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Clustering of galaxies in brane world models

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Abstract

In this paper, we analyze the clustering of galaxies using a modified Newtonian potential. This modification of the Newtonian potential occurs due to the existence of extra dimensions in brane world models. We will analyze a system of galaxies interacting with each other through this modified Newtonian potential. The partition function for this system of galaxies will be calculated, and this partition function will be used to calculate the free energy of this system of galaxies. The entropy and the chemical potential for this system will also be calculated. We will derive explicit expression for the clustering parameter for this system. This parameter will determine the behavior of this system, and we will be able to express various thermodynamic quantities using this clustering parameter. Thus, we will be able to explicitly analyze the effect that modifying the Newtonian potential can have on the clustering of galaxies. We also analyse the effect of extra dimensions on the two-point functions between galaxies.

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References

  1. Randall, L., Sundrum, R.: Phys. Rev. Lett. 83, 4690 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  2. Randall, L., Sundrum, R.: Phys. Rev. Lett. 83, 3370 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  3. Arkani-Hamed, N., Schmaltz, M.: Phys. Rev. D 61, 033005 (2000)

    Article  ADS  Google Scholar 

  4. Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: Phys. Rev. D 59, 086004 (1999)

    Article  ADS  Google Scholar 

  5. Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: Phys. Lett. B 436, 257 (1998)

    Article  ADS  Google Scholar 

  6. Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: Phys. Lett. B 429, 263 (1998)

    Article  ADS  Google Scholar 

  7. Floratos, E.G., Leontaris, G.K.: Phys. Lett. B 465, 95 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  8. Long, J.C., Chan, H.W., Churnside, A.B., Gulbis, E.A., Varney, M.C.M., Price, J.C.: Nature 421, 922 (2003)

    Article  ADS  Google Scholar 

  9. Bronnikov, K.A., Kononogov, S.A., Melnikov, V.N.: Gen. Relativ. Gravit. 38, 1215 (2006)

    Article  ADS  Google Scholar 

  10. Nicolini, P.: Phys. Rev. D 82, 044030 (2010)

    Article  ADS  Google Scholar 

  11. Gregory, C.M., Pinzul, A.: Phys. Rev. D 88, 064030 (2013)

    Article  ADS  Google Scholar 

  12. Ali, A.F., Tawfik, A.: Adv. High Energy Phys. 2013, 126528 (2013)

    MathSciNet  Google Scholar 

  13. Setare, M.R., Momeni, D., Myrzakulov, R.: Phys. Scripta 85, 065007 (2012)

    Article  ADS  Google Scholar 

  14. Nojiri, S., Odintsov, S.D.: Phys. Rev. D 77, 026007 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  15. Verlinde, E.P.: JHEP 1104, 029 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  16. Majumder, B.: Adv. High Energy Phys. 2013, 296836 (2013)

    MathSciNet  Google Scholar 

  17. Milgrom, M.: Astrophys. J. 270, 365 (1983)

    Article  ADS  Google Scholar 

  18. Milgrom, M.: Astrophys. J. 270, 384 (1983)

    Article  ADS  Google Scholar 

  19. Moffat, J.W., Toth, V.T.: Phys. Rev. D 91(4), 043004 (2015)

    Article  ADS  Google Scholar 

  20. Roshan, M., Abbassi, S.: Astrophys. J. 802(1), 9 (2015)

    Article  ADS  Google Scholar 

  21. Saslaw, W.C.: Gravitational Physics of Stellar and Galactic Systems. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  22. Ahmad, F., Hameeda, M.: Astrophys. Space Sci. 330, 227 (2010)

    Article  ADS  Google Scholar 

  23. Saslaw, W.C., Hamilton, A.J.S.: Astrophys. J. 276, 13 (1984)

    Article  ADS  Google Scholar 

  24. Ahmad, F., Saslaw, W.C., Bhat, N.I.: Astrophys. J. 571, 576 (2002)

    Article  ADS  Google Scholar 

  25. Ahmad, F., Saslaw, W.C., Malik, M.A.: Astrophys. J. 645, 940 (2006)

    Article  ADS  Google Scholar 

  26. Ahmad, F., Malik, M.A., Hameeda, M.: Astrophys. Space Sci. 343, 763766 (2013)

    Article  Google Scholar 

  27. Ahmad, F., Malik, M.A., Hameeda, M.: MNRAS 438, 438 (2014)

    Article  ADS  Google Scholar 

  28. Saslaw, W.C.: The Distribution of the Galaxies Gravitational Clustering in Cosmology. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  29. Peebles, P.J.E.: The Large Scale Structure of the Universe. Princeton University Press, Princeton (1980)

    Google Scholar 

  30. Suto, Y., Itoh, M., Inagaki, S.: Astrophys. J. 350, 492 (1990)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, S.P. College, Srinagar, Kashmir, 190001, India

    Mir Hameeda

  2. IUCCA, Pune, 41100, India

    Mir Hameeda

  3. Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Mir Faizal

  4. Department of Physics, Florida State University, Tallahassee, FL, 32306, USA

    Ahmed Farag Ali

  5. Department of Physics, Faculty of Science, Benha University, Benha, 13518, Egypt

    Ahmed Farag Ali

Authors
  1. Mir Hameeda
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  2. Mir Faizal
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  3. Ahmed Farag Ali
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Corresponding author

Correspondence to Ahmed Farag Ali.

Appendix

Appendix

We write the calculations of the first integral of Eq. (17), the second integral has similar calculations.

$$\begin{aligned} I_{1}(T,V)= & {} 4\pi V\left[ \left( \frac{Gm^{2}}{T}\right) ^{0}\int _{0}^{R_{1}}r^{2}dr+\frac{1}{1!}\left( \frac{Gm^{2}}{T}\right) ^{1}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+ \epsilon ^{2})^{1/2}}\right. \nonumber \\&+\,\frac{1}{2!}\left( \frac{Gm^{2}}{T}\right) ^{2}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+\epsilon ^{2})}dr+ \frac{1}{3!}\left( \frac{Gm^{2}}{T}\right) ^{3}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+\epsilon ^{2})^{3/2}}dr\nonumber \\&+\,\frac{1}{4!}\left( \frac{Gm^{2}}{T}\right) ^{4}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+\epsilon ^{2})^{2}}dr+ \frac{1}{5!}\left( \frac{Gm^{2}}{T}\right) ^{5}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+\epsilon ^{2})^{5/2}}dr\nonumber \\&+\,\frac{1}{6!}\left( \frac{Gm^{2}}{T}\right) ^{6}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+\epsilon ^{2})^{3}}dr+ \frac{1}{7!}\left( \frac{Gm^{2}}{T}\right) ^{7}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+\epsilon ^{2})^{7/2}}dr\nonumber \\&\left. +\,\frac{1}{8!}\left( \frac{Gm^{2}}{T}\right) ^{8}\int _{0}^{R_{1}}\frac{r^{2}}{(r^{2}+\epsilon ^{2})^{4}}dr+\cdots \right] \end{aligned}$$
(53)
$$\begin{aligned} I_{1}(T,V)= & {} 4\pi V\left[ \frac{R_{1}^{3}}{3}+\left( \frac{Gm^{2}}{T}\right) \left( \frac{R_{1}^{2}}{2}\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}} +\frac{\epsilon ^{2}}{2}ln\frac{\frac{\epsilon }{R_{1}}}{1+\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}\right) \right. \nonumber \\&+\,\frac{1}{2!}\left( \frac{Gm^2}{T}\right) ^2\left( R_{1}-\epsilon \tan ^{-1}\left( \frac{1}{\epsilon /R_{1}}\right) \right) +\frac{1}{3!}\left( \frac{Gm^2}{T}\right) ^3\nonumber \\&\times \left( -\frac{1}{\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}-ln\frac{\frac{\epsilon }{R_{1}}}{1+\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}\right) +\frac{1}{4!}\left( \frac{Gm^{2}}{T}\right) ^4\nonumber \\&\times \left( -\frac{1}{2R_{1}\left( 1+\frac{\epsilon ^2}{R_{1}^2}\right) } +\frac{1}{2\epsilon }\tan ^{-1}\frac{1}{\epsilon /R_{1}}\right) +\frac{1}{5!}\left( \frac{Gm^2}{T}\right) ^5\nonumber \\&\times \left( \frac{1}{3R_{1}^{3}\left( 1+\frac{\epsilon ^2}{R_{1}^2}\right) ^3}\right) +\frac{1}{6!}\left( \frac{Gm^2}{T}\right) ^6\nonumber \\&\times \left( -\frac{1}{4R_{1}^3\left( 1+\frac{\epsilon ^2}{R_{1}^2}\right) ^2} +\frac{1}{\epsilon ^2R_{1}\left( 1+\frac{\epsilon ^2}{R_{1}^2}\right) }\left. +\,\frac{1}{8\epsilon ^3}\tan ^{-1}\left( \frac{1}{\epsilon /R_{1}}\right) \right) \right. \nonumber \\&+\frac{1}{7!}\left( \frac{Gm^2}{T}\right) ^7 \left( \frac{1}{3\epsilon ^4\left( 1+\frac{\epsilon ^2}{R_{1}^2}\right) ^{3/2}}\right. \left. \left. - \,\frac{1}{5\epsilon ^4\left( 1+\frac{\epsilon ^2}{R_{1}^2}\right) ^{5/2}}\right) +\cdots \right] \nonumber \\ \end{aligned}$$
(54)
$$\begin{aligned} I_{1}(T,V)= & {} 4\pi V\left[ \frac{R_{1}^{3}}{3}+\left( \frac{3Gm^{2}}{2TR_1}\right) \left( \frac{2R_1}{3\epsilon }\right) \left( \frac{R_1^{3}}{3}\frac{3\epsilon }{R_1^{3}}\right) \right. \nonumber \\&\times \left( \frac{R_{1}^{2}}{2}\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}+ \frac{\epsilon ^{2}}{2}ln\frac{\frac{\epsilon }{R_{1}}}{1+\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}\right) +\frac{1}{2!}\left( \frac{3Gm^2}{2TR_1}\right) ^2\nonumber \\&\times \left( \frac{2R_1}{3\epsilon }\right) ^2\left( \frac{R_1^{3}}{3}\frac{3\epsilon ^2}{R_1^{3}}\right) \left( R_{1}-\epsilon \tan ^{-1}\left( \frac{1}{\epsilon /R_{1}}\right) \right) +\frac{1}{3!}\left( \frac{3Gm^2}{2TR_1}\right) ^3\nonumber \\&\times \left( \frac{2R_1}{3\epsilon }\right) ^3\left( \frac{R_1^{3}}{3}\frac{3\epsilon ^3}{R_1^{3}}\right) \left( -\frac{1}{\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}-ln\frac{\frac{\epsilon }{R_{1}}}{1+\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}\right) +\frac{1}{4!}\left( \frac{3Gm^{2}}{2TR_1}\right) ^4\nonumber \\&\times \left( \frac{2R_1}{3\epsilon }\right) ^4\left( \frac{R_1^{3}}{3}\frac{3\epsilon ^4}{R_1^{3}}\right) \left( -\frac{1}{2R_{1}(1+\frac{\epsilon ^2}{R_{1}^2})}+\frac{1}{2\epsilon }\tan ^{-1}\frac{1}{\epsilon /R_{1}}\right) \nonumber \\&+\,\frac{1}{5!}\left( \frac{3Gm^2}{2TR_1}\right) ^5\left( \frac{2R_1}{3\epsilon }\right) ^5\left( \frac{R_1^{3}}{3}\frac{3\epsilon ^5}{R_1^{3}}\right) \left( \frac{1}{3R_{1}^{3}\left( 1+\frac{\epsilon ^2}{R_{1}^2}\right) ^3}\right) \nonumber \\&+\frac{1}{6!}\left( \frac{3Gm^2}{2TR_1}\right) ^6\left( \frac{2R_1}{3\epsilon }\right) ^6\left( \frac{R_1^{3}}{3}\frac{3\epsilon ^6}{R_1^{3}}\right) \left( -\frac{1}{4R_{1}^3(1+\frac{\epsilon ^2}{R_{1}^2})^2}\right. \nonumber \\&\left. \left. +\,\frac{1}{\epsilon ^2R_{1}(1+\frac{\epsilon ^2}{R_{1}^2})}+\frac{1}{8\epsilon ^3}\tan ^{-1}\left( \frac{1}{\epsilon /R_{1}}\right) \right) +\cdots \right] \end{aligned}$$
(55)
$$\begin{aligned}&I_{1}(T,V)\nonumber \\&=V^2(1+x\left( \frac{1}{1!}\left( \frac{2R_{1}}{3\epsilon }\right) ^{1} F \left( \frac{3}{2},\frac{1}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right) +x^2\frac{1}{2!}\left( \frac{2R_{1}}{3\epsilon }\right) ^{2} F \left( \frac{3}{2},\frac{2}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right) \right. \nonumber \\&\quad +\,x^3\frac{1}{3!}\left( \frac{2R_{1}}{3\epsilon }\right) ^{3} F \left( \frac{3}{2},\frac{3}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right) +x^4\frac{1}{4!}\left( \frac{2R_{1}}{3\epsilon }\right) ^{4} F \left( \frac{3}{2},\frac{4}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right) \nonumber \\&\left. \quad +\,x^5\frac{1}{5!}\left( \frac{2R_{1}}{3\epsilon }\right) ^{5} F \left( \frac{3}{2},\frac{5}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right) +x^6\frac{1}{6!}\left( \frac{2R_{1}}{3\epsilon }\right) ^{6} F \left( \frac{3}{2},\frac{6}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right) +\cdots \right) \nonumber \\ \end{aligned}$$
(56)

where

$$\begin{aligned} F\left( \frac{3}{2},\frac{1}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right)= & {} \frac{3\epsilon }{R_1^3}\left( \frac{R_{1}^{2}}{2}\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}+\frac{\epsilon ^{2}}{2}ln\frac{\frac{\epsilon }{R_{1}}}{1+\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}\right) \end{aligned}$$
(57)
$$\begin{aligned} F \left( \frac{3}{2},\frac{2}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right)= & {} \frac{3\epsilon ^2}{R_1^3}\left( R_{1}-\epsilon \tan ^{-1}(\frac{1}{\epsilon /R_{1}})\right) \end{aligned}$$
(58)
$$\begin{aligned} F \left( \frac{3}{2},\frac{3}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right)= & {} \frac{3\epsilon ^3}{R_1^3}\left( -\frac{1}{\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}-ln\frac{\frac{\epsilon }{R_{1}}}{1+\sqrt{1+\frac{\epsilon ^2}{R_{1}^{2}}}}\right) \end{aligned}$$
(59)
$$\begin{aligned} F \left( \frac{3}{2},\frac{4}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right)= & {} \frac{3\epsilon ^4}{R_1^3}\left( -\frac{1}{2R_{1}(1+\frac{\epsilon ^2}{R_{1}^2})}+\frac{1}{2\epsilon }\tan ^{-1}\frac{1}{\epsilon /R_{1}}\right) \end{aligned}$$
(60)
$$\begin{aligned} F \left( \frac{3}{2},\frac{5}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right)= & {} \frac{3\epsilon ^5}{R_1^3}\left( \frac{1}{3R_{1}^{3}(1+\frac{\epsilon ^2}{R_{1}^2})^3}\right) \end{aligned}$$
(61)
$$\begin{aligned} F \left( \frac{3}{2},\frac{6}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right)= & {} \frac{3\epsilon ^6}{R_1^3}\left( -\frac{1}{4R_{1}^3(1+\frac{\epsilon ^2}{R_{1}^2})^2}+\frac{1}{\epsilon ^2R_{1}(1+\frac{\epsilon ^2}{R_{1}^2})}\right. \nonumber \\&\left. +\,\frac{1}{8\epsilon ^3}\tan ^{-1}\left( \frac{1}{\epsilon /R_{1}}\right) \right) \end{aligned}$$
(62)

In general

$$\begin{aligned} I_{1}(T,V)=V^2\left( 1+\sum _{n=1}^\infty x^n\frac{1}{n!}\left( \frac{2R_{1}}{3\epsilon }\right) ^{n} F \left( \frac{3}{2},\frac{n}{2};\frac{5}{2};-\frac{R_{1}^2}{\epsilon ^2}\right) \right) \end{aligned}$$
(63)

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Hameeda, M., Faizal, M. & Ali, A.F. Clustering of galaxies in brane world models. Gen Relativ Gravit 48, 47 (2016). https://doi.org/10.1007/s10714-016-2049-y

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