Anisotropic String Cosmological Models in Heckmann-Suchuking Space-Time

0.22 MB·

by  G. K. Goswami

Anisotropic String Cosmological Models in Heckmann-Suchuking
Space-Time
G. K. Goswami1, R. N. Dewangan2, A. K. Yadav3 & A. Pradhan4
1,2Department of Mathematics, Kalyan P. G. College, Bhilai - 490006,India
Email: [email protected]
3Department of Physics, United College of Engineering & Research, Greater Noida - 201306,India
Email: [email protected]
4Department of Mathematics, GLA University, Mathura, India
Email: [email protected]
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1
0 Abstract
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Inthepresentwork wehavesearchedtheexistenceofthelatetimeacceleration oftheuniversewithstringfluid

n assourceofmatterin anisotropicHeckmann-Suchkingspace-timebyusing287highred shift(0.3≤z≤1.4) SNIa
a dataofobservedabsolutemagnitudealongwiththeirpossibleerrorfromUnion2.1compilation. Itisfoundthatthe
J best fit values for (Ωm)0, (ΩΛ)0, (Ωσ)0 and (q)0 are 0.2820, 0.7177, 0.0002 & -0.5793 respectively. Several physical

8 aspects and geometrical properties of themodel are discussed in detail.
2

] Key words: String, SN Ia data, Heckmann-Suchking space-time.
c
q
-
r
g 1 Introduction
[

2 SN Ia observations (Riess et al. 1998; Perlmutter et al. 1999) confirmed that the observable universe is undergoing
v an accelerated expansion. This acceleration is realized due to unknown cosmic fluid - dark energy (DE) which have
3
positive energy density andnegative pressure. So, it violatethe strong energycondition(SEC). The authors ofref. [3]
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confirmed that the violation of SEC gives anti gravitationaleffect that provides an elegant description of transition of
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6 universe fromdecelerationzone to accelerationzone. In the literature, cosmologicalconstantis the simplest candidate
0 to describe the present accelerationof universe (GrØn and Hervik 2007)but it suffers two problems - the fine tunning
. and cosmic coincidence problems (Carrollet al. 1992, Copeland et al. 2006). In the early of 21 century, some authors
1
0 (Copelandetal. 2006,Alametal. 2003)havedevelopedfewscalarfieldDEmodels,namelythephantom,quintessence
6 and k-essence rather than positive Λ. In the physical cosmology,the dynamical form of DE with an effective equation
1 ofstate(EOS),ω <−1,wereproposedinsteadofpositiveΛ(Komastuetal. 2009,Riessetal. 2004,Astieretal. 2006).
: 3
v
i The substantial theoretical progress in string theory has brought forth a diverse new generation of cosmological
X
models, some of which are subject to direct observational tests. Firstly the gravitational effect of cosmic strings are
r
a investigated by Letelier (1979, 1983). The present day observations of universe indicate the existence of large scale
network of strings in early universe (Kible, 1976,1980). Recently, Pradhan et al. (2007)and Yadav et al. (2009)have
studiedstringcosmologicalmodelinnon-homogeneousspace-timeinwhichgeometricstringswereconsideredassource
of matter. After the big bang, the universe may have undergone so many phase transitions as its temperature cooled
below some critical temperature as suggested by grand unified theories (Zel’dovich et al. 1975; Kibble 1976, 1980;
Everett 1981; Vilenkin 1981). At the very early stage of evolution of universe, the symmetry of universe was broken
spontaneouslythatrisessometopologically-stabledefectssuchasdomainwalls,stringsandmonopoles(Vilenkin1981,
1985). Among all the three cosmological structures, only strings give rise the density perturbations that leads the
formationofgalaxies. Pogosianetal. (2003,2006)havesuggestedthatthe cosmicstringsarenotresponsibleforCMB
fluctuations andobservedclusteringofgalaxies. In 2011,PradhanandAmirhashchi(2011)haveconsideredtime vary-
ing scale factor that generates a transitioning universe in Bianchi -V space-time. Yadav et al. (2011) and Bali (2008)
have obtained Bianchi-V string cosmological models in general relativity. The magnetized string cosmological models
are discussed by some authors (Chakraborty 1980; Tikekar and Patel 1992, 1994; Patel and Maharaj 1996; Singh and
Singh 1999; Saha and Visineusu 2008) in different physical contexts. Recently Goswami et al (2015) have studied
ΛCDM cosmologicalmodel in absence of string. In this paper we have established the existence of string cosmological
model in anisotropic Hecmann-Suchuking space-time.The organizationof the paper is as follows: The model and field
equations are presented in section 2. In Section 3, 4 & 5 we obtain the expressions for Hubble constant, Luminosity
1
distance and apparent magnitude. Section 6 deals the estimation of present values of energy parameters. Finally the
conclusions of the paper are presented in section 7.
2 The Model and Field equations
We consider a general Heckmann-Schucking metric
ds2 =c2dt2−A2dx2−B2dy2−C2dz2 (1)
where A, B and C are functions of time only.
The energy-momentum tensor for a cloud of massive strings and perfect fluid distribution is taken as
T =(p+ρ)u u −pg −λxixj (2)
ij i j ij
with
g uiuj =1 (3)
ij
and
xixi =−1,uix =0 (4)
i
where ui is the four velocity vector, ρ is the rest energy density of the system of strings, λ is the tension density of
the strings and xi is a unit space-like vector representing the direction of strings.
In co-moving co-ordinates,
uα =0,α=1,2or3. (5)
Choosing xi parallel to ∂/∂x, we have
xi =(A−1,0,0,0). (6)
If the particle density of the configuration is denoted by ρ , then
m
ρ=ρ +λ (7)
m
The Einstein field equations are
1 8πG
R − Rg +Λg =− T (8)
ij 2 ij ij c4 ij
Choosing co-moving coordinates,the field equations (8) in terms of line element (1) can be written as
B C B C 8πG
44 + 44 + 4 4 =− (p−λ)+Λc2 (9)
B C BC c2
A C A C 8πG
44 + 44 + 4 4 =− p+Λc2 (10)
A C AC c2
A B A B 8πG
44 + 44 + 4 4 =− p+Λc2 (11)
A C AB c2
A B B C C A 8πG
4 4 + 4 4 + 4 4 = ρ+Λc2 (12)
AB BC AC c2
where A ,B and C stand for time derivatives of A,B,C respectively. the mass-energy conservation equation Tij =0
4 4 4 ;j
gives
A B C A
ρ +(p+ρ)( 4 + 4 + 4)−λ 4 =0 (13)
4
A B C A
Where A ,B and C stand for time derivatives of A,B and C respectively. Subtracting (10) from (9),(11) from (10)
4 4 4
and (11) from (9) we get
A B A C B C 8πG
44 − 44 + 4 4 − 4 4 =− λ (14)
A B AC BC c2
B C A B A C
44 − 44 + 4 4 − 4 4 =0 (15)
B C AB AC
C A B C A B 8πG
44 − 44 + 4 4 − 4 4 = λ (16)
C A BC AB c2
2
Subtracting(16) from (14), we get
B C 2B C A A C A B 16πG
44 44 4 4 44 4 4 4 4
+ + =2 + + + λ
B C BC A AC AB c2
This equation can be re-written in the following form
(BC) (BC) 2 A A2 A (BC) 16πG
4 + 4 =2 4 +2 4 + 4 4 + λ
BC BC A A2 ABC c2
(cid:18) (cid:19)4 (cid:18) (cid:19) (cid:18) (cid:19)4
This is simplified as
(BC) A (BC) A (ABC) 16πG
4 −2 4 + 4 −2 4 4 = λ (17)
BC A BC A ABC c2
(cid:18) (cid:19)4 (cid:18) (cid:19)
This equation suggests the following relation ship among A B and C
An =BC (18)
where n6=2, otherwise λ would be zero. We can assume
B =An/2D&C =An/2D−1,D =D(t) (19)
Further integrating equation (15), we get the first integral
D K
4
= (20)
D An+1
Where K is an arbitrary constant of integration. Equation(13) simplifies as
A A
ρ +(n+1) 4(p+ρ)−λ 4 =0
4
A A
Using equation(7), this becomes
A A
4 4
(ρ ) +(n+1) (ρ +p)+λ +n λ=0 (21)
m 4 m 4
A A
If matter and string tension co-exist without much interaction , this equation splits into two parts.
A
4
(ρ ) +(n+1) (ρ +p)=0 (22)
m 4 m
A
&
A
4
λ +n λ=0 (23)
4
A
This gives following expression for matter density ρ and string tension λ for p=0(dust filled universe)
m
(ρ ) (A)n+1
ρ = m 0 0 (24)
m An+1
and
(λ) (A)n
λ= 0 0 (25)
An
The Hubble’s constant in this model is
1 A B C (n+1)A
H =ui = ( 4 + 4 + 4)= 4 (26)
;i 3 A B C 3 A
Equations (9)-(12) and (20) are simplified as
n+2A n2A2 8πG Λc4 K2c2
44 + 4 =− p− + (27)
2 A 4 A2 c2 8πG 8πGA2(n+1)
(cid:16) (cid:17)
A A2 16πG λ
44 +n 4 = (28)
A A2 c2 n−2
3
n(n+4)A2 K2 8πG
4 = + ρ+Λc2 (29)
4 A2 A2(n+1) c2
We see from equation(28) that λ=0 for n=2 and other equations convert to the Einstein’s field equations of LRS
Bianchi type I model for perfect fluid without string
The average scale factor ’a’ of LRS Bianchi type I model is defined as
a=(ABC)1/3 =An+31 (30)
The spatial volume ’V’ is given by
V =a3 =ABC =An+1 (31)
Equations(27)-(29) can be re-written as
n(n+4) A n−1A2 8πG λ K2c2 Λc4
44 + 4 =− p− + − (32)
2(n+1) A 2 A2 c2 n+1 8πGA2(n+1) 8πG
(cid:16) (cid:17) (cid:16) (cid:17)
4(n+1)2 8πG Λc4 K2c2
H2 = ρ +λ+ + (33)
9n(n+4) c2 m 8πG 8πGA2(n+1)
(cid:16) (cid:17)
Wenowassumethatthestringtensionλ,cosmologicalconstantΛandthetermduetoanisotropyalsoactlikeenergies
with densities and pressures as
Λc4 K2c2
ρ =λ ρ = ρ =
λ Λ 8πG σ 8πGA2(n+1)
λ Λc4 K2c2
p =− p =− p =
λ n+1 Λ 8πG σ 8πGA2(n+1)
(34)
It can be easily verified that energy conservation laws (22) & (23) holds separately for ρ , ρ & ρ
Λ σ λ
(ρ ) +3H(p +ρ )=0, (ρ ) +3H(p +ρ )=0 & (ρ ) +3H(p +ρ )=0
Λ 4 Λ Λ σ 4 σ σ λ 4 λ λ
(35)
The equations of state for matter, λ, σ and Λ energies are as follows
p =ω ρ
m m m
1 1
p =ω ρ ∵p =− ρ ∴ω =−
λ λ λ λ λ λ
n+1 n+1
p =ω ρ ∵p +ρ =0∴ω =−1
Λ Λ Λ Λ Λ Λ
∵p =ρ ∴ω =1
σ σ σ
(36)
where ω =0 for matter in form of dust, ω = 1 for matter in form of radiation. There are certain more values of ω
m m 3 m
for matter in different forms during the coarse of evolutionof the universe. Now we use the following relationbetween
scale factor a and red shift z
a0 A0 n+1
=( ) 3 =1+z (37)
a A
The suffix(0) is meant for the value at present time.The energy density ρ comprises of following components
ρ= ρ +ρ +ρ +ρ (38)
m λ Λ σ
Integrations of energy equations(35) yield (cid:0) (cid:1)
ρ =(ρ ) (1+z)3(1+ωi) ∴ρ= (ρ ) (1+z)3(1+ωi)
i i 0 i 0
i
X
(39)
where suffix i corresponds to various energies densities. We write equations (32) and (33) as
n(n+4) A n−1A2 8πG
44 + 4 =− p +p +p +p (40)
2(n+1) A 2 A2 c2 m λ Λ σ
(cid:16) (cid:17) (cid:16) (cid:17)
4(n+1)2 8πG
H2 = ρ +ρ +ρ +ρ (41)
9n(n+4) c2 m λ Λ σ
(cid:0) (cid:1)
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3 Dust filled universe
The present stage of the universe is full of dust for which pressure p =0. We define critical density ρ as
m c
9n(n+4)c2H2
ρ = (42)
c 32(n+1)2πG
Then (41) gives
Ω +Ω +Ω +Ω =1 (43)
m λ Λ σ
Where
Ω = ρm = (Ωm)0H02(1+z)3,Ω = ρλ = (Ωλ)0H02(1+z)n3+n1
m ρ H2 λ ρ H2
c c
ρ (Ω ) H2 ρ (Ω ) H2(1+z)6
Ω = Λ = Λ 0 0,Ω = σ = σ 0 0
Λ ρ H2 σ ρ H2
c c
(44)
3.1 Expression for Hubble’s Constant
Adding all the three above and using (43), we get expression for Hubble’s constant
H2 = H2 (Ω ) (1+z)3(1+ωi)
0 i 0
i
X
= H02 (Ωm)0(1+z)3+(Ωλ)0(1+z)n3+n1 +(Ωσ)0(1+z)6+(ΩΛ)0
A A A
= H2(cid:2)(Ω ) ( 0)n+1(Ω ) ( 0)n+(Ω ) ( 0)2(n+1)+(Ω ) (cid:3)
0 m 0 A λ 0 A σ 0 A Λ 0
(cid:2) (cid:3)
(45)
Now we present two graphs on the basis of above equation(see fig. 1 & fig. 2).
These figures show that Hubble’s constant increases over red shift. We can say that Hubble’s constant decreases
over time. We also notice from fig. 1 that in the case lambda dominated universe ((Ω ) =0 or .2820 ) Hubble’s
m 0
constant vary slowly over time in comparison to matter dominated universe(Ω =0).
Λ
4 Luminosity Distance verses Red Shift Relation
If x be the spatially coordinate distance of a source with red shift z from us, the luminosity distance which determines
flux of the source is given by
D =A x(1+z) (46)
L 0
Geodesic for metric (1) ensures that if in the beginning
dy dz
=0and =0
ds ds
then
d2y d2z
=0and =0
ds2 ds2
So if a particle moves along x- direction, it continues to move along x- direction always.If we assume that line of sight
of a vantage galaxy from us is along x-direction then path of photons traveling through it satisfies
ds2 =c2dt2−A2dx2 =0
From this we obtain
x t0 dt 1 z dz
x = dx= =
ˆ0 ˆt A(t) A0H0 ˆ0 (1+z)nn−+12h(z)
1 z dz
=
A0H0 ˆ0 (1+z)nn−+12 (Ωm)0(1+z)3+(Ωλ)0(1+z)n3+n1 +(Ωσ)0(1+z)6+(ΩΛ)0
q
(cid:2) (cid:3)
5
(47)
Where we have used dt=dz/z˙ and from eqns (26) and (37)
H
z˙ =−H(1+z) & h(z)=
H
0
.
So the luminosity distance is given by
c(1+z) z dz
D = (48)
L H0 ˆ0 (1+z)nn−+12 (Ωm)0(1+z)3+(Ωλ)0(1+z)n3+n1 +(Ωσ)0(1+z)6+(ΩΛ)0
q
(cid:2) (cid:3)
5 Apparent Magnitude verses Red Shift relation for Type Ia supernova’s(SN
Ia):
The Absolute and Apparent magnitude of a source (M and m) are related to the red shift of the source by following
relation
D
m−M =5log L +25 (49)
10
Mpc
(cid:0) (cid:1)
The Type Ia supernova (SN Ia) can be observed when white dwarf stars exceed the mass of the Chandrasekhar limit
and explode. The belief is that SN Ia areformedin the samewayirrespective ofwhere they are inthe universe,which
means thatthey haveacommonabsolute magnitudeMindependent ofthe redshiftz. Thus they canbe treatedasan
ideal standardcandle. We canmeasure the apparentmagnitude m and the red shift z observationally,which of course
depends upon the objects we observe.
TogetabsoluteMagnitudeMofasupernova,weconsidersupernovaatverysmallredshift. Letusconsiderasupernova
1992P at low-red shift z = 0.026 with m = 16.08.For low red shift supernova,we have the following relation
cz
D = (50)
L
H
0
Putting values z = 0.026, m = 16.08 and D from (50),
L
Equation (49) gives M for all SN Ia as follows.
H
M =5log 0 −8.92 (51)
10
.026c
(cid:0) (cid:1)
From equations (49) and (51)
log (H D )=(m−16.08)/5+log (.026cMpc) (52)
10 0 L 10
Theequation(52)givesobservedvalueofLuminositydistanceintermofobservedvalueofapparentmagnitudewhereas
the equation (48) gives the theoretically value of luminosity distance in terms of red shift on the basis of our model.
Next, from equations (52) and (48) we get the following expression for (m,z) relations of Supernova’s SN Ia.
1+z
m=16.08+5log ×
10
.026
(cid:16) (cid:17)
z dz
(53)
ˆ0 (1+z)nn−+12 (Ωm)0(1+z)3+(Ωλ)0(1+z)n3+n1 +(Ωσ)0(1+z)6+(ΩΛ)0
q
(cid:2) (cid:3)
6 Estimation of Present values of Energy parameters (Ω ) ,(Ω ) ,(Ω ) & (Ω )
m 0 Λ 0 λ 0 σ 0
Stringtensionplayimportantroleinthestudyofearlyevolutionoftheuniverse. Wealsonoticethatafterpublicationof
WMAPdata,todaythereisconsiderableevidenceinsupportofanisotropicmodelofuniverse. Inthepast,theremight
had been certain anisotropies in the universe . Our model validates that both string tension as well as anisotropy
decreases over time. Considering it, we give very low present values to (Ω ) and (Ω ) . In our earlier work we
σ 0 λ 0
developed models with n = 2 which make string tension λ=0. So we take
n=2.1,(Ω ) =.0002&(Ω ) =.0001 (54)
σ 0 λ 0
6
inourpresentmodel. NowwepresenttwoTables(SeeAppendix-1and2)whichcontain287highredshift(0.3≤z ≤1.4
) SN Ia supernova data of observedapparentmagnitude and luminosity distances along with their possible error from
Union 2.1 compilation. These Tables also contains the corresponding theoretical values of apparent magnitudes and
luminosity distances for Ω = 0.2820,0 and 1 respectively which are obtained as per from Equation(48)and (52). As
m
statedintheintroduction,ourpurposeistoobtaintheresultsclosetoWMAPonthebasisofunion2.1compilationsfor
our model, so we have obtained the various sets of theoreticaldata’s of apparentmagnitudes and luminosity distances
correspondingtodifferentvaluesofΩ inbetween0to1. Inordertoseethatoutofthethesesetsoftheoreticaldata’s
m
of apparent magnitudes and luminosity distances which one is close to the observational values, we calculate χ2 using
following formula due to Amanullah et al.(2010). The tables[1]and [2] describe the variousvalues of χ2 againstvalues
of Ω ranging in between 0 to 1.
m
χ2 = A− B2 +log (C )
SN C 10 2π
Where
A = 60 [(m)ob−(m)th]2
σ2
i=1 i
P
B = 60 [((m)ob−((m)th]
σ2
i=1 i
P
And
60
C = 1
σ2
i=1 i
P
(55)
(Ω χ2 χ2 /287
m)0 SN SN
0 5462.2 19.03205575
0.1 4972.8 17.32682927
0.2 4822.8 16.80418118
0.21 4816.5 16.78222997
0.26 4800.2 16.72543554
0.27 4799.4 16.72264808
0.28 4799.3 16.72229965
0.28 4799.3 16.72229965
0.281 4799.3 16.72229965
0.282 4799.3 16.72229965
0.283 4799.4 16.72264808
0.29 4799.8 16.72404181
0.3 4800.9 16.72787456
0.9997 5312.2 18.50940767
Table 1 : χ2 table for best fitting theoretical and observed values of apparent magnitudes‘m’
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(Ω χ2 χ2 /287
m)0 SN SN
0 223.1983 0.777694425
0.1 203.6246 0.70949338
0.2 197.6239 0.688585017
0.21 197.3738 0.687713589
0.26 196.7195 0.685433798
0.27 196.6878 0.685323345
0.28 196.6836 0.685308711
0.281 196.6836 0.685308711
0.282 196.6836 0.685308711
0.283 196.6874 0.685321951
0.29 196.7049 0.685382927
0.3 196.7499 0.685539721
0.9997 217.2009 0.756797561
Table 2 : χ2 table for best fitting theoretical and observed values of luminosity distanceDL
The above tables depict the fact that the best fit values of Ω and therefore Ω are as follows
m Λ
Ω =0.2820 & Ω =0.7177 (56)
m Λ
6.1 Deceleration Parameter
The deceleration parameter is given by
aa n−2 3 AA
q =− 44 =− − 44
a2 n+1 n+1 A2
4 4
From equations (39)-(53)
AA n+1 n−1
44 = − ω Ω −
A2 2 i i 2
4 i
X
n+1 1 n−1
= − (−Ω − Ω +Ω )−
Λ λ σ
2 n+1 2
∴q = 3 (−(ΩΛ)0− n+11(Ωλ)0(1+z)n3+n1 +(Ωσ)0(1+z)6) + 1
2 (Ωm)0(1+z)3+(Ωλ)0(1+z)n3+n1 +(Ωσ)0(1+z)6+(ΩΛ)0 2
(cid:2) (cid:3) (57)
At z =0.7170, q =−0.0004 & at z =0.7180, q =.0002
So the universe entered the accelerating phase at z ∼ 0.7175⋍ t ∼ 0.4616H−1 ∼ 6.27×109yrs in the past before as
0
from now.
The present value of q is
q =−0.5793
0
This equation clearly show that without presence of Λ term in the Einstein’s field equation(8), one can’t imagine of
acceleratinguniverse.Thisequationalsoexpressesthefactthatanisotropyraisesthelowerlimitvalueof(Ω ) required
Λ 0
for acceleration at present. This may be seen in the following way.
For isotropic model(FRW) acceleration requires
(Ω ) ≥.33 (58)
Λ 0
For anisotropic model with out string
(Ω ) ≥.33+(Ω ) =.3302 (59)
Λ 0 σ 0
For anisotropic model with string
(Ω ) ≥.3559+(Ω ) −.0123(Ω ) =.3360 (60)
Λ 0 σ 0 λ 0
8
6.2 Age of the Universe
The present age of the universe is obtained as follows
t0 ∞ dz
t = dt=
0 ˆ ˆ H(1+z)
0 0
∞
dz
=
ˆ0 H0(1+z) (Ωm)0(1+z)3+(Ωλ)0(1+z)n3+n1 +(Ωσ)0(1+z)6+(ΩΛ)0
∞ q(cid:2) 3dA (cid:3)
=
ˆ
0 (n+1)H A (Ω ) (A0)(n+1)+(Ω ) (A0)n+(Ω ) (A0)2(n+1)+(Ω )
0 m 0 A λ 0 A σ 0 A Λ 0
q
(cid:2) (cid:3)
(61)
Where we have used dt=dz/z˙ and z˙ =−H(1+z)
We see that at large red-shift, H t becomes constant and it tends to the value 0.9505. So the Present age of the
0
universe as per our model is
t =0.9505H−1=13.2799×109years (62)
0 0
6.3 Densities of the various Energies in the universe :
The energy density is given by (39)
ρ =(Ω ) (ρ ) (1+z)3(1+ωi)
i i 0 c 0
∴ρ= (Ω )(ρ ) (1+z)3(1+ωi)
i c 0
i
X
(63)
whereistandsforvarioustypesofenergiessuchasmatterenergy,darkenergy,energiesduetoanisotropyoftheuniverse
and string tension. Taking,n=2.1,Ω =0.2820,Ω =0.7177,Ω = 0.0002&Ω =0.0001 , H =72km/sec./Mpc,h ⋍
m Λ σ λ 0 0
.72 G=6.6720e−008cm3/sec2/gm
9n(n+4)c2H2
(ρ ) = 0 =1.8795h2×10−29gm/cm3 (64)
c 0 32(n+1)2πG 0
The dust energy and its current value for flat universe is given as
(ρ )=(Ω ) (ρ ) (1+z)3, & (ρ ) =.5262h2×10−29gm/cm3 (65)
m m 0 c 0 m 0 0
dark energy (ρ ) and its current value is given as
Λ
(ρ )=(ρ ) =1.35268h2×10−29gm/cm3 (66)
Λ Λ 0 0
The string tension λ and its current value is given as
λ=(Ω ) (ρ ) (1+z)2.0323 & (λ) =(ρ ) =0.00018h2×10−29gm/cm3 (67)
λ 0 c 0 0 λ 0 0
And finally anisotropy ρ and its current value is given as
σ
(ρ )=(Ω ) (ρ ) (1+z)6 & (ρ ) =0.00036h2×10−29gm/cm3 (68)
σ σ 0 c 0 σ 0 0
9
45 40
Ωm=0.282,ΩΛ=0.7177,Ωσ=0.0002 & Ωλ=0.0001 Ωm=0.2820,ΩΛ=0.7177,Ωσ=0.0002 & Ωλ=0.0001
40 Ωm=0.9997,ΩΛ=0.0000,Ωσ=0.0002 & Ωλ=0.0001 35 Ωm=0.9997,ΩΛ=0.0000,Ωσ=0.0002 & Ωλ=0.0001
35 Ωm=0,ΩΛ=0.9997,Ωσ=0.0002 & Ωλ=0.0001 Ωm=0,ΩΛ=0.9997,Ωσ=0.0002 & Ωλ=0.0001
30
H030 H0
H/ H/ 25
Hubble Constant 122505 Hubble Constant 1250
10
10
5 5
0 0
0 2 4 6 8 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Red shift z Scale factor A/A
0
Observed datas
27 3.5 Ωm=0.2820,ΩΛ=0.7177,Ωσ=0.0002 & Ωλ=0.0001
Ωm=0.0000,ΩΛ=0.9997,Ωσ=0.0002 & Ωλ=0.0001
26 3 Ωm=0.9997,ΩΛ=0.0000,Ωσ=0.0002 & Ωλ=0.0001
Apparent magnitude m222345 minosity Distance HD/c0L12..255
Observed datas Lu 1
22 Ωm=0.2820,Ωσ=0.7177,Ωσ=0.0002 & Ωλ=0.0001
Ωm=0.0000,ΩΛ=0.9997,Ωσ=0.0002 & Ωλ=0.0001 0.5
Ωm=0.9997,ΩΛ=0.0000,Ωσ=0.0002 & Ωλ=0.0001
21 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Red shift z Red shift z
2 1.8
1.6
1.5
1.4
meter q 1 1.2
Deceleration para 0.05 Time H t000..168

−0.5 ΩΩmm==00..29890907,,ΩΩΛΛ==00..70109070,,ΩΩσσ==00..00000022 && ΩΩλλ==00..00000011 00..24 ΩΩmm==00..29892907,,ΩΩΛΛ==00..70107070,,ΩΩσσ==00..00000022 && ΩΩλλ==00..00000011
Ωm=0.0000,ΩΛ=0.9997,Ωσ=0.0002 & Ωλ=0.0001 Ωm=0.0000,ΩΛ=0.9997,Ωσ=0.0002 & Ωλ=0.0001
−1 0
0 20 40 60 80 100 0 2 4 6 8 10
Red shift z Red shift z
x 104
1.8 3
Ωm=0.2820,ΩΛ=0.7197,Ωσ=0.0002 & Ωλ=0.0001 Ωm=0.2820,ΩΛ=0.7177,Ωσ=0.0002 & Ωλ=0.0001
1.6 Ωm=0,ΩΛ=0.9997,Ωσ=0.0002 & Ωλ=0.0001
1.4 Ωm=0.9997,ΩΛ=0,Ωσ=0.0002 & Ωλ=0.0001 2.5
1.2 2
σ
Time H t00.18 Shear scalar 1.5
0.6 1
0.4
0.5
0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100
scale factor A/A red shift z
0
Figure 1: Variation of H , m, H0DL, q, H t and σ with redshift.
H0 c 0
10