Inflation and Scalar Fields

1. Inflation and Scalar Fields Adam Ginsburg

2. Guth made inflation So the big bang could start as A chill, then bubbles

3. Outline • Why do we want inflation? • Matching inflation to the power spectrum • What is it? What happens? • Vacuum Energy • Bubble nucleation • Reheating • Basic math behind scalar fields and inflation

4. Outline 2 - Papers • Most papers are reviews or general descriptions of inflation • Zel’dovich paper is approximately “Inflation for Astronomers” • Marco and Notari paper presents a specific model in the quest for a simple inflation that satisfies observations

5. Motivation Inflation • Good solution to big bang initial condition problems • Therefore inflation theories must be free from initial conditions • Began when universe emerged from quantum gravity (Planck) conditions • Based on evolution of a classical scalar field

6. Results we want • Growth/expansion of 1030, or e70 • Universe must inflate long enough to become flat, homogeneous, and isotropic • No ‘bad particles’ (e.g. magnetic monopoles) • Discontinuities in scalar fields associated with GUT phase transitions should be accounted for • Inflation must begin and end (not necessarily a trivial requirement)

7. Testing inflation? • Harrison and Zeldovich predicted a density perturbation spectrum is responsible for structure • Inflation is capable of producing said structure • During inflation, quantum fluctuations are magnified • Matter collects in “troughs” of gravitational waves • Complicated process. That’s all I’ve got on it. • Once we refine possible spectra, compare to inflationary models

8. WMAP Power Spectrum • Inflation models must match CMB power spectrum to be validated • WMAP gives n=.93 +/- .3

9. Distracting Animation • Animations show variation of the power spectrum with varying spectral index. The spectral index is one of the best constraints on inflation models

10. WMAP Power Spectrum • Just showing you again so you can get that annoying animation out of your head

11. Why does inflation happen? • As the universe cools, the scalar field supercools • For the scalar field potential, there is a true minimum. However, at a certain temperature, there is a new phase with a lower minimum • While the scalar field is supercooled, inflation happens

12. Bubble Nucleation • When false vacuum tunnels into true vacuum, a phase change occurs • Similar to ice crystals forming • Bubbles rapidly grow (percolate?) and can merge • Mergers generate heat, warm the universe

13. False Vacuum and Vacuum Energy • Normal stuff pushes outward • Vacuum Energy pulls in, exhibits tension • Why does it repulse? • The difference in pressure, not the tension itself, causes expansion • The Friedmann equation has a term e + 3p = -2e when the vacuum dominates (gravitational repulsion)

14. Bubble Nucleation – Old Guth • In some models, inflation can form bubbles but then the false vacuum repulsion pushes them away from each other before they can interact • This is the failure behind “old inflation” • Might be the idea behind many distinct universes

15. Scalar Field Energy Density and Pressure • Both consist of two terms: “Kinetic” and “Potential” • All systems attempt to minimize potential terms, Scalar Fields aren’t very good at it • Equation of state shows small kinetic energy -> w=-1 Liddle claims there is no equation of state “because the same energy density can correspond to different values of the pressure if the energy density is distributed differently between the potential and kinetic terms.''

16. Friedmann Equationn [sic] • Substitute density/energy density and pressure into Friedmann and Fluid equations • Friedmann Equation only depends on density (rho) • Expansion when potential term dominates

17. How much inflation • Recall: we require 70 e-foldings • Can calculate inflation without equations of motion • Amount of inflation doesn’t change if V is multiplied by a constant

18. What happens during inflation? • Massive growth of the scale factor (1030) • In turn means massive redshift • Energy density reduced by extremely large factor (naively, 1090). However, the false vacuum consists of vacuum energy and keeps the energy density constant (vacuum dominates during inflation) • Along with scale factor drop, matter and radiation densities go to zero. Everything gets very cold

19. Hot and cold • The universe was hot going into inflation and coming out of it • In between, it got very cold • There needs to be some mechanism to reheat the universe

20. Reheating and Density Perturbations • Quantum Field Theory describes virtual particle pair creation. If positional uncertainty exceeds the event horizon, particles become real • Energy cost is paid by the scalar field • “Quantum Fluctuations” become real • If true-vacuum bubbles are created in the false vacuum, inflation will blow them up to large scales visible as density fluctuations in the CMB

21. Model outline • No theory currently tells us what potential to use • Arbitrary, hopefully general, potentials are selected in the hope that they will represent ‘real’ potentials once a theory has been developed • Models: • Guth’s “Old Inflation” • New Inflation • “Graceful” Old Inflation (Marco and Notari) • Coleman-Weinberg Potential • Chaotic Inflation • Hybrid Inflation

22. Guth’s Inflation • Initially, at high temperature, in false vacuum • False vacuum decays in regions by quantum tunneling, forming bubbles • Problems: bubbles don’t combine well • Bubbles may actually move apart, creating empty, cold universes

23. Guth’s Old and New Inflation • Many bubbles form, separated by quickly inflating space, so they can never connect • Bubbles would remain cold (depends on interaction between bubbles to heat up) • New inflation posited that we are in just one of the bubbles, and instead of instantaneous tunneling to true vacuum there is a slow roll • New inflation is “eternal” - there’s not necessarily any zero point. • Quantum fluctuations can increase potential

24. Marco and Notari “Graceful” Old Inflation • Motivation: hoped to use a relatively simple inflationary model (and scalar potential) to explain results that commonly require more complex potentials • Hypothesis: “Just one additional ingredient” could fix Guth’s Old Inflation model

25. “Graceful” Old Inflation • Scale factor grows first exponentially and then as a power of time • Phase I (exponential) produces a good power spectrum, i.e. a flat (n=1) spectrum • Phase I exponential growth is “Einstein Gravity” • Pure exponential growth leads to graceful-exit problem • Bubbles must be produced after this period, otherwise they would grow too large and spoil CMB isotropy

26. Not graceful enough • Phase II, power law, gives a graceful exit • Pure power law growth, as in extended inflation, is limited to COBE-excluded n<.75 • The curvaton is responsible for the phase switch • Model becomes a hybrid inflation model • Generates greater CMB fluctuations • However, even the non-minimally-interacting curvaton isn’t enough to generate the WMAP predicted power spectrum – more modifications are needed for this model to be viable

27. Coleman-Weinberg Potential • At high temperatures, the zero point appears to be at phi=0 • As temperature decreases, phi goes into metastable false vacuum state, tunnels into true vacuum • Oscillations in true vacuum damped by particle creation, which leads to reheating

28. Chaotic Inflation • Proposed by Linde • Self interacting (phi4) scalar field • Phi starts at “chaotic” initial point, decays down potential well • Chaotic conditions mean different bubbles would have different initial conditions, we’re in the one that gives our set of physics • Requires kinetic energy smaller than potential • Slow roll from “friction term” instead of simply flat potential

29. Slow Roll Approximation • Assumes kinetic term small compared to potential term • Slow Roll parameters measure slope (epsilon) and curvature (eta) • Change in the scalar field is slow • More conditions beyond the scope of this talk are required for SRA to hold

30. Chaotic Potential Diagram • Two phases: • “slow roll” down a mild slope gives inflation • Faster fall into lowest energy state and oscillation -> reheating • Oscillations around minimum potential are damped by particle formation, universe gets reheated

31. Chaotic Condition for Inflation • Inflation occurs when scale factor growth is accelerating, i.e. d2a/dt2 > 0 • The Slow Roll Approximation is equivalent to the inflation condition • Therefore, Slow Roll => inflation

32. Hybrid Inflation Models • Simplest contains two fields • Interesting part is along “channel” when potential acts as a single field • Useful for interactions between “inflaton” and GUT potentials

33. Hybrid Inflation • Two non-interacting scalar fields allows for different steps without making non-natural potentials • Can step from a fast potential (e.g. the quantum tunneling that initiates inflation) to a slow, flat potential that gives slow roll

34. Conclusion • Guth’s old inflation failed • New Inflation and Chaotic Inflation (possibly variations on a theme) are legitimate models • Hybrid Inflation is a legitimate model • “Graceful” Old Inflation currently fails, though some modifications of parameters may fix it. • Further WMAP and Planck measurements will help thin down the number of inflationary models • Inflation still stands up well against alternate theories

35. References • A great place to see how different parameters affect the CMB spectrum: space.mit.edu/home/tegmark/cmb/ • Liddle, A, “An introduction to cosmological inflation”  arXiv:astro-ph/9901124v1, 1999 • Zel’dovich, YA. B., “Cosmological Field Theory for Observational Astronomers.”, Sov. Sci. Rev. E Astrophys. Space Phys., Vol. 5, 1986, pp. 1-37 • Lineweaver, C. H., “Inflation and the Cosmic Microwave Background” arXiv:astro-ph/0305179v1, 2003 • Garcia-Bellido, J. “Cosmology and Astrophysics” lecture • Narlikar, J. V., and Padmanabhan, T., “Inflation for Astronomers” Annual review of astronomy and astrophysics, 1991, 29:325-63 • Copeland, E. J, “Inflation – In the early universe and today” lecture • Marco, F. D. and Notari, A ‘”Graceful” Old Inflation’ arXiv:astro-ph/0511396v2, 2006

36. If I get to this slide before 30 minutes is up... • My speech in a witty quote: "All models are wrong, but some are useful" George E. P. Box • "Brevity is the soul of wit." William Shakespeare • "Amplification is the vice of modern oratory. It is an insult to an assembly of reasonable men, disgusting and revolting instead of persuading. Speeches measured by the hour, die with the hour." Thomas Jefferson • "Few sinners are saved after the first twenty minutes of a sermon." Mark Twain • "The secret of being a bore is to tell everything." Voltaire