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Only D = 2 Fractal is compatible with Homogeneous Radiation

A. K. Mittal∗, Daksh Lohiya† Inter University Centre for Astronomy and Astrophysics, Postbag 4, Ganeshkhind, Pune 411 007, India

PACS. 98.80-k - Cosmology PACS. 98.65Dx - Superclusters; large-scale structure of the PACS. 05.45Df - Fractals

Abstract Homogeneous radiation is included in the Fractal Cosmological model obtained earlier by assuming an isotropic fractal cosmography, General Relativity and the Copernican Principle. It is shown that the only of distribution, compatible with a coexisting homogeneous distribution of radiation, is D =2.For D = 2, both matter and radiation make comparable contribution to Einstein’s equation in governing the dynamics of the scale factor at all epochs.

∗Department of Physics, University of Allahabad, Allahabad - 211 002, India; mittal [email protected] †Department of Physics and Astrophysics, University of Delhi, New Delhi–7, India; [email protected]

1 Recently a fractal cosmological model has been proposed [1] that follows naturally if one assumes (i) that matter in the Universe is distributed like an isotropic fractal of dimension 2, (ii) the General Theory of Relativity and (iii) the Conditional Cosmological Principle [2]. In a fractal Universe, density is not defined at any point. The concept of density has to be replaced by that of a “mass measure” defined over sets. The mass measure as obtained by any observer moving with the cosmological fluid (part of the fractal) will be the same as that obtained by any other observer. However, observers in a region of void are precluded, because spheres drawn with such points (not belonging to the fractal) as centre, will be empty with probability 1 [3]. It was shown in [1] that the –time component G00 of the Einstein tensor for any observer on an occupied point of the fractal (galaxy), must be of the form: 00 00 Gfractal(t, χ, θ, ϕ)=f(χ)GFRW(t)(1) 00 where Gfractal and f(χ) are to be regarded as ansatz to compute measures to satisfy the integrated Einstein’s equations over any 3-volume in the constant time hypersurface and:

a˙ 2 k 00 =3 + (2) GFRW 2 (a a ) is the G00 obtained from the FRW metric:

ds2 = dt2 + a2(t) dχ2 +Σ2(dθ2 +sin2 θdϕ2) (3) − { } where

K Σ sin χ if k K = +1 (positive spatial curvature) (4) ≡ ≡ | |

Σ χ if k K = 0 (zero spatial curvature) (5) ≡ ≡

K Σ sinh χ if k K = 1 (negative spatial curvature) (6) ≡ ≡ | | −

2 3 Let SP (R) denote a hypersphere of radius R centered at P on the hypersur- 3 face of constant time t, and assume MP (R), the mass enclosed in SP (R), to have a fractal dimension D. Then: C(t)RD if P the fractal MP (R)= ∈ (7) ( 0 otherwise The integrated Einstein’s eqns. give:

00 D 3 GFRW(t)=6C(t)a − (t)(8) and D D 3 f(χ)= Σ − (χ)(9) 3

We denote the “fractal mass density ” C(t)byCa, because the scale factor a is a function of time t. The ‘fractal mass density ” for scale factor a(t)is related to that at the present scale factor a0 by:

3 3 a (t)Ca = aoCao (10) Eqns (2), (8) and (10) yield:

2 3 a˙ k ao D 3 3 + =6 − (11) 2 3 Cao a (a a ) a Therefore: 2 D 4 a˙ = k + µa − (12) − 3 where µ =2aoCao . For D = 2, eqn(12) has solutions: µ1/4t1/2 1/2 for k =0 a(t)= t1/2 2µ1/2 t for k =+1 (13)  − 1/2  1/2  1/2  for k = 1 t 2µ + t −    These along with  2 f(χ)= (14) 3Σ(χ) describe the fractal cosmology completely for a universe in which matter is distributed as a fractal of dimension 2 and “fractal mass density” Cao when the scale factor is ao.

3 There is evidence to suggest [4] that matter distribution in the Universe could be a fractal of dimension 2. However, there is no evidence to suggest that radiation is not homogeneous to a very good accuracy. So let us include homogeneous radiation in our model described by radi- ation density ργ which scales with scale factor a(t)as:

4 4 ργa = ργ0 a0 (15) Then the integrated Einstein’s equations will yield:

χ 00 3 2 4πGFRW(t)a (t) f(χ)Σ (χ)Σ0(χ)dχ Z0 χ D D 3 2 =8π C(t)a (t)Σ (χ)+4πργ(t)a (t) Σ (χ)Σ0(χ)dχ { Z0 } 3 4 χ Ca0 a0 D D a0(t) 2 =8 ( )Σ ( )+4 Σ ( )Σ0( ) (16) π 3 a t χ πργ0 χ χ dχ { a (t) a(t) Z0 } This equation will have solutions only for two cases: (i) D = 3, the usual homogeneous matter distribution and (ii) D = 2, fractal matter distribution. Thus it is clear form equation (16) that only the D = 2 fractal cosmology model of [1] can incorporate a homogeneous distribution of radiation. This is precisely the dimension estimated from deepest 3-D catalogues [4]. In this case eqn(16) has the solution:

6C a3 +8πρ a4 G00 (t)= a0 0 γ0 0 (17) FRW a4(t) and

χ 3 4 3 2 Ca0 a0 2 4πργ0 a0 Σ (χ) f(χ)Σ (χ)dΣ= 3 4 Σ (χ)+ 3 4 0 3 +4 3 +4 3 Z Ca0 a0 πργ0 a0 Ca0 a0 πργ0 a0 × or 3 4 2 2 2Ca0 a0Σ(χ)Σ0(χ)+4πργ0 a0Σ (χ)Σ0(χ) f(χ)Σ (χ)Σ0(χ)= 3 4 (18) 3Ca0 a0 +4πργ0 a0 Where the numerator in (17) is determined by demanding: (i) that eqn(17) will reduce to eqn(11) if there is no radiation and to the standard radiation dominated equation if there is no matter distribution, and (ii) that G00 is linear in “fractal matter density” Ca0 and radiation density ργ0 .

4 Eqn(18) gives:

3 1 2Ca0 a0 4 f(χ)= 3 4 ( +4πργ0 a0) (19) 3Ca0 a0 +4πργ0 a0 Σ(χ) Defining λ 2C a3 + 8π ρ a4 µ + µ ; eqns(2) and (17) yield: ≡ a0 0 3 γ0 0 ≡ m r λ1/4t1/2 1/2 for k =0 a(t)= t1/2 2λ1/2 t for k =+1 (20)  − 1/2  1/2  1/2  for k = 1 t 2λ + t −    Eqns(19) and (20) completely describe the fractal cosmological model which contains matter distributed as a fractal of dimension 2 with “fractal mass density” Ca0 and radiation distributed homogeneously with density ργ0 when the scale factor is a0. The value of λ can be estimated from observations in the present universe.

From the number density versus scale length graph in [4] we infer Ca0 to be about 4 galaxies per (Mpc)2. Assuming a typical galaxy to be about 2 1011 16 2 × solar masses, we estimate µm 4 10 years .Thevalueofµr would depend on the number of relativistic≈ species× in the universe. At the current epoch the microwave background radiation of effective temperature 2.75 K gives a 16 2 contribution of 10 (years) to µr. Two species of neutrinoes, together with their antiparticles,≈ (at an effective temperature 1.9 K) would give a total contribution .42 1016 (years)2 at the present epoch. This contribution × gets enhanced to 1.75 1016 (years)2 at temperatures greater than 5 109 K. Finally, three species× of neutrinoes would have corresponding contribution× 16 2 16 2 .63 10 (years) and 2.63 10 (years) to µr at the current and the high temperature× epochs respectively.× This shows that unlike in standard model, both matter and radiation make comparable contribution to the Einstein’s equation in governing the dynamics of the Universe at all epochs. In this model there is neither matter nor radiation dominance in any era. In [5], fractal matter was treated as a perturbation in a radiation dom- inated universe. The fractal cosmology of [1], which follows naturally from first principles, incorporates fractal distribution of matter a priori. It is seen that “fractal matter density” has about as much influence on scale factor as the radiation density. This model may be treated as a first approximation to the observed uni- verse and is highly falsifiable. It can therefore prove to play a significant role

5 in the “fractal debate” [6] about whether the universe becomes homogeneous at suffitiently large scales.

Acknowledgements

We thank Inter University Centre for Astronomy and Astrophysics (IUCAA) for hospitality and facilities to carry out this research.

References

[1] A. K. Mittal and D. Lohiya, “From Fractal Cosmography to Fractal Cosmology” astro-ph/0104370.

[2] Mandelbrot, B. The Fractal Geometry of Nature. Freeman, San Fran- cisco, 1983.

[3] Mandelbrot, B.; in “Current topics in Astrofundamental Physics: Pri- mordial Cosmology”; eds. N. Sanchez, A. Zichichi NATO ASI series C- Vol. 511; Kluwer Acad. Pubs. (1998)

[4] S. F. Labini, M. Montuori, L. Pietronero: Phys. Rep.293, (1998) 66

[5] Joyce, M., Anderson, P. W., Montuori, M., Pietronero, L., Sylos Labini, F. astro-ph/0002504.

[6] Pietronero L., “The Fractal debate”; http: //www. phys. uniroma1. it/ DOCS/ PIL/ pil.html

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