Origami Mathematics and Cosmology MWF 9:15-11:00am! Bloomberg 278! Instructor: Dr. Mark Neyrinck! Assoc. Research Scientist, Physics & Astronomy

Mark Neyrinck, JHU Cosmology noun 1. the science of the origin and development of the universe. Modern cosmology is dominated by the Big Bang theory, which brings together observational astronomy and particle physics. 2. an account or theory of the origin of the universe. ! ORIGIN mid 17th cent.: from French cosmologie or modern Latin cosmologia, from Greek kosmos ‘order, world’ + -logia ‘discourse.’ Cosmology Concerned with the following questions: (anciently) ! • What makes the sun rise and fall, and why does it do that? • What are stars? Where are the stars? • Why do some “wandering stars” (planets) move compared to the rest of the stars? • Was there a beginning? • If so, what was it like? • How did the “world” come to be? • Will there be an end? • If so what will happen? • What is the nature of the Universe? • How is stuff arranged in the Universe? • What role do humans play in it? Modern Cosmology Concerned with the following questions: (modernly) ! • What makes the sun rise and fall, and why does it do that? • What are stars? Where are the stars? astronomy • Why do some “wandering stars” (planets) move compared to the rest of the stars? } • Was there a beginning? • If so, what was it like? • How did the “world” come to be? • Will there be an end? • If so what will happen? } interesting, still cosmology, but not testable • What is the nature of the universe? • How is stuff arranged in the Universe? • What role do humans play in it? } philosophy? Origami

noun the Japanese art of folding paper into decorative shapes and figures. ! ORIGIN Japanese, from oru, -ori ‘fold’ + kami ‘paper.’ Origami mathematics noun The study of the underlying laws of the art and science of origami.

Example questions: - What structures can fold at all/in a particular way? (mathematical question) - How do we design a particular shape? (engineering/design question) Origami mathematics

Robert J. Lang

Mark Neyrinck, JHU Origami applications • Art, of course • Robotics: Transformers, “programmable matter” • http://vimeo.com/102049447 • Self-folding lamp: • https://www.youtube.com/watch?v=xpHr87Crn3Y • Gracias Lab, JHU: https://www.youtube.com/watch? v=GL0im9b6GgU#t=18.5 ! • Protein folding • Cosmology?! “Fold your own universe” at the end Cosmology

- Two chronological ways to present: - Historical (from us out) - Pointless, including errors along the way? No! Science is a living, breathing thing! - From the Universe’s point of view (from the big bang inward) ! - We’ll go forth and back … Historically, our knowledge kind of mirrors this journey outward. Begin with simple model, then modify when necessary: “Occam’s razor” ! - Observation: sky, sun, stars, moon, planets … - Heliocentric model: requires “epicycles”: - orbits within orbits - “epicycles” still a word used metaphorically in science: “add epicycles” Clear test of heliocentricity: what are the phases of Venus like?

✔Verified by Galileo with his telescope, reported in The Starry Messenger.

Also, Jupiter has moons, so not everything revolves around the Earth What about the stars? Is there structure in the Milky Way?

Image Credit & Copyright: Max Rive What about the stars? Is there structure in the Milky Way?

Galileo (1610): observed the Milky Way with his new telescope. "The Galaxy is nothing else than a disordered jumble of innumerable stars distributed in clusters.” ! - First observation that the Milky Way was made of many unresolved faint stars. What about the stars? Is there structure in the Milky Way?

Immanuel Kant (1755): Misread a newspaper account of Wright's model. ! Milky Way is an “Island Universe”, a lens-shaped disk of stars rotating about its center. Other "nebulae" are distant, rotating milky ways like ours. What about the stars? Is there structure in the Milky Way?

Herschels’ (William and Caroline) observations: assuming all stars are equally bright, mapped farthest stars (1785) “Kapteyn Universe” (1922): based on measurement, assuming all stars are equally bright, mapped farthest stars. Similar conclusion as Herschel parsec: “parallax second” = 3.26 light years = 200,000 AU (dist from Earth to Sun) = 2x1013 miles = 3x1013 km What about “nebulae”? ! Shapley (1921) estimated distances to globular (star) clusters - Found that they are concentrated around Sagittarius - Postulated that the center of the Galaxy is in Sagittarius (correct!) • A debate persisted until ~1925 about whether the Milky Way was the Universe, or whether there were other galaxies (“island universes”) • Hubble (1923-1925): the Andromeda “nebula” is ~ 40 kpc = 1 Mpc away. • 1 Mpc is too far away to be part of the galaxy! It is as big as the Milky Way itself Modern view: How are the galaxies arranged? This is so-called “Large-scale structure” — my personal research focus ! ! ! ! ! - Jim Gott’s map of the universe … - Exoplanet app - “Laniākea” supercluster - https://www.youtube.com/watch?v=rENyyRwxpHo

- Simple dynamically: elliptical orbits, etc. - Simple; homogeneous

- Complicated?

How do we make these maps? - “Redshift Survey”: - Redshifts = Doppler-shifts of light - Survey = same as a surveyor does! Measuring, mapping out a region. ! - One of the first redshift surveys to show deep structure: ! Refresher on Doppler shifting … ! Applied to light, this is called red/blue shifting … ! A plot of how much light is coming from an object at different frequencies is called its spectrum. ! Spectra have: - “broadband” features, and ! - “lines” (emission lines, absorption lines) ! Applied to light, this is called red/blue shifting … ! A plot of how much light is coming from an object at different frequencies is called its spectrum. ! Applied to light, Doppler shifting is called red/blue shifting … ! In a redshift survey, many galaxies are observed on the sky, and their spectra are compared to a standard galaxy spectrum. ! How do we make these maps? - “Redshift Survey”: - Redshifts = Doppler-shifts of light - Survey = same as a surveyor does! Measuring, mapping out a region. ! - One of the first redshift surveys to show deep structure: ! Hubble’s Law (1929): galaxies are receding! Based also on Vesto Slipher’s data !

The closest galaxies are coming toward us, because of gravitational pull. Still, the Universe seems to be expanding! ! Hubble’s Law is fundamental to our understanding of cosmology. But it also gives us a way to map it out! ! The closest galaxies are coming toward us, because of gravitational pull. Still, the Universe seems to be expanding! ! Coming back to the “stickman diagram,” notice that it has weird “cz (km/s)” units — this is a recession velocity, redshift times the speed of light. This way to estimate distances is not perfect — the stickman’s body is called a “finger of god” ! A more recent map, from the Sloan Digital Sky Survey (lots of JHU involvement) ! Another recent one: the VIPERS survey. At “high redshift” up to z=1 ! Q: about how far away is the edge of the observable universe, given that the universe is ~14 billion years (14 Gyr) old? ! - Facts (implicitly) packed into that question: - There is an “edge of the observable Universe!” - There is not necessarily an “edge of the Universe” A: about the speed of light times the age of the Universe, 14 billion light years — there’s a factor of 4 or so because of expansion — its radius is ~46 billion light years (13 Gpc) Principles

! • Copernican principle: The Earth (or, by extension, the Sun, or the Milky Way) is not in a special location • not entirely true — the Earth supports life! • Anthropic principle: observers exist; thus the Universe must be such that observers can exist — not an a priori explanation, thus rather unsatisfying • There are various details about the Earth, and the Milky Way, that are rare, but that does not mean they are “special.” E.g., the Milky Way is quite rare for its wall “environment”. This is not an essential component of our cosmological model, though. Principles

! • Cosmological principle: There is no special place in the universe! More specifically, the laws of physics are the same everywhere — this idea arguably started with Newton’s universal law of gravitation

• Strong cosmological principle: there is no special location, direction, or time • (Weak) cosmological principle: there is no special location, or direction. Homogeneity and Isotropy

! • Before Hubble’s Law, the strong cosmological principle was preferred • With the cosmological principle, general relativity (GR) provided a way to study the evolution (e.g. expansion) of a generic patch of the Universe, through the metric • Friedmann-Lemaitre-Robertson-Walker, solution of Einstein’s equations for a homogeneous and isotropic universe: • constant density and pressure everywhere • no preferred direction Q. Are the following homogeneous? isotropic? The History and Fate of the Universe: the Critical Density The History and Fate of the Universe: Omega (fateful Greek letter)

Ω=ρ/ρcrit The History and Fate of the Universe: Omega (fateful Greek letter)

Ω=ρ/ρcrit

Ω>1 Ω=1 Ω<1 ΩM+ΩΛ=1 • Strong cosmological principle: there is no special location, direction, or time • (Weak) cosmological principle: there is no special location, or direction.

The strong cosmological principle was the source of two “brilliant blunders”: [(c) 2013 Mario Livio, Space Telescope Science Institute] • Einstein (1917): introduced cosmological constant Λ to keep a static universe. 2 blunders: • This is unstable! • He could have predicted the expansion of the Universe found by Hubble and others! • Strong cosmological principle: there is no special location, direction, or time • (Weak) cosmological principle: there is no special location, or direction.

The strong cosmological principle was the source of two “brilliant blunders”: [(c) 2013 Mario Livio, Space Telescope Science Institute] • Fred Hoyle (1946): reconciled expanding universe with strong cosmological principle by postulating matter generation in voids ! • Made extremely unlikely by discovery of the ! Cosmic Microwave Background (1965) … • Blunder: holding onto the theory even after this and other evidence Perhaps the Strongest Pillar of Cosmology: The Cosmic Microwave Background

Penzias & Wilson (1965) Nobel prize, 1978 COBE: COBE (1991) Nobel Prize 2006 (Mather & Smoot) Back to the blunder though … there seems to be a cosmological constant!

Ω>1 Ω=1 Ω<1 ΩM+ΩΛ=1 The scale factor a(t) in different cosmological models Back to the blunder though … there seems to be a cosmological constant! Adam Riess, Perlmutter & Schmidt, 2011 Nobel Prize Back to the blunder though … there seems to be a cosmological constant! ! - Very unsettling theoretically though — the “natural” value of the cosmological constant is 10122 times larger than the observed value! - The multiverse model: there are many fields that get added together to make this cosmological constant — some of them have a low one Back to the blunder though … there seems to be a cosmological constant! ! - Not unexpected in quantum field theory — “zero- point energy” (e.g. electron in an atom, applied to the Universe!) would act like a cosmological constant ! - Unsettling theoretically though — the “natural” value 5 2 122 of ρΛ is the Planck density, ρPl = c /(ℏG ), 10 times its apparent value! - By Occam’s razor, most people assumed ρΛ=0. - “Predicted” by Steven Weinberg — to form galaxies as we know them, |ΩΛ|≈0. - ρΛ larger and +: Universe would have accelerated away before galaxies could have formed - ρΛ larger and - : Universe would have recollapsed - The multiverse model: perhaps there are many quantum fields that get added together to make this cosmological constant — some have |ΩΛ|≈0 - Anthropic principle again … - That’s dark energy … what about dark matter? - How do we know dark matter exists? - Galaxy clusters have huge mass-to-light ratios, estimated by measuring galaxy velocities in clusters (Fritz Zwicky) - So do galaxies — “flat” rotation curves ! ! - How do we know dark matter exists? - Gravitational lensing also indicates large “dark” mass ! ! - How do we know dark matter exists? - Dark matter does not seem to be made out of large lumps, e.g. lots of black holes or asteroids: - WIMPs (weakly-interacting massive particles) vs - MACHOs (MAssive Compact Halo Objects) - MACHOs could be detected with “microlensing” — were not - Dark matter seems “collisionless;” one patch of dark matter can slide right through another - Why? More structure formed than it could have with “collisional” usual matter - Good visual: “Bullet Cluster” !

- Another reason: this curve fits really well! ! Inhomogeneities! (toward origami)

A billion light years Folding in a 1D universe ...

Mark Neyrinck, JHU N-body cosmological simulation in phase space: a 2D slice

vx y x

z y x

Mark Neyrinck, JHU Halo spins in a 2D simulation

Galaxy spins? ! • To minimize # streams, haloes connected by filaments have alternating spins • A void surrounded by haloes will therefore have an even # haloes