* Biagetti et al. (2020) Biagetti M., Cole A., Shiu G., 2020, arXiv e-prints, p. arXiv:2009.04819
  * Biscio & Møller (2016) Biscio C., Møller J., 2016, arXiv e-prints, p. arXiv:1611.00630
  * Bobrowski & Borman (2012) Bobrowski O., Borman M. S., 2012, Journal of Topology and Analysis, 4
  * Bobrowski & Skraba (2020) Bobrowski O., Skraba P., 2020, Phys. Rev. E, 101, 032304
  * Bond et al. (1991) Bond J. R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440
  * Bond & Myers (1996) Bond J. R., Myers S. T., 1996, Astrophys. J. Suppl., 103, 1
  * Bubenik (2015) Bubenik P., 2015, J. Mach. Learn. Res., 16, 77
  * Buchert et al. (2017) Buchert T., France M. J., Steiner F., 2017, Classical and Quantum Gravity, 34, 094002
  * Busch & White (2020) Busch P., White S. D. M., 2020, Mon. Not. Royal Astro. Soc., 493, 5693
  * Carlsson (2009) Carlsson G., 2009, Bulletin of the American Mathematical Society, 46, 255
  * Carlsson & Zomorodian (2009) Carlsson G., Zomorodian A., 2009, Discrete & Computational Geometry, 42, 71
  * Carlsson et al. (2005) Carlsson G., Zomorodian A., Collins A., Guibas L. J., 2005, International Journal of Shape Modeling, 11, 149
  * Chazal et al. (2014) Chazal F., Fasy B. T., Lecci F., Rinaldo A., Wasserman L., 2014, in Proceedings of the Thirtieth Annual Symposium on Computational Geometry SOCG’14, Stochastic convergence of persistence landscapes and silhouettes. ACM, New York, NY, USA, pp 474:474–474:483
  * Chen et al. (2015) Chen Y.-C., Ho S., Tenneti A., Mandelbaum R., Croft R., DiMatteo T., Freeman P. E., Genovese C. R., Wasserman L., 2015, Mon. Not. Royal Astro. Soc., 454, 3341
  * Chen et al. (2015) Chen Y.-C., Wang D., Rinaldo A., Wasserman L., 2015, arXiv e-prints, p. arXiv:1510.02502
  * Cheng & Schwartzman (2015) Cheng D., Schwartzman A., 2015, arXiv:1511.06835
  * Codis et al. (2018) Codis S., Pogosyan D., Pichon C., 2018, arXiv:1803.11477
  * Cohen-Steiner et al. (2007) Cohen-Steiner D., Edelsbrunner H., Harer J., 2007, Discrete Comput. Geom., 37, 103
  * Cole & Shiu (2018) Cole A., Shiu G., 2018, Jour. Cos. and Part. Phys., 3, 025
  * Crofton (1868) Crofton M. W., 1868, Philosophical Transactions of the Royal Society of London, 158, 181
  * Di Valentino et al. (2019) Di Valentino E., Melchiorri A., Silk J., 2019, Nature Astronomy
  * Doroshkevich (1970) Doroshkevich A. G., 1970, Astrophysics, 6, 320
  * Dunkley et al. (2009) Dunkley J., Komatsu E., Nolta M. R., Spergel D. N., Larson D., Hinshaw G., Page L., Bennett C. L., Gold B., Jarosik N., Weiland J. L., Halpern M., Hill R. S., Kogut A., Limon M., Meyer S. S., Tucker G. S., Wollack E., Wright E. L., 2009, Astrophysical Journal Suppl., 180, 306
  * Edelsbrunner & Harer (2010) Edelsbrunner H., Harer J., 2010, Computational Topology: An Introduction. Applied mathematics, American Mathematical Society
  * Edelsbrunner et al. (2012) Edelsbrunner H., Ivanov A., Karasev R., 2012, Model. Anal. Inform. Sist., 19, 5
  * Edelsbrunner et al. (2002) Edelsbrunner H., Letscher J., Zomorodian A., 2002, Discrete & Computational Geometry, 28, 511
  * Efstathiou & Bond (1999) Efstathiou G., Bond J. R., 1999, MNRAS, 304, 75
  * Euler (1758) Euler L., 1758, Novi Commentarii academiae scientiarum Petropolitanae, 4, 140
  * Feldbrugge et al. (2019) Feldbrugge J., van Engelen M., van de Weygaert R., Pranav P., Vegter G., 2019, Journal of Cosmology and Astroparticle Physics, 2019, 052–052
  * Fergusson & Shellard (2009) Fergusson J. R., Shellard E. P. S., 2009, Physical Review D, 80, 043510
  * Gauss (1900) Gauss C. F., 1900, K. Gesellschaft Wissenschaft, 8
  * Gott et al. (2009) Gott J. R., Choi Y.-Y., Park C., Kim J., 2009, ApJ Letters, 695, L45
  * Gott et al. (1986) Gott III J. R., Dickinson M., Melott A. L., 1986, Astrophysical Journal, 306, 341
  * Gott et al. (2008) Gott III J. R., Hambrick D. C., Vogeley M. S., Kim J., Park C., Choi Y.-Y., Cen R., Ostriker J. P., Nagamine K., 2008, Astrophysical Journal, 675, 16
  * Gott et al. (1989) Gott III J. R., Miller J., Thuan T. X., Schneider S. E., Weinberg D. H., Gammie C., Polk K., Vogeley M., Jeffrey S., Bhavsar S. P., Melott A. L., Giovanelli R., Hayes M. P., Tully R. B., Hamilton A. J. S., 1989, Astrophysical Journal, 340, 625
  * Guth (1981) Guth A. H., 1981, Physical Review D, 23, 347
  * Guth & Pi (1982) Guth A. H., Pi S.-Y., 1982, Physical Review Letters, 49, 1110
  * Hadwiger (1957) Hadwiger H., 1957, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Grundlehren der mathematischen Wissenschaften, Springer
  * Hahn et al. (2007) Hahn O., Porciani C., et al., 2007, Mon. Not. Royal Astro. Soc., 375, 489
  * Harrison (1970) Harrison E. R., 1970, Physical Review D, 1, 2726
  * Heavens & Sheth (1999) Heavens A. F., Sheth R. K., 1999, Mon. Not. Royal Astro. Soc., 310, 1062
  * Hikage et al. (2008) Hikage C., Coles P., Grossi M., Moscardini L., Dolag K., Branchini E., Matarrese S., 2008, Mon. Not. Royal Astro. Soc., 385, 1613
  * Hikage et al. (2002) Hikage C., Suto Y., Kayo I., Taruya A., Matsubara T., Vogeley M. S., Hoyle F., Gott III J. R., Brinkmann J., 2002, Pub. Astro. Soc. of Japan, 54, 707
  * Jones (2017) Jones B. J. T., 2017, Precision Cosmology: The First Half Million Years. Cambridge University Press
  * Kac (1943) Kac M., 1943, Bulletin of the American Mathematical Society, 49, 314
  * Kahle (2014) Kahle M., 2014, in Contemp. Math., Vol. 620, Algebraic topology: applications and new directions. Amer. Math. Soc., Providence, RI, pp 201–221
  * Kannan et al. (2019) Kannan H., Saucan E., Roy I., Samal A., 2019, Scientific Reports, 9
  * Kerscher et al. (2001) Kerscher M., Buchert T., Futamase T., 2001, Astrophysical Journal Letters, 558, L79
  * Kerscher et al. (1996) Kerscher M., Schmalzing J., Buchert T., 1996, Mapping, measuring and modelling the universe. pp 247–252
  * Kolb et al. (1990) Kolb E. W., Salopek D. S., Turner M. S., 1990, Physical Review D, 42, 3925
  * Komatsu et al. (2011) Komatsu E., Smith K. M., Dunkley J., Bennett C. L., Gold B., Hinshaw G., Jarosik N., Larson D., Nolta M. R., Page L., Spergel D. N., Halpern M., Hill R. S., Kogut A., Limon M., Meyer S. S., Odegard N., Tucker G. S., Weiland J. L., Wollack E., Wright E. L., 2011, The Astrophysical Journal Supplement Series, 192, 18
  * Kono et al. (2020) Kono K. T., Takeuchi T. T., Cooray S., Nishizawa A. J., Murakami K., 2020, arXiv e-prints, p. arXiv:2006.02905
  * Liddle & Lyth (2000) Liddle A. R., Lyth D. H., 2000, Cosmological Inflation and Large-Scale Structure
  * Linde (1981) Linde A. D., 1981, Physics Letters B, 100, 37
  * Luminet & Roukema (1999) Luminet J.-P., Roukema B. F., 1999, Topology of the Universe: Theory and Observation. Springer Netherlands, Dordrecht, pp 117–156
  * Makarenko et al. (2018) Makarenko I., Bushby P., Fletcher A., Henderson R., Makarenko N., Shukurov A., 2018, J. Plasma Phys., 84, 047303
  * Makarenko et al. (2017) Makarenko I., Shukurov A., Henderson R., Rodrigues L. F. S., Bushby P., Fletcher A., 2017, ArXiv e-prints
  * Matarrese et al. (1991) Matarrese S., Lucchin F., Messina A., Moscardini L., 1991, Mon. Not. Royal Astro. Soc., 253, 35
  * Matsubara (2010) Matsubara T., 2010, Physical Review D, 81, 083505
  * Matsubara & Suto (1996) Matsubara T., Suto Y., 1996, Astrophysical Journal, 460, 51
  * Mecke et al. (1994) Mecke K. R., Buchert T., Wagner H., 1994, Astronomy & Astrophysics, 288, 697
  * Mecke & Wagner (1991) Mecke K. R., Wagner H., 1991, Journal of Statistical Physics, 64, 843
  * Milnor (1963) Milnor J., 1963, Journal of Mathematical Physics
  * Mondal et al. (2015) Mondal R., Bharadwaj S., Majumdar S., 2015, Monthly Notices of the Royal Astronomical Society, 456, 1936–1947
  * Moraleda et al. (2019) Moraleda R., Valous N., Xiong W., Halama N., 2019, Computational Topology for Biomedical Image and Data Analysis: Theory and Applications. Focus Series in Medical Physics and Biomedical Engineering, CRC Press
  * Mukhanov & Chibisov (1981) Mukhanov V. F., Chibisov G. V., 1981, ZhETF Pisma Redaktsiiu, 33, 549
  * Munkres (1984) Munkres J., 1984, Elements of Algebraic Topology. Advanced book classics, Perseus Books
  * Naselsky et al. (2005) Naselsky P., Chiang L.-Y., Olesen P., Novikov I., 2005, Physical Review D, 72, 063512
  * O’Hagan & Leonard (1976) O’Hagan A., Leonard T., 1976, Biometrika, 63, pp. 201
  * Park et al. (2001) Park C., Gott III J. R., Choi Y. J., 2001, Astrophysical Journal, 553, 33
  * Park et al. (2013) Park C., Pranav P., Chingangbam P., van de Weygaert R., Jones B., Vegter G., Kim I., Hidding J., Hellwing W. A., 2013, Journal of Korean Astronomical Society, 46, 125
  * Peacock & Heavens (1990) Peacock J. A., Heavens A. F., 1990, Mon. Not. Royal Astro. Soc., 243, 133
  * Peebles (1980) Peebles P., 1980, The Large-scale Structure of the Universe. Princeton series in physics, Princeton University Press
  * Peebles & Yu (1970) Peebles P. J. E., Yu J. T., 1970, Astrophysical Journal, 162, 815
  * Petri et al. (2014) Petri G., Expert P., Turkheimer F., Carhart-Harris R., Nutt D., Hellyer P. J., Vaccarino F., 2014, Journal of The Royal Society Interface, 11, 20140873
  * Planck Collaboration et al. (2016) Planck Collaboration Ade P. A. R., Aghanim N., Arnaud M., Arroja F., Ashdown M., Aumont J., Baccigalupi C., Ballardini M., Banday A. J., et al. 2016, Astron. & Astrophysics, 594, A17
  * Planck Collaboration et al. (2016a) Planck Collaboration Ade P. A. R., Aghanim N., Arnaud M., Ashdown M., Aumont J., Baccigalupi C., Banday A. J., Barreiro R. B., Bartlett J. G., et al. 2016a, Astron. & Astrophysics, 594, A13
  * Planck Collaboration et al. (2016b) Planck Collaboration Ade P. A. R., Aghanim N., Arnaud M., Ashdown M., Aumont J., Baccigalupi C., Banday A. J., Barreiro R. B., Bartlett J. G., et al. 2016b, Astronomy & Astrophysics, 594, A13
  * Pranav (2015) Pranav P., 2015, Persistent Holes in the Universe: A Hierarchical Topology of the Cosmic Mass Distribution. University of Groningen
  * Pranav (2021) Pranav P., 2021, IEEE Signal Processing Magazine, 38, 130
  * Pranav et al. (2019) Pranav P., Adler R. J., Buchert T., Edelsbrunner H., Jones B. J. T., Schwartzman A., Wagner H., van de Weygaert R., 2019, Astron. & Astrophysics, 627, A163
  * Pranav et al. (2017) Pranav P., Edelsbrunner H., van de Weygaert R., Vegter G., Kerber M., Jones B. J. T., Wintraecken M., 2017, Mon. Not. Royal Astro. Soc., 465, 4281
  * Pranav et al. (2019) Pranav P., van de Weygaert R., Vegter G., Jones B. J. T., Adler R. J., Feldbrugge J., Park C., Buchert T., Kerber M., 2019, Mon. Not. Royal Astro. Soc., 485, 4167
  * Press & Schechter (1974) Press W. H., Schechter P., 1974, Astrophysical Journal, 187, 425
  * Pun et al. (2018) Pun C. S., Xia K., Lee S. X., , 2018, Persistent-Homology-based Machine Learning and its Applications – A Survey
  * Reimann et al. (2017) Reimann M. W., Nolte M., Scolamiero M., Turner K., Perin R., Chindemi G., Dłotko P., Levi R., Hess K., Markram H., 2017, Frontiers in Computational Neuroscience, 11, 48
  * Rhoads et al. (1994) Rhoads J. E., Gott III J. R., Postman M., 1994, Astrophysical Journal, 421, 1
  * Rice (1944) Rice S. O., 1944, The Bell System Technical Journal, 23, 282
  * Roukema et al. (2004) Roukema B. F., Lew B., Cechowska M., Marecki A., Bajtlik S., 2004, Astron. & Astrophysics, 423, 821
  * Sahni et al. (1997) Sahni V., Sathyaprakash B. S., Shandarin S. F., 1997, Astrophysical Journal Letters, 476, L1
  * Sahni et al. (1998) Sahni V., Sathyprakash B., Shandarin S., 1998, Astrophysical Journal, 507, L109
  * Scaramella & Vittorio (1991) Scaramella R., Vittorio N., 1991, Astrophysical Journal, 375, 439
  * Schmalzing & Buchert (1997) Schmalzing J., Buchert T., 1997, Astrophysical Journal Letters, 482, L1+
  * Schmalzing & Gorski (1998) Schmalzing J., Gorski K. M., 1998, Mon. Not. Royal Astro. Soc., 297, 355
  * Sefusatti et al. (2007) Sefusatti E., Vale C., Kadota K., Frieman J., 2007, Astrophysical Journal, 658, 669
  * Seljak & Zaldarriaga (1999) Seljak U., Zaldarriaga M., , 1999, CMBFAST: A microwave anisotropy code, Astrophysics Source Code Library
  * Sheth (1998) Sheth R. K., 1998, Mon. Not. Royal Astro. Soc., 300, 1057
  * Sheth et al. (2001) Sheth R. K., Mo H. J., Tormen G., 2001, Mon. Not. Royal Astro. Soc., 323, 1
  * Sheth & Tormen (2002) Sheth R. K., Tormen G., 2002, Mon. Not. Royal Astro. Soc., 329, 61
  * Sheth & van de Weygaert (2004) Sheth R. K., van de Weygaert R., 2004, Mon. Not. Royal Astro. Soc., 350, 517
  * Shivashankar et al. (2016) Shivashankar N., Pranav P., Natarajan V., van de Weygaert R., Bos E. G. P., Rieder S., 2016, IEEE Trans. Vis. Comput. Graph., 22, 1745
  * Smoot et al. (1992) Smoot G. F., Bennett C. L., Kogut A., et. al. 1992, Astrophysical Journal Letters, 396, L1
  * Sousbie (2011) Sousbie T., 2011, MNRAS, 414, 350
  * Sousbie et al. (2011) Sousbie T., Pichon C., Kawahara H., 2011, MNRAS, 414, 384
  * Spergel et al. (2007) Spergel D. N., Bean R., Doré O., et. al. 2007, Astrophys. J. Suppl., 170, 377
  * Starobinsky (1982) Starobinsky A. A., 1982, Physics Letters B, 117, 175
  * Taylor (2006) Taylor J., 2006, Ann. Probab., 34, 122
  * Telschow et al. (2019) Telschow F., Schwartzman A., Cheng D., Pranav P., 2019, arXiv e-prints, p. arXiv:1908.02493
  * Utreras et al. (2020) Utreras J., Blanc G. A., Escala A., Meidt S., Emsellem E., Bigiel F., Glover S. C. O., Henshaw J., Hygate A., Kruijssen J. M. D., Rosolowsky E., Schinnerer E., Schruba A., 2020, Astrophysical Journal, 892, 94
  * van de Weygaert & Bertschinger (1996) van de Weygaert R., Bertschinger E., 1996, Mon. Not. Royal Astro. Soc., 281, 84
  * van de Weygaert & Bond (2008a) van de Weygaert R., Bond J. R., 2008a, in Plionis M., López-Cruz O., Hughes D., eds, A Pan-Chromatic View of Clusters of Galaxies and the Large-Scale Structure Vol. 740 of Lecture Notes in Physics, Berlin Springer Verlag, Clusters and the theory of the Cosmic Web. p. 335
  * van de Weygaert & Bond (2008b) van de Weygaert R., Bond J. R., 2008b, in Plionis M., López-Cruz O., Hughes D., eds, A Pan-Chromatic View of Clusters of Galaxies and the Large-Scale Structure Vol. 740 of Lecture Notes in Physics, Berlin Springer Verlag, Observations and Morphology of the Cosmic Web. p. 409
  * van de Weygaert et al. (2011) van de Weygaert R., Pranav P., Jones B. J. T., Bos E. G. P., Vegter G., Edelsbrunner H., Teillaud M., Hellwing W. A., Park C., Hidding J., Wintraecken M., 2011, ArXiv e-prints, p. arXiv:1110.5528
  * van de Weygaert et al. (2011) van de Weygaert R., Vegter G., Edelsbrunner H., Jones B., et al. P. P., 2011, Transactions on Computational Science, XIV, 60
  * van de Weygaert et al. (2011) van de Weygaert R., Vegter G., Edelsbrunner H., Jones B. J. T., Pranav P., Park C., Hellwing W. A., Eldering B., Kruithof N., Bos E. G. P. P., Hidding J., Feldbrugge J., ten Have E., van Engelen M., Caroli M., Teillaud M., 2011, Transactions on Computational Science, 14, 60
  * Verde et al. (2001) Verde L., Jimenez R., Kamionkowski M., Matarrese S., 2001, Mon. Not. Royal Astro. Soc., 325, 412
  * Verde et al. (2000) Verde L., Wang L., Heavens A. F., Kamionkowski M., 2000, Mon. Not. Royal Astro. Soc., 313, 141
  * Wasserman (2016) Wasserman L., 2016, Annual Reviews in Statistics
  * Wilding et al. (2020) Wilding G., Nevenzeel K., van de Weygaert R., Vegter G., Pranav P., Jones B. J. T., Efstathiou K., Feldbrugge J., 2020, arXiv e-prints, p. arXiv:2011.12851
  * Xia et al. (2011) Xia J.-Q., Baccigalupi C., Matarrese S., Verde L., Viel M., 2011, JCAP, 8, 33
  * Xu et al. (2019) Xu X., Cisewski-Kehe J., Green S., Nagai D., 2019, Astronomy and Computing, 27, 34–52
  * Zeldovich (1972) Zeldovich Y. B., 1972, Mon. Not. Royal Astro. Soc., 160, 1P
  * Zentner (2007) Zentner A. R., 2007, International Journal of Modern Physics D, 16, 763
  * Zomorodian et al. (2005) Zomorodian A. J., Ablowitz M. J., Davis S. H., Hinch E. J., Iserles A., Ockendon J., Olver P. J., 2005, Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics). Cambridge University Press, New York, NY, USA

##  Appendix A Skew-normal distribution

The skew-normal distribution with the _skewness parameter_ $\alpha$ is given
by O’Hagan & Leonard (1976); Azzalini (1985); Pranav (2015); Wilding et al.
(2020):

$f(x)=2\phi(\nu)\Phi(\alpha\nu).$ (49)

Here,

$\phi(\nu)=\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{\nu^{2}}{2}}.$ (50)

is the standard normal distribution. The function

$\displaystyle\Phi(\nu)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{x}\phi(t)\,\mathrm{d}t$ (51) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\nu}{\sqrt{2}}\right)\right],$

is the cumulative distribution function, in which _erf(x)_ is the error
function. Note that one recovers the familiar normal distribution when the
skewness parameter $\alpha=0$. he absolute value of skewness increases as the
absolute value of $\alpha$ increases. Note that the skewness parameter
$\alpha$ is different than the skewness, i.e. the third moment of the
distribution

$\gamma_{1}=\mu^{3}/\sigma^{3}.$ (52)

By definition, a curve has positive skewness if it has a more prominent tail
for increasing values of $\nu$, and a negative skewness when its balance is
shifted towards decreasing values of $\nu$.

To account for the location of the peak and the width of the curve, one
usually makes the transformation

$\mathrm{x}\to\mathrm{x}-\xi/\omega,$ (53)

where $\xi$ and $\omega$ are the location and scale parameters respectively.

The probability distribution function with location $\xi$, scale $\omega$ and
skewness parameter $\alpha$ becomes

$f(x)=\frac{2}{\omega}\phi\left(\frac{\nu-\xi}{\omega}\right)\Phi\left(\alpha\left(\frac{\nu-\xi}{\omega}\right)\right).$
(54)

Introducing the amplitude parameter $A_{0}$, this takes the form

$f(\nu)=\frac{A_{0}}{\omega\pi}\mathrm{e}^{-\frac{(\nu-\xi)^{2}}{2\omega^{2}}}\int_{-\infty}^{\alpha\left(\frac{\nu-\xi}{\omega}\right)}\mathrm{e}^{-\frac{t^{2}}{2}}\,\mathrm{d}t.$
(55)