The Perfect Cosmological Principle
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1 Cosmoloy AS7009, 0 Lecture Outline The cosmoloical principle: Isotropy Homoeneity Bi Ban vs. Steady State cosmoloy Redshift and Hubble s law Scale factor, Hubble time, Horizon distance Olbers paradox: Why is the sky dark at niht? Particles and forces Theories of ravity: Einstein vs. Newton Cosmic curvature Covers chapter + half of chapter 3 in Ryden The Cosmoloical Principle I Modern cosmoloy is based on the assumption that the Universe is: Homoeneous The cosmoloical principle Isotropic The Cosmoloical Principle II These tenets seem to hold on lare scales (>00 Mpc), but definitely not on small Voids typically 70 Mpc across The Perfect Cosmoloical Principle In this case, one assumes that the Universe on lare scales is: Homoeneous Isotropic Non-evolvin This is incompatible with the Bi Ban scenario, but the Steady State model (popular in the s) was based on this idea
2 Steady State Cosmoloy Universe continuously expands, but due to continuous creation of matter, no dilution occurs Steady State, no hot initial Bi Ban and no initial sinularity a(t)=exp(ht) Suestion for Literature Exercise: Quasi-Steady State Cosmoloy Attempt in 990s to resurrect Steady-State State and explain the: CMBR production of liht elements dark matter supernova type Ia data Lare-scale structure But this cosmoloy fails to explain the CMBR, T CMBR (z), the production of liht elements and the redshift evolution of alaxies & AGN Suestion for Literature Exercise: Quasi-Steady State Cosmoloy Cyclic creation events Lon-term steady expansion, but with short-term term oscillations To meet observational constraints, QSS requires: Strane interalactic dust Cyclic creation events ( little bans ) which cause local expansion of space Scale factor a(t) time Short-term oscillations Lon-term steady expansion Redshift Definition of redshift: λobs λ z = λ em em Redshift or velocity Hubble s law I Distance Galaxies Hubble s law: At low z: Hubble s law II v z c The Hubble constant H d c 0 z = v = H 0d Luminosity distance In observational astronomy, the term recession velocity, v, occurs frequently:
3 Expansion of the Universe I Expansion of the Universe II Redshift and distance I Low redshift (z 0) corresponds to: Small distance (local Universe) Present epoch in the history of the Universe Hih redshift corresponds to: Lare distance Earlier epoch in the history of the Universe Redshift and distance II But beware: At low redshift, Doppler components comin from peculiar motions may be substantial must be corrected for before d is derived from z or v The redshift comin from cosmic expansion is not a Doppler shift don t treat it like one! The linear version of Hubble s law is only appropriate at z<0.5 (at 0% accuracy) Scale factor Scale factor and redshift r r 3 r 3 3 Time t (earlier) Scale factor a(t) r r 3 r 3 Time t 0 (now) Scale factor a(t 0 )= r (t) = a(t)/a(t 0 ) r (t 0 ) = a(t) r (t 0 ) v (t) = a /a r (t) 3 a + z = 0 = a a Cosmic scale factor today (at t 0 ) can be set to a 0 = Cosmic scale factor when the Liht was emitted (the epoch correspondin to the redshift z) 3
4 Today The Hubble constant H 0 7±5 km s - Mpc - [s - ] Errorbars possibly underestimated Note: Sloppy astronomers often write km/s/mpc Hubble time In the case of constant expansion rate, the Hubble time ives the ae of the Universe: t H = H 0 4 Gyr In eneral: H a & a Not a constant in our Universe! In more realistic scenarios, the expansion rate chanes over time, but the currently favoured ae of the Universe is still pretty close around 3 4 Gyr. Olbers paradox I Olbers paradox II Why is the sky dark at niht? (Heinrich Olbers 96) If the Universe is: Spatially infinite (i.e. infinite volume) Infinitely old and unevolvin - then the niht sky should be briht! Olbers paradox III Olbers paradox IV Horizon distance Planet Earth surrounded by stars in an infinte, unevolvin Universe Main solution: The Universe has finite ae The liht from most stars have not had time to reach us! 4
5 Horizon distance Horizon distance = Current distance to the most faraway reion from which liht has had time to reach us This delimits the causally connected part of the Universe an observer can see at any iven time Horizon distance at time t : d hor ( t ) = c t t = 0 dt a( t) Most realistic scenarios ive: d hor (t 0 )~c/h 0 (the so-called Hubble radius) Particles and forces I The particles that make up the matter we encounter in everyday life: Protons, p MeV Neutrons, n MeV Electrons, e MeV Baryons (made of 3 quarks) Lepton Since most of the mass of ordinary matter is contributed by protons and neutrons, such matter is often referred to as baryonic. Examples of mostly baryonic objects: Planets, stars, as clouds (but not alaxies or alaxy clusters) Particles and forces II Other important particles (for this course): Photon, γ Massless, velocity: c Neutrinos, ν e ν µ ν τ Leptons ~ev (?), velocity close to c Interacts via weak nuclear force only Particles and forces III The four forces of Nature: Stron force Very stron, but has short rane (~0-5 m) Holds atomic nuclei toether Weak force Weak and has short rane Responsible for radioactice decay and neutrino interactions Electromanetic force Weak but lon-rane Acts on matter carryin electric chare Gravity Weak, very lon-rane and always attratice On the lare scales involved in cosmoloy, ravity is by far the dominant one Newtonian ravity Space is Euclidian (i.e. flat) Planet are kept in their orbits because of the ravitational force: GM F = r The acceleration resultin from the ravitational force: Inertial mass m F = m a i Gravitational mass Equivalence Principle Gravitational acceleration towards an object with mass M is: GM = r a m m Empirically M =M i (to very hih precision) The equality of ravitational mass and inertial mass is called the equivalence principle In Newtonian ravity, M =M i is just a strane coincidence, but in General Relativity, this stems from the idea that masses cause curvature of space i 5
6 General Relativity 4D space-time Mass/enery curves space-time Gravity = curvature Pocket summary: Mass/enery tells space-time how to curve Curved space-time tells mass/enery how to move Small-scale curvature Small-scale distortions caused by astronomical objects What about the lare-scale curvature? Global Curvature I In the world models of eneral relativity, our Universe may have spatial curvature (on lobal scales) Positive curvature Global Curvature II Very tricky stuff This an intrinsic curvature in 3D space Note: No need for encapsulatin our 3D space in 4D space to make this work Neative curvature Zero curvature (flat/euclidian space) This represents 3D No need for anythin here α γ Global Curvature III Anles in curved spaces β Flat: α+β+γ = 80 α γ β Neative: α+β+γ < 80 α γ β Positive: α+β+γ > 80 Metrics I Metric: A description of the distance between two points Metric in dimensional, flat space: ds + = dx dy (Pythaoras) Metric in 3 dimensional, flat space: d s = dx + dy + dz 6
7 Metrics II Metric in 3 dimesions, flat space, polar coordinates: ds = dr + r dω dω = dθ + sin θ dφ Metric in 3 dimensions, arbitrary curvature: dx ds = + x dω κx / R Flat : κ = 0, x = r Neative : Positive : κ =, x = R sinh( r / R) κ =, x = R sin( r / R) Curvature radius 7
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