How the Big Bang Ends up Inside a Black Hole

Citation: Gaztanaga, E. How the Big Bang Ends Up Inside a Black Hole. Universe 2022,8, 257. https:// doi.org/10.3390/universe8050257 Academic Editors: Mariusz P. D ˛abrowski, Adam Balcerzak, Vincenzo Salzano, Yi-Fu Cai and Vincenzo Salzano Received: 27 January 2022 Accepted: 20 April 2022 Published: 21 April 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Review How the Big Bang Ends Up Inside a Black Hole Enrique Gaztanaga 1,2 1Institute of Space Sciences (ICE, CSIC), 08193 Barcelona, Spain; [email protected] 2Institut d Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain Abstract: The standard model of cosmology assumes that our Universe began 14 Gyrs (billion years) ago from a singular Big Bang creation. This can explain a vast range of different astrophysical data from a handful of free cosmological parameters. However, we have no direct evidence or fundamental understanding of some key assumptions: Inflation, Dark Matter and Dark Energy. Here we review the idea that cosmic expansion originates instead from gravitational collapse and bounce. The collapse generates a Black Hole (BH) of mass M≃ 5 × 10 22 M that formed 25 Gyrs ago. As there is no pressure support, the cold collapse can continue inside in free fall until it reaches atomic nuclear saturation (GeV), when is halted by Quantum Mechanics, as two particles cannot occupy the same quantum state. The collapse then bounces like a core-collapse supernovae, producing the Big Bang expansion. Cosmic acceleration results from the BH event horizon. During collapse, perturbations exit the horizon to re-enter during expansion, giving rise to the observed universe without the need for Inflation or Dark Energy. Using Ockham’s razor, this makes the BH Universe (BHU) model more compelling than the standard singular Big Bang creation. Keywords: cosmology; dark energy; general relativity; black holes 1. Introduction A cosmological model predicts the evolution of the observed Universe given some initial conditions. The standard cosmological model [ 1 , 2 ], called Λ Cold Dark Matter ( Λ CDM), assumes that our Universe expansion began in a hot Big Bang creation at the very beginning of space-time. The Λ CDM model explains the formation, composition and evolution of our Universe starting from a quantum fluctuation close to Planck scale. Planck scales are so small that space-time itself has to be treated as a quantum object. Such initial conditions violate energy conservation and are very unlikely as they have a low entropy [3,4]. To address such initial conditions properly, we need a new quantum theory of space-time (Quantum Gravity), which opens the door to brane Cosmology [ 5 , 6 ] which is a very exciting idea, but it is hard to test with observations. Despite these shortfalls, the Λ CDM model is very successful. However, this is at the cost of introducing three more exotic ingredients or mathematical tricks: Inflation, Dark Matter and Dark Energy, for which we have no direct evidence or understanding at any fundamental level. Are they windows for new discoveries, such as String Theory or new forms of matter/energy, or a signal that the paradigm needs to be replaced? Can we choose some different initial conditions and reproduce the success of the Λ CDM model without those exotic fixes and within the known laws of Physics? Here, we present a brief review that summarizes several recent papers [ 7 – 13 ] that suggest a simpler explanation: the Black Hole Universe (BHU). This review also includes some new results and ideas. Some previous studies misinterpreted super horizon scales as scales that were outside the BHU. This is clarified here together with some new details regarding the Big Bounce and the observational interpretation of super horizon perturbations. In Section 2we give a brief presentation of the Λ CDM model and its observational support. In Section 3we present the BHU model using a Newtonian approach. In Cosmology, one can use Newtonian physics to model to a good approximation of both the background and Universe 2022,8, 257. https://doi.org/10.3390/universe8050257 https://www.mdpi.com/journal/universe Universe 2022,8, 257 2 of 22 its perturbations [ 14 ]. A consistent Newtonian version of the BHU solution is a good indication that we understand the physics involved. Appendices Aand Bpresent a summary of the same BHU solution in the more rigorous GR approach. Appendix Cpresents some new considerations of the possible effect of rotation in the BHU solution. We end with a Summary and Discussion that includes a review of related literature and previous results and a comparison between models. 2. Observational Evidence for ΛCDM We briefly discuss here the main observational evidence of the Λ CDM , focusing on why exactly it needs those fixes. This review is not exhaustive and does not include all the relevant references. It is just a brief introduction and further work can be found in the references within. We assume flat topology (we will explain why in Section 4). 2.1. The Expansion of the Universe and the FLRW Metric In 1929, Edwin Hubble published [ 15 ] his famous diagram or linear relation (the Hubble law): ˙ r=Hr relating the radial distance r of 46 galaxies to their radial recession velocity ˙ r≃zc , given by the redshift z and the speed of light c ( ˙ r is the time τ derivative: ˙ r≡dr dτ) . Hubble used redshift z from galaxy spectra estimated and published by Vesto Slipher (1917) [ 16 ] and Cephid distances r developed by Henrietta Leavitt [ 17 ] and calibrated by E. Opik [ 18 ]. However, it was George Lemaitre who first understood, in 1927 [ 19 ], the meaning of such a discovery [ 20 ]: that spacetime is expanding following the new theory of General Relativity (GR) by Albert Einstein [21]. However, you do not actually need GR to figure out the expansion equations. At large scales, the observable Universe looks homogeneous and isotropic. This alone tells us that a physical radial distance r has to scale as r=a(τ)χ , where a(τ) is a dimensionless scale factor and χ is a comoving coordinate, which is fixed ( ˙ χ= 0) for any comoving observer like us, moving with the expansion. This is the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric (i.e., Equation (A1)). The observed expansion law follows from derivation: ˙ r=˙ aχ=Hr where H≡˙ a/a is the Hubble expansion rate. Nowadays, H is measured to be H0≃ 70 Km/s/Mpc, so a galaxy at r≃ 300 Mpc has a recession velocity of ˙ r≃zc with a redshift z≃ 0.07. The Universe was 7% smaller at the time the light from that galaxy was emitted, τ≃r c=92 Myr ago. The expansion time is H−1 0≃14 Gyr. Consider a spherically symmetric region of space r<R with a fixed mass M (such as Lemaitre model [ 19 ]). We can use Gauss law (or the corollary to Birkhoff theorem in GR [ 22 ]) to ignore what is outside Rso the dynamics of Rwill be given by the free-fall equation: E=Φ+K=0⇒K=1 2˙ R2=1 2H2R2=−Φ=GM R=4πG 3ρR2. (1) The above equation leads to: r−2 H≡H2(τ) = 8πG 3ρ(τ), (2) which is independent of R . This simple Newtonian derivation reproduces exactly the full solution to GR field equations (i.e., Equation (A2)). At any time, the expansion rate H2 is given by ρ . In our Universe we have measured their values today ( ρ0 and H0 ) to find that they do follow: H2 0≃ 8 πGρ0/ 3. We use units where the speed of light is c= 1, and rH≡H−1 is called the Hubble Horizon because it corresponds to an expansion velocity ˙ r=Hr equal to the speed of light ( ˙ rH= 1). Energy–Mass conservation requires ρ∝a−3(1+ω) , where ω=p/ρ is the equation of the state of the fluid: ω= 0 for matter, ω= 1 / 3 for radiation and ω=− 1 for vacuum. Given a∗=a(τ∗) at some reference time τ∗and τ=0 at a=0, the solution to Equation (2) for one component is: H2=H2 ∗a a∗−3(1+ω) ⇒a(τ) = a∗3(1+ω) 2τH∗2 3(1+ω)⇒rH=3(1+ω) 2τ. (3) Universe 2022,8, 257 3 of 22 During collapse, H and τ are negative. Note that rH∝a3(1+ω)/2 , so for regular matter ( ω> 0), it grows faster than comoving scales: r=aχ (the opposite is true for ω=− 1). Thus, for ω> 0 all scales become super horizon ( r>rH ) during collapse ( H< 0) and re-enter the Hubble horizon during expansion. Note that rH increases with time τ (like the particle horizon). These equations are the exact solutions to GR for an FLRW metric, where τis the proper time for a comoving observer. Using Equations (2) and (3) we find: ρ=[(1+ω)τ]−2 6πG≃1.3 ×10−12 M Km3(1+ω)τ seconds −2 , (4) which tell us what the density is at any time τ . In general, ρ could be made of several components ρi : ρ=∑iρi , each with different ωi . The relative contributions are called cosmological parameters: Ωi≡ρi/ρ , so that ∑iΩi= 1. As τ⇒ 0, the matter density becomes very high and the radiation density dominates as temperature increases: T=T0/a . 2.2. Nucleosynthesis and CMB In 1964, Penzias and Wilson [ 23 ] accidentally found a uniform Cosmic Microwave Background (CMB) radiation of temperature T0≃ 3 K . Robert Dicke, James Peebles, P. G. Roll, and D. T. Wilkinson in the companion publication [ 24 ] interpret this radiation as a signature from the hot Big Bang: the oldest light in the Universe. This was first noticed in 1948 by R.Alpher and R.Herman [ 25 , 26 ] who developed the theory of the primordial nucleosynthesis and predicted a leftover CMB radiation of T0≃ 5 K , closed to the observed value. The idea behind it is simple. Because the universe is expanding, when you imagine going back in time the density must become higher and higher, atoms will break and the resulting plasma will be dominated by radiation, like the interior of a star. If you simulate an expansion from such initial conditions you can build a prediction for the primordial abundance of elements and radiation that we observed today. This is called primordial nucleosynthesis. Hydrogen is the most abundant element measured in our Universe. Around ∼ 75% of the total mass of the atoms (nucleons) in the Universe is in the form of hydrogen, the remaining 25% is mostly Helium. The abundance predicted by nucleosynthesis depends on the cross section of several Nuclear Physics reactions, such as neutron capture or decay. These are proportional to the ratio η=ρB/ρR of the number density of baryons ρB (protons and neutrons) to that of photons, ρR , given by the CMB background temperature T=T0/a . So a measurement of the primordial element abundance and ρB can be used to predict T0 . Nowadays, we use the more precise measured value T0= 2.726 K and the observed abundance to predict ρB . In relative units: ΩB=ρB/ρ≃ 0.05 [ 27 ]. So that only ≃ 5% of the total energy-density in our Universe is made of regular matter (i.e., made of known baryons and leptons). The rest, according to Λ CDM , is made of Dark Matter and Dark Energy. Where do these numbers come from? Why 5% and not 20%? 2.3. Cosmic Inflation and the Horizon Problem Cosmic Inflation [ 28 – 31 ] consists of a period of exponential expansion that must have happened right after the beginning of time ( τ= 0). There are over a hundred versions and variations [ 2 , 32 ], but generically the model requires some hypothetical new scalar field (the inflaton) with negligible kinetic energy ( ω=− 1) to dominate the very early universe. After expanding by a factor of e60 , inflation leaves the universe empty and we need a mechanism to stop inflation and create the matter and radiation that we observe today. This is called reheating. All these components require some fine tuning and free parameters that are hard to test because the physics involved is beyond reach by experiments [ 1 ]. However, Inflation solves some important mysteries that we do not know how to fix otherwise within Λ CDM . As mentioned below Equation (3), the structures that we observe today were not in causal contact in the past. We say that they are super horizon scales. Structures that are larger than rH cannot evolve because the time a perturbation takes to travel that distance is larger than the expansion time. How can these structures form if they were not in causal Universe 2022,8, 257 4 of 22 contact? This is the horizon problem. A clear evidence of this problem is the uniform CMB temperature across the full sky. The Hubble horizon rH at CMB times only subtends about one degree in our sky. So causality cannot explain the observed all sky CMB uniformity. The horizon problem is solved by inflation because, during inflation, structures of all scales become a super horizon. After inflation ends, they re-enter the horizon ( ω=− 1). Moreover, reheating provides a very uniform temperature background at the end of inflation. 2.4. Structure Formation and Dark Matter In 1992, NASA’s Cosmic Background Explorer (COBE) satellite detected temperature variations of very small relative amplitude δT≃ 10 −5 in the CMB [ 33 ]. We believe those were the seeds that grew under gravitational collapse from to form stars, galaxies and the cosmic web that we observe today. However, where do the seeds come from? Models of Inflation propose that these seeds come from super horizon quantum fluctuations that were exponentially inflated during inflation. Inflation predicts a power law (almost scale invariant) spectrum of fluctuations which agrees with the shape measured by later CMB missions [ 34 – 37 ] and clustering in Galaxy Surveys [ 38 – 41 ]. However, inflation does not provide a specific prediction for δT≃10−5: it is just a free parameter of the model. The measured δT≃ 10 −5 is too small to explain the observed structure in galaxy surveys today [ 42 , 43 ]. Some fix is needed: galaxy bias [ 44 , 45 ] or a Λ term [ 38 ]. The shape of the spectrum of fluctuations (including the Baryon Acoustic Oscillations, BAO [ 46 – 49 ]) in the CMB and Galaxy Surveys, also required another free component to agree with the Λ CDM model. They require a new type of matter, that we called Cold Dark Matter (CDM [ 50 ]), which is not made of regular matter (baryons) and interacts very weakly with matter or radiation (thus the name). CDM needs to be about four times more abundant than regular matter: ΩCDM ≃ 4 ΩB . Such CDM is also needed to understand the motion of galaxies in clusters [ 51 ], the galaxy rotational curves [ 52 ], gravitational lensing [ 53 ], galaxy evolution [50], cosmic flows [54] and structure in galaxy maps [38–41]. Despite enormous observational efforts in the last 30 yrs, such Dark Matter component has never been directly detected as a real particle or object [55,56]. 2.5. Cosmic Acceleration, Dark Energy and the Static Universe Usually, cosmic acceleration is defined by the adimensional coefficient q≡(¨ a/a)H−2 . Taking a derivative of Equation (3) we find q=−1 2( 1 + 3 ω) . For regular matter we have ω> 0 so we expect the expansion to decelerate ( q< 0) because of gravity. However, the latest concordant measurements from a Type Ia supernova (SN) [ 57 , 58 ], galaxy clustering and CMB all agree with an expansion that tends to ω=− 1.03 ± 0.03 [ 41 ] or q≃ 1 in our future. Dark Energy (DE) was introduced [ 59 ] to account for ω< 0. However, there is no fundamental understanding of what DE is or why ω≃ − 1. A natural candidate for DE is the cosmological constant Λ [ 60 – 63 ], which has ω=− 1 and can also be thought of as the ground state of a scalar field (the DE), similar to the Inflaton. Λ can also be a fundamental constant in GR, but this has some other complications [ 61 – 63 ]. Including DE in the Λ CDM model is also needed to complete the energy budget for our Universe: 5% baryons ( ΩB≃ 0.05), 25% Dark Matter ( ΩDM ≃ 0.25) and 70% DE ( ΩΛ≃ 0.7), so that ΩB+ΩDM +ΩΛ= 1, as needed. DE is also important for understanding the Integrated Sachs–Wolfe (ISW) effect [ 64 – 67 ], and to have a longer age estimate of 14 Gyr, which is needed both to account for the oldest stars and to have more time for structures to grow from the small CMB seeds δT≃ 10 −5 to the amplitude (and shape) we observe today in Galaxy Maps [38,42]. Note how q=1 means ˙ H=0, so that Hbecomes constant and all structures become super horizon and freeze, as in Inflation. In the physical or rest frame (see Appendix A.1) this corresponds to a static (deSitter) metric. We are used to repeating that the universe accelerates, but in the limit q⇒ 1 it is more physical to say that the universe becomes static, as proposed by Einstein [ 60 ] when he introduced Λ . This can be understood with the Twin Universe 2022,8, 257 5 of 22 paradox analogy of Especial Relativity to explain time dilation. Time happens slower for the comoving observer according to the physical observer at rest. In the limit of exponential expansion, time freezers and the expansion stops. Something that is static for the rest frame observer happens at constant velocity (Hconstant) for the comoving observer. 3. Inside a Black Hole (BH) In this section we will first present three different arguments that indicate that our Universe is inside a BH. This will lead to the BH Universe (BHU) model. How did we end up inside a BH? We will conjecture a new start for our Universe that could explain both the Big Bang expansion and why we are inside a BH, without the need to restore to a Quantum Gravity singularity. 3.1. What Is a BH? A BH is an object with a radial escape velocity ˙ R=c≡ 1. The escape velocity ˙ R is the minimum one needed to just escape the gravitational pull of a mass M . This requires: 1 2˙ R2=GM R . Thus, for ˙ R=c≡ 1 we have that R≡rS= 2 GM , which is called the Event Horizon. As events cannot travel faster than c , nothing can escape from inside rS . Thus we define a BH as an object of mass Mwhose radius is: rs=2GM ≃2.9 Km M M, (5) so that a solar mass BH has a radius of 2.9 Km. The density of a BH only depends on rS: ρBH =M V=3M 4πr3 S =3r−2 S 8πG≃9.8 ×10−3M M2M Km3. (6) This value should be compared to the atomic nuclear saturation density: ρNS ≃2×10−4M Km3, (7) which corresponds to the density of heavy nuclei and results from the Pauli exclusion principle. For a Neutron Star (NS) with M≃ 7 M both densities are the same: ρBH =ρNS . This explains why typical NS stars are not larger than M≃ 7 M , as they could collapse first into a BH. This is illustrated by Figure 1which compares the collapse density of a fix mass cold cloud as a function radius to the density of a BH. Because the star is collapsing in freefall (assuming no significant pressure support) nothing prevents the BH to form if the density reaches BH density before it reaches nuclear saturation. The maximum observed M for NS is closer to M≃ 3 M [ 68 ] which agrees with more detailed considerations that include the equation of state estimates. 3.2. Inside a Black Hole The density of a BH in Equation (6) is the exact density of our Universe in Equation (2) inside its Hubble Horizon rH= 1 /H , as for R=rH , the expansion law gives: ˙ R=HR = 1. So the Hubble volume around us ( R<rH ) is causally disconnected from the rest ( R>rH ) and has the density of a BH. Very different observations (CMB, SN, BAO, lensing and LSS) indicate [ 41 ] that H tends to a constant H2 Λ=H2 0ΩΛ=8πG 3ρΛ (i.e., ω=− 1) so the Universe asymptotically becomes static with a fixed radius ( rΛ=H−1 Λ ). Nothing can escape rΛand the mass inside is given by: M=4π 3r3 ΛρΛ=rΛ 2G, (8) Universe 2022,8, 257 6 of 22 i.e.,: rΛ= 2 GM . This is the definition of a BH. So we do live inside a BH of mass and size: M≃5×1022M;rS=rΛ=rH(a=∞)≃6×1022 km, (9) for ΩΛ≃ 0.7 and H0≃ 70 Km/s/Mpc. Figure 2compares the BHU formation with that of an NS. Inside rS≃ 6 × 10 22 km the density is very low and nothing can stop further collapse. So NS, galaxies and planets could also eventually form inside a BH. Figure 1. Illustration of the collapse of one solar mass (1 M , green dotted line) Neutron Star (NS). As the NS collapses the radius R decreases and the density increases as R−3 . The collapse stops (bounce back or explodes as a supernova) when the density reaches nuclear saturation ρns (green horizontal line). For masses larger that 7 M (red dotted line) the cold cloud collapses first into a BH and matter gets trapped inside the event horizon rS (black dashed line R−2 ). A NS could collapse inside a larger BH, but it can not escape rS. Figure 2. This is similar to Figure 1but extending the scale to include a cloud of mass M=5×1022 M(red dotted line) which corresponds to the size of our Universe. Universe 2022,8, 257 7 of 22 3.3. The Black Hole Universe (BHU) An FLRW metric with Λ is inside a trapped surface. The maximum radial distance travel by light (a null geodesic) is: r∗=aZ∞ τ dτ a(τ)=aZ∞ a dln a aH(a)<1 HΛ≡rΛ. (10) As time increase, the Hubble rate becomes constant and r∗ becomes a constant value r∗=rΛ. No signal from inside r∗can reach outside, similar to in the interior of a BH. If we use rS=2GM in Equation (1) we find: R= [r2 HrS]1/3 ⇒R(τ) = 3(1+ω) 2τ1/3 ∗τ2/3 =rSa aBH 1+ω , (11) where aBH is the scale factor when the BH event horizon is reached. For a regular star R>rS so the expansion is subluminar R<rH . Our Universe has R>rH (we observe super-horizon scales in the CMB) which, using Equation (11), requires R<rS . This is a third indication that we are inside our own BH! For ω=p= 0, R is a time-like geodesic with constant χ=R/a=rS/aBH . For a null geodesic R=r∗ ( ω6= 0) in Equation (10). Equation 11 gives the evolution of a finite FLRW cloud radius R(τ) . Compared to Equation (3) we can see that R grows slower than rH so perturbations become super-horizon during collapse and re-enter during expansion. So the collapsing phase acts like Inflation. In units of rH today c/H0≡ 1, at CMB times ( a≃ 10 −3 ): rH≃ 5 × 10 −5 , while R is about 30 times larger. For constant H=HΛ we have R=rS , which is larger than R0 today. Note how for R<rS (i.e., inside the BH) Equation (11) indicates that there is a region with no matter: rS>r>R and a region with matter outside the Hubble horizon R>r>rH. This is illustrated in Figure 3. Here we have obtained Equations (2) and (11) just using Newtonian Mechanics with the definition of a BH. This is the same solution as the BH Universe (BHU) [ 10 ], which is an exact solution to GR and corresponds to an FLRW cloud as in Equation (1). Appendix A presents this same BHU solution within GR. Figure 3. Illustration of our Universe inside the event horizon rS= 2 GM . This is a Schwarzschild (empty) metric outside ( r>R ) and an FLRW metric inside R (red dashed line) with mass M . The Hubble radius rH=c/H (dashed line) defines the volume inside causal contact (blue shading) from the center. The BHU solution in Equation (11) requires R= [r2 HrS]1/3 , so that R grows slower than rH . There is a region with matter outside the Hubble radius R>r>rH (yellow shading) with super horizon (or frozen) perturbations. This solves the horizon problem in Cosmology and is a source for perturbations that enter the horizon as the metric expands, creating LSS and BAO in Cosmic Maps, pretty much like what is usually assumed for Cosmic Inflation. Universe 2022,8, 257 8 of 22 3.4. How Did We End Up Inside a BH? Our Universe must have collapsed to form a BH. Before it collapsed, the density is so small that there are no interactions other than gravity. Even radiation escapes the cloud. The density is still very low when M approaches its corresponding event horizon R=rS= 2 GM , but the gravitational pull is still that of a BH. Radial comoving shells of matter are in free fall collapse, so they do not feel that pull. This is the Equivalence Principle. So the collapse continuous pass R=rS inside the BH. We take τ∗ in Equation (3) to correspond to the time τBH when rH=−rS, i.e., FLRW cloud formed a BH: τBH =τ∗=−2 3(1+ω)rS≃ −11 Gyrs, (12) where we have used Equation (9) and ω≃ 0 (the latest stages of the collapse could have ω≃ 1 / 3, but they last a negligible time compare to matter domination). Thus the BH forms 11 Gyr before τ= 0 (the Big Bang) or 25 Gyr ago. The cold collapse continued after the BH formation. In the last stages of the collapse atoms could ionized and part of the energy could transform into heat. This could slow down the collapse. Figure 4shows the numerical BHU solution using Equations (3) and (11). Figure 4. Physical coordinate radius R collapsing and expanding as a function of comoving time τ . A spherical cloud of radius R and mass M starts collapsing free-fall under its own gravity. When it reaches R=rS= 2 GM it becomes a BH (black sphere). The collapse proceeds inside the BH until it bounces into an expansion (the hot Big Bang). The BH Event Horizon rS behaves like a cosmological constant with Λ= 3 /r2 S so that the expansion freezes before it reaches back to rS . Blue shading ( R<rH ) indicates causal evolution of radial perturbations. White is approximated as empty space. Super horizon structures in-between R and rH (yellow shading) are “frozen” and they seed structure formation as they re-enter rH. Contrary to Inflation, the super horizon spectrum of perturbations in the BHU has a cut-off given by R. 3.5. The Big Crunch As mentioned before, there is a region outside the Hubble Horizon R>r>rH which is dynamically frozen (yellow shading in Figures 3and 4). This is the source for super horizon perturbations, which can be observed today in the CMB temperature maps. Any small irregularities δ≡∆ρ/ρ (such as the particle composition of the fluid) will grow under gravity. This is the so-called gravitational instability. The growth of δ can start early on within the FLRW cloud, well before τBH . The amplitude of δ from gravitational instability is scale invariant [ 69 – 71 ]. In the linear regime, δ follows a damped harmonic oscillator equation whose solutions [ 14 ] are D+∝a and D−∝a−3/2 , which correspond to the growing and decaying mode during expansion. In the collapsing phase the damping Universe 2022,8, 257 9 of 22 term has a negative sign and fluctuations grow faster with time because D− is the growing mode when a goes to zero. It is therefore likely that galaxy, stars, planets or life could also form during the collapsing phase. The details might depend on the original FLRW cloud composition. As the cloud collapses and the background density increases, the structures will disappear inside a hot Big Crunch, but the largest scale density perturbations will become super horizon scales (freeze out) and survive the Big Bounce, as they correspond to variations of the background over scales that are causally disconnected. 3.6. The Big Bounce The energy density ρ in Equation (4) is the same everywhere inside R . By τ≃ − 10 −4 s, ρ approaches nuclear saturation (GeV) in Equation (7). The radius of our Universe R is close to the distance between Earth and the Sun. However, the Hubble radius is only few Km (containing a few solar masses). So the physical situation is similar to the interior of a regular collapsing star. We conjecture that this leads to a Big Bounce because of the Pauli exclusion principle of Quantum Mechanics. Neutron density is the highest cold density observed in nature. The collapse is halted by neutron degeneracy pressure, causing the implosion to rebound [ 72 ]. If the neutron material is elastic enough [ 73 ] the collapse could just bounce into an expansion, pretty much like a bouncing of a ball. However, if the expansion rate is too high, the collapse could also led to a supernova (SN) explosion. Global rotation of the FLRW cloud, could slow down the expansion rate (see Appendix C) and play some role in the bounce. Stars explode as supernovas (SN) either because of runaway nuclear reactions or because of gravitational core-collapse. Protons and neutrons combine and form neutrinos by electron capture. The gravitational potential energy Φ of the collapse is converted into a neutrino burst. Neutrinos are reabsorbed by the infalling layers producing an SN explosion. For example, the Crab Nebula pulsar in Figure 5is thought to be a core collapse supernova that exploded releasing a total energy of 10 51 –10 52 ergs in the explosion. This energy is very similar to the FLRW collapsed energy of a M star within rH≃ 30 Km. Recall that the collapse speed is cfor rH, so this is also closed to the internal (or rest) energy in Einstein’s most famous equation: E=Mc2. Figure 5. ( Left ): The Crab Nebula explosion as observed in 1999 from the Hubble Space Telescope, 945 years after it exploded. A pulsar remnant could be part of the Dark Matter. ( Right ): the MICE simulation [ 74 ] of our expanding Universe. The resulting structures are related in the Big Bounce model. The bounce is synchronized at different locations because the background density is the same everywhere in the FLRW cloud. The collapse energy is converted into expansion energy ( H> 0). Neutron stars, small primordial BHs (PBHs) or quark stars [ 75 ] could result as remnants of the SN explosions and they could contribute to the Dark Matter that Universe 2022,8, 257 16 of 22 Appendix A.3. Junction Conditions We can arrive at the same BHU solution using Israel’s junction conditions [ 114 , 115 ]. We can combine two solutions with different energy content, as in Equation (A12), to find a new solution. To do that we need to find a hypersurface junction Σ to match them well. In our case, this will be given by R . The junction conditions require that the metric and its derivative (the extrinsic curvature K ) match at Σ . The join metric then provides a new solution to GR. In many cases, like in the Bubble Universes or gravastar [ 92 – 97 ], which match dS and SW metric, this does not work and the junction requires a surface term (the bubble) to glue both solutions together. For the BHU there are no surface terms [ 10 ], which shows that this is an exact solution. In the limit where the FLRW has constant H (i.e., our future), the BHU solution corresponds to match between dS and SW metric. So a Bubble Universe without bubble. To see this, consider the case where Σ is given by R in the freefall collapse of an FLRW cloud of fixed mass M . For matter domination, this corresponds to R=a(τ)χ∗ as in Equation (11), where χ∗=rS/aBH is fixed. The induced 3D metric on Σ is h− αβ with coordinates dyα= (dτ,dδ,dθ): ds2 Σ−=h− αβdyαdyβ=−dτ2+a2(τ)χ2 ∗dΩ2. (A15) For the outside SW frame, the junction Σ+ is described by r=R(τ) and t=T(τ) , where τ is the FLRW comoving time and tthe time in the physical frame. We then have: dr =˙ Rdτ;dt =˙ Tdτ, (A16) where the dot refers to derivatives with respect to τ . The metric h+ induced in the outside SW metric is: ds2 Σ+=h+ αβdyαdyβ=−Fdt2+dr2 F+r2dΩ2 =−(F˙ T2−˙ R2/F)dτ2+R2dΩ2, (A17) where F≡ 1 −rS/R . Comparing Equation (A15) with Equation (A17), the first matching conditions h−=h+are then: R(τ) = a(τ)χ∗;F˙ T=p˙ R2+F≡β(R,˙ R). (A18) For any given a(τ) and χ∗ we can find both R(τ) and β(τ) . We also want the derivative of the metric to be continuous at Σ . For this, we estimate the extrinsic curvature K± normal to Σ±from each side of the hypersurface: Kαβ =−[∂anb−ncΓc ab]ea αeb β, (A19) where ea α=∂xa/∂yα and na is the 4D vector normal to Σ . The outward 4D velocity is ua=ea τ= ( 1,0,0,0 ) and the normal to Σ− on the inside is then n−= ( 0, a ,0,0 ) . On the outside ua= ( ˙ T , ˙ R ,0,0 ) and n+= (−˙ R , ˙ T ,0,0 ) . It is straightforward to verify that: naua= 0 and nana= + 1 (for a timelike surface) for both n− and n+ . We then find that the extrinsic curvature in Equation (A19) to the Σ junction, estimated with the inside FLRW metric, i.e., K−is: K− ττ =−(∂τn− τ−aΓχ ττ)eτ τeτ τ=0 K− θθ =aΓχ θθeθ θeθ θ=−aχ∗=−R. (A20) For the SW metric: Universe 2022,8, 257 17 of 22 K+ ττ =¨ R˙ T−˙ R¨ T+˙ TrS 2R2F(˙ T2F2−3˙ R2) = ˙ β ˙ R K+ θθ =˙ TΓr θθ =−˙ TFR =−βR, (A21) where we have used the definition of β in Equation (A18). In both cases Kδδ =sin2θKθθ , so that when K− θθ =K+ θθ , it follows that K− δδ =K+ δδ . Comparing Equation (A20) with Equation (A21 ), the matching conditions K− αβ =K+ αβ require β= 1, which using Equation (A18) gives: R=r2 HrS1/3 . This reproduces the junction in Equation (11). So the two metrics and derivatives (the extrinsic curvature) are identical in the hypersurface defined by R . This completes the proof that the FLRW cloud is an exact solution of GR without surface terms. For more details see [10]. Appendix B. The Action of GR and the Λterm Consider the Einstein–Hilbert action [113,116]: S=ZV4 dV4R−2Λ 16πG+L, (A22) where dV4=√−gd4x is the invariant volume element, V4 is the volume of the 4D spacetime manifold, R=Rµ µ=gµνRµν is the Ricci scalar curvature and L the Lagrangian of the energy-matter content. We can obtain Einstein’s field equations (EFE) for the metric field gµν from this action by requiring S to be stationary δS= 0 under arbitrary variations of the metric δgµν. The solution is [21,113]: Gµν +Λgµν =8πG Tµν ≡ −16πG √−g δ(√−gL) δgµν , (A23) where Gµν ≡Rµν −1 2gµνR. For perfect fluid in spherical coordinates: Tµν = (ρ+p)uµuν+pgµν, (A24) where ρ , and p are the energy-matter density and pressure. This fluid can be made of several components, each with a different equation of state p=ωρ. Equation (A23) requires that boundary terms vanish (e.g., see [ 113 , 117 ]). If there are boundaries to the dynamic equations, we need to add a Gibbons-Hawking-York (GHY) boundary term [82–84] to the action in Equation (A22): SGHY =1 8πGI∂V4 d3y√−h K, (A25) so that the total action is S+SGHY and K is the trace of the extrinsic curvature at the boundary ∂V4 and h is the induced metric. The expansion that happens inside an isolated BH is bounded by its event horizon r<rS and we need to add the GHY boundary term SGHY . The integral is over the induced metric at ∂V4 , which for a time-like junction dχ= 0 corresponds to R=rS: ds2 ∂V4=hαβdyαdyβ=−dτ2+r2 SdΩ2. (A26) So the only remaining degrees of freedom in the action are time τ and the angular coordinates. We can use this metric and the trace of the extrinsic curvature at R=rS to estimate K=− 2 /rS from Equation (A20). This result is also valid for a null geodesic [ 10 ]. We then have: SGHY =1 8πGZdτ4πr2 SK=−rS Gτ. (A27) The Λ contribution to the action in Equation (A22) is: SΛ=−ΛV4/( 8 πG) = −r3 SΛτ/ 3 G , where we have estimated the total 4D volume V4 as that bounded by ∂V4 inside r<rS . Universe 2022,8, 257 18 of 22 i.e., V4= 2 V3τ , where the factor 2 accounts for the fact that V3= 4 πr3 S/ 3 is covered twice, first during collapse and again during expansion. Comparing the two terms we can see that we need Λ= 3 r−2 S or equivalently rΛ=rS to cancel the boundary term. In other words: expansion inside a BH event horizon induces an effective Λ term in the EFE even when there is no Λ background term to start with. Such event horizon becomes a boundary for outgoing geodesics, i.e., expanding solutions. This provides a fundamental interpretation of the observed Λas a causal boundary [7,8,11]. Appendix C. Outside Our BHU: A Rotating Cloud If the FLRW cloud is not totally isolated it could have some rotation. This could be a way to infer if there is something outside our BHU. Any rotation, no matter how small, could prevent or interfere with the cloud collapse. Can we detect such rotation? A rotating BH is a bit more difficult to model because spherical symmetry is lost and the BH becomes oblate (i.e., the Kerr metric [118]): x=qr2+r2 Jsin θcos Φ;y=qr2+r2 Jsin θcos Φ;z=rcos θ, (A28) where rJ=J/M is the ratio between the angular momentum J and the BH mass. A detailed analysis of this case is outside the scope of this review, but we will make some energy considerations to understand the possible impact of such rotation on the Big Bounce. We assume that both mass M and angular momentum J are conserved, so rJ is constant. We also assume that rJrS so during the collapse we can neglect deviations from spherical symmetry. If we start from the FLRW cloud of size R and mass M with some small initial rotation, ˙ θ, these products have to be constant: J M=rJ=R2˙ θ=r2 S˙ θBH. (A29) As R gets smaller, ˙ θ will become larger. The kinetic energy term in Equation (1) will have another contribution 2K=˙ R2+˙ θ2R2, so that Equation (2) becomes: r−2 H≡H2(τ) = 8πG 3ρ(τ)−r2 J R4=r−2 Sa aBH −3(1+ω) −r2 J r4 Sa aBH −4(1+ω) , (A30) where in the last step we have used Equations (3) and (11) for a collapsing FLRW cloud with equation of state ω . So, for ω= 0, rotation acts like a radiation term of negative energy density. Rotation is negligible, except when a⇒ 0 when rotation tends to delay the collapse, as it reduces the expansion rate H . 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