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A. Cole and G. Shiu
is the persistence landscape [16]. Coupling a differentiable persistence computation
to a neural network architecture and proving stability bounds is an active area of
research, see e.g.. [15, 20, 62, 63, 65, 84].
9.3 TDA for Cosmology
Cosmology is marching into an era of Big Data. For example, ongoing and upcoming
surveys promise to generate huge numbers of new observations of galaxies, giving
us greater access to the history of the universe than ever before. To fully leverage
this flood of data, we would benefit greatly from a general pattern recognition tool.
However, to sufficiently trust such a tool, we should understand the pipeline in terms
of familiar concepts. For example, for a given N-body simulation of Large-Scale
Structure, the simulation’s trustworthiness breaks down before the simulation’s grid
size. (See [92] for a comparison of matter power spectrum predictions with various
simulation methods.) We would like to be able to formulate a data science pipeline that
does not rely on simulation details at this scale. It appears that persistent homology
could provide such a general tool.
For example, we would like to learn about inflation. Inflation is a postulated
period of exponential expansion in the early universe that solves the flatness, horizon, and monopole problems of Big Bang cosmology [6, 52, 72, 94]. During this
inflationary phase, quantum fluctuations are stretc.hed to superhorizon scales, where
they freeze and classicalize. When these fluctuations reenter the horizon, they evolve
under gravity, with overdense regions collapsing to form the cosmic structure we
observe today. Generically, inflation predicts an almost scale-invariant primordial
density perturbation spectrum [11, 53, 58, 79–81, 95]. These perturbations are not
exactly Gaussian, but their deviation from Gaussianity is typically small for single
field slow-roll inflation [73]. A broader class of inflationary models, even within
the single field context, allows significant levels of non-Gaussianity with distinctive
“shapes” (functional dependences on momenta) [22]. These shapes reflect the underlying physics of the model, e.g.. a large non-Gaussianity in the so-called squeezed
limit can only by achieved by multi-field inflation. We therefore aim to constrain
models of inflation by studying the statistics of cosmological observables and reverse
engineering the quantum perturbations and primordial physics that generated them.
Further non-Gaussianities develop via the nonlinearity of gravitational evolution,
so that we can also access the dynamics of cosmological expansion (and e.g.. the
dark energy equation of state) via non-Gaussianities. Observables allowing us to do
this include the Cosmic Microwave Background (CMB) and the “cosmic web” of
Large-Scale Structure (LSS).
However, directly detecting primordial non-Gaussianity via higher-order moments
proves difficult in practice, as it relies on the measurement of rare events. Moreover,
since a random variable can deviate from Gaussianity in infinitely many ways, there is
no single signature of primordial non-Gaussianity. So far, primordial non-Gaussianity
in the CMB has proven elusive, with the primordial non-Gaussian component of data
A. Cole and G. Shiu
is the persistence landscape [16]. Coupling a differentiable persistence computation
to a neural network architecture and proving stability bounds is an active area of
research, see e.g.. [15, 20, 62, 63, 65, 84].
9.3 TDA for Cosmology
Cosmology is marching into an era of Big Data. For example, ongoing and upcoming
surveys promise to generate huge numbers of new observations of galaxies, giving
us greater access to the history of the universe than ever before. To fully leverage
this flood of data, we would benefit greatly from a general pattern recognition tool.
However, to sufficiently trust such a tool, we should understand the pipeline in terms
of familiar concepts. For example, for a given N-body simulation of Large-Scale
Structure, the simulation’s trustworthiness breaks down before the simulation’s grid
size. (See [92] for a comparison of matter power spectrum predictions with various
simulation methods.) We would like to be able to formulate a data science pipeline that
does not rely on simulation details at this scale. It appears that persistent homology
could provide such a general tool.
For example, we would like to learn about inflation. Inflation is a postulated
period of exponential expansion in the early universe that solves the flatness, horizon, and monopole problems of Big Bang cosmology [6, 52, 72, 94]. During this
inflationary phase, quantum fluctuations are stretc.hed to superhorizon scales, where
they freeze and classicalize. When these fluctuations reenter the horizon, they evolve
under gravity, with overdense regions collapsing to form the cosmic structure we
observe today. Generically, inflation predicts an almost scale-invariant primordial
density perturbation spectrum [11, 53, 58, 79–81, 95]. These perturbations are not
exactly Gaussian, but their deviation from Gaussianity is typically small for single
field slow-roll inflation [73]. A broader class of inflationary models, even within
the single field context, allows significant levels of non-Gaussianity with distinctive
“shapes” (functional dependences on momenta) [22]. These shapes reflect the underlying physics of the model, e.g.. a large non-Gaussianity in the so-called squeezed
limit can only by achieved by multi-field inflation. We therefore aim to constrain
models of inflation by studying the statistics of cosmological observables and reverse
engineering the quantum perturbations and primordial physics that generated them.
Further non-Gaussianities develop via the nonlinearity of gravitational evolution,
so that we can also access the dynamics of cosmological expansion (and e.g.. the
dark energy equation of state) via non-Gaussianities. Observables allowing us to do
this include the Cosmic Microwave Background (CMB) and the “cosmic web” of
Large-Scale Structure (LSS).
However, directly detecting primordial non-Gaussianity via higher-order moments
proves difficult in practice, as it relies on the measurement of rare events. Moreover,
since a random variable can deviate from Gaussianity in infinitely many ways, there is
no single signature of primordial non-Gaussianity. So far, primordial non-Gaussianity
in the CMB has proven elusive, with the primordial non-Gaussian component of data