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A. Cole and G. Shiu
In this chapter, we reviewed two such applications. In the first, persistent homology was used to characterize the effect of primordial non-Gaussianity on the Cosmic Microwave Background. A statistical pipeline based on persistent homology
demonstrated impressive increase in sensitivity compared to previous topological
observables, motivating future work into making analytic predictions. In the second application, persistent homology was used to characterize distributions of string
vacua. In this context, it proved helpful to introduce a dressed persistence diagram
that shows the relationships between cycles of different dimension.
Developing further these applications appears promising. As mentioned briefly,
applying persistent homology in a cosmological context would prove more powerful
if we had a stronger grasp on analytic properties of the persistent topology of random
fields. Moreover, in addition to the CMB and the LSS there is the 21-cm field, the
topology of which is an interesting probe of the epoch of reionization [45]. In terms
of the string landscape, it would be interesting to use persistent homology to study
data sets in dimensions higher than four, and to further understand the distribution
of vacua with phenomenologically interesting properties.
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
References
1. Abel, S., Rizos, J.: Genetic algorithms and the search for viable string vacua. JHEP 08, 010
(2014). https://doi.org/10.1007/JHEP08(2014)010
2. Acharya, B.S., Denef, F., Valandro, R.: Statistics of M theory vacua. JHEP 06, 056 (2005).
https://doi.org/10.1088/1126-6708/2005/06/056
3. Ackley, D.H., Hinton, G.E., Sejnowski, T.J.: A learning algorithm for boltzmann machines.
Cogn. Sci. 9(1), 147–169 (1985)
4. Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova,
S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: a stable vector representation
of persistent homology. J. Mach. Learn. Res. 18(1), 218–252 (2017)
5. Ade, P.A.R., et al.: Planck 2015 results XVII constraints on primordial non-gaussianity.
Astron. Astrophys. 594, A17 (2016). https://doi.org/10.1051/0004-6361/201525836
6. Albrecht, A., Steinhardt, P.J.: Cosmology for grand unified theories with radiatively
induced symmetry breaking. Phys. Rev. Lett. 48, 1220–1223 (1982). https://doi.org/10.1103/
PhysRevLett.48.1220
7. Ashmore, A., He, Y.H., Ovrut, B.A.: Machine learning Calabi-Yau metrics (2019)
8. Ashok, S., Douglas, M.R.: Counting flux vacua. JHEP 01, 060 (2004). https://doi.org/10.
1088/1126-6708/2004/01/060
9. Babich, D., Creminelli, P., Zaldarriaga, M.: The shape of non-gaussianities. JCAP 0408, 009
(2004). https://doi.org/10.1088/1475-7516/2004/08/009
10. Banks, T., Dine, M., Gorbatov, E.: Is there a string theory landscape? JHEP 08, 058 (2004).
https://doi.org/10.1088/1126-6708/2004/08/058
11. Bardeen, J.M., Steinhardt, P.J., Turner, M.S.: Spontaneous creation of almost scale-free density
perturbations in an inflationary universe. Phys. Rev. D 28(4), 679 (1983)
12. Biagetti, M., Cole, A., Shiu, G.: The persistence of large scale structures I: primordial nongaussianity 9, (2020)
A. Cole and G. Shiu
In this chapter, we reviewed two such applications. In the first, persistent homology was used to characterize the effect of primordial non-Gaussianity on the Cosmic Microwave Background. A statistical pipeline based on persistent homology
demonstrated impressive increase in sensitivity compared to previous topological
observables, motivating future work into making analytic predictions. In the second application, persistent homology was used to characterize distributions of string
vacua. In this context, it proved helpful to introduce a dressed persistence diagram
that shows the relationships between cycles of different dimension.
Developing further these applications appears promising. As mentioned briefly,
applying persistent homology in a cosmological context would prove more powerful
if we had a stronger grasp on analytic properties of the persistent topology of random
fields. Moreover, in addition to the CMB and the LSS there is the 21-cm field, the
topology of which is an interesting probe of the epoch of reionization [45]. In terms
of the string landscape, it would be interesting to use persistent homology to study
data sets in dimensions higher than four, and to further understand the distribution
of vacua with phenomenologically interesting properties.
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
References
1. Abel, S., Rizos, J.: Genetic algorithms and the search for viable string vacua. JHEP 08, 010
(2014). https://doi.org/10.1007/JHEP08(2014)010
2. Acharya, B.S., Denef, F., Valandro, R.: Statistics of M theory vacua. JHEP 06, 056 (2005).
https://doi.org/10.1088/1126-6708/2005/06/056
3. Ackley, D.H., Hinton, G.E., Sejnowski, T.J.: A learning algorithm for boltzmann machines.
Cogn. Sci. 9(1), 147–169 (1985)
4. Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova,
S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: a stable vector representation
of persistent homology. J. Mach. Learn. Res. 18(1), 218–252 (2017)
5. Ade, P.A.R., et al.: Planck 2015 results XVII constraints on primordial non-gaussianity.
Astron. Astrophys. 594, A17 (2016). https://doi.org/10.1051/0004-6361/201525836
6. Albrecht, A., Steinhardt, P.J.: Cosmology for grand unified theories with radiatively
induced symmetry breaking. Phys. Rev. Lett. 48, 1220–1223 (1982). https://doi.org/10.1103/
PhysRevLett.48.1220
7. Ashmore, A., He, Y.H., Ovrut, B.A.: Machine learning Calabi-Yau metrics (2019)
8. Ashok, S., Douglas, M.R.: Counting flux vacua. JHEP 01, 060 (2004). https://doi.org/10.
1088/1126-6708/2004/01/060
9. Babich, D., Creminelli, P., Zaldarriaga, M.: The shape of non-gaussianities. JCAP 0408, 009
(2004). https://doi.org/10.1088/1475-7516/2004/08/009
10. Banks, T., Dine, M., Gorbatov, E.: Is there a string theory landscape? JHEP 08, 058 (2004).
https://doi.org/10.1088/1126-6708/2004/08/058
11. Bardeen, J.M., Steinhardt, P.J., Turner, M.S.: Spontaneous creation of almost scale-free density
perturbations in an inflationary universe. Phys. Rev. D 28(4), 679 (1983)
12. Biagetti, M., Cole, A., Shiu, G.: The persistence of large scale structures I: primordial nongaussianity 9, (2020)