Bjoern

2007-Jan-25, 09:00 PM

One often reads that the dark energy (cosmological constant) has an equation of state of the form

p = -rho,

where rho is the energy density, and p is the pressure. However, I don't understand entirely how it is derived. In the following, I will be referring specifically to the following paper:

The Cosmological Constant and Dark Energy, P. J. E. Peebles and Bharat Ratra, astro-ph/0207347

(It's a really nice paper, explaining both the basic concepts and the available evidence; I'd recommend it to anyone interested in DE.)

The crucial equations in that paper are (13), (14), (15), (18), and (19); see also the bottom half of page 8 for more details.

(19) says that the energy-momentum tensor for dark energy is given by

T^Lambda_munu = rho_Lambda g_munu

From (13) and (15), it is easy to conclude that g_munu (the well-known Robertson-Walker metric), is a diagonal matrix with the diagonal elements (for the simplest case of a flat universe, K=0):

(1, -a^2(t), -a^2(t), -a^2(t))

where a(t) is the scale factor. We thus should have that T^Lambda_munu also is a diagonal matrix with the diagonal elements (rho_Lambda, -a^2(t) rho_Lambda, -a^2(t) rho_Lambda, -a^2(t) rho_Lambda).

Comparing this to the usual form of the energy-momentum tensor for homogenously distributed, isotropic stuff, equation (18), we get:

rho = rho_Lambda (duh)

p = -a^2(t) rho_Lambda

Thus, the equation of state seems to be not p = -rho, but p = - a^2(t) rho!

I'm really confused by this; what am I doing wrong? The paper says that if one combines equations (14), (18) and (19), where (14) is the usual Minkowski metric, one gets p = -rho. That's obviously right - but why am I allowed in using the Minkowski metric instead of the Robertson-Walker metric here???

I suspect it has something to do with the remarks made directly below (14), i. e. that a freely falling inertial observer can always choose locally Minkowski coordinates. But after all, we are not asking what the energy-momentum tensor (and hence the equation of state for dark energy) look like locally, but we want to know that globally! (or don't we???)

Please help me out......... :(

p = -rho,

where rho is the energy density, and p is the pressure. However, I don't understand entirely how it is derived. In the following, I will be referring specifically to the following paper:

The Cosmological Constant and Dark Energy, P. J. E. Peebles and Bharat Ratra, astro-ph/0207347

(It's a really nice paper, explaining both the basic concepts and the available evidence; I'd recommend it to anyone interested in DE.)

The crucial equations in that paper are (13), (14), (15), (18), and (19); see also the bottom half of page 8 for more details.

(19) says that the energy-momentum tensor for dark energy is given by

T^Lambda_munu = rho_Lambda g_munu

From (13) and (15), it is easy to conclude that g_munu (the well-known Robertson-Walker metric), is a diagonal matrix with the diagonal elements (for the simplest case of a flat universe, K=0):

(1, -a^2(t), -a^2(t), -a^2(t))

where a(t) is the scale factor. We thus should have that T^Lambda_munu also is a diagonal matrix with the diagonal elements (rho_Lambda, -a^2(t) rho_Lambda, -a^2(t) rho_Lambda, -a^2(t) rho_Lambda).

Comparing this to the usual form of the energy-momentum tensor for homogenously distributed, isotropic stuff, equation (18), we get:

rho = rho_Lambda (duh)

p = -a^2(t) rho_Lambda

Thus, the equation of state seems to be not p = -rho, but p = - a^2(t) rho!

I'm really confused by this; what am I doing wrong? The paper says that if one combines equations (14), (18) and (19), where (14) is the usual Minkowski metric, one gets p = -rho. That's obviously right - but why am I allowed in using the Minkowski metric instead of the Robertson-Walker metric here???

I suspect it has something to do with the remarks made directly below (14), i. e. that a freely falling inertial observer can always choose locally Minkowski coordinates. But after all, we are not asking what the energy-momentum tensor (and hence the equation of state for dark energy) look like locally, but we want to know that globally! (or don't we???)

Please help me out......... :(