Cosmology PDF

Cosmology Dr. Rafiee 1 Cosmology Is the study of the universe. To construct a model of the universe that make sense: Experiences Beliefs Religious reasoning Philosophical reasoning Scientific reasoning 2 Co...

Cosmology Dr. Rafiee 1 Cosmology Is the study of the universe. To construct a model of the universe that make sense: Experiences Beliefs Religious reasoning Philosophical reasoning Scientific reasoning 2 Cosmologist Everyone could be a cosmologist? How big is the universe? How old the the universe? What is the fate of the universe? etc. 3 Cosmological Principle Principle: The universe is homogenous and isotropic in large scales. Implication: Universe has no centre (or no special point in the universe) Universe has no edge. 4 Homogeneity All places look alike. No special location in the universe. 5 Isotropy All direction look alike. No special direction in the universe. 6 7 12 First Evidences for isotropy Distribution of radio galaxies observed with the Very Large Array (VLA) in New Mexico There are 40,000 bright radio galaxies in the upper panel with isotropic distribution There are the same number of fainter radio galaxies in the lower panel with isotropic distribution Proc. Natl. Acad. Sci. USAVol. 96, pp. 4756 – 4758, April 1999 13 https://www.pnas.org/content/pnas/96/9/4756.full.pdf Second Evidences for isotropy Distribution of diffused microwave background. What we see is the mean temperature of the CMB of 2.725 Kelvin. On this rough scale it looks very uniform. isotropic 14 Evidence for Homogeneity It is much harder to show it. If universe is Homogenous? If one place is isotropic, then all other places must be isotropic as well. If universe is not Homogenous? One place might be isotropic, but the other places could be anisotropic. Therefore, proving one place is isotropic doesn’t prove homogeneity Although, inhomogeneity requires the existence of “special locations” 15 Homogenous on large scales but not on small scales By Max Tegmark Universe is homogenous if one smear out the irregularities (e.g, galaxies, and clusters …) Thus inhomogeneity decreases at large distances Universe is homogenous in large scale Universe is inhomogeneous (clumpier) in small scale https://space.mit.edu/home/tegmark/sdss.html 16 Universe is homogenous in large scale (this image) Universe is inhomogeneous (clumpier) in small scale Universe is homogenous in large scale Universe is inhomogeneous (clumpier) in small scale (this image) Perfect cosmological principle? When you have a perfect symmetry in space and time. No special locations exist in space and time No special direction exist in space and time Implications: If perfect cosmological principle is valid then physical state of the universe should not change with time! 1. You should have static universe: which means it doesn’t expand! (∴ 𝐻0 = 0) (However, this is against the Hubble law observation) 2. You have steady state universe: universe expands at a constant rate (∴ 𝐻0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡). Matter continuously created. (Against the accelerating-expanding universe) Conclusion: Perfect cosmological principle is not valid since the universe is in accelerating expanding state. 19 Is time different from space? The space-time symmetry is broken! Space is reversible Time is NOT reversible Almost all fundamental theories of physics possess time reversibility Space and time is unified in special and general theory of gravity. Quantum mechanics is also fine with time reversal. However, there is no reversibility in everyday phenomena (macroscopic), like thermodynamics. a.k.a 2 law of thermodynamics. nd 20 The Second law of thermodynamics It states that There is a cost for the first law of thermodynamics (energy is conserved but can be converted to any type) heat always flow naturally from Hot reservoir to Cold reservoir. Entropy increases or remain constant in any natural processes. Time moves forward. Decreasing entropy is Not natural: Natural there is a cost for it when it happens 21 The Second law of thermodynamics when you have a closed system, total energy is conserved, which means Energy doesn’t change with time. But energy still can exchange between different part of that system, can change type inside that system. electric potential energy can change to radiation potential gravity can change to motion. But why this internal exchange is permitted? Because there is a cost for that change. that cost is the second law. Entropy of the final state is higher that initial state. Entropy Is disorder, randomness, … Is related to the amount of information Is related to the degree of disturbance or complexity Δ𝑄 It is defined as Δ𝑆 = in thermodynamics Δ𝑇 Δ𝑄 is the heat exchange during the process Δ𝑇 is the temperature change during the process Δ𝑆 is the change in entropy during the process It is defined as 𝑆 = 𝑁𝑘𝐵 𝑙𝑜𝑔(𝑊) for a microscopic system 𝑁 is he number of particles in the system 𝑘𝐵 is the Boltzman constant = 1.38 × 10−23 𝑚2 𝑘𝑔𝑠 −2 𝐾 −1 𝑊 is the number of possible states in the system 23 The shape of the universe? Is there an edge for the universe? The Quilted Multiverse What is outside of the universe? What is the universe expanding into? Penrose’s universe What was before the Big Bang? When/Where/How and why the Big Bang occur? What is the shape of the universe? How many dimensions this universe has or working with? 25 What is the Curvature of space and time? The radius of an osculating circle can be used to measure curvature of a line at a given point. 1 𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 = 𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 Curvature is in units of 1/length A straight line (zero curvature) has R=infinity. Gaussian Curvature: curved surfaces To determine the curvature of a surface: Draw two principle osculating circles at a given point on the surface Obtain two principal curvature radii, 𝑅1 𝑎𝑛𝑑 𝑅2. Gaussian curvature is given by 1 K= 𝑅1 𝑅2 K is in units of 1/area How Curved is the space and time? Homogenous and Isotropic space can be either flat, spherical or hyperbolic. 𝐾 > 0 we get spherical space 𝐾 = 0 we get flat space 𝐾 < 0 we get hyperbolic space Special Relativity (1905) Unified the space and time to spacetime distance. Spacetime is flat. No absolute time and No absolute space. https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.29.189 Two Invariants: Speed of light exact value is defined as 299 792 458 metres per second Spacetime distance 2 2 2 2 𝑑𝑠 = 𝑐 𝑑𝑡 − 𝑑𝑥 General Relativity (1916) Is an extension of special relativity, a hope to include the gravity in that theory. Equivalence Principle Equivalent principle states that inertial mass and gravitational mass are the same mass. Or acceleration and gravity are the same. 𝐺𝑀𝑔 𝑚𝑔 2 = 𝐹𝑔 ≡ 𝐹𝑖 = 𝑚 𝑖 𝑎 𝑅 Gravity causes the geometry of spacetime to become curved. Einstein Field Equation: 8𝜋𝐺 8𝜋𝐺 𝐺𝜇𝜈 = 4 𝑇𝜇𝜈 → 𝐺𝜇𝜈 +𝛬𝑔𝜇𝜈 = 4 𝑇𝜇𝜈 𝑐 𝑐 1 𝐺𝜇𝜈 is the Einstein Tensor (defines the curvature of space) 𝐺𝜇𝜈 = 𝑅𝜇𝜈 − 𝑅𝑔𝜇𝜈 2 𝑅𝜇𝜈 is the Ricci curvature tensor and R is the scalar curvature. 𝑔𝜇𝜈 is the metric curvature 2 𝜇 𝜈 𝑑𝑠 = 𝑔𝜇𝜈 𝑑𝑥 𝑑𝑥 𝑇𝜇𝜈 is the energy-momentum tensor 𝛬 is the cosmological constant.  𝑘𝑚 𝑣𝑠𝑢𝑛 = 617 𝑠 2 617 180 × 60 × 60 “ 𝜃~2 × ~1.75 “ 3 × 105 𝜋  𝜆𝑜 1+𝑧 = 𝜆𝑒 https://www.ligo.caltech.edu/video/gravitational-waves