Cosmology Project Report PDF

Observational Explanation to the accelerated expansion of the Universe, the measurement review of the contents of Universe and the way out from the big bang cosmology problems in Inflationary scenario Muhammad Usman Centre for advanced mathematics and physics (camp)...

Observational Explanation to the accelerated expansion of the Universe, the measurement review of the contents of Universe and the way out from the big bang cosmology problems in Inflationary scenario Muhammad Usman Centre for advanced mathematics and physics (camp) National university of science and technology (nust) h-12 islamabad  Cosmology Report TABLE OF CONTENTS 1 FRW Cosmology 2 1.1 Energy Densities 4 2 Dark Matter and Dark Energy 6 2.1 Dark Matter Evidence I 6 2.2 Dark Matter Evidence II 6 2.3 Dark Energy 7 3 Cosmic Microwave Background Radiation 8 3.1 Measurement of the CMB 8 Anisotropies in the CMB 8 3.1.1 The Dipole Anisotropy 8 3.1.2 Primary Fluctuations in the CMB 9 3.2 Determining Cosmological Parameters from Anisotropies in the CMB 11 4 Determining Cosmological Parameters from Type Ia Supernovae 13 4.1 Type Ia Supernovae 13 4.2 Measuring the Hubble Parameter 15 4.3 Measuring Ωm, Ω Λ and q0 16 5 A brief view to the Big Bang Theory of the Universe’s Problems 19 5.1 Inflation 19 5.1.1 The Flatness Problem 20 5.1.2 The Horizon Problem 21 5.1.3 The Monopole Problem 22 5.2 The way out from the problems of big bang cosmology in Inflationary Scenario 24 5.2.1 Exit from the Flatness problem 24 5.2.2 Exit from the Horizon Problem 25 5.2.3 Exit from the Monopole Problem 26 References 28 1  Cosmology Report 1. FRW Cosmology Over the years the relativistic hot big bang model has emerged as the most widely accepted and essentially the only tenable model of the Universe that is consistent with the observational data. In fact study of this model is now called standard cosmology. In 1929 Hubble established that the spectra of galaxies at greater distances were systematically redshifted to longer wavelengths which implies that the galaxies at distance d are receding away from us at velocities proportional to their distance from us which shows that is Universe is expanding. The mathematical description of standard cosmology is based upon the assumption that the Universe is homogeneous and isotropic at large scales this leads to the FRW metric which is dr 2 ds 2 = -c 2 dt 2 + a (t ) 2  + r 2  d 2 + sin 2 d 2 )] (1) 1- kr 2 Where a(t) is cosmic scale factor. Coordinates r, θ and φ are known as commoving coordinates. Figure 1: κ=+1 corresponds to the Closed Universe, κ=0 corresponds to Flat Universe, κ=-1 corresponds to open Universe. To get the evolution of the Universe we need to know that how the scale factor evolves with time. Einstein Field equations allow us to determine the evolution of the scale factor provided that the matter contents of Universe are known. 2  Cosmology Report The Einstein Field equations are 1 8πG Gμυ  Rμυ  g μυ R=κTμυ where κ= 2 (2) 2 C Where Gμν is the Einstein tensor, Rμν is the Ricci tensor which depends upon the metric used and R is the Ricci scalar, on the right hand side of the equation Tμν is the energy momentum tensor given by p Tμυ =( +ρ )u μu υ -pg μυ (3) C2 Where gμν is the space-time metric, ρ is the density, P is isotropic pressure and u=C2× (g00)1/2(1,0,0,0) is the velocity vector for the isotropic fluid in comoving coordinates, here it takes the form Tυμ =diag(-ρ,P,P,P) (4) Also, From the first law of thermodynamics dQ  dW  dU (5) Where Q is the energy given by any external source, W is the work done on the system and U is the internal energy of the system, d represents the change in the respective quantities. For dQ=0 d ( V )  - PdV (6) Where ρ is the energy density and V is the total volume. This gives Vd   -( P   )dV (7) Since V  a 3 This gives d  3H (   )  0 (8) dt Defining the equation of state    (9) Thus equation (8) can also be written as   3H (1   )  0 (10)  Integration of equation (10) gives 3  Cosmology Report  a 3(1 )  Constant (11) The Einstein field equations for the metric given by equation (1) become a 2 8πGρ κ H2  = - (12) a2 3 a2 a a 2 κ    -4 G (13) a a2 a2 Inserting equation (12) in equation (13) we get a 4 G -    3  (14) a 3 Equation (13) can also be written as   4πG ( ρ+Ρ )  κ H= (15) a2 Since the acceleration is positive which means that d 2a  1 0 if - or   - (16-a) dt 2 3 3 d 2a  1 0 if - or   - (16-b) dt 2 3 3 Thus the accelerated expansion of the Universe could be fuelled by an exotic form of matter of large negative pressure which makes the effective pressure of the all contents of the Universe negative which turns effective gravity into a repulsive force, or gravity does not act as we know it on the large scale structures. The choice of choosing κ=0 is discussed later (Flatness problem). 1.1. Energy Densities Critical to the geometry of the Universe is the energy density of the various components present in the Universe evolves according to equation (11).Equation (12) can also be written as tot   =1 (17) Where   a2  =  (18) c 3H 2 8 G And 4  Cosmology Report tot tot tot =  (19) c 3H 2 8 G Since the Universe is expanding and matter and radiation is self-attractive so there is some content of Universe present which is causing the accelerated expansion of the Universe so we can split the term ρtot into three parts one will correspond to the matter, one will radiation and the other one will correspond to the content which is causing the accelerated expansion of the Universe. So equation (17) can be written as m     +r =1 (20) The first term Ω m actually is m=L.M  D.M (21) Where ΩL.M corresponds to thee luminous matter and ΩD.M corresponds to the dark matter. Now, with κ=0 and cosmological constant the Friedmann equations (12) and (13) become a 2 8πG =  ρ+  (22) a2 3 a a 2 2  2  8 G      (23) a a In the present Universe the contribution of the pressure p in equation (23) is negligible, also neglecting Λ and solving the equation (23) we get 2 H(t )= (24) 3t Or 2 a (t )=a0t 3 (25) This shows that a(t) increases with t and this gives the expanding Universe. From equation (8) with p=0 the solution for ρ(t) becomes 0  (t )= (26) a3 Using (25) in we get 0  (t )= (27) a03t 2 5  Cosmology Report 2. Dark Matter and Dark Energy 2.1. Dark Matter Evidence I The age of the Universe ttoday can be determined from the present value H0~72km/Sec×1/Mpc of the Hubble constant. The age we get from H0 is ttoday ≡ t0 ∼1.4 × 1010 years → (28) Which also is approximately the age of the oldest stars and galaxies. We obtain for the matter density Ω(t0) from (27), which comes out to be ρ(t0 ) ∼ 2 × 10-27 kg m-3 Comparing this value to the density of galaxies and intergalactic dust. The ration comes out to be  (t0 ) 6 (29) known This means that, besides the known matter, there should exist an unknown form of “dark matter” (“dark” since, evidently, it does not emit light). The contribution of dark matter to the total matter density seems to be about five times the contribution of known matter. 2.2. Dark Matter Evidence II At this point we should discuss a phenomenon related to the dynamics of stars inside galaxies: the nearly circular motion of stars around the center of galaxies is caused by the gravitational attraction between the stars. From the known form of the gravitational force we can compute the rotational velocity v(r) of a star, which depends on its distance r to the center of the galaxy and the mass M(r) inside a fictitious sphere with radius r (G is Newton’s gravitational constant) Figure 2: Radius r and rotational velocity v(r) of a star rotating around the center of a galaxy GM(r ) v2  (30) r In practice we can measure the rotational velocities v(r) of stars at different distances r to the galactic center for a large number of galaxies and estimate M(r). Surprisingly, the observations do not agree with (30): either the measured values of v(r) are systematically 6  Cosmology Report too large, or the estimates of M(r) are systematically too small! (Notably for large r, where the density of stars decreases and where M(r) should hardly increase with r, v(r) does not decrease as 1/r1/2, but remains approximately constant.) This discrepancy suggested, already before cosmology, the existence of additional dark (invisible) matter, which contributes to M(r) and thus to the attractive gravitational force of galaxies. 2.3. Dark Energy In recent years, very distant supernova explosions whose light was emitted a very long time ago have been successfully observed [2–5]. Through measurements of their radial velocities, with the help of the Doppler effect, and their distances, with the help of the known luminosity of such supernova explosions, the time dependence of H(t) was determined for the first time, allowing a comparison to solutions of the Friedmann equations. It appeared that Ḣ(t) is somewhat larger than expected from the above solution, which was derived under the assumption Λ = 0. The measured value of Ḣ(t) is compatible only with a positive value of Λ (a “dark energy”) in Friedmann equations (22) and (23) Λ ∼ 4 × 10−10 kg s−2 m−1 → (31) The Nobel Prize 2011 was awarded to Saul Perlmutter, Adam Riess and Brian Schmidt for this observation. Now comparing the two terms on the right-hand side of (22) one finds that, on the one hand, both are of the same order today: Λ ∼2 × ρ(t0) → (32) However, the time dependences of the two terms are very different: ρ(t) behaves as 1/t2 but Λ is considered as constant. Figure 3: Schematic time dependences of ρ(t ) (∼1/t 2 ) and Λ (constant) Correspondingly, at earlier times, i.e., for t < < t0, ρ(t ) was much larger than Λ ,and Λ was numerically negligible. For this reason, Λ has had an impact on the evolution of the Universe only in recent times; corresponding small corrections have already been taken into account in the value of the age of the Universe. 7  Cosmology Report Because of the different time dependences of ρ(t ) and Λ, it appears a remarkable coincidence that we live in a kind of transition period and in the (still very far) future the evolution of the Universe will be determined nearly exclusively by the Λ terms in (22) and (23) where upon a(t) will increase exponentially with t. 3. Cosmic Microwave Background Radiation In the Standard model of Cosmology just a f t e r the big bang the Universe was very hot and very dense. There were no neutral atoms or even nuclei. Matter a n d radiation w e r e at thermal equilibrium at that time due to the Compton scattering and Electrons were tightly coupled to protons due to Coulomb scattering. As the Universe expanded, it cooled down. At some time the temperature of the Universe dropped below the binding energies of typical nuclei (H, He etc.) and light elements started to form. Electrons got bound into neutral atoms, this process is called recombination. Because of recombination photons could no longer scatter of free electrons. Radiation decoupled from matter. This moment is therefore called decoupling. The surface where the decopling of matter and radiation took place is called surface of last scattering. 3.1. Measurement of the CMB Anisotropies in the CMB A few sources of the anisotropy in the CMB are  The Dipole Anisotropy  The Primary Fluctuations in the CMB 3.1.1. The Dipole Anisotropy To analyze the dipole anisotropy it is useful to look at the density of photons in phase space Nγ (p) 2 1 N ( p)  (33) h exp  p c kT   1 This would be the density of photons measured by an observer at rest with respect to the radiation background. The phase space volume is Lorentz invariant and the number of photons is also Lorentz invariant. Thus, the density of photons in phase space is a scalar in the sense that a Lorentz transformation that transforms p to p` does not transform it i.e. N  ( p)  N  ( p ) (34) If the earth is moving in the three-direction with velocity β with respect to the cosmic background and we take p to be the momentum of a photon in the frame at rest in the cosmic background and p 0 to be the photon momentum in the frame of the earth, then 8  Cosmology Report  p1   1 0 0 0   p1        p2    0 1 0 0   p2  (35)  p3   0 0     p3        p  0 0     p  Where   (1   2 )1 2. The matrix multiplication gives p   (1   cos  ) p (36) Where θ` is the angle between p and the three axis. Now using (33) in (34) we get 2 1 2 1  h exp  p c kT    1 h exp  p c kT   1 2 1 2 1  (37) h  pc  h exp  p c kT   1 exp  kT    1   (1   cos  )  This gives T T  (38)  (1   cos  ) Thus we expect a direction in which the temperature of the photons is maximal and we expect the photons coming from the opposite direction to have a minimal temperature. From this the motion of the earth with respect to the cosmic background can be deduced. The WMAP satellite experiment has found this maximum temperature increase of 3.346±0.017mK to be in the direction l= 2630.85±0.10, b = 48.25 0 ±0.040. These results indicate a motion of the solar system with a velocity (0.00335)c/(2.725) =370km/s. The earth has a velocity relative to the center of the galaxy of about 215km/s, more or less in the opposite direction. Taking this into account we can find the net velocity of the local group of galaxies relative to the cosmic background. This yields 627 ± 22km/s in a direction (l = 2760 ± 30, b = 300 ± 30) [7-9]. 3.1.2. Primary Fluctuations in the CMB Small anisotropies have been found in the cosmic microwave background. These anisotropies result from small perturbations in the energy density of the very early Universe. Hence the detection of these anisotropies has provided evidence for the existence of the density perturbations in the early Universe. In order to do this it is very general to write the temperature difference ∆T (n̂) between the temperature measured in the direction of the unit vector n̂ and the mean value of the temperature T0 as the spherical harmonics Ylm(n̂). 1 T (nˆ )  T (nˆ )  To   almYl m (nˆ ), T0   d 2 nT ˆ (nˆ ) (39) lm 4 The sum over l runs over all positive integers and m runs from –l to l. Since the average of two ∆T has to be rotational invariant which gives alm alm   l,l  m , mCl (40) The average of the product of two temperature differences now becomes 9  Cosmology Report T (nˆ )T (nˆ)   Cl Yl m (nˆ )Yl  m (nˆ) (41) lm  2l  1    Cl   Pl (nˆ.nˆ) (42) lm  4  Where the sum over m is performed in the second line and yields a Legendre polynomial Pl. Inverting the second equation to get an expression for the multipole coefficients Cl which turns out to be 1 4  Cl  ˆ 2 nˆPl (nˆ.nˆ) T (nˆ )T (nˆ) d 2 nd (43) The multipole coefficients Cl are real and positive. The l is related to the angle between different directions n̂ by l ∼ 1/θ (l and θ are fourier transform of each other).Thus large l s correspond to small angles and vice versa. Since Cl is very difficult to compute so defining Clobs which can be computed easily and then calculating the fractional difference between Cl and Clobs. So, 1 Clobs  4  ˆ 2 nˆPl (nˆ.nˆ)T (nˆ )T (nˆ) d 2 nd (44)  Cl  Clobs   Cl   Clobs   2      l,l  (45)  Cl   Cl   2l  1 The R.H.S of equation (45) goes to zero for large l and hence for large l (small angle) both Cl and Clobs are nearly equal and for small l the difference is large. Figure 3: The power spectrum of the cosmic microwave background. The blue region denotes the cosmic variance. It is large for small l and it disappears at large l. 10  Cosmology Report Thus for l < 5 the cosmic variance is too big for Clobs to be of any cosmological interest. And for l > 2000 the uncertainty in Clobs becomes too large due to foreground effects to be of any interest. To obtain information about the Universe we are restricted to the region between 5 and 2000, still a rather large region. Different regions in l have different explanations shown in the figure below Figure 4: The theoretical CMB power spectrum. Details of different regions from. 3.2. Determining Cosmological Parameters from Anisotropies in the CMB The analysis of the temperature fluctuations does in fact reveal patterns corresponding to a harmonic series of longitudinal oscillations. The various modes correspond to the number of oscillations completed before recombination. The longest wavelength mode, subtending the largest angular size for the primary anisotropies, is the fundamental mode this was the first mode detected. There is now strong evidence that both the 2nd and 3rd modes have also been observed [11-13]. The distance sound waves could have traveled in the time before recombination is called the sound horizon rs. The sound horizon is a fixed physical scale at the surface of last scattering. The size of the sound horizon depends on the values of cosmological 11  Cosmology Report parameters. The distance to the surface of last scattering dsls, also depends on cosmological parameters. Together, they determine the angular size of the sound horizon. r s  s (46) d sls Analysis of the temperature anisotropies in the CMB determine and the cosmological parameters can be varied in rs and dsls to determine the best-fit results. We can estimate the sound horizon by the distance that sound can travel from the big bang t=0, to recombination t∗ t* rs ( z* ; b ,  r )   cs dt (47) 0 Where z* is the redshift parameter at recombination (z*~1100) , Ωr is the density parameter for radiation (photons), cs is the speed of sound in the photon-baryon fluid, given by. 1 2   3   cs  c 3 1  b   (48)   4r   Which depends on the baryon-to-photon density ratio. Since the energy density of radiation scales as ρr∝a - 4 , so with the addition of radiation, Friedmann equation generalizes to  da  2   a0  2  a0   a0   2    H 0 r ,0    m,0    k ,0    ,0    (49) 2  dt   a a  a   Using a0/a=1+z and equation (20), equation (49) becomes   1 2 dt  H 01 (1  z )1 (1   m,0 z ) 1  z   z  2  z  (1  z )2  r ,0    ,0  (50) 2 dz The distance to the surface of last scattering, corresponding to its angular size, is given by what is called the angular diameter distance. It has a simple relationship to the luminosity distance d. The luminosity distance is given by d  (1  z )a0 r The location of the first acoustic peak is given by l ≈ dsls/rs and is most sensitive to the curvature of the Universe Ωk. We will consider a prediction for the first acoustic peak for the case of a flat Universe. To leading order, the speed of sound in the photon-baryon fluid, equation (48), is constant cs = c/31/2. We further make the simplifying assumption that the early Universe was matter-dominated. With these assumptions, equations (49) and (47) yield (dropping the ’0’ from the density parameters) cs  rs  H 0 m 0 (1  z ) 5 2 dz (51) Which gives 2cs rs  (1  z* ) 3 2 (52) 3H 0  m The luminosity distance of surface of last scattering is given by 12  Cosmology Report d sls  2c 7 H 0 (1  z* )  m 9  23m  (53) Dividing equation (53) by (52) to get the prediction of first acoustic peak which gives d   l  sls  0.74 (1  z* ) 9  23m  221 rs (54) This result is consistent with the more detailed result that l  200 (1   k ) (55) Which tells us that l~200 implies a flat Universe. The BOOMERanG collaboration found l ≈197 ± 6, and the MAXIMA-1 collaboration measured l ≈220. It is important to recognize that the constraints on cosmological parameters obtained through this sort of analysis are correlated so that the range of possible values of ΩΛ , for example, depends on what is assumed for the possible range of values of the Hubble constant. Therefore, it is customary to incorporate results from other observational (or theoretical) work in the analysis of the CMB data. With this in mind, we use the value of the Hubble constant stated in the line above equation (28). Given this assumption, a combined study of the CMB anisotropy data from the BOOMERanG , MAXIMA-1 , and COBE-DMR collaborations suggests the following values for the two cosmological parameters being considered here :  k ,0  0.11  0.07   (56)  b ,0  0.062  0.01  4. Determining Cosmological Parameters from Type Ia Supernovae 4.1. Type Ia Supernovae “Throughout their lives, stars remain in stable (hydrostatic) equilibrium due to the balance between outward pressures (from the fluid and radiation) and the inward pressure due to the gravitational force. The enormously energetic nuclear fusion that occurs in stellar cores causes the outward pressure. The weight of the outer region of the star causes the inward pressure. A supernova occurs when the gravitational pressure overcomes the internal pressure, causing the star to collapse, and then violently explode. There is so much energy released (in the form of light) that we can see these events out to extremely large distances.” Supernovae are classified into two types according to their spectral features and light curves (plot of luminosity vs. time). Specifically, the spectra of type Ia supernovae are hydrogen-poor, and their light curves show a sharp rise with a steady, gradual decline. In addition to these spectroscopic features, the locations of these supernovae, and the absence of planetary nebulae, allow us to determine the genesis of these events. Based on these facts, it is believed that the progenitor of a type Ia supernova is a binary star system consisting of a white dwarf with a red giant companion. Other binary systems 13  Cosmology Report have been theorized to cause these supernovae, but are not consistent with spectroscopic observation. Although the Sun is not part of a binary system, approximately half of all stellar systems are. Both members are gravitationally bound and therefore revolve around each other. While a binary star system is very common, the members of the progenitor to a type Ia supernova have special properties. White dwarf stars are different from stars like the Sun in that nuclear fusion does not take place within these objects. Electron degeneracy pressure, which is related to the well-known Pauli Exclusion Principle, holds the white dwarf up against its own weight. For electron degeneracy pressure to become important, an object must be extremely dense. White dwarf stars have the mass of the Sun, but are the size of the Earth. Also, the physics of this exotic form of pressure produces a strange effect: heavier white dwarfs are actually smaller in size (Mass × Volume = Constant). Red giant stars, on the other hand, are the largest known stars and contain a relatively small amount of mass. As a result, gravity is relatively weak at the exterior region of red giant stars. In such a binary system, the strong gravitational attraction of the white dwarf overcomes the weaker gravity of the red giant. At the outer edge of the red giant, the gravitational force from the white dwarf is stronger than that from the red giant. This causes mass from the outer envelope of the red giant to be accreted onto the white dwarf. As a result, the mass of the white dwarf increases, causing its size to decrease. This process continues until the mass of the white dwarf reaches the Chandrasekhar limit (1.44 solar masses) beyond which electron degeneracy, pressure is no longer able to balance the increasing pressure due to the gravitational force. At the center of the white dwarf, the intense pressure and temperature ignites the fusion of Carbon nuclei. This sudden burst of energy produces an explosive deflagration (subsonic) wave that destroys the star. This violently exploding white dwarf is what we see as a type Ia supernova. The use of type Ia supernovae for determining cosmological parameters rests on the ability of these supernovae to act as standard candles. Standard candles have been used to determine distances to celestial objects for many years. They are luminous objects whose intrinsic (or absolute) brightness can be determined independent of their distance. The intrinsic brightness, together with the observed apparent brightness (which depends on the distance to the object), can be used to calculate distances. The distance calculated from measurements of the luminosity (power output) of an object is appropriately termed the luminosity distance. d=10(m-M-25)/5 → (57) Where m is the apparent brightness measured in magnitudes (apparent magnitude), M is the absolute magnitude, and d is the luminosity distance in units of mega parsecs. The quantity, (m− M) is commonly known as the distance modulus. As explained above, all type Ia supernovae are caused by the same process, a white dwarf reaching 1.44 solar masses by accretion from a red giant. As a result of this consistency, we not only expect to see extremely consistent light curves from these events, but we also expect that these light curves will reach the same peak magnitude. If this latter point is true, type Ia supernovae can be used as standard candles and, therefore, distance indicators. Methods for determining the absolute magnitude of a type Ia supernova can be divided into two categories depending on whether or not we know the distance to the event. If we know the distance to the host galaxy of the supernova, by means of a Cepheid variable for example, and we observe the apparent magnitude of the event ‘m’, then we can use the distance modulus to calculate the absolute magnitude 14  Cosmology Report directly m − M = 5 log(d) + 25. → (58) If the distance is not known, the peak luminosity must be inferred from observational data. The techniques for making this inference often involve corrections for many processes that would otherwise adversely affect the results. These processes include interstellar extinction within the host galaxy, redshift of the light from the expansion of the Universe, gravitational lensing, and an apparently natural scatter in the peak brightness. Once the luminosity L of a supernova has been determined, this luminosity, together with the luminosity L0 and absolute magnitude M0, of a well-known object (such as the Sun) will yield the absolute magnitude of the supernova M = M0 − 2.5 log(L/ L0) → (59) Taking all of this into account, it has been determined that the peak absolute magnitude of type Ia supernovae is. MIa = −19.5 ± 0.2 mag → (60) 4.2. Measuring the Hubble Constant Observationally, we measure the recession velocity as a redshift z, in the light from the supernova (vr=cz). Since every type Ia supernovae has about the same absolute magnitude equation (60), the apparent magnitude provides an indirect measure of its distance. Therefore, for nearby supernovae (z ≤ 0.3) the Hubble Law is equivalent to a relationship between the redshift and the magnitude. Inserting (60) into (58), using vr=cz, and applying to the current epoch, yields the redshift-magnitude relation m = MIa + 5 log(cz) − 5 log(H0 ) + 25 → (61) Defining the z = 0 intercept as M̃ ≡ MIa − 5 log(H0 ) + 25 → (62) We can write equation (27) as m = M̃ + 5 log(cz) → (63) The low-redshift data can be used to find M̃ and equation (62) to solve for the Hubble constant. Studies on type Ia supernova consistently suggest a value for the Hubble constant of about 63 km s−1 Mpc−1. The result for H0, found from low-redshift supernovae, tends to set the lower bound when compared with other methods for obtaining H0. For example, if the distances to enough galaxies can be accurately found, then the Hubble law can be used directly to obtain a value of H0. This has partly been the goal of the Hubble Space Telescope Key Project. This project has shown that a careful consideration of the type Ia supernova results in combination with the other methods for obtaining H0 produces what has become a widely accepted value for the Hubble constant H0 = 72 ± 8 km Sec−1Mpc−1 → (64) 15  Cosmology Report 4.3. Measuring Ωm, ΩΛ and q0 To obtain the general expression for the luminosity distance, consider photons from a distance source moving radially toward us. Since we are considering photons, ds2=0, and since they are moving radially, dθ 2=dφ2=0. The FRW metric, then implies a (t )dr dt  (65) (1   r 2 )1 2 The equation (50) without Ω r,0 is (because Ω r,0 is negligible today).   1 2 dt  H 01 (1  z ) 1 (1   m ,0 z ) 1  z   z  2  z    ,0 (66) 2 dz Equating (65) and (66) and integrating and then inserting in the formula of luminosity distance d = (1+z)a0r []. We get   z 1 2   sinh   ,o  1  z   1   m,o z    z (2  z )  ,o  dz 12 2  0   0   1 z  sin   1 2 1  z  2 1   z    z (2  z )  dz  for   0 z 1 2 d H 0  ,o 12      ,o   m , o  , o    (67)  0   ,o 12  0   Inserting (67) in (58) we get redshift magnitude relation m  M  5  d  z;  m ,     (68) In practice, astronomers observe the apparent magnitude and redshift of a distant supernova. The density parameters are then determined by those values that produce the best fit to the observed data according to equation (68) for different cosmological models. Under the continued assumption that the fluid pressure of the matter in the Universe is negligible (pm≈0), equation (14) with cosmological constant implies that the deceleration parameter at the present time is given by q0 = Ωm,0 /2 − ΩΛ,0 →(69) Therefore, once the density parameters have been determined by the above procedure, the deceleration parameter can then be found. Typical values for the cosmological parameters as determined by detailed analysis of the type just discussed are the following  m ,0  0.24 0.24 0.56     ,0  0.72 0.48 0.72  (70) q0  1.0  0.4   16  Cosmology Report Figure 5: SN data gives ΩΛ=0.7 and ΩM=0.3 and Ωk=0 Credit: http://abyss.uoregon.edu/~js/lectures/cosmo_101.html 17  Cosmology Report Figure 6: The combined result of CMB, Supernovae and Clusters (structure formation) Also see Figure 7. 18  Cosmology Report 5. A brief view to the Big Bang Theory of the Universe’s Problems The big bang theory of the Universe was suggested by George Gamow in 1948 and is the most acceptable theory by most of the Physicists but it still has some problems which have become very prominent in the recent time of the Universe. But it also has some unexplainable problem in the classical domain (means old fashioned way). Some of them are  The Flatness Problem (related to the curvature of the Universe)  The Horizon Problem (related to the anisotropy of the Universe)  The magnetic monopole problem 5.1. Inflation Inflation is defined as an epoch in the history of Universe when a  0. To get this we just need to have (p/ρ)a(ti) in other words the growth of the scale factor is enormous. This does solve the big bang cosmology problems. Which in turn means that the Inflationary era should lie in the very early Universe when it not far from its GUT era. We now first take the overview of these problems and then a way out from these problems in Inflationary scenario. 5.1.1. The Flatness Problem The equation (12) can be written as  1 3  (75)  8 G  a 2 If the energy density is predominantly due to non-relativistic matter   R3 then equation (75) gives  1 a (76)  If the energy density is predominantly due to the relativistic matter (radiation) then   R4 and the equation (75) gives  1  a2 (77)  As we move back in time the scale factor approaches zero and thus Ω will converge to 1, but as we move forward in time the scale factor increase and thus Ω moves away from 1. To get the clear picture we need to show that how Ω changes with time. The recent observations show that κ~0 so taking κ=0 in equation (12) we get  a  8 G  2    (78) a 3 For the non-relativistic matter dominated era equation (78) gives 2 at 3 (79) For the radiation dominated era equation (78) gives 1 at 3 (80) The results of the type Ia supernova observations and the measurements of the CMB anisotropy bound on the Ω0 is 1  0  0.2 (81) This is a small value and can be regarded as the coincidence that the initial conditions of the Universe has led the Universe to have the density parameter close to 1 but as we get back in time we run in problem. 20  Cosmology Report Figure 7: the values of current Ω m and Ω Λ permitted by the MAXIMA and COBE DMR data. The SN Ia data is also shown. The curves marked 68%, 95% and 98% gives the best fits, at the designated confidence levels, for the combined CMB and SN Ia data (taken from Stompor et al. 2001. ApJ, 561, L7) Also see figure 5. If we calculate 1   p at plank time tp~5×10-44sec or ap~2×10-32, we find that the density parameter gets the bound 1   p  1060 (82) This is not just a coincidence to have such a small bound and called Flatness problem. 5.1.2. The Horizon Problem The horizon problem is related to the homogeneity and isotropy of the Universe at large scales. Consider two points on the last scattering surface. The current proper distance to the surface of last scattering is t0 dt  p (t0 )   (83) tls a(t ) The current proper distance to the last scattering surface is dhor=1.02×dp or dp=0.98×dhor. thus the points on the sphere connected by the diameter are separated by a proper distance 1.96×dhor. since dp>dhor they have not had enough time to send messages to each other and so have not had enough time required for them to come into thermal equilibrium with each other. 21  Cosmology Report Figure 8: The current Universe in the hot big bang model we are at the center of the sphere (taken from ) Since the Hubble distance at the time of last scattering was ~ 0.2Mpc so the horizon distance is ~0.4Mpc. so the points having distance between them more than 0.4Mpc are not connected. The angular diameter distance at the last scattering surface is dA~13 Mpc. So the angle for which the points are connected is d (t ) 0.03  sls  hor sls   0.03rad  20 (84) dA 13 Thus the points on the surface of last scattering separated by the angle >20 were out of contact at the time when CMB was formed yet the temperature fluctuations of the CMB is of the order of 10-5 even for θ>20 (l