Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://xray.sai.msu.ru/~popova/papers/lit12.ps Äàòà èçìåíåíèÿ: Wed Jul 26 16:18:56 2006 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:42:54 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 2

FRIEDMANN COSMOLOGY IN ALTERNATIVE DIMENSIONS
A.D. Popova
Sternberg Astronomical Institute, Moscow, Russia
Introduction
If one breaks up a habit to consider integer dimensionalities, he can open new possibilities
for at least theoretical studying our Universe. A popular subject is considering multidi-
mensional models with integer dimensions. However, we think that dimensionality may
be not only noninteger, but also variable at different scales: It may change with relative
distances between bodies. Moreover, we prefer to speak not of extra but of lacking di-
mensionalities. The dimensionality which is equal (or approximately equal) to 3 at the
laboratory scales should be less than 3 at larger scales. It seems that there exist some
convincing arguments. First, passing to greater and greater scales, from galaxies to clus-
ters and superclusters of galaxies, the discrepancy between the luminous matter and its
dynamics becomes stronger and stronger. To avoid this difficulty, one usually introduces
the concept of dark matter that is required in increasing amounts when passing to larger
scales [1] or try to modify the Newtonian dynamics [2]. Second, there exist the discrepancy
between the age of the galaxies [3] and globular clusters [4] and that of the relatively small
age of our Universe in the standard cosmology [5]. These discrepancies can be resolved if
one recalculates the above dynamics and age for the case of a space dimensionality less
than 3.
However, in refusing of the integer and constant spatial dimensionality, we can do the
same with the temporal dimensionality because general relativity learns us that space and
time should be considered at equal grounds. We have no consistent theory to describe
our propositions in full although there exists an approach to general relativity involving
noninteger dimensionalities [6], but we construct here Friedmann-like cosmological models
which are in principle solvable due to their simplicity and symmetry. Our results have
some preliminary meaning. We present also traditional cosmological tests for arbitrary
spatial dimensions.
Homogeneous and Isotropic World in n Spatial Dimen-
sions
Consider the metric interval in the form
ds 2 = c 2 dt 2
- a 2 (t)[dr 2 + ”
# 2 (r) d# 2
(n) ] (1)
which corresponds to the Riemannian space which is a topological product of the time axis
and an isotropic n-dimensional space. The quantity d# (n) is the angle-distance element at
the surface of n-dimensional Euclidean sphere, the but it is not required explicitly. The
function ”
#(r) is the same as in the 3-dimensional case [5]:
”
#(r) =
# # # # # # #
sin r, 0 # r # #, k = +1;
r, r # 0, k = 0;
sh r, r # 0, k = -1.
1

The parameter k = -1, 0, +1 corresponds to the sign of the Gauss curvature.
Let the space-time be filled by the hydrodynamic matter with the stress-energy tensor
T 0
0 = #, T i
0 = 0, T k
i = -p# k
i , (2)
(i, k, ... = 1, 2, ...n), note that the trace of (2) now is T = # - np.
A standard way of calculating the Einstein equations gives (a dot denotes differenti-
ation with respect to ct):
1
2
n(n - 1) # —
a 2
a 2
+
k
a 2
# =
# (n)
c 2
#, (3)
(n - 1) # ˜ a
a
+
n - 2
2
# —
a 2
a 2 +
k
a 2
## = -
# (n)
c 2 p, (4)
and from the conservation law T #
µ;# = 0 (µ, #, ... = 0, 1, ..., n) it follows that
—
# + n
—
a
a
(# + p) = 0. (5)
We also present the equation with the "eliminated curvature"which is the combination
of Eqs. (3) and (4)
n(n - 1)
˜
a
a
= - # (n)
c 2 [(n - 2)# + np]. (6)
The equation of state is taken in the habitual form
p = q# (7)
with q some parameter: q = 0 for the dust matter and q = 1/n for pure radiation. We
assume q #= -1 in this Section because q = -1 is an isolated case.
From here, we can forget that n is an integer and supply it with real values n > 2.
In eliminating the quantities # and p from Eqs. (3), (5) and (6), we obtain the general
solution for the scale factor a(t) in the implicit form when k = -1 and k = +1
2
n(q + 1)
A # a
A
# n(q+1)
2
2 F 1
# 1
2
, #; # + 1; k # a
A
# n(q+1)-2
# = ct (8)
where 2 F 1 is the Gauss hypergeometric function, A is some integration constant,
# =
n(q + 1)
2[n(q + 1) - 2]
.
For k = +1, the solution (8) gives a branch which increases until the maximal value
of the scale factor A: a max = A; the time of reaching a max is
t (n)
max =
2 # #
n(q + 1) ·
#(# + 1)
#(# + 1/2)
A
c
, (9)
here and below # is the gamma function. The periodic solution can be composed from
matching together the branches similar to the above and similar to those obtained by an
operation of reflecting the above with respect to the line t = t (n)
max = Const.
2

The asymptotics of the solution (8) for small t is given by the formula
a # t
2
n(q+1) . (10)
The flat solution k = 0 is represented explicitly by the same formula (10) for the whole
time axis.
We should remark that, first, the expressions (8), (9) and (10) are reducible to the
same known expressions for the case n = 3. Second, it is interesting to note that the
parameter n appears in the above formulae in the only combination n(q + 1). Thus, the
latter plays the role of an effective spatial dimensionality: for different but suitably chosen
n and q the solution can be the same. For example, the dust solution in n+ 1 dimensions
is equivalent to the pure-radiation solution in n dimensions. Third, the restriction n > 2
is caused by the fact that for q = 0 and k = +1 an analytic limit of (8) when n tends to
2 does not exist, however this limit exists for k = -1.
As mentioned, the case n = 2 is distinct. It is also interesting by that there exists the
stationary solution in the dust case (q = 0) where the scale factor is connected with the
matter energy density:
a = c (# (2) #) -1/2 . (11)
It is worth recalling that the Einstein equations without a cosmological constant in n #= 2
dimensions have no physically acceptable stationary solutions. Besides (11), the linear
solutions (a # t) take place for all k in the 2-dimensional dust case.
In order to derive cosmological tests in the next section, we introduce the dimensionless
density parameter
# (n) =
#
# c
where the critical density # c is determined as usual by settling k equal to zero in Eq.(3):
# c =
n(n - 1)
2
H 2
# (n)
with H = c —
a/a the Hubble parameter. The deceleration parameter q (n) can be defined as
usual and expressed
via# (n) with the help of Eq.(6):
q (n)
# -
c 2 ˜ a
aH 2 =
n(q + 1) - 2
2#
(n) .
Cosmological Tests in n Spatial Dimensions and the Age
of the Universe
We present here the three traditional cosmological tests [7], in n = 3 these tests were
derived in [5]: visual magnitude, angular size and the number of sources vs. redshift derived
for the dust-filled Universe. We keep the only two orders in the redshift z; although the
exact expressions for the tests have different analytic forms for different values of k, the
above two orders surprisingly coincide analytically for all n like in the known case n = 3.
Visual magnitude vs. redshift. This test is very sensitive to the possible change of the
dimensionality because it depends on the volume of the region between a source and an
observer who measures the luminosity. Assume conditionally that the dimensionality is
3

3 if the distance between any bodies is less than R 0 and it is n otherwise. If the ratio
R 0 H 0 /cz is small for small z, then the distance modulus m-M is
m-M = 5z # 5 - n
4 -
(n - 1)(n - 2)
8
# lg e - 5+
2.5 # (n - 1)lg
cz
H 0
+ (3 - n)lg
R 0
# 2 - lg# (n) # +
5
16
(n - 1)(3 - n)
n + 1
R 0 H 0
cz
where, apart from R 0 , the subscript "0"means that a given quantity is taken at the modern
epoch, # (n) is some constant:
# (n) =
2 (n+1)/2 #(n/2)
(n - 1) # # #((n - 1)/2)
.
Angular size vs. redshift. The ratio ”
R of the source linear size l to its angular size #
can be expressed as follows:
”
R #
l
#
=
c
H 0
# z - # 3
2
+
n - 2
4#
(n)
0 # z 2 # =
c
H 0
# z -
1
2
(3 + •
q (n)
0 )z 2 #
where • q (n)
0 = (n
-2)# (n)
0 /2 is the deceleration parameter for the dust-filled Universe.
This test is insensitive to a possible change of dimensionality inside a given z because we
suppose that light rays propagate uniformly in spaces with any dimensionalities. Unfortu-
nately, in the first two orders in z the contributions of n
and# (n)
0 are undistinguishable.
We can also see that the expression of ”
R via •
q (n)
0 is formally independent of n.
The number of sources vs. redshift. The differential of the number of sources at a
given z is
dN = # 0 s (n) # c
H 0
# n
z n-1 # 1 -
n + 1
2
(1 + q (n)
0 )z #
where # 0 is the density of the sources, its physical dimensionality is (length) -n , s (n) is the
area of an n-dimensional sphere:
s (n) =
2# n/2
#(n/2)
.
This test could be the most powerful one in order to determine n unless it was mostly
contaminated by evolution and selection effects.
We should add that the presented tests hardly make easier the tasks for observers
because they involve one or more additional parameters: n and/or R 0 . The above tests
evidently reduce to familiar ones in the case n = 3.
The age of the Universe is given by the following integral which cannot be expressed
via elementary functions except of the cases n = 2, 3
T (n) = H -1
0 # dz (1 + z) -2 [1
-# (n)
0
+# (n)
0 (1 + z) n(q+1)-2 ] -1/2 .
For n = 3 the resulting expression is in agreement with an expression in [5]. We can see
that for n varying from 3 to 2, T (n) increases from some value T (3) < H -1
0 to exactly H -1
0 :
T (2) = H -1
0 independently of the value
of# (n)
0 . For k = 0
(# (n)
0 = 1), T (n) = 2H -1 0 /n(q+1).
4

Homogeneous and Isotropic World in n Spatial Dimen-
sions and m Time Dimensions
Let now the Riemannian space be the topological product of m-dimensional flat "tempo-
ral"space and n-dimensional isotropic space (m and n are integers as yet). Let the metric
interval have the form [8]
ds 2 = c 2 (dt 2
1 + dt 2
2 + ... + dt 2
m ) - a 2 (t)[dr 2 + ”
# 2 (r) d# 2
(n) ]
where the radial coordinate in the temporal space has the meaning of time:
t = (t 2
1 + t 2
2 + ... t 2
m ) 1/2 . (12)
As before, let i, k, ... = 1, ..., n and let the capital Latin letters be referred to the temporal
space: A, B... = 1, ..., m.
In order to make our cosmological model free of possible restrictions we have to intro-
duce the new parameter p # which describes a matter tension in the temporal space leading
to the stress-energy tensor written in the form
T B
A = # uA u B
- p # (# B
A - uA u B ), T i
A = 0, T k
i = -p # k
i . (13)
Clearly, the parameter p # does enter T B
A when m = 1. The vector uA = t A /t is the unit
vector field in the temporal space. Moreover, the quantities uA = t A /t represent the m
components of the (m+n)-dimensional vector field which has the meaning of the (m+n)
velocity. As usual, its spatial components are zero: u i = 0. In the case m = 1, it is the
habitual 4-velocity vector.
Now we can calculate the generalized Einstein equations by introducing some coupling
constant # (m,n) and taking (13) as the source. We obtain the following set of equations
where a dot denotes differentiation with respect to ct with the quantity (12) taking as
the time:
n # m- 1
ct
—
a
a
+
n - 1
2
# —
a 2
a 2 +
k
a 2
## =
# (m,n)
c 2 #, (14)
n # ˜ a
a
+
m- 2
ct
—
a
a
+
n - 1
2
# —
a 2
a 2
+
k
a 2
## = -
# (m,n)
c 2
p # , (15)
(n - 1) # ˜
a
a
+
m- 1
ct
—
a
a
+
n - 2
2
# —
a 2
a 2
+
k
a 2
## = -
# (m,n)
c 2
p. (16)
The conservation law T #
µ;# = 0 (µ, #, ... = A, B, ... # i, k, ...) is, as usual, the integrability
condition of Eqs. (14)-(16). It has now the form
—
# +
m- 1
ct
(# + p # ) + n
—
a
a
(# + p) = 0.
As before, from now we consider n and m not integers but real numbers: n > 1 and m > 0.
In order to make the set of Eqs. (14)-(16) solvable, we employ the equation of state (7).
Then, in multiplying (14) on q and taking the sum with (16) using (7), we obtain the
equation for the scale factor:
˜ a
a
+
m- 1
m cr - 1
1
ct
—
a
a
+
1
2
# m cr - (2 - n)
m cr - 1 - 1 # # —
a 2
a 2
+
k
a 2
# = 0 (17)
5

where we have denoted by
m cr = 2 -
nq
n(q + 1) - 1
some critical value of the temporal dimensionality. After that, Eq.(15) will determine
the parameter p # after substituting there a solution a(t) found from Eq.(17). Unfortu-
nately, Eq.(17) has no general solutions in elementary functions. We present here the
only solutions for some physically interesting cases.
1)The flat n-dimensional space (k = 0). In this case Eq.(17) with k = 0 is reducible
to an equation in full differentials. Consider the following subcases:
1a) m #= m cr and q #= -1. The solution is
a = C t # 1a , # 1a =
2(m cr -m)
m cr - (2 - n)
(18)
where C is an arbitrary number. Let n > 1 and q > -(n - 1)/n, then m cr > (2 - n) and
the behavior of (18) is mainly determined by the sign of m cr - m. If m < m cr , then the
scale factor increases from zero to infinity: The derivative —
a is zero at the point t = 0
for m < [m cr - (2 - n)]/2 and it is equal to infinity for [m cr - (2 - n)]/2 < m < m cr . If
m > m cr , then the scale factor collapses from infinity to zero during infinite time. The
time dependence of the matter parameters is as follows
#, p # , p # t -2 . (19)
1b) m = m cr and q #= -1. The solution is logarithmic and starts from the point t = 1:
a = C (lnt) # 1b , # 1b =
2
n(q + 1)
.
In this case
#, p # , p # (t lnt) -2 . (20)
Now we turn to the case q = -1. It is interesting by its physical content giving rise to
inflationary solutions [9] as well as by that it is an isolated case for Eq.(17). The solutions
below contain the dependence on n only via m cr , m cr = 2 - n here.
1c) m #= m cr and q = -1
a = C 1 exp(C 2 t # 1c ), # 1c =
m cr -m
m cr - 1
=
m- (2 - n)
n - 1
. (21)
Due to that we have assumed before n > 1, m cr < 1 and the behavior of (21) is mainly
determined by the signs of m cr - m and the constant C 2 . If m > m cr , then all the
solutions start at the point t = 0, a = C 1 ; for C 2 > 0 a solution infinitely increases
and C 2 < 0 a solution decreases up to zero. In this case, for m cr < m < 1 and m > 1
all the solutions start from t = 0 with the infinite derivative —
a and the zero derivative,
respectively. If m < m cr , the stationary regime a = C 1 is reached for infinite time and the
above derivative at t = 0 has always infinite modulus: For C 2 > 0 a solution decreases
from infinity and for C 2 < 0 a solution increases from zero. As to the matter parameters,
#, p # , p # t 2(# 1c -1) .
1d) m = m cr and q = -1. In this simple subcase the solution is linear: a = Ct, and
the time dependence of the matter parameters is again (19).
6

2) The stationary case. This case generalizes the n = 2 stationary solution (11). Here,
any dependence on m disappears, if a = Const, then
# =
n(n - 1)
2
k
a 2 , p # = -# = -
n(n - 1)
2
k
a 2 , p = - (n - 1)(n - 2)
2
k
a 2 .
meaning that q = -(n - 2)/n. For n > 2, the stationary Universe cannot be filled by
ordinary matter: it must possess negative energy density or negative pressure.
3)The 2-dimensional (n = 2) dust-filled (q = 0) spacetime. It is interesting that the
solution is the same for all k:
a = # C t 2-m , m #= 2;
C ln t, m = 2.
Note that here m cr = 2 and the time dependence of the matter parameters is the same as
in (19) and (20) for m #= 2 and m = 2, respectively.
References
1. M.Davies et al., Astron. J. 238 (1980) l113.
2. M.Milgrom, Astroph. J. 270 (1983) 365.
3. A.G.Brusual, Astroph. J. 273 (1983) 105.
4. T.S. van Olbada et al., Monthly Notes 196 (1981) 823.
5. Ya.B.Zeldovich and I.D.Novikov, The Structure and Evolution of the Universe, Moscow,
Nauka, 1975, in Russian.
6. V.Yu.Koloskov, Stochastic Metrization of Spaces with Noninteger Dimensionalities,
Master Theses, Moscow, 1990.
7. A.D.Popova, to appear in Astron. and Astroph. Trans..
8. A.D.Popova and A.N.Kulik, to appear in Astron. and Astroph. Trans..
9. A.D.Dolgov, Ya.B.Zeldovich and M.V.Sazhin , The Cosmology of the Early Universe,
Moscow, Moscow State University, 1988, in Russian.
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