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Linear Stability of Einstein and de Sitter Universes in the Quadratic Theory of Modified Gravity
by Mudhahir AlAjmi
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Mudhahir AlAjmi 
Submission information  

Preprint Link:  scipost_202010_00001v2 (pdf) 
Date accepted:  20210430 
Date submitted:  20210322 17:52 
Submitted by:  AlAjmi, Mudhahir 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider the Einstein static and the de Sitter universe solutions and examine their instabilities in a subclass of quadratic modified theories for gravity. This modification proposed by Nash is an attempt to generalize general relativity. Interestingly, we discover that the Einstein static universe is unstable in the context of the modified gravity. In contrast to Einstein static universe, the de Sitter universe remains stable under metric perturbation up to the second order.
Author comments upon resubmission
Dear Editor, Thank you very much for your valuable comments and remarks. I tried hard to make the paper up to the level of your satisfaction. List of the changes are explained below: 1. The typos mentioned are corrected plus others, 2. The change has been made from "nonisotroposity" to "anisotropy" as suggested. 3. The term "arrive at" is used instead of "yield to" as suggested thankfully. 4. The word "redeuced" to "reduced". 5. Equation mentioned (eq 28) has been removed to clear our to confusion that it may cause. 6. Equation 43 is corrected. We have done a thorough revision of equations as follows: a. Equations in the previous versions were too long and somewhat boring. The equations (starting from Equation 6) are written in a shorter format and the repeated terms are written separately (equation 9, 17, 21, 2729 in the updated version). b. In Section 6 in equation 27 (in the previous version, equation 24 in the updated version), there was a typo of h as mentioned by the referees. This also has been corrected. This leads to formulating the subsequent equations again accordingly (equations 2538). 7. The parameters A_1, A_2, C_1, C_2, E_1, E_2 are rewritten again (as explained in point 6b above). The constant A,B,C A_1, A_2, C_1, C_2, E_1, E_2 are replaced by Greek letters instead to avoid the confusion with the earlier notations used for the scale factors. An example of our claim is given in a subsection. The above mentioned parameters are rewritten again because of the reason mentioned in 6b. 8. I rechecked equation 58. They seem to be OK. We think that, after substituting the perturbation terms and because we are omitting the higher order perturbation terms, somewhat the mentioned symmetry in the equation has been lost specially we can see that the Lagrangian (eq 5) is not highly symmetric. 9. The conclusion is modified and few sentences are added as future works. The updated version of the file is attached. Thank you, again, for your fruitful comments which led, undoubtfully, to the improvement of the search. Mudhahir AlAjmi
List of changes
1. The typos mentioned are corrected plus others,
2. The change has been made from "nonisotroposity" to "anisotropy" as suggested.
3. The term "arrive at" is used instead of "yield to" as suggested thankfully.
4. The word "redeuced" to "reduced".
5. Equation mentioned (eq 28) has been removed to clear our to confusion that it may cause.
6. Equation 43 is corrected. We have done a thorough revision of equations as follows: a. Equations in the previous versions were too long and somewhat boring. The equations (starting from Equation 6) are written in a shorter format and the repeated terms are written separately (equation 9, 17, 21, 2729 in the updated version). b. In Section 6 in equation 27 (in the previous version, equation 24 in the updated version), there was a typo of h as mentioned by the referees. This also has been corrected. This leads to formulating the subsequent equations again accordingly (equations 2538).
7. The parameters A_1, A_2, C_1, C_2, E_1, E_2 are rewritten again (as explained in point 6b above). The constant A,B,C A_1, A_2, C_1, C_2, E_1, E_2 are replaced by Greek letters instead to avoid the confusion with the earlier notations used for the scale factors. An example of our claim is given in a subsection. The above mentioned parameters are rewritten again because of the reason mentioned in 6b.
8. I rechecked equation 58. They seem to be OK. We think that, after substituting the perturbation terms and because we are omitting the higher order perturbation terms, somewhat the mentioned symmetry in the equation has been lost specially we can see that the Lagrangian (eq 5) is not highly symmetric.
9. The conclusion is modified and few sentences are added as future works. The updated version of the file is attached. Thank you, again, for your fruitful comments which led, undoubtfully, to the improvement of the search.
Published as SciPost Phys. Proc. 4, 012 (2021)
Reports on this Submission
Anonymous Report 1 on 2021414 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202010_00001v2, delivered 20210414, doi: 10.21468/SciPost.Report.2792
Report
The author has addressed the main concerns raised in the first submission and in particular provided a more detailed discussion of the requirements for the fluctuation exponents to be negative, including an example as requested. The conclusions were slightly modified accordingly to indicate that stability in the de Sitter case depends on certain choices of constants (which perhaps can be further investigated in the future).
In addition, several typos were corrected as requested and many clarifications were made.
Therefore the submission can now be published in SciPost Physics Proceedings.