Planck 2018 and Brane Inflation Revisited

We revisit phenomenological as well as string-theoretical aspects of D-brane inflation cosmological models. Phenomenologically these models stand out on par with $\alpha$-attractors, as models with Planck-compatible values of $n_s$, moving down to the sweet spot in the data with decreasing value of $r$. On the formal side we present a new supersymmetric version of these models in the context of de Sitter supergravity with a nilpotent multiplet and volume modulus stabilization. The geometry of the nilpotent multiplet is evaluated in the framework of string theory.


Introduction
D-brane inflation models have two aspects, a phenomenological and a string-theoretical. Both became very interesting after the investigation of inflationary models in the Planck 2018 data release [1]. The Planck 2018 n s -r plane is shown in Fig. 1. The dark (light) blue regions describe the 1σ (2σ) confidence level for the CMB related data obtained by Planck 2018 and Bicep/Keck2014, additionally including the baryon oscillations (BAO) data. Meanwhile the comparison of predictions of inflationary models with the data in [1] was based on the CMB related data only, excluding BAO. The corresponding 1σ and 2σ regions are shown in red in Fig. 1.
The main part of the left hand side of the 1σ region in this plane is described by the simplest α-attractor model [2], with the predictions bounded by the two yellow lines corresponding to the α-attractor prediction with the number of e-foldings N e = 50 and N e = 60. The lower part of the α-attractor yellow band covers also the predictions of the Starobinsky model [3], the GL supergravity model [4], and the Higgs inflation model [5,6]. The values of n s and r for α-attractor models were shown in [2] and [7], see Fig. 3 here. Special cases with α = 1 correspond to Starobinsky, Higgs and conformal inflation [8], α = 1/9 corresponds to the GL model [4], α = 2, 1/2 correspond to fibre inflation [9,10]. Models with 3α = 7, 6, 5, 4, 3, 2, 1 originate from theories with maximal supersymmetry [11,12].
As one can see from [1], the two yellow lines corresponding to α-attractors cover the left hand side of the 1σ dark blue (and dark red) areas in Fig. 1. However, the Planck 2018 analysis of inflationary models presented in Table 5 in [1] reveals yet another class of models, which may match the observations equally well, in a complementary way. It is a class of D-brane inflation models with the potential proportional to 1 − m φ 4 + · · · , studied in Appendix C of the KKLMMT paper [13]. Predictions of such models cover the right hand side of the 1σ dark blue (and dark red) areas in Fig. 1.
As an example, let us simultaneously plot the predictions of α-attractors and of the simplest D-brane inflationary model with V ∼ 1 − m φ 4 in figures representing the Planck 2018 data for r on log 10 r scale, which is more suitable for illustration of the predictions of the models in the limit of small r, where both of these classes of models exhibit attractor The reason why the predictions of α-attractors and of the simplest D-brane inflation match each other in Fig. 2 so perfectly is very simple. According to the investigation in Appendix C of [13], the value of n s in the D-brane inflationary model with V ∼ 1 − m φ 4 in the small m (small r) limit is given by (1. 2) The cosmological evolution in these models was studied in detail in [14] and in Sect. 5.19 of [15], see Figs. 159-166 there.
Magically, n s for α-attractors shown by the right yellow line corresponding to N e = 60 in Fig. 2 exactly coincides with n s for the simplest D-brane inflation models for the left red line with N e = 50: This explains why α-attractors and D-brane inflation match Planck results so well in Fig. 2, and why both models provide a very good match to the Planck 2018 result n s = 0.9649 ± 0.0042 [1].
Of course, one should remember that exact predictions of these models depend on details of the models, mechanism of reheating, etc., and the position of the 1σ region for Planck 2018 depends on the data set (e.g. with or without BAO). Nevertheless the perfect match shown in Fig. 2 is quite striking.
Note, that the predictions of α-attractors also allow some variability, converging to (1.1) in the limit of large N e , and small α, which corresponds to small r, see Fig. 3.

α-attractors
Simple Fanned E-models Maximal supergravity Maximal superconformal theory Thus, phenomenologically D-brane inflation model after Planck 2018 has acquired a new significance, even independent of its string theory origin. But we will show here that a new progress can be made with regard to string theory implementation of the phenomenologically attractive versions of D-brane inflation model.
The string theory origin of D-brane inflation model is often attributed to KKLMMT model [13], where D3-brane-D3-brane interaction was studied in the context of the volume modulus stabilization. Earlier proposals for D-brane inflation relevant to our current discussion were made in [16][17][18]. The inflationary potentials corresponding to Dp-brane-Dpbrane interaction were proposed in the form In [15] a detailed cosmological analysis of both potentials was performed, with emphasis on the cases p = 3 and p = 5, i.e. for inverse quartic and quadratic potentials. The first potential in (1.4) was called BI (Brane Inflation), the second one was called KKLTI (KKLT Inflation).
The earlier proposals in [16][17][18] were made without addressing the volume stabilization issue. A consistent cosmological evolution of Dp-branes in string theory with the fundamental ten-dimensional geometry can be interpreted as an evolution in the four-dimensional Einstein equations under condition that the six-dimensional internal space has a constant volume, not a runaway behavior which destroys the consistency of the four-dimensional cosmology.
In [13] D3-brane-D3-brane interaction was studied simultaneously with the volume stabilization. The concrete example in string theory was based on strong warping, which is dual to an almost conformal four-dimensional field theory. Therefore the scalar field describing the motion of the brane was conformally coupled to gravity. This was consistent with the choice of the Kähler potential and the KKLT superpotential in the form However, it is well known that it is difficult to achieve inflation in the theories with a conformally coupled inflaton. This was one of the the main reasons to consider a generalized KKLT superpotential (1.5) where the dependence on the distance between branes Φ was included in A(Φ). A specific explicit model of inflation which followed from (1.5) was presented in [19]. The corresponding potential, an inflection point potential, has n s ≈ 0.93, which is ruled out by the data, see for example Fig. 1. More about the derivation of this model can be found in a review paper on string cosmology [20].
The reason for us to revisit the KKLMMT paper [13] after Planck 2018 is that, in addition and independently of an example of a particular form of the Kähler potential and superpotential (1.5), Ref. [13] provided a basis for other, more general approaches to string cosmology. It can be used at present in a form which is in fact supported by the latest data, using in particular equation (1.2) for the tilt of the spectrum as derived in Appendix C of [13]. We will therefore start with phenomenological properties of D-brane inflation models following [13][14][15], and then we will discuss a possibility to implement such models in supergravity and string theory.

KKLMMT scenario and inverse quartic and quadratic models
A short preview of the phenomenology of the simplest D-brane inflationary models is given in Figure 2. In this section we will study a broad class of phenomenological models of D-brane inflation in greater detail. As in Encyclopedia Inflationaris [15], we will call the corresponding potentials either BI (Brane Inflation) for a Coulomb-type interaction, or KKLTI (for KKLT Inflation) when the potential takes a form of the inverse harmonic function. The case of the inverse quadratic Coulomb-type interaction is whereas the potential in a form of the inverse harmonic function is (2. 2) At small m φ the 'exact' non-singular at φ = 0 potential V KKLT I takes the form of V BI . At very large values of m the potential tends to a quadratic one, in the area where φ m, In the limit m 1, predictions of this model coincide with the predictions of the simplest chaotic inflation model V ∼ φ 2 , which is ruled out. However, as we will see soon, in the case m 1 this model provides a good fit to Planck 2018 data, see Fig. 4.
The case of the inverse quartic Coulomb-type interaction is whereas the potential in a form of the inverse harmonic function is (2.5) Here again, at small m φ the 'exact' non-singular at φ = 0 potential V KKLT I takes the form of V BI . At very large values of m the potential tends to a quartic one, in the area where φ m, The α-attractor models have For Dp-brane-Dp-brane inflation models with V = A − B φ 7−p , the general formula for small r is This was also given in [17] and in [15] in slightly different notation. For example in [15] the formula is in terms of k = 7 − p and is given as n s ≈ 1 − 2(k+1) (k+2)Ne . Note that for the polynomial potentials φ 2n This means that the brane inflation spectral index n s at small r coincides with n s of inflation in a theory with a polynomial potential φ 2n with (2.10) The quartic brane inflation for D3-D3 model at small r has the same n s [13] as the one for φ 4/3 : The quadratic brane inflation model, with D5-D5 potential, at small r has the same n s [18] as inflation in a theory with a linear potential φ, On the left side in Fig. 4 we have an α-attractor band which starts at φ 2 and moves down in a straight line. Next is the inverse quartic brane inflation model which at small r is in the position corresponding to φ 4/3 , both for BI and KKLTI models, and continues straight. The inverse quadratic brane inflation at small at the right panel in Fig. 4 at small r is in a position corresponding to φ, both for BI and KKLTI models, and continues straight. These three models pretty much cover all admissible area in the n s − r plane below r < 10 −2 .
To understand the reason of similarity between predictions of α-attractors and D-brane inflation, we show in Fig. 5 two potentials. One of them is the potential of the α-attractor model with V = tanh 4 φ √ 6α with α = 1/6. The second one is of the inverse quartic D-brane inflation type, V = φ 4 φ 4 +m 4 for m = 1. This figure shows that in both cases we have plateau potentials at large fields, and for some choice of parameters (e.g. for α = 1/9 and m = 1) one can even make these potentials look even more similar, even though predictions of these models for n s continue to be slightly different. with α = 1/6. The dark blue line shows the KKLTI potential V = φ 4 φ 4 +m 4 for m = 1. In both cases we have plateau potentials at large φ. For some choices of parameters (e.g. for α = 1/9 and m = 1) these potentials almost coincide, which explains similarity of predictions of these models.
The value of r for α-attractors depends on α, (2.13) Meanwhile in brane inflation models the value of r depends on m. In quartic case, using [13] for m 1 one finds . (2.14) In particular, for m = 1, N e = 50 we find r ∼ 10 −3 . Using eq. (5.332) of [15], we find a more general expression for r for For an inverse quadratic potential with k = 2 we find (2.16) In particular, for m = 1, N e = 50 we find r ∼ 4 × 10 −3 .
It is instructive to compare these results with the ones presented in Fig. 4. One can see that with the decrease of r the results for n s in the BI and KKLTI models converge to each other at the values of r approximately corresponding to m ∼ 1, just as one could expect by comparing to each other the potentials of these models.
Thus here we have presented our analysis of quartic and quadratic BI and KKLTI models inside 2σ region of Planck 2018. As we see, the combination of these models covers the main part of the 2σ area in the Planck 2018 data in Fig. 4. This is explained by the similarity of potentials of these models illustrated by Fig. 5. Our results are compatible with the ones found in [15].

Inverse linear case with D6-D6 potential
In the previous discussion we concentrated on investigation of inverse quadratic and inverse quartic mode, as in the Planck 2018 analysis in [1]. Both cases are associated with type IIB string theory, where moduli stabilization was viewed as possible due to KKLT and LVS constructions. However, there was a progress recently with regard to an uplifting role of the D6 brane in [25] and de Sitter vacua in type IIA string theory in [26].
Therefore we would like to add here an example of D6-D6 potential using the general equations above to find out the phenomenology of these models. This is the case of Note that the variable φ is not a coordinate, but a distance in the moduli space, φ = |φ| × e iθ , which is why the potentials depend on |φ|.
At large m, the predictions of the 1 V KKLT I model converge to the predictions of the theory with a simple linear potential V ∼ φ. However, at small m it predicts the same value of n s as the theory with V ∼ φ 2/3 . At large N e and small r (for m < 1), the predictions are .

(2.19)
For N e = 50 this gives n s ≈ 0.973, which is within the 2σ Planck area for small r.

String theory analysis
To understand the D-brane dynamics in string theory in application to inflation it is useful to consider the original Polchinski's computation of the energy between two parallel Dirichlet p-branes at the distance Y from each other [27,28], see Fig. 6. In eq. (90) in [27] there is an answer for two D-branes 2 . Here we present it both for two D-branes, with the negative where q = e −πt and all functions f i (q) are defined in eq. (49) in [27] and t is world sheet modulus. This takes into account that the world-sheet fermions that are periodic around the cylinder correspond to R-R exchange, while the ones which are anti-periodic come from NS-NS exchange.
These three terms for two Dp-branes sum to zero by the 'abstruse identity,' since in this case the open string spectrum is supersymmetric. But in case of D-brane-anti-D-brane, when R-sector has a different sign, they sum up since Thus for two D-branes In terms of the closed string exchange, this reflects the fact that D-branes are BPS states, the net forces from NS-NS and R-R exchanges canceling. For D-brane-anti-D-brane We are interested in the limit t → 0 which is dominated by the lowest lying modes in the closed string spectrum. In such case (3.5) Here G 9−p (Y 2 ) is a massless propagator in the Euclidean 9−p dimension. The corresponding equation is (3.6) For example, for D3 case it is ∆ 2 6 G 6 (Y 2 ) = Cδ 6 (Y ) and Thus for two D-branes the net force between Ramond-Ramond repulsion and gravitational plus dilaton attraction cancels. For D-brane-anti-D-brane the force between Ramond-Ramond attraction and gravitational plus dilaton attraction doubles.
At large distances Y one can use an approximated expression for the cylinder amplitude at (3.5). At short distances b 1 − b 2 Y 4 blows up, but it means that our approximation where only low lying string states are taken into account is not valid and the full tower of string states contributes as shown in eq. (3.4).
From the perspective of string theory computation with the result in (3.4) it would be confusing to use the concept of the brane-anti-brane annihilation. D-branes and anti-Dbranes are not particles with the opposite charge which annihilate and disappear. When the distance between them is small one should not use the approximation t → 0 which allows to use the harmonic function in eq. (3.5) and the corresponding potential energy of the brane-anti-brane system. It is the difference between particles and strings which becomes essential at small distances which has to be taken into account in attempts to provide an interpretation of this physical system.
In application to cosmology we will be interested in both potentials in eq. (3.7). But we will be particularly interested in the region where the difference between these two is not significant.

On D3/D3 potential in the space-time picture
Now we follow the strategy in KKLMMT paper [13] where in Sec. 2 and in Appendix B a computation of the D3/D3 potential in warped geometries is proposed. This is also related to a discussion above were we followed [27,28] in their computation of the cylinder diagram directly in string theory. It is also useful to follow [14] where the same procedure is explained conveniently, not necessarily requiring a warped 5d geometry. The review of inflation in string theory in [20] is also helpful here. Note that here we assume that we already have a space-time picture. This means that the full string theory computation in (3.4) is already approximated by the case where t → 0 limit is taken. But we will see a different way how the harmonic function in the 9 − p = 6 Euclidean space shows up and defines the potential of the D3/D3 system. D3-brane is perturbing the background and we calculate the resulting energy of the D3-brane in this perturbed geometry. This gives the same answer for the potential energy of the brane-anti-brane pair. We start with 10d geometry In case that the moving D3-brane is at a position r 1 at a radial location in the six-dimensional space and the D3-brane is at a fixed position r the corresponding harmonic function in the six-dimensional space is given in KKLMMT as Here R is a characteristic length scale of the AdS 5 geometry, and N is the five-form charge. This expression is valid for strongly warped geometries. In [14] a general form of a harmonic function is used, namely for a position of the moving brane φ, which has a canonical kinetic term following from the D-brane actioñ where c 1 , c 2 are some constants. This choice is more in the spirit of the analysis in the previous section where an inverse harmonic function in a six-dimensional Euclidean space for the D3/D3 potential is an approximation to the stringy computation of the cylinder diagram in Fig. 6.
The Born-Infeld and Chern-Simons actions of the D3-brane in the background of a moving D3-brane are given by the following expression: Here Therefore, the action representing D3/D3 interaction for slow velocities leads to the action of the inflaton field where (3.14) From the D-brane action V = T 3 h(φ) . Thus At φ = 0 the only potential which is left is a KKLT potential corresponding to an uplift of dS vacuum, as defined in (5.12).

D-brane Inflation in String Theory/de Sitter Supergravity
We propose to use a shift symmetric Kähler potential of the form This kind of shift symmetric Kähler potentials were used in the past for a class of string theory cosmological models in the context of K3 × T 2 Z 2 compactification [30][31][32][33][34]. See also review of string cosmology models with unwarped branes in [20]. It is interesting that the relevant stringy D3/D7 models of inflation, called D-term inflation in its supergravity version, generically has n s = 0.98 and is now practically ruled out. However, its investigation gave us a useful tool: shift symmetric Kähler potential (4.1).

Inflaton shift symmetry and quantum corrections
It is known that at the classical level in string theory/supergravity one can find situations, like a choice of compactification manifold, when the shift symmetry as shown in eq. (4.1) is possible. It was actually derived in [32] in case of K3 × T 2 Z 2 compactification, using the special Kähler geometry and the corresponding holomorphic section Ω = X Λ , F Λ = ∂F ∂X Λ , which, in the special coordinate symplectic frame, is expressed in terms of a prepotential F depending on closed string moduli (s, t, u) as well as open string moduli, including a position of D3 brane.
However, quantum corrections may break this symmetry to some extent. The corresponding studies were performed in [35,36], mostly in the context of the stringy version of the D3/D7 brane inflation. The situation there may be summarized as follows with regard to detailed studies in D3/D7 brane inflation [34] in notation of that paper, where s = C 4 − iV ol(K3), t is the torus complex structure, and u is the axion-dilaton. With account taken of quantum corrections to gauge coupling of D7 brane, the non-perturbative superpotential dependence on the position of D3 brane y 3 was given in the form where quadratic and quartic corrections to where , and Σ(t 0 ) is a function of ϑ and E 2 , depending on complex structure modulus t, given in eq. (F.18) of [34]. Here c is some group theoretical factor, ϑ(ν, t) represent string theory theta functions. The function E 2 (t) was introduced in [37]. In [34] the effect of this dependence of the superpotential on the mobile D3 position y 3 was taken into account in the potential. As a result, in addition to a standard supergravity D-term potential, there are quantum corrections of the form of a mass term with a parameter m 2 and a quartic term with the parameter λ. Few examples were studied and plotted in Figs. 4 and 7 in [34]. These examples have shown some region of complex structure modulus t where the corrections to m 2 as well as to λ are small. The conclusion was made that with some fine-tuning of the value of t, defined by fluxes, it was possible to make quantum corrections to the potential of D3/D7 brane inflation small. But the problem of D3/D7 brane inflation with and without quantum corrections is that none of these models fit the data from Planck 2018.
For D-brane inflation models with K3 × T 2 Z 2 compactification, quantum corrections to superpotential have not been studied. If the calculations of quantum corrections associated with gaugino condensation in D3/D7 brane inflation would apply also to D-brane inflation models, one would expect that fine tuning is necessary to make these corrections small. It would be interesting to investigate this issue and describe the situations when such corrections are small, since without quantum corrections D-brane inflation models fit the data from Planck 2018 so well.
In particular, one may study the case where KKLT-type volume stabilization proceeds via a superpotential generated by Euclidean D3-branes [38], not by gaugino condensation. The nonperturbative effects in absence of a background flux require that the relevant fourcycle satisfies a topological condition derived in [38]. However, it was shown in [39,40] that these topological conditions are changed in the presence of flux. In this case the one-loop correction comes from an instanton fluctuation determinant, which has not been computed in the context of the cosmological models that we study here. It would be important to find out whether such computation can be performed and what it would entail.

A nilpotent multiplet in α-attractors
An additional tool we use here is a nilpotent multiplet S, representing an uplifting D3brane, [41][42][43][44]. de Sitter supergravity [45,46] is a local version of non-linearly realized global Volkov-Akulov supersymmetry [47]. Using the nilpotent multiplet we will build de Sitter supergravity for the brane inflation models compatible with the data. An important ingredient of cosmological models in dS supergravity is the Kähler metric K SS = ∂ S ∂SK of the nilpotent superfield S, which depends on other moduli It has been observed in the past [48,49] that the Kähler metric of the nilpotent superfield might carry the information about the inflationary potential. For example, it was shown in [49] that the simplest α-attractor model with the potential can be presented by a particular dS supergravity with the following nilpotent superfield geometry where F S = D S W . Here Z is a disk variable of the hyperbolic geometry, m 2 ZZ = m 2 tanh 2 φ √ 6α = V inf (Z,Z) and the cosmological constant at the exit from inflation is given by the difference between two constants, Λ = |F S | 2 − 3W 2 0 > 0.
In case of Dp-brane inflationary models we will show below that the dependence of the nilpotent field geometry K SS on the inflaton superfield (Φ,Φ) has an interesting explanation. It comes in the context of the KKLMMT construction combined with the recent investigation in [25] of the dictionary between string theory models with local sources in ten dimensions and the four-dimensional de Sitter supergravity.
5 On stringy origin of the nilpotent geometry K SS (Φ,Φ) Recently the dictionary between string theory models and K and W for dS supergravities with closed string moduli was established in [25]. In case of open string moduli the analogous analysis was not performed yet. Here we will consider a very particular situation, known from cosmology, where it is possible to identify the relevant geometry of the nilpotent multiplet from the first principles of string theory with D-branes.
Consider modifications of K and W due to the presence of the nilpotent multiplet, Here the superpotential has a simple dependence on S as in (5.2). When z i ,z i are closed string moduli we have shown in [25] why K SS (z i ,z i ) is computable: for each set of ingredients in 10d of the so-called 'full-fledged string theory models' one can compute K SS (z i ,z i ) in 4d as a function of the overall volume, the dilaton and the volume moduli of the supersymmetric cycles on which the Dp-branes are wrapped.
Since the nilpotent multiplet does not have a scalar component, the new potential has an additional term, but it still depends on the same closed string moduli. The new F-term potential acquires an additional nowhere vanishing positive term, associated with Volkov-Akulov non-linearly realized supersymmetry is the standard supergravity potential without the nilpotent multiplet (without the D3 brane in string theory). It is important to stress here that there is a dictionary between string theory models in ten dimensions described by supergravity with fluxes and local sources, Dp-branes and Op-planes. Upon compactification on calibrated manifolds these string theory models lead to specific choices of K and W in four-dimensional supergravity, see [25] and references therein.
The reason why the nilpotent field metric, K SS (z i ,z i ) is computable in string theory is that on one hand, the corrected potential due to presence of Dp-brane has a simple dependence on moduli, K SS (z i ,z i ) under condition that D S W | S=0 = µ 2 , as shown in eqs. (5.2), (5.4). Thus, the extra potential has a simple dependence on geometry of the nilpotent superfield On the other hand, the addition to potential due to Dp-brane action can be inferred from the knowledge of the bosonic Dp-brane action. By comparing these two we have identified in [25] the values of K SS (z i ,z i ) as functions of closed string moduli, Here the corresponding action for the Dp brane wrapped on a p − 3 supersymmetric cycle is given by the following expression, and it depends on various closed string moduli, including the volume of the supersymmetric cycles, More details about this action can be found in [25]. This leaves us with the dictionary between the nilpotent field geometry in presence of a pseudo-calibrated Dp-brane and string theory models with closed string moduli, Here we study a particular case of the computation of K SS (z i ,z i ; Φ,Φ), where Φ is an open string moduli. The new relation between the energy and geometry is an analog of eq. (5.6) Thus if we know the dependence of the potential on moduli which is added to the standard supergravity action via a nilpotent field, we can find the geometry using eq. (5.10). The corresponding geometry of the nilpotent superfield is determined by the total potential (5.11)

KKLT uplift
A manifestly supersymmetric version of the KKLT uplifting was proposed in the form in which the D3-brane is represented by a nilpotent multiplet S with S 2 = 0, corresponding to Volkov-Akulov non-linearly realized supersymmetry [41][42][43][44]. In this case the new K and W are given by (in unwarped case) and This is in agreement with using only closed string moduli and V D3 action. At present there is a consensus that eqs. (5.12) and (5.13) represent a manifestly supersymmetric version of the KKLT uplift. It involves a nilpotent multiplet representing an D3 brane in the framework of de Sitter supergravity with a non-linearly realized supersymmetry.

Inflationary uplift
Here we consider the situation where at the end of inflation the uplifting energy is due to an D3 brane which is at some fixed point in the manifold [13,50], for example on a top of an O3-plane, as discussed more recently in [44].
Our new proposal here is to look for a combination of potentials due to KKLT uplift to dS vacua, and an additional uplift by the inflationary energy depending on open string modulus. In case of D3-brane inflation, the new term depends on the energy of D3/D3 interaction. Our proposal means that the corresponding geometry of the nilpotent superfield will be defined by the total potential (5.14) In addition to V D3 we have now added the energy of the V D3/D3 system depending on open string modulus. In such case, it follows from (5.11) that .
This is our definition of the inflationary uplift in the case of D3/D3 inflation. We will demonstrate below that it is very useful for inflationary model building.
6 Brane Inflation with volume stabilization in dS supergravity

D-brane inflaton potentials
Our proposal for a supersymmetric version of D-brane inflationary models is based on shift-symmetric Kähler potential (4.1) and on an inflationary potential of the type (3.14) It could also be a potential of the form where the terms with · · · have to be added to remove the singularity at Φ −Φ. We will see below that the cosmological evolution with strong volume modulus stabilization as proposed in [51][52][53] works well for the inflationary models we study below. Namely we can use either KKLT type volume stabilization assuming that m 3/2 > H to avoid volume destabilization during inflation [54], or using the KL mechanism with two exponents [54][55][56]. In both cases the process of inflation does not affect the volume modulus stabilization and vice versa, inflation is not affected by the volume modulus stabilization. One should note that the geometric approach used in our paper may impose certain constraints on the gravitino mass, which should be taken into account in the model building [57].

Unifying inflation and strong volume stabilization
We consider a general theory of volume stabilization in combination of the inflationary potential and discuss the back-reaction of the inflaton potential on the moduli. We use the following set of Kähler and superpotential where T = ρ + iσ is a volume modulus multiplet, Φ = χ + iφ is an inflaton multiplet, S is an D3 nilpotent multiplet, respectively. β = 0, 1 depends on where the D3 is, and in the warped case, β = 1, in the unwarped case, β = 0. The Kähler coupling f (Φ,Φ) gives rise to inflaton potential, see (5.15). The scalar potential is given by In all models to be considered, inflation occurs along a stable inflationary trajectory σ = χ = 0. We will also consider f (Φ,Φ) = F (−i(Φ −Φ)/2). In that case, the general expression for the potential of the inflaton field φ and of the volume modulus ρ is given by In what follows, we will consider two models where the potential V (ρ) ensures strong stabilization of the volume modulus ρ near its post-inflationary value ρ 0 , such that during inflation one has ρ ≈ ρ 0 . Also, one can ensure that the post-inflationary vacuum energy V (ρ 0 ) = Λ ∼ 10 −120 is many orders of magnitude smaller than the inflaton potential. In that case, the potential during inflation can be represented by a very simple expression This expression shows that one can easily combine strong moduli stabilization with construction of inflationary models with arbitrary potentials (6.7). Similar, but slightly more complicated methods were used in the past in the phenomenological inflationary models without nilpotent superfield [51][52][53].

D-brane inflation and strongly stabilized KKLT model
We consider the KKLT moduli stabilization with inflationary potential and discuss the back-reaction of the inflaton potential on the moduli. We use the following set of Kähler and superpotential 3 : , (6.8) where T = ρ + iσ being a volume modulus multiplet, Φ = χ + iφ being an inflaton multiplet, S being an D3 nilpotent multiplet, respectively. β = 0, 1 depends on where the D3 is, and in the warped case, β = 1, in the unwarped case, β = 0. The Kähler coupling f (Φ,Φ) gives rise to inflaton potential, see (6.7).

Figure 7:
The potential of the volume modulus δρ and the inflaton φ in KKLT case, (6.8), (6.9). V inf is chosen to be the form V inf = V0(1 + µ 4 φ 4 ) −1 . Inflation is realized along the straight valley of the φ direction, while the volume δρ is stabilized. In realistic models one should take V0 ∼ 10 −10 ; here we took V0 = 1 just for illustration. The shape of the potential along the inflationary valley in the φ direction is shown by the dark blue line in Fig. 5.
Let us first consider the simplest case f = 0 and χ = 0. In this case, the scalar potential is simply given by where we have minimized with respect to σ and ρ 0 is the value at which D T W = 0 is satisfied. We denote the minimum of this potential as ρ = ρ 0 , which satisfies We assume that f (χ) = 0 at the minimum of χ. From (almost) vanishing cosmological constant condition, we find . (6.12) For simplicity, let us choose β = 1, and one finds Tuning on the inflaton dependence f , the potential minimum of ρ is no longer ρ = ρ 0 . Let us call the deviation from ρ 0 as δρ. We expand the potential with respect to δρ and minimize it, which gives the following effective potential, (6.14) Here we have set χ = 0, which is justified below, and defined The second term is regarded as the deformation of potential from the back-reaction of heavy modulus ρ. We find that the back-reaction term has the factor V inf /m 2 3/2 ∼ H 2 inf /m 2 3/2 . We remind here that KKLT uplift is consistent with inflation only if the height of the barrier is higher than the scale of inflation, m 3/2 > H inf [54]. Thus, in our case H 2 inf /m 2 3/2 must be small (i.e. SUSY breaking must be higher than inflation scale), which also protects the system from back-reaction.
Finally, let us comment on the stability of sinflaton χ. Near the minimum δρ ∼ 0, the mass of sinflaton is given by Thus, the sinflaton mass is always greater than inflation scale, and the sinflaton can be set at its origin.

KL model
Next, we consider inflation coupled to KL moduli stabilization [54]. The system is given by , (6.16) We focus on the vacuum where In particular, we assume the relation This relation leads to W = 0 when δw = 0 = W T is satisfied. One can check that σ = 0 is the minimum.
As previous case, we expand the potential in δρ = ρ − ρ 0 up to the quadratic order, where ρ 0 is the value of ρ for the case with χ = f = 0. By minimizing the potential with respect to δρ, we find the following effective potential, is the mass of ρ at δw = µ = 0. The ellipses denote the higher order terms suppressed by the factor m 3/2 /M . For M m 3/2 , the leading part of the effective potential is V inf = 3m 2 3/2 f . We rewrite the effective potential as The moduli stabilization during inflation requires H inf M , and this condition means that the second term in the effective potential is much smaller than the leading term. Thus, we can safely ignore the correction coming from the back-reaction of the heavy modulus ρ. Note that in KL model the mass M is not related to the mass of gravitino. This allows much greater flexibility with respect to the strength of SUSY breaking.
Finally, we note that the mass of χ near δρ = 0 is given by m 2 χ = (8 + 12f )m 2 3/2 = 8m 2 3/2 + 4V inf . (6.20) Therefore, χ is stabilized at its origin during inflation with a sufficiently large mass. The shape of the potential in this model is very similar to the one shown in Fig. 7.

Discussion
During the last 15 years the accuracy of determination of the spectral parameter n s increased dramatically. In 2003, after the first WMAP data release, the combination of all available data suggested that n s = 0.93±0.03 [58]. 9 years later, in the 9 year WMAP data release, the result was n s = 0.972 ± 0.013 [59]. In the Planck 2013 data release the corresponding number was n s = 0.9603 ± 0.0073 [60]. Finally, the Planck 2018 result is n s = 0.965 ± 0.004 [61].
In view of phenomenological importance of the D-brane inflation models, we revisited their stringy origin. Based on the original Polchinski's computation of D-brane-D-brane (vanishing) potential as well as the studies of this in [13], we find that D-brane-anti-D-brane potential leads to a specific dependence on the open string modulus of the geometry of the nilpotent multiplet K SS (z i ,z i ; Φ,Φ) = µ 4 e K(z i ,z i ;Φ,Φ) . This is our generalization, to the case of the open string moduli, of the dictionary between string theory models with local sources in ten dimensions and the four-dimensional de Sitter supergravity studied for closed string theory moduli in [25].
We combined this geometric information with the shift symmetric Kähler potential K = −3 log(T +T − (Φ +Φ) 2 ) of the kind known for K3 × T 2 Z 2 compactification. The resulting de Sitter type supergravity models, with either KKLT or KL volume stabilization, were studied in the regime of strong volume stabilization. The combined potential of the inflaton and the volume modulus is shown in Fig. 7. One can see a nearly flat inflationary χ direction, with the D-brane inflaton potential remaining practically unchanged by the presence of the strongly stabilized volume modulus.
To complete this construction, it would be important to study quantum corrections of the type discussed in fiber inflation in [9,10] and in D3/D7 model in [34], reviewed here in Sec. 4.1. It will also be important to study quantum corrections in case that KKLT-type volume stabilization proceeds via a superpotential generated by Euclidean D3-branes [38][39][40], not by gaugino condensation. This is a very challenging task, but the phenomenological success of the simple D-brane inflation models considered in this paper suggests that these theories deserve a detailed investigation.
where M, m and χ 0 are constants. First, we integrate out the heavy moduli χ, and find the following effective potential Here we have neglected higher derivative corrections since they are smaller compared to the corrections to potential [23].

A.2 Supergravity model
Next we consider a supergravity model with where X is a nilpotent superfield and Y is a constrained superfield satisfying XY = 0. Both X and Y do not have dynamical scalars, and F X = 0 for consistency. 4 Φ = χ + iφ and χ is the inflaton. The scalar potential becomes where Λ 4 = µ 4 − 3W 2 0 is a cosmological constant which is fine-tuned to be O(10 −120 ). We expand the potential with respect to χ, and find Thus, the sinflaton χ is stabilized at χ = 0 during inflation, and we find a single-field inflation with the KKLTI potential.