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Вестник Челябинского государственного университета. 2015. № 7 (362). Физика. Вып. 20. С. 7-10.
АСТРОФИЗИКА
M. Ghosh, A. S. Vaganov, A. A. Yanbekov, E. A. Penkina, E. R. Zhdanov, R. N. Izmailov
MODELS COMPARISON IN THE DARK MATTER PROBLEM
Perfect fluid model and braneworld model are two important models associated with dark matter. It is shown that the solution of a static spherically symmetric spacetime has a number of physically interesting properties. A brief comparison between these two models is also given.
Keywords: Dark matter, perfect fluid model, braneworld model.
Dark matter (DM) is at the core of the modern cosmology and astrophysics. Since DM cannot be made of any of the usual standard model particles, DM is also a central focus of elementary-particle physics. DM is only detectable through their gravitational influence. The evidence for DM is the flatness of galactic rotation curves. These measurements are particularly important now not only for establishing the existence of DM, but particularly for fixing the local DM density, relevant for impact of DM.
One of idea of solving flat rotation curves problem is to incorporate DM is the braneworld theory. Braneworld theory assumes the nature of DM as an effect of extra dimensions in our four dimensional (4D) Universe [1] The DM lives in an extra dimension of the Universe in a hidden brane and gravity is the only candidate by which DM interacts with the fields in the visible brane. A problem with this model is what to identify as a four-dimensional observable quantity. Effective 4D quantities should be arbitrary due to the finite thickness of the brane. The simplest prescription one can consider is to define the 4D effective quantity associated to a 5D quantity as its spatial average over the brane thickness ([2]). The above prescription is lawful only if we deal with branes with well-defined thickness. There is not a unique way to choose the brane thickness when the brane is smoothly spread over the extra-space. This misguides us in the computation of 4D measurable quantities [3].
Perfect fluid can be a well approximation of DM because the evolution of our homogeneous and iso-tropic Universe is well described by Friedmann's equations that come from general relativity and the main components of our Universe can be described by fluids ([4]).
Static spherically symmetric solution in a perfect fluid and a braneworld regime
The general static spherically symmetric spacetime is represented by the following metric
ds2 = -ev(r)dt2 + eMr)dr2 + r2(d02 + sin2 0d^2)(1)
where the functions u(r) and 1(r) are the metric potentials. For the perfect fluid, the matter energy momentum tensor T^ is given by T = Tee = = p. Considering rotation curve to be flat, an exact solution of Einstein field equations (c = 1) is given by [5],
r > = B0rl,
e~*= С + K,
where B > 0, K are integration constants and
a = ■
4(1 +1) -12 2 +1
4
c = ■
2 +1
l = 2(v^ )2 = 10-6.
(2)
(3)
(4)
(5)
(6)
The static and approximately spherically symmetric gravitational field of a galaxy is represented by the space-time metric [6]:
ds1 = -е2Ф (r) dt2
dr2
2m(r)
+
--
+r2(d02 + sin2 0dф2),
(7)
which is completely determined by the two metric functions ®(r), m(r).
Comparing (1) and (7) and with the help of (2), we have,
Ф(г) =
log( Bo) +1 log(r) 2 '
(8)
and
m(r) =
Hi - < - K
a ra 2
(9)
The potentials obtained from rotation curve and lensing observations [7], ®RC(r) and <&lens(r), are given by
Фйс (r) =
ф(г) = log(+llog(r), (10)
2
r
®lens (r ) =
®(r ) +1 f m(r) dr = log(B0) +1 log(r ) 2 2f r2 r 4
K(2 +1)r 2+l + (-4 +1)l log(r)
(11)
4(-4 - 4/ +12) The mass which is inferred by rotation curve measurements mRC, is given by
Ir
mRC = rZ®'RC (r) = —.
(12)
Another pseudo-mass mlnn(f) obtained from lens-ing measurements has been defined as
miers (r ) =
rRC (r) , m(r)
2 2 r(-c + a(1 +1 - Kr)) 4a '
(13)
The differences between 0RC(r) and ®lem(r), mRC and m, are small for small and positive values of
lens i
K but the differences are big for large negative values of K (e.g. K = -1).
The first order approximations of Einstein's equations yield:
p(r) = ''ens (r) " m'RC (r) _
4nr
= rl-2-a) (-cra + a(ra - K) + a2K) (14) 8na
Pr (r ) + 2 pt (r ) = -
2(m'rc (r) - m'lens (r))
4nr2
r(-2-a) (cra - a2K + a((-1 +1)ra + K))
8na
. (15)
For small values of K (K = 10-12), pr + 2pt = 10"1p . But for large negative values of K (e.g., K = -1), we see not only negative pressure but also the density and pressure are of opposite sign and of same order. Defining the dimensionless quantity
Pr (r ) + 2 Pt (r )
ra(r ) = -
3p(r )
we have
(^(r ) _ 2 ^ RC (r ) ^ /ens
(r )
3 'lens (r ) _ ™ 'RC (r )
_ cra _ a2K + a((_1 + l)ra + K) _ 3(_xra + a(ra _ K) + a2K) '
Newtonian mass (M(r)) is given by
.2. r (a - c - ar ~aK )
M(r) = 4nfp(r)r2dr ■■
2a
(16)
(17)
and the mass in the first post Newtonian approximation is:
M
pN
t
(r ) = 4nf(p
+ Pr
!r
+ 2 pt )r 2dr = y. (18)
Note that MpN is an integrated quantity and comes from the equation of state and mRC = r20'RC(r) totally depends on pseudo potential O(r), but our results shows that MpN = mRC.
The conditions must be fulfilled for dark matter to be nonexotic: ra > 0,M > 0, p > 0. While counting it is set, that -10-18 < K < 1,33 10-12 — the condition must be satisfied for existence of nonexotic matter in halo (all the figures content K = 10-12).
It can be also noted that the difference of rotation curve potential (0RC(r)) and lensing potential (®lens(r)) is of order 10-6; the difference of rotation curve mass (mRC) and lensing mass (mlens) is of order 10-4, i.e. they are negligible in the region 100 < r < 500. The Newtonian mass (M(r)) and the mass in the first post Newtonian approximation MpN(r) are of same order. The pressure and density are also so. So we expect a Newtonian nature of the dark matter.
Assuming the known flat rotation curve condition from [5] obtained solution is as follows:
,V(r) _
= Bo r',
C0 + K A> rA
A) =
4 + 2l +1
2
C =
4+l 4
4+7 '
(19)
(20)
(21) (22)
> 1. So,
where K0 is an integration constant.
We know that for a valid metric the constant K0 have to satisfy the condition -100 < K0 < 0,00005.
In the braneworld regime, the rotation curve potential (Orc), lensing potential (Olens), rotation curve mass (inRC), lensing mass (inlens) are given by:
log( B0) +1 log(r)
(r ) = -
rn (r ) = -
1 -
2
C0 - K Ao rA
(23)
(24)
®RC (r) = ôir) = lQg( ^ +1 l0g(r (25) ) = +
2
1
+— 4
4+2l+t
'2 A
l(2 + l)log(r) + (4 +1)K0r 4+1 4 + 21 +12 4 + 21 +12
r
2
Models comparison in the dark matter problem
9
mRC
m
lens(r) 4
l (6 + 3l +12)r 4 + 2l +12
- *or
l (l+1) A
" 4+l
(27)
(28)
One can see that as like the perfect fluid, the difference of potentials (ORC and Olens) and two masses (m and in, ) is too small to take into account in the
v lens'
region 100 < r < 500, but it increases with r .
The density p(r), pressure ~t(r), Newtonian mass (M(r)), first post-Newtonian approximation of mass
(M (r)) are as follows:
P (r) =
8nr2
i (i+i) A
2 +1 (1 +1) K0 r
4+l
4 + 2l +12
4 +1
(29)
8nr2
P r (r) + 2 p t (r) = 2 +1 +12 (1 +1) K0 r-
i(i+i) \
4+l
4 + 2l +12
4 +1
(31)
_ (r) = Pr (r) + 2pt (r) = l x 3 p r(r) 3
2 3 l+ls+D
I2 _i_ 1 4+l
(8 + 6l + 5l2 + l3)r 4+l -K0(4 + 6l + 3l2 +13) (31)
i (i+l)
(8 + 6l + l2)r 4+l + K0(4 + 6l + 3l2 +13)
i i(i+i) A
mm (r) = 2
(2 +1)r K0r 4+l
4 + 2l +12
l
M pN (r) =
(32)
(35)
Conclusion
It is well known that observation of flat rotation curve suggests that a substantial amount of non-luminous dark matter is hidden in the galactic halo. Here we have found that the flat rotation curve suggests also the background geometry of the universe. Here is presented a comparison between two models.
l
Fig. 1. Plot of O, - O, vs. r shows the evidence Fig. 2. Plot of m, - m, vs. r shows the evidence
^ lens lens ^ lens lens
of small differences of lensing potential of small differences of lensing mass
Fig. 3. Plot of p,p vs. r shows differences Fig. 4. Plot of M,M(r) vs. r shows significant differences
of density of two models of Newtonian mass of two models
The solution (3) of equation (1) provides a static and approximately spherically symmetric solution. From the above discussions, we recommended that for the existence of non-exotic matter in the halo, the value of K have to satisfy the condition -10-18 < K < 1,33-10~n. We can say that cosmological constants play a crucial role for the determination of equation of state.
From the solution (20) of the equation (1) in a braneworld regime, we have the usual equation of state for radiation pr + 2pt = p for a suitable value of K0, while in a perfect fluid regime from the equation (3) we have the equation of state of "curvature fluid" p + 3p = 0 for a suitable value of K. Calculations indicate that the pressure in the braneworld model is much more effective than in the perfect fluid model .
Acknowledgement: One of us (Ramil Izmailov) was supported by the Ministry of Education and science of Russian Federation. This work was supported in part by an internal grant of M. Akmullah Bashkir State Pedagogical University on natural sciences field.
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