String Cosmology Gary Shiu 1 September 28, 2009 1 These lecture notes are based on four lectures given at the New Perspectives in String Theory School , 08 June - 19 June 2009, Florence, Italy. The notes were typed by Timm Wrase (wrase@mppmu.mpg.de).
1 Introduction These notes will provide a brief overview of string inflation. After a motiva- tion for inflation in the context of string theory, we will start out by reviewing inflation. Then we will discuss warped D-brane inflation and conclude with a few comments about DBI-inflation. Throughout the notes we will give only very few references and refer the reader to previous review articles about string cosmology [1, 2, 3, 4, 5, 6, 7] and the references therein. Appendix A summarizes conventions and notation. 2 Motivation String theory is the best understood candidate for a UV completion of grav- ity. It therefore should become relevant at energies of the order of the Planck mass M p . While it is virtually impossible to directly test this energy scale, one might hope that cosmological observation might provide some insight. As we will see in more detail below, inflationary models are UV sensitive. Although it is unlikely that we can make direct tests of string theory through cosmolog- ical observations, measurements of the cosmic microwave background (CMB) can exclude certain string inflation models and can rule out part of the land- scape. To further motivate the relevance of string theory for inflation let us look at three examples: Example 1: For inflation to last sufficiently long (see below) we need generically that the two slow-roll parameters � V ′ � 2 V ′′ ǫ = 1 2 M 2 η = M 2 , (1) p p V V are sufficiently small ( ǫ, | η | ≪ 1). As will be elaborated below, corrections to the inflaton potential up to dimension 6 lead to order 1 corrections to the V p φ 2 . Therefore, a theory of η parameter since they are of the form δV = M 2 inflation needs to know about at least dimension 6 terms in the potential. Calculating such Planck suppressed corrections requires a theory of quantum gravity whose prime candidate is string theory. 1
Example 2: As we will see below an epoch of inflation will generate scalar and tensor perturbations whose ratio we denote by r . Lyth [8] derived a lower bound on the variation in the inflaton field during inflation known as the Lyth Bound � r ∆ φ . 01 O (1) , = (2) M p Future experiment are sensitive to values of r for which � r . 01 ∼ 1. If they would detect tensor fluctuations of this magnitude, then from the expansion of the inflaton potential � φ ∞ � n � V ( φ ) = V renormalizable + φ 4 c n (3) M p n =1 we see that we not only need to know the renormalizable part of the potential but also infinitely many other terms. To argue for the absence of this terms would require a fine tuning of infinitely many terms. String theory, however, could allow one to calculate these terms or might provide a symmetry that leads to their absence. Example 3: Future experiments like the recently started Planck satellite are also search- ing for non-Gaussianity in the density fluctuations generated by quantum fluctuations in the inflaton field. Those can be generated from higher deriva- tive corrections to the action which we can write as � 1 � � d 4 x √− g 2 M 2 S = p R + P ( X, φ ) , (4) where P ( X, φ ) is a polynomial in X = − 1 2 g µν ∂ µ φ∂ ν φ and φ . Again we need string theory to determine the form of the polynomial P . One particular example of higher derivative corrections that can be nicely summed up to a closed form is the DBI action that can be used for DBI inflation as sketched at the end of these notes. 2
3 Review of inflation 3.1 Slow-roll inflation The precise definition of inflation is ⇒ d 1 Inflation ⇐ aH < 0 . (5) dt 1 Because aH is the comoving Hubble length, the condition for inflation is that the comoving Hubble length is decreasing with time. This means that in coordinates fixed with the expansion, the observable universe actually becomes smaller during inflation. An alternative definition (for an expanding universe i.e. for ˙ a > 0) is simply an epoch during which the scale factor of the universe is accelerating Inflation ⇐ ⇒ ¨ a > 0 . (6) If we use the Raychaudhuri equation ¨ a 1 a = − ( ρ + 3 p ) , (7) 6 M 2 p which can be derived from Einstein’s equations for the Robertson-Walker metric (60), we find Inflation ⇐ ⇒ ρ + 3 p < 0 . (8) Because we always assume that the energy density ρ is positive, it is necessary for the pressure p to be negative. This condition can be fulfilled by a scalar field with Lagrangian 1 L = − 1 2 g µν ∂ µ φ∂ ν φ − V ( φ ) . (9) From T µν = ∂ µ φ∂ ν φ + g µν L (10) = ∂ µ φ∂ ν φ + g µν ( − 1 2 g ρσ ∂ ρ φ∂ σ φ − V ( φ )) , 1 Although inflation is an intrinsically quantum mechanical process, we are treating the scalar field classically, i.e. consider the expectation value � φ � . Quantum effects are negligible if we demand that V ≪ M 4 p . 3
we can read off = 1 φ 2 + 1 2 a 2 ( ∇ φ ) 2 + V ( φ ) ˙ T 00 ρ = and (11) 2 p = a 2 T ii = 1 φ 2 − 1 � 6 a 2 ( ∇ φ ) 2 − V ( φ ) . ˙ (12) 3 2 i If the spatial inhomogeneities in the inflaton field φ are small and the poten- tial V ( φ ) is much bigger than the square of the time derivative of the inflaton field, i.e. 1 a 2 ( ∇ φ ) 2 ≃ 0 and (13) V ( φ ) ≫ ˙ φ 2 , (14) we have the condition ρ = − p (15) and the universe is in accordance with (8) in an inflationary phase. The standard technique for analyzing inflation is the slow-roll approximation, from which we obtain some restrictions on the potential V ( φ ). ∂ √ ∂ √ | det g |L | det g |L From the Euler-Lagrange Equation ∂ µ = we can derive ∂ ( ∂ µ φ ) ∂φ the equation of motion for φ � � | det g | g µν ∂ ν φ ] = − | det g | V ′ ( φ ) − ∂ µ [ ∂ t [ a 3 ( t ) ˙ φ ] − a ( t ) ∇ 2 φ = − a 3 ( t ) V ′ ( φ ) φ + 3 ˙ a φ − 1 ¨ ˙ a 2 ∇ 2 φ = − V ′ ( φ ) a � �� � ≃ 0 φ + 3 H ˙ ¨ φ + V ′ ( φ ) = 0 . (16) We now make the slow-roll approximation that | ¨ φ | is negligible in comparison with | 3 H ˙ φ | and | V ′ ( φ ) | . This step is required in order that inflation can happen 2 and leads to the slow-rolling form for the equation of motion 3 H ˙ φ ≃ − V ′ ( φ ) . (17) 2 If | ¨ φ | is comparable to | 3 H ˙ φ | , ˙ φ would change considerably and condition (14) is not satisfied. If we assume that there is a characteristic temporal scale T for the inflaton field, φ 2 ∼ φ 2 /T 2 that dV/dφ ∼ V/φ ≫ φ/T 2 ∼ ¨ we get from V ( φ ) ≫ ˙ φ . 4
From the Friedman equation that follows from Einstein’s equations for the Robertson-Walker metric and using (11) we find ρ − κ V H 2 = a 2 ≃ . (18) 3 M 2 3 M 2 p p Thus, we can rewrite the condition (14) in form of two dimensionless param- eters � V ′ � 2 ǫ ≡ 1 2 M 2 ≪ 1 (19) p V � V ′′ � η ≡ M 2 ≪ 1 , (20) p V where we have differentiated the expression for ǫ 1 M p | V ′′ | ≪ | V ′ | ≪ | V | (21) M p to get η . These two criteria make perfect intuitive sense: the potential must be flat in the sense of having small derivatives, if the field is to roll slowly enough for inflation to be possible. Similar arguments could be made for the spacial part. However, they are less critical. Since a ( t ) increases very rapidly, spacial perturbations are damped away: assuming V is large enough for inflation to start in the first place, inhomogeneities rapidly become negligible. As we argued above, spatial derivatives of the inflaton field can be neglected. This is not always true for time derivatives. Although they may be negligible initially, the relative importance of time derivatives increases as φ rolls down the potential and V approaches zero 3 . Even if the potential does not steepen, sooner or later we will have ǫ ≃ 1 or | η | ≃ 1 and the inflationary phase will cease. Instead of rolling slowly ’downhill’, the field will oscillate about the bottom of the potential. Due to the coupling of the inflaton field to matter fields, which we have neglected so far, the rapid oscillatory phase will produce particles, leading to the reheating of the universe. Thus, even if the 3 We are leaving aside the subtle question why the potential minimum is so close to zero. Note however that if the minimum would not be close to zero, the universe would continue to inflate without end and not be able to bear life. 5
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