Inflation By The amazing sleeping man, Dan the Man and the Alices
AIMS Introduction to basic inflationary cosmology. Solving the rate of expansion equation both analytically and numerically using different potentials. Perturbations: a basic introduction to perturbations and the origin of structure.
The 3 problems with the Hot Big Bang Theory Horizon problem: The universe is nearly isotropic and homogeneous today and was even more so in the past. Flatness problem: The universe is nearly flat today and was even more so in the past. (At the Planck time (-43s), Ω, the matter density parameter, should coincide with 1 with an accuracy of -60 i.e. completely flat). Monopole problem: The universe is apparently free of magnetic monopoles.
Solution: Inflation The Theory of Inflation solves all these problems and provides a mechanism to generate the seeds from which galaxies and large scale structures develop.
Inflation Any time in the evolution of the universe in which the scale factor, a, describing the size of the universe, is accelerating, leading to a very rapid expansion of the universe: ä>0 where d 1 0 dt ah (where 1/aH is the co-moving Hubble length which must decrease with time for inflation to occur). Generally thought to last between GUT time -36 s and -34 s, during which time all distances in the universe would have expanded by a factor 50.
Inflation solving the problems Horizon problem : Inflation allows dramatic reduction in co-moving Hubble length, allowing observable universe to originate from a tiny region within the Hubble radius in which the universe was all in causal contact and was allowed to reach thermal equilibrium before inflation. All initial in-homogeneities stretched to sizes larger than the observable universe. Monopole problem : The density of primordial monopoles becomes exponentiallydiluted by inflation.
Flatness problem : The condition for inflation is the condition that Ω is driven towards one with inflation. Inflating the universe by a huge factor ensures that the curvature of spacetime becomes indistinguishably close to zero, (k 0).
Scalar fields The condition for inflation can be rewritten as a requirement on the material driving the expansion. Directly from the acceleration equation, can be seen: && a 4 G = + 3p a 3 + 3p < 0 Where ρ is the energy density and p is the pressure. This requires a negative pressure. A scalar field is a material with this property an associated potential energy V(φ). The scalar field thought to drive inflation is called the inflaton.
Slow-roll This project was analysed both analytically and numerically. Numerically, 2 simple potential energy functions were used: m 2 2 V = 2 V = 4 Both these functions have a minimum at φ = 0. In a rapidly expanding universe the scalar field moves down very slowly until it reaches close to the minimum where it begins oscillating (in the reheating phase).
Slow-roll approximations dv && + 3H t & + hc = 0 Equation for inflation rate of universe d 3 In Cosmology 2 ï a& ï 8 G ï1 & 2 ï Hubble s constant in full form 2 H =ï = + V ï ï ï2 a 3 ï ï ï ï Approximation made in analytical analysis, since 1 2 & < V 2
Slow roll parameters: ï¾ ¾ ï 2 ïv ï = M Pl ï ï ïv ï Planck mass 2 M Pl2 ïv ï¾ ï = ï ï 2 ïv ï Reduced Planck mass Slow roll conditions where:
Models of Inflation Three different models are analysed analytically: Using the Slow roll approximation each is solved analytically using the following equation: The solution can also be found in terms of N (No. of e-folds). (Change of variables)
Polynomial Solution Slow Roll conditions: - 12 4 Hence: 3 2 1 For slow roll conditions to be satisfied. 0.5 1 The solution for phi as a function of t and as a function of N: 1.5 2 2.5 3
2.975 2.95 2.925 2.9 2.875 2.85 0.2 0.4 0.6 0.8 1 t - 40 mpl 8 6 4 2 20 30 40 50 60 N
Slow Roll conditions: - 12 35 30 25 20 15 5 1 Solutions for phi: 2 3 4
0.2 0.4 0.6 0.8 1 t - 40 mpl 50 60 0.99 0.98 0.97 0.96 4 3 2 1 20 30 40 N
Length of Inflation: Where is defined when The Number of e-folds (N approx 60) gives an estimation of the scale of :
Power-Law inflationslow roll conditions: 0.025 0.02 0.015 0.01 The slow roll parameters are independent of phi. 0.005 1 Solutions for phi: Power law inflation never ends! 2 3 4 5
5.5 5.25 5 4.75 4.5 4.25 8 8 8 8 8 8 0.2 0.4 0.6 0.8 1 t - 40 mpl 35 30 25 20 15 5 20 30 40 50 60 N
The Hubble parameter and Scale factor and H(t) and a(t) can be found for each potential using: 6 5 4 3 2 1 0.2 0.4 0.6 0.8 1 t - 40 mpl
6 1 800000 600000 400000 200000 0.2 0.4 0.6 0.8 0.999999 0.2 0.4 0.6 0.8 1 1 t 0.999998 0.999997 0.999996 (using p=2) t - 40 mpl - 40 mpl
Unknown constants? Each solution for phi (in terms of time) has unknown constants, the scale of these constants are found using perturbation theory
Numerical analysis && + 3H t & + dv =0 d Full equation used and analysed in MATLAB 8 G ï 1 2 ï & H = + V ï ï 3 ï2 ï 2 of Hubble s parameter used Full-form The following relations were used in order to find φ in terms of N, the number of e-foldings of inflation, as opposed to the time. N = ln a dn H= dt a tf =e a ti N a& H= a About 50-70 e-foldings are required to solve the horizon and flatness problems. We shifted certain parameters in order to reach approximately 60 e-foldings by the end of inflation.
Numerical analysis The full equation to be solved becomes: m 2 2 V = 2 V = 4 3 && & = -3H t - 4 2 && & = -3H t - m Initially the slow roll approximated form of H was used, then the equations were solved without any slow roll approximations in order to compare the results. These equations were solved using the ODE45 integrator in MATLAB to find φ and then plotting φ vs N. -6 The values used for the constants m andλ were found analytically by Alis to be mpl and 13 respectively (as you will see later). 2 unknown parameters which we could change each time; m or λ and an initial value for φ.
Results- With Slow roll approximation Phi vs. Number of e-foldings, N 3 2.5 Phi vs. Number of e-foldings, N 2 0.5 0.4 1.5 Phi 0.3 1 0.2 0.1 Phi 0.5 0-0.5 i =4. 3 m pl 0-0.1 0 i =3 m pl 20-0.2 30-0.3 40 50 60 70 N -0.4-0.5 56 56.5 57 Φ scales in terms of planck mass, 57.5 m pl 58 58.5 N 59 59.5 60 60.5 61
Phi vs. Number of e-foldings, N 4.5 4 3.5 Phi vs. Number of e-foldings, N 3 0.5 Phi 2.5 0.4 2 0.3 1.5 0.2 1 0.1 Phi 0.5 0-0.5 0-0.1 0 20-0.2 30 40 50 60 70 N -0.3-0.4 i =4.3 m pl -0.5 57 58 Φ scales in terms of planck mass, 59 60 61 N m pl 62 63 64
Results- Without slow roll approximation Phi vs. Number of e-foldings, N 9 8 Phi vs. Number of e-foldings, N 7 0.5 6 0.4 5 4 0.2 3 0.1 i =8. 5 m pl 2 Phi Phi 0.3 0 1-0.1 0-0.2-1 0 20-0.3 30 40 50 60 70 N -0.4-0.5 60.5 i 61 61.5 62.5 N =8.5 m pl Φ scales in terms of planck mass, 62 m pl 63 63.5 64
Phi vs. Number of e-foldings, N 12 8 Phi vs. Number of e-foldings, N 0.5 6 Phi 0.4 4 0.3 0.2 2 0.1-2 Phi 0 0 i =11.5 m pl -0.1 0 20 30-0.2 40 50 60 70 N -0.3-0.4 i =11.5 m pl -0.5 60 60.5 Φ scales in terms of planck mass, 61 m pl 61.5 62 N 62.5 63 63.5 64
Perturbations Quantum fluctuations in the vacuum during inflation are responsible for large scale structure. Usually these fluctuations have very small wavelengths, but a period of inflation can increase the wavelengths. When the wavelength of the perturbation becomes greater than 1/H, the fluctuation stops oscillating and becomes frozen in at some non-zero value δφ(x). The amplitude of the perturbation then remains constant but the wavelength continues to grow exponentially with inflation. Spectrum of inflaton field perturbation: H P φ = A δφ 2 >= 2π 2 k =ah Where =mean quantum fluctuation squared and at k=ah the scale crosses the horizon.
If φ is perturbed matter clumps Density Perturbations form curvature perturbed The perturbation in the curvature can be expressed (under the slow roll approximation)as: Where this function determines the initial density perturbations. It contains all the information needed to analyze the density perturbations. It is called a Transfer Function. The Power Spectrum can be related to the curvature perturbation by: P R k =< R k. R k
Using the Transfer Function, the power spectrum can be related to the spectrum of the inflaton field: Hence the Power Spectrum can be expressed as: Using and Where
Finding the scale of the constants from the power spectrum: From COBE measurements: δ H 2 x 5 (Where H denotes δ when crossing the horizon) Using this result and the previous results of the scale of phi (for 60 e-folds of inflation) the forms of the potentials can be constrained. However the constant in Power-Law inflation (p) can only be constrained to be greater than 1. The slow roll parameters have no phi dependence and hence p cannot be found in the same way as lambda and m.
Spectral Index n is used to characterize the density power spectrum: n can be found by using the slow-roll conditions Firstly the following change of variables was made*: The derivative of the slow roll parameter, epsilon was then calculated:
*Additional slide (as the rate of change of H is negligible compared to a) So using and
Using these results the derivative of the power density with respect to k can be found:
Hence the spectral index can be related to the slow roll parameters: Spectral index for each potential: and and
For slow roll conditions p>>1
First peak - Corresponds to matter density of the Universe,Ωm Second peak - The relative height of the first peak to the second gives baryonic density,ωb Third peak - Largely corresponds to the density of cold dark matter in the universe,ωcdm The location of the peaks give the geometry of the Universe. The first peak occurs at about l=220 implying a flat geometry.
Numerical Results for spectral index - With slow roll approximation -8 0.985 0.98 0.975 P=k n s P(K) 0.97-9 0.965 ln P n s 1 ln k 0.96 n s 1 0.955 0.95 0.945-0.94-5 -5-4 -4-3 -3 KK -2-2 n s 1-1 -1 n s 0.95 0.96
Numerical Results for spectral index - With slow roll approximation -12.79 0.985 0.98-12.8 0.975 s P(K) n 0.97 P=k -12.81 0.965 0.96 ln P n s 1 ln k n s 1-12.82 0.955 0.95 0.945-12.83 0.94-5 -4-4 -3-3 K -2-2 n s 1-1 n s 0.98
Comparison to WMAP data The 3 year WMAP data gives a value of n~0.95. The n values found were very close to this but the Power Spectrums do not match... WHY? Only one parameter was changed!there are many other parameters (such as the matter density, dark matter content etc) which will effect the form of the power spectrum. An example of a parameter which was not considered is the Running Spectral Index. If n depends on k the dependence of P on k is given by: P R k k n 1 1 dn 2 d ln k dn Where d ln k is the running spectral index, obviously the addition of this parameter will change the form of the power spectrum and in many models this will give a better fit.
Conclusions There is as yet no conclusion on the correct functional form of the inflationary potential. Many models are quite successful, but the complete picture is not yet known. Already of the two potentials used in this project is looking unlikely as the spectral index is too high and is only just acceptable for the values given by WMAP
Problems? The single scalar field potentials are beginning to be ruled out! The scales of the constants in these potentials pose problems why are their scales much smaller than the scale? m pl Improvements/alternate theories Hybrid Inflation: a theory with multiple scalar fields. Eternal inflation: implies that an infinite number of pocket universes are produced. Supersymmetry and supergravity predicting the form of the scalar field. Open inflation models. Extended inflation, natural inflation, topological inflation. list goes on!
Future Developments In this project only Scalar perturbations were considered, Tensor perturbations can give rise togravitational Wavesthese are beginning to be detected! See if the LHC detects the Higgs boson will be the first detected fundamental scalar particle (Spin-0) and will mean a scalar field exists called the Higgs field Better Microwave background data from newer, better satellites yet to be invented could narrow the allowed range theorised for the spectral index, further ruling out possible forms of potential.
Thank you for listening with Special thanks to Lisa Hall PS. Lisa, Marry me, I am a Duke! Love Dan xxx