Foundations of Cosmology

1. Foundations of Cosmology Dr Timothy Clifton QMUL Goldsmiths AstroCourse

2. From Newton to Einstein • Created the foundations of the Classical Laws of Motion • Provides the Newtonian Law of Universal Gravitation • Allows a simple model for the expansion of the Universe • Not consistent with the invariance of the speed of light • Not consistent with modern observations of gravity • Created a new theory of space and time, compatible with an observer independent speed of light • Extended this to include gravity, in the General Theory of Relativity • Allows for sophisticated modelling of the Universe • Not consistent with Quantum Field Theory • Requires large amounts of Dark Matter & Dark Energy

3. Space, Time & Gravity in Newton’s Theory • Newton envisages a fixed space, in which physics plays out: • He provides us with laws of motion, and a law of gravitation: time time space space

4. Newtonian Cosmology I • Consider some evenly distributed matter as a set of concentric spheres: • Apply Newton’s laws to each sphere: Newton’s sphere theorem

5. Newtonian Cosmology II • Consider the force on some particles on a sphere of radius r: • The total energy of each particle is: Newtonian Hubble rate

6. Problems with Newtonian Cosmology • It’s based on a theory which is known to be only an approximation to nature, valid at low velocities and energies. • It provides no real insight into the way space and time behave on the scale of the observable Universe. • For (spatially) constant densities, the velocity goes like . This exceeds the speed of light at large enough distances. We require Einstein’s theory!

7. Space and Time in Einstein’s theory • There is no absolute space. What appears as space to one observer is not the same as what appears as space to another in relative motion: • It is only the union of space and time (space-time) which has any observer independent meaning: Required for everyone to measure the same speed of light! time space “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” Hermann Minkowski

8. Gravity in Einstein’s Theory • Space-time is not flat, but is curved by energy and momentum: • In the absence of external forces, particles follow the shortest paths available (c.f. Newton’s 1st law). • What we experience as gravity in our everyday lives is, in reality, an artefact of our habit of interpreting physics as occurring in flat space.

9. The Geometry of Flat Space • In a flat space we use Pythagoras’ theorem: • In three dimensions this becomes: [or ]

10. The Geometry of Curved Space • Use an infinitesimal version of Pythagoras’ theorem: where D is the number of dimensions. (2 for a plane, 3 for space, and 4 for space-time). • The object is called the “line-element”, and tells you about the length of a line in a curved space (c.f. the hypotenuse). • The object is called the “metric”, and tells you everything about the curvature of the space.

11. Cosmology in Einstein’s Theory • To perform cosmology in Einstein’s theory we need to know about the geometry of space-time. • To find the geometry of space-time we need to solve Einstein’s field equations (a set of differential equations that relate the curvature of space-time to the matter content of the Universe): • Some knowledge (either from astronomers, or assumed) about the matter content of the Universe. • An understanding of what specific matter distributions imply about the geometry of space-time. energy & momentum of matter curvature (2nd derivative of metric) This requires:

12. Observations of the Universe I • The distribution of galaxies in the Universe is being mapped: Credit: M. Blanton and the Sloan Digital Sky Survey.

13. Observations of the Universe II • The background radiation in the Universe is being mapped: Credit: ESA and the Planck Collaboration.

14. From Observations to Geometry • Large galaxy surveys are suggestive of a space-time geometry that is close to homogeneous (the same at every point) and isotropic (the same in every direction). • The background radiation of the Universe suggests isotropy of the Universe around us, and homogeneity at early times. Proof, however, requires knowledge of the peculiar velocities (i.e. small-scale motions), lensing of light (i.e. how light bends as it passes massive objects), and luminosity distances (i.e. how bright distant objects are). [R. Maartens and D. R. Matravers, Class. Quantum Grav. 11 (1994) 2693] This can be formulated as an exact proof if one assumes that the radiation was emitted with a black-body spectrum in the early universe, and if it is measured to have a black-body spectrum today. [T. Clifton, C. Clarkson and P. Bull, Phys. Rev. Lett. 109 (2012) 051303]

15. Homogeneous & Isotropic Space-Times I • If a space-time is spatially homogeneous and isotropic then its line-element can always be written as: where is known as the “scale factor”, and is the curvature of space:

16. Homogeneous & Isotropic Space-Times II • The evolution of the “scale” of space is given by Einstein’s equations as: • Solutions to this equation behave like this: Compare with the corresponding equation in Newtonian cosmology! • If k>0 then the Universe will eventually re-collapse • If k=0 then the Universe has spatial sections that are expanding flat planes • If k<0 then the Universe eventually expands with a scale factor a(t)=t

17. Homogeneous & Isotropic Space-Times III • The global structure of space-times that do not re-collapse looks like this: time space a(t)=0 at this point (the big bang!)

18. Inhomogeneous Space-Times I • The real Universe is not exactly homogeneous and isotropic. We must therefore check that inhomogeneous space-times behave in a similar way to homogeneous space-times, if we are to use them as models of the real universe. • Inhomogeneous cosmological solutions of Einstein’s equations are notoriously difficult to find. • One can make progress by considering simple, regular distributions of discrete massive objects. [A. Krasinski, Inhomogeneous Cosmological Models, Cambridge University Press 1997] [H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers & E. Herlt, Exact Solutions of Einstein’s Field Equation – 2nd Edition, Cambridge University Press 2003] [T. Clifton, K. Rosquist and R. Tavakol, Phys. Rev. D 86 (2012) 043506] [T. Clifton, D. Gregoris, K. Rosquist and R. Tavakol, JCAP 11 (2013) 010]

19. Inhomogeneous Space-Times II • By condensing all the matter in the Universe into a small number of masses, we can check that the large-scale behaviour is similar to that of a homogeneous model with the same mass. More masses means more spherical, and closer to the behaviour of a homogeneous universe! [T. Clifton, K. Rosquist and R. Tavakol, Phys. Rev. D 86 (2012) 043506]

20. Summary • One can perform cosmology in Newton’s theory, but while the equations that govern the large-scale behaviour may be similar to those of Einstein’s theory, the interpretation and compatibility with observations is found to be lacking. • By using Einstein’s theory we can gain a deep understanding of the nature of space and time in our Universe, as well as how space-time behaves as the Universe evolves. • The homogeneous and isotropic solutions of Einstein’s equations are well motivated by observations, and provide a working model within which observations can be interpreted. • There remain a number of open questions at the heart of cosmology. How did the early Universe behave? What provided the seeds of structure in the Universe? To what degree can Einstein’s theory be reliably applied to the Universe as a whole? How should we best model the Universe? What are Dark Matter and Dark Energy? Etc. etc.

21. Thank you