String Cosmology Gary Shiu1 September 28, 2009 1These 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 ([email protected]). 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 Mp. 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 su±ciently long (see below) we need generically that the two slow-roll parameters µ ¶ 1 V 0 2 V 00 ² = M 2 ; ´ = M 2 (1) 2 p V p V are su±ciently small (²; j´j ¿ 1). As will be elaborated below, corrections to the inflaton potential up to dimension 6 lead to order 1 corrections to the V 2 ´ parameter since they are of the form ±V = 2 Á . Therefore, a theory of Mp 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 ¯eld during inflation known as the Lyth Bound r ¢Á r = O(1); (2) Mp :01 p r Future experiment are sensitive to values of r for which :01 » 1. If they would detect tensor fluctuations of this magnitude, then from the expansion of the inflaton potential µ ¶ X1 Á n V (Á) = V + Á4 c (3) renormalizable n M n=1 p we see that we not only need to know the renormalizable part of the potential but also in¯nitely many other terms. To argue for the absence of this terms would require a ¯ne tuning of in¯nitely 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 ¯eld. Those can be generated from higher deriva- tive corrections to the action which we can write as Z µ ¶ p 1 S = d4x ¡g M 2R + P (X; Á) ; (4) 2 p 1 ¹º where P (X; Á) is a polynomial in X = ¡ 2 g @¹Á@ºÁ and Á. 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 de¯nition of inflation is d 1 Inflation () < 0: (5) dt aH 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 ¯xed with the expansion, the observable universe actually becomes smaller during inflation. An alternative de¯nition (for an expanding universe i.e. fora _ > 0) is simply an epoch during which the scale factor of the universe is accelerating Inflation () aÄ > 0: (6) If we use the Raychaudhuri equation aÄ 1 = ¡ 2 (½ + 3p); (7) a 6Mp which can be derived from Einstein's equations for the Robertson-Walker metric (60), we ¯nd Inflation () ½ + 3p < 0: (8) Because we always assume that the energy density ½ is positive, it is necessary for the pressure p to be negative. This condition can be ful¯lled by a scalar ¯eld with Lagrangian1 1 ¹º L = ¡ 2 g @¹Á@ºÁ ¡ V (Á): (9) From T ¹º = @¹Á@ºÁ + g¹ºL ¹ º ¹º 1 ρσ (10) = @ Á@ Á + g (¡ 2 g @½Á@σÁ ¡ V (Á)); 1Although inflation is an intrinsically quantum mechanical process, we are treating the scalar ¯eld classically, i.e. consider the expectation value hÁi. Quantum e®ects are 4 negligible if we demand that V ¿ Mp . 3 we can read o® 1 1 ½ = T 00 = Á_2 + (rÁ)2 + V (Á) and (11) 2 2a2 a2 X 1 1 p = T ii = Á_2 ¡ (rÁ)2 ¡ V (Á): (12) 3 2 6a2 i If the spatial inhomogeneities in the inflaton ¯eld Á are small and the poten- tial V (Á) is much bigger than the square of the time derivative of the inflaton ¯eld, i.e. 1 (rÁ)2 ' 0 and (13) a2 V (Á) À Á_2; (14) we have the condition ½ = ¡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 onp the potentialp V (Á). @ j det gjL @ j det gjL From the Euler-Lagrange Equation @ = we can derive ¹ @(@¹Á) @Á the equation of motion for Á p p ¹º 0 ¡@¹[ j det gjg @ºÁ] = ¡ j det gjV (Á) 3 _ 2 3 0 @t[a (t)Á] ¡ a(t)r Á = ¡a (t)V (Á) a_ 1 Ä _ 2 0 Á + 3 Á ¡ 2 r Á = ¡V (Á) a |a {z } '0 ÁÄ + 3HÁ_ + V 0(Á) = 0: (16) We now make the slow-roll approximation that jÁÄj is negligible in comparison with j3HÁ_j and jV 0(Á)j. This step is required in order that inflation can happen2 and leads to the slow-rolling form for the equation of motion 3HÁ_ '¡V 0(Á): (17) 2If jÁÄj is comparable to j3HÁ_j, Á_ would change considerably and condition (14) is not satis¯ed. If we assume that there is a characteristic temporal scale T for the inflaton ¯eld, we get from V (Á) À Á_2 » Á2=T 2 that dV=dÁ » V=Á À Á=T 2 » ÁÄ. 4 From the Friedman equation that follows from Einstein's equations for the Robertson-Walker metric and using (11) we ¯nd 2 ½ · V H = 2 ¡ 2 ' 2 : (18) 3Mp a 3Mp Thus, we can rewrite the condition (14) in form of two dimensionless param- eters µ ¶ 1 V 0 2 ² ´ M 2 ¿ 1 (19) 2 p V µ ¶ V 00 ´ ´ M 2 ¿ 1; (20) p V where we have di®erentiated the expression for ² 00 0 1 MpjV j ¿ jV j ¿ jV j (21) Mp to get ´. These two criteria make perfect intuitive sense: the potential must be flat in the sense of having small derivatives, if the ¯eld 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 ¯rst place, inhomogeneities rapidly become negligible. As we argued above, spatial derivatives of the inflaton ¯eld 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 Á rolls down the potential and V approaches zero3. Even if the potential does not steepen, sooner or later we will have ² ' 1 or j´j ' 1 and the inflationary phase will cease. Instead of rolling slowly 'downhill', the ¯eld will oscillate about the bottom of the potential. Due to the coupling of the inflaton ¯eld to matter ¯elds, which we have neglected so far, the rapid oscillatory phase will produce particles, leading to the reheating of the universe. Thus, even if the 3We 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 minimum of V is at V = 0, the universe is left containing roughly the same energy density as it started with, but now in the form of normal matter and radiation - which starts the usual FRW phase, albeit with desired special initial conditions, as we will see now. 3.2 Inflation solves three problems The very attracting feature of inflation is that a certain amount of inflation solves the most serious problems of the standard cosmology. This leads to a lower bound on the number of e-foldings the universe had to expand during the inflationary phase. 3.2.1 The flatness problem We can rewrite the Friedmann equation as an equation for the density pa- rameter · ­ ¡ 1 = ; (22) a2H2 where the cosmological constant is included in ­.