baryons+dm radiation+neutrinos vacuum curvature

baryons+dm radiation+neutrinos vacuum curvature

DENSITY ISN T DESTINY! The cosmologist T-shirt (if Λ>0):

ΩM,0=0.9 Ω Λ,0 =1.6 ΩM,0=2.5 ΩM,0=2.0 Ω Λ,0 =0.1 ΩM,0=2.0

http://map.gsfc.nasa.gov/resources/camb_tool/cmb_plot.swf

ECHOS FROM THE BIG BANG COBE As the Universe expands and cools down, the energy spectrum of the CMB continues to correspond to a thermal distribution, but one with even lower temperature.. launched Nov 89 6 270.42 C 454.76 F The CMB is the most accurate Planck curve ever measured. Physicists in their labs cannot make a better blackbody!

WMAP cold spot: T=2.7262 K launched June 01 hot spot: T=2.7266 K The WMAP all sky map, after removal of the radiation coming from Milky Way disk.

A NEARLY PERFECT UNIVERSE How to Measure the CMB Power Spectrum 1.Measure <T> inside a circular region of diameter θ degrees centered on a random spot. 2.Repeat this for many random spots. 3.Measure the variance (`scatter ) in these average temperatures. This is the power on a scale of θ degrees. Planck 4.Repeat steps 1-3 for different values of θ. The power as a function of θ is called the power spectrum.

Planck

FLAT LIKE A PANCAKE

DARK MATTERS Baryons 10 t0 68% Dark Energy 5% Dark Matter 27% 13.7 GYR AGO Dark Energy 100% TODAY 24% Radiation Baryons 12% Dark Matter 64%

Best cosmological model (Planck+ext, k=0) as of 2015! EdS

Note that, if dark energy Λ, then: can get acceleration with an energy density that decreases with time! phantom energy Big Rip! singularity in the future! If w 1, the coincidence between the observed vacuum energy and the current matter density appears completely unnatural, as the relative balance of vacuum and matter changes rapidly as the Universe expands:

As a consequence, at early times the vacuum energy was negligible in comparison to matter, while at late times matter is negligible. There is only a brief epoch of the Universe's history during which it would be possible to witness the transition from domination by one type of component to another. It seems remarkable that we live during the short transitional period between these two eras. Let us compute Ωi(a) as

Who ordered this? ΩR ΩM ΩΛ Selection Effect: AP? e

The approximate coincidence between matter and vacuum energies in the current Universe is one of several puzzling features of the composition of the total energy density. Another great surprise is the comparable magnitudes of the baryon density and the density of cold non-baryonic dark matter, and perhaps also that in massive neutrinos. In our current understanding, these components are relics of completely unrelated processes in the very early Universe, and there seems to be no good reason why they should be of the same order of magnitude. The real world seems to be a more rich and complex place than Occam s razor might have predicted. It is important to keep in mind, however, the crucial distinction between the coincidences relating the various matter components and that relating the matter and vacuum energy: the former are set once and for all by primordial processes and remain unchanged as the Universe evolves, while the latter holds true only during a certain era.

Written in 300 BC, Euclid s Elements is the most influential works in the history of mathematics: 13 books, studied for 24 centuries, 2nd only to the Bible in the number of editions published (>1000)!

It is impossible to derive the Parallel Postulate from the first four. The numerous (and failed) attempts to do that gave rise to a slew of statements equivalent to the postulate itself: 1. There exists a pair of similar noncongruent triangles. 2. There exists a pair of straight lines everywhere equidistant from one another. 3. For any three noncollinear points, there exists a circle passing through them. 4. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. 5. If a straight line intersects one of two parallels it will intersect the other. 6. Straight lines parallel to a third line are parallel to each other. 7. Two straight lines that intersect one another cannot be parallel to a third line. 8. There is no upper limit to the area of a triangle. The last one seems especially intuitive. The reverse holds in non- Euclidean geometries of Lobachevsky and Riemann. Lewis Carroll (mathematicians and author of Alice in Wonderland) could not accept this assertion and considered it as a proof of the contradictory nature of non-euclidean geometries.

Euclidean flat geometry: 1) the angles of a triangle add up to 180 o. 2) the circumference of a circle of radius r is equal to 2πr. Riemann (a student of Gauss) On the hypotheses which lie at the foundation of geometry (1868): 1) Euclid s final axiom was an arbitrary choice. 2) founded non-euclidean geometries which are the mathematical foundation of Einstein s GR. Simplest non-euclidean geometry is spherical (or elliptical) k>0: 1) Unlike the case of a flat geometry (k=0), the spherical surface is finite in extent (its area being 4πr 2 ), and yet there is no boundary or edge. If we draw parallel lines on the surface of the Earth, then they violate Euclid's final axiom.

2) The definition of a straight line is the shortest distance between two points (geodesic), which means that the straight lines in spherical geometry are segments of great circles, such as the equator or the lines of longitude. 3) If you draw a triangle on a sphere, we find that the angles do not add up to 180 o either: start at the North Pole, draw two great circles down towards the equator, 90 o apart, and then join them with a line at the equator. You have drawn a triangle in which all 3 angles are 90 o. 4) The circumference of a circle of radius r is smaller than 2πr.

Hyperbolic (saddle-shaped) geometry k<0: 1) In a hyperbolic geometry, parallel lines never meet in fact they break Euclid's axiom by diverging away from one another. Because parallel lines never meet, such a Universe must be infinite in extent, just an in the flat case. 2) The angles of a triangle add up to less than 180 o. 3) The circumference of a circle of radius r is greater than 2πr.

THE WAY OF NEWTON: Matter tells gravity how to exert a force (F=-GMm/r 2 ) Force tells mass how to accelerate (F=ma) THE WAY OF EINSTEIN: Matter-energy (stress-energy tensor) tells spacetime how to curve Curved space-time tells mass-energy how to move

Global Positioning System. The satellites making up the GPS system contain atomic clocks which broadcast a time signal, and the rate at which those clocks tick has to be adjusted to take general relativity into account. Without the relativistic correction, the clocks would drift by some +38 (+45 from GR -7 from SR) microseconds a day, corresponding to 11 km of position uncertainty. As the system works to give you your position on the Earth to within a few meters, we know that the relativistic correction works, and thus general relativity is correct.

4.2 METRIC A metric defines how a distance can be measured between two nearby events in spacetime in terms of the coordinate system. The metric is a formula which describes how displacements through a curved manifold can be translated into distances relates coordinate separations to lengths. Whether we use the term or not, we are all familiar with metrics. For example, using the Pythagorean theorem, we can write the distance between two points in ordinary 3D space as This is the so called Euclidean metric. In the special theory of relativity this becomes generalized to the distance between two spacetime points: We can write the infinitesimal version of this as dx0 cdt gij = metric tensor = 4x4 symmetric matrix

where the only non-zero component of gij are the diagonal terms (-1, 1, 1, 1). This is the Minkowski spacetime. Note that the line element is invariant under Lorentz transformations v as well as under coordinate transformations

Observers in relative motion disagree on spatial separation Observers in relative motion disagree on time separation Observers in relative motion agree on spacetime separation Metric turns observer-dependent coordinates into invariants! If two points x i and x i +dx i can be connected by a light ray, then photons propagate along null trajectories

Example: In a homogeneous and isotropic Universe there are no off-diagonal terms in gij. I shall give the argument that proves that there are no terms of the form, say, dxdt. Let s set dy=dz=0 so that ds 2 =dx 2 c 2 dt 2. If we now set ds=0, we have dx/dt=±c, which is the equation of a light ray propagating along the positive or negative x-axis ( null trajectory''). Let us suppose now we have a metric of the form ds 2 =dx 2 + dxcdt c 2 dt 2. If we set ds 2 =0 for this metric, there will be two very different solutions for dx/dt: dx/dt=(c/2) ( 1 ± 5). In this new case, because of the dxcdt term, the symmetry of the flat space case is destroyed.

Another way of thinking about the metric: when handed a vector (say connecting two grid points) we think of a line with an arrow (direction) attached, the length of the line corresponding to the length of the vector. This notion is rooted too firmly in flat Euclidean space! In actuality, the length of the vector depends on the metric. In the figure, the number of lines crossed by a vector is a measure of the vertical distance traveled by a hiker. Vector of the same apparent 2D length corresponding to identical coordinate separations corresponds to different physical distances. Contour map of Mauna Kea. Closely spaced contours near the center corresponds to rapid elevation gains. The two thin lines correspond to hikes of very different difficulty even though they appear to be of the same length.

The great advantage of the metric is that it can incorporate gravity. Instead of thinking of gravity as an external force and talking about particles moving in a gravitational field, we can include gravity in the metric and talk of particles moving freely in a distorted or curved spacetime. metric on the surface of a 2D sphere: R R R R dl The polar angle θ is measured from a fixed zenith direction, and the azimuth angle (in a reference plane that passes through the origin and is orthogonal to the zenith) is measured from a fixed reference direction on that plane. new radial coordinate: r= R sinθ flat space k=1/r 2 >0 Gaussian curvature

R R R R

negative Gaussian curvature

ROBERTSON-WALKER (FRW) METRIC The metric for an expanding spacetime that has homogeneous and isotropic spatial sections takes the Robertson-Walker form where (r, θ, φ) are time-independent (comoving) spherical coordinates and t is the (cosmic) time since the Big Bang measured by comoving observers who are at rest with respect to the matter around them. The curvature constant k (with dimension length 2 ) determines the geometry of the metric: it is positive if the universe is closed, zero if it is flat, and negative if it is open. The metric is non static because of the time dependence of the scale (or expansion ) factor a(t). k>0 k<0 k=0

The non static character of the metric can be made more explicit by calculating the physical (or proper) distance at time t from an observer at the origin to a point at comoving radial coordinate r, Since r is time-independent, the proper distance increases with a(t). The rate of change of the proper distance between two comoving observer is then Hubble law NB This statement is independent of GR. We ve only used symmetry so far!

The FRW metric describes all of the possible geometries of a homogenous isotropic Universe, but not all possible topologies. Geometry has local (visible Universe) structure, while topology only has global structure. Definition. A surface s geometry consists of those properties which do change when the surface is deformed. For example, curvature, area, distances, and angles are all geometric properties. k>0 Λ=0 k=0 k<0

The concordance cosmological model assumes that the Universe possesses a simply-connected topology It is a common misconception to describe a flat or hyperbolic Universe as necessarily open (i.e. infinite). topology

Take a flat Euclidean surface. That means, if we draw a triangle on the surface, its angles will sum to 180 deg. Now roll that surface up into a cylinder. The surface has now acquired extrinsic curvature because of the particular way it is embedded in a higher dimension. However its intrinsic curvature (that belonging to the surface alone) has not changed; it is still intrinsically flat. To see this consider any figure that you might have drawn on the surface. Within the surface, nothing about the figure is disturbed. If the figure conformed to Euclidean geometry before being rolled up, it will conform to Euclidean geometry after being rolled up. k=0 k=0 Gauss's Theorema Egregium: When surfaces are bent (but not stretched!), because measurements of lengths and angles on them remain unchanged, their Gaussian curvatures will not change either. In technical terms, the Gaussian curvature is invariant under isometries.

4.3 COSMOLOGICAL REDSHIFT The FRW line element vanishes for two events connected by a light signal; for photons moving along a radial trajectory (dθ=dφ=0), ds 2 =0 implies Without loss of generality, we can place the observer at the origin of the coordinate system. A light pulse leaving a source at comoving coordinate re at time te will arrive at the origin r=0 at a later time t0 given by Photons emitted at a later time te +δte will arrive at time t0 +δt0 after traveling the same comoving distance, so the integral will not change:

For small δte and δt0, the previous equation implies time dilation! This time dilation also applies to wavelengths (think of light pulses separated by one period), so not due to Doppler! The expansion of the universe therefore stretches photon wavelengths by a(t0)/a(te), a factor generally denoted with (1+z): where we have used again the usual convention a(t0) = 1.

In terms of the emitted and observed frequencies, the relation is NB By measuring z we obtain no information on when the light was actually emitted! Notice how the redshift we observe for a distant object depends only on the relative scale factors at the time of emission and observations, not on the rate of change of the scale factor at those times! re Cosmological time dilation effects can be directly observed in the light curves of Type 1a SNe.

Time dilation of supernova light curves. The left panel shows light curve points from high-redshift (blue) and nearby (black) supernovae. The right panel shows the same after removing the time dilation expected from redshift. From Goldhaber et al. 2001, ApJ, 558, 359.

A rapidly star-forming galaxy 700 million years after the Big Bang at z=7.5! 700 Myr 1000 Rest Wavelength (Å) 500 1000 2000 4000 24 25 Flux Density (njy) 100 Observed Flux Densities Observed 2σ Limit 26 27 28 AB Magnitude 10 Best fit Model (z=7.51) Best fit Model (z=1.78) 29 0.5 1 2 3 4 5 Observed Wavelength (µm) Finkelstein et al. 2013