In the paper Backreaction in late-time cosmology by Thomas Buchert and Syksy Rasanen, Annual Review of Nuclear and Particle Science 62 (2012) 57-79, in eq .2.2 the covariant divergence:

$$\nabla_\alpha u^\alpha$$

is written as the volume expansion rate. Now just from dimensional analysis:

$$[\nabla_\alpha u^\alpha]=[\partial_\alpha u^\alpha]=[\frac{\partial}{\partial x^\alpha}\frac{dx^\alpha}{d\tau}]=[\frac{ d^2 x^\alpha}{d\tau^2}\frac{\partial \tau}{\partial x^\alpha}]=[\frac{a^\alpha}{u^\alpha}]$$

Using this the volume expansion rate of FLRW metric comes as:

$$\frac{\ddot a(t)}{\dot a(t)}$$

For all three spatial directions:

$$3\frac{\ddot a(t)}{\dot a(t)}$$

From Friedmann equation for a dust (pressureless) and spatially flat:

$$2\frac{\ddot a}{ a}+H^2=0$$

but for FLRW it is $3H(t)$. Am I missing something?

  • $\begingroup$ Okk from the continuity equation we find $$\nabla •v=\frac{\partial \rho}{\rho \partial t}$$ ,in this case $\rho$ is the mass per unit volume so ,$$\nabla•v=\frac{\partial M(t)}{M(t)\partial t}$$ ,and for 4-D spacetime it is the covariant divergence.is it?? $\endgroup$ Mar 7, 2019 at 5:35
  • $\begingroup$ Your equation $2\ddot{a}/a+H^2=0$ seems to me like it's either wrong or else you need more context to explain what you're talking about. In the Friedmann equations, $\ddot{a}/a$ is not generically predicted to be a constant. Do you have in mind some specific equation of state, such as a vacuum-dominated cosmology? $\endgroup$
    – user4552
    Mar 7, 2019 at 22:03
  • $\begingroup$ The equation is $$2\frac{\ddot a}{a}+(\frac{\dot a}{a})^2+\frac{k}{a^2}=-8\pi Gp$$ and for pressureless dust in a spatially flat universe it reduces to the above,it is obtained for FLRW metric from the Einstein's field equations.$$R_{\mu \nu}-\frac{Rg_{\mu \nu}}{2}=8\pi GT_{\mu \nu}$$ $\endgroup$ Mar 8, 2019 at 3:53
  • 1
    $\begingroup$ Rather than answering in comments, please edit your question to provide the relevant information. $\endgroup$
    – user4552
    Mar 8, 2019 at 14:03

1 Answer 1


Buchert's formalism is worked out in comoving coordinates $x$, in which we have that $u^\alpha=a\dot{y}^\alpha$ with $y$ the standard Eulerian spatial coordinates on the spatial hypersurface defined by $t=\text{constant}$. Incorporating this into your dimensional analysis should fix the problem. This is a matter of 'accounting', and seeing where the scale parameter $a$ pops up in the variables expressed in the comoving coordinates.

This is in line with their derivation. As from that Equation (2.2) in Buchert & Rasanen (2012) tells us directly $$ \nabla_\alpha u^\alpha = \frac{1}{3} {h_\alpha}^\alpha \Theta + {\omega_\alpha}^\alpha, $$ as the shear tensor $\sigma_{\alpha \beta}$ is traceless, and the vorticity tensor $\omega_{\alpha \beta}$ is dimensionless.


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