Cosmology: Friedmann-Lemaître Universes


Cosmology starts by assuming that the large-scale evolution of spacetime can be determined by applying Einstein’s field equations of Gravitation everywhere: global evolution will follow from local physics. The standard models of cosmology are based on the assumption that once one has averaged over a large enough physical scale, isotropy is observed by all fundamental observers (the preferred family of observers associated with the average motion of matter in the universe). When this isotropy is exact, the universe is spatially homogeneous as well as isotropic. The matter motion is then along irrotational and shearfree geodesic curves with tangent vector ua, implying the existence of a canonical time-variable t obeying ua = −t,a. The Robertson-Walker (‘RW’) geometries used to describe the large-scale structure of the universe embody these symmetries exactly. Consequently they are conformally flat, that is, the Weyl tensor is zero:

Cijkl := Rijkl + 1/2(Rikgjl + Rjlgik − Ril gjk − Rjkgil) − 1/6R(gikgjl − gilgjk) = 0 —– (1)

this tensor represents the free gravitational field, enabling non-local effects such as tidal forces and gravitational waves which do not occur in the exact RW geometries.

Comoving coordinates can be chosen so that the metric takes the form:

ds2 = −dt2 + S2(t)dσ2, ua = δa0 (a=0,1,2,3) —– (2)

where S(t) is the time-dependent scale factor, and the worldlines with tangent vector ua = dxa/dt represent the histories of fundamental observers. The space sections {t = const} are surfaces of homogeneity and have maximal symmetry: they are 3-spaces of constant curvature K = k/S2(t) where k is the sign of K. The normalized metric dσ2 characterizes a 3-space of normalized constant curvature k; coordinates (r, θ, φ) can be chosen such that

2 = dr2 + f2(r) dθ2 + sin2θdφ2 —– (3)

where f (r) = {sin r, r, sinh r} if k = {+1, 0, −1} respectively. The rate of expansion at any time t is characterized by the Hubble parameter H(t) = S ̇/S.

To determine the metric’s evolution in time, one applies the Einstein Field Equations, showing the effect of matter on space-time curvature, to the metric (2,3). Because of local isotropy, the matter tensor Tab necessarily takes a perfect fluid form relative to the preferred worldlines with tangent vector ua:

Tab = (μ + p/c2)uaub + (p/c2)gab —– (4)

, c is the speed of light. The energy density μ(t) and pressure term p(t)/c2 are the timelike and spacelike eigenvalues of Tab. The integrability conditions for the Einstein Field Equations are the energy-density conservation equation

Tab;b = 0 ⇔ μ ̇ + (μ + p/c2)3S ̇/S = 0 —– (5)

This becomes determinate when a suitable equation of state function w := pc2/μ relates the pressure p to the energy density μ and temperature T : p = w(μ,T)μ/c2 (w may or may not be constant). Baryons have {pbar = 0 ⇔ w = 0} and radiation has {pradc2 = μrad/3 ⇔ w = 1/3,μrad = aT4rad}, which by (5) imply

μbar ∝ S−3, μrad ∝ S−4, Trad ∝ S−1 —– (6)

The scale factor S(t) obeys the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ + 3p/c2) + Λ —– (7)

, where κ is the gravitational constant and Λ is the cosmological constant. A cosmological constant can also be regarded as a fluid with pressure p related to the energy density μ by {p = −μc2 ⇔ w = −1}. This shows that the active gravitational mass density of the matter and fields present is μgrav := μ + 3p/c2. For ordinary matter this will be positive:

μ + 3p/c2 > 0 ⇔ w > −1/3 —– (8)

(the ‘Strong Energy Condition’), so ordinary matter will tend to cause the universe to decelerate (S ̈ < 0). It is also apparent that a positive cosmological constant on its own will cause an accelerating expansion (S ̈ > 0). When matter and a cosmological constant are both present, either result may occur depending on which effect is dominant. The first integral of equations (5, 7) when S ̇≠ 0 is the Friedmann equation

S ̇2/S2 = κμ/3 + Λ/3 – k/S2 —– (9)

This is just the Gauss equation relating the 3-space curvature to the 4-space curvature, showing how matter directly causes a curvature of 3-spaces. Because of the spacetime symmetries, the ten Einstein Filed Equations are equivalent to the two equations (7, 9). Models of this kind, that is with a Robertson-Walker (‘RW’) geometry with metric (2, 3) and dynamics governed by equations (5, 7, 9), are called Friedmann-Lemaître universes (‘FL’). The Friedmann equation (9) controls the expansion of the universe, and the conservation equation (5) controls the density of matter as the universe expands; when S ̇≠ 0 , equation (7) will necessarily hold if (5, 9) are both satisfied. Given a determinate matter description (specifying the equation of state w = w(μ, T) explicitly or implicitly) for each matter component, the existence and uniqueness of solutions follows both for a single matter component and for a combination of different kinds of matter, for example μ = μbar + μrad + μcdm + μν where we include cold dark matter (cdm) and neutrinos (ν). Initial data for such solutions at an arbitrary time t0 (eg. today) consists of,

• The Hubble constant H0 := (S ̇/S)0 = 100h km/sec/Mpc;

• A dimensionless density parameter Ωi0 := κμi0/3H02 for each type of matter present (labelled by i);

• If Λ ≠ 0, either ΩΛ0 := Λ/3H20, or the dimensionless deceleration parameter q := −(S ̈/S) H−20.

Given the equations of state for the matter, this data then determines a unique solution {S(t), μ(t)}, i.e. a unique corresponding universe history. The total matter density is the sum of the terms Ωi0 for each type of matter present, for example

Ωm0 = Ωbar0 + Ωrad0 + Ωcdm0 + Ων0, —– (10)

and the total density parameter Ω0 is the sum of that for matter and for the cosmological constant:

Ω0 = Ωm0 + ΩΛ0 —– (11)

Evaluating the Raychaudhuri equation (7) at the present time gives an important relation between these parameters: when the pressure term p/c2 can be ignored relative to the matter term μ (as is plausible at the present time, and assuming we represent ‘dark energy’ as a cosmological constant.),

q0 = 1/2 Ωm0 − ΩΛ0 —– (12)

This shows that a cosmological constant Λ can cause an acceleration (negative q0); if it vanishes, the expression simplifies: Λ = 0 ⇒ q = 1 Ωm0, showing how matter causes a deceleration of the universe. Evaluating the Friedmann equation (9) at the time t0, the spatial curvature is
K0:= k/S02 = H020 − 1) —– (13)
The value Ω0 = 1 corresponds to spatially flat universes (K0 = 0), separating models with positive spatial curvature (Ω0 > 1 ⇔ K0 > 0) from those with negative spatial curvature (Ω0 < 1 ⇔ K0 < 0).
The FL models are the standard models of modern cosmology, surprisingly effective in view of their extreme geometrical simplicity. One of their great strengths is their explanatory role in terms of making explicit the way the local gravitational effect of matter and radiation determines the evolution of the universe as a whole, this in turn forming the dynamic background for local physics (including the evolution of the matter and radiation).

++ Occult/Esoteric ++


The Greeks were adept in the use of imagery to convey profound esoteric truths, often using the form of sport; or, for instance, they would read into the exercises of the stadium inner significance. One of the best known examples of this was their portrayal through the torchbearer race of the mystic line of succession of great teachers.

In the torch-race, the torch-bearer ran from post to post. On reaching the end of his stage he handed the lighted torch which he carried to the one there waiting, who immediately took up the race and in his turn handed it to the one waiting for him. This exercise of the arena or stadium was taken by many Greek and Latin writers as symbolizing the carrying on of Light from age to age, and as pointing to the spiritual Torch-bearers who pass the Torch of Truth from hand to hand throughout unending time…. This handing on of the light of truth “throughout unending time” has formed the theme of many Mystery parables. The Greeks also referred to this spiritual succession as the Golden Chain of Hermes which they believed to stretch far into the realms of Olympus, to “Father Zeus downwards through a series or line of spiritual beings and then through certain elect and lofty human beings to ordinary men” (Esoteric Tradition).

Purucker described this mystic succession as the guruparampara. This is a Sanskrit compound literally meaning “teacher beyond beyond.” The term signifies a line of teachers reaching beyond the beyond, through past, present, and into the distant future, whose sublime purpose is ever the same: the work of spiritualization.

The ancient Mystery-Schools of every country of the globe and of whatever epoch in time, have had each one a Succession of Teachers trained and authorized by their training to teach in their turn; and as long as this transmission of the light of Truth was a reality in any one country, it was in every sense a truly spiritual institution.

An outstanding example of this ancient transmission is the succession of “living buddhas” of Tibet, which “is a real one, but of a somewhat special type, and it is by no means what Occidental scholars mistake it to be or have frequently misunderstood it to be” (Esoteric Tradition).

Further, in the Eleusinian Mysteries of Greece,

hierophants were drawn from one family, the Eumolpidae, living in Athens, and the torchbearers were drawn from another family, the Lycomidae, living in Athens; and we have reason to believe that the Mysteries of Samothrace, the seat of an older rite, and which were, like the Mysteries of Eleusis, a State function, were also conducted in the same manner by the passing on of the tradition held sacred and incommunicable to outsiders; and the bond of union between the initiates of these so-called Mysteries was considered indissoluble, impossible of dissolution, for death merely strengthened the tie.– Fundamentals of the Esoteric Philosophy

In Persia as well as Egypt, we find this line of succession manifesting in another form. For example, there were the thirteen or more Zoroasters whose esoteric contribution to Persia’s history was the inspiration of that once mighty civilization:

The number of Zoroasters who have appeared from time to time is confusing, so long as we consider, and wrongly consider, these Zoroasters to be reimbodiments of one single ego, instead of different egos imbodying what we may interpret from the occult records as the “Zoroaster-spirit.” The truth of the matter is that in the scheme and terminology of Zoroastrianism, every Root-Race and sub-race, and minor race of the latter, has its own Zoroaster or Zoroasters. The term Zoroaster means in Zoroastrianism, very much what the term Buddha does in Buddhism, or Avatara does in Brahmanism. Thus there were great Zoroasters, and less Zoroasters — the qualificatory adjective depending upon the work done by each Zoroaster, and the sphere of things. Hence we can speak of the Zoroasters as being thirteen in number from one standpoint, or fourteen from another; or like the Manus in Brahmanism, or like the Buddhas in Buddhism, we can multiply each of these by seven again, or even fourteen if we take in every little branchlet race with its guiding Zoroaster-spirit.– Studies in Occult Philosophy.

In Egypt, Hermes Trismegistus (“Hermes the thrice greatest”) stands out from the long Hermes line, whose writings and teachings were founded on the ancient Mystery doctrine. In Greece also we find the Orphic Mysteries, from whose halls of esoteric instruction came forth many who bore the name of Orpheus.

What impelled these pupils to take the names of their teachers? Why did they sign their work, or give oral instruction, in the name of Orpheus, Hermes, or Zoroaster? Was it a kind of spiritual plagiarism, or was it rather because of a compelling gratitude to the teacher who had given them ALL, who had lighted the flame of esoteric fire in their hearts? Surely the latter, for whatever message they had of inspiration and light they deemed not theirs, but “his who sent me” — “As we have received it, thus shall we pass it on.” This practice is distressing to later historians who struggle always to attach correct labels to things, yet one cannot help but love these old disciples for that loyalty of soul which banishes all thought of individual greatness.

The relationship between disciple and teacher is a most sacred bond of spiritual intimacy. Gratitude wells up from the disciple commensurate with greatness of soul: the little of heart feel only resentment when guidance and protection are offered; but the large of heart burn with the flame of loving and inextinguishable gratitude. The links in this Golden Chain of Hermes are joined by gratitude. As each link is coupled with its brother link, heart with heart, teacher with pupil, pupil with teacher — each teacher a pupil to the one above, each pupil a teacher to the one below — all bonded by unbreakable links of love, fidelity, and gratitude to the teacher, to the Brotherhood, to the esoteric wisdom:

Like signal-fires of the olden times, which, lighted and extinguished by turns upon one hill-top after another, conveyed intelligence along a whole stretch of country, so we see a long line of “wise” men from the beginning of history down to our own times communicating the word of wisdom to their direct successors. Passing from seer to seer, the “Word” flashes out like lightning, and while carrying off the initiator from human sight forever, brings the new initiate into view. — Isis Unveiled

This “long line of `wise’ men” has been kept unbroken since the middle of the third root-race by two methods: (a) the actual reincarnation of adepts, and (b) the birth of the initiate out of the disciple. In this way the Brotherhood revitalizes its membership through the rebirth of hierophants, and the “second birth” of recruits from the ranks of the Mystery chambers. The “Passing of the Word” was the final rite of the solar initiation: without it no transmission of occult authority could be made from initiator to disciple.

Hence the line of esoteric authority and wisdom advances in serial order through grade after grade of chelaship to the adepts; from adepts to high mahatmas; from high mahatmas to buddhas; from buddhas to dhyani-buddhas; from dhyani-buddhas to the spiritual guide and protector of the planetary chain of earth; from the earth planetary spirit to the heart of the sun. Truly a line of luminous glory linking the humblest of disciples of wisdom with the solar logos.