Upper and Lower Bounds on Gravitational Entropy

S. Weinberg, Relativity and Cosmology

1972

Earlier work this paper cites.

R.C. Tolman, Relativity, Thermodynamics and Cosmology

1987

Earlier work this paper cites.

L. Susskind, The World as a Hologram

1995

Earlier work this paper cites.

T.W. Kephart and Y.J. Ng, JCAP 0311

2003

Cited alongside, same era.

Some further references on cosmological entropy are listed here: F. Tipler, Nature 280,

Original

2003

Cited alongside, same era.

J.D. Bekenstein, Contemp. Phys. 45,

2005

Cited alongside, same era.

G. ’t Hooft, Dimensional Reduction in Quantum Gravity

Cited in the paper.

Taking the log of Eq. (6) and holding everything but p p fixed we can write log 10 ⁡ S = a + b ​ p \log_{10}S=a+bp or equivalently S = 10 a + b ​ p S=10^{a+bp} . Now requiring a a and b b be chosen to satisfies the two boundary conditions set by Eq. (10) gives the result. Note that p → ∞ p\rightarrow\infty corresponds to no additional gravitational entropy from the dark matter

Cited in the paper.

These bounds are implicit in the discussions of [ 7 ] but they were neither explored in the detail we have provided here, nor were they modeled. The lower bound was also pointed out in [ 8 ]

Cited in the paper.

No less an authority than Sir Roger Penrose has written about entropy and gravitational clumping. He has suggested that gravitational degrees of freedom are nonactive in a homogeneous Big Bang, but become active once clumping begins, i.e., as density perturbations become important. See R. Penrose, The Emperor’s New Mind

2006

Later among the works it cites.

L. Baum and P.H. Frampton, Turnaround in Cyclic Cosmology

2007

Closest in time.