**[Cosmological Principle]**^{**} xix-

In the following short review, we shall try to display the structure and logic of modern cosmology. Our philosophy is that the history of the Universe is infinitely more interesting than the history of the study of the Universe.

The most important fact of cosmology is the observed isotropy of the Universe. This applies to the radio emission penetrating the entire Universe (the "relic radiation" 𝑅𝑅). These forms of radiation reach us from great distances, yet the deviation from isotropy in their intensity does not exceed 0.1-1.0%.

The next step in understanding the structure of the Universe consists of a bold extrapolation. We know hat we are located in a spiral galaxy, which in turn part of a cluster of galaxies. However, the isotropy of 𝑅𝑅 and of the other forms of radiation incident from afar is more fundamental than the local inhomogeneity of our immediate surrounding. The deviation from homogeneity turn out to be less than 0.1-1.0% on a scale of ~10^{10} light years. [The assumption that the Universe is spatially homogeneous and isotropic, i.e. the cosmological principle; empirically, this is justified on scales larger than ~100Moc.]

It is possible to assume that the Earth is located at the center and the average density of the matter in the Universe depends on the distance from the center, but not on the direction. Finally, an in-depth analysis of all of the observational data leads to the conclusion that the Universe is homogeneous.

**[General Theory of Relativity]** xx-

Such a change in the laws of physics over large scales did occur when the general theory of relativity 𝐺𝑇𝑅, with its notion of spacetime curvature, was developed. Now 𝐺𝑇𝑅 is so designed that the correspondence principle is satisfied: it gives the known laws of physics identically over small scales. Besides the demonstrated existence of spacetime curvature, there is still the unanswered question that applies specifically to large scale-the question of the cosmological constant.

The cosmological constant describes the (positive or negative) energy density and corresponding pressure attributed to the vacuum. This leads to the appearance of an additional relative acceleration between any two particles, an acceleration depending only on the distance between the particles and not on their mass. The cosmological constant describes a universal forces of repulsion or attraction proportional to distance. For the dynamics of the Universe as a whole, because of the enormous distances involved, the effects of the cosmological constant can be substantial.

When prequantum physics collided with the ultraviolet catastrophe in the radiation law and with the unexplained stability of atoms. So it was, when a relativistic generalization of Schrödinger's equation was required for a description of high-velocity quantum phenomena. However, there are no observational data suggesting a limitation on the applicability of 𝐺𝑇𝑅 to the scale of the Universe.

Thus, the aggregate of theoretical, experimental, and observational facts stand in favor of the applicability of the physical laws and 𝐺𝑇𝑅 to a description of the evolution of the Universe from *almost* the very beginning of the expansion. They apply from times when the matter density is much greater than the density of nuclear matter, (𝜌 > 10^{14} g cm^{-3}), up to the present time. In short, we adopt the viewpoint that the homogeneous and isotropic Universe can be examined within the realm of 𝐺𝑇𝑅.

**[Hubble's Law]** xxi-

Observation show that we live in an expanding, indeed an evolving, Universe. The latest measurement is confirm the proportionality between the velocity and the distance: 𝑣 = 𝐻_{0}𝑟, where 𝐻_{0} ≈ 50 km s^{-1} Mpc^{-1} = (6 ⨯ 10^{17} s)^{-1} = (2 ⨯ 10^{10} yr)^{-1}.^{***} The quantity with the dimensions of time inside the parentheses corresponds roughly to the time for expansion from the high-density state. The quantity 𝐻_{0} "Hubble's constant" (the present value of the "Hubble parameter")-is known from observations to an accuracy of at least 50%.

Many solutions of the equations of 𝐺𝑇𝑅 agree with the linear relation 𝑣 = 𝐻_{0}𝑟, At the same time one also assumes that the density and pressure of the matter are everywhere the same. For a complete solution to the problem of specifying a solution of the equation, it is also necessary to know the numerical values of matter density and pressure today, and the value of the cosmological constant 𝛬.
**[Matter Density and Cosmological Constant]** xxii-

Very likely, 𝛬 ≈ 0. In this case the future evolution depends essentially on 𝜌_{0}, the present-day value of the matter density only. There is a critical value of the density, 𝜌_{c}, depending on 𝐻_{0}. 𝜌_{c} ≈ 0.5 ⨯ 10^{-29} g m^{-3} [𝜌_{c} ≈ 10^{-29} g cm^{-3}; 𝜌_{c} = 3𝐻_{0}^{2}/8π𝐺, 𝐺: gravitational constant]. For 𝜌_{0} < 𝜌_{c} the expansion will continue without bond, while 𝜌_{0} > 𝜌_{c} the expansion will change to contraction. Thus the most important property of the Universe-the geometrical structure-also depends on the density alone (if 𝛬 ≈ 0.) For 𝜌_{0} < 𝜌_{c}, the Universe is infinite and, in this sense, does not differ from classical, three dimensional , Euclidean space. Thus the density, the future, and the geometrical structure of the Universe are related to one another.

The case of nonzero cosmological constant is more complicated. For example, given a definite relation 𝛬 and 𝜌_{0}, the finite Universe case is possible, though in the future, unlimited expansion is in prospect. We omit a complete classification of such solutions here.
In principle, one can determine the structure of the Universe from the curvature of space by observing how the brightness of sources with known absolute luminosity changes in relation to their distance or to their red shift. However, this redshift-distance dependence varies from model to model.

The theory of light propagation and the connections between the properties of distant objects and observable quantities are those aspects of cosmology which have been worked out most extensively. It would, in principle, be sufficient to measure the redshift and the light flux of two distant objects with known absolute luminosity in order to determine 𝐻_{0} and the density 𝜌_{0}. This is true only when 𝛬 = 0; however, if 𝛬 ≠ 0, an additional observation is necessary. [Using the value known in 2018 𝛬 ≈ 10^{-48} cm^{-2}] In order to establish the structure of the Universe, one can measure in place of luminosity the number of objects of a given type in a given element of solid angle and in a specified interval of redshift. Again though, this must be for objects whose density in space is known from independent considerations.

Rather, proof has been obtained that the specific number density (or the intrinsic luminosity) of the most powerful sources energy in the Universe-quasars and radio sources-was substantially higher in the past than it is today. Therefore the structure of the Universe has not been established through observations of ordinary galaxies because of small distance and evolutionary effect.

**[Density Parameter]** xxiii-

The average density of visible matter in galaxy today is known approximately. If one "spreads" it over all of space, then the resulting density is about 10^{-31} g cm^{-3} [Smoothed density of galactic material throughout universe: 2 ⨯ 10^{-31} g cm^{-3}], i.e., about 50 times less than the critical density. [Density parameter 𝛺 ≡ 𝜌/𝜌_{c}, the ratio of the actual density 𝜌 to the critical density 𝜌_{c} of the Friedmann universe] Very recently it has been shown that massive elliptical galaxies are possibly surrounded by halos consisting of starts of low luminosity. Taking account of this, the average density can be increased several-fold. Now the mass of galaxies is determined from an analysis of the motions of stars and gas clouds under the natural assumption that the age of the galaxies is many times larger than the period of a star in a galactic orbit. Such a method permits one to determine the total mass of a galaxy, a mass including that of invisible forms of material such as extinct stars. Since the mass of a galaxy is approximately equal to the sum of the rest masses of baryons of which the galaxy is composed, one can thereby obtain the average number density of baryon in the Universe today-a density whose order of magnitude is ∼6 ⨯ 10^{-8} cm^{-3}. [2-2.5 ⨯ 10^{-7} cm^{-3}]

**[Charge Asymmetry]** xxiv-

Special searches have been undertaken for the effects of the annihilation of particles and antiparticles between matter galaxies and antimatter galaxies. These searches have yielded a negative result. Therefore, at the present time the assumption of charge asymmetry in the Universe is more probable.

**[Relic Radiation-Cosmic Background Radiation]**

Observations show the presence of intense radio emission in the centimeter and millimeter bands, which has been assumed that the primordial and arise in the very early stages of the Universe. Using as a basis the observational data and a minor theoretical extrapolation, the number density of the primordial photons-𝑅𝑅 [also known as cosmic background radiation 𝐶𝑀𝐵 or 𝐶𝑀𝐵𝑅] has been calculated to be about 400 cm^{-3}. The average energy of the photons is about 0.0007 eV, [Planck-Einstein relation 𝛦 = 𝘩𝑐/𝜆, 𝜆 ≈ 0.2 cm] while energy density is 𝜖 = 4 ⨯ 10^{-13} ergs cm^{-3}. The corresponding mass density is 𝜌_{rad} = 𝜖/𝑐^{2} = 5 ⨯ 10^{-34} g cm^{-3}. Thus the contribution of the radiation to the total density is quite small at the present time.

**[Intergalactic Gas]**

The numerous theoretical and observational studies of the intergalactic gas are still incomplete. The neutral hydrogen content of this gas is quite low, for any such gas must be entirely ionized. Its temperature presumably lies within the limits 2 ⨯ 104 K < 𝑇 < 3 ⨯ 108 K. Such gas is practically transparent at all wavelengths; indeed, the gas itself would emit somewhere between the ultraviolet and X-ray part of the spectrum. The best estimates give the inequality 𝜌_{gas} ≲ 𝜌_{c}. [Warm-hot intergalactic medium 𝑊𝐻𝐼𝑀 which was proposed solution to the missing baryon problem and was detected since 2010 by Chandra X-ray Observatory along the galaxies some million light-years from Earth.]

**[Neutrinos and Gravitational Waves]**

A direct determination of the density of neutrinos and gravitational waves is quite difficult. Their density could exceed by many times the density of ordinary matter (baryons). [Refer to "dark matter" which is theoretically supposed to be ∼27% of the University while normal matter is ∼5% since 1998 discovery of the accelerated expansion of the Universe] Even so, direct physical methods would not have sufficient sensitivity to detect these neutrinos and gravitons. Indirect estimates suggest that the density of such particles is substantially less that the density of ordinary matter. [Neutrinos and gravitions can be among many candidates for dark matter though.]

Thus, there is as yet no answer to the question of whether or not 𝜌_{o} is larger than 𝜌_{c}, and consequently, whether or not the Universe is finite or infinite.

**[Theory of Hot Universe]** xxv-

Knowing the present content of the Universe, one can trace the earlier stages of evolution. The most important fact is that of the expansion of the Universe. As the expansion proceeds, the 𝑅𝑅 temperature falls. The value of this temperature today is 2.7 K;in the past it was much higher.Early in the expansion, the dense matter is opaque to radiation. The matter and radiation are in thermodynamic equilibrium and have a very high temperature. Hence the name, the "theory of the hot Universe."

This picture entails first of all the homogeneous and isotropic. Second, it entails a Universe filled with material that is numerically dominated by photons. Thus, knowing the expansion law and making use of the laws of physics, one can compute the past history of the matter and determine the ongoing physical process.

**[Radiation-Dominated Era]**

The mass density of ordinary matter (baryons) falls with the expansion inversely as the volume 𝜌_{b} ∼ 𝑉-1. The 𝑅𝑅 energy density falls faster than 𝜌_{b}: 𝜌_{rad} ∼ 𝑉^{-4/3}. Consequently, in the past, the photons predominate at an early stage, not only with respect to number but also respect to mass. At this stage, called the radiation-dominated 𝑅𝐷 era, the ordinary matter consists of completely ionized helium and hydrogen.

From the equations of mechanics and from the known interactions between photons and atoms, one can find the temperature and composition as a function time (where we will reckon the time 𝑡 from the moment when 𝜌 = ∞ in the Friedmann solution). For example, at the time 𝑡 = 1 s, the temperature is about 1 megavolt or 1010 K, while the density is 106 g cm^{-3}. [Boltzmann constant: 𝑇 = 𝐸/𝑘_{B}, 𝑘_{B} ≈ 8.6 ⨯ 10^{-5} eV K^{-1}.] Besides photons, there are at this time almost as many pairs of electrons and positrons of present.However, complex nuclei cannot exist at this time. Protons and neutrons exist in almost equal numbers due to the collisions with electrons and positron, collisions which lead to mutual transformations of the protons and neutrons into each other.

As the expansion proceeds, the positrons disappear. A portion of the neutrons decay, while the remaining neutrons combine with the protons, yielding finally 70% hydrogen and 30% helium by mass. Traces of deuterium and of helium-3 may result, but there is almost a complete absence of heavier elements. This prediction of the theory does not contradict the meager evidence on the possible content of primordial matter (matter not processed by stellar nucleosynthesis). Neutrinos and antineutrinos from this period must also remain. those remaining must be roughly equal in number to the number of photons and must have an average energy today corresponding to a temperature of several degrees.

**[Epoch of Recombination]** xxvi-

During the expansion following the 𝑅𝐷 era the mass density of ordinary matter exceeds the mass density of photons. Moreover, the matter is in the form of neutral atoms.

These includes a treatment of deviation from homogeneity and a treatment of the earliest stages, when, along with the photons, diverse particles and antiparticles are present and quantum phenomena are important. There is a definite connection between the question of inhomogeneity and that of the evolution during the earliest stages.

From the observational point of view, the possibility of studying the early stages is limited by scattering of electromagnetic waves by electrons. All 𝑅𝑅 received today is repeatedly scattered by electrons when the scale of the universe 1000 times smaller than that now, i.e.at a redshift 𝑧 ≡ 𝛥𝜆/𝜆 ≈ 1000. [update: 𝑧 ≈ 1100; (1 + 𝑧) = 𝑎_{observe}/𝑎_{emit} = 1/𝑎, 𝑎: cosmic scale factor] at that time the matter is in the form of an ionized plasma at a temperature of 3000 K. only in this epoch nearest of ours the temperature fall sufficiently so that the recombination of hydrogen occurs. When the plasma is neutral, it becomes transparent to radiation to radiation. One cannot "see" the more distant past with the help of electromagnetic waves. This is the same as ∼10^{6} yr from the beginning of the expansion of the Universe. [about 380,000 years after the Big Bang]

We call "observable" that part of the universe from which electromagnetic waves can reach the observer without being absorbed or scattered en route. In principle, neutrinos, which are very penetrating particles, could enable one to observe an even larger part of the Universe. Particles emitted at the singularity, moving with the speed of light, could reach the observer; the corresponding distance is called the "horizon".

the "observable" radius is large-about 97%, [99.9% according to the scale of the Universe at 𝑧 ≈ 1100 though], of the "horizon" radius. The consequent volume accessible to observations with the help of 𝑅𝑅 is 90% of the maximum possible volume. Homogeneity on a large scale is thereby established. However, not excluded is the possibility of inhomogeneity and of anisotropy of the expansion before the recombination time of 𝑧 ≈ 1100. What is more, the present-day structure of the Universe with its separate galaxies and clusters of galaxies, demonstrates the existence of and the requirement for deviations from the ideal picture of homogeneity and isotrophy on all scales.

**[More Subtle Questions]** xxvii-

The general picture pd an expanding "hot Universe" set force above is reliably established and is one of the most important achievements of 20th-century science. Now we shall pass to more subtle questions, such as the problem of the origin of galaxies and the problem of the beginning of the cosmological expansion. Not surprisingly, we will speak in a less definite tone and will enumerate various hypotheses.

**[Theory of Small Perturbation]**

Producing a through analysis of the deviations from the ideal picture of a homogeneous, isotropic Universe is the most important task confronting cosmology.The theory of small perturbations provides a solid foundation for the research. the most general property of a Universe close to the ideal is that one can classify the deviations from the ideal (perturbations) into separate forms ("modes") that evolve independently of one another. Such a theory is of great value in an analysis of the observations.

At each instant the ideal model is characterized by several numbers in all (the radius of the Universe, the expansion rate, the density, the temperature, and so forth).The complete evolutionary picture in his approximation is then dependence of these several numbers on time. This simplification is achieved by limiting the problem to the homogeneous case, or, as is said in mathematics, by considering a "degenerate" case. In distinction, we must consider a least four variables-time and the three spatial coordinates. The difficulty is further aggravated by our knowledge of the initial conditions-the conditions at the beginning of the expansion. It is necessary either to investigate the large number of variants of the initial state or, having an (incomplete!) idea of the present-day states, to solve the equations for the behavior in the past. Both approaches encounter great difficulties.

The theory of small perturbations is remarkable in that it combines generality of formulation with mathematical simplicity. On the other hand, by virtue of the small amplitude of the perturbations or by virtue of the linearity of the problem. an examination of each mode reduces simply to finding several functions of time. In other words, the problem is of the same class of difficulty as the problem of the motion of a point. This remarkable combination of generality and simplicity explain why a significant part of the book is devoted to a study of the theory of small perturbations of the homogeneous, isotropic Universe.

In the theory of small perturbations, one considers the evolution of separate modes. However, one must take this quantity from outside the theory-from observation or from other theoretical considerations. The theory tells us, for example, that a density perturbation encompassing a mass greater than 10^{15} 𝑀_{⊙} grows by a factor of 100 in the time that the scale of the Universe also grows by a factor of 100 in the time that the scale of the Universe also grows by a factor of 100 (for example from 𝑡 = 10^{6} yr to 𝑡 = 10^{9} yr). However, whether or not this kind of perturbation occurs in nature, its amplitude remains unknown in terms of this theory alone. But what do observations tell us? Direct observations of the matter distribution tell us that on scales of the order of 10 Mpc and less, the inhomogeneity is large. This maximum scale is roughfly the average distance between neighboring clusters of galaxies. For 𝜌_{0} between 10^{-31} g cm^{-3} and 𝜌_{c} ≈ 1/2 ⨯ 10^{-29} g cm^{-3} the mass in a sphere whose diameter is 10 Mpc lies between 7 ⨯ 10^{11} 𝑀_{⊙} and 3 ⨯ 10^{13} 𝑀_{⊙}. The density perturbations on this scale today satisfy (𝛿𝜌/𝜌)_{0} ≈ 1. When do such large perturbations form? Apparently in the recent past, at a redshift 𝑧 ≈ 5.

**[Epoch of Formation]** xxviii-

Indeed, on can compute roughly that the matter density in the clusters is of the order of the average density at just that epoch when the clusters-to-be separate from the expanding background and transform into bound configurations. One can clearly imagine that before some definite redshift 𝑧_{1} (i.e. for 𝑧 > 𝑧_{1}), the matter is expanding, while thereafter (for 𝑧 < 𝑧_{1}), it divides into separate parts-the clusters of galaxies. The distance between these parts continues to increase, but the parts themselves are "preserved." They are maintained in a state of dynamic equilibrium without a change in their average density. Therefore, the present average density of the matter in clusters is roughly characteristic of the average matter density at the epoch of formation. It is this fact which permits one to estimate the formation time.

**[Amplitude of Density Perturbations]**

Using perturbation theory, one can draw from this a conclusion concerning the amplitude of density perturbations and etc.in the distant past. For the plasma state in the 𝑅𝐷 stage, both 𝛿𝜌/𝜌 ≈ 10^{-3} is obtained for density oscillations. Correspondingly, both the velocity of the separate elements of matter with respect to the background of general expansion and the dimensionless perturbations of the spacetime metric are of the same order, 10^{-3}. All of this relates to scales characteristic of the present-day scale of clusters of galaxies.

**[The Spectrum of Perturbations]** xxix-

Upon the inclusion of theoretical considerations, radio observations of the fluctuations of 𝑅𝑅 and of its spectrum permits one to estimates the magnitudes of the perturbations as functions of the scale (or mass). i.e., to estimate the spectrum of perturbations. We will discuss this in detail later.

Summarizing, the picture of the Universe sketched here represents a weakly perturbed (amost homogeneous), expanding Universe with a definite initial (and large) entropy. In support of this picture are the measurements of the spectrum and spatial distribution of the relic radio emission. This is the most important result from the observations and theoretical developments of recent years.

**[Some Doubts]**

So the formation of galaxies and of other structural peculiarities is explained by deviations from homogeneity. Such deviations are sufficiently small that they do not invalidate the desirable features of the homogeneous model. However, such a formulation is not universally accepted. On the one hand, doubts have been raised as to whether this formulation is capable of explaining several observed properties of the Universe. We refer particularly to the observed rotation of galaxies [V. Rubin et al. measured the velocity curve of edge-on spiral galaxies with greater accuracy in 1978 and the result showed most galaxies must contain about six times as much dark as visible mass.] and the presence of galactic-scale magnetic fields. Perhaps one should also add the question of the origin of quasars. For this we must know the nature of quasars-their structure and sources and energy.

**[Magnetic Fields on Cosmic Scales]**

The problem of the origin and intensification of magnetic fields on cosmic scales was very troublesome. The properties causing the difficulties are the well-known high electrical conductivity of a plasma and the consequent conservation of a magnetic flux. It is the latter effect to which one refers when it is said that magnetic fields are "frozen into" the plasma. Maintenance of a magnetic field requires no external sources of electric current because the decay time of a current in the plasma is on cosmic scales. The problem was that it seemed for a time that this same "frozen in" property would prevent an ordered magnetic field from growing in a plasma originally field free.

In this connection, "magnetic" cosmological models with a primordial field have been examined. Inasmuch as the field picks out a definite direction in space, though, such models are distinguished by primordial anisotropy.

Among the three-dimensional motions of the plasma during the formation of rotating galaxies are motions which are capable of generating and intensifying magnetic fields to the values observed today. Hence the hypothesis of a primordial magnetic field has become unnecessary. The question remain unsettled, howeve, because a demonstration of the absence of a small initial field (corresponding to a 10^{-10} gauss field after the expansion to the present state0 would be very difficult.

**[Rotation of Galaxies]** xxx-

The rotation of galaxies is among their most important features, for it is through such rotation, or due to the centrifugal force, that their disklike shape is produced. Moreover, the origin of distinct arms and of spiral structure is a further consequence of the instability of a flat rotating disk. Thus, there is no doubt that rotation exists and plays an important role. A theory ignoring rotation would be quite incomplete and unacceptable.

There is a property of rotational motion analogous to the concept of a frozen-in magnetic field. We refer here to the conservation of angular momentum of an isolated system. If galaxies arise from "potential" or vortex-free (irrotational) perturbations, then it is difficult at first glace to understand the origin of the rotation of galaxies. The simplest explanation of the rotation is that elements of the medium present in the 𝑅𝐷 stage are endowed with primordial rotation. It is in just this way that the hypothesis of a turbulent Universe and of initial photon vortices is formulated. A later development is to use the modern theory of turbulence to obtain more detailed relationships among the angular momentum of a galaxy, the mass of a galaxy, and so forth.

However, one can also explain the origin of the rotation of galaxies within the confines of a vortex-free theory. At the time of the formation of galaxies from nearly homogeneous matter, neighboring protogalaxies interact in the strongest possible way. Thus, the law of angular momentum conservation does not apply to each galaxiy individually. It is therefore by no means excluded that rotation can arise spontaneously in the evolution of perturbations which are originally vortex-free.

**[Vortex Theory]**

The heated discussion between the proponents of the "potential" and the proponents of the "vortex" theories relate to two difficulties with the vortex theory. The first is that the theory apparently leads to an excessively early isolation of the galaxies from the expanding matter. This isolation occurs when the average density of the Universe is still rather high-about 10^{-20} g cm^{-3}. This in turn leads to several properties for galaxies that are difficult to reconcile with observations. The second problem is related to the assumption concerning the existence of primordial vortex motions. This assumption requires a change in the character of the early stages of the cosmological expansion itself. Even modest, subsonic vortex motions require a break with the theory of an isotropic expansion and of a weakly perturbed, homogeneous model. According to the revised theory, at least the first 10^{5} s [∼28 hr] of the expansion are described by the homogeneous, isotropic model. In addition, by admitting the vortex theory, one must also alter the theory of primordial nulceosynthesis, for this synthesis occurs during the first 100 s of the expansion.

**[Explosion Theory]** xxxi-

In this theory galaxies form not from the rarefied, expanding gas by means of the gravitational condensation of small inhomogeneities, but by means of explosions of hyperdense bodies. But the proponents of this theory must address themselves to the violations of the fundamental laws of physics implied by the existense of pregalactic bodies. For example, the conservation laws for angular momentum and energy appear to be violated in such bodies. To be sure we do not think that the solution to the problem of galaxy formation lies along a path involving a rejection of the foundations of modern physics.

**[The Very Beginning]**

Earlier we outlined in general terms a picture of the hot, expanding Universe from the first second to the time of galaxy formation. We are thus led to a problem of great importance: the question of the very beginning of the cosmological expansion. From what does the expansion begin? How does the Universe expand at the very beginning? What happens before the start of the observable expansion?

**[Singularity Itself]**

Quite recently, the question was discussed of whether there is actually a singularity-of whether there is actually infinite density t the start of the expansion. Is it possible that a singularity is characteristic only of the ideal homogeneous, isotropic model, and that for a general, asymmetric, nodegenerate solution, contraction of the matter in some preceding epoch changes into expansion without the occurrence of a singularity? What we know now is that there is strict proof of the existence of a singularity in the past even it the behavior of the early stages differs sharply from the presumed homogeneous, isotropic expansion. Suppose we examine (1) the various types of cosmological models laying some claim to a description of the Universe close to the start of the expansion and (2) the specific theories of various types of perturbations possibly existing at the very early stages. This examination would leads us to a very fundamental question: Given the data of all possible: can one unambiguously establish the past history of the Universe,viz., the behavior of the very early stages of expansion, of the singularity itself, and of the epoch "before the expansion".

**[Forgotten Initial State]** xxxii-

To approach this question, think of thermodynamics. It is easy to imagine that under certain conditions a wide class of initial states quickly leads to the same final state, which in turn serves as an initial state for further evolution. The actual state is thus *forgotten*. In this spirit, we perceive a new direction in cosmology-theoretical research of the unobserved early stage. It therefore becomes a question of finding that cosmological model which can be obtained from a wide class of initial early states.

**[Quantum Cosmology]**

It turns out, for example, that a whole group of anisotropic initial states leads to isotropic expansion. Is this the expansion of why the Universe expands isotropically? Perhaps this means that, a large number of of *in principle possible initial Universe*, many become identical to the Universe observed and studied by us. But there is only one Universe, and therefore statistical arguments do not seem applicable.
Why is the entropy of the Universe large? Why is the Universe hot at the start of the expansion? Finally, why are the perturbations of such magnitude that they leads to galaxy formation in our epoch and not much earlier or much later? It is possible that a new theoretical formulation of these questions will arise from a consideration of quantum cosmology.

**[Theoretical Cosmology]**

In conclusion we emphasize the basis-the physical laws upon which theoretical cosmology is constructed. It would be tempting to proclaim that new laws are necessary to describe phenomena on the scale of the Universe-similar to what occurred when quantum laws are developed to describe the microcosm.

The microcosm called attention to the insufficiency of classical theory 50 years before the creation of quantum theory. In cosmology, however, we have not yet encountered any insoluble contradiction between theory and experiment or any internal logical difficulty in the theory. The theory of the expanding Universe was created before the actual discovery of galaxy recession. Similarly, the theory of hot Universe was created long before the discovery of the 𝑅𝑅. These factors support the correctness of our theories and the applicability of them to scales characteristic of the Universe. indeed, extraordinary and seemingly new phenomena arise in cosmology as a result of the application of existing theory.

𝐺𝑇𝑅, for example, leads to the new situation of relativistic collapse. New phenomena also arise from combining the quantum theory of matter with theory of gravity. The particle creation creation occurs throughout space time-in violation of the laboratory laws of physics. Such theories of continuous matter creation are refuted by observations. However, there is a period in the evolution-close to the singularity-when particles are created in the then very strong gravitational field. In this case, thogh, it turns out that intense particle creation proceeds only when the expansion is anisotropic-a result which is by no means obvious.

**[Quantization of Metric]** xxxiii

Finally, new phenomena may arise in this same period when account is taken of the quantization of the metric. This is a possible theoretical development applicable to conditions definitely unknown to modern physics.

Thus, the fundamental propositions of modern physics (based on experiment and self-consistency) plus the application of them to the new conditions at the singularity itself plus the observational data obtained with the help of both terrestrial instruments as well as experiments in space form the foundation upon which modern cosmology is,or must be, based.

**[Historical Remark]**

In conclusion, we want to make a historical remark. It is well known that A Friedmann created the theory of evolving Universe in 1922-1924. The famous discussion with Einstein on the pages of the *Zeitschrift für Physik* (Magazine for Physics) called attention to Friedmann's work. We therefore cannot agree that Lemaitre (1927) "independently" establisihed the laws of the expanding Universe.

After Hubble's discovery (1929) of the linear redshift-distance relation, the mathematical solutions attained the status of established theory. In reviewing the situation in 1931, Einstein wrote: "Friedmann was the first to follow this way ... ".

* Textbook: Ya. B. Zel'dovich & I. D. Novikov *The Structure and Evolution of the Universe* (University of Chicago Press 1983) Introduction

** Supplements in [ ] according to Wikipedia or *Handbook of Space Astronomy and Astrophysics* M. V. Zombeck, Cambridge University Press 2007

*** 𝐻_{0} ≈ 70 km s^{-1} Mpc^{-1} = (4.2 ⨯ 10^{17} s)^{-1} = (1.4 ⨯ 10^{10} yr)^{-1} [Planck 2018, HST 2019]