Cmjv02i01p0061 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. Vol.2 (1), 61-67 (2011)

Mathematical Solution ooff Friedmann Equations Determining tthe Time Evolution and the Expansion of Universe R. K. DUBEY*, AJAI KUMAR TRIPATHI1 and ANAND SHANKAR DUBEY2 *Associate Professor Govt. P.G. College Teonther, M.P. India email: rkdubey2004@yahoo.co.in 1 Department of Mathematics Govt. P. G. College Shahdol, M.P. India email: aktarmy@rediffmail.com 2 Department of Mathematics Govt.P.G. College Shahdol, M.P. India email: mail: anand1277_au@rediffmail.com ABSTRACT The time evolution of the universe from the Big bang until today is described ribed by the general relativity i.e. by Einsteinâ&#x20AC;&#x2122;s gravitational field equations. In this paper the mathematical solution of Friedmann equation is presented which relate the scale radius R (t), energy density and the pressure P for flat, open and closed universes. This paper will describe scale radius R (t) by a parameter which define expansion of universe in the flat model, open model and closed model. Key Words Words: cosmology, matterdeceleration parameter, scale radius.

1. INTRODUCTION It seems established that at the epoch the universe expands with acceleration .it follows directly from the observation of high red-shift shift supernovae1 and indirectly from the measurement of angular fluctuation of cosmic microwave background(CMBR)2.The knowledge of the function R(t) is essential for the determining the rate of expansion of the universe and the other ther physical properties of the expanding universe .In order to find the solution for R(t) , we need a theory of cosmic dynamics

dominated

universe,

based on the gravitational field equation. We therefore combine the homogeneous isotropic Robertson-Walker Walker metric with the gravitational itational field equations to obtain the dynamics equations satisfied by the scale radius R (t). These equations are called Friedmann equations. The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, iso empirically this is justified on scales larger than 100 Mpc. The cosmological principal implies that the metric of the universe must be of the form

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169 169)

R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (1), 61-67 (2011)

ds2 = R (t) 2 ds32 – dt2 Where ds32 is a three dimensional metric that must be one of (i) Flat space (ii) A sphere of constant positive curvature (iii) Hyperbolic space with constant negative curvature The parameter k takes the values 0, 1,-1 in these cases respectively. The Friedmann equations for the matterdominated epoch of the universe as follows.

&& R& 2 + kc 2 R 2 + =0 R R2 R& 2 + kc 2 8π Gρ 0 R03 = 3 R2 R3 2.

H2+

8π Gρ 0 R kc = 2 3 R R

(2)

kc 2 8π Gρ 0 = 3 R02

8 π Gρ 0 H 02 k = − 2 R02 3c 2 c 8π G (ρ 0 − ρ c ) , = 3c 2

(5)

From Friedmann equations (A) and (B) and equation (5) We obtain

4π Gρ 0 R03 3H 2 R 3

q0 = 2.1

ρ 4π G ρ 0 = 0 2 2ρ c 3H 0

(6)

(7)

The Flat Model (k=0)

For the flat model with k=0 equation (3) gives ρ 0 = ρ c and equation (7) we find q0 =

At the present epoch t= t0 with R=R0 and H=H0 equation (2) becomes

H 02 +

&&(t ) R(t ) && R 1 R =− 2 2 R& (t ) H R

(B)

(1)

3 0 3

q (t) = −

The present epoch value of the deceleration parameter is given by

Substituting (1) in to (B) we obtain 2

Let us also recall the definition of deceleration parameter q(t) given as

(A)

In order to solve equations (A) and (B) let us recall the Hubble constant given by

R& (t ) R(t )

(4)

q (t ) =

SOLUTION OF THE FRIEDMANN EQUATIONS

H (t) =

3H 02 ρc = 8π G

1 Using ρ 0 = ρ c and equation 2

(4) can be written as

H 02 =

8 π Gρ 0 3

(8)

Friedmann equation (B) becomes

8π Gρ 0 R03 H 02 3 2 & R = = R0 3 R R (3)

Where ρc is the critical matter density of the universe defined by

A2 or R& 2 = R

, where A = H 0 R03 / 2

Equation (9) can be rewritten as

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

(9)

R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (1), 61-67 (2011)

dR R& = = A R −1 / 2 dt ∫ R dR = A ∫ dt + c

B = 2 q0

At R (0) = 0, c = 0

 3A    2 

R(t) = 

2/3

t 2/3

(14)

 R   R0

  

H 02 R03 c2

H 02 1 = R02 c2

3/ 2

(11)

At present epoch equation (11)

(15)

Furthermore using equation (3) for k=1 We may write

(10)

2 3/ 2 R 3A

2 t= 3H 0

H 02 R03 c2

Using the definition of the present – epoch deceleration parameter (7), we obtain

2 3/2 = At + c R 3 So

4 π Gρ 0 3H 02

 4 π Gρ 0  2 − 1  2  3H 0  

H 02 1 = 2 (2q0 − 1 ) R02 c

(16)

becomes

t0 =

2 3H 0

(12)

R0 =

c H0

(2q0 − 1 ) −1 / 2

(17)

The solution (10) for the scale radius R (t) with the choice of the curvature constant k=0 is some time called the Einstein –de sitter solution.

Substituting equation (17) into (15) we obtain

2.2

Equation (13) may be rewritten in the form

The Closed Model (k=1)

For the closed model with k=1 equation (3) gives ρ 0 > ρ c and from equation (7) we find

q0 >

1 The Friedman 2

equation (B) then becomes

8π Gρ 0 R03 R& 2 +1= R c2 3c 2 2 & R B +1= 2 c R Where

2 q0 c . 3/ 2 ( 2q0 − 1) H0

B−R R

1 dR = c dt

∫

(18)

dt =

∫

R dR B − R

Let us now introduce an angular parameter η as follows (13)

8π G ρ 0 3 R0 3c 2

R = B sin2 dR = B sin

η 2

= B / 2 (1 − cosη ) cos

η 2

dη

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

R. K. Dubey,, et al., J. Comp. & Math. Sci. Vol.2 (1), 61-67 (2011)

ct =

∫

B sin 2

∫

B cos 2

sin

= B = B/2

cos

∫

sin

∫

η 2

Taking q0 = 1

B sin

cos

η 2

dη

η η

B sin

η 2

cos

η 2

dη

From equation (17) R0 =

2c H0

c H0

It mean that in the closed model for q0 =1 the present universe will expand to twice its present size before it starts contracting. 2.3 The Open Model (k= -1) 1)

η dη 2

(1 − cos η ) d η

2π H0

From equation (18) we have B =

2 2

We have t L =

= B/2 (η − sin η )

For the open model with k= -1 equation (3) gives ρ 0 < ρ c and from

We get two parameter equations R = B/2(1 – cos η) dη, ct = B/2 (η - sinη) (19)

equation (7) we find q0 <

Equation (19) define the scale radius R(t) . The function R (t) defined above is a cycloid. The universe starts expansion at the cosmic time t=0 (η = 0) with scale radius R=0.and expands to a maximum size R=B at the cosmic time t= /2 (η = π). The time in the closed model is called life span of the universe. Taking η = 2π in equation (19)

tL =

B (2π − sin 2π ) 2c

tL =

2 π q0 1 . 3/ 2 (2q 0 − 1) H0

( From equation (18))

The Friedmann equation equation (B) becomes

R& 2 c2

−1 =

8 π Gρ 0 R03 R 3c2

B R

R& 2 2

B +1 R

B+R R

c R& 2 c

B = 2π ) 2c t L = Bπ / c

1 . 2

 dR  2  B+R   =c    dt   R  dR B+R =c dt R 2

(20)

c dt =

R B+R

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169 169)

R. K. Dubey,, et al., J. Comp. & Math. Sci. Vol.2 (1), 61-67 (2011)

On integrating

∫

c dt =

∫

R dR B+R

(21)

Taking an angular parameter η as

R = B sin h 2 R=

η 2

B (cos hη − 1) 2

dR = B sin h

(22)

cos h

η 2

dη

B sinh 2

R 2 dR = η B+R B + B sinh 2 2 B sinh

η 2

sinh 2

1 + sinh

= B sinh =

cosh

η 2

2 2

B sinh

Ω=

η 2

cosh

η 2

dη

2 dη

From equation (21) we get

B (sinh η −η ) 2

kc 2 = (23) i.e.

And from (22)

B ( cos hη −1) 2

ρ ρc

(25)

Which is just the ratio of the mean density of the universe to the critical density. Evidently, Ω = 1 if the density of matter in the universe is exactly equal to the critical density. Notice from equation (3) that the curvature k of the universe can be related to the value of the density parameter at some som reference time (say, at the present time, Ω0),

B (cosh η −1) dη 2

ct =

3H 02 8π G

Of course, the actual density of the universe may be greater than or less than the critical density and so it is useful to define a density parameter

dη

The function R (t) defined by (24) is growing indefinitely as t → ∞ (η → ∞) .Like the flat Einstein –de de Sitter solution ,the open solution continues to expand forever .it should also be noted that the expansion of the universe in the open model is faster than that of the flat model because of the exponential function in the parameter . The future evolution therefore depends on the mean density of the universe. When the universe has the critical density

ρc =

From equation (21)

(24)

8 π GR02 (ρ − ρ ) , o c 3

kc 2 = (Ω0 − 1) H 02 R02

(26)

This is an important equation because it links the geometry of the universe (via the curvature k)to the density of the universe.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169 169)

66 If

R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (1), 61-67 (2011)

Ω0 < 1, k < 0, and the universe has negative curvature, (an open universe). Ω0 = 1, k = 0, and the universe is spatially flat, (Einstein-de sitter universe).

Ω <1

Ω0 > 1, k > 0, and the universe has positive curvature, (closed universe). The solution for R (t) for above three cases are sketched below

(open)

Ω =1

(Einstein de sitter )

R(t)

Ω > 1 (closed)

Big Bang

Big Crunch

Figure: Schematic evolution of the scale radius in closed, open and Einstein de-Sitter universes.

CONCLUDING REMARKS In this paper we have described appropriate solutions for the scale radius R(t) for the different values of the curvature constant k .The curvature constant play an important role in theoretical models describing the expansion of the universe. The scale radius R (t) with choice of the curvature constant k=0 is called the Einstein de-sitter solution, for k=1 the scale radius define a cycloid and the scale radius is a cyclic function of the cosmic time. The time tL in the closed model is called the lifespan of the present universe. We obtained the present scale radius is equal to one half of

the maximum scale radius. It means that in the closed model for q0=1, the present universe will expand to twice its present size before it starts contracting. The function R (t) for k= -1 is growing indefinitely as t → ∞ . The expansion in the open model is faster than of flat model. The analysis of the three possible models for the expansion of the universe presented above does not resolve the question whether the present universe is open or closed. The answer to that question must be looked for in astronomical observations and estimates of the various parameters of the model.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

R. K. Dubey, et al., J. Comp. & Math. Sci. Vol.2 (1), 61-67 (2011)

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Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)