperature as dark energy. *)

(* Average excitation level (in Joules, one degree of freedom): *)

1/2*deTemp*epToJoules*1.0

(* What fraction of the proton mass does this imply, in excitation? *)

((1/2*deTemp*epToJoules)/JoulesPerKg)/protonMassKg*1.0

(* Estimated mass of observable universe in Planck masses: *)

1.5×10^53/mpToKg

(* If baryons are in thermal equilibrium with background temperature, how much baryonic rest mass excitation does this imply? *)

(((1/2*deTemp*epToJoules)/JoulesPerKg)/protonMassKg)*(1.5×10^53/mpToKg)

(* Far too little to explain dark matter. However, the appearance of a constant intrinsic "dark energy" density to spacetime volume might be explained as a phase transition buffered by black holes smaller than approximately 170 solar masses, as given above. *)