A Small Cosmological Constant from a Large Number of Extra Dimensions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

There is an important point in this paper which seems to be contradictory, and which needs a clear discussion. The point is the following.

The cosmological model considered here is based on the action (3) which, for the (4+n)-dimensional metric (1), and for a constant value of the scale factor b of the n internal dimensions, leads to the three field equations (11), (12), (13). Such equations, as explicitly stressed by the author, are all independent. One then needs an additional equation, describing the matter equation of state, in order to solve the system of four independent equations for the four variables a, rho, p, q. Up to this point I completely agree with the author.

As a subsequent step the author considers the equation describing the energy-momentum conservation law which follows from the previous field equations, and which takes the explicit form (17) for the case of a constant internal scale factor, b = cost. The equation (17) is then used to replace one of the previous field equations, in particular eq. (12) for the spatial components of the 3d metric. And this, again, is correct. But still we have three independent equations (11), (13), (17) to be solved, with the addition of an appropriate equation of state (14) or (18).

So, it turns out to be quite incomprehensible and unmotivated the statement of the author, who says: "We do not need to consider Eq. (13)  any more".
Indeed, even in the limiting case in which one would like to impose on the extra-dimensional pressure the condition q=0, one would obtain, from Eq. (13), an additional constraint on the 3d scale factor a, which has to be satisfied quite independently  from all other equations. And it is not clear at all whether or not such a constraint is imposed and satisfied by the model consider by the author.

I cannot recommend the publication of this paper until this point has not been correctly discussed and clarified.

Author Response

\emph{Comment}: {There is an important point in this paper which seems to be contradictory, and which needs a clear discussion. The point is the following.  
    The cosmological model considered here is based on the action (3) which, for the (4+n)-dimensional metric (1), and for a constant value of the scale factor b of the n internal dimensions, leads to the three field equations (11), (12), (13). Such equations, as explicitly stressed by the author, are all independent. One then needs an additional equation, describing the matter equation of state, in order to solve the system of four independent equations for the four variables a, rho, p, q. Up to this point I completely agree with the author.

As a subsequent step the author considers the equation describing the energy-momentum conservation law which follows from the previous field equations, and which takes the explicit form (17) for the case of a constant internal scale factor, b = cost. The equation (17) is then used to replace one of the previous field equations, in particular eq. (12) for the spatial components of the 3d metric. And this, again, is correct. But still we have three independent equations (11), (13), (17) to be solved, with the addition of an appropriate equation of state (14) or (18).

So, it turns out to be quite incomprehensible and unmotivated the statement of the author, who says: "We do not need to consider Eq. (13)  any more".
Indeed, even in the limiting case in which one would like to impose on the extra-dimensional pressure the condition q=0, one would obtain, from Eq. (13), an additional constraint on the 3d scale factor a, which has to be satisfied quite independently  from all other equations. And it is not clear at all whether or not such a constraint is imposed and satisfied by the model consider by the author.

I cannot recommend the publication of this paper until this point has not been correctly discussed and clarified.}
 
 
\emph{Answer}: We clarify this point as follows. In practice, Eq~(11), Eq.~(13),  Eq.~(17) plus the equation of state (18)
constitute the closed system of equations of motion. On the other hand, Eq~(11),  Eq.~(17) plus the equation of state (18) also
constitute a closed system of equations of motion from which the evolution of scale factor $a$, the energy density $\rho$ and the pressure $p$ can be obtained. Then substituting the expression of scale factor into Eq.(13), we obtain the expression of pressure $q$.  It is not allowed to impose on the extra-dimensional pressure $q$ the condition of $q=0$ because Eqs. (11,13,17,18) are already closed. Understanding this point in another way, a constraint on $q$ is equivalent to assuming the second equation of state. The first equation of state is  (18). One can not assume two equations of state. We have made clarification on this point in the revision in blue.    

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In the manuscript entitled ``A small cosmological constant from a large number of extra dimensions'' the author, Changjun Gao, proposes an explanation for the observed value of the cosmological constant based on the
Lovelock gravity contribution from curved extra dimensions. As the (total) cosmological constant gives rise to the curvature of the maximally-symmetric space-time (entire space), projection onto the three-dimensional
subspace would correspond to ``our'' cosmological constant (Lambda-term), so that the value of the Lambda-term could be traced to the specifics of the maximally-symmetric solution, and that is exactly what the
author is doing, with the number of Lovelock contributions and number of extra dimensions being among the parameters. However, this requires fine-tuning of the parameters, so under this fine-tuning correct value
of the Lambda-term is predicted but the cosmological constant problem is not truly solved.

Despite that, I found the manuscript to be of reasonable significance for publication. So that I would recommend acceptance after minor revision - once listed below points are taken into account:

- the author cites relevant publications on the cosmological dynamics in Lovelock gravity but since the spatial curvature of the extra dimensions is an important part of the problem under consideration, the author
should add citations to the appropriate papers (which focuses of the dynamics and/or properties of the solutions with spatial curvature of extra dimensions) with an additional paragraph within Introduction to describe them;

- I would say that requirement (30) is too strong - limitations from current accelerator experiments on Kaluza-Klein theories give much higher scales like 10^{-7} cm or similar. I would suggest either consider several values or a range and plot a graph(s) with resulting n and/or N depending on the scale.

Author Response

Comment: In the manuscript entitled ``A small cosmological constant from a large number of extra dimensions'' the author, Changjun Gao, proposes an explanation for the observed value of the cosmological constant based on the Lovelock gravity contribution from curved extra dimensions. As the (total) cosmological constant gives rise to the curvature of the maximally-symmetric space-time (entire space), projection onto the three-dimensional subspace would correspond to ``our'' cosmological constant (Lambda-term), so that the value of the Lambda-term could be traced to the specifics of the maximally-symmetric solution, and that is exactly what the author is doing, with the number of Lovelock contributions and number of extra dimensions being among the parameters. However, this requires fine-tuning of the parameters, so under this fine-tuning correct value of the Lambda-term is predicted but the cosmological constant problem is not truly solved.

Despite that, I found the manuscript to be of reasonable significance for publication. So that I would recommend acceptance after minor revision - once listed below points are taken into account:

\emph{Question One}:  the author cites relevant publications on the cosmological dynamics in Lovelock gravity but since the spatial curvature of the extra dimensions is an important part of the problem under consideration, the author should add citations to the appropriate papers (which focuses of the dynamics and/or properties of the solutions with spatial curvature of extra dimensions) with an additional paragraph within Introduction to describe them;

\emph{Answer}: We agree with the referee on this point. We have added a new paragraph in the revision and several references are added. The new paragraph are written in blue. 

 \emph{Question Two}: I would say that requirement (30) is too strong-limitations from current accelerator experiments on Kaluza-Klein theories give much higher scales like $10^{-7}$ cm or similar. I would suggest either consider several values or a range and plot a graph(s) with resulting $n$ and/or N depending on the scale.

\emph{Answer}: We also agree with the referee on this point. We have added a table in the revision and several values are calculated. The table is presented in blue.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

After the performed improvements the paper can be accepted for publication.