Towards A Falsifiable Black-Hole Cosmos

 by Steven Gussman

THE CONCEPT¹

        It is sometimes supposed that what is inside of black holes is another universe (and therefore that our universe may be inside of a black hole within a parent universe). Both our universe and black holes change sizes: the former during the inflationary epoch (sometimes blamed on a scalar “inflaton” field), and less rapidly during the current accelerated expansion epoch (labeled “dark energy” and associated with Albert Einstein's cosmological constant); the latter because it grows when it is impinged upon with the cosmic microwave background (or, less often but more dramatically, when other stellar objects collide with the black hole),² and because it shrinks when Hawking radiation dominates, sending energy outward. The Schwarzschild radius of a black hole is defined as:³

r_s = 2Gm / c²

One can see that the radius and the mass are directly proportional. Yet the CMB impinges on the black hole always from all sides: the amount of mass increase (that is, the flux) is proportional to the surface area, or the square of the radius, which in turn increases the radius / surface area. One notices that the increasing size of a black hole is super-linear, just as the expansion of the universe has tended to be. The concept is a physical dual between (internal) cosmology and (external) black hole astrophysics.

        One might wonder what, given this interpretation, would be enough to quickly expand a black hole the way the universe seems to have doubled 85 times in a fraction of a second during the inflationary epoch.⁴ Most obvious would be to set up a precarious situation in which a very many black holes of various sizes were orbiting each other, unstably, and when they finally collided, the smallest would have experienced 85 doublings in quick succession. This is of course an unrealistic setup, nothing the likes of which has been observed. But one could instead imagine something more like the CMB case, but at a much higher temperature / background density. Black holes are known to form when stars past a certain mass contract after exhausting the last bout of nuclear fusion in their core. The reason that fusion takes place in the core is because it is the densest area, and it is the densest area because it experiences the greatest gravity (because the force of gravity is inversely proportional to the square of the distance from this core). Therefore, during gravitational collapse, it is perhaps not unrealistic to argue that a small proportion of the core of the star meets the Schwarzschild radius, first—from there, this small black hole inside of its sun would rapidly expand, eating up all of the material of the star, the black hole growing and the star in-falling, simultaneously:

This inflationary process naturally shuts off when all of the stellar material is inside of the black hole, and it must switch from consuming stellar material to the in-falling CMB—a stark switch to a medium of a much smaller density (physical cosmologists have little idea, as it is, what could have switched the inflationary epoch on or off so suddenly). It is thought that stars can collapse into black holes even without supernovae in under a second (meanwhile, the inflationary epoch is thought to have lasted well under a second).⁵

        At this point, the black hole settles into a still-accelerating, but much less rapid, expansionary phase. It is not clear to me at present whether the competing Hawking radiation would temporarily keep the expansion somewhat linear (as our universe appears to have experienced a five billion year period of linear expansion prior to the current accelerated phase, something physicists also cannot explain)⁶, yet I do know that the Hawking radiation rate is inversely proportional to a black hole's size, whereas the CMB-expansion is directly proportional to the black hole's size, suggesting that, given expansion has an edge, it will be a better first-approximation on its own (and so expand at a more accelerated rate) the larger the black hole gets (holding the CMB temperature constant). What is clear is that it will exist in a long stage of accelerated expansion, the longest of the three phases, in keeping with the fact that we find ourselves in a universe in this phase (perhaps allowing for an invocation of the soft anthropic principle).

        Assuming that our parent universe is also expanding, then its CMB is also decreasing in temperature, meaning that at some point, it will be so sparse that the black hole will be expanding from the CMB at a slower rate than it will be Hawking radiating: in our homology from black holes to universes, this predicts a contraction phase as the black hole containing our universe shrinks (and actually loses mass) until it finally evaporates entirely back into the parent universe in a big crunch. Typically, cosmologies with a big bang and a big crunch are eternal and cyclical in nature: this cosmology may be somehow eternal at the level of the hierarchical multiverse, but it is novel that at the level of the individual universe, it would not be cyclical: a big bang would be the beginning of a given child universe, and a big crunch would be the genuine end of that universe.

        This set of hypotheses does not break the law of conservation of energy as one might worry, because it simply argues that universes are not closed systems—the hierarchical multiverse would be the closed system in which energy is perfectly conserved in totality, with universes leaking energy into their children and parents through CMB expansion and Hawking radiation.⁷ One genuine issue with this model is that it is non-terminating (or otherwise does not comment on the termination of the mechanism): it supposes universes within universes, but the expansion of a given universe is caused by that universe's black hole counterpart in its parent universe consuming the CMB around it—if the parent universe is also expanding, this is due to its parent universe (the grandparent universe to the universe we were originally considering) has a CMB, which is also shrinking, and so on in both directions. An issue with the distribution arises, too: how does the in-falling CMB energy mix in with the universe inside of the black hole? Should it be concentrated out towards the edges of the universe where it is difficult-to-impossible to observe (which would indicate an un-falsifiable hypotheses worthy of ignoring)?⁸ Would it overcome the expansion of the universe, at least in certain epochs, and race towards observers, nonetheless? Wouldn't this model violate the Copernican Principle and classical spatial relativity, naturally arguing that the universe does have a center—the singularity center of the black hole? Or does physicist Leonard Susskind's holographic principle hypothesis come to the rescue, arguing we exist on the (center-less) surface of the event horizon of an expanding black hole? If so, would that not argue that density would not change at all with expansion, since the universe's size is then proportional to the area (not the volume), same as the impinging CMB? If the universe is modeled as the interior volume of the black hole's event horizon, could the sponge-like cosmic filament structure be the structure stable enough to expand and contract without a singularity point of infinite density (though with a center after all)?

        If its weaknesses could be fixed, this concept has the potential to unify physical cosmology and black hole astrophysics while providing a mechanism for the universe's different expansionary epochs (unifying dark energy and inflation), as well as the transitions between them.

THE PREDICTIONS

The main non-exotic, falsifiable prediction of this model is that it predicts a different density fall-off rate for an expanding universe than the standard view. Normally, when one has a container full of gas and one increases the radius of the container, the contained mass remains constant, and so the density falls off with the volume, or the cube of the radius. It is assumed that this is what is happening with our universe. But if universes are contained inside of black holes, and their expansion is explained by the Schwarzschild radius response to in-falling CMB energy (a concomitant prediction of this hypothesis is a Schwarzschild-relationship between the radius and mass of the universe), then the black hole / universe is gaining mass proportional to its surface area, as it gains volume, reducing the rate at which the density decreases. Because the volume scales with the cube of the radius, and the mass scales with the square of the radius, the density is then predicted only to fall off with the square of the radius:⁹

        r_s = 2Gm / c²

        A_sphere = 4πr²

        r = (A/4π)^1/2 = 2Gm / c²

        :. r ∝ A^1/2 ∝ m

Longitudinal measurements of the density of the universe could yield whether it has been decreasing with the square or with the cube of the radius during expansion, and therefore distinguish whether our universe exists inside of a black hole. When one checks quasar-density, one finds that over 13.8 billion years, the quasar density has fallen from 10⁶ quasars per unit volume to 100 galaxies per unit volume, which is in line with a decrease by the square of the radius. This would tend to support the hypothesis that black holes contain universes.

Another measure may be something like the CMB, directly measured during two distant epochs (though this is difficult to come by). At one point, I was under the impression that the CMB density dropped off closer to the cube of the radius, in support of the classical picture.

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Conversely, the predicted mass-increase of the universe can be empirically tested by comparing like objects over time, say the average mass of quasars, galaxies, galaxy clusters or filaments between two epochs (over the 13.8 By since the big bang, the universe expanded one-thousand-fold; classically, this occurred with no change in total mass, but if that expansion was due to accretion of material from a parent-universe, then the mass too should have grown one-thousand-fold as all objects uniformly collected this material). None of these are particularly good empirical tests of the hypothesis, and I am hoping future measurements (or those I am unaware of) can be brought to bear more definitely on this empirical question.

        Another consequence of this model would be a cooler early universe, because the universe would begin with a very low mass and grow in size as it grew in mass during the inflationary epoch; if this does not take some of the focus off of high-energy particle physics and particle-accelerator experiments, it at least argues that they don't need to be as high-energy as once thought. In addition, though rare, if we were to observe a sudden, significant increase in the size of the universe (that is, if distant galaxies were suddenly much more distant), the only known explanation would be that the black hole which contains our universe collided with another significant mass (such as a neutron star or another black hole) out in our parent universe, near-instantly expanding our universe.

THE MODEL

One can derive the equations which govern the evolution of a black-hole's mass (and therefore radius) over time:¹⁰

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Equipped with these equations, the dynamics of black-hole-universes can be predicted, though the initial and final conditions are not obvious.

STELLAR-COLLAPSE / INFLATION

        When a stellar-mass black-hole collapses in this model, a smallest-possible black hole forms at its core, first, and expands outward rapidly. But how small is small, in this case? The smallest possibility is to imagine a black hole whose Schwarzschild radius is the hypothetical Planck length. But this length is not as well-motivated as physicists seem to believe, and even if it were, it is likely that the minimum black-hole radius would need to be some multiple of the Planck length. The mass of a quark is arguably the mass of the smallest matter particle; the mass and radius of a Hydrogen atom are common “small” units in the universe; and Iron atoms are what actually churn out at the core of stars which will collapse into black-holes. Sometimes, scientists will even (for some reason) say that the universe was, before inflation, the size of a grapefruit! But perhaps most realistic is the theoretically derived minimum black hole radius, limited by the Heisenberg uncertainty principle.¹¹ Below are tables of possibilities when different initial masses / radii and final masses / radii are examined in terms of how many doublings they produce, and whether they result in a black hole:

For this model to work, the final radius and mass should be 1,000 times and 100 time smaller than the values known of the current observable universe. But the most realistic scenario on offer, here, is that in which the initial size is dictated by the theoretical prediction for the smallest possible black-hole (5.85E-8 kg, 9E-35 m) which doubles about 575 times, leading to a universe of mass 3.5E52 kg and radius 4.4E24 m, two orders of magnitude smaller than that required for a black hole. Better it be a black hole than not, but the model would seem to predict that the radius of the observable universe should equal that of the Schwarzschild radius (although the known size of the universe is somewhat larger than the observable size, it appears the difference is not enough).¹²

        These kinds of doublings are certainly reminiscent of the inflationary epoch, but how long did the process take? To model this, we will use the equation for black-hole growth from the CMB and scale it such that we are dealing with the high density of an iron-forming star, rather than the low-density of the CMB.¹³ This yields absurdly long lengths of time, the exact value depending on whether you merely change the temperature (10¹⁰ s) or scale-by-the-density (10² s), tending to falsify the hypothesis as it pertains to black-hole formation. The timing can be got down to sub-second timing only if you both scale the temperature and the fourth power of the density (10^-38 s), but it is not at all clear why one would do that.

All of this tends to falsify the hypothesis as it pertains to black-hole formation (which should have been the “easy” hurdle to pass).

        We can also work our way backwards from what is empirically known from our universe, in its present state. A star of what mass would be needed to collapse into the black hole with a Schwarzschild radius 1,000x smaller than that of our visible universe's radius? It turns out it would need to be 9E16 kg—quite unrealistic for a star (in our universe, anyway).¹⁴ The inflationary stage would require 147 doublings and take perhaps too long, at 8E-15 seconds.

CMB-INTAKE / EXPANSION

        From here, the universe's radius would not increase in size by a factor of 1,000 in 13.8 billion years (4.35 x 10¹⁷ s) thereafter like ours appears to have—this would take 10⁵⁰ - 10⁵² seconds (far longer than 10¹⁷ seconds). Meanwhile, in 13.8 billion years, these equations predict almost no change in the the radius / mass of the universe (quite less than the known thousand-fold increase over that duration). At best, this suggests that parent universes must be much hotter than child universes (which is pretty perverse, as they should also be larger); far more likely: it falsifies the model.

HAWKING-RADIATION / BIG CRUNCH

        Though there seems to be no way to test this, another prediction is that this universe would switch into a contraction phase by 7.2E38 (not 2E143?) seconds later, and it would then take 10¹⁴³ seconds (10¹²⁵x longer than the current age of the universe) until the big crunch (but of course the failure of this method to properly predict the previous values suggests this too is meaningless).

        Much of this tends to falsify the idea that universes exists inside of black holes, at least if the outer universe is much like ours (the idea is already exotic enough that I do not feel comfortable pushing it further and speculating about the strange characteristics our speculative parent universe might have to make this model work, free-parameters being an ignoble cheat).

CONCLUSION

        It should be noted that I do not generally like multiverse hypotheses, the use of the hard anthropic principle, nor even the idea that perhaps universes are contained inside of black holes. My leaning has always been that we live in a singular universe and that we will one day discover that the singularity in a black hole is some kind of exotic physical object or process. Yet the observation that CMB consumption by black holes could provide a mechanism for the inflationary-and-dark-energy expansions of our universe remains interesting, particularly given that it makes a straightforward empirical prediction in the universe's density drop-off rate.

It is not unlikely that the new James Webb Space Telescope will make empirical measurements which bear on this prediction.

ADDITIONAL ODDITIES

• (Kerr) black holes are known generally to have a charge and rotation; what would this mean for the universe within?

• Wouldn't the universe squash-and-stretch in a periodic, measurable way if it were contained in a black hole which had co-orbiting bodies? The universe would also be losing energy in the form of gravitational waves sent out into the parent universe as well, in this case: a characteristic increasing loss before a very large gain in mass and radius.

• Couldn't it alternatively be that many small black hole universes form when a star collapses, but they end up expanding into each other, making the singular large black hole universe in the end (each experiencing this as rapid inflation)?

• Why would the space inside of a black hole uniformly expand at all points rather than just grow to include more space at the outer limits? Is it under spring-like pressure, perhaps the cosmic filament structure like those old children's toys, the bar-balls that shrink or expand?

• Is the filament structure uniquely stable under high-energies such that it may avoid full singularity-collapse? Interestingly, squeezed sponges, like a spring, do tend to re-inflate to reach their equilibrium point. A compressed sponge fills its container almost like a fluid. Is expansion a spring-like restorative force (something proponents of cyclical cosmologies are used to asking)? Perhaps the return-to equilibrium is a moving target?

• How does the holographic principle fit into this?

• The size (and therefore mass) of a black hole is allowed to grow arbitrarily quickly, because it's a region of spacetime, not a material object whose dimensions are subject to relativistic constraints; that is crucial to the dual hypothesis because space-time itself is also allowed to expand superluminously in the cosmological context.

• Can the outstanding dynamic-Hubble-constant mystery be explained through the CMB-intake / expansion mechanism?

• This hypothesis predicts that the radius of the universe is always r_s[t]

Appendix: Selected Journal Pages

Footnotes:

1. I originally explored this idea in my personal journal in the fall of 2021.

2. Interestingly, this appears similar to Sir Fred Hoyle's infamous steady-state model which included the hypothesis, which seemed to break the law of conservation of energy in its singular universe form, that as the universe expands, more energy is created so as to keep it homogeneous. Here, the universe is not necessarily kept homogeneous, and the law of conservation of energy is obeyed at the level of the multiverse, but matter-energy does come into the universe at its edges when it expands (in this case, this impinging matter causes the expansion). I originally noted some of these dynamics when critiquing physicist Lee Smolin's evolutionary multiverse hypothesis, see “A Profusion Of Place | Part I: Of Unity And Philosophy” by Steven Gussman (Footnote Physicist) (2020) (footnote 32) (https://footnotephysicist.blogspot.com/2020/03/a-profusion-of-place-part-i-of-unity.html). I first heard of the effect of black-hole growth from CMB-intake from mathematician John Carlos Baez, see his January 24^th, 2020 tweet: https://twitter.com/johncarlosbaez/status/1220777945158975489?s=20&t=ZvNNC7S9XeuhTzvlL4hANQ.

3. See the “Schwarzschild Radius” entry in Google Equations (https://www.google.com/search?q=schwarzchild+radius&rlz=1C1CHBF_enUS775US775&oq=schwarzchild+radius&aqs=chrome..69i57j0i10i433j0i10l8.5819j1j7&sourceid=chrome&ie=UTF-8#wptab=si:ANhW_Np-M9IG6f0Jmofyq9Z_porxo80E_iKsIhZdu5t1S_osyoHGo7xh54d_Eue51zvx1iNLLsccYzjq9uEoeTyDWoUi_tHzCR2Z5ZsC8yoYecnHsfdXeUBvODZ3ksnlqJQKAolLWNcLGVMsrB0dttC2kwtRqxcrOg%3D%3D).

4. See the "Inflation" section of "Cosmology: Astronomy" by Frank H. Shu (Encyclopedia Britannica) (1998 / 2023) (https://www.britannica.com/science/cosmology-astronomy/Inflation), which reads that during inflation, the universe was, "doubling its size roughly once every 10⁻⁴³ or 10⁻³⁵ second. After at least 85 doublings, the temperature, which started out at 10³² or 10²⁸ K, would have dropped to very low values near absolute zero," (though I do not believe have not read this entire section).

5. It is believed that the LBT, Hubble, and Spitzer telescopes observed such a clean collapse with N6946-BH1's lights going out without a trace, see “Collapsing Star Gives Birth to a Black Hole” edited by Karl Hille (NASA) (2017) (https://www.nasa.gov/feature/goddard/2017/collapsing-star-gives-birth-to-a-black-hole). One can estimate the time for a gas cloud to collapse with t ≈ (3 / 2πGp)^1/2 , see “Gravitational Collapse” by Chris Mihos (Case Western Reserve University) (http://burro.astr.cwru.edu/Academics/Astr221/LifeCycle/collapse.html). The density of a pre-black-hole star is p ≈ 1410 kg/m³, see Google's “Sun / Density” information entry (https://www.google.com/search?q=density+of+sun&rlz=1C1CHBF_enUS775US775&oq=density+of+sun&aqs=chrome..69i57j35i39j0i512l8.6206j0j9&sourceid=chrome&ie=UTF-8). Therefore t ≈ 38 min. Elizabeth Landau has reported that stellar collapse black holes “can likely form in seconds”, see “10 Questions You Might Have About Black Holes” by Elizabeth Landau (NASA) (2019) (https://solarsystem.nasa.gov/news/1068/10-questions-you-might-have-about-black-holes/). Jeff Mangum claims that direct stellar collapse takes t < 0.5 s, see “When Does a Neutron Star or Black Hole Form After a Supernova?” by Jeff Mangum (NSF / NRAO) (2020) (https://public.nrao.edu/ask/when-does-a-neutron-star-or-black-hole-form-after-a-supernova/).

6. I cannot quickly find the Wikipedia article where this was mentioned of cosmological expansion.

7. In fact, this concept was stimulated by a criticism I made of physicist Lee Smolin's evolutionary cosmology in which he envisions black holes as child universes with mutations in the biological sense—I pointed out that due to Hawking radiation, these universes would technically not obey the law of conservation of energy, locally, see “A Profusion Of Place | Part I: Of Unity And Philosophy” by Steven Gussman (footnote 32) (https://footnotephysicist.blogspot.com/2020/03/a-profusion-of-place-part-i-of-unity.html) which further cites...

8. Interestingly, this looks something like Sir Fred Hoyle's proposal of matter-energy being created at the margins as the universe expands, although here it does not do so in such a way as to produce a static universe, and the law of conservation of energy is not broken when you take into account the multiverse as the closed system.

9. For the spherical area equation, see Google (https://www.google.com/search?q=area+of+a+sphere&rlz=1C1CHBF_enUS775US775&oq=area+of+a+sphere&aqs=chrome.0.35i39j0i512l9.1959j1j7&sourceid=chrome&ie=UTF-8).

10. See "Let's Get Funky" by Kevin Moore and John Fuhrman (Harvey Mudd College) (1997) (https://www.physics.hmc.edu/student_projects/astro62/hawking_radiation/derivation.html) and Black Holes: A Traveler's Guide by Clifford A. Pickover (Wiley) (1997) (pp. Y).

11. See Brief Answers To The Big Questions by Stephen Hawking (Bantam Books) (2018) (pp. 110-111).

12. See “Cosmological Horizon” (Wikipedia) (accessed 1/31/23) (https://en.wikipedia.org/wiki/Cosmological_horizon) and “Particle Horizon” (Wikipedia) (accessed 1/31/23) (https://en.wikipedia.org/wiki/Particle_horizon) (though I have read neither article in its entirety), which place the difference at 46.9 Gly – 13.8 Gly = 33.1 Gly (a factor of 3.4x rather than the needed 100x).

13. See "Lecture 18: Supernovae" by Richard Pogge (The Ohio State University) (2006) (https://www.astronomy.ohio-state.edu/pogge.1/Ast162/Unit3/supernova.html) which gives the following values for iron-core stars: T > 10¹⁰ K and p = 10¹³ kg/m³ – 10¹⁷ kg/m³. (Using the gravitationally collapsing gas equation from above with these higher densities, one gets t = 10^-4 s - 10^-2 s).

14. The initial mass of the universe is:
r_s = 2GM/c²
M = r_sc²/2G = (1.3E-10 m)(9E16 m²/s²)/2(6.67E-11) = 9E16 kg
or 10⁴⁰x more massive than the current mass of the visible universe. The scale factor between the mass of the universe before and after inflation is (5E31 kg)/(9E16 kg) = 6E14. This means the doubling is log₂(6E14) x 3 ≈ 147 volume-doublings (some 62 more than typically thought).

Some additional references: NASA for the T_CMB value; physics.NIST.gov for a p_CMB value; NASA for the figures for stellar-mass black holes as between 3M_Sol – 10M_Sol; chandra.harvard.edu for the belief that stellar-mass black holes are thought to form 1/10^th – ½ s; some may be missing due to the smeared-out nature of trying so many angles over multiple notebooks.