Analytical Scaling Solutions for the Evolution of Cosmic Domain Walls in a Parameter-Free Velocity-Dependent One-Scale Model

Round 1

Reviewer 1 Report

The authors discuss  evolution of the characteristic length and velocity distribution width of frictionless domain wall network in FRW universe in their VOS model.  They choose a power-law evolution for the scale factor.  I agree with the overall analysis and results.  I think, however, the paper will be better formed if the following points are discussed:

1) The paper assumes power-law growth throughout. But, it would be good to comment also on the exponential phase. What aspect would be most affected if the growth were exponential? 

2) Conclusion can be expanded by commenting on future applications (including condensed matter and biological systems).

3) The paper gives no mention of the domain wall tension. This is an important point as it can cause cosmological problems depending on the scenario used. It needs be discussed for spherical and cylindirical cases comparatively. 

Author Response

Dear Editor and Reviewers,

Please find enclosed the revised version of the manuscript "Analytical scaling solutions for the evolution of cosmic domain walls in a parameter-free velocity-dependent one-scale model" (symmetry-1850122) by P.P. Avelino, David Gruber and Lara Sousa which we now resubmit for publication in Symmetry. We thank the three reviewers for the positive reports and for valuable comments and suggestions. We have made changes to the manuscript to accommodate them, and comment on the issues raised by the reviewers 1 and 3 below.

___reviewer 1___

The points made by Referee 1 (queries Q1 to Q3, in order of appearance in the referee report, are addressed in answers A1 to A3).

Q1. The paper assumes power-law growth throughout. But, it would be good to comment also on the exponential phase. What aspect would be most affected if the growth were exponential?

A1. When we have an exponential growth of the scale factor, the network is no longer able to attain a linear scaling regime. As a matter of fact, the domain wall network is frozen in comoving coordinates, experiencing a regime during which the characteristic length grows proportionally to the scale factor and the velocity rapidly goes to zero. The network then is rapidly diluted as a result of the accelerated expansion --- with \zeta=0 and \sigma_v=0 --- and there is no energy loss as a result of domain wall collapse. This may be seen as the limiting case of the non-relativistic regime (\lambda\to 1) described in the manuscript.

Q2. Conclusion can be expanded by commenting on future applications (including condensed matter and biological systems).

A2. We have added one sentence at the end of the conclusions following the recommendation made by the referee. 

Q3. The paper gives no mention of the domain wall tension. This is an important point as it can cause cosmological problems depending on the scenario used. It needs be discussed for spherical and cylindrical cases comparatively.

A3. We have added a few sentences discussing current bounds on the energy scale of domain wall formation at the end of the first paragraph of the introduction (which may be translated into bounds on domain wall tension, since \sigma~\eta^3). Moreover, in the non-relativistic limit the cosmological evolution of the domain wall network is independent on the shape of the domains. This results in similar constraints on domain wall tension for spherical and cylindrical non-relativistic domain walls for \lambda \ge 0.5 (the most interesting regime from a cosmological point of view).

___reviewer 3___

The points made by Referee 1 (queries Q1 to Q2, in order of appearance in the referee report, are addressed in answers A1 to A2).

Q1. Their analytical estimate is consistent with numerical findings for two limiting cases λ \to 1 and λ \to 0. Authors should give relevant examples of the cosmological models that represent a(t) \propto t^0 and a(t) \propto t^1 with proper references. This is important since the authors have claimed the usefulness of their analytical calculation in future studies of the cosmological consequences of domain wall networks.

A1. Although we have derived an analytical approximation for the non-relativistic regime and an exact solution for \lambda=0, these were used to construct a fit (Eq. (34)) to the solutions of the parameter-free model for any \lambda such that 0<\lambda<1 that provides an excellent description of the numerical results of the model. Thus, our results may be used to describe the predictions of the parameter-free VOS model not just in the two limiting cases mentioned. In any case, the \lambda \to 1 limit may be used to describe the evolution towards dark-energy domination, which is very relevant in cosmology.

Q2. The author should explicitly mention the differences obtained in their analytical calculation with the numerical one for radiation, and matter dominated Universe since these two special cases are essential part of the standard Big bang cosmology.

A2. As pointed out in the previous answer, the analytical fit derived in this manuscript provides an excellent fit to the numerical results of the parameter-free model for all \lambda, including the radiation- and matter-dominated eras. However, although the predictions of the parameter-free VOS model provide an excellent description of field theory simulations in the non-relativistic limit, there are discrepancies in the relativistic limit. Note however that the origin of these differences is currently not clear: the results of numerical simulations in this limit may be affected by a number of systematic errors. For instance, simulations, in order to maintain resolution of the domain walls, artificially enforce a fixed comoving thickness which affects the dynamics of the scalar radiation emitted by the domain walls and can have an (indirect) impact on domain wall evolution; also, the accuracy of the methods used to determine the characteristic length and RMS velocity may be compromised by the copious amounts of scalar radiation emitted in the relativistic limit (just to name a few). As mentioned in the introduction, until these issues are clarified, there is no way of saying whether simulations or theoretical predictions provide the best description of the evolution of realistic domain wall networks.

Best Regards,

Pedro Avelino
David Gruber
Lara Sousa

Reviewer 2 Report

The authors consider analytical approximation for the formula describing the linear scaling evolution of the characteristic length L and the root-mean-squared velocity $\sigma_v$ of standard frictionless domain wall network in FRW universe.

They provides original approximated method to calculate those values of domain wall network using a recent parameter-free velocity dependent one-scale model. Their method is compared with other methods like the one using exact solution with $\lambda =0$ and numerical computation; $\lambda$ is parameter in scale factor evolution in FRW universe $a \propt t^{\lambda}$. It is then found that their approximation is good for $\lambda \sim 1$.

Thus their results provide a new way to calculate the values of domain wall network approximately. It can help to simplify the calculation and to get results easier especially for $\lambda \sim 1$. The results are appropriately summarized with relevant references and I think it can be published.

 

Author Response

Dear Editor and Reviewers,

Please find enclosed the revised version of the manuscript "Analytical scaling solutions for the evolution of cosmic domain walls in a parameter-free velocity-dependent one-scale model" (symmetry-1850122) by P.P. Avelino, David Gruber and Lara Sousa which we now resubmit for publication in Symmetry. We thank the three reviewers for the positive reports and for valuable comments and suggestions. We have made changes to the manuscript to accommodate them, and comment on the issues raised by the reviewers 1 and 3 below.

___reviewer 1___

The points made by Referee 1 (queries Q1 to Q3, in order of appearance in the referee report, are addressed in answers A1 to A3).

Q1. The paper assumes power-law growth throughout. But, it would be good to comment also on the exponential phase. What aspect would be most affected if the growth were exponential?

A1. When we have an exponential growth of the scale factor, the network is no longer able to attain a linear scaling regime. As a matter of fact, the domain wall network is frozen in comoving coordinates, experiencing a regime during which the characteristic length grows proportionally to the scale factor and the velocity rapidly goes to zero. The network then is rapidly diluted as a result of the accelerated expansion --- with \zeta=0 and \sigma_v=0 --- and there is no energy loss as a result of domain wall collapse. This may be seen as the limiting case of the non-relativistic regime (\lambda\to 1) described in the manuscript.

Q2. Conclusion can be expanded by commenting on future applications (including condensed matter and biological systems).

A2. We have added one sentence at the end of the conclusions following the recommendation made by the referee. 

Q3. The paper gives no mention of the domain wall tension. This is an important point as it can cause cosmological problems depending on the scenario used. It needs be discussed for spherical and cylindrical cases comparatively.

A3. We have added a few sentences discussing current bounds on the energy scale of domain wall formation at the end of the first paragraph of the introduction (which may be translated into bounds on domain wall tension, since \sigma~\eta^3). Moreover, in the non-relativistic limit the cosmological evolution of the domain wall network is independent on the shape of the domains. This results in similar constraints on domain wall tension for spherical and cylindrical non-relativistic domain walls for \lambda \ge 0.5 (the most interesting regime from a cosmological point of view).

___reviewer 3___

The points made by Referee 1 (queries Q1 to Q2, in order of appearance in the referee report, are addressed in answers A1 to A2).

Q1. Their analytical estimate is consistent with numerical findings for two limiting cases λ \to 1 and λ \to 0. Authors should give relevant examples of the cosmological models that represent a(t) \propto t^0 and a(t) \propto t^1 with proper references. This is important since the authors have claimed the usefulness of their analytical calculation in future studies of the cosmological consequences of domain wall networks.

A1. Although we have derived an analytical approximation for the non-relativistic regime and an exact solution for \lambda=0, these were used to construct a fit (Eq. (34)) to the solutions of the parameter-free model for any \lambda such that 0<\lambda<1 that provides an excellent description of the numerical results of the model. Thus, our results may be used to describe the predictions of the parameter-free VOS model not just in the two limiting cases mentioned. In any case, the \lambda \to 1 limit may be used to describe the evolution towards dark-energy domination, which is very relevant in cosmology.

Q2. The author should explicitly mention the differences obtained in their analytical calculation with the numerical one for radiation, and matter dominated Universe since these two special cases are essential part of the standard Big bang cosmology.

A2. As pointed out in the previous answer, the analytical fit derived in this manuscript provides an excellent fit to the numerical results of the parameter-free model for all \lambda, including the radiation- and matter-dominated eras. However, although the predictions of the parameter-free VOS model provide an excellent description of field theory simulations in the non-relativistic limit, there are discrepancies in the relativistic limit. Note however that the origin of these differences is currently not clear: the results of numerical simulations in this limit may be affected by a number of systematic errors. For instance, simulations, in order to maintain resolution of the domain walls, artificially enforce a fixed comoving thickness which affects the dynamics of the scalar radiation emitted by the domain walls and can have an (indirect) impact on domain wall evolution; also, the accuracy of the methods used to determine the characteristic length and RMS velocity may be compromised by the copious amounts of scalar radiation emitted in the relativistic limit (just to name a few). As mentioned in the introduction, until these issues are clarified, there is no way of saying whether simulations or theoretical predictions provide the best description of the evolution of realistic domain wall networks.

Best Regards,

Pedro Avelino
David Gruber
Lara Sousa

Reviewer 3 Report

Please address the couple of points while preparing the updated manuscript.

Comments for author File: Comments.pdf

Author Response

Dear Editor and Reviewers,

Please find enclosed the revised version of the manuscript "Analytical scaling solutions for the evolution of cosmic domain walls in a parameter-free velocity-dependent one-scale model" (symmetry-1850122) by P.P. Avelino, David Gruber and Lara Sousa which we now resubmit for publication in Symmetry. We thank the three reviewers for the positive reports and for valuable comments and suggestions. We have made changes to the manuscript to accommodate them, and comment on the issues raised by the reviewers 1 and 3 below.

___reviewer 1___

The points made by Referee 1 (queries Q1 to Q3, in order of appearance in the referee report, are addressed in answers A1 to A3).

Q1. The paper assumes power-law growth throughout. But, it would be good to comment also on the exponential phase. What aspect would be most affected if the growth were exponential?

A1. When we have an exponential growth of the scale factor, the network is no longer able to attain a linear scaling regime. As a matter of fact, the domain wall network is frozen in comoving coordinates, experiencing a regime during which the characteristic length grows proportionally to the scale factor and the velocity rapidly goes to zero. The network then is rapidly diluted as a result of the accelerated expansion --- with \zeta=0 and \sigma_v=0 --- and there is no energy loss as a result of domain wall collapse. This may be seen as the limiting case of the non-relativistic regime (\lambda\to 1) described in the manuscript.

Q2. Conclusion can be expanded by commenting on future applications (including condensed matter and biological systems).

A2. We have added one sentence at the end of the conclusions following the recommendation made by the referee. 

Q3. The paper gives no mention of the domain wall tension. This is an important point as it can cause cosmological problems depending on the scenario used. It needs be discussed for spherical and cylindrical cases comparatively.

A3. We have added a few sentences discussing current bounds on the energy scale of domain wall formation at the end of the first paragraph of the introduction (which may be translated into bounds on domain wall tension, since \sigma~\eta^3). Moreover, in the non-relativistic limit the cosmological evolution of the domain wall network is independent on the shape of the domains. This results in similar constraints on domain wall tension for spherical and cylindrical non-relativistic domain walls for \lambda \ge 0.5 (the most interesting regime from a cosmological point of view).

___reviewer 3___

The points made by Referee 1 (queries Q1 to Q2, in order of appearance in the referee report, are addressed in answers A1 to A2).

Q1. Their analytical estimate is consistent with numerical findings for two limiting cases λ \to 1 and λ \to 0. Authors should give relevant examples of the cosmological models that represent a(t) \propto t^0 and a(t) \propto t^1 with proper references. This is important since the authors have claimed the usefulness of their analytical calculation in future studies of the cosmological consequences of domain wall networks.

A1. Although we have derived an analytical approximation for the non-relativistic regime and an exact solution for \lambda=0, these were used to construct a fit (Eq. (34)) to the solutions of the parameter-free model for any \lambda such that 0<\lambda<1 that provides an excellent description of the numerical results of the model. Thus, our results may be used to describe the predictions of the parameter-free VOS model not just in the two limiting cases mentioned. In any case, the \lambda \to 1 limit may be used to describe the evolution towards dark-energy domination, which is very relevant in cosmology.

Q2. The author should explicitly mention the differences obtained in their analytical calculation with the numerical one for radiation, and matter dominated Universe since these two special cases are essential part of the standard Big bang cosmology.

A2. As pointed out in the previous answer, the analytical fit derived in this manuscript provides an excellent fit to the numerical results of the parameter-free model for all \lambda, including the radiation- and matter-dominated eras. However, although the predictions of the parameter-free VOS model provide an excellent description of field theory simulations in the non-relativistic limit, there are discrepancies in the relativistic limit. Note however that the origin of these differences is currently not clear: the results of numerical simulations in this limit may be affected by a number of systematic errors. For instance, simulations, in order to maintain resolution of the domain walls, artificially enforce a fixed comoving thickness which affects the dynamics of the scalar radiation emitted by the domain walls and can have an (indirect) impact on domain wall evolution; also, the accuracy of the methods used to determine the characteristic length and RMS velocity may be compromised by the copious amounts of scalar radiation emitted in the relativistic limit (just to name a few). As mentioned in the introduction, until these issues are clarified, there is no way of saying whether simulations or theoretical predictions provide the best description of the evolution of realistic domain wall networks.

Best Regards,

Pedro Avelino
David Gruber
Lara Sousa

Round 2

Reviewer 1 Report

The authors properly addressed the points raised by me. I recommend publication of this manuscript in this revised form.