Dilaton-Axion Inflation with PBHs and GWs

In the simple case the one-inflaton T- and E-models Kallosh:2013yoa with exponential approach to the plateau have $n_{s}$ and $r$ independent of the properties of the large class of potentials. However, $r$ depends on the curvature of the Kähler geometry $\mathcal{R}_{K}=-{2\over 3\alpha}$. At large values of $\varphi$ the potential is $V_{0}(1-e^{-\varphi/\mu}+\dots)$. The tensor to scalar ratio $r$ depends on the parameter $\mu=\sqrt{3\alpha/2}$ describing the approach of the canonical field to the plateau.

In polynomial approach to the plateau one-inflaton $\alpha$-attractor models Kallosh:2022feu the potential at large values of $\varphi$ is $V_{0}\big{(}1-\big{(}{\mu\over\varphi}\big{)}^{k}+\dots\big{)}$. In these models $n_{s}$ depends on $k$, whereas $r$ depends both on $k$ and $\mu$, Kallosh:2018zsi ; Martin:2013tda . Both $n_{s}$ and $r$ do not depend on the curvature of the Kähler geometry $\mathcal{R}_{K}=-{2\over 3\alpha}$.

Here we will describe the two-stage dilaton-axion models of inflation developed in Linde:2018hmx , as well as their their generalizations. The original E-model dilaton potential with the exponential approach to plateau studied in Linde:2018hmx can be replaced by a polynomial attractor Kallosh:2022feu , i.e. we can replace an E-model potential by the one which has a power law approach to the plateau, a quadratic KKTTI attractor Kallosh:2018zsi ; Martin:2013tda .

Both types of models have the same dilaton-axion kinetic term

$${\cal L}_{\rm kin}={1\over 2}\Big{[}(\partial\varphi)^{2}+e^{2\sqrt{2\over 3\alpha}\varphi}(\partial a)^{2}\Big{]}\ .$$

(1)

The potentials studied in Linde:2018hmx are

$$V(\varphi,a)=V_{0}\Big{(}1-e^{-\sqrt{2\over 3\alpha}(\varphi-\varphi_{0})}\Big{)}^{2}+{1\over 2}m_{a}^{2}a^{2}\ .$$

(2)

The dilaton potential in (2) is the E-model exponential $\alpha$-attractor with the position of the minimum at $\varphi=\varphi_{0}$. The axion has a mass term potential³³3The axion potential used in Linde:2018hmx at $\phi=\phi_{0}$ is more general, it is proportional to $\cos^{2}{\theta\over 2}$ where $\theta-\pi=\sqrt{2\over 3\alpha}a$. For small deviation of the axion from the minimum the model becomes the one shown in (2)..

We now replace the potential $V_{0}\Big{(}1-e^{-\sqrt{2\over 3\alpha}(\varphi-\varphi_{0})}\Big{)}^{2}$ with an exponential approach to the plateau used in Linde:2018hmx by a polynomial attractor Kallosh:2022feu , which has a power law approach to the plateau, and find the dilaton-axion potential of the form

$$V(\varphi,a)=V_{0}{\varphi^{2}\over\varphi^{2}+\mu^{2}}+{1\over 2}m_{a}^{2}a^{2}\ .$$

(3)

In a class of models considered in Braglia:2020eai ; Pi:2021dft , where the PBHs can be produced, a kinetic term was chosen in the form

$${\cal L}_{\rm kin}={1\over 2}\big{[}\partial_{\mu}\varphi\partial^{\mu}\varphi+f(\varphi)\partial_{\mu}a\partial^{\mu}a\big{]}\ .$$

(4)

One of the models studied in Braglia:2020eai has

$$f(\phi)=e^{2b_{1}\varphi}\ ,$$

(5)

but it was not clear whether this choice may have a fundamental geometric interpretation, and whether it is any better than another choice $f(\phi)=e^{2b_{2}\varphi^{2}}$ also studied in Braglia:2020eai .

Now we can relate it to the hyperbolic geometry of $\alpha$-attractors and incorporate this model in supergravity with the corresponding Kähler geometry iff

$$f(\phi)=e^{2b_{1}\varphi}=e^{2\sqrt{2\over 3\alpha}\varphi}\,,\qquad{\rm i.e.}\qquad b_{1}=\sqrt{2\over 3\alpha}\ .$$

(6)

Thus, if we choose the kinetic term in Braglia:2020eai as in (5), we can relate their parameter $b_{1}$ to the Kähler curvature,

$$b_{1}^{2}=-\mathcal{R}_{K}={2\over 3\alpha}\,,\qquad e^{2b_{1}\varphi}=e^{2\sqrt{|\mathcal{R}_{K}|}\varphi}\ .$$

(7)

The phenomenology of this cosmological model introduced in Braglia:2020eai , which we now embedded into hyperbolic geometry, was studied extensively with regard to PBHs and GWs. We will describe the microscopic origin of both models in (2), and in (3) in supergravity, and extract an interesting information about phenomenology from the fact that the model in (3) is a dilaton-axion inflationary model in hyperbolic geometry which has attractor properties of polynomial $\alpha$-attractors.

Importantly, one of the parameters in this model defining the properties of PBHs and GWs is actually a curvature of the hyperbolic geometry, as shown in (6). Other features of this model following from the fact that it is embedded into hyperbolic geometry with inflationary plateau potential approaching polynomially Kallosh:2022feu will be identified. We will show that $n_{s}$ and $r$ are defined by their plateau potential attractor values Kallosh:2018zsi ; Kallosh:2022feu .

The point of transition to the second stage of inflation is characterized by the effective mass square of the isocurvature perturbations $m^{2}_{\rm eff}$. It can abruptly and temporarily become large and negative; we will find out that $m^{2}_{\rm eff}\sim\mathcal{R}_{K}$.

A simple way to introduce this class of metrics is to start with equation (5) in Kallosh:2007ig , where the Kähler potential defining the Kähler geometry metric is given by

$$K=-3\alpha\ln(T+\bar{T})\,,$$

(8)

in notation ($c=3\alpha$) adapted to the definition of $\alpha$-attractor E-models. The Kähler metric is $g_{T\bar{T}}=\partial_{T}\partial_{\bar{T}}K$ and the geometry defines the kinetic term for a complex scalar

$${\cal L}_{\rm kin}=3\alpha{\partial_{\mu}T\partial_{\nu}\bar{T}g^{\mu\nu}\over(T+\bar{T})^{2}}\ .$$

(9)

It has an $SL(2,\mathbb{R})$ symmetry. It was explained around equations (5) and (6) in Kallosh:2007ig that the case $3\alpha=3$ corresponds to the dilaton-axion in string theory Kachru:2003aw where the total volume is defined by the dilaton, and no-scale supergravity Cremmer:1983bf ; Ellis:1983sf . The case $3\alpha=1$ is a single dilaton-axion case. Now we know more examples of discrete $3\alpha=7,6,5,4,3,2,1$ Ferrara:2016fwe as well as continuous $3\alpha$ in the context of $\mathcal{N}=1$ supergravity Ferrara:2013rsa ; Kallosh:2013yoa . Following Kallosh:2007ig we present our complex field $T$, a half-plane coordinate of the hyperbolic geometry, as a dilaton and an axion:

$$T(x)=e^{-\sqrt{2\over 3\alpha}\varphi(x)}+i\sqrt{2\over 3\alpha}\,a(x)\,,\qquad T+\bar{T}>0\ .$$

(10)

The real part of the $T$-field is an exponent of the dilaton $\varphi$, therefore clearly positive, which explains the ‘half-plane’ coordinate name. The dilaton-axion kinetic term (9) is

$${\cal L}_{\rm kin}={1\over 2}\Big{[}\partial_{\mu}\varphi\partial_{\nu}\varphi g^{\mu\nu}+e^{2\sqrt{2\over 3\alpha}\varphi}\partial_{\mu}a\,\partial_{\nu}ag^{\mu\nu}\Big{]}.$$

(11)

Here $\varphi$ is a dilaton and $a$ is the axion whose kinetic term couples exponentially to the dilaton.

$$\varphi=-\sqrt{3\alpha\over 2}\ln{1\over 2}(\bar{T}+T)\,,\qquad a={1\over 2i}\sqrt{3\alpha\over 2}(\bar{T}-T)\ .$$

(12)

The total non-gravitational Lagrangian, including the potential, in geometric variables is

$${\cal L}=3\alpha{\partial_{\mu}T\partial_{\nu}\bar{T}g^{\mu\nu}\over(T+\bar{T})^{2}}-V(T,\bar{T})\ .$$

(13)

The supergravity version of the hyperbolic geometry models with both dilaton and axion evolving during inflation was developed in Kallosh:2015zsa ; Achucarro:2017ing ; Yamada:2018nsk and Linde:2018hmx . In addition to the dilaton-axion multiplet $T$ we need a nilpotent one, $X^{2}=0$. The Kähler potential and the superpotential for models of our interest here were proposed and studied in Yamada:2018nsk and in Linde:2018hmx .

We generalize supergravity models in Yamada:2018nsk and in Linde:2018hmx by introducing two different parameters. One is describing the breaking of supersymmetry in the $X$ directions, we call it $F_{X}$, the other describing the breaking of supersymmetry in the $T$ directions will be $W_{0}$ as in Yamada:2018nsk and in Linde:2018hmx . Our model is now defined as follows, for $\alpha<1$

$$K=-3\alpha\ln(T+\bar{T})+G_{X\bar{X}}X\bar{X}\,,\qquad W=W_{0}+F_{X}X\ .$$

(14)

$$G_{X\bar{X}}={F_{X}^{2}\over(T+\bar{T})^{3\alpha}V(T,\bar{T})+3W_{0}^{2}(1-\alpha)}\ .$$

(15)

The value of the potential in this model, at $X=0$ is

$$V_{\rm final}=V(T,\bar{T})\ .$$

(16)

Thus, in terms of the dilaton and axion bosonic fields of our supergravity models (14), (15), the non-gravitational part of the action is

$${\cal L}(\varphi,a)={1\over 2}\large[(\partial\varphi)^{2}+e^{2\sqrt{2\over 3\alpha}\varphi}(\partial a)^{2}\large]-V\big{(}T(\varphi,a),\bar{T}(\varphi,a)\big{)}\,,\quad(\partial\varphi)^{2}\equiv\partial_{\mu}\varphi\partial_{\nu}\varphi g^{\mu\nu}\ .$$

(17)

This expression is the same as the one in eq. (13), for any potential $V(T,\bar{T})$.

Now we can present these models in geometric hyperbolic variables $T$ where the action is given in eq. (13) and $T$ is defined in eq. (10). Using

$${1\over 2}(\bar{T}+T)\equiv t\ ,\qquad{1\over 2i}\sqrt{3\alpha\over 2}(\bar{T}-T)\equiv a\ ,$$

(18)

we can present the potentials of the models in Linde:2018hmx ; Braglia:2020eai which can be used for the supergravity version of the model.

The supergravity version of the Hypernatural inflation model (2) requires a potential depending on geometric fields $T,\bar{T}$. It is given by

$$V(T,\bar{T})=V_{0}\big{(}1-t\big{)}^{2}+{1\over 2}m_{a}^{2}a^{2}\ .$$

(19)

where $t(T,\bar{T})$ and $a(T,\bar{T})$ are given in eq. (18). Replacing $t$ by the canonically normalized field $\varphi$ results in (2) Linde:2018hmx .

The cosmological model developed in Braglia:2020eai is

$${\cal L}={1\over 2}\bigl{[}-(\partial_{t}\phi)^{2}-e^{2b_{1}\phi}(\partial_{t}\chi)^{2}\bigr{]}-V_{0}{\phi^{2}\over\mu^{2}+\phi^{2}}-{1\over 2}m_{\chi}^{2}\chi^{2}\ .$$

(20)

It is now easy to recognize it as the one we presented in eqs. (1), (3) under condition

$$b_{1}=\sqrt{2\over 3\alpha}\,,\qquad\varphi\rightarrow\phi\,,\qquad a\rightarrow\chi\ .$$

(21)

It can also be given in the form with the $SL(2,\mathbb{R})$ invariant metric (9). The potential breaks the $SL(2,\mathbb{R})$ symmetry of the kinetic term and is given by the expression where the inflaton part of the potential was presented in Kallosh:2022feu . Namely, the potential of model Braglia:2020eai in geometric variables is

$$V(T,\bar{T})=V_{0}{\ln^{2}t\over c^{2}+\ln^{2}t}+{1\over 2}m_{a}^{2}a^{2}\,,\qquad c^{2}=\mu^{2}|\mathcal{R}_{K}|\ .$$

(22)

where $t(T,\bar{T})$ and $a(T,\bar{T})$ are given in eq. (18). This also means that the supergravity version of the model in Braglia:2020eai with (5) is now available in eqs. (14), (15), (16), (22). It is a dilaton-axion cosmological attractor model of the type described in Kallosh:2022feu .

The $\alpha$-attractor models with stabilized axion have stable attractor predictions. In case of the polynomial $\alpha$-attractors which we have here, we can first look at numerical examples studied in Braglia:2020eai and see if they are supported by the attractor values presented in eq. (2.14) in Kallosh:2022feu . For slow roll parameters we have

$$n_{s}=1-{3\over 2N_{e}},\qquad r={\sqrt{2}\mu\over N_{e}^{3/2}}\ .$$

(23)

We compare this prediction with the numerical cases studied in Braglia:2020eai and displayed in their Fig. 1, we show them in our Fig. 4.

We can explain the main features of the model based on its attractor properties as well as numerical solutions supporting them, see Figs. 4, 5. The effective mass of the axion is suppressed by the exponential factor $e^{-\sqrt{2\over 3\alpha}\varphi}$ due to kinetic term coupling. The axion is exponentially light at large $\varphi$ at the plateau, and starts moving only when the field $\varphi$ approaches the minimum of its potential at $\varphi=0$, and the exponential suppression of its effective mass disappears. This is the ‘rolling on the ridge effect’ effect found in Linde:2018hmx ; Achucarro:2017ing . Thus the dilaton stage of inflation ends at $a\approx a_{i}$. After $\varphi$ reaches $\varphi=0$, the axion in models with $a_{i}\neq 0$, undergoes a stage of chaotic inflation due to the axion potential ${1\over 2}m_{a}^{2}a^{2}$. The number of e-folds at this stage depends on how far is $a_{i}$ from its minimum $a=0$ when it rolls along the valley with the quadratic potential shown as the elongated red area at $\varphi=0$ in Fig. 5. We plot the trajectory of the fields during the two-stage inflation. At the turning point we found that $\epsilon$ jumps to $1$ and $\eta$ jumps down to $-17$ and rises up to $+10$ before settling to the slow roll axion inflation stage.

For the total number of e-folds to be 57, the number of e-foldings $N_{e}$ responsible for observations in eq. (23) should be smaller than 57, in view of additional e-folds at the axion stage:

$$N_{\rm total}=N_{e}+N_{\rm axion}\approx 57\ .$$

(24)

Note that if initially at $\phi=7$ the axion would be placed at $a=0$, in the middle of the plateau, it would be only one stage of inflation with $N_{e}=57$, $n_{s}\approx 0.97$.

We will consider 4 different cases which differ by the choice of the initial position of the axion, as we see in Table 1 in Braglia:2020eai and reproduced here in Fig. 6.

The number of e-foldings $N_{e}$ which we have to use in our attractor formula (23) can be identified by the values of $n_{s}$ or $r$ in the table. We get $N_{e}$ from $r$ using the table, and calculate $n_{s}$ with the same $N_{e}$ we find a good agreement with the table of numerical solutions given in Braglia:2020eai . Namely, we find for SKA, LISA, BBO, ET examples from (23)

	$\displaystyle N_{e}=19$		$\displaystyle n_{s}=0.921\qquad r=0.042$		(25)
	$\displaystyle N_{e}=32$		$\displaystyle n_{s}=0.952\qquad r=0.020$		(26)
	$\displaystyle N_{e}=35$		$\displaystyle n_{s}=0.957\qquad r=0.017$		(27)
	$\displaystyle N_{e}=39$		$\displaystyle n_{s}=0.961\qquad r=0.014$		(28)

One should take into account also that the parameters $\mu=\sqrt{6}$ and $r>10^{-2}$ are above the values of these parameters where the attractor regime is reached in these models, as explained in Kallosh:2018zsi . One can also see it in Fig. 3 in Kallosh:2022feu , which shows that the attractor regime is reached only for $r\lesssim 10^{-2}$.

And we definitely see the trend which is also clear from the properties of the dilaton-axion model of inflation. In particular, the attractor values of $n_{s}$ and $r$ in eq. (23) are $\alpha$-independent, which clearly explains why in Fig. 4 almost all curves for different values of $b_{1}=\sqrt{2\over 3\alpha}$ coincide.

Here in Fig. 7 we describe the transition area at the fixed value of Kähler curvature. The kinetic term for the axion field is $e^{2\sqrt{|\mathcal{R}_{K}|}\varphi}(\partial a)^{2}$. It means that the ‘physical distance’ is $e^{\sqrt{|\mathcal{R}_{K}|}\varphi}\partial a$, which is the reason why at large positive $\varphi$ the axion field $a$ is not moving for a long time. We can see it in Fig. 4 at the upper left corner where $\chi=a$ remains at his initial position for a long time during the first stage of the dilaton inflation. When the field $\varphi$ reached his minimum at $\varphi=0$ this protection of the axion position at its initial value vanishes. Moreover, when $\varphi$ becomes negative during the oscillation near the minimum, this factor becomes $e^{-\sqrt{|\mathcal{R}_{K}|}|\varphi|}\partial a$. This forces the axion first to change dramatically from its initial position and after a while a slow roll axion inflation takes place.

Note that this effect is significant for large curvatures $|\mathcal{R}_{K}|$ and less significant for smaller ones, which explains the color dependence of the value of $m^{2}_{\rm eff}/H^{2}$ in the blow-up region in the upper right plot in Fig. 4. The color there is codifying $b_{1}=\sqrt{2\over 3\alpha}$, it changes from blue to red when $b_{1}$ is increasing. Note that in all figures in in Fig. 4 there is no color dependence. The only exception is a blow-up region in the upper right plot of the effective mass of isocurvature perturbations $m^{2}_{\rm eff}/H^{2}$. This color-dependence of $m^{2}_{\rm eff}/H^{2}$ is practically absent during the first and the second stage of inflation, it is only present at the point of transition. Our Fig. 7 compliments a blow-up region in the upper right plot of the effective mass of isocurvature perturbations $m^{2}_{\rm eff}/H^{2}$ in Fig. 4. The effective mass of isocurvature perturbations $m^{2}_{\rm eff}$ becomes temporarily negative at the transition between the two stages of inflation and leads to a transient tachyonic amplification of the isocurvature perturbations leading to a large peak in the power spectrum. We found an additional explanation of the color dependence of $m^{2}_{\rm eff}$ defined in Braglia:2020eai at the region of transition where $\phi_{tran}\approx 0$. We have found that the effective mass to Hubble ratio depends on the Kähler curvature $\mathcal{R}_{K}=-b_{1}^{2}$ as follows

$${m^{2}_{\rm eff}\over H^{2}}\Rightarrow 5\,\mathcal{R}_{K}\Big{(}{\partial a\over\partial N}\Big{)}^{2}+\dots$$

(29)

The $\dots$ in (29) are for terms which mildly depend on $\mathcal{R}_{K}$, only in the form $e^{\sqrt{|\mathcal{R}_{K}|}\phi_{tran}}$ with $\phi_{tran}\approx 0$, or are independent on $\mathcal{R}_{K}$.

The first term is linear in Kähler curvature, depends on how fast the axion $a$ changes as a function of $N$, and it is dominant at the transition, the second term depends on Kähler curvature times $\phi_{tran}\approx 0$. This means that at the region of transition the negativity of the effective isocurvature mass is due to the negativity of the Kähler curvature as the first term in eq. (29) shows.

The upper right plot in Fig. 4 shows that the most negative effective mass of isocurvature perturbations in the blow-up region can be explained by the formula (29) which says that for bigger $|\mathcal{R}_{K}|$ the most negative value is reached (the red part of the figure at $N\approx 68$). This property of the ratio $m^{2}_{\rm eff}/H^{2}$ explains why the phenomenology of the PBHs and GWs strongly depends on the curvature. In Figs. 8 and 9 we reproduce figures from Braglia:2020eai where one can see that the change in $b_{1}=\sqrt{2\over 3\alpha}$ leads to dramatic changes in phenomenology. In particular, when $6.4\leq b_{1}\leq 8.4$ we need very small $\alpha$ so that $0.028\leq 3\alpha\leq 0.049$ and very high Kähler curvature $-\mathcal{R}_{K}=b_{1}^{2}\gtrsim 40$.

An additional set of models with parameters ${V_{0}\over m_{a}^{2}}=R=30,1050,3800$ was also studied in Braglia:2020eai , see Figure 10 here.

It was observed in Aragam:2021scu that rapid-turn inflation models in hyperbolic geometry tend to have high-curvature. This underscores the difficulty of obtaining such models from string theory. For example, the ones associated with M-theory and type IIB string theory in Kallosh:2021vcf without rapid terms have $3\alpha=7,6,5,4,3,2,1$.

Interestingly, with increasing $R={V_{0}\over m_{a}^{2}}$ which corresponds to lighter axions, the required value of $b_{1}$ is decreased and $\alpha$ is increased. We can provide here a qualitative explanation of this feature, comparing the cases with $R=500$ and $R=3800$ in Braglia:2020eai .

The mass squared of the axion is decreasing from the case described in details with $m_{a}^{2}={V_{0}\over R}$ to the case $\tilde{m}_{a}^{2}$ with different $V_{0}$ and $R$ and (case 2 and case 4 in Fig. 10). Also the initial conditions change from $a_{i}$ to $\tilde{a}_{i}$. We can compare the effective mass of the axion in case 2 and case 4

$$e^{-2b_{1}\phi}m_{a}^{2}a_{i}^{2}\,,\qquad e^{-2\tilde{b}_{1}\tilde{\phi}}\tilde{m}_{a}^{2}\tilde{a}_{i}^{2}\ .$$

(30)

It follows that

$$e^{2b_{1}\phi_{c}-\ln C^{2}}\approx e^{2\tilde{b}_{1}\tilde{\phi}_{c}}\ .$$

(31)

and numerically $\ln C^{2}\approx 1.57$. If we assume that we can compare these two models at the point $\phi_{c}\approx 0.49$ which is approximately the same in both cases, we find that

$$b_{1}-\tilde{b}_{1}\approx 1.55\ ,$$

(32)

to be compared with the difference between $b_{1}=7.837$ to $\tilde{b}_{1}=6.233$, which is $1.6$.

This is desirable modification of the parameters in the model since increasing $\alpha$ would be a step towards smaller Kähler curvature. For example, with $3\alpha=1$ the Kähler curvature is $-\mathcal{R}_{K}=2/3\alpha=2$. This is a smallest Poincaré disk associated with string theory. Smaller $\alpha$ are possible in $\mathcal{N}=1$ supergravity but the ones like $3\alpha=1,2,3,4,5,6,7$ originate from string theory and maximal $\mathcal{N}=8$ supergravity.

It would be also interesting if the models with smaller values of $\mu$ can be investigated, so that the system gets closer to the attractor regime and maybe works for smaller Kähler curvature.