Setting up the initial condition for the hot big-bang Universe

Transcription

1 Chapter 6 Inflation Setting up the initial condition for the hot big-bang Universe We have studied various thermal events happen at early times in the expanding universe. In this chapter, we will consider the very early epoch of the history, the time that is so early that we can consider it as the initial time of the Universe. As we do not have much (if not at all) direct observational evidence of what had happened at this time, the guiding principle in this chapter is naturalness. The question that we want to address first is, therefore, what should be the natural initial condition for the hot big-bang Universe? Before going into the details, we should clarify what we mean by initial time. Let us define the initial time to the time at which we think there is a big-bang singularity, the time when the Universe was in a hot, dense state. Of couse, there is a possibility that the Universe may be now in the n-th cycle of the bouncing cosmology, or Ekpyrotic scenario. In that case our initial time is when the size of the Universe is minimum. That is, we are not discussing what could possibly happens before the rebounce. The highest energy scale of defining initial condition may be around the anck scale above which we may need a completely new physics of quantum gravity. For many physicists, energy scale around the anck scale seems to be the natural choice of the initial energy scale. But, we can be super-conservative (or liberal) to call any scale above TeV as an initial condition because that is the scale above which we have not probed yet. The exact energy scale does not matter for the purpose of our discussion here. 6. Motivation In order to deduce the natural initial conditions, let us review the observational propoeries of the Universe as listed earlier in this course. We listed that the Universe is old big approximately homogeneous and isotropic expanding approximately flat

2 2 CHAPTER 6. INFLATION have a nearly scale invariant seed fluctuation with δρ/ρ 5 that follows nearly Gaussian statistics. What should the initial condition of the Universe requres to make the Universe that we see today? expanding and old General relativity tells us that Universe is dynamical and therefore it either expands or contracts. From the Friedmann equation, 3H 2 + 3κ = 8πGρ, (6.) we see that if ρ and κ initially then the Universe will expand forever and to yield something like what we are observing. For κ >, if expanding at the beginning, the Universe will keep expanding as long as 8πGρ 3κ >, but will recollapse when 8πGρ = 3κ (let s assume that H(t) is the monotonic function). Therefore, the condition must be that 8πGρ > 3κ (6.2) for all cosmic time until now to keep the Universe expanding. In terms of the energy density parameter, the condition is Ω i (t) > Ω k (t), (6.3) i that is, the spatial curvature should never dominates the other energy components at any time until now in order to keep the Universe expanding. Like our Universe, at any given time, the universe is dominated by one energy component. To satisfy this condition, we have to have (setting a (i) = ) Ω k = Ω Ω i (i) k Ω (i) i a 2+3(+wi) = Ω(i) k Ω (i) i a +3w i <, (6.4) which is satisfied either w i < /3 for all components (that is, dominated by, e.g., dark energy or domain wall from the initial time until now!), or the initial Ω k is extrememly small. The second condition is related to the flattness that we will discuss later. big Let us consider a natural Universe: a anck size universe expanding with the anck rate H L (in the anck unit). The current expansion rate is, in the anck unit, H = h GeV = h. (6.5) If the Universe is dominated by either radiation or matter from the anck time to now, the current size of the Universe would be, respectively, L = L a a =L =L H 2/3 = = 57. h 2/3 km (matter dominated) (6.6) H /2 = =.39 h /2 mm (radiation dominated). (6.7) H Remember Einstein s greatest blunder problem? H

3 6.. MOTIVATION 3 Ha! That s just a little bit smaller than the size of the Universe now: L = c/h c 28 /h cm (c >, order unity number depending on what you mean by size of the Universe)! To start from LH and end up with L H >, we need the acceleration at some point in the history! d d t (LH) = d (ah) = ä, (6.8) d t approximately flat The observed curvature density parameter is smaller than % of the critical density right now: Ω k <.. In terms of the spatial curvature, it is κ <.H 2 = h 2! (6.9) Extrapolating this to the anck time assuming that the Universe has been dominated by matter or radiation, we have the initial curvature of κ = κ a a 2 = κ = κ 4/3 < h 2/3 (matter dominated) H < (radiation dominated). (6.) H H H Why is that value so small compared to the other natural values (ρ L H )? In other words, why should our Universe start with such a special (finely-tuned) value of the spatial curvature? This problem in the statdard big-bang cosmology is called flatness problem. To avoid the problem, we need something other than the matter or radiation that dominates the expansion to dynamically change κ H 2 at the beginning to κ H 2 at later time: or 2 d ln H d ln a d ln H d ln a d ln κ d ln a From the definition of the Hubble constant, we find d ln H d ln a = d d ln a (ln ȧ ln a) = ȧ dȧ a = 2, (6.). (6.2) da = aä ȧ that the condition is equivalent to (a > ), again, the acceleration, 2 (6.3) ä. (6.4) Therefore, in order to explain the simple observational fact that our Universe is big and approximately flat, we need that the mean acceleration of the cosmic expansion must be positive. From the Raychaudhuri equation ä a = 4πG (ρ + 3P), (6.5) 3 we know that much of the recent expansion of the Universe occurred with ä <, and so require a sufficiently long earlier epoch with ä >! That is called cosmic inflation. The remaining two properties of the Universe can be explained as a consequence of having cosmic inflation at early time.

4 4 CHAPTER 6. INFLATION approximately homogeneous and isotropic The observed CMB temperature is isotropic to one part in hundred-thousand, 5. Is that something we naturally expect from the hot big-bang? The CMB photons that we observe today lastscattered at z 9 (anck23 best-fitting value). The particle horizon at z is the maximum radial distance within which particle could communicate from the big-bang to the last-scattering time (with Ω M =.3): d max (t ) = a a As the angular diameter distance to the last-scattering surface is D A (z ) = z + z da a 2 H(a) a H H. (6.6) dz H(z) = a a da a 2 H(a) a 3.8 =.29H H, (6.7) this makes that the angle subtended by the particle horizon at the time of photon s last-scattering is θ = d max(t ) D A (z ).2 rad.2. (6.8) Do you see the problem now? The angular size corresponding to the causally connected patch on the sky is.2, but yet we see the same temperature over the full sky! How come the one part of the sky knows anything about the area outside of the particle horizon to synchronize its temperature to 5 accuracy? Yeap! It s pretty un-natural, and is worth a name: horizon problem. We need an extreme amount of fine-tuning unless we have some earlier events that synchronize the temperature. Then, what does it take to make a whole sky within a causal contact prior to the last-scattering of CMB? The condition for that is, to make, say d max (t ) > 2D A (z ). Here, what is relevant is the comoving particle horizon size, as a is commong in d max (t ) and D A (z ): d max,c (t ) = t a a d t a(t) = da a ah = da a d c.h.(a). (6.9) For the final equality, we use the comoving horizon d c.h = /(ah), that is the rate of increase of comoving particle horizon d max,c in the logarithmic interval of the scale factor 2. As we have discussed earlier, the time derivative of the comoving horizon is ḋc.h. = ä/ȧ 2, which means that the particle horizon size at earlier time can be larger than the comovingi distance to that time iff ä > (i.e. inflation) so that the comoving horizon size decreases in time. What does that mean? Decreasing comoving horizon means that the size of the causually connected patch was larger at earlier time than later. In other words, if we fix the horizon size, more and more comoving scales were in causual contact at earlier time, then it left outside of horizon as time went by. Thus, with cosmic inflation happened earlier in the history, things that are outside of horizon at the time of the CMB last scattering were in causal contact before. This should be contrasted with the case for the radiation and matter dominated Universe where the comoving horizon size increases in time. That is, in the decelerating Universe, we only gain more and more comoving mode inside of the comoving horizon, and the number of modes inside of the horizon always decreases as going to the earlier times. 2 This is consistent with our earlier definition of the comoving horizon: the comoving size of the region within which particles can communicate within one hubble time.

5 6.2. INFLATION: DEFINITION AND REQUIREMENT 5 As we will see later in this section, the standard inflationary scenarios assume some degree of homogeneity and isotropy to realize the cosmic inflation. That is, we may need to have the homogeneity and isotropy from the first place 3, and inflation does NOT actually solve the homogeneity problem. But, it is fair to say that inflation makes homogeneous and isotropic Universe more homogeneous and isotropic. One more thig that inflation does by making things more homogeneous and isotropic is that it had cleaned up unwanted relics possibly generated from some high energy physics. For example, many physics theories, e.g. grand unified theory, have some higher symmetries at high-energy, which underwent some phase transitions as the energy scale of the Universe droped. Depending on the nature of the symmetry and the way it is broken, these phase transitions typically generate topological defects like domain walls, cosmic strings or monopole. If dominating the energy density of the Universe, these topological defects can be harmful as, e.g. they can overclose the Universe before even radiation dominated epoch starts. Another example of unwanted relics are some exotic particles that a theory predicts, but has never been observed in our Universe like dragons. Yes, if you have those particles before inflation, then inflation will wash them out for you. Primordial black holes that might exist pre-inflationary epoch can also be diluted out by inflation. Again, the key feature here is the acceleration. As the comoving horizon size shrunk during the acceleration phase, the density of exotica in the horizon kept decreasing during inflation and eventually goes below unity. Therefore, the number of exotica per standard model particles, that will be created after the inflation from the process called reheating, becomes much smaller! have a nearly scale invariant seed fluctuation with δρ/ρ 5 that follows nearly Gaussian statistics: When a given scale is stretched outside of the comoving horizon, the quantum (or thermal) fluctuations existed on that scales can be magnified into classical perturbations and get frozen out on superhorizon scales. Predictions from simplestic inflation theory is that the perturbation generated in this way should be nearly scale invariant and follow nearly Gaussian statistics (as observed from the large-scale structure of the Universe)! 6.2 Inflation: definition and requirement As we have seen from many of the basic observational properties of the Universe, it is natural to expect that we have a sufficiently long epoch of acceleration ä > at early time, which defines inflation: infation = An accelerated expansion at early time, ä > Then, during inflation, given comoving scales leave the horizon: d d t λphysical /H and Hubble parameter decreases sufficiently slowly: d ln H d ln a 3 Note that this is NOT the only possibility, but certainly the simplest one. >, (6.2) <, (6.2)

6 6 CHAPTER 6. INFLATION but curvature decreases relative to the Hubble parameter square: and the pressure must be negative (from Raychandhuri equation): d ln H2 d ln a < d ln κ d ln a, (6.22) ρ + 3P <. (6.23) How much infltion do we need to provide the natural initial condition small curvature and large comoving horizon at the end of the inflation? Often, we parameterize the amount of inflation by ȧ f af H f = ln = ln. (6.24) ȧ i a i H i During inflation, the Hubble expansion rate H usually stays constant, so is approximately the same as the e-folding of the scale factor between the beginning and the end of the inflation: af N = ln = a i t f t i H(t)d t, (6.25) which is usually used ot describe the time during inflation. Let s see how much e-folding we need to resolve the flatness problem and the horizon priblem. flatness problem As for a natural initial condition, let us consider the case where κ = H 2 at the beginning of the inflation. Then, at the end of inflation, the density parameter of curvature can be very small, as small as 43 that we have trouble before: κ e H 2 e = e 2 κ H 2 = sign(κ)e 2. (6.26) How large should be to resolve the flatness problem? We relate the calculation above to the density parameter of curvature now as Ω k = κ H 2 = κ e κ H 2 e He 2 κ e H 2 = e 2 ae H 2 e <., (6.27) a H that makes > ln a eh e. (6.28) a H Assuming that a e H e a H, where the subscript denotes the time at the beginning of the radiation dominated epoch, where H = H(eq) a 2 eq a 2 2 a 2 = H Ω m (a eq /a ) 3 eq H a 2 a = Ωm a eq a 3 /4 (6.29) H we find > ln = ln a H a H = ln Ω r ρ ρ,crit H /4 H Ω m a eq a /4 H = ln Ω /4 r H ln ρ GeV 4, (6.3)

7 6.3. INFLATION: HOW TO ACCELERATE THE UNIVERSE 7 where we use Ω r = h 2 ρ crit = h 2 GeV 4 with h =.7. If ρ /4 M.22 9 GeV, then it gives > 7. But, the energy scale at the beginning of the radiation dominated epoch can be much smaller than that; the absolute minimum density is ρ > ( MeV) 4 in order not to spoil the success of the BBN. For this lower limit, we just need > 2. IF the BICEP2 result still holds, then I might just say that the energy scale of inflation is ρ 6 GeV (GUT scale), that gives > 63. horizon problem As we have discussed earlier, after inflation, the comoving horizon size get only increasing. Therefore, if we have inflation prior to the radiation dominated epoch, then the comoving particle horizon must dominate by what had happened before the radiation epoch: d max,c (t ) = t d t a te t i d t a = a e H e e N a e H e e N, (6.3) where we have used a(t) = a i e H(t ti) = a e e H(te t) for the definiteness in the last equality. To explain he homogeneity problem, we need d max,c (t ) > 2d A (z ) 6.36/(a H ), or e N > 6.36 N > ln 6.36 a eh e, (6.32) a e H e a H a H which gives the similar bound to the flatness problem! 6.3 Inflation: how to accelerate the Universe What have we learn so far? In order to provide a natural initial condition for the hot big-bang Universe, we need an inflationary, the accelerated expansion, period prior to the radiation dominated epoch, and inflation must be done for about 6 e-folding if the energy scale of inflation is above the GUT scale. How do we realize such a long duration of the inflation? The requirement for an acceleration is ρ + 3P <, but as we have discussed already, the matter or radiation component cannot satisfy this condition. There are two popular ways of accelerating the Universe: modifying the theory of gravity, and introducing a new energy component (called inflaton) Modification of the theory of gravity Modification of the theory of gravity can be done in a way that the Universe accelerates without the condition ρ + 3P <. There are many classes of modified gravity theories, but almost all, if not all, models known thus far do not pass the solar system test by themselves. Making a viable theory of modification of gravity is an active research field as of now, and we will leave out this possibilities in this course. But, for the purpose of broadening your education, I just highlight a few points here. there are scalartensor theory f (φ)r theory, f (R) theory, and brane-world models, etc. In order to satisfy the solar system test, most of these theories shield the solar system from the modification, so that the theory will go back to the general relativity in a solar-system like (e.g., high density) neighborhood. To do so, they include some non-linear screaning mechanisms called Symmetron (coupling constant β), Chameleon (mass m), or Vainshtein (kinetic term Z) mechanism. These modify the usual /r 2 -law of gravity as F = Gm m 2 r 2 + β 2 Zcs e mr, (6.33)

8 8 CHAPTER 6. INFLATION outside of the solar-system like environment. Usually these theories are to explain the recent cosmic acceleration (without dark energy), but the same theory should be applicable to the early inflation (without inflaton field). I will end the discussion with one interesting general point on the modification of gravity. Namely, the f (R) modification of gravity is equivalent to the scalar-tensor theory. That is, for a generic f (R) theory S = we can construct a equivalent scalar-tensor f (φ)r theory S = d 4 x g [ f (R) + M ], (6.34) d 4 x g f (φ) + f (φ)(r φ) + M. (6.35) The equation of motion for the scalar field in the scalar-tensor theory Lagrangian is δ δl δφ = µ f (φ) + f (φ)(r φ) f (φ) =. (6.36) δ( µ φ) For a non-trivial stable f (φ), it is reasonable to assume f (φ), that reads R = φ and the action becomes identical to the f (R) theory A new energy component: inflaton field If we fix the theory of gravity, we need to find out the way to make the Universe accelerating: ρ+3p <. The most popular way of realizing this is by introducing a scalar field ϕ(t, x), called inflaton field, that has a single value at every point in the spacetime. Classical scalar field in curved spacetime: A brief introduction In the flat spacetime, the scalar field has a Lagrangian density = (ϕ, µ ϕ), which gives the Lagrangian L = d 3 x and action S = d t L = d 4. When the kinetic energy density of the scalar field is /2 ϕ 2, the gradient-energy density is /2( ϕ) 2 (the choice of above kinetic and gradient energy is called canonical 4 ), and the potential-energy density is V (ϕ), the Lagrangian density is given by = /2[ ϕ 2 ( ϕ) 2 ] V (ϕ), which can be written in the Lorentz-covariant form as S flat ϕ = d 4 x 2 ηµν ( µ ϕ)( ν ϕ) V (ϕ). (6.38) 4 One useful way of understanding the scalar field is thinking of the surface of the mattress, we can denote the height of the mattress at a two-dimensional location x i = (x i, y i ) as ϕ(x i ). Imagine a real mattress that every point on the mattress moves up and down, and the adjacent points are connected by a springs. The kinetic energy at each point is /2m ϕ 2, and we can writhe the Lagrangian as L = i [/2m ϕ2 i j i V (x j; x i )]. Then, we go to the continuous limit, and assuming that the potential energy is dominated by the neighboring springs, L = i 2 m ϕ2 i V (x j ; x i ) 2 m ϕ2 i 2 k i jϕ(x j )ϕ(x i ) 2 m ϕ2 i 2 k i j (ϕ i ϕ j ) 2 ϕ 2 ϕ 2 i j i j i i i j i i i j i d 2 x ϕ 2 ϕ 2 ϕ 2 σ ρ + τϕ 2 +, (6.37) 2 t x y where σ = m/a is the surface mass density, and ρ, σ is related to the coupling matrix k i j (although we use the statistical isotropy to have one ρ for two directions). With the Lagrangian above, the phase velocity of the free mattress theory is given by v = ρ/σ, which guarantees the Lorentz invariance on the mattress (with the phase velocity playing a role of the speed of light). We can trivially extend the picture above to the three dimension and get the Lagrangian we discuss in the main text.

9 6.3. INFLATION: HOW TO ACCELERATE THE UNIVERSE 9 Then, we extend the action to the general curved space-time as S ϕ = d 4 x g 2 gµν µ ϕ ν ϕ V (ϕ). (6.39) Note that we made three changes: ) including g in order to make the integration measure coordinate independent (scalar), 2) changing η µν to the inverse metric tensor g µν, and 3) replacing the partial derivative µ to the covariant derivative µ (although it does not make any difference when acting on the scalar function). When the action is given, we calculate the energy momentum tensor (see, chapter ) as T µν 2 δ( g ϕ ) g δg µν = µ ϕ ν ϕ g µν 2 αϕ α ϕ + V (ϕ), (6.4) from which we read off (see, homework ) ρ ϕ T = 2 ϕ2 + ( ϕ) 2 + V (ϕ) 2 a 2 P ϕ 3 T i i = 2 ϕ2 ( ϕ) 2 V (ϕ). (6.4) 6 a 2 The calculate equation of motion of the scalar field from the Euler-Lagrange equation δ( g ) δϕ as following. First, calculating the derivatives: and = α δ( g ) δ( α ϕ), (6.42) δ( g ) = g dv δϕ dϕ δ( g ) = g g µν δ( α ϕ) 2 δ µα ν ϕ + µ ϕδ να = g g αν ν ϕ, (6.43) Then, the equation of motion becomes Here, we use the D Alembertian operator δ α = α g g αν ν ϕ g ϕ. (6.44) δ( α ϕ) In the FRW background, the D Alembertian becomes ϕ dv =. (6.45) dϕ f = g µν µ ν f = g ν g g µν ν f. (6.46) ϕ = g µν µ ν ϕ = ϕ 3H ϕ + ϕ, (6.47) a2 with ϕ = g i j i j ϕ is the Laplacian in the curved hyperspace. Using this, the equation of motion for the scalar field is ϕ + 3H ϕ dv ϕ + =. (6.48) a2 dϕ

10 CHAPTER 6. INFLATION Accelerated expansion with inflaton Using the condition in Eq. (6.4), the condition for the acceleration now reads or ρ ϕ + 3P ϕ = 2 ϕ 2 2V <, (6.49) ϕ 2 < V (ϕ). (6.5) That is, as long as the potential-energy density dominates over the kinetic-energy density, the expansion of universe accelerates! Although it is not absolutely necessary, in order to simplify the analysis, from now on, we shall assume that the inflation happens in a spatially homogeneous patch so that ϕ = ϕ(t) 5. Also, let us assume that the spatially flat Universe, as the spatial curvature will become very small during inflation. Then, the Friedmann equation becomes 3H 2 = 8πGρ ϕ = 4πG ϕ 2 + 2V. (6.5) From now on, we will then discuss the necessary configuration of the scalar field that accelerates the Universe for about N > 6 e-folding.. An inflaton at the true vacuum, V (ϕ) > : de Sitter expansion First, let us consider the case when the inflaton field sits at the local minimum of the potential V (ϕ) >. In this case, ϕ < V, and this is the de Sitter expansion with scale factor a(t) = a(t i )e H(t t i), (6.52) where H = 8πG/3V is a constant Hubble expansion rate. In this case, we can get as many e-folding as we want, but the inflation never ends, as the inflaton is sitting at the true vacuum. 2. Inflation with a first-order phase transition: Old inflation Then, what if the potential minimum V in the example above is a false vacuum, and we have a true vacuum state V true around zero? Then, the first order phase transition happens so that the inflaton undergoes transition between the false vacuum (inflating phase) to the true vacuum (non-inflating phase). In this case, we can make the inflaton stays at the false vacuum long enough time to have a N > 6 e- folding before the phase transition. This is the idea of Guth in his seminal paper and called old inflation model. The idea of having the first-order phase transition has a problem called graceful exit problem. It is becasue the phase transition that ends the inflation happens quite stochastically, and there is no synchronized clock over the whole Universe. Typical size of the true vacuum patch that underwent the tunneling is the size of the comoving horizon at the end of the inflation, which is pretty small. Each true vacuum patch is surrounded by the false vacuum that still is inflating, and this will separate the true vacuum patches further away as time goes. Thus, there is no chance for the true vacuum patches to merge together to build up a large homogeneous and isotropic patches that we need to solve the naturalness problems of the hot big-bang. 5 This is what almost all inflationary models assume anyway. But, as you have seen earlier, a universe can still accelerate with a non-zero field gradient ( ϕ ) as this term cancels when calculating ρ ϕ +3P ϕ combination. It is presumably because any small scale classical curvature inside of the horizon will eventually be diluted away during inflation. But, still, I think there could be some classical field gradient, that is, ϕ = ϕ(t, x)) (you may call this a classical curvature perturbations) stretched outside of the horizon during the inflation.

11 6.3. INFLATION: HOW TO ACCELERATE THE UNIVERSE 3. Inflation with a second-order phase transition: New inflation The problem with the first order phase transition happens because that the tunneling happens not at the same time. Then, what about the second order phase transition? That is, inflaton was at the origin (false vacuum) before the phase transition and the phase transition smoothly transits the inflaton away from the origin to the true vacuum. This avoid the graceful exit problem of the old inflation, and called new inflation models. This model is suggested by Linde (982) and Albreicht and Steinhardt (982). 4. Randomly placed initial conditions: Chaotic inflation Why should the inflaton stayed at the false vacuum state initially? There is no a good reason for that, except it can give us as many e-folding as we want. Alternatively, it is also conceivable that inflaton field has widely different value here and there and inflation occurs at the pathces that satisfies the condition for accelerating expansion ρ ϕ + 3P ϕ <. This idea is called Chaotic inflation and due to Linde (983). For the new inflation and chaotic inflation, inflation happens while the inflaton field slowly rolling down the potential hill: slowly because acceleration requires ϕ 2 < V (ϕ). Let us analyze the necessary condition for inflation in terms of the infltan field dynamics. The governing equations here are Friedmann equations: and the equation of motion for the inflaton field: H 2 = 4πG ϕ 2 + 2V 3 (6.53) ä a = 4πG 2 ϕ 2 2V 3 (6.54) ϕ + 3H ϕ + V (ϕ) =. (6.55) Here, we use V (ϕ) = dv /dϕ. Combining the two Friedmann equations, we find Ḣ = ä a H2 = 4πG ϕ 2. (6.56) To get a nearly exponential expansion close to the de-sitter Universe where H is constant, we need that the fractional change of the Hubble expansion rate is much smaller than the unity during one hubble time /H: H Ḣ which is, in terms of inflaton field and its potential, H Ḣ = 4πG ϕ 2 4πG 3 = Ḣ, (6.57) H2 ϕ 2 + 2V = H 2. (6.58) That gives ϕ 2 V. (6.59) The condition is the same as what we have seen before. If the potential energy density dominates over the kinetic energy density, then the expansion of the Universe accelerates. Now, the Friedmann equation can be approximated as H 2 8πG V, (6.6) 3

12 2 CHAPTER 6. INFLATION or Ḣ 4πG 3H V ϕ. (6.6) To maintain a sufficiently long duration of the inflation, it is not enough to have an acceleration at one epoch, but maintin the small kinetic energy density for a long time. That is, the fractional change of the field velocity must be small: ϕ. (6.62) H ϕ Using this condition, we can approximate the equation of motion for the inflaton field as ϕ V 3H. (6.63) Coming back to the condition for the nearly exponential expansion, the condition now reads Ḣ H 4πG 2 3H V ϕ 4πG 3 3H 3 V 2 V 2 V 3H = 4πG 4πG 3H 2 8πGV 2 = 6πG V 2. (6.64) The second condition for the longevity of the accelerated expansion becomes ϕ H ϕ 3 V ϕh V Ḣ = V ϕh V 3H 2 V H V Ḣ 2 V H 2 = V 3H + Ḣ 2 H 2 = V 8πG V + Ḣ H 2. (6.65) Because Ḣ /H 2, it must be The arguments above motivates us to define the slow-roll parameters as V. (6.66) 8πG V ε V 2 = M 2 V 2 (6.67) 6πG V 2 V η V V = M 2, (6.68) 8πG V V where M GeV is the reduced anck mass. These slow-roll parameters must be small in order to have inflation (ε) and to have it persisted for a long time (η). In particular, with the slow-roll condition, the expansion of the Universe follows nearly exponential expansion (called quasi-de Sitter expansion). The number of e-folding in this case can be calculated as V t f ϕf N = Hd t = t i ϕ i ϕf H ϕ dϕ ϕ i ϕi 3H2 dϕ 8πG V ϕi V ϕ V dϕ 2 4πGdϕ ϕ. (6.69) f ϕ M f This means that the field excursion during the inflation ϕ must satisfy ϕ N 2 M, (6.7) or if the field excursion is of order ϕ (4πG) /2.282 m pl, the number of e-folding is guaranteed to be large.

13 6.3. INFLATION: HOW TO ACCELERATE THE UNIVERSE Slow-roll inflation: two examples. V (ϕ) = 2 m2 ϕ 2 model The simplest stable potential is V (ϕ) = /2m 2 ϕ 2. The slow-roll conditions of this potential are which is satisfied when ε = η = 2M 2 ϕ 2, (6.7) ϕ 2M ϕ f. (6.72) This means that inflation will happen as long as ϕ > ϕ f, close to the anck energy scale. But, don t be scared. It does not necessarily mean that we have to know quantum gravity to describe the inflation with the quadratic potential. It is not the field value or excursion, but the energy density of inflaton ρ ϕ V (ϕ) that should be smaller than the planck scale 6 That is, the condition that we need to worry about the quantum gravity effect is or M 4 pl V (ϕ) = 2 m2 ϕ 2, (6.73) ϕ 2 M m M. (6.74) Therefore, as long as m M, the field excursion can transpass the anckian value. As long as the slow-roll condition is satisfied, the equation of motion that we solve gets greatly simplitied: which reads ϕ + 3H ϕ + V (ϕ) = 3H ϕ + V (ϕ) = (6.75) H 2 = 4πG ϕ 2 + 2V (ϕ) H 2 = 8πG V (ϕ), 3 3 (6.76) That reduces to the simple equation: 3H ϕ + m 2 ϕ =, H 2 = m2 ϕ 2. (6.77) 6M 2 2 ϕ = 3 mm, H d ln a 3 m = ϕ (6.78) d t 2 M with solution: 2 ϕ(t) =ϕ i 3 mm (t t i ) (6.79) 3 t m a(t) =a i exp ϕ(t)d t. (6.8) 2 M t i 6 Some people may not agree on the statement above. In the effective theory s point of view, φ/m is required because it is the dimensionless quantity enters in the Lagrangian. If the simple potential is a low-energy effective potential, then the natural parameter for the effectiveness of the description is φ/m, and the true theory must be included when φ/m. In particlular, some people think of it seriously as /2m 2 ϕ 2 is the lowest order taylor expansion of many, if not all, theories. Yes, harmonic oscillators are everywhere!

14 4 CHAPTER 6. INFLATION The total number of e-folding is N = ϕi M 2 ϕ f V V dϕ = 2 M 2 ϕ 2 i ϕ 2 ϕi f = 2. (6.8) M The condition to get N > 6 e-folding is now ϕ i > 62M 7.8M. (6.82) We can think of the anck energy scale as the initial time: ϕ i 2(M /m)m, which then gives the limit on the inflaton mass from Eq. (6.82) as 2 m < 7.8 M.8M. (6.83) Then, how long had the universe been inflated? With ϕ i and ϕ f given above and from Eq. (6.79), we find 3 ϕ t = 8 2 mm m 2.M 4 sec, (6.84) m just enough to give acceleration to expand e 6 26 times! Finally, let us consider the field value when the quantum mechanical fluctuation (Hawking radiation) (δϕ) 2 H 2π = 2πM mϕ 6 (6.85) is similar to the classical field excursion during one Hubble time. The Hubble time is and the classical field excursion during this time is t H H = 6M mϕ, (6.86) Therefore, for ϕ = 2M 2 ϕ. (6.87) ϕ 2 > 4π 6M 3, (6.88) m quantum fluctuations of ϕ becomes significant, and as large (or larger) than the classical field excursion! At this energy, we expect to have an eternal inflation, where there exist many Hubble patches outside of our Universe that is still undergoing inflation! Then, what happens after the inflation ends? Inflation ends at ϕ = ϕ f = 2M, when H 2 f 3M 2 V (ϕ) = 3M 2 2 m2 ϕ 2 f = 3 m2 H m, (6.89) 3 then the Hubble expansion rate drops even further afterwards. Therefore, H m after the end of the inflation, and the the equation of motion can be approximated by for the simple harmonic oscillator: ϕ + m 2 ϕ =, (6.9)

15 6.3. INFLATION: HOW TO ACCELERATE THE UNIVERSE 5 and the inflaton oscillates around the potential minimum to the first approximation. Then, let us consider the kinetic term: ϕ 2 = d d t (ϕ ϕ) ϕ ϕ = d d t (ϕ ϕ) + m2 ϕ 2. (6.9) When averaging over the time, the time derivative in the right hand side vanishes because ϕ ϕ is bounded. Therefore, we find 7 ϕ 2 = m 2 ϕ 2 (6.92) which gives ρϕ = 2 ϕ2 + V (ϕ) = ϕ 2 (6.93) Pϕ = 2 ϕ2 V (ϕ) =. (6.94) Thus, the scalar field oscillating around the minimum of the harmonic potential behave just like a matter that will give rise to the decelerating expansion with a t 2/3. Therefore, inflation ends gracefully! To be more precise, taking the Hubble friction back into account, the oscillation is a damped harmonic oscillation with ϕ(t) e ( 3 2 H±im)t. (6.95) Let us consider this case more in depth. The Friedmann equation becomes 6H 2 = ϕ 2 M 2 + m 2 ϕ 2. (6.96) As this is the equation for the ellipse in the phase space, we define a new variable θ so that ϕ = 6H M sin θ, mϕ = 6H M cos θ. (6.97) The equation of motion then gives Ḣ m ϕ = m 6H M sin θ = 6M cos θ H sin θ θ (6.98) Ḣ ϕ = 3H ϕ m 2 ϕ = 6H M [ 3H sin θ m cos θ] = 6M sin θ + H cos θ θ, (6.99) which reads Ḣ cos θ H sin θ θ = mh sin θ (6.) Ḣ sin θ + H cos θ θ = 3H 2 sin θ mh cos θ. (6.) We can solve the coupled linear equation above by multiplying the rotation matrix to both side to get Ḣ =mh sin θ cos θ 3H 2 sin 2 θ mh cos θ sin θ = 3H 2 sin 2 θ, (6.2) H θ = mh sin 2 θ 3H 2 sin θ cos θ mh cos θ 2 = mh 3 2 H2 sin 2θ. (6.3) From the first equation, we see that the Hubble expansion rate decays. Then, the second term in the second equation can be neglected as a oscillation with decaying amplitude H 2, and we approximate 7 Do you notice that I just proved the virial theorem? θ(t) mt + θ = mt (6.4)

16 6 CHAPTER 6. INFLATION where θ is a phase that we set to zero. Feeding this back into the Hubble equation, we find that or Ḣ H = d = 3 sin 2 ( mt) = 3 sin 2 (mt), (6.5) 2 d t H t H(t) = 3 sin 2 (mt )d t = 2 t cos( 2mt ) d t = 2 sin(2mt). (6.6) 3 3t 2mt This gives the scale factor as a(t) e Hd t exp t 2/3 cos(2mt) 6m 2 t 2 t d t 2 3t + 2 3t sin(2mt ) 24m 2 t mt. (6.7) Therefore, up to decaying oscillatory correations, the universe expands like a matter-dominated universe with zero pressure as we have expected from the simple analysis above. 2. V (ϕ) = ge λϕ model Our second example is the power law inflation with the exponential potential, V (ϕ) = ge λϕ. (6.8) This potential is usual in the extra-dimensional theories (e.g. Kaluza-Klein model and supergravity/superstring models) after dimensional reduction mechanisms have been applied. What concerns us here is that the exact solution for the full equations is known: ϕ(t) = 8πG gε 2 λ ln t 2 (6.9) 3 ε for this model with ε λ2 6πG. (6.) and the FRW scale factor evolves with power-law (hence the name power law inflation) a t /ε, then the Hubble expansion rate is H = /(εt). Note that from the functional form of the solution, we can see that the solution exists only for ε = λ 2 M 2 /2 < 3. Let us verify the exact solution above by setting λϕ(t) = A + ln t 2. (6.) Then, λ ϕ = 2 t, λ ϕ = 2 t 2 (6.2) and the equation of motion becomes 2 t 2 + 3H 2 t ge A λ t 2 =, (6.3)

17 6.3. INFLATION: HOW TO ACCELERATE THE UNIVERSE 7 and the Friedmann equation becomes Here, we define H 2 = 4πG 3t 2 4 λ + 2 2ge A ε 2 t. (6.4) 2 ε 4πG λ + 2 2ge A = 6πG + 2 3λ 2 λ2 ge A Feeding this into the equation of motion, we have another equation for ε and A: from which we find then, which gives Then, as desired to demonstrate ε ge A λ 2 = ε = 3 (6.5) + 2 λ2 ge A, (6.6) ε = 6πG. (6.7) λ 2 2 λ2 ge A = 3 ε 2(3 ε) 8πGε 2 g A = ln = ln ε εgλ 2 3 ε (6.8) ϕ(t) = 8πG gε 2 λ ln t 2. (6.9) 3 ε H = t f t f tε a(t) = a i exp H(t)d t = a i exp t i t Now, let us do the slow-roll analysis. The slow-roll parameters in this model are εt d t t /ε, (6.2) ε = M 2 2 λ2 = η, (6.2) therefore slow-roll condition is satisfied as long as λ 2/M regardless of the field value (thus, the inflation never ends). This is understandable as exponential potential is self-similar. Therefore, one must end the inflation by introducing other mechanisms (e.g. energy transfer to radiation sector). What concerns us here is how good the slow-roll equations are to describe the dynamics of the inflaton field in this case. The slow-roll equations are ϕ = V 3H = gλe λϕ 3H (6.22) 3H 2 =M 2 ge λϕ, (6.23) which reads then we find 9H 2 = 3M 2 ge λϕ = g2 λ 2 e 2λϕ ϕ 2 = gλ2 2 M e λϕ = 2gε ϕ e λϕ (6.24) ϕ(t) = λ ln gελ 2 6 t2 = 8πG gε 2 λ ln t 2. (6.25) 3

18 8 CHAPTER 6. INFLATION That is very close to the real dynamics as long as ε! The number of e-folding with solw-roll approximation is t f ϕ f ϕf H N = Hd t = t ϕ dϕ = i ϕ i M 2 λ dϕ = λ ϕ, (6.26) 2ε while the exact solution is Viva slow roll! ϕ i a f N = ln = t a i ε ln f = λ ϕ. (6.27) t i 2ε Non slow-roll inflation Slow-rolling scalar field guarantees to give an inflationary epoch, but the converse is not always true: inflation can still happen even though slow-roll condition is violated! An example is a fast-roll inflation where ϕ V (ϕ) = ln. (6.28) for ϕ ϕ c (and ϕ c < M ). Then, connect the potential with V (ϕ) = /2m 2 (ϕ) 2 continuously for ϕ < ϕ c. With this potential, the slow-roll condition is from ϕ c V = ϕ, V = ϕ ϕ 3, (6.29) that ε = M ϕ ln( ϕ /ϕ c ) (6.3) η =M 2 ϕ ϕ 3 ln( ϕ /ϕ c ) (6.3) or (let s consider when ϕ > for simplificity) ϕ ϕ ln ϕ c ϕ c M 2ϕc >. (6.32) Therefore, slow-roll condition is violated when ϕ ϕ c. On the other hand, the time-averaged kinetic energy is 2 ϕ2 ϕv,ϕ = 2 2. (6.33) Therefore, as long as ϕ 2 = < V (ϕ), (6.34) or ϕ/ϕ c > e, the universe accelerates without the slow-roll condition! 6.4 From Inflation to hot big-bang Universe During inflation, the energy density and Hubble expansion rate of the Universe stays almost the same while the comoving wave modes continuously strethed outside of the comoving horizon to loose causal contact.

19 6.4. FROM INFLATION TO HOT BIG-BANG UNIVERSE 9 As a result of the accelerated expansion with N > 6 e-folding, inflation generate a homogeneous and isotropic expanding universe, but without any other particles. So, we need a mechanism that convert the energy density stored in the inflaton field to the other fields, eventually to the standard model particles to give rise to a universe like ours. This procedure is called reheating of the Universe. Because we do not know the nature of the inflaton field, the way that inflaton couples to other fields and standard model particles is also unknown, and the theory of reheating is far from complete in this sense. What people does here is, again, introduce a effective scalar field that supposedly model the energy transfer from inflaton to other fields/particles. We then can study how the energy of inflaton was transfered to the other fields/particles. Depending on the mass and coupling of the new scalar field, usually χ, sometimes a resonance can happen to enhance the energy transfer rate. This is an active research area that has still many un-answered questions. During inflation, the comoving modes continuously stretches outside of horizon, and so do the quantum fluctuation of amplitde δϕ = H/2π and quantum metric fluctuation of δh = H/2π. While staying outside of comoving horizon, the quantum fluctuations underwent the quantum to classical transition to form a classical curvature (gravitational potential) perturbations. During the later epoch of radiation domination and matter domination, these comoving modes come inside of horizon then seed the cosmic structures. Although initially quite small 5, these seed fluctuations grow due to gravitational instability, and eventually turn themselves to the temperature fluctuations and polarizations of cosmic microwave background radiation, and large-scale distribution of galaxies. Conversely, from a in-depth study of the observed large-scale structure, we can study the properties of the initial condition, and possible inflation models that would make such a initial condition possible. This is the basic idea behind the way that we study inflationary period from the large-scale structure observations.