We present an overview of the thermal history of the Universe and the sequence of objects (e.g., protons, planets, and galaxies) that condensed out of the background as the Universe expanded and cooled. We plot (i) the density and temperature of the Universe as a function of time and (ii) the masses and sizes of all objects in the Universe. These comprehensive pedagogical plots draw attention to the triangular regions forbidden by general relativity and quantum uncertainty and help navigate the relationship between gravity and quantum mechanics. How can we interpret their intersection at the smallest possible objects: Planck-mass black holes (“instantons”)? Does their Planck density and Planck temperature make them good candidates for the initial conditions of the Universe? Our plot of all objects also seems to suggest that the Universe is a black hole. We explain how this depends on the unlikely assumption that our Universe is surrounded by zero density Minkowski space.

## I. INTRODUCTION

### A. Condensation of objects

The early Universe was a hot plasma of fundamental relativistic particles: quarks, leptons, photons, and gluons. There were no composite objects such as protons, atoms, planets, or galaxies.^{1–5} As the Universe cooled, composite objects condensed out of the background much as droplets of steam condense out of hot water vapor as it cools. This condensation happened when the binding energy of an object exceeded the background energy. For example, as the Universe expanded and cooled during the quark-hadron transition, the binding energy of the strong force overcame the background energy as the quark-gluon plasma condensed into protons, neutrons, and other hadrons. With further expansion and decrease in temperature, during the epoch of big bang nucleosynthesis, the binding energy of the residual strong force overcame the background energy as the hot plasma of protons and neutrons condensed into atomic nuclei. Further expansion and cooling led to the formation of helium and then hydrogen atoms when the binding energy of coulomb forces overcame the background energy. With further cooling, chemical bond energies overcame the kinetic energy of atoms as they condensed into molecules. Further cooling allowed matter-overdensities to form stars, planets, galaxies, and clusters of galaxies as their gravitational binding energy overcame their kinetic energy.^{1,2}

As a result of this sequence of condensations, due to the strong force, electromagnetism, and gravity, the Universe is now filled with protons, atoms, molecules, stars, planets, black holes, and galaxies whose densities are higher than the current average density of the Universe. These condensations can also be described as first-order symmetry-breaking phase transitions from a disordered higher symmetry hot phase to a more ordered lower symmetry cooler phase.^{3,4} To help quantify the context for this sequence of transitions, we compute and plot (Fig. 1) the time dependence of the decreasing density and temperature of the Universe.

### B. Changing dominant densities in the Universe

Starting with inflation, the dominant densities have been the densities of the false vacuum energy of inflation ( $ \Omega \Lambda i$), radiation (Ω_{r}), matter (Ω_{m}), and finally today, vacuum energy or dark energy ( $ \Omega \Lambda $). The three transitions among these four epochs are known, respectively, as reheating, matter-radiation equality, and the beginning of vacuum energy domination.

The details of inflation are largely unknown.^{3,12} For simplicity, we assume the initial condition at the Planck time that the Universe was at the Planck temperature and the Planck density (*t _{p}*,

*T*, and

_{p}*ρ*, respectively). We assume the Universe underwent inflationary expansion

_{p}^{13–15}that ended at the grand unified theory (GUT) scale ( $ t \u223c 10 \u2212 32 \u2009 \u2009 s$) when reheating produced a radiation-dominated Universe with an energy density equal to the energy density during inflation: $ \rho GUT = \rho \Lambda i$.

^{16}Following Refs. 1 and 17, we also assume radiation domination before inflation. These assumptions constrain inflation to start at $ t \u223c 10 \u2212 36 \u2009 s$.

As the Universe expanded, the scalefactor (*a*) increased. Since the density of radiation $ \rho r \u221d a \u2212 4$, while the density of matter $ \rho m \u221d a \u2212 3$, expansion led to matter-radiation equality: $ \rho r \u223c \rho m$. After equality, the Universe became matter dominated and gravity, like the other stronger forces before it, could begin to condense or accrete objects out of the background.

### C. Relativistic degrees of freedom in the early Universe: $ g *$

^{1,3,18}

^{19}The $ g *$ in Eq. (2) can be thought of as a measure of the heat capacity of the hot relativistic plasma. It is analogous to the number of degrees of freedom of a polyatomic gas. As the temperature increases, more vibrational and rotational degrees of freedom become available. Energy added to the system has to be partitioned among the increasing number of degrees of freedom, rather than directly increasing the temperature of the system. With more degrees of freedom, the heat capacity of the gas increases.

Similarly, as we go back in time (before $ t \u223c 10 2 \u2009 s$) to the increasingly high energies of the early Universe, $ g *$ increases as the two degrees of freedom of photons are joined in thermal equilibrium by the degrees of freedom of the increasingly numerous relativistic particles. Hot relativistic particles act like massless photons since their energy, $ E = ( p 2 c 2 + m 2 c 4 ) 1 / 2$ is dominated by their momentum and can be well-approximated by $ E \u2248 p c$. As we go back in time, getting closer to the big bang, $ g *$ increases. Thus, we need to replace Eq. (1) with Eq. (2), from which we can see that as we get closer to the big bang, *T* does not increase as fast as $ \u223c a \u2212 1$. In the lower panel of Fig. 1, we can see that $ g *$ begins to increase for $ t \u2272 10 2 \u2009 s$. If photons are the only form of radiation, Eqs. (1) and (2) are identical since $ g * = 2$ (one degree of freedom for each of the two photon spin states).

Currently, neutrinos are not in thermal equilibrium with the 3 K photons of the cosmic microwave background. The relativistic degrees of freedom of neutrinos are not included in our $ g *$ for temperatures $ T \u2272 10 10 K$ when they are decoupled from photons.

*ρ*of a relativistic gas also depends on $ g *$. If we only have photons, the energy density is given in Eq. (3). However, if there are other relativistic particles in thermal equilibrium with photons at a common temperature

_{r}*T*, to compute their combined energy density we need to multiply Eq. (3) by $ g * / 2$ to obtain the generalization Eq. (4).

^{1,7,8}Finally, using Eq. (2), we substitute for

*T*in Eq. (4) and obtain Eq. (5): the energy density in all relativistic degrees of freedom (in thermal equilibrium with photons) as a function of scale factor and $ g *$,

^{3,12}

^{3}Comparing Eqs. (2) and (5), we see that both temperature and density have the same $ g * \u2212 1 / 3$ dependence. Inserting the $ g *$ of the lower panel of Fig. 1 into Eqs. (2) and (5) enables us to plot in the upper panel of Fig. 1 the time dependence of the background temperature and density during the condensation of objects in the Universe.

## II. PLOT OF ALL OBJECTS

### A. Objects and isodensity lines

In Fig. 2, we plot all the composite objects in the Universe: protons, atoms, life forms, asteroids, moons, planets, stars, galaxies, galaxy clusters, giant voids, and the Universe itself. Humans are represented by a mass of 70 kg and a radius of 50 cm (we assume sphericity), while whales are represented by a mass of 10^{5} kg and a radius of 7 m. Objects with uniform density *ρ* are described by $ m \u221d \rho \u2009 r 3$. Thus, in a log(m)–log(r) plot such as Fig. 2, all objects of the same density fall along the same isodensity line of slope 3. For example, atoms and objects made of atoms, such as life on Earth (viruses, bacteria, fleas, humans, and whales) asteroids, moons, planets, and main sequence stars, lie close to the atomic density line $ \rho atomic \u223c \rho water = 1 \u2009 gm / cm 3$. At the top of the plot, this line is labeled “atomic $ 10 3 s$,” because objects along this isodensity line have the density of water, and because the entire Universe had this density at the end of Big Bang Nucleosynthesis, $ \u223c 10 3$ s after the big bang. Protons, neutrons, and neutron stars are found along the slope $ = 3$, nuclear density line which is $ \u223c 14$ orders of magnitude more dense than anything made of atoms: $ \rho nuclear / \rho atomic \u223c 10 14$. It is labeled “nuclear $ 10 \u2212 6 \u2009 s$” because the entire Universe was at this nuclear density a millionth of a second after the big bang.

The largest objects in the upper right are super-clusters of galaxies with densities approximately 20% larger than the current matter density of the Universe. For completeness, we have also plotted the largest known voids. The current matter density is the longest diagonal isodensity line on the right labeled at the top “now $ 10 17 \u2009 s$”). This density is the value in Fig. 1 of the black ( $ \rho r + \rho m$) line at *t* = *now*.

### B. Black holes and the zone forbidden by gravity

^{29}

### C. Compton wavelengths and the zone forbidden by quantum uncertainty

*m*is the mass of the particle. The higher the velocity

*v*of a particle, the smaller its deBroglie wavelength. In the relativistic limit when $ v \u2192 c$, the deBroglie wavelength asymptotes to the smaller Compton wavelength

*λ*,

_{c}*λ*, the concept of a single quantum mechanical particle (“object”) breaks down and we must switch to a field description in which particle creation and annihilation occur, preventing further spatial localization. In other words, localization of a wave packet to constrain a particle to a size less than its Compton wavelength is prevented by pair-production. Since the Compton wavelength is the lower limit beyond which object size and position are conflated by quantum uncertainty, we take the Compton wavelength as the effective minimum radius of a particle. This produces the $ m \u221d r \u2212 1$ line [Eq. (7)] delimiting the triangular “quantum uncertainty” region in Fig. 2.

_{c}In addition to composite particles, we also plot fundamental structureless particles, e.g., quarks and leptons. As examples, we plot the top quark, electron, and neutrinos. These all lie along the Compton wavelength boundary. For completeness, we would also like to plot massless photons. However, since the Compton wavelength of a massless particle (photons, gluons, and gravitons) is infinity, we plot photons at $ ( m eff , size ) = ( E / c 2 , \lambda \gamma )$ where their angular wavelengths $ \lambda \gamma = \u210f c / E$. Thus, photons of the entire electromagnetic spectrum can be plotted. They fall along the Compton limit line since $ m eff \u223c \lambda \gamma \u2212 1$. The narrow rainbow at $ E \u223c 10 \u2212 9 \u2009 GeV$ is the entire visible spectrum, while the entire electromagnetic spectrum extends from the shortest wavelength gamma ray $ \lambda \gamma = l p$ to the longest radio waves extending off the plot beyond the size of the observable Universe.

### D. Stellar mass black holes and degeneracy pressure

Figure 3 illustrates some important features of stellar evolution. When a main sequence star (right side of Fig. 3) runs out of fuel, it can no longer maintain the thermal radiation pressure *P _{rad}*, to counteract gravitational pressure

*P*: $ ( P rad \u2192 0 < P g )$. It collapses and becomes a white dwarf held up mostly by electron degeneracy pressure ( $ P e \u223c P g$). Counter-intuitively, more massive white dwarfs are smaller than less massive ones because as gravity compresses massive particles, temperatures increase, velocities increase, and the deBroglie wavelengths

_{g}*λ*of the electrons decrease and at relativistic energies asymptote to their smaller Compton wavelengths [Eq. (7)]. Gravity cannot compress the sizes of the electrons to be less than their Compton wavelengths. This size limit is the source of the electron degeneracy pressure that holds up white dwarfs. However, if a white dwarf can accrete more mass than the Chandrasekhar limit $ \u223c 1.4 \u2009 M \u2299$,

_{deB}^{30}gravitational pressure at the center is enough to overcome electron degeneracy pressure (

*P*>

_{g}*P*). Electrons are pushed into protons producing neutrons, and thus, white dwarfs collapse into neutron stars held up by neutron degeneracy pressure.

_{e}^{31}If a neutron star can accrete more mass than the Volkoff–Oppenheimer–Tolman limit of $ \u223c 3 \u2009 M \u2299$,

^{31}the star will continue to collapse, overcoming neutron degeneracy pressure and collapsing into a black hole.

## III. SOME FUNDAMENTAL QUESTIONS

### A. Is the Universe a black hole?

*r*called the Hubble radius, the recession velocity is equal to the speed of light,

_{H}*r*centered on us and is often taken as the size of the Universe. At present, $ r H \u2248 14$ Gly.

_{H}^{33}In Fig. 2, the most massive point on the black hole line is labeled “Hubble radius” at the point $ ( m U , r H )$ where the mass of the Universe is the critical density times its volume: $ m U = \rho c \u2009 ( 4 / 3 ) \u2009 \pi \u2009 r H 3$.

*r*=

_{s}*r*), we can use Eqs. (6) and (8) in Eq. (10) to obtain

_{H}*ρ*is the critical density. Thus, a Schwarzschild black hole the same size as our Universe has the same mass and density as our Universe. This seems to suggest that the entire Universe is a black hole. Although this idea has been explored in Refs. 34–37, Fig. 4 illustrates why this is not the case.

_{c}### B. Where exactly do the black hole and Compton boundaries cross?

^{28}We want to verify that this instanton crossing point happens at $ ( l p , m p )$. The Compton wavelength [Eq. (7)] of a Planck-mass particle is

*l*is the Planck length and the Planck mass $ m p = ( \u210f c / G ) 1 / 2$. Thus, the Compton wavelength of a Planck-mass particle equals the Planck length: $ \lambda c ( m p ) = l p$. However, what about the black hole diagonal line? Is the Schwarzschild radius of a Planck-mass black hole equal to the Planck length?

_{p}*L*per unit mass $ r L = ( L / m ) \u2009 c$ [Ref. 16, p 60, Eq. (2.100)]. Then, in the equatorial plane of the rotating black hole, we have singularity solutions

*r*can take on values in the range $ r L \u2208 [ 0 , r s / 2 ]$. These two solutions are called the inner (Cauchy) horizon and the outer horizon. For

_{L}*r*= 0 (non-rotating), we recover the Schwarzschild solution $ r + = r s$ as the outer horizon. However, we also have a solution for the inner (Cauchy) horizon $ r \u2212 = 0$ that is often ignored. Importantly, $ r + < r s$ for all non-zero values of

_{L}*r*. For all values of

_{L}*r*, the average of the two solutions equals $ r s / 2$. For a maximally rotating black hole, the two horizons merge, $ r \xb1 = r s / 2$. The inner (Cauchy) and outer horizons for a maximally rotating Planck-mass black hole are both equal to the Planck length: $ r \xb1 = l p = \lambda c ( m p )$.

_{L}The Reissner–Nördstrom metric for a charged (non-rotating) black hole leads to analogous solutions: maximally charged Planck-mass black holes have $ r \xb1 ( m p ) = r s / 2 = l p$.^{39,40} Thus, the most fundamental length in both the Kerr and Reissner-Nördstrom metrics for a Planck-mass black hole is the Planck length $ l p = r s / 2$. In Fig. 2, if we had represented the radius of a black hole by the average of the outer horizon and the inner (Cauchy) horizon: $ r B H = ( r + + r \u2212 ) / 2$, the black hole line and the Compton wavelength line would cross exactly at the instanton point $ ( l p , m p )$.

## IV. DISCUSSION

The Planck-mass instanton is the smallest mass a black hole can have without entering the region of quantum uncertainty. Instantons seem to be the smallest objects in the Universe (white dot in Fig. 2).^{20} On the upper left side of Fig. 1, we have assumed the initial condition that the Universe started out at the Planck time with the Planck density and Planck temperature. In Fig. 2, the intersection point of the vertical white line at the Planck length and the diagonal dashed white line at the Planck density is an instanton. The Hawking temperature of an instanton is the Planck temperature.^{16} Thus, we have assumed that the initial conditions of the Universe are that of an instanton. Instantons seem to be an essential ingredient for quantum cosmology, and their study is an active field of research that is beyond the scope of this paper.^{20,26,27,41–47}

It is possible that some kind of quantum degeneracy pressure holds up the core of a black hole and prevents it from becoming a Schwarzschild singularity.^{48,49} If so, the cores of black holes could be Planck-density objects located in the “forbidden by gravity” region along the Planck isodensity line. Or the cores could have sizes corresponding to the inner Cauchy horizons [ $ r \u2212$ in Eq. (14)], also located in the “forbidden by gravity” region.

*r*of an object violates both general relativity: $ r < r s = 2 G m / c 2$ and quantum uncertainty: $ r < \lambda c = \u210f / m c$. In terms of mass

*m*, gravity and quantum uncertainty prevent the mass of an object from satisfying

^{27,50,51}

Carr and collaborators have raised some fundamental issues about the orthogonal symmetry of the black hole line ( $ m \u221d r$) and Compton line ( $ m \u221d r \u2212 1$) around the horizontal dashed line in Fig. 2. They refer to this symmetry as the “Compton–Schwarzschild correspondence,” which plays a fundamental role in quantum gravity.^{20,27,51,52}

The history of objects in the Universe can be seen as a history of condensations of composite objects from an undifferentiated background. Although composite objects condensed when the binding energy of the object exceeded the background energy, notice in Fig. 2 that no known objects condense before the electroweak (EW) energy scale at $ 10 \u2212 10 \u2009 s$, because the binding energies of all known composite objects are less than the background energy at these early times. Perhaps there are composite objects embedded in the quark-gluon plasma (QGP) held together by the unified strong, weak and electromagnetic forces. Two important open questions are: What were the first composite objects? and If we consider virtual particles to be objects, where do they belong in the diagram?

## V. CONCLUSIONS

There is a long inspiring pedagogical tradition in physics of putting everything into one log-log plot. This tradition includes a logarithmic overview of all space (powers of ten^{53}), a logarithmic overview of all time (time in powers of ten^{54}), and “the complete history of the Universe” (Fig. 3.7 of Ref. 1). Okun's “the physical theories cube” (Fig. 2 of Ref. 55) is a powerful pedagogical tool that enables us to imagine the variation of three fundamental constants $ 1 / c$, *G,* and $\u210f$. Each of the eight vertices of his cube corresponds to different physical theories.

Here, we provide an overview of the history of the Universe and the sequence of composite objects (e.g., protons, planets, galaxies) that condensed out of the background as the Universe expanded and cooled. We describe the role of the effective number of relativistic degrees of freedom ( $ g *$) needed to understand the thermal history of the Universe during the first few minutes after the big bang. We compute and plot the background density and temperature of the Universe (Fig. 1). To extrapolate into the first billionth of a second, we make some common, explicit, but speculative assumptions.

We then make the most comprehensive pedagogical plot of the masses and sizes of all the objects in the Universe (Fig. 2). This plot draws attention to the unphysical regions forbidden by general relativity and quantum uncertainty—regions bounded by black holes and the Compton limit. The Compton limit creates an ambiguous region beyond which object size and position are conflated by quantum uncertainty, thus undermining the classical notion that the size of an object can be arbitrary small. Figure 2 also helps navigate the relationship between gravity and quantum mechanics and helps formulate some fundamental questions about the limits of physics: How can we interpret the regions forbidden by general relativity and quantum uncertainty? How should we interpret the fact that the two boundaries of the forbidden regions intersect at the instanton (Planck-mass black holes)? Are instantons the smallest possible objects? Do their size, density and temperature make them the best candidates for the initial conditions of the Universe (Fig. 1)? Is the Schwarzschild radius the minimum size for an object of a given mass? Or might the non-singular cores of black holes be objects with the Planck density?

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

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