Open Universe Model: Description by Mathieu Functions

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by  A. M. Baranov

Gravitation & Cosmology, Vol. 8 (2002), Supplement II,pp. 12–13
Proceedingsofthe5thInernational ConferenceonGravitationandAstrophysicsofAsian-PacificCountries(ICGA-2001),
Moscow,1-7October2001
(cid:13)c 2002 RussianGravitational Society
OPEN UNIVERSE MODEL: DESCRIPTION BY MATHIEU FUNCTIONS 1
A.M.Baranov2 and D.A.Baranov3
Dep. of Theoretical Physics, Krasnoyarsk State University, 79 Svobodny Av., Krasnoyarsk, 660041, Russia
2 A model of the open Universe described by a conformally flat 4-metric is considered. The gravitational equations
1 withaperfectPascalfluidasasourcearereducedtothenonlinearequationofoscillations. Itisproposedtoconsider
0 this equation as the Mathieu equation encompassing some cosmological models. It is shown that the behaviour of
2 theUniversestate function is in qualitative accordance with theBig Bang scenario.

n
a
J
The interest in cosmological models does not fall in
1
2the recentdecades,in particular,due to multiple obser- dy ·(xdy −y)= y6 κε; (4.1)
vational data, which in accordance with perfecting the dx dx 12x3

]instruments and methods of observation are constantly
c d2y y5
qimproved. Accordingtotheacceptedmodernviews,the =− κp, (4.2)
dx2 4x4
-matter filling our Universe changed its state during the
grevolution: from a stage of physical vacuum to an ultra- where the new function y =eσ/2 and the new variable
[relativisticstageandfurther tothe Friedmannstage. It x=1/S have been introduced.

means that topical in the theory is a search for exact Thelastequationcanbereducedtoarepresentation
1
vsolutionsof the gravitationalequations with the matter of Newton’s second law
2equation of state depending on a space-time point.
3 Let us consider a model of the open Universe of d2y
=F (5)
4Friedmann type, which can be described by the con- dx2
4
.formally flat metric for the potential force F =−dU/dy and the potential
1
function U = Ω2(x)y2/2, if we define the right-hand
0 ds2 =e2σ(S)·δ dxµdxν =
2 µν side of Fq.(4.2) as the force F [1].
1 e2σ(S)(dt2−dx2−dy2−dz2) (1) Thus we have the differential equation of nonlinear
: oscillations
vwith δ =diag(1,−1,−1,−1);S2=t2−r2 is a square
i µν
Xof distance in the Minkowski world, r2 =x2+y2+z2. d2y
+Ω2(x)·y =0, (6)
r The Einstein gravitationalequations are written as dx2
a
G =−κT , (2) where the function Ω2(x) is the squared frequency.
αβ αβ
To solve the problem of finding a solution to this
with the Einstein tensor G and the energy-momen- equation it is suggested to reduce this equation to the
αβ
tum tensor (EMT) of the Pascal perfect fluid Mathieu equation, extending some special cases ob-
tained earlier,
T =ε·u u +p·b , (3)
αβ α β αβ
d2y
where ε is the energy density, p is the pressure, b = +B2[1+hcos(γx)]·y =0, (7)
αβ dx2
(u u − g ) is a projector onto the 3-space, g is
α β αβ αβ
the metric 4-tensor, u = exp(σ)·S is the 4-velosity, where B, h, γ are constants.
α ,α
If Ω2 = 0, i.e. the pressure is absent (p = 0),
Greek indices take the values 0,1,2,3. The velocity
weobtaintheFriedmannsolutionfortheopenUniverse
of light and the Newtonian gravitational constant are
filled with incoherent dust,
chosen to be equal to unity.
Substituting the metric (1) into Eqs.(2), we obtain
twodifferentialnonlinearequations(forthe energyden- yF =1−Ax, (8)
sity and pressure)
where A=const is determined the by modern value of
1Talkpresentedatthe5thInt. Conf. onGrav. andAstrophys. dust density in the Universe
ofAcian-PacificCountries(ICGA-5), Moscow,2001.
2e-mail: alex m [email protected] A
3e-mail: [email protected] κεdust ≈12S3. (9)
Open Universe Model: Description by Mathieu Functions 13
When Ω=B2 =const, we haveanaxactcosmolog- The numerical value of the parameter µ = 0.56
ical solution [2] for the open Universe filled with dust is found for Eq.(13) using a computational method of
and electromagnetic equilibrium radiation, a characteristic exponent from Hill’s determinant [3].
Other parameters may be selected so that we had the
yBS =p1+(A/B)2cos(Bx−α0), (10) qualitativepicture ofmodelbehaviourofthe Universe’s
functionofstateaccordingtotheBigBangscenario. At
where tgα =A/B,theconstantvalue B isconnected
0 thefirstinstants(whenwehaveacosmologicalsingular-
withthedensityofequilibriumcosmologicalelectromag-
ity) the Universe,being initially in the state of physical
netic radiation vacuum (S = 0,β ≈ −1), in an inflationary phase is
warmedupandreachestheradiationphase(β ≈+1/3)
B2
κεel−mag ≈12S4, (11) and then, being expanded and cooled, slowly evolves to
the modern epoch (β ≈0).
and the solution (10) at large times (S → ∞; x → 0)
passes through the Friedmann solution (8).
References
The Mathieu equation (7) may be rewritten in a
dimensionless form by introducing the variable ζ =
[1] A.M.Baranov and E.V.Saveljev, Izv. Vuzov (Fizika),
B/S =Bx:
No.1,pp.89-94(1994);RussianPhysicsJournal,37,No
7. pp. 640-644 (1994).
d2y/dζ2+[1+hcos(γζ/B)]·y =0. (12)
[2] A.M.Baranov, and E.V.Saveljev, Izv. Vuzov (Fizika),
For solutions of the Mathieu equation ther are reso- No.7, pp. 32-35 (1984); Russian Physics Journal, 27,
nance regions in accordance with the values of the pa- No 7. pp.569-572 (1984).
rameter γ = 2B/n, n = 1,2,... Here we shall choose [3] M.J.O.Strutt, ”Lamesche-Mathieusch und verwandte
the first resonance region (γ =2B). Then Eq.(12) will Funktionen in Physik und Tecknik”, State Sci/ Tech.
be rewritten as Press of Ukraine, Kiev-Kharkov,1935 (in Russian).
d2y/dζ2+[1+hcos(2ζ)]·y =0. (13)
In this case, according to Floquet’s theorem the so-
lution of the Mathieu equation may be substituted as
y =eµζce (ζ,h)−C·e−µζse (ζ,h), (14)
1 1
where µ is a chracteristic parameter, ce and se are
1 1
Mathieu’s functions having the form
ce1(ζ)=XAkcos(2k+1)ζ; (15.1)
k
se1(ζ)=XAksin(2k+1)ζ. (15.2)
k
The Mathieu functions, the signs and constant C =
µ+A/B arechosenheresothatinthetransitiontothe
modern epoch (S →∞; x →0) the solution (14) will
pass through the Friedmann solution (8), and at µ =0
and k =1 it will be the same as the solution (10).
Bymeansofparameterselectionitispossibletocon-
struct the physical behaviour of the function of state
p 1 x·y·d2y/dx2
β(x)= =− , (16)
ε 3(dy/dx)·(xdy/dx−y)
which is at each instant S an equation of state of the
Universe.
Itisclearthatthefirstapproximationofthesolution
(14) will be written as
y ≈eµζcosζ−C·e−µζsinζ
=eµB/Scos(B/S)−C·e−µB/Ssin(B/S). (17)