November 2022

HU-EP-22/34-RTG

An $\alpha^{\prime}$-complete theory of cosmology
and its tensionless limit

Tomas Codina†, Olaf Hohm† and Diego Marques∗

† Institute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

∗ Instituto de Astronomía y Física del Espacio,
Casilla de Correo 67 - Suc. 28 (C1428ZAA), Buenos Aires, Argentina

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We explore the exactly duality invariant higher-derivative extension of
> double field theory due to Hohm, Siegel and Zwiebach (HSZ) specialized to
> cosmological backgrounds. Despite featuring a finite number of derivatives
> in its original formulation, this theory encodes infinitely many
> $\alpha^{\prime}$ corrections for metric, B-field and dilaton, which are
> obtained upon integrating out certain extra fields. We perform a
> cosmological reduction with fields depending only on time and show
> consistency of this truncation. We compute the $\alpha^{\prime 4}$
> coefficients of the general cosmological classification. As a possible model
> for how to deal with all $\alpha^{\prime}$ corrections in string theory we
> give a two-derivative reformulation in which the extra fields are kept. The
> corresponding Friedmann equations are then ordinary second order
> differential equations that capture all $\alpha^{\prime}$ corrections. We
> explore the tensionless limit $\alpha^{\prime}\rightarrow\infty$, which
> features string frame de Sitter vacua, and we set up perturbation theory in
> $\frac{1}{\alpha^{\prime}}$.

###### Contents

  1. 1 Introduction
  2. 2 Consistent truncation to cosmological backgrounds
    1. 2.1 Review of HSZ theory
    2. 2.2 The cosmological truncation
    3. 2.3 The cosmological action
  3. 3 Higher-derivative corrections at order $\alpha^{\prime\,4}$
    1. 3.1 From the double to the generalized metric
    2. 3.2 Integrating out the extra fields
  4. 4 Two-derivative reformulation
    1. 4.1 Two-derivative theory
    2. 4.2 Tensionless limit
  5. 5 Friedmann Universe and tensionless limit
    1. 5.1 $\alpha^{\prime}$-exact Friedmann equations
    2. 5.2 Tensionless limit, de Sitter solution and $\frac{1}{\alpha^{\prime}}$ expansion
  6. 6 Conclusions and Outlook
  7. A Field Redefinitions

##  1 Introduction

A basic obstacle for understanding string theory, let alone confronting it
with observation, is that we do not know the defining equations of string
theory. We know the low-energy effective theories given by (super)gravity, but
even classical string theory restricted to the massless fields features an
infinite number of higher-derivative $\alpha^{\prime}$ corrections going
beyond conventional gravity. While these corrections have been determined to a
few orders in $\alpha^{\prime}$, a computation of all $\alpha^{\prime}$
corrections is out of reach. But it is precisely in regimes where a
gravitational theory beyond general relativity is almost certainly needed
(singularities of cosmology or black holes), that one plausibly expects all
$\alpha^{\prime}$ corrections to be important. (See [1, 2, 3, 4, 5, 6, 7, 8]
for classic and more recent string cosmology proposals requiring the inclusion
of all $\alpha^{\prime}$ corrections.)

In this paper we explore a particular spacetime theory based on double field
theory [9, 10, 11, 12], due to Hohm, Siegel and Zwiebach (HSZ) [13], and apply
it to cosmological backgrounds. Our goal is two-fold, namely first to compute
the $\alpha^{\prime 4}$ coefficients of the general cosmological
classification, thereby going beyond the current state of the art [14, 15,
16], and second and perhaps more intriguingly, to have a model for a theory
that is ‘$\alpha^{\prime}$-complete’ in a sense made precise below. The
construction of HSZ theory was based on a non-standard chiral CFT and is thus
not a conventional string theory. (It also appears closely related to the
‘chiral string theory’ of [17] and/or to the ambitwistor string [18], but to
our knowledge the precise connections have not been established.)
Nevertheless, HSZ theory shares crucial features of any string theory, such as
1) duality invariance under $O(d,d,\mathbb{R})$ for backgrounds with $d$
Abelian isometries, and 2) the presence of infinitely many higher-derivative
corrections for the massless fields. While in its original formulation HSZ
theory carries only up to six derivatives, it also features extra massive
fields, in addition to the familiar massless fields of string theory (metric,
B-field and dilaton), and integrating out these extra fields induces an
infinite tower of $\alpha^{\prime}$ corrections for the massless fields that
are kept. These higher-derivative corrections include a Green-Schwarz-type
deformation at order $\alpha^{\prime}$ and a Riemann-cube invariant at order
$\alpha^{\prime 2}$ [19, 20, 21], but beyond that only very little is known.

In order to understand the theory in a simplified setting, and to relate to
the physically interesting case of cosmology, we specialize HSZ theory to
cosmological backgrounds, i.e. with fields depending only on time. We then
show how to integrate out the extra fields so as to compute the order
$\alpha^{\prime 4}$ coefficients of the general cosmological classification
[22, 23]. As a second main result of this paper, we then show that, rather
than integrating out the extra fields, we can keep them and reformulate the
theory so that it features only up to two derivatives. Such a reformulation
was given in [24] for HSZ theory without reduction, but restricted to
quadratic order in a background field expansion about flat space. Here we
exhibit the corresponding reformulation in the cosmological reduction but
including all non-linear couplings. Intriguingly, in this formulation the
tensionless limit $\alpha^{\prime}\rightarrow\infty$ can be taken smoothly. We
also present the equations of motion for Friedmann-Robertson-Walker
backgrounds, which by construction capture all $\alpha^{\prime}$ corrections.

It has sometimes been speculated that the tensionless limit of string theory,
if it were accessible, would be a powerful tool for learning about the
fundamental degrees of freedom of string theory [25, 26]. We thus take the
opportunity and explore the cosmological Friedmann equations in the
tensionless limit, in which we find the most general solutions. Remarkably,
these solutions include string frame de Sitter vacua (that are also de Sitter
vacua in Einstein frame for $d=4$, i.e., in five spacetime dimensions).
Interestingly, the de Sitter length scale here is not a bare parameter in the
action but rather appears as an integration constant. In order to find
solutions of the full equations we then use perturbation theory, but not in
$\alpha^{\prime}$ as would be appropriate for small $\alpha^{\prime}$, but
rather in $\frac{1}{\alpha^{\prime}}$ as is appropriate for perturbations
around infinite $\alpha^{\prime}$. We find that the de Sitter vacua survive
the first order corrections in $\frac{1}{\alpha^{\prime}}$.

We close this introduction by discussing the $\alpha^{\prime}$-completeness of
this theory. Gravity theories with a finite number of higher derivatives
generically suffer from various pathologies, as the presence of unphysical
ghost modes in the spectrum.111Counter examples are the Starobinsky model
[27], which augments the Einstein-Hilbert term by the square of the Ricci
scalar, and ‘new massive gravity’ [28], with a particular curvature-squared
modification of 3D gravity. For this reason, the usual view is that higher-
derivative modifications only make sense in perturbation theory, with features
of the two-derivative theory being corrected by a small parameter like
$\alpha^{\prime}$ and carrying an infinite number of such higher-derivative
corrections. Another perspective showing that a finite number of higher-
derivative corrections is problematic is that of $O(d,d,\mathbb{R})$
invariance for backgrounds with $d$ Abelian isometries, in the following
referred to as ‘duality’, which string theory must possess to all orders in
$\alpha^{\prime}$ [29]. While in the two-derivative truncation this duality is
easily recognized and realized exactly, once higher-derivative terms are
included the situation becomes more subtle [30]. Adding the known four-
derivative (order $\alpha^{\prime}$) terms in the bosonic string action and
performing dimensional reduction along $d$ dimensions, the resulting theory
does not possess the expected duality invariance. It is, however, invariant
under higher-derivative deformations of the duality transformations up to
terms of order $\alpha^{\prime 2}$ [30]. Thus, within a theory with infinitely
many $\alpha^{\prime}$ corrections this shows compatibility with duality, but
a finite higher-derivative modification is generally incompatible with
duality. The HSZ theory to be investigated here is exactly duality invariant
and in this sense $\alpha^{\prime}$-complete.

The rest of this paper is organized as follows. In sec. 2 we introduce the
cosmological ansatz for HSZ theory, which is non-trivial in that certain off-
diagonal field components need to be kept. We prove consistency of this
truncation and give the cosmological action. This is used in sec. 3 in order
to integrate out the extra fields up to and including order $\alpha^{\prime
4}$. The resulting action is then brought to the canonical form of the
cosmological classification [23] by means of field redefinitions. In sec. 4 we
present a reformulation of the cosmological HSZ action, including the extra
fields, with only two derivatives. In sec. 5 we give the corresponding
$\alpha^{\prime}$-complete Friedmann equations, take the tensionless limit and
find the most general solutions which include a de Sitter vacua as a
particular case. We finish this section by computing the first
$\frac{1}{\alpha^{\prime}}$ correction to this specific solution. We conclude
with a brief outlook in sec. 6.

##  2 Consistent truncation to cosmological backgrounds

In this section we review the double $\alpha^{\prime}$-geometry by Hohm,
Siegel and Zwiebach [13], here referred to as HSZ theory. We will introduce
the field content, symmetries and action. We then perform the truncation that
will take us to the time-dependent cosmological action in terms of (duality
covariant) scalar fields.

###  2.1 Review of HSZ theory

The field content of HSZ theory includes the ‘double metric’
$\mathcal{M}_{\hat{M}\hat{N}}$, with $O(D,D)$ indices
$\hat{M},\hat{N}=1,\dots,2D$, and a generalized dilaton field $d$. They are
defined on a double space with coordinates $X^{\hat{M}}$, but all products of
fields and gauge parameters are annihilated by the duality invariant Laplacian

$\eta^{\hat{M}\hat{N}}\partial_{\hat{M}}\partial_{\hat{N}}\cdots=0\;,$ (2.1)

where $\eta_{\hat{M}\hat{N}}$ is the $O(D,D)$ invariant metric that raises and
lowers indices. This is known as the strong constraint. In contrast to the
standard generalized metric formulation of double field theory (DFT)[9, 10,
11, 12], where the generalized metric ${\cal H}_{\hat{M}\hat{N}}$ obeys the
constraint ${\cal H}\eta^{-1}{\cal H}=\eta$, the double metric is symmetric
but otherwise unconstrained. Under infinitesimal generalized diffeomorphisms
with parameter $\xi^{\hat{M}}$, the generalized dilaton transforms as in
conventional DFT

$\delta
d=\xi^{\hat{P}}\partial_{\hat{P}}d-\frac{1}{2}\partial_{\hat{P}}\xi^{\hat{P}}\,,$
(2.2)

while the double metric receives linear and quadratic corrections in
$\alpha^{\prime}$

$\displaystyle\delta\mathcal{M}_{\hat{M}\hat{N}}=\widehat{\mathcal{L}}_{\xi}\mathcal{M}_{\hat{M}\hat{N}}+\alpha^{\prime}\mathcal{J}^{(1)}_{\hat{M}\hat{N}}+\alpha^{\prime}{}^{2}\mathcal{J}^{(2)}_{\hat{M}\hat{N}}\,,$
(2.3)

where the standard generalized Lie derivative is defined by

$\widehat{\mathcal{L}}_{\xi}\mathcal{M}_{\hat{M}\hat{N}}\equiv\xi^{\hat{P}}\partial_{\hat{P}}\mathcal{M}_{\hat{M}\hat{N}}+\mathcal{K}_{\hat{M}}{}^{\hat{P}}\mathcal{M}_{\hat{P}\hat{N}}+\mathcal{K}_{\hat{N}}{}^{\hat{P}}\mathcal{M}_{\hat{M}\hat{P}}\;,\qquad\mathcal{K}_{\hat{M}\hat{N}}\equiv
2\partial_{[\hat{M}}\xi_{\hat{N}]}\;,$ (2.4)

and the higher-derivative contributions are given by

$\displaystyle\mathcal{J}^{(1)}_{\hat{M}\hat{N}}$
$\displaystyle\equiv-\frac{1}{2}\partial_{\hat{M}}\mathcal{M}^{\hat{P}\hat{Q}}\partial_{\hat{P}}\mathcal{K}_{\hat{Q}\hat{N}}-\partial_{\hat{P}}\mathcal{M}_{\hat{Q}\hat{M}}\partial_{\hat{N}}\mathcal{K}^{\hat{Q}\hat{P}}+(\hat{M}\rightleftarrows\hat{N})\;,$
(2.5) $\displaystyle\mathcal{J}^{(2)}_{\hat{M}\hat{N}}$
$\displaystyle\equiv-\frac{1}{4}\partial_{\hat{M}\hat{K}}\mathcal{M}^{\hat{P}\hat{Q}}\partial_{\hat{N}\hat{P}}\mathcal{K}_{\hat{Q}}{}^{\hat{K}}+(\hat{M}\rightleftarrows\hat{N})\;.$

These transformations close under a deformation of the C-bracket

$\big{[}\xi_{1},\xi_{2}\big{]}^{\hat{M}}_{(C)}\equiv
2\xi^{\hat{P}}_{[1}\partial_{\hat{P}}\xi^{\hat{M}}_{2]}-\xi^{\hat{P}}_{[1}\partial^{\hat{M}}\xi_{2]\hat{P}}\,,$
(2.6)

which we call the $C^{\prime}$-bracket:

$\displaystyle\big{[}\xi_{1},\xi_{2}\big{]}^{\hat{M}}_{(C^{\prime})}\equiv\big{[}\xi_{1},\xi_{2}\big{]}^{\hat{M}}_{(C)}+\alpha^{\prime}\partial_{\hat{P}}\xi^{\hat{Q}}_{[1}\partial^{\hat{M}}\partial_{\hat{Q}}\xi^{\hat{P}}_{2]}\,.$
(2.7)

The dynamics of the theory is encoded in an action, that can be written
compactly as

$I_{\rm HSZ}=\int
d^{2D}\,X\,e^{-2d}\,\left\langle\mathcal{M}\,\left|\,\eta-\frac{1}{6}\mathcal{M}\star\mathcal{M}\right.\right\rangle\;.$
(2.8)

The definitions for the inner product
$\left\langle\,\cdot\,|\,\cdot\,\right\rangle$ and star-product $\star$
involve long expressions in terms of $\mathcal{M}$, $d$ and combinations of
them up to and including six derivatives. Since these explicit expressions are
not very illuminating, we refer the reader to equations (2.11), (2.12) and
(2.14) of [31] where the definitions are given in detail, or to appendix D
therein where the complete action is given. The exact gauge symmetry under
(2.2) and (2.3) can be checked once the definitions of the products are used,
together with the strong constraint (2.1).

###  2.2 The cosmological truncation

We will now truncate the above theory to a cosmological ansatz in which fields
depend only on time, and we will prove that the truncation is consistent. To
this end we assume a split of the coordinates and indices as follows

$X^{\hat{M}}=\big{(}t\,,\;\tilde{t}\,,\;Y^{M}\big{)}\;,\qquad{}_{\hat{M}}=\big{(}_{0}\,,\;{}^{0}\,,\;_{M}\big{)}\;,\qquad
M=1,\dots,2(D-1)\;.$ (2.9)

This ansatz breaks the manifest $O(D,D)$ invariance to $O(1,1)\times
O(D-1,D-1)$. Furthermore, we will solve the strong constraint by selecting a
frame in which the fields do not depend on $\tilde{t}$ nor $Y^{M}$, which
breaks the $O(1,1)$ factor and importantly preserves the internal
$O(D-1,D-1)$. We will thus set

$\partial^{0}=\partial_{M}=0\;,$ (2.10)

everywhere in the field equations and gauge transformations.

Let us now turn to the decompositions of the fields and the $O(D,D)$ metric,
which are given by

$\eta_{\hat{M}\hat{N}}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&\eta_{MN}\end{pmatrix},\qquad\mathcal{M}_{\hat{M}\hat{N}}=\begin{pmatrix}-n^{2}B&A&0\\\
A&-\frac{1}{n^{2}}&0\\\ 0&0&\mathcal{Z}_{MN}\end{pmatrix},\qquad
d=\frac{1}{2}\Phi-\frac{1}{2}\ln n\;,$ (2.11)

where all fields depend only on time $t$. Apart from the exclusive time
dependence, the only truncation applied above is given by the vanishing of
some components of the double metric. We explain latter why setting these
‘vector’ components to zero is a consistent choice. Additionally, one would be
tempted to further reduce the external $2\times 2$ block in the double metric
to be diagonal and $O(1,1)$ valued by setting $A=0$, $B=1$, but this turns out
to be inconsistent. By inspecting the action of generalized diffeomorphisms
(2.3) for this ansatz, with the gauge parameter decomposed as

$\xi^{\hat{M}}=\big{(}\xi^{0},\xi_{0},\xi^{M}\big{)}\equiv\big{(}\xi,0,{0}\big{)}\;,$
(2.12)

we find the following gauge transformations for the component fields

$\displaystyle\delta n$ $\displaystyle=\xi\dot{n}+n\dot{\xi}\,,$ (2.13)
$\displaystyle\delta A$
$\displaystyle=\xi\dot{A}-3\alpha^{\prime}\ddot{\xi}\frac{\dot{n}}{n^{3}}\,,$
$\displaystyle\delta B$
$\displaystyle=\xi\dot{B}+\alpha^{\prime}\frac{1}{n^{2}}\ddot{\xi}\dot{A}+\alpha^{\prime}{}^{2}\frac{1}{n^{5}}\dddot{\xi}\left(\ddot{n}-3\frac{\dot{n}^{2}}{n}\right)\,,$
$\displaystyle\delta\mathcal{Z}_{MN}$
$\displaystyle=\xi\dot{\mathcal{Z}}_{MN}\,,$ $\displaystyle\delta\Phi$
$\displaystyle=\xi\dot{\Phi}\,,$

where here and in the following the dot denotes time derivatives, i.e.
$\partial_{t}\Psi=\partial_{0}\Psi\equiv\dot{\Psi}$. To zeroth order in
$\alpha^{\prime}$ we recognize the familiar transformations under time
reparametrizations $t\rightarrow t-\xi(t)$, but these transformations receive
$\alpha^{\prime}$-corrections. The fact that the corrections to the
transformations of $A$ and $B$ contain corrections not depending on $A$ and
$B$ themselves, prevents us from setting them to a constant, so both $A$ and
$B$ must be kept for consistency of the truncation.

Instead, setting to zero the vectorial components of the double metric ${\cal
M}_{0M}={\cal M}{}^{0}{}_{M}=0$ is perfectly consistent. In fact, the
transformations (2.12) acting on these components vanish when the components
themselves are set to zero, contrary to what happens with $A$ and $B$.
Moreover, and equally important, the equations of motion of these components
also vanish when the components do. Put differently, the information contained
in the equations of motion of all fields after setting ${\cal M}_{0M}={\cal
M}{}^{0}{}_{M}=0$, is the same as the one obtained by setting these components
to zero in the action and then computing the equations of motion of the fields
we kept. As a consequence of this truncation, the full original gauge symmetry
of HSZ (2.3) is now broken to time reparametrizations (2.12).

It is instructive to inspect also the gauge algebra under this cosmological
reduction. One obtains with (2.7)

$\displaystyle\big{[}\xi_{1},\xi_{2}\big{]}_{0\,(C^{\prime})}$
$\displaystyle=2\xi_{[1}\dot{\xi}_{2]}\;,$ (2.14)
$\displaystyle\big{[}\xi_{1},\xi_{2}\big{]}^{0}_{(C^{\prime})}$
$\displaystyle=\alpha^{\prime}\dot{\xi}_{[1}\ddot{\xi}_{2]}=\left[\sqrt{\tfrac{\alpha^{\prime}}{2}}\dot{\xi}_{1},\sqrt{\tfrac{\alpha^{\prime}}{2}}\dot{\xi}_{2}\right]_{0\,(C^{\prime})}\;,$
(2.15) $\displaystyle\big{[}\xi_{1},\xi_{2}\big{]}^{M}_{(C^{\prime})}$
$\displaystyle=0\;.$ (2.16)

Given the relationship in the second line, one may suspect that the only
surviving algebra is that of standard one-dimensional diffeomorphisms,
suggesting that there should be a field basis in which this symmetry is
realized in the standard way. Indeed, we can find an explicit field
redefinition that removes the higher-derivative terms in $\delta A$ and
$\delta B$. To this end, it is convenient to introduce the derivative operator

${\cal D}\equiv\frac{1}{n}\partial_{t}\,,$ (2.17)

which is covariant under conventional time reparametrizations. Specifically,
writing the original fields in terms of new primed fields as

$\displaystyle A$ $\displaystyle=A^{\prime}-\frac{3}{2}\alpha^{\prime}({\cal
D}\ln n)^{2}\;,$ (2.18) $\displaystyle B$
$\displaystyle=B^{\prime}+\alpha^{\prime}({\cal D}\ln n){\cal
D}A^{\prime}-\alpha^{\prime}{}^{2}\left[\frac{1}{4}({\cal D}\ln n)^{4}+({\cal
D}^{2}\ln n)({\cal D}\ln n)^{2}-\frac{1}{2}({\cal D}^{2}\ln n)^{2}\right]\,,$

it is straightforward to verify that for $A^{\prime}$ and $B^{\prime}$ being
reparametrization scalars, the higher-derivative terms induce precisely the
higher-order corrections in (2.13). Furthermore, it is immediate that the
above relations can be inverted hence proving that this is a legal field
redefinition. All in all, we can express the theory in terms of fields given
by the lapse function $n$ and a number of reparametrization scalars, with
transformation rules

$\delta n=\xi\dot{n}+n\dot{\xi}\,,\ \ \ \delta A=\xi\dot{A}\,,\ \ \ \delta
B=\xi\dot{B}\,,\ \ \ \delta\mathcal{Z}_{MN}=\xi\dot{\mathcal{Z}}_{MN}\,,\ \ \
\delta\Phi=\xi\dot{\Phi}\,,$ (2.19)

where we removed the primes from $A^{\prime}$ and $B^{\prime}$.

###  2.3 The cosmological action

We now give the HSZ action in this cosmological reduction, which is obtained
by plugging the ansatz (2.11) together with the field redefinitions (2.18)
into (2.8). More precisely, we performed the reduction at the level of the
inner and star products using their explicit definitions as given in [31] and
then combined these results back into the form of the action (2.8). As a
consistency check we can use that in this field basis the diffeomorphisms act
in the usual way, which implies that the derivatives of the lapse function
should combine to form covariant derivatives ${\cal
D}=\frac{1}{n}\partial_{t}$ of the scalar fields.222As a second consistency
check, we also performed the reduction directly at the level of the
$\alpha^{\prime}$-expanded action as given in Appendix D of [31]. We used
Cadabra2 [32] for the reductions and consistency checks. We find that the
final manifestly gauge invariant action is given by

$\displaystyle I$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left[\frac{1}{\alpha^{\prime}}\mathcal{L}^{(-1)}+\mathcal{L}^{(0)}+\alpha^{\prime}\mathcal{L}^{(1)}+\alpha^{\prime}{}^{2}\mathcal{L}^{(2)}\right]\;,$
(2.20)

where

$\displaystyle\mathcal{L}^{(-1)}$
$\displaystyle=A-\frac{1}{3}A^{3}-AB+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\,,$ (2.21)
$\displaystyle\mathcal{L}^{(0)}$ $\displaystyle=-\frac{31}{12}({\cal
D}A)^{2}-\frac{3}{2}A^{2}\left[({\cal D}\Phi)^{2}-2{\cal
D}^{2}\Phi\right]+\frac{35}{6}{\cal D}\Phi A{\cal D}A-\frac{17}{6}A{\cal
D}^{2}A-\frac{5}{12}{\cal D}^{2}B+\frac{11}{12}{\cal D}\Phi{\cal D}B$
$\displaystyle-\frac{1}{2}B\left[({\cal D}\Phi)^{2}-2{\cal
D}^{2}\Phi\right]-\frac{1}{24}{\rm Tr}{({\cal
D}\mathcal{Z})^{2}}+\frac{1}{12}{\rm Tr}{\mathcal{Z}{\cal
D}^{2}\mathcal{Z}}-\frac{1}{12}{\cal D}\Phi{\rm Tr}{\mathcal{Z}{\cal
D}\mathcal{Z}}+\frac{3}{2}({\cal D}\Phi)^{2}-3{\cal D}^{2}\Phi\,,$
$\displaystyle\mathcal{L}^{(1)}$ $\displaystyle=-\frac{1}{4}{\cal
D}^{4}A-\frac{3}{4}({\cal D}\Phi)^{2}{\cal D}^{2}A+\frac{1}{4}({\cal
D}\Phi)^{3}{\cal D}A-4({\cal D}^{2}\Phi)^{2}A+\frac{3}{4}{\cal D}\Phi{\cal
D}^{3}A+3{\cal D}^{2}\Phi{\cal D}^{2}A$ $\displaystyle\quad+\frac{9}{2}{\cal
D}^{3}\Phi{\cal D}A+\frac{5}{2}{\cal D}^{4}\Phi A+\frac{5}{2}({\cal
D}\Phi)^{2}{\cal D}^{2}\Phi A-\frac{11}{2}{\cal D}\Phi{\cal D}^{2}\Phi{\cal
D}A-5{\cal D}\Phi{\cal D}^{3}\Phi A\,,$ $\displaystyle\mathcal{L}^{(2)}$
$\displaystyle=\frac{3}{2}({\cal D}^{2}\Phi)^{3}-2({\cal
D}^{3}\Phi)^{2}+\frac{1}{4}{\cal D}^{6}\Phi-({\cal D}\Phi)^{2}({\cal
D}^{2}\Phi)^{2}+\frac{3}{4}({\cal D}\Phi)^{2}{\cal
D}^{4}\Phi-\frac{1}{4}({\cal D}\Phi)^{3}{\cal D}^{3}\Phi$
$\displaystyle\quad-\frac{3}{4}{\cal D}\Phi{\cal D}^{5}\Phi-\frac{11}{4}{\cal
D}^{2}\Phi{\cal D}^{4}\Phi+\frac{19}{4}{\cal D}\Phi{\cal D}^{2}\Phi{\cal
D}^{3}\Phi\,.$

By performing integration by parts one may verify that (2.21) is equivalent to
the simpler action of the same structural form (2.20) but with Lagrangian

$\displaystyle\mathcal{L}^{(-1)}$ $\displaystyle=\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm
Tr}{\mathcal{Z}^{3}}+A-\frac{1}{3}A^{3}-AB\,,$ (2.22)
$\displaystyle\mathcal{L}^{(0)}$ $\displaystyle=-\frac{1}{8}{\rm Tr}{({\cal
D}\mathcal{Z})^{2}}-\frac{3}{2}({\cal D}\Phi)^{2}+\frac{1}{4}({\cal
D}A)^{2}+\frac{3}{2}A^{2}{\cal D}^{2}\Phi+\frac{1}{2}B{\cal D}^{2}\Phi\,,$
$\displaystyle\mathcal{L}^{(1)}$ $\displaystyle=\frac{1}{2}A\left[{\cal
D}^{4}\Phi-{\cal D}\Phi{\cal D}^{3}\Phi-3({\cal D}^{2}\Phi)^{2}\right]\,,$
$\displaystyle\mathcal{L}^{(2)}$ $\displaystyle=\frac{1}{4}({\cal
D}^{3}\Phi)^{2}+\frac{1}{2}({\cal D}^{2}\Phi)^{3}\,.$

Finally, we can bring the full action to its simplest form by performing the
following field redefinition

$\displaystyle B^{\prime}\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!B-1+\frac{1}{3}A^{2}-\alpha^{\prime}\left[\frac{4}{3}A{\cal
D}^{2}\Phi+\frac{1}{4}{\cal D}\Phi{\cal D}A-\frac{1}{4}{\cal
D}^{2}A\right]-\frac{\alpha^{\prime}{}^{2}}{2}\left[\frac{3}{4}{\cal
D}^{4}\Phi-\frac{3}{4}{\cal D}\Phi{\cal D}^{3}\Phi-\frac{5}{3}({\cal
D}^{2}\Phi)^{2}\right],$ $\displaystyle A^{\prime}\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\\!-A+\frac{\alpha^{\prime}}{2}{\cal
D}^{2}\Phi\,,$ (2.23)

to get (omitting the primes for $A^{\prime}$ and $B^{\prime}$)

$\begin{split}I=\int
dt\,n\,e^{-\Phi}\,\Big{\\{}&\frac{1}{\alpha^{\prime}}\Big{[}AB+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\Big{]}-\frac{1}{8}{\rm
Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal D}\Phi)^{2}\\\
&+\frac{\alpha^{\prime}{}^{2}}{4}\Big{[}\frac{1}{4}({\cal
D}^{3}\Phi)^{2}+\frac{1}{3}({\cal
D}^{2}\Phi)^{3}\Big{]}\Big{\\}}\;.\end{split}$ (2.24)

We observe that after the above series of field redefinitions $A$ and $B$
completely trivialize in the sense that their equations of motion simply set
them to zero. Therefore, the original theory given by (2.22) is equivalent to
an effective theory for $\mathcal{Z},n$ and $\Phi$ only, whose action is given
by (2.24) after setting $A=B=0$. As a consequence, from now on we can use this
effective action to explore the physics of the $\mathcal{Z},n,\Phi$ system
alone, which in the following section we take as a starting point to get an
effective action for standard gravity fields. In section 4, however, we will
reintroduce $A$ and $B$ to see that, even though they encode no extra
information for the physical fields, they can still be used to bring the
theory to a formulation without positive powers of $\alpha^{\prime}$, i.e.,
without more than two derivatives.

##  3 Higher-derivative corrections at order $\alpha^{\prime\,4}$

Our goal in this section is to relate the cosmological reduction of HSZ theory
to a conventional gravity theory featuring only the metric, B-field and
dilaton. To this end, setting $d=D-1$, we express the internal double metric
$\mathcal{Z}$ in terms of an $O(d,d)$ valued generalized metric, encoding the
metric and the $B$-field, plus extra fields. Integrating out these extra
fields then induces higher-derivative $\alpha^{\prime}$ corrections for the
remaining fields that we compute to order $\alpha^{\prime 4}$. We end the
section by showing how to bring the resulting action to the minimal form
classified by Hohm and Zwiebach [22].

###  3.1 From the double to the generalized metric

We start from the action in the form (2.24), setting $A=B=0$, as implied by
their own field equations. Thus, we consider the following action for the
remaining fields $\mathcal{Z},n$ and $\Phi$:

$\displaystyle I$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{\frac{1}{2\alpha^{\prime}}\left[{\rm
Tr}{\mathcal{Z}}-\frac{1}{3}{\rm Tr}{\mathcal{Z}^{3}}\right]-\frac{1}{8}{\rm
Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}+\frac{\alpha^{\prime}{}^{2}}{4}\left[\frac{1}{4}({\cal
D}^{3}\Phi)^{2}+\frac{1}{3}({\cal D}^{2}\Phi)^{3}\right]\right\\}\,.$ (3.1)

From the general variation of this action

$\delta I=\int dt\,n\,e^{-\Phi}\left[{\rm
Tr}{\delta\mathcal{Z}E_{\mathcal{Z}}}+\delta\Phi E_{\Phi}+\frac{\delta
n}{n}E_{n}\right]\,,$ (3.2)

one obtains the equations of motion

$\displaystyle 0=E_{\mathcal{Z}}$
$\displaystyle\equiv\frac{1}{2\alpha^{\prime}}\left(1-\mathcal{Z}^{2}\right)-\frac{1}{4}{\cal
D}\Phi{\cal D}\mathcal{Z}+\frac{1}{4}{\cal D}^{2}\mathcal{Z}\,,$ (3.3a)
$\displaystyle 0=E_{\Phi}$
$\displaystyle\equiv-\frac{1}{2\alpha^{\prime}}\left[{\rm
Tr}{\mathcal{Z}}-\frac{1}{3}{\rm Tr}{\mathcal{Z}^{3}}\right]+\frac{1}{8}{\rm
Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal D}\Phi)^{2}+2{\cal D}^{2}\Phi$
$\displaystyle\quad-\frac{\alpha^{\prime}{}^{2}}{8}\left[\vphantom{\frac{1}{2}}{\cal
D}^{6}\Phi-3{\cal D}\Phi{\cal D}^{5}\Phi-7{\cal D}^{2}\Phi{\cal
D}^{4}\Phi+3({\cal D}\Phi)^{2}{\cal D}^{4}\Phi\right.-\frac{9}{2}({\cal
D}^{3}\Phi)^{2}+11{\cal D}\Phi{\cal D}^{2}\Phi{\cal D}^{3}\Phi$
$\displaystyle\quad\quad\quad\quad-(D\Phi)^{3}{\cal
D}^{3}\Phi\left.\quad+\frac{8}{3}({\cal D}^{2}\Phi)^{3}-2({\cal
D}\Phi)^{2}({\cal D}^{2}\Phi)^{2}\right]\,,$ (3.3b) $\displaystyle 0=E_{n}$
$\displaystyle\equiv\frac{1}{2\alpha^{\prime}}\left[{\rm
Tr}{\mathcal{Z}}-\frac{1}{3}{\rm Tr}{\mathcal{Z}^{3}}\right]+\frac{1}{8}{\rm
Tr}{({\cal D}\mathcal{Z})^{2}}+({\cal D}\Phi)^{2}$
$\displaystyle\quad-\frac{\alpha^{\prime}{}^{2}}{8}\left[\vphantom{\frac{1}{2}}{\cal
D}\Phi{\cal D}^{5}\Phi-{\cal D}^{2}\Phi{\cal D}^{4}\Phi-2({\cal
D}\Phi)^{2}{\cal D}^{4}\Phi\right.+\frac{1}{2}({\cal D}^{3}\Phi)^{2}-4{\cal
D}\Phi{\cal D}^{2}\Phi{\cal D}^{3}\Phi$
$\displaystyle\quad\quad\quad\quad+(D\Phi)^{3}{\cal
D}^{3}\Phi\left.+\frac{4}{3}({\cal D}^{2}\Phi)^{3}+2({\cal D}\Phi)^{2}({\cal
D}^{2}\Phi)^{2}\right]\,,$ (3.3c)

where for the first equation we used a matrix notation for
$\mathcal{Z}_{\bullet}{}^{\bullet}$. As a consistency check, we verified that
these equations satisfy the Bianchi/Noether identity implied by gauge
invariance under time reparameterizations. To this end one specializes in
(3.2) the variations to the one-dimensional diffeomorphism transformations

$\delta\Phi=\xi\dot{\Phi}\,,\quad\quad\delta n=\partial_{t}(\xi
n)\,,\quad\quad\delta\mathcal{Z}=\xi\dot{\mathcal{Z}}\,,$ (3.4)

which yields

$\delta_{\xi}I=\int dt\,n\,e^{-\Phi}\xi\left[{\rm
Tr}{\dot{\mathcal{Z}}E_{\mathcal{Z}}}+\dot{\Phi}E_{\Phi}-e^{\Phi}\partial_{t}\left(e^{-\Phi}E_{n}\right)\right]=0\,.$
(3.5)

Gauge invariance implies that this must hold for arbitrary $\xi$, from which
we infer:

${\cal D}\Phi(E_{\Phi}+E_{n})+{\rm Tr}{{\cal
D}\mathcal{Z}E_{\mathcal{Z}}}-{\cal D}E_{n}=0\,.$ (3.6)

This is indeed identically satisfied by (3.3).

Even though discussing solutions of (3.3) is beyond the scope of this work,
there are two of them that can be easily found. The first one is the trivial
configuration

$\mathcal{Z}(t)_{M}{}^{N}=(S_{0})_{M}{}^{N}\,,\quad\Phi(t)=\Phi_{0}\,,$ (3.7)

where $S_{0}$ is the standard generalized metric but with constant entries,
and $\Phi_{0}$ is also a constant. In that case (3.3) is trivially satisfied.
More surprisingly, however, is the non-trivial and non-perturbative solution

$\mathcal{Z}(t)_{M}{}^{N}=-\delta_{M}{}^{N}\,,\quad\Phi(t)=\sqrt{\frac{2d}{3\alpha^{\prime}}}\,t+\Phi_{0}\,.$
(3.8)

Note that $\mathcal{Z}(t)_{M}{}^{N}=\delta_{M}{}^{N}$ is not a solution for a
real dilaton field. We consider this solution to belong to a separate and
somewhat pathological branch that is incompatible with the interpretation as a
generalized metric featuring a conventional spacetime metric plus B-field. For
this reason we will focus in later sections on solutions with
$\mathcal{Z}\neq-{\bf 1}$.

We now turn to the decomposition of the double metric into a generalized
metric that encodes conventional gravity fields plus extra fields that will be
integrated out perturbatively[33]. We write

$\mathcal{Z}=S+F\,,$ (3.9)

where $S\in O(d,d)$ is the internal generalized scalar metric satisfying
$S^{2}=1$. A generalized metric can be parametrized in terms of symmetric and
antisymmetric tensors $g$ and $b$, respectively, as follows

$S\equiv\begin{pmatrix}bg^{-1}&g-bg^{-1}b\\\ g^{-1}&-g^{-1}b\end{pmatrix}.$
(3.10)

The matrix $F$ encodes the extra fields and can be assumed to obey some
constraints, which we now describe. We first recall that with the generalized
metric one can build projectors

$P\equiv\frac{1}{2}\left(1-S\right)\,,\quad\quad\bar{P}\equiv\frac{1}{2}\left(1+S\right)\,,$
(3.11)

satisfying

$P^{2}=P\,,\quad\quad\bar{P}^{2}=\bar{P}\,,\quad\quad P\bar{P}=\bar{P}P=0\,.$
(3.12)

Furthermore, we have the following useful identities

$PS=SP=-P\,,\quad\quad\bar{P}S=S\bar{P}=\bar{P}\,.$ (3.13)

A generic matrix $A=A_{\bullet}{}^{\bullet}$ then can be projected into $\pm$
components, defined as

$A_{+}\equiv PAP+\bar{P}A\bar{P}\,,\quad\quad A_{-}\equiv
PA\bar{P}+\bar{P}AP\,,$ (3.14)

such that

$A=A_{+}+A_{-}\,.$ (3.15)

We also have the relations

$A_{\pm}=\frac{1}{2}\left(A\pm SAS\right)\,,\quad\quad A_{\pm}S=\pm
SA_{\pm}\,.$ (3.16)

Such relations are useful in order to show that certain traces vanish:

${\rm Tr}{A_{-}}={\rm Tr}{A_{-}S^{2}}=-{\rm Tr}{SA_{-}S}=-{\rm
Tr}{A_{-}S^{2}}=-{\rm Tr}{A_{-}}=0\,,$ (3.17)

where we used $S^{2}=1$, (3.16) and the cyclicity of the trace. Thus, traces
of minus-projected tensors vanish.

While the decomposition (3.9) is totally generic, in perturbation theory we
think of $F$ as being of one order in $\alpha^{\prime}$ higher than $S$. This
is motivated by the fact that, in this case, the leading order contribution to
(3.1) does not depend on $F$ and it corresponds to the standard Neveu-Schwarz
sector of supergravity in cosmological backgrounds. We can then assume $F$ to
be a constrained field belonging to the $+$ subspace, i.e. $F=F_{+}$, because
any part in $F$ belonging to the $-$ subspace can be removed by a field
redefinition $S\rightarrow S+\delta S$ since $\delta S=[\delta S]_{-}$ as
follows by taking the variation of $S^{2}=1$. Thus, without loss of
generality, in perturbation theory we can write

$\mathcal{Z}=S+F_{+}\;.$ (3.18)

Let us now inspect the equation of motion (3.3a) after using this
decomposition of $\mathcal{Z}$. Inserting (3.18) into (3.3a) one obtains

$SF_{+}=\frac{\alpha^{\prime}}{4}\Box_{\Phi}(S+F_{+})-\frac{1}{2}F_{+}^{2}\,,$
(3.19)

where here and in the following it is convenient to introduce the linear
operator

$\Box_{\Phi}\equiv{\cal D}^{2}-{\cal D}\Phi{\cal D}\,.$ (3.20)

From this we can obtain the equations for $F_{+}$ or $S$ by projecting into
$\pm$ subspaces. For the generalized metric we project to the $-$ subspace,
and use $[F_{+}]_{-}=[F_{+}^{2}]_{-}=0$. This yields

$\left[\Box_{\Phi}(S+F_{+})\right]_{-}=0\;.$ (3.21)

For the extra fields we take the $+$ projection:

$F_{+}=\frac{\alpha^{\prime}}{4}[\Box_{\Phi}S]_{+}S+\frac{\alpha^{\prime}}{4}[\Box_{\Phi}F_{+}]_{+}S-\frac{1}{2}F_{+}^{2}S\;.$
(3.22)

Since this is the equation of motion for $F_{+}$, we can solve for it
perturbatively in $\alpha^{\prime}$ following the iterative procedure
described in the following section.

###  3.2 Integrating out the extra fields

Equation (3.22) is the starting point for integrating out the extra fields
$F_{+}$. We assume a perturbative expansion in $\alpha^{\prime}$, namely

$F_{+}=\sum_{i\geq 1}\alpha^{\prime}{}^{i}F_{+}^{(i)}\;.$ (3.23)

Inserting this expansion into (3.22) we obtain the recursive relations

$\displaystyle F_{+}^{(1)}$
$\displaystyle=\frac{1}{4}[\Box_{\Phi}S]_{+}S=-\frac{1}{4}({\cal D}S)^{2}\,,$
(3.24) $\displaystyle F_{+}^{(i)}$
$\displaystyle=\frac{1}{4}[\Box_{\Phi}F_{+}^{(i-1)}]_{+}S-\frac{1}{2}\sum_{j=1}^{i-1}F_{+}^{(j)}F_{+}^{(i-j)}S\,,\quad\quad
i\geq 2\,.$

By solving these equations recursively, we can express all $F_{+}^{(i)}$ in
terms of $S$, $\Phi$ and $n$. Plugging these expressions back into the action
(3.1) yields the effective action for the conventional fields.

We perform this computation explicitly to order $\alpha^{\prime 4}$ by first
inserting the decomposition $\mathcal{Z}=S+F_{+}$ into the action (3.1), which
then splits as

$I=I^{(0)}+\alpha^{\prime}I^{(1)}+\alpha^{\prime}{}^{2}I^{(2)}+\alpha^{\prime}{}^{3}I^{(3)}+\alpha^{\prime}{}^{4}I^{(4)}+\mathcal{O}(\alpha^{\prime}{}^{5})\,,$
(3.25)

with

$\displaystyle I^{(0)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{8}{\rm Tr}{({\cal D}S)^{2}}-({\cal
D}\Phi)^{2}\right\\}\,,$ (3.26) $\displaystyle I^{(1)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{2}{\rm
Tr}{S(F_{+}^{(1)})^{2}}-\frac{1}{4}{\rm Tr}{{\cal D}S{\cal
D}F_{+}^{(1)}}\right\\}\,,$ $\displaystyle I^{(2)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{6}{\rm
Tr}{(F_{+}^{(1)})^{3}}-\frac{1}{8}{\rm Tr}{({\cal
D}F_{+}^{(1)})^{2}}+\frac{1}{16}({\cal D}^{3}\Phi)^{2}+\frac{1}{12}({\cal
D}^{2}\Phi)^{3}\right.$ $\displaystyle\left.\hskip 68.28644pt-{\rm
Tr}{SF_{+}^{(1)}F_{+}^{(2)}}-\frac{1}{4}{\rm Tr}{{\cal D}S{\cal
D}F_{+}^{(2)}}\right\\}\,,$ $\displaystyle I^{(3)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{2}{\rm
Tr}{S(F_{+}^{(2)})^{2}}-\frac{1}{2}{\rm
Tr}{F^{(2)}_{+}(F_{+}^{(1)})^{2}}-\frac{1}{4}{\rm Tr}{{\cal D}F_{+}^{(1)}{\cal
D}F_{+}^{(2)}}\right.$ $\displaystyle\left.\hskip 68.28644pt-{\rm
Tr}{SF_{+}^{(1)}F_{+}^{(3)}}-\frac{1}{4}{\rm Tr}{{\cal D}S{\cal
D}F_{+}^{(3)}}\right\\}\,,$ $\displaystyle I^{(4)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{2}{\rm
Tr}{F_{+}^{(1)}(F_{+}^{(2)})^{2}}-\frac{1}{8}{\rm Tr}{({\cal
D}F_{+}^{(2)})^{2}}\right.$ $\displaystyle\hskip 68.28644pt-{\rm
Tr}{SF_{+}^{(1)}F_{+}^{(4)}}-\frac{1}{4}{\rm Tr}{{\cal D}S{\cal
D}F_{+}^{(4)}}$ $\displaystyle\left.\hskip 68.28644pt-{\rm
Tr}{SF_{+}^{(2)}F_{+}^{(3)}}-\frac{1}{2}{\rm
Tr}{(F_{+}^{(1)})^{2}F_{+}^{(3)}}-\frac{1}{4}{\rm Tr}{{\cal D}F_{+}^{(1)}{\cal
D}F_{+}^{(3)}}\right\\}\,.$

With these expressions, naively it seems that we need to solve the recursive
equation up to quartic order to get the final expression of the action in
terms of $S$. However, now we proceed to show that (3.26) can be simplified
and in particular we can remove any appearance of $F_{+}^{(3)}$ and
$F_{+}^{(4)}$. We describe in detail how to simplify $I^{(1)}$, since the
exact same simplifications can be performed in the higher orders. We find

$\displaystyle I^{(1)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{2}{\rm
Tr}{S(F_{+}^{(1)})^{2}}-\frac{1}{4}{\rm Tr}{{\cal D}S{\cal
D}F_{+}^{(1)}}\right\\}$ (3.27) $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{2}{\rm
Tr}{S(F_{+}^{(1)})^{2}}+\frac{1}{4}{\rm Tr}{\Box_{\Phi}SF_{+}^{(1)}}\right\\}$
$\displaystyle=\int dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{2}{\rm
Tr}{S(F_{+}^{(1)})^{2}}+\frac{1}{4}{\rm
Tr}{[\Box_{\Phi}S]_{+}F_{+}^{(1)}}\right\\}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,{\rm
Tr}{\left[-\frac{1}{2}SF_{+}^{(1)}+\frac{1}{4}[\Box_{\Phi}S]_{+}\right]F_{+}^{(1)}}$
$\displaystyle=\frac{1}{2}\int dt\,n\,e^{-\Phi}\,{\rm
Tr}{S(F_{+}^{(1)})^{2}}\,,$

where in the second equality we integrated by parts and used the definition
(3.20). From the second to third line we use that only the $+$ projection
survives inside the trace due to (3.17). For the final step one uses the
leading solution (3.24) for $F_{+}^{(1)}$. Following these steps in the higher
corrections leads to the following simplifications

$\displaystyle I^{(0)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{8}{\rm Tr}{({\cal D}S)^{2}}-({\cal
D}\Phi)^{2}\right\\}\,,$ (3.28) $\displaystyle I^{(1)}$
$\displaystyle=\frac{1}{2}\int dt\,n\,e^{-\Phi}\,{\rm
Tr}{S(F_{+}^{(1)})^{2}}\,,$ $\displaystyle I^{(2)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{6}{\rm
Tr}{(F_{+}^{(1)})^{3}}-\frac{1}{8}{\rm Tr}{({\cal
D}F_{+}^{(1)})^{2}}+\frac{1}{16}({\cal D}^{3}\Phi)^{2}+\frac{1}{12}({\cal
D}^{2}\Phi)^{3}\right\\}\,,$ $\displaystyle I^{(3)}$
$\displaystyle=\frac{1}{2}\int dt\,n\,e^{-\Phi}\,{\rm
Tr}{S(F_{+}^{(2)})^{2}}\,,$ $\displaystyle I^{(4)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{2}{\rm
Tr}{F_{+}^{(1)}(F_{+}^{(2)})^{2}}-\frac{1}{8}{\rm Tr}{({\cal
D}F_{+}^{(2)})^{2}}\right\\}\,.$

This rewriting tells us that only $F_{+}^{(1)}$ and $F_{+}^{(2)}$ are needed,
which are given by (3.24) for $i=1,2$,

$\displaystyle F_{+}^{(1)}$ $\displaystyle=-\frac{1}{4}({\cal D}S)^{2}\,,$
(3.29) $\displaystyle F_{+}^{(2)}$
$\displaystyle=-\frac{1}{16}[\Box_{\Phi}({\cal D}S{\cal
D}S)]_{+}S-\frac{1}{32}({\cal D}S)^{4}S\,.$

In order to evaluate the action (3.28) for these solutions we found it
convenient to write the result in terms of

$\mathcal{L}\equiv{\cal D}SS\,,$ (3.30)

which satisfies

$\mathcal{L}=[\mathcal{L}]_{-}\ ,\ \ \quad\quad{\cal D}\mathcal{L}=[{\cal
D}\mathcal{L}]_{-}\,.$ (3.31)

In particular, we will see that the zeroth order equations for $S$ take a
simpler form when written in terms of $\mathcal{L}$. The lowest order
solutions for the extra fields then read

$\displaystyle F^{(1)}_{+}$ $\displaystyle=\frac{1}{4}\mathcal{L}^{2}\,,$
(3.32) $\displaystyle F^{(2)}_{+}$
$\displaystyle=-\frac{1}{32}\mathcal{L}^{4}S+\frac{1}{16}[\Box_{\Phi}(\mathcal{L}^{2})]_{+}S\,.$

Inserting these expressions into (3.28) one obtains an action of the form
(3.25) with

$\displaystyle I^{(0)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{\frac{1}{8}{\rm Tr}{\mathcal{L}^{2}}-({\cal
D}\Phi)^{2}\right\\}\,,$ (3.33) $\displaystyle I^{(1)}$ $\displaystyle=0\,,$
$\displaystyle I^{(2)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{3.2^{7}}{\rm
Tr}{\mathcal{L}^{6}}-\frac{1}{2^{7}}{\rm Tr}{{\cal D}(\mathcal{L}^{2}){\cal
D}(\mathcal{L}^{2})}+\frac{1}{16}({\cal D}^{3}\Phi)^{2}+\frac{1}{12}({\cal
D}^{2}\Phi)^{3}\right\\}\,,$ $\displaystyle I^{(3)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\mathbb{L}_{3}({\cal D}\mathcal{L},{\cal D}\Phi,{\rm
Tr}{\mathcal{L}^{2}})\,,$ $\displaystyle I^{(4)}$ $\displaystyle=\int
dt\,n\,e^{-\Phi}\,\mathbb{L}_{4}({\cal D}\mathcal{L},{\cal D}\Phi,{\rm
Tr}{\mathcal{L}^{2}})\,,$

where we used

${\rm Tr}{S\mathcal{L}^{n}}=0\,,\quad n\in\mathbb{N}_{0}\,,$ (3.34)

which follows by anticommuting $S$ with $\mathcal{L}$ and using the cyclicity
of the trace. The notation $\mathbb{L}_{i}({\cal D}\mathcal{L},{\cal
D}\Phi,{\rm Tr}{\mathcal{L}^{2}})$ refers to functions of the given arguments,
whose specific form is irrelevant for our purposes. As discussed in previous
papers [23, 15], terms containing derivatives of $\mathcal{L}$, dilaton terms
or ${\rm Tr}{\mathcal{L}^{2}}$ can be removed by field redefinitions at the
expense of introducing higher order terms. We then call such terms
“removable”. The process of implementing field redefinitions is cumbersome and
has been discussed previously [15], so we move the general procedure to the
Appendix. Here we just summarize our findings. A redefinition of order $4$
eliminates $I^{(4)}$ because it is purely removable. Of course it might induce
higher order terms, that we are neglecting here. A redefinition of order $3$
eliminates $I^{(3)}$ for the same reason. This could reintroduce new terms in
$I^{(4)}$, but it does not because there is no $I^{(1)}$ and so the first
order correction to the equations of motion vanishes. So we can then start by
eliminating $I^{(3)}$ and $I^{(4)}$ completely to order $\alpha^{\prime 4}$.
When it comes to removing the last three terms in $I^{(2)}$, the second order
equations of motion reintroduce terms in $I^{(4)}$, some which are removable
at no cost, plus an extra term that defines the new coefficients found in this
paper: $c_{5,0}$ and $c_{5,1}$. The final result of this systematic procedure
is the HSZ action in the cosmological classification

$I=\int dt\,n\,e^{-\Phi}\,\left\\{\frac{1}{8}{\rm Tr}{\mathcal{L}^{2}}-({\cal
D}\Phi)^{2}-\alpha^{\prime}{}^{2}\frac{1}{3\cdot 2^{7}}{\rm
Tr}{\mathcal{L}^{6}}+\alpha^{\prime}{}^{4}\frac{1}{3\cdot 2^{13}}{\rm
Tr}{\mathcal{L}^{4}}{\rm
Tr}{\mathcal{L}^{6}}+\mathcal{O}(\alpha^{\prime}{}^{5})\right\\}\,.$ (3.35)

By coming back to the original basis in terms of ${\cal D}S$ and the
classification given in [23], according to which the action must take the form

$\begin{split}I=\int dt\,e^{-\Phi}\Big{[}&-\dot{\Phi}^{2}+c_{1,0}\,{\rm
Tr}{({\cal D}S)^{2}}+\alpha^{\prime}\,c_{2,0}\,{\rm Tr}{({\cal
D}S)^{4}}+\alpha^{\prime}{}^{2}\,c_{3,0}\,{\rm Tr}{({\cal D}S)^{6}}\\\
&+\alpha^{\prime}{}^{3}\left(c_{4,0}\,{\rm Tr}{({\cal D}S)^{8}}+c_{4,1}\,({\rm
Tr}{({\cal D}S)^{4}})^{2}\right)\\\ &+\alpha^{\prime}{}^{4}\left(c_{5,0}\,{\rm
Tr}{({\cal D}S)^{10}}+c_{5,1}\,{\rm Tr}{({\cal D}S)^{6}}\,{\rm Tr}{({\cal
D}S)^{4}}\right)\vphantom{\frac{1}{8}}\Big{]}\ ,\end{split}$ (3.36)

we obtain by comparison of coefficients

$c_{1,0}$ | $c_{2,0}$ | $c_{3,0}$ | $c_{4,0}$ | $c_{4,1}$ | $c_{5,0}$ | $c_{5,1}$
---|---|---|---|---|---|---
$-\frac{1}{8}$ | $0$ | $\frac{1}{3\cdot 2^{7}}$ | $0$ | $0$ | $0$ | $-\frac{1}{3\cdot 2^{13}}$

Up to and including $c_{4}$ the coefficients coincide with the ones obtained
in [15], whereas $c_{5,0}$ and $c_{5,1}$ are new results.

##  4 Two-derivative reformulation

In [24], HSZ theory was considered up to quadratic order in field
perturbations about flat space. In this limit the higher-derivative terms can
be removed by introducing certain auxiliary fields. One can then take the
tensionless limit $\alpha^{\prime}\rightarrow\infty$ smoothly, for which one
finds an enhanced gauge symmetry whose corresponding gauge fields are the
auxiliary fields. The objective of this section is two-fold: First, we will
show that in the cosmological setting the auxiliary fields introduced in [24]
are not required in order to remove higher orders in $\alpha^{\prime}$, since
the already present $A$ and $B$ can be used to accomplish this. Second, for
completeness, we show that the auxiliary fields are nevertheless needed in
order to make contact with the formulation of [24], but we will see that for
the cosmological setting the enhanced gauge symmetries can be trivialized in
the sense that the fields can be rearranged into gauge invariant variables.

###  4.1 Two-derivative theory

The action (2.24) contains only two terms at higher orders in
$\alpha^{\prime}$ given by the terms in the second line. Remarkably, both
terms can be removed by exact field redefinitions of $A$ and $B$. Explicitly,
by performing the transformations

$\displaystyle B$
$\displaystyle=B^{\prime}-\frac{2}{3}A^{\prime}{}^{2}+\alpha^{\prime}\left[\frac{1}{4}{\cal
D}A^{\prime}{\cal D}\Phi-\frac{1}{4}{\cal
D}^{2}A^{\prime}+\frac{1}{3}A^{\prime}{\cal D}^{2}\Phi\right]$ (4.1)
$\displaystyle\hskip
62.59596pt-\frac{\alpha^{\prime}{}^{2}}{2}\left[-\frac{1}{4}{\cal
D}^{4}\Phi+\frac{1}{4}{\cal D}\Phi{\cal D}^{3}\Phi+\frac{1}{3}({\cal
D}^{2}\Phi)^{2}\right]\,,$ $\displaystyle A$
$\displaystyle=A^{\prime}+\frac{\alpha^{\prime}}{2}{\cal D}^{2}\Phi\,,$

the action (2.24) can be brought to the following two-derivative form
(omitting the primes)

$\begin{split}I=\int
dt\,n\,e^{-\Phi}\,\Big{\\{}&\frac{1}{\alpha^{\prime}}\Big{[}AB-\frac{2}{3}A^{3}+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\Big{]}\\\
&-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}+\frac{1}{4}({\cal D}A)^{2}+\frac{1}{2}B{\cal
D}^{2}\Phi\Big{\\}}\,.\end{split}$ (4.2)

In this formulation, all higher orders in $\alpha^{\prime}$ are ‘hidden’ in
the on-shell value of $A$ and $B$. Indeed, we note that $B$ enters in the
action just linearly, so it plays the role of a Lagrange multiplier, which
imposes a condition on the scalar field $A$:

$\frac{\delta I}{\delta B}=0\quad\Rightarrow\quad
A=-\frac{\alpha^{\prime}}{2}{\cal D}^{2}\Phi\,.$ (4.3)

Reinserting this value of $A$ into the action (4.2), the $A^{3}$ and $({\cal
D}A)^{2}$ couplings are replaced by the $\alpha^{\prime}{}^{2}$ terms $({\cal
D}^{2}\Phi)^{3}$ and $({\cal D}^{3}\Phi)^{2}$, respectively. The resulting
action is exactly the effective action for $\mathcal{Z},n$ and $\Phi$ obtained
from (2.24) after fixing $A=B=0$.

In order to relate to the results in [24] it is instructive to go to a more
democratic formulation in terms of two ‘massive’ scalar fields $A^{+}$ and
$A^{-}$. To do so, we first integrate by parts the term $B{\cal D}^{2}\Phi$
and perform the following field redefinitions:

$\displaystyle A$ $\displaystyle=-(A^{+}+A^{-})\,,\qquad B=2(A^{+}-A^{-})\,,$
(4.4)

to arrive at

$\begin{split}I=\int
dt\,n\,e^{-\Phi}\,\Big{\\{}&\frac{1}{\alpha^{\prime}}\Big{[}-2(A^{+})^{2}+2(A^{-})^{2}+\frac{2}{3}(A^{+}+A^{-})^{3}+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\Big{]}\\\
&-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}+\frac{1}{4}({\cal D}A^{+})^{2}+\frac{1}{4}({\cal D}A^{-})^{2}\\\
&+\frac{1}{2}{\cal D}A^{+}{\cal D}A^{-}-{\cal D}(A^{+}-A^{-}){\cal
D}\Phi+(A^{+}-A^{-})({\cal D}\Phi)^{2}\Big{\\}}\,.\end{split}$ (4.5)

Finally we remove the mixed terms with a field redefinition of the dilaton,

$\Phi=\Phi^{\prime}-\frac{1}{2}(A^{+}-A^{-})\,.$ (4.6)

Omitting the primes for the transformed dilaton, the final action reads

$\begin{split}I=\int
dt\,n\,e^{-\Phi+\frac{1}{2}(A^{+}-A^{-})}\,&\left\\{\frac{1}{\alpha^{\prime}}\left[\frac{2}{3}(A^{+}+A^{-})^{3}+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\right]\right.\\\
&-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal D}\Phi)^{2}\\\
&+\frac{1}{2}({\cal
D}A^{+})^{2}-\frac{2}{\alpha^{\prime}}(A^{+})^{2}+\frac{1}{2}({\cal
D}A^{-})^{2}+\frac{2}{\alpha^{\prime}}(A^{-})^{2}\\\
&\left.+(A^{+}-A^{-})\left[{\cal
D}(\Phi-\frac{1}{2}(A^{+}-A^{-}))\right]^{2}\right\\}\,.\end{split}$ (4.7)

Here we recognize from the quadratic terms in the third line that $A^{\pm}$
correspond to two ‘massive’ scalar fields with masses
$(M^{\pm})^{2}=\mp\frac{4}{\alpha^{\prime}}$, respectively.

Once we have the action in a form with no positive $\alpha^{\prime}$ powers we
can take the tensionless limit $\alpha^{\prime}\rightarrow\infty$. Then all
$\frac{1}{\alpha^{\prime}}$ terms drop from the action, in particular the mass
terms for $A^{\pm}$. The resulting action is given by

$\displaystyle I_{\infty}=\int dt\,n\,e^{-\Phi+\frac{1}{2}(A^{+}-A^{-})}$
$\displaystyle\left\\{-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}+\frac{1}{2}({\cal D}A^{+})^{2}+\frac{1}{2}({\cal
D}A^{-})^{2}\right.$ (4.8) $\displaystyle\left.+\,(A^{+}-A^{-})\left[{\cal
D}(\Phi-\frac{1}{2}(A^{+}-A^{-}))\right]^{2}\right\\}\,.$

In contrast to the enhanced gauge symmetries uncovered in [24], here the
tensionless limit does not appear to exhibit any additional gauge symmetries.
However, it does have a symmetry in the form of a constant shift
$A^{\pm}\rightarrow a=$ const. because, apart from the kinetic term, $A^{\pm}$
appear in the combination $A^{+}-A^{-}$, in contrast to the full theory (4.7)
with finite $\alpha^{\prime}$. We will show in the next section that upon
introducing two extra auxiliary fields, motivated by [24], we can display the
expected enhanced gauge symmetries, albeit in a somewhat trivial fashion.

###  4.2 Tensionless limit

In the previous section we have seen that introducing additional auxiliary
fields is not required in order to remove higher orders in $\alpha^{\prime}$.
Nevertheless, here we will introduce two extra fields with the goal of
identifying the enhanced symmetry found in [24], after taking the tensionless
limit.

Returning to (2.24) let us ‘integrate-in’ two auxiliary fields $\varphi$ and
$\bar{\varphi}$ in order to remove the terms of order $\alpha^{\prime}{}^{2}$,
mimicking the procedure in [24]:

$\displaystyle\int dt\,n\,e^{-\Phi}$
$\displaystyle\left\\{\frac{1}{\alpha^{\prime}}\left[AB-2\varphi^{2}+2\bar{\varphi}^{2}+\frac{2}{3}(\varphi+\bar{\varphi})^{3}+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\right]\right.$ (4.9)
$\displaystyle\left.-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}+\frac{1}{4}({\cal D}\varphi)^{2}+\frac{1}{4}({\cal
D}\bar{\varphi})^{2}+\frac{1}{2}{\cal D}\varphi{\cal D}\bar{\varphi}\right.$
$\displaystyle\left.-{\cal D}(\varphi-\bar{\varphi}){\cal
D}\Phi+(\varphi-\bar{\varphi})({\cal D}\Phi)^{2}\right\\}\,.$

In order to show that this action is equivalent to (2.24) we compute the
equations of motion for $\varphi$ and $\bar{\varphi}$, which yields

$\displaystyle\varphi$ $\displaystyle=\frac{\alpha^{\prime}}{4}{\cal
D}^{2}\Phi+\eta\,,\qquad\bar{\varphi}=\frac{\alpha^{\prime}}{4}{\cal
D}^{2}\Phi-\eta\,,$ (4.10) $\displaystyle\eta$
$\displaystyle\equiv\frac{1}{2}(\varphi+\bar{\varphi})^{2}+\frac{\alpha^{\prime}}{8}\left[{\cal
D}(\varphi+\bar{\varphi}){\cal D}\Phi-{\cal
D}^{2}(\varphi+\bar{\varphi})\right]\,.$

Upon reinsertion into (4.9), all $\eta$ contributions cancel, and the only
surviving terms are exactly the two dilaton contributions at order
$\alpha^{\prime}{}^{2}$ of (2.24). Another simpler way to see that this theory
is equivalent to the original one is by noticing that (4.9) is equal to (4.5)
after identifying $\varphi\rightarrow A^{+}$ and $\bar{\varphi}\rightarrow
A^{-}$ and setting $A=B=0$. Moreover, as we did in the previous section, we
can perform a dilaton field redefinition

$\Phi=\Phi^{\prime}-\frac{1}{2}(\varphi-\bar{\varphi})\,,$ (4.11)

to arrive at the analogue of (4.7), namely

$\displaystyle I=\int dt\,n\,e^{-\Phi+\frac{1}{2}(\varphi-\bar{\varphi})}$
$\displaystyle\left\\{\frac{1}{\alpha^{\prime}}\left[AB+\frac{2}{3}(\varphi+\bar{\varphi})^{3}+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\right]\right.$ (4.12)
$\displaystyle-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}$ $\displaystyle+\frac{1}{2}({\cal
D}\varphi)^{2}-\frac{2}{\alpha^{\prime}}\varphi^{2}+\frac{1}{2}({\cal
D}\bar{\varphi})^{2}+\frac{2}{\alpha^{\prime}}\bar{\varphi}^{2}$
$\displaystyle\left.+(\varphi-\bar{\varphi})\left[{\cal
D}(\Phi-\frac{1}{2}(\varphi-\bar{\varphi}))\right]^{2}\right\\}\,.$

In contrast to (4.7), here we still have $A$ and $B$, and we can perform the
following field redefinitions333Here $\bar{A}$ plays the role of a
$B^{\prime}$.

$\displaystyle A$
$\displaystyle=A^{\prime}+\bar{A}-(\varphi^{\prime}-\bar{\varphi}^{\prime})\,,\quad$
$\displaystyle
B=A^{\prime}-\bar{A}-(\varphi^{\prime}+\bar{\varphi}^{\prime})\,,$ (4.13)
$\displaystyle\varphi$
$\displaystyle=\varphi^{\prime}-\frac{1}{2}A^{\prime}\,,\quad$
$\displaystyle\bar{\varphi}=\bar{\varphi}^{\prime}+\frac{1}{2}\bar{A}\,,$

to arrive at the action

$\displaystyle I=\int
dt\,n\,e^{-\Phi+\frac{1}{2}(\varphi-\bar{\varphi})-\frac{1}{4}(A+\bar{A})}$
$\displaystyle\left\\{\frac{1}{\alpha^{\prime}}\left[\frac{2}{3}(\varphi+\bar{\varphi}-\frac{1}{2}(A-\bar{A}))^{3}+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\right]\right.$ (4.14)
$\displaystyle-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}$ $\displaystyle+\frac{1}{8}({\cal D}A)^{2}+\frac{1}{2}({\cal
D}\varphi)^{2}-\frac{1}{2}{\cal D}A{\cal
D}\varphi+\frac{1}{\alpha^{\prime}}\left[\frac{1}{2}A^{2}-\varphi^{2}\right]$
$\displaystyle+\frac{1}{8}({\cal D}\bar{A})^{2}+\frac{1}{2}({\cal
D}\bar{\varphi})^{2}-\frac{1}{2}{\cal D}\bar{A}{\cal
D}\bar{\varphi}-\frac{1}{\alpha^{\prime}}\left[\frac{1}{2}\bar{A}^{2}-\bar{\varphi}^{2}\right]$
$\displaystyle\left.+\left[\varphi-\bar{\varphi}-\frac{1}{2}(A+\bar{A})\right]\left[{\cal
D}(\Phi-\frac{1}{2}(\varphi-\bar{\varphi})+\frac{1}{4}(A+\bar{A}))\right]^{2}\right\\}\,,$

omitting again the primes of the transformed fields. One may verify that the
quadratic approximation for field perturbations around the classical solution
$A=\bar{A}=\varphi=\bar{\varphi}=0$ coincides with the one obtained in
equation (5.1) of [24], once reduced to a cosmological background, where the
components $a_{00},\bar{a}_{00},\varphi_{00}$ and $\bar{\varphi}_{00}$
correspond to the fluctuations of $A,\bar{A},\varphi$ and $\bar{\varphi}$,
respectively.

Given this match, we expect to recover the enhanced gauge symmetry in the
tensionless limit. To this end consider the following field redefinition of
the lapse function:

$n=n^{\prime}e^{-\Phi+\frac{1}{2}(\varphi-\bar{\varphi})-\frac{1}{4}(A+\bar{A})}\,,$
(4.15)

which removes completely the exponential factor for the two-derivative part of
the action, at the expense of modifying the measure of the
$\frac{1}{\alpha^{\prime}}$ terms. But the latter terms disappear in the limit
$\alpha^{\prime}\rightarrow\infty$, and we obtain

$\displaystyle I_{\infty}=\int dt\,n$ $\displaystyle\left\\{-\frac{1}{8}{\rm
Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal D}\Phi)^{2}\right.$ (4.16)
$\displaystyle+\frac{1}{8}({\cal D}A)^{2}+\frac{1}{2}({\cal
D}\varphi)^{2}-\frac{1}{2}{\cal D}A{\cal D}\varphi+\frac{1}{8}({\cal
D}\bar{A})^{2}+\frac{1}{2}({\cal D}\bar{\varphi})^{2}-\frac{1}{2}{\cal
D}\bar{A}{\cal D}\bar{\varphi}$
$\displaystyle\left.+\left[\varphi-\bar{\varphi}-\frac{1}{2}(A+\bar{A})\right]\left[{\cal
D}(\Phi-\frac{1}{2}(\varphi-\bar{\varphi})+\frac{1}{4}(A+\bar{A}))\right]^{2}\right\\}\,.$

This action is invariant under the Stückelberg gauge invariance, with two
independent gauge parameters $\chi$ and $\bar{\chi}$,

$\begin{split}\delta A&=2\chi\,,\quad\quad\delta\bar{A}=-2\bar{\chi}\,,\\\
\delta\varphi&=\chi\,,\qquad\;\;\delta\bar{\varphi}=\bar{\chi}\;.\end{split}$
(4.17)

Upon setting

$\chi=-{\cal D}\zeta\,,\quad\quad\bar{\chi}={\cal D}\bar{\zeta}\,,$ (4.18)

these transformations agree with (the cosmological reduction of) the gauge
symmetries found in [24], but we observe that the one-dimensional setting is
somewhat degenerate in that the gauge symmetries become mere Stückelberg
transformations. Consequently, one may immediately pass over to the gauge
invariant quantities

$A^{+}\equiv\varphi-\frac{1}{2}A\,,\quad\quad
A^{-}\equiv\bar{\varphi}+\frac{1}{2}\bar{A}\,,$ (4.19)

obeying $\delta A^{\pm}=0$, for which the action (4.16) reads

$\displaystyle I_{\infty}=\int dt\,n$ $\displaystyle\left\\{-\frac{1}{8}{\rm
Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal D}\Phi)^{2}+\frac{1}{2}({\cal
D}A^{+})^{2}+\frac{1}{2}({\cal D}A^{-})^{2}\right.$ (4.20)
$\displaystyle\left.+\,(A^{+}-A^{-})\left[{\cal
D}(\Phi-\frac{1}{2}(A^{+}-A^{-}))\right]^{2}\right\\}\,.$

This manifestly gauge invariant action agrees precisely with (4.8) obtained
above.

However, as to be expected, the symmetry (4.17) does not survive for the full
theory (4.14). This can be seen by trying to rewrite it in terms of the gauge
invariant quantities $A^{\pm}$, (4.19)

$\displaystyle I=\int dt\,n\,e^{-\Phi+\frac{1}{2}(A^{+}-A^{-})}$
$\displaystyle\left\\{\frac{1}{\alpha^{\prime}}\left[\frac{2}{3}(A^{+}+A^{-})^{3}+\frac{1}{2}{\rm
Tr}{\mathcal{Z}}-\frac{1}{6}{\rm Tr}{\mathcal{Z}^{3}}\right]\right.$ (4.21)
$\displaystyle-\frac{1}{8}{\rm Tr}{({\cal D}\mathcal{Z})^{2}}-({\cal
D}\Phi)^{2}$ $\displaystyle+\frac{1}{2}({\cal
D}A^{+})^{2}+\frac{1}{\alpha^{\prime}}\left[\varphi^{2}-4\varphi
A^{+}+2(A^{+})^{2}\right]$ $\displaystyle+\frac{1}{2}({\cal
D}A^{-})^{2}-\frac{1}{\alpha^{\prime}}\left[\bar{\varphi}^{2}-4\bar{\varphi}A^{-}+2(A^{-})^{2}\right]$
$\displaystyle\left.+(A^{+}-A^{-})\left[{\cal
D}(\Phi-\frac{1}{2}(A^{+}-A^{-})\right]^{2}\right\\}\,,$

and observing that the mass-like terms in the third and fourth line explicitly
depend on $\varphi$ and $\bar{\varphi}$. Curiously, however, (4.21) is
invariant under the non-linear gauge symmetry given by

$\delta\varphi=(\bar{\varphi}-2A^{-})\eta\,,\quad\delta\bar{\varphi}=(\varphi-2A^{+})\eta\,,\quad\delta
A^{(\pm)}=0\,,$ (4.22)

where $\eta(t)$ is an arbitrary gauge parameter.

We finish this analysis by making a final consistency check. We note that in
the form (4.21), $\varphi$ and $\bar{\varphi}$ can be integrated out directly
since their equations of motion are given by

$\varphi=2A^{+}\,,\quad\bar{\varphi}=2A^{-}\,,$ (4.23)

and reinserting this into the action yields precisely (4.7).

##  5 Friedmann Universe and tensionless limit

In this section we analyze the two-derivative equations for FRW backgrounds
with a single scale factor. To this end we bring the equations into the form
of string cosmology with ‘matter fields’, which here are the extra fields
$A,B$ and $F$. While the equations in general are hard to solve, it is
straightforward to give the general solutions in the tensionless limit
$\alpha^{\prime}\rightarrow\infty$, which include string frame de Sitter
vacua. We then outline perturbation theory in $\frac{1}{\alpha^{\prime}}$ and
show how these solutions get corrected.

###  5.1 $\alpha^{\prime}$-exact Friedmann equations

We begin by considering the 2-derivative form of the action (4.2) and
expanding $\mathcal{Z}=S+F$, which yields

$I=I^{(0)}+I_{m}\,,$ (5.1)

with

$\displaystyle I^{(0)}$ $\displaystyle\equiv\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{8}{\rm Tr}{({\cal D}S)^{2}}-({\cal
D}\Phi)^{2}\right\\}\,,$ (5.2) $\displaystyle I_{m}$ $\displaystyle\equiv\int
dt\,n\,e^{-\Phi}\,\left\\{\frac{1}{\alpha^{\prime}}\left[AB-\frac{2}{3}A^{3}-\frac{1}{2}{\rm
Tr}{SF^{2}}-\frac{1}{6}{\rm Tr}{F^{3}}\right]\right.$ $\displaystyle\hskip
71.13188pt\left.+\frac{1}{4}({\cal D}A)^{2}+\frac{1}{2}B{\cal
D}^{2}\Phi-\frac{1}{4}{\rm Tr}{{\cal D}S{\cal D}F}-\frac{1}{8}{\rm Tr}{({\cal
D}F)^{2}}\right\\}\,.$

In this split $I^{(0)}$ is the standard lowest order, two-derivative gravity
action, and $I_{m}$ encodes the matter content parameterized by $A,B$ and $F$.
It is important to point out that here we are keeping both projections of the
extra fields $F=F_{+}+F_{-}$, and we make no a priori assumptions on their
dependence on $\alpha^{\prime}$. This is different from the perturbative setup
of section 3.1. From $I^{(0)}$ one obtains the equations of motion (EOM) for
the massless fields, see equation (A.3), while the variation of the matter
action is conveniently encoded in the $O(d,d)$-covariant energy momentum
tensor, energy density and dilatonic charge[34, 35] as444All quantities are
defined with a $\sqrt{|g|}e^{\Phi}$ re-scaling compared to standard
definitions.

$\displaystyle\mathcal{T}_{M}{}^{N}$
$\displaystyle\equiv-2\frac{e^{\Phi}}{n}S_{M}{}^{P}\frac{\delta I_{m}}{\delta
S^{P}{}_{N}}\,,$ (5.3a) $\displaystyle\rho$
$\displaystyle\equiv-e^{\Phi}\frac{\delta I_{m}}{\delta n}\,,$ (5.3b)
$\displaystyle\sigma$ $\displaystyle\equiv-2\frac{e^{\Phi}}{n}\frac{\delta
I_{m}}{\delta\Phi}\,.$ (5.3c)

The EOM following from (5.1) for $S,\Phi$ and $n$ are then given by

$\displaystyle\left[\Box_{\Phi}S\right]_{-}S$
$\displaystyle=-2[\mathcal{T}]_{-}\,,$ (5.4a) $\displaystyle 2{\cal
D}^{2}\Phi-({\cal D}\Phi)^{2}+\frac{1}{8}{\rm Tr}{({\cal D}S)^{2}}$
$\displaystyle=\frac{1}{2}\sigma\,,$ (5.4b) $\displaystyle({\cal
D}\Phi)^{2}+\frac{1}{8}{\rm Tr}{({\cal D}S)^{2}}$ $\displaystyle=\rho\,,$
(5.4c)

with

$\displaystyle[\mathcal{T}]_{-}$
$\displaystyle=S\left[-\frac{1}{2}\Box_{\Phi}F+\frac{1}{\alpha^{\prime}}F^{2}\right]_{-}\,,$
(5.5a) $\displaystyle\rho$ $\displaystyle=\frac{1}{2\alpha^{\prime}}{\rm
Tr}{SF^{2}}+\frac{1}{6\alpha^{\prime}}{\rm Tr}{F^{3}}-\frac{1}{4}{\rm
Tr}{{\cal D}S{\cal D}F}-\frac{1}{8}{\rm Tr}{({\cal D}F)^{2}}$
$\displaystyle\quad-\frac{1}{\alpha^{\prime}}\left(AB-\frac{2}{3}A^{3}\right)+\frac{1}{4}({\cal
D}A)^{2}+\frac{1}{2}B({\cal D}\Phi)^{2}-\frac{1}{2}{\cal D}B{\cal D}\Phi\,,$
(5.5b) $\displaystyle\sigma$ $\displaystyle=-\frac{1}{\alpha^{\prime}}{\rm
Tr}{SF^{2}}-\frac{1}{3\alpha^{\prime}}{\rm Tr}{F^{3}}-\frac{1}{2}{\rm
Tr}{{\cal D}S{\cal D}F}-\frac{1}{4}{\rm Tr}{({\cal D}F)^{2}}$
$\displaystyle\quad+\frac{2}{\alpha^{\prime}}\left(AB-\frac{2}{3}A^{3}\right)+\frac{1}{2}({\cal
D}A)^{2}+2B{\cal D}^{2}\Phi-B({\cal D}\Phi)^{2}+2{\cal D}B{\cal D}\Phi-{\cal
D}^{2}B\,.$ (5.5c)

In addition, the EOM for the matter fields are given by

$\displaystyle 0$
$\displaystyle=\frac{1}{4}\Box_{\Phi}(S+F)-\frac{1}{2\alpha^{\prime}}\left(SF+FS\right)-\frac{1}{2\alpha^{\prime}}F^{2}\,,$
(5.6a) $\displaystyle A$ $\displaystyle=-\frac{\alpha^{\prime}}{2}{\cal
D}^{2}\Phi\,,$ (5.6b) $\displaystyle B$
$\displaystyle=2A^{2}+\frac{\alpha^{\prime}}{2}\Box_{\Phi}A=\alpha^{\prime}{}^{2}\left[-\frac{1}{4}{\cal
D}^{4}\Phi+\frac{1}{2}({\cal D}^{2}\Phi)^{2}+\frac{1}{4}{\cal D}\Phi{\cal
D}^{3}\Phi\right]\,,$ (5.6c)

where in the last equality of (5.6c) we used the on-shell value of $A$.
Indeed, we see here that $A$ and $B$ can be eliminated completely, but we find
it convenient to keep them in order to be able to work with second-order
equations.

We also note that reparametrization invariance implies the following Noether
identity or continuity equation:

${\cal D}\rho+\frac{1}{2}{\rm Tr}{S{\cal D}S\mathcal{T}}-{\cal
D}\Phi(\rho+\frac{1}{2}\sigma)=0\,.$ (5.7)

Let us now specify to FRW backgrounds, which are characterized by a single
scale factor $a(t)$, such that

$S=\begin{pmatrix}0&a^{2}\\\ a^{-2}&0\end{pmatrix}\,.$ (5.8)

It is also convenient to define the following quantities

$\mathcal{J}\equiv\begin{pmatrix}0&a^{2}\\\
-a^{-2}&0\end{pmatrix}\,,\qquad\mathcal{K}\equiv\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,,$ (5.9)

which, together with $S$, satisfy identities that will become useful below:

$\displaystyle\mathcal{J}^{2}$
$\displaystyle=-1\,,\qquad\mathcal{J}=[\mathcal{J}]_{-}\,,$ (5.10)
$\displaystyle\mathcal{K}^{2}$
$\displaystyle=1\,,\qquad\,\mathcal{J}S=\mathcal{K}\,,$ $\displaystyle{\rm
Tr}{S}$ $\displaystyle={\rm Tr}{\mathcal{K}}={\rm Tr}{\mathcal{J}}=0\,.$

Moreover, for the Hubble parameter defined as $H(t)\equiv\frac{{\cal D}a}{a}$
we have

${\cal D}S=2H\mathcal{J}\,,\quad{\cal D}\mathcal{J}=2HS\,.$ (5.11)

Using (5.10) and (5.11) we can write out the l.h.s of (5.4) for the FRW
background

$\displaystyle\left[\Box_{\Phi}S\right]_{-}S$ $\displaystyle\ \rightarrow\
2({\cal D}H-{\cal D}\Phi H)\mathcal{K}\,,$ (5.12) $\displaystyle 2{\cal
D}^{2}\Phi-({\cal D}\Phi)^{2}+\frac{1}{8}{\rm Tr}{({\cal D}S)^{2}}$
$\displaystyle\ \rightarrow 2\ {\cal D}^{2}\Phi-({\cal D}\Phi)^{2}-dH^{2}\,,$
$\displaystyle({\cal D}\Phi)^{2}+\frac{1}{8}{\rm Tr}{({\cal D}S)^{2}}$
$\displaystyle\ \rightarrow\ ({\cal D}\Phi)^{2}-dH^{2}\,.$

For the r.h.s of (5.4) we need to choose a parameterization for $F$. Instead
of considering the extra fields in full generality, however, we will truncate
the theory to the subsector $F_{-}=0$. Since now we are exploring the theory
also in non-perturbative regimes, the argument of section 3.1 to fix $F=F_{+}$
using field redefinitions does not hold anymore. There the theories with or
without $F_{-}$ were equivalent, but here we need to treat $F_{-}=0$ as a
genuine truncation of the theory. While we find that in general this is not a
consistent truncation of the full theory (5.2), it is a consistent truncation
for FRW backgrounds. To see this we split $F=F_{+}+F_{-}$ in (5.2), compute
the EOM for $S,F_{+}$ and $F_{-}$ separately and then set $F_{-}=0$:555When
computing the variation of the action w.r.t. $F_{\pm}$, one should remember
that they are constrained fields. As a consequence, part of their variations
are determined by $\delta S$ which induce extra terms for the EOM of $S$.

$\displaystyle\left.\frac{\delta I}{\delta S}\right|_{F_{-}=0}$
$\displaystyle=\frac{1}{8}S\left\\{(S+F_{+})\left[\Box_{\Phi}(S+F_{+})\right]_{-}-\left[\Box_{\Phi}(S+F_{+})\right]_{-}(S+F_{+})\right\\}=0\,,$
(5.13a) $\displaystyle\left.\frac{\delta I}{\delta F_{+}}\right|_{F_{-}=0}$
$\displaystyle=\frac{1}{4}\left[\Box_{\Phi}(S+F_{+})\right]_{+}-\frac{1}{\alpha^{\prime}}SF_{+}-\frac{1}{2\alpha^{\prime}}F_{+}^{2}=0\,,$
(5.13b) $\displaystyle\left.\frac{\delta I}{\delta F_{-}}\right|_{F_{-}=0}$
$\displaystyle=\frac{1}{4}\left[\Box_{\Phi}(S+F_{+})\right]_{-}=0\,.$ (5.13c)

Even though we set $F_{-}=0$ we still have to satisfy (5.13c), which is a
constraint on $S$ and $F_{+}$. While (5.13c) implies (5.13a), the reverse need
not to be true which means that $F_{-}=0$ is not a consistent truncation in
general. In other words, in principle there could be solutions to (5.13a) that
are not of the form (5.13c). For FRW, however, $F_{-}=0$ is a consistent
truncation because in this case one can show that both equations are
equivalent. Thus, from now on we consider $F_{-}=0$ and we proceed to
parameterize $F_{+}$. The most general ansatz for a +-projected
$O(d,d,\mathbb{R})$ tensor consistent with the FRW background (5.8) is given
in terms of two symmetric matrices $f_{1}{}_{m}{}^{n}(t)$ and
$f_{2}{}_{m}{}^{n}(t)$,

$F_{+}=\begin{pmatrix}f_{2}{}&f_{1}{}a^{2}\\\
a^{-2}f_{1}{}&f_{2}{}\end{pmatrix}\,,$ (5.14)

where the indices are raised and lowered with $\delta_{mn}$. However, here we
want to implement homogeneity and isotropy, as for standard cosmology, by
setting

$f_{1}{}_{mn}=f_{1}{}\delta_{mn}\,,\quad f_{2}{}_{mn}=f_{2}{}\delta_{mn}\,,$
(5.15)

with $f_{1}$, $f_{2}$ being functions of time. This can be seen to be a
consistent truncation of the most general ansatz.666To this end one considers
the generic ansatz until getting the final expressions for the EOM (5.4) and
(5.6) in terms of $f_{1}{}_{m}{}^{n}$ and $f_{2}{}_{m}{}^{n}$ and then
verifies that a diagonal ansatz is consistent.

In this simplified scheme the matter content is described by just four scalar
fields, $A,B,f_{1}{},$ and $f_{2}{}$. $F_{+}$ then takes the simple form

$F_{+}=\begin{pmatrix}f_{2}{}&0\\\
0&f_{2}{}\end{pmatrix}+\begin{pmatrix}0&f_{1}{}a^{2}\\\
a^{-2}f_{1}{}&0\end{pmatrix}=f_{1}{}S+f_{2}{}{\bf 1}\,,$ (5.16)

where ${\bf 1}$ denotes the unit matrix with components $\delta_{M}{}^{N}$. By
taking derivatives of (5.16) and using the definition for the generalized
energy momentum tensor (5.5a) with $F=F_{+}$ it follows

$\displaystyle\left[\mathcal{T}\right]_{-}=-\frac{1}{2}S\left[\Box_{\Phi}F_{+}\right]_{-}=\left[2H{\cal
D}f_{1}{}+\left({\cal D}H-{\cal D}\Phi H\right)f_{1}{}\right]\mathcal{K}\;,$
(5.17)

where we used the matrix defined in (5.9).

In order to give a physical interpretation to (5.17) notice that for FRW
backgrounds (5.3a) we have

$\mathcal{T}_{M}{}^{N}=-2\frac{e^{\Phi}}{n}S_{M}{}^{P}\frac{\delta
I_{m}}{\delta S^{P}{}_{N}}=\begin{pmatrix}-T_{m}{}^{n}&0\\\
0&T^{m}{}_{n}\end{pmatrix}\,,$ (5.18)

with

$T_{m}{}^{n}=p\,\delta_{m}{}^{n}\,,$ (5.19)

which describes a perfect fluid with pressure $p$. Comparison with (5.17) then
motivates us to view the extra fields of HSZ as describing an effective
perfect fluid with an effective pressure determined by $f_{1}$,

$\left[\mathcal{T}\right]_{-}=-p\,\mathcal{K}\,,\quad p=-2H{\cal
D}f_{1}{}-\left({\cal D}H-{\cal D}\Phi H\right)f_{1}{}\,.$ (5.20)

For the energy density and the dilatonic charge we just need to insert our
expressions for $S,\mathcal{L},F_{+}$ and their derivatives in terms of
$f_{1}{}$ and $f_{2}{}$ into the definition of $\rho$ and $\sigma$ in (5.5b)
and (5.5c). With the help of the identities (5.10) and (5.11) one obtains

$\displaystyle\rho$
$\displaystyle=\frac{d}{\alpha^{\prime}}\left(2f_{2}{}f_{1}{}+\frac{1}{3}f_{2}{}^{3}+f_{2}{}f_{1}{}^{2}\right)+dH^{2}f_{1}{}(f_{1}{}+2)-\frac{d}{4}\left(({\cal
D}f_{1}{})^{2}+({\cal D}f_{2}{})^{2}\right)$ (5.21)
$\displaystyle\quad-\frac{1}{\alpha^{\prime}}\left(AB-\frac{2}{3}A^{3}\right)+\frac{1}{4}({\cal
D}A)^{2}+\frac{1}{2}B({\cal D}\Phi)^{2}-\frac{1}{2}{\cal D}B{\cal D}\Phi\,,$
$\displaystyle\sigma$
$\displaystyle=-\frac{2d}{\alpha^{\prime}}\left(2f_{2}{}f_{1}{}+\frac{1}{3}f_{2}{}^{3}+f_{2}{}f_{1}{}^{2}\right)+2dH^{2}f_{1}{}(f_{1}{}+2)-\frac{d}{2}\left(({\cal
D}f_{1}{})^{2}+({\cal D}f_{2}{})^{2}\right)$
$\displaystyle\quad+\frac{2}{\alpha^{\prime}}\left(AB-\frac{2}{3}A^{3}\right)+\frac{1}{2}({\cal
D}A)^{2}+2B{\cal D}^{2}\Phi-B({\cal D}\Phi)^{2}+2{\cal D}B{\cal D}\Phi-{\cal
D}^{2}B\,.$

With (5.20) and (5.21) the right-hand sides of the equations of motion (5.4)
are completely determined.

Next, for the EOM of the extra fields we can consider just the $+$ projection
of (5.6a), because the minus projection is exactly the EOM of $S$ in (5.4a)
and so it vanishes on-shell. By taking $F_{-}=0$, the $+$ projection of (5.6a)
reduces to

$\displaystyle 0$
$\displaystyle=-\frac{\alpha^{\prime}}{4}\left[\Box_{\Phi}(S+F_{+})\right]_{+}+SF_{+}+\frac{1}{2}F_{+}^{2}$
(5.22)
$\displaystyle=\left[\frac{1}{2}f_{1}{}^{2}+\frac{1}{2}f_{2}{}^{2}+f_{1}{}-\frac{\alpha^{\prime}}{4}\Box_{\Phi}f_{2}{}\right]{\bf
1}$
$\displaystyle\quad+\left[(f_{2}{}-\alpha^{\prime}H^{2})(f_{1}{}+1)-\frac{\alpha^{\prime}}{4}\Box_{\Phi}f_{1}{}\right]S\,,$

which imply two inequivalent equations (one for each scalar field)

$\displaystyle\frac{1}{2}f_{1}{}^{2}+\frac{1}{2}f_{2}{}^{2}+f_{1}{}-\frac{\alpha^{\prime}}{4}\Box_{\Phi}f_{2}{}$
$\displaystyle=0\,,$ (5.23)
$\displaystyle(f_{2}{}-\alpha^{\prime}H^{2})(f_{1}{}+1)-\frac{\alpha^{\prime}}{4}\Box_{\Phi}f_{1}{}$
$\displaystyle=0\,.$

When combining (5.12), (5.20), (5.21) and (5.23), we end up with the following
non-linear system of coupled second-order differential equations for
$a,\Phi,n,A,B,f_{1}{}$ and $f_{2}{}$

$\displaystyle p$ $\displaystyle={\cal D}H-{\cal D}\Phi H\,,$ (5.24a)
$\displaystyle\frac{1}{2}(\frac{1}{2}\sigma+\rho)$ $\displaystyle={\cal
D}^{2}\Phi-dH^{2}\,,$ (5.24b) $\displaystyle\rho$ $\displaystyle=({\cal
D}\Phi)^{2}-dH^{2}\,,$ (5.24c) $\displaystyle 0$
$\displaystyle=(f_{2}{}-\alpha^{\prime}H^{2})(f_{1}{}+1)-\frac{\alpha^{\prime}}{4}\Box_{\Phi}f_{1}{}\,,$
(5.24d) $\displaystyle 0$
$\displaystyle=\frac{1}{2}f_{1}{}^{2}+\frac{1}{2}f_{2}{}^{2}+f_{1}{}-\frac{\alpha^{\prime}}{4}\Box_{\Phi}f_{2}{}\,,$
(5.24e) $\displaystyle A$ $\displaystyle=-\frac{\alpha^{\prime}}{2}{\cal
D}^{2}\Phi\,,$ (5.24f) $\displaystyle B$
$\displaystyle=2A^{2}+\frac{\alpha^{\prime}}{2}\Box_{\Phi}A\,,$ (5.24g)

with the ‘effective matter sources’

$\displaystyle p$ $\displaystyle=-2H{\cal D}f_{1}{}-\left({\cal D}H-{\cal
D}\Phi H\right)f_{1}{}\,,$ (5.25a) $\displaystyle\rho$
$\displaystyle=\frac{d}{\alpha^{\prime}}\left(2f_{1}{}+f_{1}{}^{2}+\frac{1}{3}f_{2}{}^{2}\right)f_{2}{}+dH^{2}f_{1}{}(f_{1}{}+2)-\frac{d}{4}\left(({\cal
D}f_{1}{})^{2}+({\cal D}f_{2}{})^{2}\right)$
$\displaystyle\quad-\frac{1}{\alpha^{\prime}}\left(AB-\frac{2}{3}A^{3}\right)+\frac{1}{4}({\cal
D}A)^{2}+\frac{1}{2}B({\cal D}\Phi)^{2}-\frac{1}{2}{\cal D}B{\cal D}\Phi\,,$
(5.25b) $\displaystyle\frac{1}{2}(\frac{1}{2}\sigma+\rho)$
$\displaystyle=dH^{2}f_{1}{}(f_{1}{}+2)-\frac{d}{4}\left(({\cal
D}f_{1}{})^{2}+({\cal D}f_{2}{})^{2}\right)$
$\displaystyle\quad+\frac{1}{4}({\cal D}A)^{2}+\frac{1}{2}B{\cal
D}^{2}\Phi+\frac{1}{4}{\cal D}B{\cal D}\Phi-\frac{1}{4}{\cal D}^{2}B\,.$
(5.25c)

It is instructive to check that these equations are invariant under duality
transformations, which in the case of FRW backgrounds reduce to simple full
factorized T-dualities

$a\rightarrow a^{-1}\quad\Rightarrow\quad H\rightarrow-H\,,$ (5.26)

while the rest of the fields, including $f_{1}$ and $f_{2}$, behave as
scalars. It is also worth performing some consistency checks before
proceeding: the first one comes from the observation that, if
$f_{1}=\mathcal{O}(\alpha^{\prime})$ and $f_{2}=\mathcal{O}(\alpha^{\prime})$,
the whole system (5.24) reduces to the standard Friedmann equations in vacuum
upon neglecting higher orders in $\alpha^{\prime}$:

$\displaystyle{\cal D}H-{\cal D}\Phi H$
$\displaystyle=\mathcal{O}(\alpha^{\prime})\,,$ (5.27a) $\displaystyle{\cal
D}^{2}\Phi-dH^{2}$ $\displaystyle=\mathcal{O}(\alpha^{\prime})\,,$ (5.27b)
$\displaystyle({\cal D}\Phi)^{2}-dH^{2}$
$\displaystyle=\mathcal{O}(\alpha^{\prime})\,.$ (5.27c)

As a second consistency check we corroborated that the continuity equation
(5.7), which for FRW backgrounds reads

${\cal D}\rho+2dHp-{\cal D}\Phi(\rho+\frac{1}{2}\sigma)=0\,,$ (5.28)

is indeed satisfied for the quantities given in (5.25).

Remarkably, the above system represents a non-perturbative and
$\alpha^{\prime}$-complete set of equations for a consistent truncation of a
theory sharing many features of genuine string theory. Unfortunately, finding
analytic solutions of (5.24) seems to be difficult, but as an example of a
rather degenerate solution, we can check the one found in (3.8). Specifically,
using the decomposition $\mathcal{Z}=S+F_{+}$ and $F_{+}=f_{1}S+f_{2}{\bf 1}$,
the solution $\mathcal{Z}=-{\bf 1}$ corresponds to

$f_{1}(t)=f_{2}(t)=-1\,,$ (5.29)

where $H(t)$ can be arbitrary. While this latter point may be hidden in the
way the equations (5.24) are written, one can see that with (5.29) all
contributions related to the scale factor just disappear. In this case (5.24d)
and (5.24e) are automatically satisfied and (5.25) take the form

$\displaystyle p$ $\displaystyle={\cal D}H-{\cal D}\Phi H\,,$ (5.30a)
$\displaystyle\rho$
$\displaystyle=\frac{2d}{3\alpha^{\prime}}-dH^{2}-\frac{1}{\alpha^{\prime}}\left(AB-\frac{2}{3}A^{3}\right)+\frac{1}{4}({\cal
D}A)^{2}+\frac{1}{2}B({\cal D}\Phi)^{2}-\frac{1}{2}{\cal D}B{\cal D}\Phi\,,$
(5.30b) $\displaystyle\frac{1}{2}(\frac{1}{2}\sigma+\rho)$
$\displaystyle=-dH^{2}+\frac{1}{4}({\cal D}A)^{2}+\frac{1}{2}B{\cal
D}^{2}\Phi+\frac{1}{4}{\cal D}B{\cal D}\Phi-\frac{1}{4}{\cal D}^{2}B\,.$
(5.30c)

By inserting these results into (5.24a), (5.24b) and (5.24c) it is easy to see
that all Hubble parameters cancel out. Finally, by picking the same dilaton
value as in (3.8),

$\Phi(t)=\sqrt{\frac{2d}{3\alpha^{\prime}}}\,t+\Phi_{0}\,,$ (5.31)

with $\Phi_{0}$ constant, $A=B=0$, the rest of the equations in (5.24) are
trivially satisfied.

Apart from this simple case, looking for exact and more complex solutions of
the system (5.24) is a complicated task and one should look for
simplifications. One option would be to restrict to backgrounds whose matter
content describes a barotropic fluid $p=w\rho$ and/or no dilatonic charge,
$\sigma=0$. A second option is to study particular configurations for the
dilaton and Hubble parameter and to ask if there exist any configuration of
the extra fields $A(t),B(t),f_{1}(t)$ and $f_{2}(t)$ such that the equations
are satisfied. Another possibility is to study the system perturbatively. In
the remainder of this section we will focus on the last two approaches. For
the perturbative computation in particular we will not consider the typical
low-energy case of small $\alpha^{\prime}$, but rather perform an expansion in
$\frac{1}{\alpha^{\prime}}$ around the tensionless limit
$\alpha^{\prime}\rightarrow\infty$.

In order to study these two particular paths, we first make some assumptions
on the fields and rewrite the system in a more convenient way. We begin by
ruling out the somewhat degenerate branch of solutions by demanding
$f_{1}\neq-1$. On top of that we also exclude the flat Minkowski background by
demanding $H\neq 0$. From now on we will gauge fix the lapse to $n(t)=1$ and
adopt the following notation for the extra fields

$x\equiv 1+f_{1}\neq 0\,,\quad y\equiv f_{2}\,.$ (5.32)

We now observe that under these assumptions equation (5.24a) can be solved
exactly. To see this we begin by noting the simpler expression for the
pressure

$\displaystyle p$ $\displaystyle=-2H\dot{x}-(\dot{H}-\dot{\Phi}H)(x-1)$ (5.33)
$\displaystyle=-2H\dot{x}-p(x-1)$ $\displaystyle=-2H\frac{\dot{x}}{x}$
$\displaystyle=-2H\partial_{t}\ln x\,,$

where in the second equality we used (5.24a) and in the third one we isolated
$p$ by assuming $x\neq 0$. Plugging (5.33) into (5.24a) we get

$\displaystyle-2H\partial_{t}\ln x$ $\displaystyle=\dot{H}-\dot{\Phi}H\,,$
(5.34) $\displaystyle\partial_{t}\ln(x^{-2})$ $\displaystyle=\partial_{t}(\ln
H-\Phi)\,,$

where in the second line we inverted $H$, which is valid since we are assuming
$H\neq 0$. This equation can be integrated exactly to arrive at

$H(t)=Qe^{\Phi(t)}x(t)^{-2}\,,\qquad Q=\text{const.}\neq 0\,.$ (5.35)

This relation tells us that the Hubble parameter is completely determined from
the dilaton and one of the extra fields. For the rest of the system we cannot
do much without considering particular truncations or certain limits of the
theory and so here we just rewrite them in terms of $x$ and $y$:

$\displaystyle H$ $\displaystyle=Qe^{\Phi}x^{-2}\,,$ (5.36a)
$\displaystyle\ddot{\Phi}$
$\displaystyle=dH^{2}x^{2}-\frac{d}{4}\left(\dot{x}^{2}+\dot{y}^{2}\right)+\frac{1}{4}(\dot{A})^{2}+\frac{1}{2}B\ddot{\Phi}+\frac{1}{4}\dot{B}\dot{\Phi}-\frac{1}{4}\ddot{B}\,,$
(5.36b) $\displaystyle\dot{\Phi}^{2}$
$\displaystyle=\frac{d}{\alpha^{\prime}}\left(x^{2}-1+\frac{1}{3}y^{2}\right)y+dH^{2}x^{2}-\frac{d}{4}\left(\dot{x}^{2}+\dot{y}^{2}\right)$
$\displaystyle\quad-\frac{1}{\alpha^{\prime}}\left(AB-\frac{2}{3}A^{3}\right)+\frac{1}{4}(\dot{A})^{2}+\frac{1}{2}B(\dot{\Phi})^{2}-\frac{1}{2}\dot{B}\dot{\Phi}\,,$
(5.36c) $\displaystyle 0$
$\displaystyle=-\frac{\alpha^{\prime}}{4}\ddot{x}+\frac{\alpha^{\prime}}{4}\dot{\Phi}\dot{x}+xy-\alpha^{\prime}H^{2}x\,,$
(5.36d) $\displaystyle 0$
$\displaystyle=-\frac{\alpha^{\prime}}{2}\ddot{y}+\frac{\alpha^{\prime}}{2}\dot{\Phi}\dot{y}+x^{2}+y^{2}-1\,,$
(5.36e) $\displaystyle A$
$\displaystyle=-\frac{\alpha^{\prime}}{2}\ddot{\Phi}\,,$ (5.36f)
$\displaystyle B$
$\displaystyle=2A^{2}+\frac{\alpha^{\prime}}{2}\ddot{A}-\frac{\alpha^{\prime}}{2}\dot{\Phi}\dot{A}\,.$
(5.36g)

Most of the complexity of the system comes from the terms involving $A$ and
$B$. This is because they are the only ones implicitly encoding up to order
six in derivatives of the dilaton, as can be seen by solving (5.36f) and
(5.36g) to express $A$ and $B$ in terms of $\Phi$. We did not succeed in
finding solutions of the full theory for specific ansätze. In the following we
summarize the backgrounds we studied that turn out not to be exact
solutions:777We are always considering $x(t)\neq 0$.

  * •

There are no solutions with constant dilaton $\Phi(t)=\Phi_{0}={\rm const.}$

  * •

There are no solutions with $H(t)=H_{0}={\rm const.}$ and $\Phi$ being a
Laurent polynomial
$\Phi(t)=\sum_{k\in\mathbb{Z}}\Phi_{k}(t-t_{0})^{k}\in\mathbb{R}[t,t^{-1}]$
where all $\Phi_{k}$ are constant and there are only finitely-many non-
vanishing of them.

In particular, this result already rules out the possibility of having de
Sitter backgrounds in Einstein frame with constant dilaton
$\phi(t)=\phi_{0}=\text{const}$. To see this we remember that the original
dilaton $\phi(t)$ coming from the 2-dimensional string sigma model is related
to the duality invariant one $\Phi(t)$ by

$\Phi(t)=2\phi(t)-d\ln a(t)\,,\quad\dot{\Phi}=2\dot{\phi}-dH\,.$ (5.37)

A constant dilaton then imposes the condition

$\dot{\Phi}=-dH\,.$ (5.38)

On top of that, one can show that for constant dilaton, the Hubble parameter
in Einstein frame is constant if and only if $H(t)=H_{0}=\text{const.}$ in
string frame. In other words, for constant dilaton, de Sitter in string frame
is equivalent to de Sitter in Einstein frame. Since (5.38) with $H(t)=H_{0}$
implies a linear dilaton, by ruling out all possible Laurent polynomial forms
for $\Phi(t)$, de Sitter solutions in Einstein frame with constant dilaton are
also excluded.

  * •

We previously observed that by assuming $x=1+\mathcal{O}(\alpha^{\prime})$ and
$y=\mathcal{O}(\alpha^{\prime})$ and performing an $\alpha^{\prime}$
expansion, the leading order equations of the system (5.36) reduce to standard
(string) Friedmann equations in vacuum (5.27). The well-known exact solution
to this system is given by

$H(t)=\pm\frac{\text{sign}(\omega)}{\sqrt{d}}\frac{1}{(t-t_{0})}\,,\quad\Phi(t)=-\log(\omega(t-t_{0}))\,.$
(5.39)

We checked whether this background corresponds to a consistent truncation of
the full theory, i.e. upon including the complete $\alpha^{\prime}$
dependency. We found that this is only the case if

$x(t)=1\,,\quad y(t)=2\alpha^{\prime}\frac{1}{(t-t_{0})^{2}}\,,\quad
d=\frac{1}{2}\,,$ (5.40)

with the final equality of course rendering this unphysical.

This analysis shows that some of the simplest backgrounds one can propose for
the standard fields $H$ and $\Phi$ are not solutions of the HSZ equations for
any configuration of the extra fields $A,B,x$ and $y$. It is worth mentioning,
however, that this study just scratches the surface of the whole landscape of
possible backgrounds one could propose, and we expect that upon a more
exhaustive analyses exact (analytic or numerical) solutions could be found.

###  5.2 Tensionless limit, de Sitter solution and $\frac{1}{\alpha^{\prime}}$
expansion

While at the present moment we cannot solve the full system (5.36)
analytically, in the tensionless limit $\alpha^{\prime}\rightarrow\infty$ the
equations are simple enough to obtain the general exact solutions. In this
limit, all non-derivative contributions disappear, and (5.36) reduces to

$\displaystyle H$ $\displaystyle=Qe^{\Phi}x^{-2}\,,\quad Q=\text{const.}\neq
0\,,$ (5.41a) $\displaystyle\ddot{\Phi}$
$\displaystyle=dH^{2}x^{2}-\frac{d}{4}\left(\dot{x}^{2}+\dot{y}^{2}\right)+\frac{1}{4}(\dot{A})^{2}+\frac{1}{2}B\ddot{\Phi}+\frac{1}{4}\dot{B}\dot{\Phi}-\frac{1}{4}\ddot{B}\,,$
(5.41b) $\displaystyle\dot{\Phi}^{2}$
$\displaystyle=dH^{2}x^{2}-\frac{d}{4}\left(\dot{x}^{2}+\dot{y}^{2}\right)+\frac{1}{4}(\dot{A})^{2}+\frac{1}{2}B(\dot{\Phi})^{2}-\frac{1}{2}\dot{B}\dot{\Phi}\,,$
(5.41c) $\displaystyle 0$
$\displaystyle=\ddot{x}-\dot{\Phi}\dot{x}+4H^{2}x\,,$ (5.41d) $\displaystyle
0$ $\displaystyle=\ddot{y}-\dot{\Phi}\dot{y}\,,$ (5.41e) $\displaystyle 0$
$\displaystyle=\ddot{\Phi}\,,$ (5.41f) $\displaystyle 0$
$\displaystyle=\ddot{A}-\dot{\Phi}\dot{A}\,.$ (5.41g)

We now turn to (5.41f), which implies a linear dilaton profile:

$\Phi(t)=-\omega(t-t_{0})\,,$ (5.42)

where $\omega$ is an integration constant. From now on the solutions are
different depending whether $\omega$ vanishes or not. Since the procedure to
get both family of solutions is almost identical, we will describe in detail
only the $\omega\neq 0$ case and just give the final result for vanishing
$\omega$.

Equations (5.41g) and (5.41e) take exactly the same form and, upon using
(5.42), they can be solved exactly by

$\displaystyle A(t)$ $\displaystyle=A_{0}+A_{1}e^{-\omega(t-t_{0})}\,,$ (5.43)
$\displaystyle y(t)$ $\displaystyle=y_{0}+y_{1}e^{-\omega(t-t_{0})}\,.$

Then, by subtracting (5.41c) from (5.41b), using (5.42) and reordering terms
we get a second order differential equation for $B$,

$\ddot{B}+3\omega\dot{B}+2\omega^{2}B-4\omega^{2}=0\,,$ (5.44)

which is exactly solved by

$\displaystyle B(t)$
$\displaystyle=2+B_{1}e^{-\omega(t-t_{0})}+B_{2}e^{-2\omega(t-t_{0})}\,.$
(5.45)

At this point we have two remaining equations for $x(t)$, namely (5.41d) and
(5.41b) (or equivalent (5.41c)). We found it easier to solve (5.41b) because
it is a first order differential equation, and then check (5.41d). Inserting
(5.41a), (5.42), (5.43) and (5.45) into (5.41b) we arrive at the first order
equation

$\dot{x}^{2}-\left(4Q^{2}x^{-2}+C_{1}\right)e^{-2\omega(t-t_{0})}=0\,,\qquad
C_{1}\equiv\frac{\omega^{2}}{d}\left(A_{1}^{2}-dy_{1}^{2}-2B_{2}\right)\,,$
(5.46)

where we defined the constant $C_{1}$ to simplify the notation. By multiplying
both sides with $x^{2}$ and changing variables to $z(t)\equiv x(t)^{2}$ we
arrive at the equation

$\dot{z}^{2}-\left(16Q^{2}+4C_{1}z\right)e^{-2\omega(t-t_{0})}=0\,,$ (5.47)

which has different solutions depending whether $C_{1}$ vanishes or not,

$\displaystyle z(t)$
$\displaystyle=\pm\frac{4Q}{\omega}e^{-\omega(t-t_{0})}+x_{0}\quad\text{if}\quad
C_{1}=0\,,$ (5.48) $\displaystyle z(t)$
$\displaystyle=\frac{C_{1}}{\omega^{2}}\left(e^{-\omega(t-t_{0})}+x_{0}\right)^{2}-\frac{4Q^{2}}{C_{1}}\quad\text{if}\quad
C_{1}\neq 0\,.$ (5.49)

Returning to the original variable $x(t)=\pm\sqrt{z(t)}$ and plugging the
result together with (5.41a) and (5.42) into (5.41d) one can verify that the
last equation of the system is also satisfied.

All in all, combining the above results we conclude that, for $\omega\neq 0$,
the most general solution to the system (5.41) is given by:

$\displaystyle\Phi(t)=$ $\displaystyle-\omega(t-t_{0})\,,\quad\omega\neq 0\,,$
(5.50a) $\displaystyle H(t)=$ $\displaystyle\
Qe^{-\omega(t-t_{0})}x(t)^{-2}\,,\quad Q\neq 0\,,$ (5.50b) $\displaystyle
A(t)=$ $\displaystyle\ A_{0}+A_{1}e^{-\omega(t-t_{0})}\,,$ (5.50c)
$\displaystyle B(t)=$ $\displaystyle\
2+B_{1}e^{-\omega(t-t_{0})}+B_{2}e^{-2\omega(t-t_{0})}\,,$ (5.50d)
$\displaystyle y(t)=$ $\displaystyle\ y_{0}+y_{1}e^{-\omega(t-t_{0})}\,,$
(5.50e) $\displaystyle x(t)=$ $\displaystyle\begin{cases}\pm
2\sqrt{\pm\frac{Q}{\omega}e^{-\omega(t-t_{0})}+x_{0}}&\text{if}\quad
C_{1}=0\,,\\\
\pm\sqrt{\frac{C_{1}}{\omega^{2}}\left(e^{-\omega(t-t_{0})}+x_{0}\right)^{2}-\frac{4Q^{2}}{C_{1}}}&\text{if}\quad
C_{1}\neq 0\,,\end{cases}\
C_{1}\equiv\frac{\omega^{2}}{d}\left(A_{1}^{2}-dy_{1}^{2}-2B_{2}\right)\,.$
(5.50f)

Repeating identical steps for the $\omega=0$ case, we get a second set of
solutions:

$\displaystyle\Phi(t)=$ $\displaystyle\ 0\,,$ (5.51a) $\displaystyle H(t)=$
$\displaystyle\ Qx(t)^{-2}\,,\quad Q\neq 0\,,$ (5.51b) $\displaystyle A(t)=$
$\displaystyle\ A_{0}+A_{1}(t-t_{0})\,,$ (5.51c) $\displaystyle B(t)=$
$\displaystyle\ B_{0}+B_{1}(t-t_{0})\,,$ (5.51d) $\displaystyle y(t)=$
$\displaystyle\ y_{0}+y_{1}(t-t_{0})\,,$ (5.51e) $\displaystyle x(t)=$
$\displaystyle\begin{cases}\pm 2\sqrt{\pm Q(t-t_{0})+x_{0}}&\text{if}\quad
C_{2}=0\,,\\\
\pm\sqrt{C_{2}[(t-t_{0})+x_{0}]^{2}-\frac{4Q^{2}}{C_{2}}}&\text{if}\quad
C_{2}\neq 0\,,\end{cases}\quad
C_{2}\equiv\frac{1}{d}\left(A_{1}^{2}-dy_{1}^{2}\right)\,.$ (5.51f)

Each of these families is parameterized by several independent free
parameters. In particular, for the $\omega\neq 0$ case one can analyze the
simplest solution of this family obtained by taking
$A_{0}=A_{1}=B_{1}=B_{2}=y_{0}=y_{1}=x_{0}=0$ and so arriving at

$\displaystyle\Phi(t)$ $\displaystyle=-\omega(t-t_{0})\,,\quad
H(t)=\text{sgn}(Q)\frac{|\omega|}{4}\,,\quad x(t)=\pm
2\sqrt{\left|\frac{Q}{\omega}\right|}e^{-\frac{1}{2}\omega(t-t_{0})}\,,$
(5.52a) $\displaystyle A(t)$ $\displaystyle=y(t)=0\,,\quad B(t)=2\,,$ (5.52b)

where we kept only the real $x(t)$ branch. Remarkably the Hubble parameter is
constant, and hence this solution corresponds to a de Sitter background in
string frame. Note that the de Sitter scale here is simply an integration
constant and not determined by a bare parameter in the action, which means
that $H$ is fixed by the initial conditions. Furthermore, (5.52) also admits a
de Sitter solution in Einstein frame with constant dilaton for the particular
case of $d=4$ (see (5.38)), corresponding to five spacetime dimensions.

The tensionless limit can be interpreted as the zeroth order of a perturbative
expansion in small $\frac{1}{\alpha^{\prime}}$. Therefore, in the remainder of
this section we explore the first order correction in
$\frac{1}{\alpha^{\prime}}$ to the system (5.41). More precisely, we return to
the full system (5.36), write for all fields

$\Psi(t)=\Psi^{(0)}(t)+\frac{1}{\alpha^{\prime}}\Psi^{(1)}(t)+\mathcal{O}\left(\frac{1}{\alpha^{\prime}{}^{2}}\right)\,,$
(5.53)

and expand all equations up to first order in $\frac{1}{\alpha^{\prime}}$. By
doing so each equation will split in two, one for each order, the leading one
corresponding to the tensionless limit studied in (5.41). We will not consider
corrections to the most general zeroth-order solutions found in (5.50) and
(5.51) but we will restrict to the particular case of (5.52) with $Q>0$ and
$\omega>0$ for simplicity. However, the following steps should be equally
applicable to the general solutions.

Rather than going into each detail, here we show some examples of the
procedure described above for the simplest equations. Taking (5.36f) as an
example, we expand and keep only up to first order in the string’s tension,

$\displaystyle 0$ $\displaystyle=\ddot{\Phi}+\frac{2}{\alpha^{\prime}}A\,,$
(5.54) $\displaystyle 0$
$\displaystyle=\ddot{\Phi}^{(0)}+\frac{1}{\alpha^{\prime}}\left(\ddot{\Phi}^{(1)}+2A^{(0)}\right)+\mathcal{O}\left(\frac{1}{\alpha^{\prime}{}^{2}}\right)\,.$

This splits into two equations, one for the tensionless limit and a new first
order equation that determines $\Phi^{(1)}$ in terms of $A^{(0)}$. Inserting
the solution (5.52) we see that (5.54) is solved by

$\Phi^{(1)}(t)=-\omega_{1}(t-t_{0})\,,$ (5.55)

where $\omega_{1}$ is a new integration constant, and we omitted a possible
constant shift for simplicity. For the second and last explicit calculation,
we consider equation (5.36g)

$\displaystyle 0$
$\displaystyle=\ddot{A}-\dot{\Phi}\dot{A}+\frac{1}{\alpha^{\prime}}\left(4A^{2}-2B\right)\,,$
(5.56) $\displaystyle 0$
$\displaystyle=\ddot{A}^{(0)}-\dot{\Phi}^{(0)}\dot{A}^{(0)}+\frac{1}{\alpha^{\prime}}\left(\ddot{A}^{(1)}-\dot{\Phi}^{(0)}\dot{A}^{(1)}-\dot{\Phi}^{(1)}\dot{A}^{(0)}+4(A^{(0)})^{2}-2B^{(0)}\right)+\mathcal{O}\left(\frac{1}{\alpha^{\prime}{}^{2}}\right)\,.$

By inserting (5.50) the leading order is automatically solved while the
$\frac{1}{\alpha^{\prime}}$ contribution determines $A^{(1)}$ in terms of the
zeroth-order solutions to be

$A^{(1)}(t)=A_{0}+A_{1}e^{-\omega(t-t_{0})}+\frac{4}{\omega}(t-t_{0})\,.$
(5.57)

Following the same procedure, $y^{(1)}(t)$ can be determined by expanding
equation (5.36e) and $B^{(1)}(t)$ by expanding the combination of (5.36b) and
(5.36c). Inserting these and all previous results into the expansion of
(5.36b) we get a first order differential equation for $x^{(1)}(t)$ which can
be solved exactly. Finally, at this point all first order corrections were
determined, yet we still have to check that the expansion of (5.36d) holds up
to first order in the string’s tension. We performed all these steps and we
found that the extension to the solution (5.52) (with positive $Q$ and
$\omega$) up to and including first order in $\frac{1}{\alpha^{\prime}}$ is
given by

$\displaystyle\Phi(t)$
$\displaystyle=-\left(\omega_{0}+\frac{1}{\alpha^{\prime}}\omega_{1}\right)(t-t_{0})\,,$
(5.58a) $\displaystyle H(t)$
$\displaystyle=\frac{1}{4}\left(\omega_{0}+\frac{1}{\alpha^{\prime}}\omega_{1}\right)+\frac{1}{\alpha^{\prime}}\left[\frac{B_{2}\omega_{0}^{2}}{8dQ}e^{-\omega_{0}(t-t_{0})}-\frac{x_{1}}{4}\sqrt{\frac{\omega_{0}^{3}}{Q}}e^{\omega_{0}(t-t_{0})}\right]\,,$
(5.58b) $\displaystyle A(t)$
$\displaystyle=\frac{1}{\alpha^{\prime}}\left[A_{0}+A_{1}e^{-\omega_{0}(t-t_{0})}+\frac{4}{\omega_{0}}(t-t_{0})\right]\,,$
(5.58c) $\displaystyle B(t)$
$\displaystyle=2+\frac{1}{\alpha^{\prime}}\left[B_{1}e^{-\omega_{0}(t-t_{0})}+B_{2}e^{-2\omega_{0}(t-t_{0})}\right]\,,$
(5.58d) $\displaystyle y(t)$
$\displaystyle=\frac{1}{\alpha^{\prime}}\left[y_{0}+y_{1}e^{-\omega_{0}(t-t_{0})}-\frac{8Q}{\omega_{0}^{2}}e^{-\omega_{0}(t-t_{0})}(t-t_{0})-\frac{2}{\omega_{0}}(t-t_{0})\right]\,,$
(5.58e) $\displaystyle x(t)$ $\displaystyle=\pm
2\sqrt{\frac{Q}{\omega_{0}}}e^{-\frac{1}{2}\omega_{0}(t-t_{0})}$ (5.58f)
$\displaystyle\quad\,\pm\frac{1}{\alpha^{\prime}}e^{-\frac{1}{2}\omega_{0}(t-t_{0})}\left[x_{1}e^{\omega_{0}(t-t_{0})}-\sqrt{\frac{Q}{\omega_{0}}}\omega_{1}\left(\frac{1}{\omega_{0}}+(t-t_{0})\right)-\sqrt{\frac{\omega_{0}}{Q}}\frac{B_{2}}{2d}e^{-\omega_{0}(t-t_{0})}\right]\,,$

where we renamed the $\omega$ appearing in (5.52) as $\omega_{0}$. For generic
integration constants the solutions (5.58) are not de Sitter vacua. However,
there are particular integration constants for which the zeroth order de
Sitter solution (5.52) is indeed preserved. For example, by choosing
$A_{0}=A_{1}=B_{1}=B_{2}=y_{0}=y_{1}=x_{1}=0$ and combining the dilaton
integration constants into a renormalized
$\omega=\omega_{0}+\frac{1}{\alpha^{\prime}}\omega_{1}$, (5.58) reduces to

$\displaystyle\Phi(t)$ $\displaystyle=-\omega(t-t_{0})\,,\quad
H(t)=\frac{\omega}{4}\,,\quad x(t)=\pm
2\sqrt{\frac{Q}{\omega}}e^{-\frac{1}{2}\omega(t-t_{0})}\,,$ (5.59a)
$\displaystyle A(t)$
$\displaystyle=\frac{4}{\alpha^{\prime}\omega}(t-t_{0})\,,\quad B(t)=2\,,$
(5.59b) $\displaystyle y(t)$
$\displaystyle=-\frac{8Q}{\alpha^{\prime}\omega^{2}}e^{-\omega(t-t_{0})}(t-t_{0})-\frac{2}{\alpha^{\prime}\omega}(t-t_{0})\,,\quad\omega>0\,,\
Q>0\,.$ (5.59c)

For $\Phi(t),H(t),x(t)$ and $B(t)$ the solutions take the same structural form
as in the tensionless limit (5.52), except that the integration constants are
$\frac{1}{\alpha^{\prime}}$-corrected. It remains as an important open
question whether the de Sitter vacua are preserved perturbatively at higher
orders in $\frac{1}{\alpha^{\prime}}$.

##  6 Conclusions and Outlook

In this paper we have explored HSZ theory for the purely time-dependent
backgrounds of cosmology. While HSZ theory, being based on a non-standard
chiral CFT, is not a conventional string theory, it shares many features of
string theory such as the presence of a fundamental parameter
$\alpha^{\prime}$ governing higher-derivative corrections and exact duality
invariance under $O(d,d,\mathbb{R})$. We hope that this theory may thus be a
model for how to deal with string theory in the regimes we are most interested
in, such as close to the singularities of the big bang and black holes.
Perhaps most intriguingly, we were able to provide a two-derivative yet
$\alpha^{\prime}$-exact reformulation in which the tensionless limit
$\alpha^{\prime}\rightarrow\infty$ can be taken smoothly, and we set up
perturbation theory with $\frac{1}{\alpha^{\prime}}$ as the small expansion
parameter.

Possible future directions for research include the following:

  * •

There are now two known limits of HSZ theory in which a two-derivative
reformulation is possible: for the quadratic theory in a background field
expansion around flat space [24] and for the reduction to one dimension
(cosmic time) displayed here. It is thus natural to inquire whether the full
HSZ theory admits a reformulation as a two-derivative theory. Such a
formulation could provide the tensionless limit without any truncations.

  * •

In modern cosmology it is equally important to have control not only over the
dynamics of the homogenous and isotropic backgrounds but also over their
fluctuations that generally break the symmetries. Recently, the cosmological
perturbation theory was explored for a so-called weakly constrained double
field theory to cubic order in fluctuations, in order to analyze the dynamics
of genuine winding modes [36]. It would be interesting to extend this analysis
to HSZ theory and, assuming the program of the previous item can be
accomplished, to see how the modes behave in the tensionless limit.

  * •

It is important to find non-perturbative solutions for the
$\alpha^{\prime}$-complete Friedmann equations, ideally upon adding genuine
matter, in order to find semi-realistic string cosmology models. In
particular, one should further explore the de Sitter solutions that we found
here in the tensionless limit, for instance in relation to their stability, as
recently explored in a related context in [37].

  * •

While HSZ theory is not a genuine string theory one might hope that the
$\alpha^{\prime}$-complete formulation introduced here could serve as a model
for a genuine string theory, with the massive string modes playing the role of
the extra fields employed here. Although genuine string theories feature an
infinite number of massive string modes, one might still hope that there are
formulations not too dissimilar to the one discussed in this paper.

  * •

Even if genuine string theories do not allow for such reformulations one may
wonder whether there are other, simpler UV completions of gravity, that
display features of string theory, yet lead to a version of ‘string cosmology’
that is as manageable as the model explored here.

### Acknowledgements

We thank Robert Brandenberger, Heliudson Bernardo, Roberto Bonezzi, Christoph
Chiaffrino, Felipe Diaz-Jaramillo, Allison Pinto, Eric Lescano, Nahuel Miron-
Granese and Barton Zwiebach for comments and discussions.

This work is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant agreement
No 771862). D. M. is supported by CONICET. T. C. is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer
417533893/GRK2575 “Rethinking Quantum Field Theory”.

##  Appendix A Field Redefinitions

In this appendix we spell out the details of the field redefinitions performed
in sec. 3. In the action (3.33) we have integrated out the extra fields,
leaving an effective action purely in terms of the fields $S,\Phi$ and $n$. In
[23] it was shown that any action can be brought to a minimal form by
performing field redefinitions. In this scheme, there are no higher
derivatives of $S$, nor any appearance of ${\cal D}\Phi$ or ${\rm Tr}{{\cal
D}S{\cal D}S}$ at any higher order in $\alpha^{\prime}$. We call any term
removable if it contains powers or derivatives of at least one of the
following objects: $\\{{\cal D}\mathcal{L},{\cal D}\Phi,{\rm
Tr}{\mathcal{L}^{2}}\\}$. Inside a Lagrangian these will be denoted as
$\mathbb{L}$, while for removable terms in the equations of motion and field
redefinitions we will use $\mathbb{E}$ and $\mathbb{D}$, respectively. The
remaining unremovable couplings are composed of traces of single powers of
$\mathcal{L}$.

The procedure to ‘clean’ removable terms order by order was explained in
detail in [23], and here we summarize the main steps that are needed. The
method relies on field redefinitions of the effective action (3.33), for which
we need the general variation

$\delta I=\int dt\,n\,e^{-\Phi}\left[{\rm Tr}{E_{S}\delta S}+E_{n}\frac{\delta
n}{n}+E_{\Phi}\delta\Phi\right]\,,$ (A.1)

where $E_{S}$ is the projected version of the original variation, namely
$E_{S}=[\frac{\delta I}{\delta S}]_{-}$. The equations of motion have an
$\alpha^{\prime}$ expansion, which in our case contains no $\alpha^{\prime}$
order because $I^{(1)}=0$,

$E_{\Psi}=E^{(0)}_{\Psi}+\alpha^{\prime}{}^{2}E^{(2)}_{\Psi}+\alpha^{\prime}{}^{3}E^{(3)}_{\Psi}+\alpha^{\prime}{}^{4}E^{(4)}_{\Psi}+\mathcal{O}(\alpha^{\prime}{}^{5})=0\;.$
(A.2)

Here we denoted all fields collectively as $\Psi=\\{S,\Phi,n\\}$. The highest
orders in our case $E_{\Psi}^{(3)}$ and $E_{\Psi}^{(4)}$ will not be needed to
perform field redefinitions. However, we do need $E_{\Psi}^{(0)}$ and
$E_{\Psi}^{(2)}$, given by

$\displaystyle E_{S}^{(0)}$ $\displaystyle=\frac{1}{4}{\cal
D}\mathcal{L}S-\frac{1}{4}{\cal D}\Phi\mathcal{L}S\,,$ (A.3) $\displaystyle
E_{n}^{(0)}$ $\displaystyle=({\cal D}\Phi)^{2}-\frac{1}{8}{\rm
Tr}{\mathcal{L}^{2}}\,,$ $\displaystyle E_{\Phi}^{(0)}$ $\displaystyle=2{\cal
D}^{2}\Phi-({\cal D}\Phi)^{2}-\frac{1}{8}{\rm Tr}{\mathcal{L}^{2}}\,,$

and for $E_{\Psi}^{(2)}$ by

$\displaystyle E_{S}^{(2)}$ $\displaystyle=\mathbb{E}_{S}\,,$ (A.4)
$\displaystyle E_{n}^{(2)}$ $\displaystyle=\frac{5}{3.2^{7}}{\rm
Tr}{\mathcal{L}^{6}}+\mathbb{E}_{n}\,,$ $\displaystyle E_{\Phi}^{(2)}$
$\displaystyle=\frac{1}{3.2^{7}}{\rm
Tr}{\mathcal{L}^{6}}+\mathbb{E}_{\Phi}\,.$

As explained before, we collected all removable terms inside
$\mathbb{E}_{\Psi}=\mathbb{E}_{\Psi}({\cal D}\mathcal{L},{\cal D}\Phi,{\rm
Tr}{\mathcal{L}^{2}})$ whose explicit form is not needed.

Let us now consider a general field redefinition

$\Psi\rightarrow\Psi^{\prime}=\Psi+\delta\Psi\,,$ (A.5)

which we take to be perturbative in $\alpha^{\prime}$, so that we can expand

$\delta\Psi=\alpha^{\prime}{}^{2}\delta^{(2)}\Psi+\alpha^{\prime}{}^{3}\delta^{(3)}\Psi+\alpha^{\prime}{}^{4}\delta^{(4)}\Psi+\cdots\,.$
(A.6)

Here we take an expansion of the redefinitions in terms of powers of
$\alpha^{\prime}$, skipping the first order because there are no such terms to
remove from the action. The action expands as follows:

$I^{\prime}[\Psi^{\prime}]\equiv
I[\Psi+\delta\Psi]=I[\Psi]+\Delta_{1}I\cdot\delta\Psi+\frac{1}{2}\Delta_{2}I\cdot(\delta\Psi)^{2}+\cdots\,,$
(A.7)

where we defined $\Delta_{n}I\equiv\frac{\delta^{n}I}{\delta\Psi^{n}}$, the
first of which is nothing but the equation of motion,
$\Delta_{1}I\equiv\frac{\delta I}{\delta\Psi}\equiv E_{\Psi}$. Each of these
variations should be expanded in $\alpha^{\prime}$ as well

$\Delta_{n}I=\Delta_{n}I^{(0)}+\alpha^{\prime}{}^{2}\Delta_{n}I^{(2)}+\alpha^{\prime}{}^{3}\Delta_{n}I^{(3)}+\alpha^{\prime}{}^{4}\Delta_{n}I^{(4)}+\cdots\,.$
(A.8)

Plugging this expansion into the redefined action $I^{\prime}$, we obtain

$\begin{split}I^{\prime}=I^{(0)}&+\alpha^{\prime}{}^{2}\Big{(}I^{(2)}+E_{\Psi}^{(0)}\cdot\delta^{(2)}\Psi\Big{)}\\\
&+\alpha^{\prime}{}^{3}\Big{(}I^{(3)}+E_{\Psi}^{(0)}\cdot\delta^{(3)}\Psi\Big{)}\\\
&+\alpha^{\prime}{}^{4}\Big{(}I^{(4)}+E_{\Psi}^{(2)}\cdot\delta^{(2)}\Psi+\frac{1}{2}\Delta_{2}I^{(0)}\cdot\big{(}\delta^{(2)}\Psi\big{)}^{2}+E_{\Psi}^{(0)}\cdot\delta^{(4)}\Psi\Big{)}+\cdots\,,\end{split}$
(A.9)

where the order $\alpha^{\prime}{}^{3}$ does not receive any induced terms
from lower orders as a consequence of not having $I^{(1)}$ in the original
action. Moreover, $\delta^{(3)}\Psi$ does not propagate to the next order,
which will become useful in the following analysis. In order to bring the
action into canonical form we apply the following recipe: first we pick a
particular $\delta^{(2)}\Psi$ to bring the action to second order in
$\alpha^{\prime}$ to canonical form. This in turn induces new terms
proportional to $E_{\Psi}^{(2)}$ and $\Delta_{2}I^{(0)}$ into the action of
order $\alpha^{\prime}{}^{4}$. For the third order we only need to pick some
$\delta^{(3)}\Psi$ that brings $I^{(3)}{}^{\prime}$ to canonical from, yet
this step induces no terms at order $\alpha^{\prime}{}^{4}$. The induced
terms, together with the original $I^{(4)}$, can be brought to canonical form
by picking a suitable $\delta^{(4)}\Psi$. Obviously these steps induce terms
to all orders in $\alpha^{\prime}$ but, since we are neglecting orders higher
than $\alpha^{\prime}{}^{4}$, the above mentioned effects are the only
relevant for our work.

In principle, from the above procedure it seems that we must keep track of the
field redefinitions $\delta^{(2)}\Psi$ and $\delta^{(4)}\Psi$ which can become
rather tedious. However, let us show how we can easily obtain
$\delta^{(2)}\Psi$ and how $\delta^{(3)}\Psi$ and $\delta^{(4)}\Psi$ are not
needed at all. In order to see this, let us suppose that the action contains a
term at second order in $\alpha^{\prime}$ which multiplies the lowest order
equations of motion, i.e.,

$I=\cdots+\alpha^{\prime}{}^{2}E_{\Psi}^{(0)}\cdot X(\Psi)+\cdots\;,$ (A.10)

where $X(\Psi)$ is an arbitrary function of the fields $\Psi$ with four
derivatives and the ellipsis denote the remaining terms in the action. By
performing a field redefinition of the form

$\delta^{(2)}\Psi=-X(\Psi)\;.$ (A.11)

from (A.9) we then infer that in the redefined action $I^{\prime}$ the term in
(A.10) is replaced by

$\begin{split}I^{\prime}=&\cdots-\alpha^{\prime}{}^{4}E_{\Psi}^{(2)}\cdot
X(\Psi)+\alpha^{\prime}{}^{4}\frac{1}{2}\Delta_{2}I^{(0)}\cdot\big{(}-X(\Psi)\big{)}^{2}+\cdots\;,\end{split}$
(A.12)

where the ellipsis denote the same terms as in (A.10), which are unaffected by
the redefinition. By comparing (A.10) and (A.12) we see that we managed to
remove a term at order $\alpha^{\prime}{}^{2}$ in the original action, at
expenses of inducing higher order contributions. In particular, we learn two
important lessons:

  1. 1.

We can simply use the equations of motion (A.2) as a replacement rule in the
form $E_{\Psi}^{(0)}=-\alpha^{\prime}{}^{2}E_{\Psi}^{(2)}+\dots$, where higher
order terms can be ignored as we are considering orders up to an including
$\alpha^{\prime}{}^{4}$.

  2. 2.

We can get access to the induced $\delta^{(2)}\Psi$ by looking at what is left
together with $E_{\Psi}^{(2)}$ in the induced terms. For instance, in the
previous example we see that in (A.12) this is exactly
$\delta^{(2)}\Psi=-X(\Psi)$.

A similar argument follows at $\alpha^{\prime}{}^{3}$ and
$\alpha^{\prime}{}^{4}$ orders. Therein we should pick suitable
$\delta^{(3)}\Psi$ and $\delta^{(4)}\Psi$ respectively to eliminate any term
proportional to the EOM. The crucial point here is that we do not need their
explicit form, because the induced effects of these field redefinitions will
become visible at order $\alpha^{\prime}{}^{5}$ and higher. In other words, at
order $\alpha^{\prime}{}^{3}$ and $\alpha^{\prime}{}^{4}$ we can just freely
use the zeroth order EOM to eliminate any removable contribution.

One can then proceed order-by-order in $\alpha^{\prime}$ by freely using the
equations of motion in the action at each order in $\alpha^{\prime}$, as long
as one keeps track of the induced terms and $\delta^{(2)}\Psi$. To this end we
will use the EOM in the form

$\displaystyle{\cal D}\mathcal{L}$ $\displaystyle={\cal
D}\Phi\mathcal{L}-\alpha^{\prime}{}^{2}4E_{S}^{(2)}S\;,$ (A.13a)
$\displaystyle({\cal D}\Phi)^{2}$ $\displaystyle=\frac{1}{8}{\rm
Tr}{\mathcal{L}^{2}}-\alpha{}^{2}E_{n}^{(2)}\;,$ (A.13b) $\displaystyle{\cal
D}^{2}\Phi$ $\displaystyle=\frac{1}{2}({\cal D}\Phi)^{2}+\frac{1}{16}{\rm
Tr}{\mathcal{L}^{2}}-\alpha^{\prime}{}^{2}\frac{1}{2}E_{\Phi}^{(2)}\;.$
(A.13c)

By using these equations in a systematic way, one can bring $I^{(2)},I^{(3)}$
and $I^{(4)}$ to canonical form where all removable terms were redefined away.
This systematic approach was explained several times already in [23] and [15].
We will not repeat these steps in detail here but just summarize the main
ideas:

  1. 1.

As a first step one removes all higher derivatives of $\mathcal{L}$ by using
(A.13a) as many times as needed.

  2. 2.

Using (A.13b) we remove any higher powers of ${\cal D}\Phi$.

  3. 3.

At this point we make use of (A.13c) to eliminate higher derivatives of the
dilaton. In this step higher powers of ${\cal D}\Phi$ and higher derivatives
of $\mathcal{L}$ could be generated in which case one has to repeat the first
two steps. It could be possible that this third step needs to be iterated a
few times.

  4. 4.

Reached this point, the only possible contribution from the dilaton comes from
a linear term in ${\cal D}\Phi$. This can be eliminated by an integration by
parts together with (A.13a).

  5. 5.

Finally, in order to eliminate terms with ${\rm Tr}{\mathcal{L}^{2}}$ one has
to use (A.13b) to make a $({\cal D}\Phi)^{2}$ factor reappear, followed by an
integration by parts of the dilaton. This will induce higher derivatives of
${\cal D}\Phi$ and $\mathcal{L}$ that can be traded for $({\cal D}\Phi)^{2}$
and ${\rm Tr}{\mathcal{L}^{2}}$ by using (A.13a) and (A.13c). Finally, one
uses (A.13b) once more to bring everything to exactly the same form as the
original term that we started from, but with a different coefficient.

We applied this systematic procedure to the second order action

$I^{(2)}=\int dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{3.2^{7}}{\rm
Tr}{\mathcal{L}^{6}}-\frac{1}{2^{7}}{\rm Tr}{{\cal D}(\mathcal{L}^{2}){\cal
D}(\mathcal{L}^{2})}+\frac{1}{16}({\cal D}^{3}\Phi)^{2}+\frac{1}{12}({\cal
D}^{2}\Phi)^{3}\right\\}\,,$ (A.14)

and we managed to redefine away all removable terms at expenses of inducing
$\alpha^{\prime}{}^{4}$ orders

$\displaystyle\alpha^{\prime}{}^{2}I^{(2)}+\alpha^{\prime}{}^{2}E_{\Psi}^{(0)}\cdot\delta^{(2)}\Psi$
$\displaystyle=\alpha^{\prime}{}^{2}\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{3.2^{7}}{\rm
Tr}{\mathcal{L}^{6}}\right\\}$ (A.15)
$\displaystyle\quad+\alpha^{\prime}{}^{4}\int
dt\,n\,e^{-\Phi}\,\left\\{\frac{1}{2^{8}}\left(E_{n}^{(2)}-E_{\Phi}^{(2)}\right){\rm
Tr}{\mathcal{L}^{4}}+\mathbb{L}\right\\}$
$\displaystyle=\alpha^{\prime}{}^{2}\int
dt\,n\,e^{-\Phi}\,\left\\{-\frac{1}{3.2^{7}}{\rm
Tr}{\mathcal{L}^{6}}\right\\}$ $\displaystyle\quad+\alpha^{\prime}{}^{4}\int
dt\,n\,e^{-\Phi}\,\left\\{\frac{1}{3.2^{13}}{\rm Tr}{\mathcal{L}^{4}}{\rm
Tr}{\mathcal{L}^{6}}+\mathbb{L}\right\\}\,,$

where we used the the second order EOM (A.4) for $n$ and $\Phi$ and packaged
all removable terms appearing at order $\alpha^{\prime}{}^{4}$ in
$\mathbb{L}=\mathbb{L}({\cal D}\mathcal{L},{\cal D}\Phi,{\rm
Tr}{\mathcal{L}^{2}})$. From the first line we can read $\delta^{(2)}\Psi$ as
explained in point 2. below equation (A.12). These variations are given by

$\displaystyle\delta^{(2)}S$ $\displaystyle=\mathbb{D}_{S}\,,$ (A.16)
$\displaystyle\frac{1}{n}\delta^{(2)}n$ $\displaystyle=\frac{1}{2^{8}}{\rm
Tr}{\mathcal{L}^{4}}+\mathbb{D}_{n}\,,$ $\displaystyle\delta^{(2)}\Phi$
$\displaystyle=-\frac{1}{2^{8}}{\rm Tr}{\mathcal{L}^{4}}+\mathbb{D}_{\Phi}\,.$

Where $\mathbb{D}_{\Psi}$ denote removable terms that we do not write
explicitly.

For $I^{(3)}$ the situation is much simpler since there are no induced terms
to keep track of. Therefore, we can just simply eliminate all removable
contributions for free, knowing that behind the scenes we picked a suitable
$\delta^{(3)}\Psi$. Moreover, from (3.33) we see that $I^{(3)}$ is purely
removable which leads to

$\alpha^{\prime}{}^{3}I^{(3)}+\alpha^{\prime}{}^{3}E_{\Psi}^{(0)}\cdot\delta^{(3)}\Psi=0\,.$
(A.17)

Finally, for the fourth order we need the induced terms coming from the
cleaning process at order $\alpha^{\prime}{}^{2}$. Together with (A.15), we
also require the quadratic terms
$\Delta_{2}I^{(0)}\cdot(\delta^{(2)}\Psi)^{2}$, which can be computed by
taking two consecutive variations of the first order action and using (A.16).
However, we can notice that the only non-removable variations we have are
$\frac{1}{n}\delta^{(2)}n\propto\delta^{(2)}\Phi\propto{\rm
Tr}{\mathcal{L}^{4}}$. Then, whatever $\Delta_{2}I^{(0)}$ is, it needs to be
an $O(d,d)$ plus 1-dimensional diffeo invariant object with 2 derivatives.
There are only three possible objects with these characteristics: ${\cal
D}^{2}\Phi,({\cal D}\Phi)^{2}$ and ${\rm Tr}{\mathcal{L}^{2}}$, all of them
removable. As a consequence, without computing it, we know already that
$\Delta_{2}I^{(0)}\cdot(\delta^{(2)}\Psi)^{2}$ with $\delta^{(2)}\Psi$ given
by (A.16), is removable at order $\alpha^{\prime}{}^{4}$.

Adding to $I^{(4)}$ (given in (3.33)) the induced terms from (A.15) together
with the (removable) $\Delta_{2}I^{(0)}\cdot(\delta^{(2)}\Psi)^{2}$ we get

$\alpha^{\prime}{}^{4}\Big{(}I^{(4)}+E_{\Psi}^{(2)}\cdot\delta^{(2)}\Psi+\frac{1}{2}\Delta_{2}I^{(0)}\cdot\big{(}\delta^{(2)}\Psi\big{)}^{2}\Big{)}=\alpha^{\prime}{}^{4}\int
dt\,n\,e^{-\Phi}\,\left\\{\frac{1}{3.2^{13}}{\rm Tr}{\mathcal{L}^{4}}{\rm
Tr}{\mathcal{L}^{6}}+\mathbb{L}\right\\}\,.$ (A.18)

Finally, we can finish the process of taking the action to minimal form by
choosing a particular $\delta^{(4)}\Psi$ (that we do not need to keep track
of) to eliminate all removable terms at order $\alpha^{\prime}{}^{4}$,
$\mathbb{L}$. The final result of this systematic procedure is the HSZ action
in the cosmological classification (3.35).

## References

  * [1] Robert H. Brandenberger and C. Vafa.  Superstrings in the Early Universe.  Nucl. Phys. B, 316:391–410, 1989.
  * [2] Arkady A. Tseytlin and C. Vafa.  Elements of string cosmology.  Nucl. Phys. B, 372:443–466, 1992.
  * [3] G. Veneziano.  Scale factor duality for classical and quantum strings.  Phys. Lett. B, 265:287–294, 1991.
  * [4] M. Gasperini, M. Maggiore, and G. Veneziano.  Towards a nonsingular pre - big bang cosmology.  Nucl. Phys. B, 494:315–330, 1997.
  * [5] M. Gasperini and G. Veneziano.  The Pre - big bang scenario in string cosmology.  Phys. Rept., 373:1–212, 2003.
  * [6] Heliudson Bernardo, Robert Brandenberger, and Guilherme Franzmann.  String cosmology backgrounds from classical string geometry.  Phys. Rev. D, 103(4):043540, 2021.
  * [7] Heliudson Bernardo, Robert Brandenberger, and Guilherme Franzmann.  Solution of the Size and Horizon Problems from Classical String Geometry.  JHEP, 10:155, 2020.
  * [8] Carmen A. Núñez and Facundo Emanuel Rost.  New non-perturbative de Sitter vacua in $\alpha^{\prime}$-complete cosmology.  JHEP, 03:007, 2021.
  * [9] W. Siegel.  Superspace duality in low-energy superstrings.  Phys. Rev. D, 48:2826–2837, 1993.
  * [10] Chris Hull and Barton Zwiebach.  Double Field Theory.  JHEP, 09:099, 2009.
  * [11] Olaf Hohm, Chris Hull, and Barton Zwiebach.  Generalized metric formulation of double field theory.  JHEP, 08:008, 2010.
  * [12] Olaf Hohm, Dieter Lüst, and Barton Zwiebach.  The Spacetime of Double Field Theory: Review, Remarks, and Outlook.  Fortsch. Phys., 61:926–966, 2013.
  * [13] Olaf Hohm, Warren Siegel, and Barton Zwiebach.  Doubled $\alpha^{\prime}$-geometry.  JHEP, 02:065, 2014.
  * [14] Tomas Codina, Olaf Hohm, and Diego Marques.  String Dualities at Order $\alpha^{\prime\,3}$.  Phys. Rev. Lett., 126(17):171602, 2021.
  * [15] Tomas Codina, Olaf Hohm, and Diego Marques.  General string cosmologies at order $\alpha$’3.  Phys. Rev. D, 104(10):106007, 2021.
  * [16] Roberto Bonezzi, Tomas Codina, and Olaf Hohm.  Duality invariant string beta functions at two loops.  JHEP, 02:109, 2022.
  * [17] Yu-tin Huang, Warren Siegel, and Ellis Ye Yuan.  Factorization of Chiral String Amplitudes.  JHEP, 09:101, 2016.
  * [18] Eduardo Casali and Piotr Tourkine.  On the null origin of the ambitwistor string.  JHEP, 11:036, 2016.
  * [19] Olaf Hohm and Barton Zwiebach.  Green-Schwarz mechanism and $\alpha^{\prime}$-deformed Courant brackets.  JHEP, 01:012, 2015.
  * [20] Olaf Hohm and Barton Zwiebach.  Double field theory at order $\alpha^{\prime}$.  JHEP, 11:075, 2014.
  * [21] Usman Naseer and Barton Zwiebach.  Three-point Functions in Duality-Invariant Higher-Derivative Gravity.  JHEP, 03:147, 2016.
  * [22] Olaf Hohm and Barton Zwiebach.  T-duality Constraints on Higher Derivatives Revisited.  JHEP, 04:101, 2016.
  * [23] Olaf Hohm and Barton Zwiebach.  Duality invariant cosmology to all orders in $\alpha$’.  Phys. Rev. D, 100(12):126011, 2019.
  * [24] Olaf Hohm, Usman Naseer, and Barton Zwiebach.  On the curious spectrum of duality invariant higher-derivative gravity.  JHEP, 08:173, 2016.
  * [25] David J. Gross.  High-Energy Symmetries of String Theory.  Phys. Rev. Lett., 60:1229, 1988.
  * [26] Joseph J. Atick and Edward Witten.  The Hagedorn Transition and the Number of Degrees of Freedom of String Theory.  Nucl. Phys. B, 310:291–334, 1988.
  * [27] Alexei A. Starobinsky.  A New Type of Isotropic Cosmological Models Without Singularity.  Phys. Lett. B, 91:99–102, 1980.
  * [28] Eric A. Bergshoeff, Olaf Hohm, and Paul K. Townsend.  Massive Gravity in Three Dimensions.  Phys. Rev. Lett., 102:201301, 2009.
  * [29] Ashoke Sen.  O(d) x O(d) symmetry of the space of cosmological solutions in string theory, scale factor duality and two-dimensional black holes.  Phys. Lett. B, 271:295–300, 1991.
  * [30] Krzysztof A. Meissner.  Symmetries of higher order string gravity actions.  Phys. Lett. B, 392:298–304, 1997.
  * [31] Eric Lescano and Diego Marques.  Second order higher-derivative corrections in Double Field Theory.  JHEP, 06:104, 2017.
  * [32] Kasper Peeters.  Introducing Cadabra: A Symbolic computer algebra system for field theory problems.  arXiv:0701238 [hep-th].
  * [33] Olaf Hohm and Barton Zwiebach.  Double metric, generalized metric, and $\alpha$’ -deformed double field theory.  Phys. Rev. D, 93(6):064035, 2016.
  * [34] M. Gasperini and G. Veneziano.  O(d,d) covariant string cosmology.  Phys. Lett. B, 277:256–264, 1992.
  * [35] Heliudson Bernardo, Robert Brandenberger, and Guilherme Franzmann.  O$(d,d)$ covariant string cosmology to all orders in $\alpha^{\prime}$.  JHEP, 02:178, 2020.
  * [36] Olaf Hohm and Allison F. Pinto.  Cosmological Perturbations in Double Field Theory.  arXiv:2207.14788 [hep-th].
  * [37] Przemyslaw Bieniek, Jan Chojnacki, Jan H. Kwapisz, and Krzysztof A. Meissner.  Stability of the non-perturbative $O(D,D)$ de-Sitter spacetime. The isotropic case.  arXiv:2208.06010 [hep-th].