On cosmological models

and some misunderstandings about them

Andrzej Krasinski

N. Copernicus Astronomical Center, Polish Academy of Sciences

Warsaw, Poland

Contents

1 What is “cosmological principle”? 2

2 Why consider generalised cosmological models? 4

3 Geometry of the cosmological models 6

3.1 The Robertson – Walker (RW) models . . . . . . . . . . . . . . . . . . . . 6

3.2 The Lemaıtre – Tolman (LT) model . . . . . . . . . . . . . . . . . . . . . . 7

4 Explaining away “accelerated expansion” of the Universe by inhomoge-

neous matter distribution 8

5 A possible observational test of homogeneity: Non-repeatable light paths 12

6 Errors and misconceptions 13

7 A brief conclusion 17

A Formulae for the RW models 18

B Formulae for the LT models 20

C Formulae for the dimming of supernovae in LT models 21

D Problems with approximations. 22

1

1 What is “cosmological principle”?

The “cosmological principle” derives from Copernicus’ discovery

that can be briefly stated as follows: when the origin of coordinates

is moved from the centre of the Earth to the centre of the Sun, the

description of motions of planets becomes simpler.

In later centuries further discoveries indicated that the position

of the Sun in the Universe is not in any way privileged.

Ultimately, this conclusion assumed a fundamentalist form: all

positions in the Universe are equivalent; every observer, no matter

where he/she is, will see the same large-scale image of the Universe.

2

This “cosmological principle” is not a summary of knowledge

based on observations, but a postulate and an ex post justification

of the first theoretical models of the Universe. Just as elements of

other theories, it requires observational verification.

Progress in observing technology, with still farther regions com-

ing into view, produced no evidence for this principle: only more

structures were becoming visible. Nevertheless, we are told that

the Universe is homogeneous “at a sufficiently large scale”. The

definition of this “sufficient scale” is far from precise (“a few” hun-

dred megaparsecs). This is the size of the “fundamental cell” of the

Universe, which should be repetitive – but it is so large that details

of mass distribution at its edges and beyond are fuzzy.

3

2 Why consider generalised cosmological models?

There is no compelling evidence for the cosmological principle, and

we are free to explore models that do not obey it.

Traditionally, structures are described by solutions of Einstein’s

equations linearised around homogeneous models. However, the

range of applicability of this approach is just impossible to deter-

mine by clean mathematical methods. Using it, we are permanently

uncertain about the degree of precision of the results.

4

Exact inhomogeneous models are reliable in their predictions as

long as we believe that general relativity is a valid theory of grav-

itation. They are not alternatives, but generalisations

(“exact perturbations”) that reduce to the traditional

ones in the limit of spatial homogeneity. The relation is

similar to that between a globe and a map of a region of the Earth.

A globe portrays the Earth as a perfect sphere, but a map of a

small area will show mountains and other features.

One should not expect that future precise observations will tell

us definitively ”homogeneous models good, inhomogeneous bad”,

or the reverse. The right expectation is that we will learn, by how

much the parameters of the inhomogeneous models can differ from

their homogeneous limits.

5

3 Geometry of the cosmological models

3.1 The Robertson – Walker (RW) models

Big Bang

flow lines of matter

position

time

Big Bang

flow lines of matter

position

time

Figure 1: Expansion in the RW models. Upper picture: The velocity of expansion

is rigidly coupled to the particle’s positon: it is proportional to the distance from the

observer at any fixed instant, but changes with time (v = H(t)d, this is the “Hubble

law”). Lower picture: The initial explosion, in the natural cosmological synchronisation,

occurs simultaneously =⇒ all matter particles have the same age at any later instant.

6

3.2 The Lemaıtre – Tolman (LT) model

Big Bang

flow lines of matter

position

time

Big Bang

flow lines of matter

position

time

Figure 2: Expansion in the LT model. Upper picture: The velocity of expansion is not

correlated with the position of a matter shell. The spatial distribution of velocity is an

arbitrary function of the radial coordinate. Lower picture: The initial explosion is, in

the natural cosmological synchronisation, non-simultaneous =⇒ the age of matter particles

depends on their radial coordinate. The “timetable” of the initial explosion is a second

arbitrary function of the radial coordinate.

The LT model is spherically symmetric. It does not represent the

whole Universe, but a single local structure embedded in an RW

background. One RW background can contain several LT regions.

7

4 Explaining away “accelerated expansion” of the Uni-

verse by inhomogeneous matter distribution

The hypothesis of accelerated expansion of the Universe arose from

observations of type Ia supernovae. A supernova of type Ia is a

final stage of evolution of a white dwarf. The maximal absolute

luminosity of all supernovae of this class is assumed to be the same.

Note the statements in italics in the next page. They empha-

sise the stages at which a model is a priori assumed to explain

the observations.

8

By measuring the redshift of these supernovae, one can calculate

the distance to each of them, assuming that the Universe

we live in is RW with known parameters, and so the

Hubble law is exactly fulfilled – and then calculate the ex-

pected flux of radiation through a unit surface area to be observed

on the Earth. It turned out that the actually observed maximal flux

is smaller than expected, as if the supernovae were farther from us

than we thought. In order to explain this discrepancy, the previ-

ously used Universe model had to be modified. The only

modification allowed by the original research team was to change

the parameters of the RW model; going beyond the RW class was

not contemplated. Attempts to fit various RW models to

the observed luminosities led to the best fit achieved when

Ωk = 0, Ωm ≈ 0.3, ΩΛ ≈ 0.7.

ΩΛ > 0 means that the Universe has to expand with acceleration.

This gave rise to the puzzle of “dark energy” (that would propel

the acceleration) and to research programs aimed at solving it.

But is this the only possible explanation of the “dim-

ming of supernovae”?

9

The conclusion that the Universe had to expand with acceleration

followed from the assumption that we have to use the RW

models.

This last remark must be exactly understood because its mis-

taken understanding created a false legend. What we have to ex-

plain is the relation between the observed luminosity of the type Ia

supernovae and their redshifts (DL(z)) in our model. The “accel-

erated expansion” of the Universe is not an observed

phenomenon, but an element of interpretation of the

observations, forced upon us by the RW models. If we

can re-create the observed function DL(z) in a decelerating inho-

mogeneous model, then the “accelerated expansion” becomes an

illusion.

This is indeed possible in the LT models, in more than one way.

The illustration on the next page is just one example.

10

central past light cone

Friedmann Big Bang

LT Big Bang

LT F

∆t

LT F

LT F

central past light cone

Friedmann Big Bang

LT Big Bang

LT F

∆t

LT F

LT F

With a non-constant tB of suitable shape, each LT matter shell

that intersects the past light cone of the central observer is older by

∆t than a Λ = 0 RW shell that would intersect the light cone at the

same point. Therefore, at this intersection, the LT shell expands

slower than an RW shell would do. The ∆t increases toward the

past, and so does the difference between the expansion velocities.

This means the LT model imitates the acceleration of expansion

relative to the Λ = 0 RW model.

Had we allowed the LT models, instead of using ex-

clusively RW, the ideas of “accelerated expansion” and

of “dark energy” would not have occurred to anyone.

11

5 A possible observational test of homogeneity: Non-

repeatable light paths

2.95

2.96

2.97

2.98

2.99

3

0 0.1 0.2 0.3 0.4

r [G

pc]

Θ [rad]

In a generic situation in an inhomogeneous model, two light rays

sent from the same source at different times to the same observer

pass through different sequences of intermediate matter particles

[4] =⇒ distant objects should change their positions in the sky.

The expected drift rate is 10−7 to 10−6 arc second per year in a

favourable configuration. This should become detectable after≈ 10

years of monitoring a given source, using devices that are already

under construction. This drift does not exist in the RW models.

12

6 Errors and misconceptions

The astrophysics community easily tolerates a loose approach to

mathematics. Papers written in this style planted errors in the

literature, which were then uncritically cited and came to be taken

as established facts. In this section examples of such errors are

presented (marked by •) together with their explanations (marked

by ∗).

• The LT models that explain away dark energy using matter

inhomogeneities contain a “weak singularity” at the centre [9],

where the scalar curvature R has the property gµνR;µν → ∞.

∗ gµνR;µν → ∞ is not a singularity by any accepted cri-

terion in general relativity [10]. It only implies a discontinuity

in the gradient of mass density – a thing quite common in Nature

(e.g. on the surface of the Earth). At the centre, gµνR;µν → ∞ im-

plies a conical profile of density – also a nonsingular configuration.

13

• Decelerating inhomogeneous models with Λ = 0 cannot be fit-

ted to the same distance–redshift relation that implies acceleration

in ΛCDM. This is because the following equation connecting the

deceleration parameter q4 to the Hubble parameter H , density ρ

and shear σ of the cosmic medium

H2q4 = 4πρ/3 + 14σ2/15. (6.1)

prohibits q4 < 0 [11].

∗ Equation (6.1) is based on approximations that are

not explicitly spelled out [10]. An approximate equation cannot

determine the sign of anything. Refs. [12, 8, 7] provide explicit

counterexamples to (6.1). If the approximations are taken as exact

constraints imposed on the LT model, they imply zero mass density,

i.e. the vacuum limit.

14

• There is a “pathology” in the LT models that causes the

redshift-space mass density to become infinite at a certain location

(called “critical point”) along the past light cone of the central

observer [9].

∗ The “critical point” is the apparent horizon (AH),

at which the past light cone of the central observer begins to re-

converge toward the past. This re-convergence had long been known

in the RW models [13, 14], and the infinity in density is a

purely numerical artifact – a consequence of trying to inte-

grate past AH an expression that becomes 0/0 at the AH. Ways to

handle this problem are known [15, 16, 8].

15

• Fitting the LT model to cosmological observations, such as

number counts or the Hubble function along the past light cone,

results in predicting a huge void, at least several hundred Mpc in

radius, around the centre (too many papers to be cited, literature

still growing).

∗ The implied huge void is a consequence of hand-

picked constraints imposed on the arbitrary functions

of the LT model, for example a constant bang time tB. With

no a priori constraints, the giant void is not implied [8].

16

7 A brief conclusion

The theory of relativity has much more to offer to cosmology than

the extremely simplistic RW models. During the 90 years after they

were found relativistic cosmology has made a lot of progress. The

inhomogeneous models allow us to explain most of the observed

phenomena without introducing any “new physics”.

The alleged pathological properties of the LT models follow from

hastily contrived reasonings that contain plain errors in computa-

tion or in interpretation of the results.

17

A Formulae for the RW models

The metric of this model results from assumptions made a priori

about the symmetries of spacetime

ds2 = dt2 − S2(t)

[dr2

1− kr2+ r2

(dϑ2 + sin2 ϑdφ2

)]. (A.1)

If the matter in the model has zero pressure, then S(t) obeys

S,t2 =

2GM

c2S− k +

1

3ΛS2, (A.2)

where k and M are arbitrary constants and Λ is the cosmological

constant.

The redshift z is defined by

z =emitted frequency

observed frequency− 1 ≡ νe

νo− 1. (A.3)

For the RW models the redshift is:

z = S(to)/S(te)− 1. (A.4)

The luminosity distance between an observer at (t, r) = (to, 0)

and the source of light at (te, re) is defined as the flat-space distance

to a source that would give the same observed flux of radiation. In

the RW models this is

DL = reS(te)(1 + z)2. (A.5)

18

The formula for DL is written in terms of observable quantities:

the redshift z, the Hubble coefficient at the present instant to:

H0 = S,t /S|t=to. (A.6)

and the dimensionless parameters of density, curvature and cosmo-

logical constant

(Ωm,Ωk,ΩΛ)def=

1

3H02

(8πGρ0,−3k/S0

2,Λ), (A.7)

Ωm + Ωk + ΩΛ ≡ 1.

(S0 and ρ0 are the present values of S and of the average matter

density in the Universe). In these variables:

DL(z) =1 + z

H0

√Ωk

sinh

∫ z

0

√Ωkdz

′√Ωm(1 + z′)3 + Ωk(1 + z′)2 + ΩΛ

.

(A.8)

This formula applies with both signs of Ωk (sinh(ix) ≡ i sin x) and

also in the limit Ωk → 0.

The current favourite model of a majority of cosmologists is the

special case of (A.8) corresponding to k = Ωk = 0.

19

B Formulae for the LT models

The metric of the Lemaıtre – Tolman [2, 3] model is

ds2 = dt2 − R,r2

1 + 2E(r)dr2 −R2(t, r)

(dϑ2 + sin2 ϑdφ2

), (B.1)

where R(t, r) obeys (from the Einstein equations):

R,t2 = 2E(r) +

2M(r)

R− 1

3ΛR2. (B.2)

M(r) and E(r) are arbitrary functions. The integral of (B.2) con-

tains one more arbitrary function, tB(r) – the “timetable” of the

initial explosion. For example, when E = 0 = Λ the solution of

(B.2) is

R =

(9M

2

)1/3

(t− tB(r))2/3 , (B.3)

which shows that the instant of the initial singularity t = tB(r),

where R = 0, depends on the position.

The RW limit of LT is

M = const · r3, E = −kr2/2, tB = const, R = rS(t).

(B.4)

Since the velocity of expansion of the matter shells is uncorrelated

with their positions, effects occur that were absent in the RW mod-

els.

20

C Formulae for the dimming of supernovae in LT

models

Let us see how the“dimming of supernovae” can be explained using

the LT model [7]. We assume that the observer is at the symmetry

centre and that E/r2 = E0 = const, (the same E as in the RW

model). For tB(r) take the implicit definition:

DL(z)

(1 + z)2≡ R|lr =

1

H0(1 + z)

∫ z

0

dz′√Ωm(1 + z′)3 + ΩΛ

, (C.1)

where Ωm = 0.3 and ΩΛ = 0.7, and H0 is the present value of the

Hubble coefficient. Equations for tB(r) can now be solved numer-

ically. The trick is that the quantities DL, z and H0 are

taken from observations, but the definition of DL(z)

is no longer (A.5). Instead, the formula for DL(z) is

taken from an LT model with Λ = 0, and this defines

tB(r).

Comparison of (C.1) with (A.8)

DL(z) =1 + z

H0

√Ωk

sinh

∫ z

0

√Ωkdz

′√Ωm(1 + z′)3 + Ωk(1 + z′)2 + ΩΛ

.

shows that (C.1) defines the same relation between the luminosity

distance and the redshift z as in the “standard” RW model with

Ωk = 0, Ωm = 0.3 and ΩΛ = 0.7. However... we achieved

this with Λ = 0, i. e. without the “dark energy”, and

with decelerated expansion – as dictated by the familiar laws

of gravitation.

Had we used the LT model rather than RW to interpret the

observations, there would be no need for the “dark energy” and

“accelerated expansion”.

21

D Problems with approximations.

Even when the value of the variable in a power series is within the

convergence radius, the convergence may become actually “visible”

only at high orders. Take a simple example: the function ex is

represented by the following power series that is convergent for

every |x| < ∞:

ex = 1 + x +x2

2+

x3

3!+ · · · + xn

n!+ . . .

But when x is large, the first orders increase very quickly. For

example, let x = 10 (→ e10 ≈ 22026.46579). Then

n = 1 → x = 10

n = 2 → x2

2= 50

n = 3 → x3

3!≈ 166.6667

and only at n = 10 each next term becomes smaller than the

preceding one (but the “aproximation” to e10 at n = 10 is still

far from good, being equal to 12444.29).

22

In practice, one expects that the perturbative calculation will

give an acceptable result when its parameter is smaller than 1. The

example below shows that this is not a sufficient condition.

Take the function

f (x) =1

1/2 + x.

It is represented by the series

∞∑n=0

(−1)n2n+1xn = 2[1− 2x + (2x)2 − (2x)3 ± . . .

],

whose radius of convergence is 1/2. Therefore, “approximating”

f (x) with this series at x = 1/2 + 1/16 we get nonsense results.

The exact values are

f (0) = 2, f (9/16) = 16/17,

while the consecutive “approximations” give

2 at order 0,

−1/4 at order 1,

73/32 ≈ 2.28 at order 2.

We will not notice that something is wrong when computational

difficulties do not allow us to go beyond the first order, and in

addition we do not know what function we approximate.

23

[1] J. Plebanski, A. Krasinski, An introduction to general rela-

tivity and cosmology. Cambridge University Press (2006).

[2] G. Lemaıtre, Ann. Soc. Sci. Bruxelles A53, 51 (1933); En-

glish translation with historical comments: Gen. Rel. Grav.

29, 637 (1997).

[3] R. C. Tolman, Proc. Nat. Acad. Sci. USA 20, 169 (1934);

Reprint with historical comments: Gen. Rel. Grav. 29, 931

(1997).

[4] Krasinski A, Bolejko K 2011 Phys. Rev. D 83 083503

[5] Celerier M N 2000 Astron. Astrophys. 353 63

[6] Bolejko K, Krasinski A, Hellaby C, Celerier M N 2009 Struc-

tures in the Universe by exact methods: formation, evolu-

tion, interactions (Cambridge: Cambridge University Press)

[7] H. Iguchi, T. Nakamura and K. Nakao, Progr. Theor. Phys.

108, 809 (2002).

[8] M.-N. Celerier, K. Bolejko and A. Krasinski, Astronomy and

Astrophysics 518, A21 (2010).

[9] Vanderveld R A, Flanagan E E, Wasserman I 2006 Phys. Rev.

D 74 023506

[10] Krasinski A, Hellaby C, Bolejko K, Celerier M N 2010 Gen.

Rel. Grav. 42 2453-75

[11] Hirata C M and Seljak U 2005 Phys. Rev. D72 083501

24

[12] Bolejko K, Hellaby C, Alfedeel A H A 2011 The Metric

of the Cosmos from Luminosity and Age Data Preprint

arXiv:1102.1449

[13] Ellis G F R 1971 in Proceedings of the International School of

Physics ‘Enrico Fermi’, Course 47: General Relativity and

Cosmology, ed. R. K. Sachs. Academic Press, New York and

London, pp. 104 – 182; reprinted, with historical comments, in

Gen. Rel. Grav. 41, 581 (2009).

[14] McCrea W H 1934 Z. Astrophys. 9 290-314

[15] Lu T H C, Hellaby C 2007 Class. Quantum Grav., 24 4107

[16] McClure M L, Hellaby C 2008 Phys. Rev. D 78 044005

25

Figure 3: (Copy from: Volker Springel, Carlos S. Frenk, Simon D. M. White, The large-

scale structure of the Universe, arXiv:astro-ph/0604561v1). The arrow shows the size of

the smaller image with respect to the larger one.

26