Preprint typeset in JHEP style - HYPER VERSION Classical Dynamics of Inflation Daniel Baumann School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 http://www.sns.ias.edu/ dbaumann/ dbaumann@ias.edu Abstract: This is the first of a series of notes on inflationary cosmology. I review the horizon problem, discuss the inflationary solution to it and present the basic elements of the physics of inflation.
Contents 1. Introduction. The Horizon Problem 3.1 FRW Spacetimes 3. Causal Structure 4.3 (Conformal) Time is Finite 4.4 Shock in the CMB 5.5 Quantum Gravity Hocus-Pocus? 6 3. A Simple Solution 6 3.1 The Shrinking Hubble Sphere 6 3. (Conformal) Time is Infinite 6 3.3 Causal Superhorizon Correlations 7 3.4 Conditions for Inflation 8 4. The Physics of Inflation 10 4.1 Scalar Field Dynamics 10 4. Slow-Roll Inflation 11 4.3 Case Study: m φ Inflation 1 4.4 The Eta Problem 14 4.5 Generalizations 15 5. Outlook 16 A. Particle Horizon and Causality 18 1
1. Introduction Running the expansion of the universe back in time, the uniformity of the cosmic microwave background (CMB) is quite puzzling. It is a famous fact that in the conventional Big Bang cosmology the CMB at the time of decoupling consisted of about 104 causally independent patches. Two points on the sky with an angular separation exceeding 1 degree, should never have been in causal contact, yet they are observed to have the same temperature to extremely high precision. This puzzle is the horizon problem. However, as we will see, the horizon problem in the form stated above assumes that no new physics becomes relevant for the dynamics of the universe at early times (and extremely high energies). In these notes I will explain how a specific form of new physics may lead to a negative pressure component and quasi-exponential expansion. This period of inflation produces the apparently acausal correlations in the CMB and hence solves the horizon problem.
Remarkably, inflation also explains why the CMB has small inhomogeneities: quantum mechanical zero-point fluctuations during inflation are promoted to cosmic significance as they are stretched outside of the horizon. When the perturbations re-enter the horizon at a later time, they seed the fluctuations in the CMB. Through explicit calculation one finds that the primordial fluctuations sourced by inflation are just of the right type (Gaussian, scale-invariant and adiabatic) to explain the observed spectrum of CMB fluctuations. This remarkable story will be told in a separate set of notes. 1 The present notes will be setting the stage by explaining how the classical dynamics during inflation solves the horizon problem. An effort was made to keep these notes as concise as possible. More details may be found in my previous lectures on the topic.. The Horizon Problem.1 FRW Spacetimes A homogeneous and isotropic universe is described by the Friedmann-Robertson-Walker (FRW) metric 3 [ ] dr ds = dt + a (t) 1 kr + r (dθ + sin θ dφ ), (.1) where k = 0, k = +1 and k = 1 for flat, positively curved and negatively curved spacelike 3-hypersurfaces, respectively. For ease of notation we will restrict most of our discussion to the case k = 0. 4 In that case, the Friedmann equations for the evolution of the scale factor a(t) are H = 1 ρ and Ḣ + H = 1 (ρ + 3p), (.) 3Mpl 6Mpl where H t ln a is the Hubble parameter and ρ and p are the density and pressure of background stress-tensor (here assumed to be a perfect fluid). To study the propagation of light (and hence the causal structure of the FRW universe) it is convenient to define conformal time dτ = dt a(t). (.3) 1 D. Baumann, Quantum Field Theory in de Sitter. D. Baumann, TASI Lectures on Inflation. 3 Throughout these notes I will set the speed of light equal to unity, c 1. 4 A flat universe is in fact favored by present observations (see Figure 7). Furthermore, as we will explain, it is a fundamental prediction of 60 e-folds of inflationary expansion. 3
The FRW metric then factorizes into a static Minkowski metric η µν multiplied by a time-dependent conformal factor a(τ). Causal Structure ds = a (τ) [ dτ + ( dr + r (dθ + sin θ dφ ) )] (.4) a (τ) η µν dx µ dx ν. (.5) In an isotropic universe the radial propagation of light (dθ = dφ 0) is characterized by the following two-dimensional line element ds = a (τ) [ dτ + dr ]. (.6) Just like in Minkowski space the null geodesics of photons (ds 0) are straight lines at ±45 angles in the τ-r plane r(τ) = ±τ + const. (.7) The maximal distance a photon (and hence any particle) can travel between an initial time t i and later time t > t i is r = τ τ τ i = t t i dt a(t ), (.8) i.e. the maximal distance travelled is equal to the amount of conformal time during the interval t = t t i. The initial time is often taken to be the origin of the universe, t i 0, defined by the initial singularity a i a(t i = 0) 0. We then get r max (t) = t 0 dt a(t ) We call this the comoving (particle) horizon..3 (Conformal) Time is Finite = τ(t) τ(0). (.9) The integral defining conformal time may be re-written in the following interesting way dt τ a(t) = (ah) 1 d ln a. (.10) This shows that the elapsed conformal time depends on evolution of the comoving Hubble radius (ah) 1. For example, for a universe dominated by a fluid with equation of state w p/ρ, we have (ah) 1 a 1 (1+3w). (.11) 4
Note the dependence of the exponent on the combination (1+3w). All familiar (meaning observed) matter sources satisfy the strong energy condition 1 + 3w > 0, so it was reasonable for the post-hubble physicists to assume that the comoving Hubble radius increases as the universe expands. Performing the integral in (.10) gives τ a 1 (1+3w). (.1) For conventional matter sources the initial singularity is therefore at τ i = 0, τ i a 1 (1+3w) i = 0, for w > 1 3. (.13) The comoving horizon (.9) is hence finite in the standard Big Bang cosmology..4 Shock in the CMB Conformal Time τ 0 Past Light-Cone Last-Scattering Surface τ rec τ i =0 Recombination Big Bang Singularity Particle Horizon Figure 1: Conformal diagram for the standard FRW cosmology. A moment s thought will convince you that the finiteness of the conformal time elapsed between t i = 0 and the time of CMB decoupling t rec implies a serious problem: most spots in the CMB have non-overlapping past light-cones (Figure 1) and hence never were in causal contact. Why aren t there order-one fluctuations in the CMB temperature? 5
.5 Quantum Gravity Hocus-Pocus? Let me digress briefly to make an important qualifier: when we inferred that the total conformal time between the singularity and decoupling is finite and small, we included times in the integral in (.10) that were arbitrarily close to the initial singularity: τ = t 0 dt δt a(t ) = 0 dt a(t ) } {{ } δτ t + δt dt a(t ) }{{} inflation? (.14) In the first integral after the second equality in (.14) we have no reason to trust the classical geometry (.1). By stating the horizon problem as we did, we were hence implicitly assuming that the breakdown of general relativity in the regime close to the singularity does not lead to large contributions to the conformal time: δτ τ. This assumption may be incorrect and there may be no horizon problem in a complete theory of quantum gravity. In the absence of an alternative solution to the horizon problem this is a completely reasonable attitude to take. However, I will now show that inflation provides as very simple and computable solution to the horizon problem. Effectively, this is achieved by modifying the scale factor evolution in the second integral in (.14), i.e. in the classical regime. I then leave it to the reader to decide if they prefer this solution over a version of quantum gravity hocus-pocus. 3. A Simple Solution 3.1 The Shrinking Hubble Sphere What if the dominant energy component in the early universe did not satisfy the strong energy condition? The comoving Hubble radius would then be decreasing in time rather than increasing; cf. Eqn. (.11): d dt (ah) 1 < 0. (3.1) The consequences of this qualitative change in the behavior of the evolution of the Hubble sphere are dramatic. 3. (Conformal) Time is Infinite We notice immediately that the Big Bang singularity is pushed to negative conformal time, τ i a 1 (1+3w) i =, for w < 1 3. (3.) 6
Conformal Time τ 0 Past Light-Cone Last-Scattering Surface τ rec 0 Recombination Reheating Particle Horizon Inflation causal contact τ i = Big Bang Singularity Figure : Conformal diagram for inflationary cosmology. This implies that there was much more conformal time between the singularity and decoupling than we had thought! Fig. shows the new conformal diagram. The past light cones of widely separated points in the CMB now had time to intersect before the singularity. 3.3 Causal Superhorizon Correlations A decreasing comoving horizon means that large scales entering the present universe were inside the horizon before inflation (see Fig. 3). Causal physics before inflation therefore established spatial homogeneity. With a period of inflation, the uniformity of the CMB is not a mystery. 7
comoving scales horizon exit horizon re-entry (ah) 1 sub-horizon super-horizon sub-horizon k 1 INFLATION reheating time [ln a] CMB recombination today Figure 3: Solution of the horizon problem. Scales of cosmological interest were larger than the Hubble radius until a 10 5. However, very early on, before inflation operated, all scales of interest were smaller than the Hubble radius and therefore susceptible to microphysical processing. Similarly, at very late time, scales of cosmological interest came back within the Hubble radius. 3.4 Conditions for Inflation There are three equivalent conditions for inflation (all related by the Friedmann equations): d dt (ah) 1 < 0 d a > 0 ρ + 3p < 0. (3.3) dt Decreasing comoving horizon I like to use the shrinking Hubble sphere as the fundamental definition of inflation since it most directly relates to the flatness and horizon problems and is key for the mechanism of generating fluctuations. Accelerated expansion From the relation d dt (ah) 1 = d dt (ȧ) 1 = ä (ȧ), (3.4) we see immediately that a shrinking comoving Hubble radius implies accelerated expansion d a dt > 0. (3.5) 8
This explains why inflation is often defined as a period of accelerated expansion. Slowly-varying Hubble Alternatively, we may write d + aḣ dt (ah) 1 = ȧh = 1 Ḣ (1 ε), where ε (ah) a H > 0. (3.6) The shrinking Hubble sphere therefore also corresponds to ε = Ḣ ln H = d H dn < 1 (3.7) Here, we have defined dn d ln a = Hdt, which measures the number of e-folds N of inflationary expansion. Eqn. (3.7) implies that the fractional change of the Hubble parameter per e-fold is small. To solve the cosmological problems we want inflation to last for a sufficiently long time (usually at least N 40 to 60 e-folds). To achieve this requires that ε remains small for a large number of Hubble times. This condition is measured by a second parameter η ε Hε = d ln ε dn. (3.8) For η < 1 the fractional change of ε per Hubble time is small and inflation persists. Negative pressure What stress-energy can source acceleration? Consulting the Friedmann equations (.), Ḣ + H = 1 ( (ρ + 3p) = H 1 + 3p ) 6Mpl ρ (3.9) we find that ε = Ḣ H = 3 ( 1 + p ) < 1 p < 1 ρ 3 ρ, (3.10) i.e. inflation requires negative pressure or a violation of the strong energy condition (SEC). How this can arise in a physical theory will be explained in the next section. We will see that there is nothing sacred about the SEC and it can easily be violated. 9
4. The Physics of Inflation A given FRW background with time-dependent Hubble parameter H(t) corresponds to prolonged cosmic acceleration if (and only if) ε = Ḣ and η = ε H Hε are small: {ε, η } < 1. What microscopic physics can lead to these conditions? 4.1 Scalar Field Dynamics (4.1) Consider a scalar field φ, the inflaton, with action S = d 4 x [ M pl g R 1 ] µφ ν φg µν V (φ), (4.) where R is the Ricci scalar derived from the metric g µν and V (φ) is an arbitrary function. The time evolution of the inflaton in the FRW background (.1) is governed by the Klein-Gordon equation φ + 3H φ = V, (4.3) where the size of the Hubble friction is determined by the Friedmann equation H = 1 [ ] 1 3Mpl φ + V. (4.4) From (4.3) and (4.4) we derive the continuity equation Ḣ = 1 φ M pl Substituting this into the definition of ε we find. (4.5) 1 ε = φ. (4.6) Mpl H Inflation therefore occurs if the potential energy, V, dominates over the kinetic energy, 1 φ. For this condition to persist we want the acceleration of the scalar field to be small. To assess this, we define the dimensionless acceleration per Hubble time Taking the time-derivative of (4.6) we find δ φ. (4.7) H φ δ = ε 1 η. (4.8) Hence, if {ε, δ } 1 then both H and ε have small fractional changes per e-fold: {ε, η } 1. 10
4. Slow-Roll Inflation So far, we ve not made any approximations. We ve just noted that in a regime where {ε, δ } 1, inflation persists. We now use these conditions to simplify the equations of motion. This is called the slow-roll approximation: 1st slow-roll condition φ The condition ε = 1 1 implies 1 φ V and hence leads to the following Mpl H simplification of the Friedmann equation H nd slow-roll condition The condition δ = V 3M pl (SR approximation) (4.9) φ 1 simplifies the Klein-Gordon equation to H φ 3H φ V (SR approximation) (4.10) Substituting (4.9) and (4.10) into (4.6) gives ε = Ḣ H M ( ) pl V ɛ v. (4.11) V Furthermore, taking the time-derivative of (4.10), leads to 3Ḣ φ + 3H φ = V φ, (4.1) δ + ε = φ H φ + Ḣ H M V pl V η v. (4.13) Hence, a convenient way to assess a potential V (φ) is to compute the potential slow-roll parameters 5 ɛv M pl ( ) V and η v M V pl V V. (4.14) When these are small, slow-roll inflation occurs. The amount of inflation is measured by the number of e-folds of accelerated expansion N tf t i H(t) dt, (4.15) 5 In contrast, the parameters ε and η are often called the Hubble slow-roll parameters. During slow-roll the parameters are related as follows: ɛ v ε and η v ɛ 1 η. 11
where t i and t f are defined as the times when ε(t i ) = ε(t f ) 1. In the slow-roll regime we can use Hdt = Ḣ dφ 3H Hdφ 1 dφ (4.16) φ V ɛv M pl to write (4.15) as an integral in the field space of the inflaton N = φf dφ, (4.17) ɛv M pl φ i 1 where φ i and φ f are defined as the boundaries of the interval where ɛ v < 1. The largest scales observed in the CMB are produced some 40 to 60 e-folds before the end of inflation φf dφ N cmb = 40 60. (4.18) ɛv M pl φ cmb 1 Successful inflation requires at least N cmb e-folds. 4.3 Case Study: m φ Inflation As an example, let us give the slow-roll analysis of arguably the simplest model of inflation: single field inflation driven by a mass term V (φ) = 1 m φ. (4.19) Figure 4: m φ inflation. The slow-roll parameters are ( ) Mpl ɛ v (φ) = η v (φ) =. (4.0) φ 1
To satisfy the slow-roll conditions ɛ v, η v < 1, we therefore need to consider super- Planckian values for the inflaton φ > M pl φ f. (4.1) The relation between the inflaton field value and the number of e-folds before the end of inflation is N(φ) = φ 4M pl Fluctuations observed in the CMB are created at 1. (4.) φ cmb = N cmb M pl 15M pl. (4.3) Finally, let us comment that slow-roll inflation for the m φ potential is an attractor solution. To see this you should study the phase space diagram using 6 d φ dφ = The result is portrayed in Figure 5. 7 3 1 ( φ + m φ ) 1/ φ M + m φ pl φ φ. (4.4) 3 mmpl φ 3 mmpl Attractor Figure 5: Phase space diagram of m φ inflation. 6 To arrive at Eqn. (4.4) we substituted φ = dφ into the Klein-Gordon equation. 7 Figure reproduced from V. Mukhanov, Physical Foundations of Cosmology. φ d φ 13
4.4 The Eta Problem E M pl UV-completion e.g. string theory M s M KK M X M susy Λ M Y low-energy EFT H m φ Figure 6: Spectrum of states and the effective theory of inflation. Above we described the simplest models of inflation by postulating that the lowenergy dynamics of the inflaton is governed by the action (4.). We now briefly discuss the challenge of finding a more fundamental description of the physics of inflation in which the action (4.) arises in the low-energy limit. If we begin with a UV-complete theory 8 (like string theory), then we can indeed derive an effective field theory (EFT) (valid below a high-energy cutoff Λ) by integrating out all massive fields with M > Λ (see Fig. 6). In the absence of detailed knowledge of the UV-completion, we can still make progress if we parameterize our ignorance of the UV theory by writing the most general EFT consistent with (postulated) symmetries. In both cases, this results in new non-renormalizable interactions among the light fields (m < Λ), L = O φ 6 ( φ) 4 e.g., (4.5) Λ 4 Λ Λ 4 where O are operators of dimensions constructed from the light fields. So far these were just general statements about effective field theory. What is special about the effective field theory of inflation is that in inflation even Plancksuppressed interactions cannot be ignored!, i.e even if we can take the cutoff to be the 8 By a UV-complete theory we here mean a theory that gives us access to all couplings of the inflaton to other fields with masses up to the Planck scale (see Fig. 6). 14
Planck mass, Λ = M pl, corrections to the action (4.) cannot be ignored. Specifically, the inflation mass is sensitive to dimension-six, Planck-suppresed operators V = O 6 M pl V 0 φ M pl η = O(1). (4.6) If such corrections are included in the action, inflation generically ends prematurely. This is the eta problem. It is rare that Planck-suppressed operators are significant in our theories of lowenergy physics. When this happens, any satisfactory theory must come with a sufficient degree of UV-completeness to estimate such corrections or to explain their absence. In recent years significant progress has been made in understanding and computing these UV-corrections in string theory. 9 4.5 Generalizations The simplest inflationary actions (4.) may be extended in a number of obvious ways: 1. Non-minimal coupling to gravity. The action (4.) is called minimally coupled in the sense that there is no direct coupling between the inflaton field and the metric. In principle, we could imagine a non-minimal coupling between the inflaton and the graviton, however, in practice, non-minimally coupled theories can be transformed to minimally coupled form by a field redefinition.. Modified gravity. Similarly, we could entertain the possibility that the Einstein-Hilbert part of the action is modified at high energies. However, the simplest examples for this UVmodification of gravity, so-called f(r) theories, can again be transformed into a minimally coupled scalar field with potential V (φ). 3. Non-canonical kinetic term. The action (4.) has a canonical kinetic term L φ = X V (φ), X 1 gµν µ φ ν φ. (4.7) Inflation can then only occur if the potential V (φ) is very flat. More generally, however, we could imagine that the high-energy theory has fields with noncanonical kinetic terms (i.e. higher-derivative corrections like in (4.5) are significant) L φ = F (φ, X) V (φ), (4.8) 9 D. Baumann and L. McAllister, Advances in String Inflation. 15
where F (φ, X) is some function of the inflaton field and its derivatives. For actions such as (4.8) it is possible that inflation is driven by the non-trivial kinetic term and occurs even in the presence of a rather steep potential. We note that F (φ, X) is the most general Lorentz-invariant Lagrangian as a function of φ and its first derivative. Going beyond actions of the form of Eqn. (4.8) requires either breaking Lorentz-invariance and/or adding terms with higher spacetimederivatives (which cannot be eliminated by partial integrations) e.g. ( φ). Cheung et al. derived the most general effective theory of single-field inflation that includes all these possibilities. 10 4. More than one field. If we allow more than one field to be dynamically relevant during inflation, then the possibilities for the inflationary dynamics (and the mechanisms for the production of fluctuations) expand dramatically and the theory often loses a lot of its predictive power. However, work on an effective field theory of multi-field inflation that might restore a more systematic approach is under way. 11 5. Outlook In these notes, I discussed the classical dynamics of inflation ( = 0) and explained how it provides a simple solution to the horizon problem. Inflation therefore explains the large-scale homogeneity, isotropy and flatness of the universe. In a separate set of notes I will present the quantum limit of inflation ( 0) and show that it provides a beautiful mechanism to explain the observed CMB fluctuations. Quantum zero-point fluctuations of the inflaton lead to primordial density fluctuations of precisely the right type to account for the observed CMB anisotropies. A compact representation of the millions of pixels of a CMB map is in terms of the angular power spectrum (see Figure 7) C l 1 l + 1 a lm, where m T (θ, φ) T = l,m a lm Y lm (θ, φ). (5.1) All the predictions of inflation are directly (or indirectly) encoded in the CMB power spectrum: On large scales the universe is 1a) homogenous: the temperature fluctuations are small: T 10µK, 10 Cheung et al., The Effective Theory of Single Field Inflation. 11 Senatore and Zaldarriaga, work in progress. 16
flat 90 Angular Scale 0.5 0. homogeneous adiabatic Gaussian scale-invariant superhorizon Multipole moment isotropic Figure 7: The observational evidence for inflation. 1b) isotropic: little information is lost by the sum over a`m s in (5.1), 1c) flat: the first peak of the power spectrum is at ` 00. Its small-scale fluctuations are a) superhorizon: the power doesn t vanish for θ >, b) scale-invariant: the primordial power is nearly independent of scale, c) Gaussian: little information is lost by reducing the data to the power spectrum, d) adiabatic: the presence of the acoustic peaks constrains isocurvature fluctuations. The quantum mechanical origin of the primordial seeds for cosmological perturbations is a beautiful, quantitative story that I will tell in a separate set of notes.1 1 D. Baumann, Quantum Field Theory in de Sitter. 17
A. Particle Horizon and Causality In this appendix we compute the angle subtended by the comoving horizon at recombination. This is defined as the ratio of the comoving particle horizon at recombination and the comoving angular diameter distance from us (an observer at redshift z = 0) to recombination (z 1090) (cf. Figure 1) θ hor = d hor d A. (A.1) A fundamental quantity is the comoving distance between redshifts z 1 and z τ τ 1 = z z 1 dz H(z) I(z 1, z ). (A.) The comoving particle horizon at recombination is d hor = τ rec τ i I(z rec, ). (A.3) In a flat universe, the comoving angular diameter distance from us to recombination is d A = τ 0 τ rec = I(0, z rec ). (A.4) The angular scale of the horizon at recombination therefore is θ hor d hor d A = I(z rec, ) I(0, z rec ). (A.5) Using H(z) = H 0 Ω m (1 + z) 3 + Ω γ (1 + z) 4 + Ω Λ, (A.6) where Ω m = 0.3, Ω Λ = 1 Ω m, Ω γ = Ω m /(1 + z eq ) and z eq = 3400, we can numerically evaluate the integrals I(0, z rec ) and I(z rec, ), to find θ hor = 1.16. (A.7) Causal theories should have vanishing correlation functions for θ > θ c θ hor =.3. (A.8) Inflation explains why we observe correlations in the CMB for θ. 18