Lecture 27

After a last few comments about Big Bang Nucleosynthesis and observations, we considered extensions forward and back of the standard Big Bang model: an inflationary epoch before radiation domination, and our current $\Lambda$ -dominated era after matter domination.

First we discussed motivations and content for inflation, the theory widely accepted to describe the era before the standard thermal history discussed last time.

The motivations for inflation are many, coming from both particle physics and cosmology. The most immediate motivator was the monopole problem, from particle physics. We introduced the paradigm of force unification in particle physics. Here fundamental physics, or the lagrangian, is symmetric; however, observations become asymmetric after spontaneous symmetry breaking, when an asymmetric ground state develops and biases all observations. (The example to picture here is the ferromagnet, since spin interactions are rotationally symmetric, simply trying to align any neighbor spins; yet the low temperature ground state, of aligned spins, is not rotationally symmetric and instead involves one favored direction, whose magnetization biases any observations one might make.) The particle physics heuristic is that all particle interactions were symmetric, or unified, at high temperature, but that gradually more and more asymmetric ground states have developed, introducing the asymmetries we observe today between weak, strong and electromagnetic interactions. Electroweak unification at the TeV scale is quite well established (although the particle associated with the asymmetric ground state, the Higgs, has not yet been detected). Grand unification, of strong and electroweak forces at a scale $10^{16}$ GeV, is anticipated. A problem, however, is that we can show (from topology of the forces' symmetry groups) that in the breaking of a grand unified theory to asymmetric strong and electroweak interactions, monopoles must form, where the ground state winds spherically. By causality, about 1 monopole forms per horizon volume, carrying a mass (or energy) of about $10^{16}$ GeV. Yet cosmology disallows the presence of these monopoles: their energy density would drive the expansion of the universe too fast, so that a universe with the parameters we observe today would be too young (that is, observed objects would be older than the inferred age of the universe). Particle physics' prediction of relic monopoles whose high density was ruled out by cosmology was called the ``monopole problem''.

Another motivator for inflation was the flatness problem, which says, as discussed in prior homework, that $\Omega -1$ grows in both RD and MD eras. For $\Omega -1$ as small as it is now measured to be, it must have been fine-tuned within a tiny factor of 1 in the past: within $10^{-3}$ at decoupling, $10^{-6}$ at matter-radiation equality, $10^{-16}$ at big bang nucleosynthesis, $10^{-28}$ at the electroweak phase transition. We believe we understand physics quite well back to the electroweak phase transition, and we have detailed and verified cosmological understanding of big bang nucleosynthesis. Yet we rely on an extreme fine-tuning of $\Omega$ to describe the universe at that time.

Finally, another longstanding cosmological problem was the horizon problem. For a RD Friedmann-Robertson-Walker universe the particle horizon is finite; at time , only points out to a finite comoving distance $\sigma_p$ can have influenced conditions at a given origin. This means that only regions of size $\sigma_p$ are causally connected; in particular, we only expect regions up to size $\sigma_p$ to have established thermal equilibrium and a uniform temperature. This becomes a problem when we consider the cosmic microwave background (CMB). Photons observed in the CMB have been en route to us since decoupling, without any further interaction; thus they give us a picture of the universe at the time of decoupling. The observed universe is times the size of a causal region $\sigma_p$ at decoupling, meaning that we'd expect the microwave background to have originated from causally distinct domains, each with an uncorrelated temperature. We thus expect a very inhomogeneous CMB, when in fact what we observe is very homogeneous, with a CMB temperature of K across the entire sky, with fluctuations of only 1 part per million. The horizon problem is, by what causal mechanism could the universe have been brought to such a uniform temperature over distinct causal domains, at the time of decoupling?

A solution to all these problems is inflation. As we discussed earlier, when the equation of state is $p = -\rho$ , the universe undergoes exponential expansion, with constant . This equation of state arises for a cosmological constant, or for a scalar field not in its ground state, contributing bulk potential energy to the universe. We noted that an era of exponential expansion makes $\Omega -1$ , the temperature, and any monopole density decay exponentially, while exponentially increasing the size of causal domains, solving the flatness, monopole, and horizon problems. The only problem is that of ending inflation. We discussed how dynamics of a scalar field, in tunneling or rolling to the true ground state (old, new, or chaotic inflation) could turn off the bulk potential energy driving exponential expansion. We described reheating as the transfer of that former bulk potential energy into particle creation, populating and reheating the universe to nonzero temperature and creating lots of entropy, from which point the universe is RD and evolves as in the standard thermal history. We also noted the last bonus of inflation: during inflation the scalar field has quantum fluctuations, which are stretched to macroscopic size by inflation. These fluctuations can be calculated and serve as inhomogeneities, seeds for future gravitational collapse to form structure (galaxies, etc). Quantum fluctuations from inflation currently provide the only viable model for the origin of structure in the universe.

We then noted some current problems concerning our universe as currently observed, with $\Omega_\Lambda = 0.73$ . First, this nonzero value of $\Lambda$ is 120 orders of magnitude below what one might predict for $\Lambda$ , from zero point energy of quantum fields, from particle theory. This is known as the cosmological constant problem; this once was the problem of why $\Lambda = 0$ , but now is the more acute aesthetic problem of how $\Lambda$ can be nonzero yet still pathologically small. A second set of problems comes from the unnatural blend of energy density in radiation, matter, and $\Lambda$ in our universe, with each giving contributions to the energy density scaling as $R^{-4}$ , $R^{-3}$ , and constant. The question of how we managed to achieve a brief matter-dominated era before $\Lambda$ -domination, and how we managed to turn up to measure the universe at such an unlikely point as just after matter- $\Lambda$ equality, is known as the cosmic coincidence problem. Like the radiation-dominated universe, the $\Lambda$ -dominated universe with its repulsive effect allows no structure formation, so the brief matter-dominated phase is essential to obtaining structure formation. Quintessence, which makes $\Lambda$ dynamical by obtaining it from scalar field dynamics, with evolving equation of state, is a current research direction trying to address these problems.

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2004-04-22