Lattice Universe: examples and problems

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We consider lattice Universes with spatial topologies ......, ......, and ....... In the Newtonian limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the ...... model, we demonstrate that the gravitational potential has no definite values on the straight lines joining identical masses in neighboring cells, i.e. at points where masses are absent. Clearly, this is a nonphysical result, since the dynamics of cosmic bodies is not determined in such a case. The only way to avoid this problem and get a regular solution at any point of the cell is the smearing of these masses over some region. Therefore, the smearing of gravitating bodies in ......-body simulations is not only a technical method but also a physically substantiated procedure. In the cases of ...... and ...... topologies, there is no way to get any physically reasonable and nontrivial solution. The only solutions we can get here are the ones which reduce these topologies to the ...... one.

Eur. Phys. J. C (2015) 75:217DOI 10.1140/epjc/s10052-015-3445-2

Regular Article - Theoretical Physics

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Web End = Lattice Universe: examples and problems

Maxim Brilenkov1,a, Maxim Eingorn2,b, Alexander Zhuk3,c

1 Department of Theoretical Physics, Odessa National University, Dvoryanskaya st. 2, Odessa 65082, Ukraine

2 Physics Department, North Carolina Central University, Fayetteville st. 1801, Durham, NC 27707, USA

3 Astronomical Observatory, Odessa National University, Dvoryanskaya st. 2, Odessa 65082, Ukraine

Received: 11 April 2015 / Accepted: 8 May 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We consider lattice Universes with spatial topologies T T T , T T R, and T R R. In the Newtonian

limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the T T T model,

we demonstrate that the gravitational potential has no denite values on the straight lines joining identical masses in neighboring cells, i.e. at points where masses are absent. Clearly, this is a nonphysical result, since the dynamics of cosmic bodies is not determined in such a case. The only way to avoid this problem and get a regular solution at any point of the cell is the smearing of these masses over some region. Therefore, the smearing of gravitating bodies in N-body simulations is not only a technical method but also a physically substantiated procedure. In the cases of T T R and T R R

topologies, there is no way to get any physically reasonable and nontrivial solution. The only solutions we can get here are the ones which reduce these topologies to the T T T

one.

1 Introduction

Papers devoted to the lattice Universe can be divided into two groups. The rst group includes articles (see, e.g., [111]) offering alternative cosmological models. Despite the great success of the standard CDM model, it has some problematic aspects. The main one is the presence of dark energy and dark matter which constitute about 96% of the total energy density in the Universe. However, the nature of these components is still unknown. Another subtle point is that the conventional model is based on the FriedmannLemaitre

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RobertsonWalker (FLRW) geometry with the homogeneous and isotropic distribution of matter in the form of a perfect uid. Observations show that such an approximation works well on rather large scales. According to simple estimates made on the basis of statistical physics, these scales correspond to 190Mpc [12], which is in good agreement with observations. This is the cell of uniformity size. Deep inside this cell, our Universe is highly inhomogeneous. Here, we clearly see galaxies, dwarf galaxies, groups, and clusters of galaxies. Therefore, it makes sense to consider matter on such scales in the form of discrete gravitational sources. In this case, we arrive at the question how this discrete distribution inuences global properties and dynamics of the Universe. This problem was investigated in the above mentioned papers (see also [13]). Here, gravitating masses are usually distributed in a very simplied and articial way. They form either periodic structures of identical masses with proper boundary conditions or correspond to Einstein equation solutions (e.g., Schwarzschild or Schwarzschildde Sitter solutions) matching with each other with the help of the Israel boundary conditions. Usually, such models do not rely on the CDM background solution and do not include observable parameters (e.g., the average rest-mass density

of matter in the Universe). As a result, these models have nothing in common with the observable Universe. Their main task is to nd new phenomena following from discretization and nontrivial topology.

Papers from the second class are devoted to numerical N-body simulations of the observable Universe. Here, the lattice is constructed as follows. In the spatially at Universe, we choose a three-dimensional cell with N arbitrarily distributed gravitating masses mi and suppose periodic boundary conditions for them on the boundary of the cell. Such models rely on a background FLRW geometry with a scale factor a. It is supposed that the background solution is the CDM model with the perfect uid in the form of dust with the average rest-mass density . Discrete inhomogeneities with

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217 Page 2 of 10 Eur. Phys. J. C (2015) 75:217

the real rest-mass density = Ni=1 mi(r ri) perturb

this background. The gravitational potential inside the cell is determined in the Newtonian limit by the following Poisson equation [1416]:

(r) = 4GN

N

i=1

mi(r ri)

, (1.1)

where GN is the Newtonian gravitational constant, and r, ri belong to the cell, e.g., xi [l1/2, l1/2], yi

[l2/2, l2/2], zi [l3/2, l3/2]. Here, the Laplace oper

ator = 2/x2 + 2/y2 + 2/z2, and the coordi

nates x, y, and z, the gravitational potential , and the rest-mass densities and correspond to the comoving frame. All these quantities are connected with the corresponding physical ones as follows: Rphys = ar, phys = /a, and

phys = /a3. Equation (1.1) is the basic equation for the

N-body simulation of the large scale structure formation in the Universe [16]. The same equation can also be obtained in the Newtonian limit of General Relativity [12,17,18]. If we know the gravitational potential, then we can investigate the dynamics of the inhomogeneities/galaxies taking into account both gravitational attraction between them and the cosmological expansion of the Universe [17,19,20].

It can easily be seen that in the case of a nite volume (e.g., the volume of the cell) Eq. (1.1) satises the superposition principle. Here, for each gravitating mass mi we can determine its contribution to the average rest-mass density: i = mi/(l1l2l3), = Ni=1 i. Therefore, we can solve

Eq. (1.1) for each mass mi separately.

If we do not assume periodic boundary conditions, at least for one of the directions, there is no lattice in these directions and space along them is not compact (in the sense of the lack of a nite period of the lattice). Obviously, in innite space the number of inhomogeneities must also be innite: N

. This case has a number of potentially dangerous points. First, the superposition principle does not work here because we cannot determine i for each of the masses mi. Second, it is known that the sum of an innite number of Newtonian potentials diverges (the NeumannSeeliger paradox [21]). Therefore, in general, the considered model can also suffer from this problem if we do not distribute masses in some specic way. Third, we can easily see from Eq. (1.1) that the presence of will result in quadratic (with respect to the noncompact distance) divergence. Hence, to avoid it, we should cut off gravitational potentials in these directions. This also may require a very specic distribution of the gravitating masses.

In the present paper, we investigate Eq. (1.1) for different topologies of space which imply different kinds of lattice structures. First, in Sect. 2, we consider the T T T topol

ogy with periodic boundary conditions in all three spatial dimensions. For point-like sources, we obtain a solution in

the form of an innite series. This series has the well-known Newtonian type divergence in the positions of the masses. However, we show that the sum of the series does not exist on the straight lines joining identical particles in neighboring cells. Therefore, there is no solution in points where masses are absent. This is a new result. To avoid this nonphysical property, in Sect. 3, we smear point-like sources. We present them in the form of uniformly lled parallelepipeds. In this case, the innite series has denite limits on the considered straight lines. Therefore, smearing of the gravitating masses in N-body simulations plays a dual role: rst, this is the absence of the Newtonian divergence in the positions of the masses, second, this is the regular behavior of the gravitational potential in all other points. Thus, in the present paper we provide a physical justication for such a smearing.

In Sects. 4 and 5 we consider a possibility to get reasonable solutions of Eq. (1.1) in the case of absence of periodicity in one or two spatial directions. In Sect. 4, we investigate a model with the spatial topology T T R, i.e. with one

noncompact dimension, let it be z. As we mentioned above, due to noncompactness, the gravitational potential may suffer from the NeumannSeeliger paradox and additionally has a divergence of the form z2 + for |z| +. In this

section we try to resolve these problems with the help of a special arrangement of gravitating masses in the direction of z. A similar procedure in the at Universe with topology R3 was performed in [17]. Unfortunately, in the case of the topology T T R, there is no possibility to arrange the

masses in such a way that the gravitational potential is a smooth function in any point z. We have the same result for the Universe with topology T R R, which is considered

in Sect. 5. Here we also demonstrate the impossibility of constructing a smooth potential. The main results are briey summarized in Sect. 6.

2 Topology T T T: point-like masses

Obviously, for topology T T T the space is covered by

identical cells, and, instead of an innite number of these cells, we may consider just one cell with periodic boundary conditions. As we mentioned in Introduction, due to the nite volume of the cell, we can apply the superposition principle. It means that we can solve Eq. (1.1) for one arbitrary gravitating mass, and the total gravitational potential in a point inside the cell is equal to a sum of gravitational potentials (in this point) of all N masses. Without loss of generality, we can put a gravitating mass m at the origin of coordinates. Then the Poisson equation (1.1) for this mass reads

= 4GN

m (r)

m l1l2l3

. (2.1)

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Eur. Phys. J. C (2015) 75:217 Page 3 of 10 217

Taking into account that delta functions can be expressed as1

(x) =

1 l1

GNm

+

dkx

+

dky

+

dkz

+

k1=

cos 2k1

l1 x

, (2.2)

we get2

= 4GN

GNm

cos(2xkx) cos(2yky) cos(2zkz)k2 =

r , (2.7)

where r = (x2 + y2 + z2)1/2, as it should. A good feature of

the potential (2.6) is that its average value (integral) over the cell is equal to zero: = 0.3 This is a physically reasonable

result, because = 0.

Clearly, in the case of a point-like gravitating source, we have the usual divergence at the point of its location. Now, we want to demonstrate that there is also a problem at the points where gravitating masses are absent. More precisely, we will show that the sum (2.6) is absent on straight lines which connect identical masses in neighboring cells. In our particular example, they are lines of intersection (pairwise) of the planes x = 0, y = 0, and z = 0. Let us consider

the potential (2.6) on the straight line y = 0, z = 0. The

numerical calculation of the potential on this straight line at the point x = l1/2 for different values of the limiting

number n (being the maximum absolute value of the summation indices: |k1,2,3| n) is presented in the following

table for the cubic cell case l1 = l2 = l3 l. This table

clearly demonstrates that the potential does not tend (with the growth of n) to any particular nite number.

m l1l2l3

+

k1=

+

k2=

+

k3=

cos 2k1

l1 x

cos 2k3

l3 z

1

. (2.3)

Therefore, it makes sense to look for a solution of this equation in the form

=

+

k1=

cos

2k2 l2 y

+

k2=

+

k3=

Ck1k2k3

, (2.4)

where the unknown coefcients Ck1k2k3 can easily be found from Eq. (2.3):

Ck1k2k3 =

GNm

l1l2l3

cos 2k1 l1 x

cos 2k2

l2 y

cos 2k3

l3 z

1

n(l/2,0,0)

GNm/l

40 0.73371 41 0.89453

60 0.72869 61 0.89969

80 0.72614 81 0.90229

To understand the reason for this, let us analyze the structure of the expression (2.6) in more detail. For z = 0 the

gravitational potential reads

(x, y, 0) =

GNm

l1l2l3

k21

l21 + k

2 2

l22 + k

2

3

l23

, k21 + k22 + k23 = 0.

n n(l/2,0,0)

GNm/l n

(2.5)

Hence, the desired gravitational potential is

1

=

GNm

l1l2l3

+

k1=

+

k2=

+

k3=

k21

l21 + k

2 2

l22 + k

2

3

l23

, (2.6)

where k21 + k22 + k23 = 0. If x, y, z simultaneously tend to

zero, then the gravitational potential (2.6) has the Newtonian limit

1 This is the standard Dirac delta-function representation for the considered geometry of the model (see, e.g., [10]).

2 In the expression below, instead of the product cos(2k1x/l1) cos(2k2y/l2) cos(2k3z/l3) we can write cos(2k1x/l1 +

2k2y/l2 + 2k3z/l3), and with the help of the well-

known formulas for the cosine of the sum this expression will give only the contribution of the above mentioned form, cos(2k1x/l1) cos(2k2y/l2) cos(2k3z/l3). In fact, all terms containing, e.g., sin(2k1x/l1) (being an odd function of k1), will disappear from the sum with symmetric limits (i.e. k1 varying from

to +).

cos 2k1 l1 x

cos 2k2

l2 y

cos 2k3

l3 z

2k2l2 y

k21

l21 + k

2

2

l22 + k

2

3

l23

+

k1=

+

k2=

+

k3=

cos 2k1

l1 x cos

=

GNml3

3l1l2

4GNm

l1l2

+

k1=1

+

k2=1

1 k21l21 + k

2

2

l22

cos 2k1 l1 x

cos 2k2

l2 y

coth

l23k21

l21 +

l23k22

l22

3 It is worth noting that in [12,17,22] the concrete mass distribution in the Universe with topology R3 is cited as an example of the case of nonzero average value = 0. From a purely mathematical point

of view this case is inadmissible in the framework of the rst-order

perturbation theory.

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217 Page 4 of 10 Eur. Phys. J. C (2015) 75:217

2GNm

l2

+

k1=1

1k1 cos

2k1 l1 x

coth l3k1 l1

fn+1(x) = 2

n+1

k1=1

n+1

k2=1

cos 2k1

l1 x

, (2.8)

where we used the tabulated formulas for sums of series (see, e.g., [23], 5.1.25). All sums in this expression are potentially dangerous. To show it, we can drop the hyperbolic cotangents because coth k 1 for k +. The two last sums are

divergent, depending on which straight line we consider: x =

0 or y = 0, respectively. For example, on the straight line y =

0, the sum +

k1=1 cos (2k1x/l1) /k1 = ln [2 sin(x/l1)]

(see [23], 5.4.2) is convergent for any ratio x/l1 = 0, 1,

while +

k2=1(1/k2) limk2+ ln k2 is logarithmically

divergent. The rough estimate of the double sum also leads to a divergent result. To be more precise, we investigate now nite sums4 of the suspect terms on the straight line y = 0:

fn(x) = 2

n

k1=1

2GNm

l1

+

k2=1

1k2 cos

2k2 l2 y

coth l3k2 l2

1k2 . (2.12)

Evidently, if the expression (2.9) is convergent for n +, then in this limit the difference fn+1(x) fn(x) 0.

After some simple algebra we get (for l1 = l2)

fn+1(x) fn(x)

= 2

n

k1=1

1 l22l21 k21 + k22

+

n+1

k2=1

cos 2k1

l1 x

1

k21 + (n + 1)2

+2

n

k2=1

cos 2(n + 1)

l1 x

1

(n + 1)2 + k22

+

n

k2=1

cos 2k1

l1 x

1 l22l21 k21 + k22

+

n

k2=1

1 + 2 cos (2(n + 1)x/l1) n + 1

fn(x) +

1 k2 .

(2.9)

It is worth noting that in the case l1 = l2 and x/l1 = 1/2

the logarithmically divergent terms exactly cancel each other. In fact, it follows directly from the following estimates:

+

k1=1

1 + 2 cos (2(n + 1)x/l1) n + 1

. (2.13)

Here, the last term in the third line vanishes for n +.

Therefore, the problem of the convergence of (2.9) is reduced now to the analysis of fn(x). In Fig. 1, we show the graph of fn(x) (for x/l1 = 1/2) as a function of n. Each point gives

the value of fn(x) for the corresponding number n. This

picture clearly demonstrates that the difference fn+1(x)

fn(x) does not tend to zero for growing n. Even more, it does not go to any denite value.

It can also be veried that a similar result takes place for any other point on any of the straight lines and holds also for l1 = l2. Therefore, we have proven that in the case of point-

like gravitating masses in the considered lattice Universe the gravitational potential has no denite values on the straight lines joining identical masses in neighboring cells. Clearly, this is a nonphysical result, since the dynamics of cosmic bodies is not determined in such a case.

+

k2=1

cos (k1)

2 k21 + k22

= 2

+

k1=1

+

k2=1

(1)k1 k21 + k22

1 (2m)2 + k22

= 2

+

m=1

+

k2=1

1

(2m 1)2 + k22

2

+

m=1

+

k2=1

2m

(2m)2 + k22 3/2

2

+

+

2x

1 1 (2x)2 + y2 3/2

dxdy

lim

R+

ln R = (2.10)

and

+

k2=1

fn

1.5

1.0

1k2

+

1

dxx lim

R+

ln R = +. (2.11)

Therefore, both of these logarithmically divergent terms cancel each other. Nevertheless, the expression (2.9) does not have a denite limit for n +. To demonstrate it, along

with (2.9) let us introduce the function

4 Obviously, the inclusion of the hyperbolic cotangents does not effect the main results but makes calculations more complicated.

ODD NUMBERS

0.5

50 100 150 200 n

0.5

EVEN NUMBERS

1.0

1.5

Fig. 1 The graph of fn(l1/2) as a function of the number n

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Eur. Phys. J. C (2015) 75:217 Page 5 of 10 217

3 Topology T T T: smeared masses

Can the smearing of gravitating masses resolve the problem found in the previous section? To answer this question, we present gravitating masses as uniformly lled parallelepipeds. This representation of the masses looks a bit articial. However, such a form is the most appropriate for the considered cells, and the most important point is that the form of smearing does not matter for us at the moment. We just want to get a principal answer to the question of the possibility to avoid the problem with the help of smearing. So, let the mass m be uniformly smeared over a parallelepiped (with the lengths of the edges a, b, and c) which we put, without loss of generality, in the middle of the cell. It is convenient to introduce a function f1(x) equal to 1 for x [a/2, a/2]

and 0 elsewhere inside [l1/2, l1/2]. We can write this func

tion

f1(x) =

al1 +

+

n=1

+

k=1

4 2nk sin

an l1

cos 2n

l1 x

+

1 l2ac

sin ck l3

cos 2k

l3 z

+

1 l3ab

+

n=1

+

j=1

4 2nj sin

an l1

cos 2n

l1 x

sin b j l2

cos 2 j

l2 y

+

1 abc

+

n=1

+

j=1

+

k=1

8 3njk

sin an l1

sin b j

l2

sin ck

l3

cos 2 j

l2 y

. (3.5)

This equation implies that it makes sense to look for a solution in the following form:

(r) =

m l1l2c

cos 2k

l3 z

cos

2n l1 x

+

n=1

2n sin

an l1

cos 2n

l1 x

. (3.1)

Similarly,

f2(y) =

bl2 +

+

j=1

2 j sin

b j l2

cos 2 j

l2 y

(3.2)

+

k=1

Ck sin ck

l3

cos 2k

l3 z

+

m l1l3b

+

j=1

C j sin

b j l2

cos 2 j

l2 y

and

f3(z) =

. (3.3)

Therefore, the rest-mass density of the mass under consideration is

(r) =

mabc f1(x) f2(y) f3(z)

cl3 +

+

k=1

2k sin

ck l3

cos 2k

l3 z

+

m l2l3a

+

n=1

C n sin

an l1

cos 2n

l1 x

+

m l1bc

+

j=1

+

k=1

C jk sin b j

l2

cos 2 j

l2 y

mabc f (r). (3.4)

Then Eq. (1.1) for this mass reads

= 4GN

mabc f (r)

sin ck l3

cos 2k

l3 z

+

m l2ac

+

n=1

+

k=1

C nk sin

an l1

cos 2n

l1 x

m l1l2l3

sin ck l3

cos 2k

l3 z

= 4GNm

1 l1l2c

+

k=1

2k sin

ck l3

cos 2k

l3 z

+

n=1

+

j=1

C nj sin

an l1

cos 2n

l1 x

+

m l3ab

+

1 l1l3b

+

j=1

2 j sin

b j l2

cos 2 j

l2 y

sin b j l2

cos 2 j

l2 y

+

1 l2l3a

+

n=1

2n sin

an l1

cos 2n

l1 x

+

m abc

+

n=1

+

j=1

+

k=1

Cnjk

+

j=1

+

k=1

42 jk sin

b j l2

cos 2 j

l2 y

+

1 l1bc

sin an l1

sin b j

l2

sin ck

l3

sin ck l3

cos 2k

l3 z

cos 2n l1 x

cos 2 j

l2 y

cos 2k

l3 z

. (3.6)

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217 Page 6 of 10 Eur. Phys. J. C (2015) 75:217

Substitution of this expression into the Poisson equation (3.5) gives

Ck =

2GN

2k

l3 k

2 , C j =

2GN

2 j

l2 j

2

,

C n =

2GN

2n

l1 n

2

,

C jk =

4GN 3 jk

1 j2

l22 + k2l23

, C nk =

4GN 3nk

1 n2

l21 + k2l23

,

C jn =

4GN 3nj

1 n2

l21 + j

,

2

l22

1

. (3.7)

Let us choose the same straight line as in the previous section, that is, y = 0, z = 0, and the same point x = l1/2.

The numerical calculation of the gravitational potential in this point for different values of the limiting number n is presented in the following table for the cubic cell case under the additional condition a = b = c = (3/7)l. In contrast

to the previous case of a point-like source, here the potential apparently tends to a particular nite value. Therefore, in the case of smeared gravitating masses the gravitational potential has a regular behavior at any point inside the cell (including, e.g., the point x = y = z = 0).

Cnjk =

8GN 4njk

n2

l21 + j

2

l22 + k2l23

m l1l2

+

k1=

+

k2=

cos 2k1

l1 x

n n(l/2,0,0)

GNm/l n

n(l/2,0,0)

GNm/l

15 0.028717 16 0.028443

19 0.028536 20 0.028222

23 0.028368 24 0.028223

cos

2k2 l2 y

(z)

. (4.2)

Evidently, we can look for a solution of this equation in the form

=

+

k1=

+

k2=

Ck1k2(z) cos 2k1

l1 x

cos 2k2

l2 y

,

(4.3)

4 Topology T T R

The T T R topology implies one noncompact dimension;

say z. Therefore, there is a lattice structure in the directions x and y and an irregular structure in the direction z. In a column x [l1/2, l1/2], y [l2/2, l2/2], z (, +) there

is an innite number of gravitating masses. To obtain a nice regular solution, we will try to arrange the masses in the z direction in such a way that in each point z the gravitational potential is determined by one mass only. There are two possibilities to do that. Let this mass be at z = 0. In the rst

scenario, the potential and its rst derivative (with respect to z) should vanish at some distance z0 (which we determine below). Then the next mass should be at a distance (in the

z direction) equal to or greater than z0 + z1, where z1 is

a distance at which the gravitational potential and its rst derivative vanish for the second mass. Similarly, we should shift in the direction of z the third mass with respect to the second one and so on. In this scenario, we can arrange strips z between masses where the potential is absent. It occurs, e.g., between the rst and second masses if the second mass is situated at distances greater that z0 +z1. In the strip, we place

a uniform medium with the rest-mass density . The coordinates x [l1/2, l1/2] and y [l2/2, l2/2] of masses

are arbitrary. In the second scenario, we should determine distances z0, z1, z2, . . . where potentials of neighboring (in the z direction) particles are smoothly matched to each other. This means that at these distances the potentials are generally nonzero. Moreover, we suppose that their rst derivatives are zero at the points of matching, i.e. the potentials have extrema in these points. In this scenario, the neighboring (in the z direction) masses should have the same coordinates x and y. Now let us consider these scenarios in detail. For both of them, we need to look for a solution just for one particle. Let this particle be in the point x = y = z = 0. Then Eq.

(1.1) reads

= 4GN (m(r) ) . (4.1)

Keeping in mind the regular structure in the x and y directions, we can represent the delta functions (x) and (y) in the form (2.2). So, Eq. (4.1) is reduced to

= 4GN

and from the Poisson equation (4.2) we get

GN

=

+

k1=

+

k2=

cos 2k1

l1 x

cos 2k2

l2 y

k22

l22 +

k21

l21

Ck1k2(z)+

mGNl1l2 (z)

C k1k2(z)

42

.

(4.4)

In this section, the prime denotes the derivative with respect to z. Now, we should determine the unknown functions Ck1k2(z). First, we nd the zero mode C00(z), which satises the equation

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Eur. Phys. J. C (2015) 75:217 Page 7 of 10 217

C 00(z)

4 =

mGNl1l2 (z) GN

. (4.5)

This equation has the solution

C00(z) = 2GN

z2 +

2l1l2 mGN|z| + B, (4.6) where B is a constant of integration. This solution is a function growing with z. Therefore, we must cut it off at some distance z0.

Let us consider the rst scenario. From the condition C 00(z0) = 0 we obtain

z0 =

m 2

l1l2

=

has a drawback inherent in the T T T model with the

point-like source.

In the second scenario with identical masses m, all of them have the same coordinates x, y and are separated by the same distance 2z0 l3 in the direction of z. Here, the function

C00(z) still has the form (4.6). Since we require C 00(z0) = 0,

the boundary z0 is determined by (4.7). However, the constant B is not given now by (4.8), because the condition C00(z0) =

0 is absent. This constant can by found from the condition = 0 over the period l3 = 2z0. That is,

+z0

z0

C00(z)dz = 0

B =

2GNmz03l1l2 =

m2z0l1l2 . (4.7)

The second condition C00(z0) = 0 provides the value of

B:

B =

GNml3

3l1l2 . (4.14)

The functions Ck1k2(z) for k21 + k22 = 0 are given by

Eq. (4.12), where the constant

B follows from the boundary condition C k1k2(z0) = 0:

B =

m2GN

2

l21l22

. (4.8)

Now, we want to determine the form of Ck1k2(z) when k21 + k22 = 0. In this case Eq. (4.4) is consistent only if the

following condition holds true:

k22

l22 +

k21

l21

Ck1k2(z) +

mGNl1l2 (z)

C k1k2(z)

mGN2l1l2 e2z0 [sinh(2z0)]1 . (4.15)

It can easily be veried that Ck1k2(z) can be rewritten in the form

Ck1k2(z) =

GNm

42 = 0,

k21 + k22 = 0. (4.9)We look for a solution of this equation in the form

Ck1k2(z) =

Ae2|z| +

Be2|z|,

k22

l22 +

l1l2 sinh(2z0) cosh [2(|z| z0)] .

k21

l21

,

(4.16)

Therefore, in the second scenario the gravitational potential is

=

+

k1=

(4.10)

+

k2=

Ck1k2(z) cos 2k1

l1 x

cos 2k2

l2 y

where

A and

B are constants. The substitution of this function into Eq. (4.9) gives

A

B =

mGNl1l2 . (4.11)

Therefore,

Ck1k2(z) = 2

B cosh(2z)

Ck10(z) cos 2k1

l1 x

= C00(z) + 2

+

k1=1

+

k2=1

2 C0k2(z) cos 2k2

l2 y

mGNl1l2 e2|z|,

k21 + k22 = 0. (4.12)From the boundary condition Ck1k2(z0) = 0 we get

B =

mGN2l1l2 e2z0 [cosh(2z0)]1 . (4.13)

It can easily be veried that the function (4.12) (with

B from (4.13)) does not satisfy the boundary condition C k1k2(z0) = 0. Hence, we cannot determine the gravitational

potential in accordance with the rst scenario.

Now, we intend to demonstrate that there is a possibility to nd the potential in the framework of the second scenario in the case of identical masses. However, this construction

+

k1=1

+

k2=1

4 Ck1k2(z) cos 2k1

l1 x

cos 2k2

l2 y

= GNm

2l1l2l3 z2

2l1l2 |z| +

l3 3l1l2

+

2 l2

+

k1=1

cos (2k1x/l1)

k1

cosh[2k1(|z| z0)/l1]

sinh(2k1z0/l1)

+

2 l1

+

k2=1

cos (2k2y/l2)

k2

cosh[2k2(|z| z0)/l2]

sinh(2k2z0/l2)

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217 Page 8 of 10 Eur. Phys. J. C (2015) 75:217

+

4 l1l2

+

k1=1

+

k2=1

cos (2k1x/l1) cos (2k2y/l2) k21l21 + k

2 2

l22

GN

=

+

k1=

cos 2k1

l1 x

2 2

l22 (|z| z0)

GN ml1 (y)(z) +

k21

l21

Ck1(y, z)

cosh 2 k21

l21 + k

sinh 2 k21

l21 + k

C k1y(y, z)

. (4.17)

When z = 0 and x, y simultaneously go to zero, the potential

GNm/ x2 + y2, as it should. From the physical

point of view, it is clear that this scenario should coincide with the T T T case. In fact, the triple sum (2.6) can

be rewritten in the form (4.17) with the help of [23] (see5.4.5). It can also easily be seen that on the plane z = 0 the

expression (4.17) exactly coincides with Eq. (2.8). Therefore, in the second scenario we again arrive at the nonphysical result that the gravitational potential has no denite values on straight lines y = 0 and x = 0.

5 Topology T R R

In this section we consider a model with a periodic boundary condition in one direction only. Two other spatial dimensions are noncompact. Here, in analogy with the previous section, we also suppose that the gravitational potential in the vicinity of a particle is determined by its mass only. The shape of such a domain is dictated by the symmetry of the model and will be described below. On the boundary (in the direction of noncompact dimensions) of this domain the potential and its rst derivative are equal to zero, and between such domains the potential is absent: = 0. There

fore, this model is similar to the rst scenario in the previous section.

Let the mass be at the point x = y = z = 0 and the

periodic boundary condition (with the period l1) be along the x coordinate. Then the Poisson equation (4.1) for this topology can be written as follows:

= 4GN

m l1

, (5.3)

2 2

l22 z0

4

C k1z(y, z) 4

where

C k1y(y, z)

2Ck1(y, z)

y2 , C k1z(y, z)

2Ck1(y, z)

z2 .

(5.4)

For the zero mode k1 = 0 this equation gives

GN

= GN

C 0y(y, z)

ml1 (y)(z)

C 0z(y, z)

4 . (5.5)

Following the geometry of the model, it makes sense to turn to polar coordinates:

y = cos , z = sin . (5.6)

Then the two-dimensional Laplace operator is

4

+

1 2

22 + . (5.7)

It is clear that due to the symmetry of the problem the functions Ck1 do not depend on the azimuthal angle . Therefore,

Eq. (5.5) reads

GN

= GN

ml1 ()

=

1

C04 . (5.8)

This equation has the solution

C0 = GN

2 + 2GN

+

k1=

cos 2k1

l1 x

(y)(z) 4GN

.

(5.1)

ml1 ln + B, (5.9) where we took into account that ln = 2 ( ). Simi

lar to the previous model with one noncompact dimension, here the solution is also divergent in some directions. In the present case, it grows with the polar radius . So, we must cut off this solution at some distance 0. Clearly, this boundary represents the cylindrical surface = 0. The domain in

which we put the mass m0 = m is a cylinder with the radius

0 and the generator is parallel to the x-axis. The length of the cylinder along the x-axis is l1. The mass m is in the center of the cylinder (with the coordinate x = 0 for the considered

case). The next particle of the mass m1 is inside its own cylinder with the generator along the x-axis and the radius 1. This particle may have a coordinate x different from the rst particle. All these cylinders have the periodic (with the period l1) boundary conditions along the x-axis. On the other hand, they should not overlap each other in the transverse (with respect to the x-axis) direction. Moreover, it is impossible to match them smoothly via cylindrical surfaces. Therefore, we

. (5.2)

Then from the Poisson equation (5.1) we get

We look for a solution of this equation in the form

=

+

k1=

Ck1(y, z) cos 2k1

l1 x

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Eur. Phys. J. C (2015) 75:217 Page 9 of 10 217

demand that the gravitational potential outside the cylinders is equal to zero. Hence, on the boundaries of the cylinders ( = 0 for the rst mass) the gravitational potential and

its rst derivative (with respect to ) are equal to zero. These boundary conditions enable us to determine the radius 0 and the constant B in Eq. (5.9). For example, from the condition

dC0(0)/d = 0 we get

0 =

m

l1

=

ms0l1 , (5.10)

where s0 = 20 is the cross-sectional area of the cylinder.

From the second boundary condition C0(0) = 0 we get the

value of B:

B =

GNm

l1 (1 2 ln 0) . (5.11)

Obviously, in the case k1 = 0, Eq. (5.3) is consistent only

if the following condition holds true:

GN ml1 ()+

k21

l21

m 20l1

C k1y(y, z)

C k1z(y, z)

Ck1(y, z)

4

4 = 0,

(5.12)

which for > 0 can be rewritten in the form

d2Ck1 d2 +

dCk1

d

42k21

l21

Ck1 = 0. (5.13)

The solutions of this equation are the modied Bessel functions:

Ck1() = C1I0

2|k1| l1

2GN

ml1 K0

2|k1| l1

,

(5.14)

where C1 is a constant of integration. We took into account that the function K0() ln for 0, so the two-

dimensional Laplacian acting on this function provides the necessary delta function in Eq. (5.12). The function (5.14) should satisfy the same boundary conditions at = 0 as

the function C0(). It can easily be veried that we cannot simultaneously reach both equalities Ck1(0) = 0 and

dCk1(0)/d = 0. Hence, we cannot determine the gravita

tional potential in accordance with the proposed scenario.

To conclude this section, it is worth noting that we can construct the potential in the scenario similar to the second one from the previous section. This is the case of identical masses distributed regularly in all directions. Obviously, this case is reduced to the T T T model from Sect. 2 with

the drawback inherent in it.

Therefore, similar to the previous section, we also failed in determining a physically reasonable gravitational potential in the model with the topology T R R.

6 Conclusion

Our paper was devoted to cosmological models with different spatial topologies. According to the recent observations, our Universe is spatially at with rather high accuracy. So, we restricted ourselves to this case. However, such a spatially at geometry may have different topologies depending on the number of directions/dimensions with toroidal discrete symmetry. These topologies result in different kinds of lattice Universes. There are a lot of articles exploring the lattice Universes (see, e.g., [111] and references therein). One of their main motivations is to provide an alternative (compared to the standard CDM model) explanation of the late-time accelerated expansion of the Universe. Another important point is that our Universe is highly inhomogeneous inside the cell of uniformity with the size of the order of 190Mpc [12]. Hence, it is quite natural to consider the Universe on such scales lled with discrete sources rather than a homogeneous isotropic perfect uid.

On the other hand, N-body simulations of the evolution of structures in the Universe are based on the dynamics of discrete sources in chosen cells. To perform such simulations, we should know the gravitational potentials generated by these sources. Therefore, the main purpose of our paper was the determination of the gravitational potentials in the cases of three different spatial topologies: T T T , T T R,

and T R R. The potential satises the corresponding

Poisson equations of the form (1.1). These equations can be obtained as a Newtonian limit of General Relativity [12,17]. So, to determine the potential, we should solve them. One of the main features of the analyzed Poisson equations is that they contain the average rest-mass density, which represents a constant in the comoving frame. This results in two problems. First, we cannot, in general, apply the superposition principle. Second, the presence of such a term leads to divergences in the directions of the noncompact dimensions. We tried to avoid these problems arranging masses in special ways. Our investigation has shown that the T T T model

is the most physical one. Here, due to the discrete symmetry in all three directions, we can represent the innite Universe as one nite cell with periodic boundary conditions in all dimensions. The nite volume of the cell enabled us to use the superposition principle and solve the Poisson equation for a single mass. The total potential in an arbitrary point of the cell is equal to the sum of potentials of all particles in the cell. Unfortunately, in the case of point-like gravitating sources the obtained solution has a very important drawback. Usually, it is expected that potentials diverge at the positions of the masses. However, in the model under consideration the gravitational potential has no denite values on the straight lines joining identical masses in neighboring cells, i.e. at the points where masses are absent. Clearly, this is a nonphysical result, since the dynamics of cosmic bodies is not determined

123

217 Page 10 of 10 Eur. Phys. J. C (2015) 75:217

in such a case. Then, looking for a more physical solution, we smeared the gravitating masses over some regions and showed that in this case the gravitational potential shows a regular behavior at any point inside the cell. Therefore, smearing represents the necessary condition of getting a regular gravitational potential in the lattice Universe. Usually in N-body simulations some sort of smearing is used to avoid divergences at the positions of the masses. Now, we have demonstrated that this procedure helps to avoid problems on the above mentioned straight lines as well. Therefore, the smearing of gravitating bodies in numerical simulations is not only a technical method but also a physically substantiated procedure, and in the present paper we provide a physical justication for such a smearing.

In the T T T model, particles in the cell may have

different masses and may be distributed arbitrarily. In the cases of the topologies T T R and T R R, we cannot

do this. We have shown that the only way to get a solution here is to suppose the periodic (in all dimensions) distribution of identical masses. However, such a solution is reduced to the one obtained in the case of the T T T topology.

Therefore, rst, this solution has a drawback inherent in this model, and, second, the distribution of the masses looks very articial.

Acknowledgments The work of M. Eingorn was supported by NSF CREST award HRD-1345219 and NASA Grant NNX09AV07A.

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