proofs JCAP_024P_0113

ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Energy of Einstein’s static universeand its implications for the ΛCDMcosmology

Abhas Mitra

Astrophysical Sciences Division, Bhabha Atomic Research Centre,Mumbai-400085, India

E-mail: amitra@barc.gov.in

Received January 14, 2013Revised February 7, 2013Accepted February 14, 2013Published ???, 2013

Abstract. The total Einstein energy (P0) of a homogeneous and isotropic universe can becomputed by using an appropriate superpotential (Rosen 1994) and also by a direct method(Mitra 2010). Irrespective of the physical significance of P0, its eventual numerical valuemust be same in both the cases because both are derived from the same Einstein pseudotensor and by employing the same coordinates. It follows then that the static isotropic andhomogeneous universe, i.e., Einstein’s static universe (ESU), must have an infinite radius andwhich tantamounts to a spatially flat case. The physical significance of this result is thatthe cosmological constant, Λ, is actually zero and ESU is the vacuous Minkowski spacetime.It is the same result which has recently been obtained in a completely independent manner(Mitra, Bhattacharyya & Bhatt 2013). Thus even though, mathematically, one can conceiveof a static 3-sphere for the foundation of relativistic cosmology, physically, no such 3-sphereexists. On the other hand, the spatial section of the universe could essentially be an Euclideanspace with local curvature spikes due to presence of lumpy matter. Since the “Dark Energy”is associated with Λ in the ΛCDM model, the result obtained here suggests that it is anartifact of departure of the lumpy and fractal universe from the ideal Friedmann RobertsonWalker model (Jackson et al. 2012, Cowley et al. 2013).

Keywords: superclusters, Lyman alpha forest, galaxy clusters, cosmic flows

c© 2013 IOP Publishing Ltd and Sissa Medialab srl doi:xxxxxxx

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Contents

1 Introduction 1

2 Application of pseudopotentials 52.1 Einstein-Tolman formula 62.2 Application for the ESU 7

3 Simplified calculation invoking superpotential 8

4 Results 9

5 Discussions 9

6 Conclusions 11

1 Introduction

For several decades, modern cosmology has been based on the hypothesis that the observeduniverse looks isotropic and homogeneous to all fundamental (comoving) observers. Thespace time geometry of such an universe is given by the Friedmann Robertson Walker (FRW)metric [1]:

ds2 = dt2 − S(t)2

[dr2

1− kr2+ r2dΩ2

](1.1)

where S(t) is the scale factor of the universe, t is the universal cosmic time, r is a comovingradial coordinate, dΩ2 = dθ2 + sin2 θdφ2, and the spatial curvature parameter k can assumevalues of 0, +1, or−1. Further, following the observation that the distant Type IA supernovaeappear to be dimmer compared to the expected brightness as supposed “standard candles”,most of the cosmologists now believe that the universe is not only expanding but also doing soin an accelerated manner. If one would include “Cosmological Constant” Λ into the Einsteinequations (G = c = 1), the dynamical equation for the scale factor becomes

S

S= −4π

3(ρe + 3pe) (1.2)

where an overdot denotes differentiation by the comoving time t. Here, the effective matterdensity in the presence of Λ is

ρ→ ρe = ρ(t) +Λ

8π(1.3)

while the effective isotropic pressure is

p→ pe = p(t)− Λ

8π(1.4)

In principle, for a fine tuned Λ > 0, one may have pe < 0 and ρe + 3pe < 0. In such acase, it is possible to have S/S > 0; and this was the reason that Λ got reinstated in mod-ern cosmology giving birth to the so-called “concordance ΛCDM” cosmology where “CDM”stands for “Cold Dark Matter”. The most likely form of “Dark Energy”, the hypothetical

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proofs JCAP_024P_0113

entity which is supposed to drive the supposed accelered expansion is thus a positive Λ.On the other hand, one may generalize the concept of “Dark Energy” (DE) as some timedependent scalar field/quintessence rather than a fixed value classical Λ. One might alsodepart from standard general relativity, in which, G and Λ are fundamental constants, andpresume Λ = Λ(t), G = G(t). But even such generalizations cannot resolve the puzzles (i)why the required value of DE is almost 10−120 times less that what would be suggested bycanonical quantum gravity arguments, (ii) if any vacuum fluctuations are responsible for DE,why they leave such an incredibly tiny residue rather than just cancel one another exactly,and (i) why this incredibly tiny residue conspires to be of the order of magnitude of thematter energy density of the universe at the present epoch rather than any other epoch. Anhonest introspection would show that any attempt to resolve such puzzles either in terms ofclassical or quantum gravity is destined to be contrived and rather “cooked up”. Thus anyproof which can set DE to be zero must be explored with earnestness.

Recall that the concept of Λ was first necessitated by Einstein for having a staticisotropic and homogeneous universe; in other words, Einstein’s static universe (ESU) is astatic form of the FRW universe. By setting matter pressure p = 0, Einstein found that forhis universe [2]:

ρ =Λ

4π(1.5)

In general, a centrally symmetric static homogeneous spacetime metric has the form [1,3, 4]

ds2 = eνdt2 − dR2

1−R2/R20

−R2dΩ2 (1.6)

where R is the circumference coordinate, a scalar, and R0 is the radius of curvature theuniverse. The constant spatial curvature K (not the normalized parameter k) is related toR0 in the following way:

K =1

R20

(1.7)

Here it may be noted that for the static case, the curvature time coordinate T = t, andit is seen that [3–6]

1

R20

=8πρe

3(1.8)

By combining equations (1.1) and (1.6)–(1.8), we find that

R = Sr (1.9)

and

R20 =

S2

k; K =

k

S2(1.10)

Following this, it is found that, for the ESU

k =8π

3ρeS

2 (1.11)

It is interesting to see that, for a static case the signature of k is the same as thatof the effective matter density ρe. As is known, ESU presumes, k = +1 and ρe > 0. Onthe other hand, for an assumed case of ρe < 0, in principle, one can have an ESU with

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proofs JCAP_024P_0113

k = −1. However, since a negative matter density is unphysical, we rule out the possibility ofa negatively curved homogeneous 3-space even if the same is allowed by differential geometry.

Further if one will introduce a new coordinate χ through

sinχ = R/R0 (1.12)

one can writeds2 = eνdt2 −R2

0(dχ2 + sin2 χ dΩ2) (1.13)

Thus the spacetime geometry of ESU has two important parameters, ν and χ. Andby Einstein’s equation spacetime geometry must be determined by matter energy momentumtensor apart from relevant coordinates. And it was found that both ν and χ depend only onρe, pe and not on any coordinates [5], i.e.,

ν = ν(ρe, pe) (1.14)

andχ = χ(ρe, pe) (1.15)

While ESU was constructed, the fact ν = ν(ρe, pe) was taken into consideration. But the en-ergy momentum dependence of χ was overlooked. This could be so because energy momentumdependence of χ becomes apparent only by the application of Tolman-Oppenheimer-Volkoff(TOV) equation, and in 1917, the TOV equation for hyrdostatic balance was not known. Ofcourse, for any homogeneous and isotropic cosmology, by definition, ρe and pe are homoge-neous; and then eqs. (1.14) and (1.15) tell that, for the ESU, one must have

ν = constant (1.16)

andχ = constant (1.17)

By using the fact that ν = constant, one can set ν = 0 without any loss of generality (actuallyby a relabelling of the time parameter):

ds2 = dt2 −R20(dχ2 + sin2 χ dΩ2) (1.18)

And by using eq. (1.17) i.e., sinχ = constant, it is seen from eq. (1.12) that, for theESU intrinsically, one has

R0 =∞ (1.19)

It must be so because by definition R is a variable while R0 is a constant and we rule outthe possibility R0 = 0 or ρe =∞. Then the general ESU equation [5]

1

R20

= −8πpe =8πρe

3(1.20)

gets extended and not contradicted to [5]

1

R20

= −8πpe =8πρe

3= 0 (1.21)

i.e.,ρe = pe = 0 (1.22)

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proofs JCAP_024P_0113

and which shows that, intrinsically, both ρ = p = Λ = 0 for any ESU. And if one would setp = 0 at the very beginning, the ESU equation (1.5) gets extended to and not contradictedto [5]

ρ =Λ

4π= 0 (1.23)

which leads to the same result ρ = Λ = 0 for the ESU.This obviously means that, ESU is actually vacuous with no matter at all, ρ = 0,

and the ad hoc cosmological constant is actually non-existent, Λ = 0. Hence although wecan mathematically conceive of a neatly curled up 3-space in the form of a 3-sphere byextending the idea of tangible globes or 2-spheres, in reality there is no such exotic 3-sphere.Instead, there is a flat Eucledian 3-space which mathematically may be considered as aflattened 3-sphere with R0 = ∞. However such an overall flat space must be undulatedwith local spatial curvatures on various scales due to presence of planets, stars, compactobjects, galaxies etc, something like a flat park is strewn with (curved) pebbles, trees andmay be boulders. Now note that the problems associated with the assumption of smoothhomogeneous distribution in relativistic cosmology is somewhat similar to the one faced bythe static Newtonian cosmology:

For a continuous homogeneous and isotropic distribution of static matter having finitedensity, the gravitational field may tend to blow up. Thus such a boundless static distrubu-tion of matter is not allowed, i.e., one should have ρ = 0. In ESU too, aleast in the absence ofa positive Λ, initially, one obtains ρ = 0 and χ = 0. And this was the reason a (supposedly)Λ > 0 was introduced. However, despite the presence of a hypothetical Λ, with respect to agiven observer, GR potential tends to blow up at R = R0 = 2GMe(R0)/c2 or at tanχ = ∞(i.e., χ = π/2), where the effective enclosed gravitational mass is [5, 6]

Me =4πρe

3R3 (1.24)

More precisely, the scalar acceleration needed to keep an object fixed at R = R0 is seento blow up just like the case of the event horizon of a Schwarzschild black hole a = ∞ [6].Note, while a positive Λ with its repulsive effect may appear at first sight to tame theattractive runaway matter gravity, it actually adds to the matter energy density rather thanoffset the same. The blowing up of the acceleration a = ∞ at R = R0 or equivalently attanχ = tanπ/2 = ∞ happens irrespective of the presence or absence of Λ. And in GR, apositive Λ would actually accentuate this catastrophe by reducing the value of R0. This resulthas an important physical implication: there are indeed “coordinate singularites” associatedwith gθθ and gφφ. But these are innocuous and occur even for a Minkowski metric expressedin spherical coordinates. On the other hand, the singuarity in gRR in metric (1.6) is a physicalsingularity even if one would like to hide it in the form of (1.13) by means of a coordinatetransformation (1.12).

In view of the importance of this conclusion, we would like to further analyze thephysics of ESU from an independent and different perspective. We will compute the totalmass energy of the ESU (P0) by using Einstein’s canonical energy momentum tensor throughtwo equivalent clannels, by (i) using a direct brute force computation without introducingany explicit superpotential [7], and then by (ii) using an intermediate superpotential [8].

And though the algebric expressions of P0 for the two cases may look different, theinherent (numerical) value in both cases must be same because both expressions rely on thesame pseudopotential and the same set of coordinates. From this consideration, we will againfind that, for self-consistency, the ESU indeed has ρ = 0 and Λ = 0.

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proofs JCAP_024P_0113

In this context, for the sake of argumentation, some readers may say that (i) pseuopo-tentials are not reliable, (ii) one may consider other superpotentials too, (iii) whether gravi-tational energy is localizable or not and so on.

Such objections would actually be irrelevant because here we are not interested in eval-uating any absolute value of P0 and claim that “this must be the total energy of ESU”. Infact, our eventual conclusion would not even depend on whether P0 can be considered a trueexpression for total energy or not. This is so because we will only compare two differentalgebric expressions for P0 irrespective of its physical meaning and actual numerical value.In other words, we will only bank on the fact the eventual numerical value of P0 (whateverit might be) must be same in the two cases since they are derived using the same canonicalprescription.

And though not important here, let us mention that, atleast for spherically symmetriccase, there is an unanimity that gravitational energy is very much localizable [9–11]. Further,the psedopotential ansatz is actually equivalent to the quasi-local approach of defining totalmatter plus gravitational field energy [11, 12].

2 Application of pseudopotentials

We expect a flat Minkowskian space time to possess zero energy. And this expectation isrealized only when one uses quasi-Cartesian coordinates for computing P0. This has led tothe unanimity that for computing an absolute value of P0 one must use such coordinatesonly. On the other hand, if one would use spherical coordinates, P0 would blow up even foran empty Minkowskian vacuum, a fact related to the blowing up of space-time connectioncoefficients in the spherical coordinates. In a strict sense, one might still obtain a finite valueof P0 for the matter content even in spherical coordinates by subtracting two infinities

P0(matter) = P0(matter + vacuum)− P0(vacuum) (2.1)

in the spirit of normalization in quantum field theories. However, in view of the much simplerquasi-Caresian route where P0(vacuum) = 0, nobody seems to have ever (unnecessarily)trodden this path laced with infinites.

The total energy of the gravitating system may be defined as [3, 4, 7, 8, 10, 11]

P0 =

∫θ0

0 d3x (2.2)

where θba is the appropriate energy momentum complex (EMC) and the integration extendsover entire 3-space. This is also the energy of a system having a finite boundary [1, 3, 4].Here we will be mostly concerned with Einstein EMC

θba =√−g (T ba + tba) (2.3)

and in particular its component:

θ00 =√−g (T 0

0 + t00) (2.4)

Here tba is the gravitation field energy momentum (pseudo) tensor, T ba is the matter energymomentum tensor generating tba and the notation 0 denotes time coordinate.

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proofs JCAP_024P_0113

2.1 Einstein-Tolman formula

The field part in the Einstein EMC is given by [1, 3, 4]

√−g tab =

1

16π

(δab [L+ 2

√−gΛ]− ∂L

∂gpqagpqb

)(2.5)

where gab is the metric tensor and Λ is the cosmological constant. Here the Lagrangiandensity is given by [2, 3, 5]

L =√−g gab

(ΓpabΓ

qpq − ΓpaqΓ

qbp

)(2.6)

where the Christoffel symbols are

Γabc =1

2gad (∂cgdb + ∂bgcd − ∂dgbc) (2.7)

Also, the symbol ∂a denotes differentiation with the coordinate xa and [2–4]

∂L

∂gabc= −Γcab +

1

2δca Γdbd +

1

2δcb Γdad (2.8)

Nore, from eq. (2.5) we obtain,

√−g t00 =

1

16π

(L+ 2

√−gΛ− ∂L

∂gpq0

gpq0

)(2.9)

Further recall that L is related to Einstein Lagrangian density LG =√−gR, where R is the

Ricci scalar in the following way [1, 3, 4]:

L = ∂c

(gab

∂L

∂gabc

)−√−gR (2.10)

Also recall that for the asuumed perfect fluid in its comoving frame, one has

R = 8πT = 8π(T 0

0 + T 11 + T 2

2 + T 33

)(2.11)

where T is trace of T ba . Using these equations, Tolman showed [2, 3], in a most general way,that one can write

P0 =

∫1

2

√−g

(T 0

0 − T 11 − T 2

2 − T 33 −

Λ

4π

)d3x (2.12)

+

∫1

16πgab∂c

(∂L

∂gabc

)d3x

Since no assumption or precondition has been imposed for the derivation of the above formula,it is valid for arbitrary system, whether it has a boundary or not, whether it is static or notand whether it is spherical or not.

In particular, Tolman showed that (by dropping Λ) [2, 3] that if one would consideran isolated system resting in an asymptotically flat spacetime and use (quasi) Cartesiancoordinates with x1 = x, x2 = y and x3 = z, one would obtain

P0 =

∫ √−g

(T 0

0 − T 11 − T 2

2 − T 33

)dxdydz (2.13)

even if the system is not spherically symmetric. Interestingly, Landau & Lifshitz [13] (pp.348,eq. (100.19)) obtained exactly the same relation for the total matter plus field of a staticsystem without invoking any pseudo tensor at all. This shows the physical superiority of theEinstein-Tolman formalism, and the merit in the comment that Einstein’s EMC could bebest amongst many other alternatives [10].

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proofs JCAP_024P_0113

2.2 Application for the ESU

For working out the Einstein-Tolman energy for the ESU, we need to first express it inisotropic quasi-Cartesian form. And introducing a coordinate transformation [1, 3, 4]

r =r

1 + kr2/4(2.14)

the metric (1.1) may be expressed in an isotropic form:

ds2 = dt2 − S2

(1 + kr2/4)2

[dr2 + r2dΩ2

](2.15)

Further, this form can easily be written in terms of Cartesian coordinates:

ds2 = dt2 − S2

f2(dx2 + dy2 + dz2) (2.16)

wheref(r) = 1 + kr2/4 (2.17)

andr2 = x2 + y2 + z2 (2.18)

Extension of the Einstein-Tolman ansatz for a boundary-less and further non-staticsystem is quite complex. For the FRW metric such a tedious exercise was carried out forthe first time by the present author [7]. For the corresponding static case, it is seen that(eq. (99) of ref. [7] )

P0 =−k

16R20

[∫12r2dr + k

∫r4dr

](2.19)

For the k = 0 and R0 =∞ case, obviously one has

P0 = 0; k = 0; R0 =∞ (2.20)

For the k = +1 case, one can easily integrate eq. (2.19). by recalling that the range of r inthis case is 0,∞:

P0 =−∞16R2

0

(2.21)

If one would indeed interpret P0 as the total energy of the k = +1, ESU, the occurrenceof the −ve value of P0 would appear by be contradictory, because a k = +1 ESU correspondsto +ve effective matter density ρe > 0.

Also recall that, if one would assume k = +1, the proper 3-volume of the universewould be

v = 2π2R30 (2.22)

so that the ESU would have a mean effective matter density:

< ρe >=P0

v=−∞

32π2R50

(2.23)

Since all physically meaningful densities must be bounded, the above expression again showsthat one must have R0 = ∞. Thus even if one would assume k = +1, eventually, K = 0implying S =∞.

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proofs JCAP_024P_0113

From such a physical reasoning, one would immediately realize that, intrinsically, theESU should have R0 = ∞ so that neither P0 nor < ρe > could be −∞. Yet, here we shallnot appeal to any physical significance of P0 for arriving at the eventual conclusion.

3 Simplified calculation invoking superpotential

For direct computation of P0, one obviously needs to explicitly introduce T ba and the computa-tions are excessively tedious. But it was noted long ago that one can avoid such complexitiesby using appropriate supperpotentials. In fact whether it is, Einstein’s or Landau-Lifstitz’sor Weinberg’s pseudo tensor, in every case, there is an appropriate “superpotential”. Theinteresting thing about this detour is that here one need not explicitly introduce T ba . Instead,in this approach, one is concerned only with gab and its derivatives. And it was done not byEinstein but by Møller [14]:

θab =1

16πHacb,c (3.1)

where a comma denotes partial diffentiation and

Hacb =

1√−g

gap

[−g(gbpgcq − gcpgbq)

],q

(3.2)

For our interest, we may consider only one component of θab :

θ00 =

1

16πH0c

0,c (3.3)

The first author to calculate the energy a closed k = +1 universe using this Einstein-Møller superpoterpotential was Rosen [8]. As is the convention, Rosen [8] worked out theEinstein super-potential in quasi-Cartesian coordinates by setting k = 1, where

H0a0 =

2Sxa

(1 + r2/4)2(3.4)

It can be easily verified that, if one would instead retain the k term in f(r), one would obtain

H0a0 =

2Skxa

(1 + kr2/4)2(3.5)

In such a case, by using the fact that

∂af =1

2kxa (3.6)

one would obtain

θ00 =

kS

8π

[3

(1 + kr2/4)2− kr2

(1 + kr2/4)3

](3.7)

For k = 0, one directly obtains θ00 = 0 and hence P0 = 0. But when one uses k = +1

(as Rosen did), one still obtainsP0 = 0 (3.8)

It may be also mentioned that Cooperstock [15], and Cooperstock and Israelit [16] alsoargued that both the local and global energy of the FRW universe should be zero.

Then by comparing the equations, (2.19), (2.20) and (3.8), we arrive at the same result,R0 =∞, for the ESU, a fact which effectively means k = 0 even if one would initially assumek = +1!

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proofs JCAP_024P_0113

4 Results

It is important to note that the above result R0 = ∞ and Λ = 0 did not at all depend onany physical interpretation of P0. On the other hand, it followed from comparison of twomathematical expressions of P0 which must have same numerical value. As emphazied at thebeginning, the same result was ealier obtained from a completely different consideration [5].

As far the latter conclusion Λ = 0 is concerned, it may be obtained from from additionalindependent considerations.

For the de-Sitter model, the Einstein-Tolman energy is found to be [7]

P dS0 =Λ

6r3eΘt (4.1)

where the expansion scalar associated with the de-Sitter metric is

Θ =√

3Λ (4.2)

On the other hand, from Einstein-Møller superpotential based on the same Einsteinpseudo tensor, for any FRW k = 0 metric including the de-Sitter metric, one has

P dS0 = 0 (4.3)

And from eqs. (4.1) and (4.3), one obtains the same result Λ = 0. Had we appealedto the physical interpretation of P0 as the total energy, we could have obtained the sameconclusion from eq. (4.1) alone by demanding that total energy must be conserved ratherthan blow up indefinitely [7, 17].

Further, if the expansion scalar Θ =√

3Λ would indeed be non-zero, the de-Sitterexpansion must not look static to any observer. However, for the de-Sitter expansion, theobservers at fixed curvature cooordinates do not see any expansion. And this self-contractioncan be removed only by realizing that Λ = 0.

As far as de-Sitter model is concerned, probably, the result Λ = 0 could be obtainedfrom a very basic and fundamental reason: it is known that the vacuum energy momentumtensor T ab is Lorentz Invariant. Then from the Einstein’s equation, Gab = −8πT ab , it followsthat, the metric of the vacuum must be Lorentz Invariant too. But the only metric which isglobally Lorenz Invariant is the vacuum Minkowski space time not containing any Λ at all.

5 Discussions

Though cosmic distance ladder calibartion may be reasonably accurate and even if the Type1A supernove might be assumed to be standard candles, we may note the argument thatLyman Alpha clouds may be absorbing lights from the distant supernovae, and the “acceler-ation” could be an apparent effect [18].

It has also been argued that the “acceleration” is an apparent effect arising from thevery large proper motion of Milkyway with respect to the cosmic rest frame [19].

Even if one would ignore such suggestions, there is no denying that the obersed universeis lumpy and inhomogeneous on various scales in contrast to the FRW picture where matterdistribution is perfectly homogeneous on any appreciable scale. When inhomogeneities occurvery much within the observable scales, it has been argued that metric perturbations could

– 9 –

proofs JCAP_024P_0113

give rise to back reactions [20–25]. Very crudely, it has been claimed that, even in the absenceof a Λ or DE, such back reactions might modify the ideal FRW equation (1.2) to

S

S= −4π

3(ρ+ 3p) +Back Reaction Terms (5.1)

And if the additional terms on the r.h.s. are positive with right values, they may causeS > 0 and simulate a “DE”.

Recall here that the ideal Hubble’s law follws only when one assumes perfect isotropyand homogeneity. In this ideal scenario, in the comoving frame one “galaxy” finds all other“galaxies” to move radially away with no collision, no intersection of geodesics. On the otherhand, the very concepts of “temperature” and “pressure” are related to random motions andmutual collisions. Thus for the ideal Hubble flow, the fluid motion should correspond to niltemperature and pressure. Even if the radiation fluid would follow the ideal Hubble flow,the radiation quanta would behave as a null dust having zero radial pressure. Indeed thisintutitive physical expectation has been verified in a rigourous manner: for the ideal FRWfluid pe = 0 [26, 27]. In such a case, the dynamic equation (1.2) would assume its perfectNewtonian form

S

S= −4π

3ρe (5.2)

even though one can formally derive the FRW metric by assuming the underlying fluid tobe “perfect” [28]. Thus even if one would assume Λ > 0, there cannot be any acceleratedexpansion for an idealized FRW fluid.

In the ΛCDM cosmology or even the earlier Big Bang cosmology, one would ideally likethe universe to be homogeneous on the scale of ∼ 10Mpc. But from the latest studies ofSloan Digital Survey, it transpires that, the lumpiness of the observed universe happens evenon ∼ 1 Gpc scale [29]. And very recently, a cosmic structure has been discovered whose lineardimension in one direction is close to ∼ 1.2 Gpc [30]. Such obervations seriously challengethe assumption of perfect “homogeneity” inherent in the ΛCDM paradigm (even if it canexplain the cosmic microwave fluctutations). Further, the conceivable time required for thevery formation of such a 1.2 Gpc structure is likely to be much larger than 14 Billion years.

Even before this recent observation of a 1.2 Gpc structure, by studying the spatialdistribution of ultra compact radio sources on largest scale, Jackson et al. concluded that [31]

“This is interpreted as meaning that the Universe is not spatially homogeneous on thelargest scales, and is better represented at late times by a spherically symmetric model witha density enhancement at its centre.”

Let us now recall the motivations for introducing relativistic cosmology by Einstein in1917: (i) likely infinite gravitational field and (ii) a preferred center in case of a finite Newto-nian cosmos. But a decade before Einstein’s paper, the Swedish astronomer Carl Charlier [32]had explicitly constructed the model of a hierarchic cosmos following earlier speculations ofFournier D’Albe. In this hierarchial model, Charlier was able to produce a mass distributionwhich avoided the Newtonian divergence of gravitational field. Most interestingly, it avoidedhaving a preferred center. Charlier’s model had a fractal structure: stars are grouped intospherical galaxies, galaxies into spherical metagalaxies, and so on. Charlier derived con-straints on how densely the systems at one level could be packed into a system of the nextlevel up without producing divergences. He showed that it was possible to build up a uniformcosmos in this way with an infinity of total (baryonic) mass, and yet an average mass den-sity of zero, and a convergent gravitational potential. Newtonian gravitational theory could

– 10 –

proofs JCAP_024P_0113

be consistently applied to such a hierarchic universe without leading to either of Einstein’sproblems [33].

In 1922, Franz Selety spelled out how Charlier’s hierarchic model avoided all the prob-lems associated with convention Newtonian cosmology in painstaking detail [34], and com-municated the same to Einstein. However, Einstein was struck by the philosophical as wellas mathematical elegance of a “closed spherical universe” and ignored Charlier’s importantresearch. But here we found that, Einstein “closed spherical universe” actually never existedeven in theory.

Further note that even in an overall static Newtonian hierarchial cosmos, no matterreally should be static. As we know, whether it is the atoms, atmosphere, oceans, or planets inthe solar system, everywhere there is localized motion; stars move around respective galacticcenters, galaxies move within galaxy clusters and so on. The Newtonian kinetic energyassociated with such motions might almost cancel the local Newtonian potential energies orat the best leave a residual gravitational acceleration which need not blow up. On the otherhand, if such localized random and organized motions would be ignored in any continuoushydrodynamic model, the gravitational potential may tend to blow up. And as first shown byCharlier, Newtonian potential may be finite for the idealized static hierarchial cosmos [32]even when one would ignore the sources of localized kinetic energy.

It has indeed been claimed that the Newtonian gravitational and potential energiesare of the same order of magnitude for typical globular clusers, galaxies, galaxy clustersat the best leaving a residual gravitational acceleration of ∼ 1.10−10 m/s2 [35, 36]. For acontinuous distribution of matter, on the other hand, gravitational potential must keep onincreasing whether one would analyze the problem by Newtonian gravitation or GR. But fora hierachial fractal distribution comprising discrete matter clumps, it might be possible thateven on the scale of ∼ 100 Mpcs, one may obtain the same residual gravitational acceletation∼ 10−10m/s2.

In fact, painstaking analyes of galaxy distrution by appropriate statistical tools haverevealed that galaxy distributions follow fractal structure upto ∼ 100 Mpc with hardly anysign of homogeneity beyond [37–42] in broad agreement with the central idea of Charlier.Such conclusions are also in agreement with the latest actual observations [30–32]. It isentirely possible that a hierarchial fractal distribution of luminous matter obeys “CopernicalPrinciple” rather than exact cosmological principle envisaged in relativistic cosmology [40,41, 43–45]. Note, there have been various efforts to describe fractal cosmic models as well astheir problems [43–48].

6 Conclusions

The best studied and best developed cosmology is the inflationary ΛCDM paradigm. Onceone would accept the implicit and explicit assumptions behind the paradigm, bulk of thepresent day observations would appear to be in agreement with it. However it may beborne in mind that cosmology is far from usual science deveoped through experimentionsand laboratory tests. And there are instances in the history of science where a paradigmhas eventually been discarded even if it could explain many natural phenomena. Note evenbefore the advent of the Heliocentric theories of “cosmos”, the Ptolemaic system was asophisticated astronomical system, and it was able to calculate the positions for the planetsto a fair degree of accuracy. Occurrences of tides and some astronomical events like eclipsescould be predicted by even more primitive theories of “cosmos”.

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The result Λ = 0 invalidates the basis of the ΛCDM cosmology. Mathematicians mightargue that for a dynamic universe, Λ = Λ(t) and even G = G(t). If so, the atomic finestructure constant α too could be time dependent; but as of now, there is no evidence for anytemporal dependence of either G or α. Of course, one can think of more complex forms of DElike “quintessence” or decaying scalar fields; one may also generate reasonable theories aboutDE by using f(R) gravity [49]. But here we would like to stick to the standard GR, and inview of the Λ = 0 result, and in view of much more contrived nature of other contendingtheories of DE, it may be concluded that there is no DE, and the supposed “accelerationof the observed universe” is an apparent effect. Further since for idealized radial and non-colliding Hubble flow pe = 0 [27, 28], even the ΛCDM model cannot explain any cosmicacceleration.

Though differential geometry allows us to conceive of positively and negatively curvedhomogeneous 3-spaces, introduction of physics rules out their existence atleast for a staticspace-time. The reason that ESU had to be vacuous is that, otherwise, the singularity ingRR in (1.6) would demand infinite scalar acceleration for matter to be at rest at R = R0.This strongly suggests that in view of the singularity in grr in the FRW metric, it too shouldeventually correspond to only k = 0.

The Big Bang model is plagued with multitude of conceptual problems starting withthe notion of “Expansion of Space” (EOS) and a singular origin. Such problems are stillhotly debated and there is hardly any likelihood of their real resolution [50–54]. Further, forthe expanding universe, the principle of energy conservation appears to be badly violated [8].

Since any exponential de-Sitter like expansion in the comoving frame can be transformedaway into perfectly static picture in the curvature frame, and since genuine physical phe-nomenon must be apparent to all observers, one should have Λ = 0 and no-Sitter phase [17].This observation almost rules out the bizarre hypothesis of “inflation”. In contrast a modestexpansion of say S(t) ∼ t1/2 cannot be transformed away and seems to be acceptable.

The observed universe has fractal structure up to ∼ 100 Mpc and may be all the way.In particular for an infinite hierachial fractal structure, the average density < ρ >→ 0 asR→∞ suggesting the problems of infinite gravitational field, typical of both Newtonian andrelativistic models based on continuous homogeneous distributions of matter may vanish.Further, the internal kinetic energy associated with both organized and random motions ofvarious cosmic structures might be neutralizing the negative gravitational potential energiesat all levels even for a cosmos which as a whole is not expanding into anything else.

If Hubble flow is genuine, there is one puzzle which even a non-static hierarchial modelsmay not resolve, and it is known as “Hubble de Vaucouleurs” paradox [43–45]. It is well knownthat exact Hubble’s law is a consequence of perfect homogeneity and isotropy inherent in theFRW model. But it is now certain that the universe is highly discrete and inhomogeneous ona scale of few Mpc. Accordingly, the Hubble flow should be very non-linear and overwhelmedby peculiar velocities of galaxies. But surprisingly, Hubble flow is quite linear despite acuteinhomogeneities! The ideal resolution of this paradox would be that the galactic red-shiftsare due to some yet unknown “tired light” effect rather than due to any kinematic effects.

From time to time, claims have been made that many cosmological data can be betterexplained in terms of a static universe rather than an expanding universe [55, 56]. In par-ticular, it has been claimed that the Gamma Ray Burst data do not show the EOS effect oreven can be better explained by a static model. However, as of now, there is no well acceptedtheory for non-kinemartical “tired light” origin of cosmic redshifts; and even if there may beno “Dark Energy”, the interpretation of cosmic redshifts in terms of “Expansion of Space”hypothesis looks elegant.

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Acknowledgments

I thank the anonymous referee for pointing out several minor errors in the intial version. Theclosely related ref. [6] was earlier about to be published in JCAP. However the referee DrN. Spyrou insisted that I highlight one paper by him which claimed that Λ is simulated byincreased pressure of warm Dark Matter. As I refused, he eventually refused. Similarly, onePRL editor, Dr E. Weinberg, refused to process the ref. [27] presumably because its conclusionwas against his academic beliefs. In this context, I especially appreciate the academic andprofessional integrity of the present referee.

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