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A. Cole and G. Shiu
contain more information about a set than the Euler character. This was exploited
in [23], which showed on a set of CMB simulations that the Betti numbers were
quantitatively more sensitive to a particular form of primordial non-Gaussianity than
the Euler character. Moreover, in Sect. 9.2 we saw that the persistence diagram in
fact contains more information than the persistent Betti numbers. One might then
ask how much of the information ignored by persistent Betti numbers is useful for
statistical discrimination. As such, in [29] the present authors studied the ability of
persistent homology to identify primordial local non-Gaussianity. More recently, [85]
computed the persistent Betti numbers of Planck data and, comparing to the persistent
Betti numbers of Planck best-fit simulations, identified some mild anomalies.
An important theme in recent work is that persistence-based statistics would be
much more powerful with a strong analytic foothold. While analytic expressions
for the MFs are known for mildly non-Gaussian fields [49, 75], we do not have
analogous expressions for even the persistent Betti numbers of a Gaussian random
field. These quantities were studied numerically in [87]. Additionally [47] proposed
a semi-analytic formula for the persistent Betti numbers of a Gaussian random field.
A full derivation, however, is still lacking. A higher aim would be predictions for the
persistence diagrams of these random fields.
Some analytic progress for other geometric observables has been made in [25],
which used DisPerSE [93] to extract the “persistent skeleton” (i.e. network connecting maxima and saddle points of a 3D field) of Gaussian random fields and
cosmological simulations, and found agreement regarding the connectivity of this
skeleton with predictions made via peak theory.
For progress from the “point-like” perspective, [86] computed persistence diagrams and persistent Betti numbers for simulations of some toy cosmological models, see also [86, 101]. Similar methods were applied to SDSS data in [66]. One
intriguing aspect of this approach is one can probe the dynamics of cosmological
evolution (including for example the dark energy equation of state) by studying LSS
at different redshifts. More recently, [12] initiated a program for detecting primordial
non-Gaussianity and constraining cosmological parameters via the persistent homology of LSS, taking the first steps towards making persistent homology a standard
tool for computational cosmologists.
9.3.2 TDA for CMB [29]
In this subsection, we describe in more detail some relevant aspects of applying persistent homology to cosmology, following [29]. We review the ansatz of primordial
local non-Gaussianity and describe how this primordial imprint propagates to cosmological observables. We describe some publicly available simulations [46] and
the statistical sensitivity of a persistence-based pipeline to non-Gaussianity in these
simulations. We find that a combination of the persistence diagrams is nearly twice
as sensitive as the persistent Betti numbers to non-Gaussianity in these simulations.
Throughout, we emphasize several pitfalls to avoid when studying the persistent
homology of cosmologically-relevant non-Gaussian fields.
A. Cole and G. Shiu
contain more information about a set than the Euler character. This was exploited
in [23], which showed on a set of CMB simulations that the Betti numbers were
quantitatively more sensitive to a particular form of primordial non-Gaussianity than
the Euler character. Moreover, in Sect. 9.2 we saw that the persistence diagram in
fact contains more information than the persistent Betti numbers. One might then
ask how much of the information ignored by persistent Betti numbers is useful for
statistical discrimination. As such, in [29] the present authors studied the ability of
persistent homology to identify primordial local non-Gaussianity. More recently, [85]
computed the persistent Betti numbers of Planck data and, comparing to the persistent
Betti numbers of Planck best-fit simulations, identified some mild anomalies.
An important theme in recent work is that persistence-based statistics would be
much more powerful with a strong analytic foothold. While analytic expressions
for the MFs are known for mildly non-Gaussian fields [49, 75], we do not have
analogous expressions for even the persistent Betti numbers of a Gaussian random
field. These quantities were studied numerically in [87]. Additionally [47] proposed
a semi-analytic formula for the persistent Betti numbers of a Gaussian random field.
A full derivation, however, is still lacking. A higher aim would be predictions for the
persistence diagrams of these random fields.
Some analytic progress for other geometric observables has been made in [25],
which used DisPerSE [93] to extract the “persistent skeleton” (i.e. network connecting maxima and saddle points of a 3D field) of Gaussian random fields and
cosmological simulations, and found agreement regarding the connectivity of this
skeleton with predictions made via peak theory.
For progress from the “point-like” perspective, [86] computed persistence diagrams and persistent Betti numbers for simulations of some toy cosmological models, see also [86, 101]. Similar methods were applied to SDSS data in [66]. One
intriguing aspect of this approach is one can probe the dynamics of cosmological
evolution (including for example the dark energy equation of state) by studying LSS
at different redshifts. More recently, [12] initiated a program for detecting primordial
non-Gaussianity and constraining cosmological parameters via the persistent homology of LSS, taking the first steps towards making persistent homology a standard
tool for computational cosmologists.
9.3.2 TDA for CMB [29]
In this subsection, we describe in more detail some relevant aspects of applying persistent homology to cosmology, following [29]. We review the ansatz of primordial
local non-Gaussianity and describe how this primordial imprint propagates to cosmological observables. We describe some publicly available simulations [46] and
the statistical sensitivity of a persistence-based pipeline to non-Gaussianity in these
simulations. We find that a combination of the persistence diagrams is nearly twice
as sensitive as the persistent Betti numbers to non-Gaussianity in these simulations.
Throughout, we emphasize several pitfalls to avoid when studying the persistent
homology of cosmologically-relevant non-Gaussian fields.