# Tilted non-spatially-flat inflation

Bharat Ratra<EMAIL_ADDRESS>Department of Physics, Kansas State
University, Manhattan, Kansas 66506, USA

###### Abstract

We construct non-linear inflaton potential energy densities that describe not-
necessarily very-slowly-rolling closed and open inflation models, and compute
tilted primordial spatial inhomogeneity power spectra that follow from quantum
mechanical fluctuations during inflation in these models. These tilted power
spectra differ from those that have previously been used to study cosmological
data in non-flat cosmological models.

##  I Introduction

If general relativity provides an adequate description of gravity on
cosmological scales — and there is no strong evidence indicating otherwise —
dark energy is the dominant contributor to the current cosmological energy
budget and powers the observed late-time accelerating cosmological expansion.
Earlier on, prior to a redshift $z\sim 0.75$, non-relativistic (cold dark and
baryonic) matter was the dominant contributor to the energy budget and was
responsible for the observed earlier-time decelerating cosmological expansion.
The simplest model consistent with these observations is the flat $\Lambda$CDM
model (Peebles1984, ), the current “standard” cosmological model. In this
model spatial hypersurfaces are chosen to be flat and the dark energy is the
cosmological constant $\Lambda$, with the next biggest contributor to the
current cosmological energy budget being cold dark matter (CDM). For reviews
see Refs. (CosmoRev, ).

The flat $\Lambda$CDM model is consistent with a variety of observational
constraints, including cosmic microwave background (CMB) anisotropy
observations (Planck2020, ), baryon acoustic oscillation (BAO) data
(eBOSS2020, ), Hubble parameter ($H(z)$) measurements (Farooqetal2016, ), and
Type Ia supernova (SNIa) apparent magnitude observations (Scolnicetal2018, ).
The standard model is also consistent with more recent constraints from probes
of the intermediate redshift Universe, that include data between $z\sim 2.3$
of the highest redshift BAO observations and $z\sim 1100$ of the CMB data.
However, the intermediate redshift data constraints are not yet as restrictive
as the lower-redshift BAO, $H(z)$, and SNIa ones, nor as restrictive as the
higher-redshift CMB anisotropy ones. These intermediate-redshift constraints
include those from HII galaxy apparent magnitude versus redshift data (HII, ),
angular size as a function of redshift measurements (AngSize, ), quasar X-ray
and UV flux observations (QSO, ), and gamma-ray burst data (GRB, ).

While most current measurements are not inconsistent with the spatially-flat
$\Lambda$CDM standard model, they also do not rule out mildly curved spatial
hypersurfaces or, weakly varying in time and space, dynamical dark energy.
Near-future measurements are anticipated to provide significantly more
restrictive constraints that should help distinguish between the options and
better determine cosmological parameter values (nearfuture, ).

There are however some suggestions of inconsistencies between observations and
the standard flat $\Lambda$CDM model. For example, differences between (model-
dependent) measurements of the Hubble constant $H_{0}$ could be an indication
of a problem with the standard model.111For recent reviews see Ref. (CosmoRev,
). An early median statistics estimate, (ChenRatra2011, ), $H_{0}=68\pm 2.8$
km s-1 Mpc-1, is consistent with a number of more recent $H_{0}$ measurements
made using a variety of methods (H0, ), including from CMB anisotropy data,
$H_{0}=67.36\pm 0.54$ km s-1 Mpc-1 (Planck2020, ). On the other hand, some
local measurements of the expansion rate favor a value significantly larger
than the CMB one, $H_{0}=73.2\pm 1.3$ km s-1 Mpc-1 (Riessetal2021, ).222We
note that some other local expansion rate determinations of $H_{0}$ are
slightly lower and have larger error bars (localH0, ). Similar issues affect
measurements of other parameters but the difference between $H_{0}$
measurements is the most significant.

Given such potential inconsistencies, and given the improving quality and
amount of data, there is significant interest in studying cosmologies that
have a free parameter or two more than the flat $\Lambda$CDM model. A widely
considered option is dynamical dark energy that mildly varies in time and
space. Scalar field dynamical dark energy ($\phi$CDM) models are a popular
example (phiCDM, ). Allowing for non-zero spatial curvature is another option
now under study. For recent discussions of observational constraints on
spatial curvature and dark energy dynamics from a variety of different data
sets, see Refs. (curvatureconstraints, ; DEconstraints, ).333Compared to the
cosmological constant, dynamical dark energy density evolves more similarly to
spatial curvature energy density and this results in weaker constraints on
both new parameters when compared to the case when either only non-zero
spatial curvature or only dark energy density dynamics is assumed
(timevaryingnonflatconstraints, ). For recent discussions of non-flat
cosmological models, see Refs. (Baumgartner, ).

CMB anisotropy data provide the most restrictive constraints on cosmological
models. To use these data to constrain cosmological parameters of a model
requires knowing the primordial power spectrum of spatial inhomogeneities as a
function of wavenumber for the model. In the inflation scenario
(InflationGuth, ; InflationOthers, ) quantum mechanical zero-point
fluctuations in the inflaton field during inflation generate the spatial
inhomogeneities InflationInhomogeneities ; Fischleretal1985 . If inflation
lasts for a long time spatial curvature is redshifted away to insignificance
(and this is by far the most commonly considered case). In this case, if the
inflaton slow-rolls down a relatively-flatter inflaton potential energy
density the cosmological scale factor grows exponentially in time (this is
spatially-flat de Sitter inflation) and the resulting primordial power
spectrum is close to scale invariant, (HPYZ, ), with very little tilt. It is
possible to increase the power spectral tilt by choosing an inflaton potential
energy density that causes the inflaton to evolve more rapidly during
inflation and makes the scale factor grow only as a power of time (this is
spatially-flat power-law inflation) (LucchinMatarrese1985, ; Ratra1992, ;
Ratra1989, ).

Gott (Gott1982, ) generalized inflation to the open cosmological model. In
this open-bubble inflation model a spatially-open bubble nucleates and the
interior inflates for only a limited time so as to not redshift away all
spatial curvature. If necessary, an earlier, pre open-bubble nucleation, epoch
of less-limited spatially-flat inflation can be used to produce spatial
homogeneity. Alternately, a slow enough open-bubble nucleation process might
ensure that the interior of the open bubble is sufficiently spatially
homogeneous.

Hawking’s prescription for the initial quantum state of the universe
(Hawking1984, ) — that the functional integral include only those field
configurations which are regular on the Euclidean section — suggests that the
universe nucleated as a closed de Sitter Lanczos (inflation) model on the
Lorentzian section (Hawking1984, ; Ratra1985, ). The equator of the Euclidean
(de Sitter Lanczos) four sphere is identified with the waist of the Lorentzian
de Sitter Lanczos hyperboloid, which is where the nucleation occurs
(Hawking1984, ; Ratra1985, ).444For variants of this picture see Refs. (Linde,
; Grattonetal2002, ; LasenbyDoran2005, ; Gordonetal2021, ). A slow enough
nucleation process might ensure a sufficiently spatially homogeneous closed
inflating de Sitter Lanczos model. Again, inflation can occur for only a
limited amount of time so as to not redshift away all spatial curvature.555For
a discussions of the nucleation of open and closed cosmological models, see
Refs. (Cespedesetal2020, ) and references therein.

To compute the power spectrum of spatial inhomogeneities generated by quantum
fluctuations during inflation in a given model requires the solution of the
spatial inhomogeneities linear perturbation equations and a set of initial
conditions. Spatially flat, open, and closed de Sitter spacetimes have as
large a symmetry group as Minkowski spacetime; spatially-flat power-law
inflation and the non-flat inflation models we study here have less symmetry
than the flat, open, and closed de Sitter inflation models. The initial
condition prescription we utilize here give inflaton scalar field two-point
functions in flat and closed de Sitter inflation models that have the
symmetries of those spacetimes, (Ratra1985, ; Fischleretal1985, ; Ratra2017,
). This initial condition prescription has also been used to compute the
primordial power spectrum in the spatially-flat power-law inflation model
(Ratra1989, ; Ratra1992, ), giving a tilted primordial power spectrum that is
in agreement with the result of Ref. (LucchinMatarrese1985, ). All these
inflation model spacetimes, as well as the two we consider in this paper, are
conformally flat, i.e., when expressed in terms of conformal time their line-
elements are proportional to the Minkowski spacetime line-element.

The initial condition prescription we use here is that during inflation, for
large wavenumbers, inside the horizon, at early time, the conformally rescaled
inflaton scalar field modes, as a function of conformal time, should be
quantum-mechanically normalized simple harmonic oscillators.

In the open-bubble inflation model this initial condition prescription
(Ratra1994, ) results in a late-time energy density inhomogeneity power
spectrum (RatraPeebles1995, ; RatraPeebles1994, ) that is the generalization
to the open inflation case (LythStewart1990, )666Lyth and Stewart
(LythStewart1990, ) also use this initial condition prescription in their more
approximate computation of the wavenumber dependence of the power spectrum of
the scale-invariant spectrum of the flat model (HPYZ, ). In a variant of the
open-bubble inflation model, that includes an initial epoch of spatially-flat
de Sitter inflation, applying the initial condition prescription in the first
epoch and following the computation through the bubble nucleation process,
results in a primordial power spectrum that is observationally
indistinguishable from that of the case when the initial conditions are
applied inside the second, open-bubble epoch, (BucherYamamotoetal1995, ;
Gorskietal1998, ).777For other early papers on open inflation, see Refs.
(OpenInflation, ).

This initial condition prescription has also been used in a computation of the
primordial power spectrum in the closed de Sitter inflation model (Ratra1985,
; Ratra2017, ). In this case this initial condition prescription is equivalent
to Hawking’s (Hawking1984, ) prescription of only including field
configurations regular on the Euclidean section (Ratra1985, ). It also leads
to a de Sitter invariant ground state inflaton scalar field two-point
correlation function (Ratra1985, ). The wavenumber dependence of the resulting
primordial power spectrum is the generalization of the scale-invariant
spectrum in the spatially-flat case (HPYZ, ) to the closed universe
(closedscaleinvariant, ).888There are other computations of primordial spectra
in the closed de Sitter inflation model (LasenbyDoran2005, ; Massoetal2008, ;
AsgariAbbassi2015, ; Bongaetal2016, ; Handley2019, ; KieferVardanyan2021, ),
using different initial conditions compared to our prescription. We emphasize
that the initial conditions we use in the closed de Sitter case results in a
scalar field perturbation two-point function that is de Sitter invariant
(Ratra1985, ).

The open and closed inflation model primordial power spectra computations of
Refs. (RatraPeebles1995, ; Ratra2017, ) were done in models that had an
inflaton ($\Phi$) potential energy density $\propto(1-\epsilon\Phi)$, where
$\epsilon$ is a small constant. These are very-slow-roll models and so the
resulting power spectra are untilted on small scales for infinitesimal
$\epsilon$. When these power spectra were used in a non-flat $\Lambda$CDM
model analysis of the Planck 2015 CMB anisotropy data (Planck2016, ) it was
found that these data favored closed spatial geometry (Oobaetal2018, ;
ParkRatra2019, ), even in combination with BAO, $H(z)$, SNIa, and other non-
CMB data, where these data jointly favored about a 1% spatial curvature energy
density contribution to the cosmological energy budget at 5$\sigma$
significance (ParkRatra2020, ).

A more correct analysis of the CMB anisotropy data requires tilted open and
closed model primordial power spectra. Such spectra would be generated by
quantum mechanical inflaton fluctuations in open and closed inflation models
with non-linear inflaton potential energy densities, unlike the linear
potential energy density function assumed in the analyses of Refs.
(RatraPeebles1995, ; Ratra2017, ). Pending such a computation, data analyses
have been performed utilizing the primordial power spectra of Refs.
(RatraPeebles1995, ; Ratra2017, ) multiplied by $k^{n-1}$, where $k$ is the
wavenumber and $n$ is the power spectral index (with $n=1$ being the scale-
invariant case in the flat model) (LesgourguesTram2014, ). Using this
phenomenological primordial power spectrum to define a tilted non-flat
$\Lambda$CDM model for the analysis of CMB anisotropy data, in this model the
Planck 2018 data (Planck2020, ) favors positive spatial curvature contributing
about 1% to the cosmological energy budget at 1.6$\sigma$, but when BAO data
are added to the mix the result is consistent with flat spatial hypersurfaces
(Planck2020, ); a similar result was originally found from the Planck 2015
data (Planck2016, ). It is of interest to determine whether the power spectra
(LesgourguesTram2014, ) used in these analyses (Planck2016, ; Planck2020, )
can be generated by inflaton quantum fluctuations in non-flat non-very-slow-
roll inflation models that are closed but very close to flat, deviating from
flatness at only the $\sim 1$% level. A recent numerical study in closed
inflation models that computes power spectra generated for a few different
initial conditions finds that it is possible to generate spectra of the form
assumed in Refs. (Planck2016, ; Planck2020, ), (Guthetal2022, ), at least in
the closed case.

Here we consider open and closed inflation models with non-linear inflaton
potential energy densities. In this paper we generalize the exponential
potential energy density of the flat-space power-law inflation model
(LucchinMatarrese1985, ; Ratra1992, ; Ratra1989, ) to inflaton potential
energy densities that allow for not-necessarily very-slow-roll inflation in
open and closed models. In the very-slow-roll limit these potential energy
densities reduce to that $\propto(1-\epsilon\Phi)$ used in the open and closed
inflation models of Refs. (RatraPeebles1995, ; Ratra2017, ), while at large
$\Phi$ they become the exponential potential energy density used in the
spatially-flat power-law inflation model of Refs. (LucchinMatarrese1985, ;
Ratra1992, ; Ratra1989, ). Here we are interested in a non-very-slow-roll
limit of these models, and potentially in the parameter-space range where they
deviate from spatial flatness at the $\sim 1$% level.

Our computed primordial power spectra of spatial inhomogeneities — that result
from quantum mechanical zero-point fluctuations during the inflation epoch in
these tilted closed and open models — differ from power spectra that have
previously been used to analyze observational data in closed and open
cosmological models. These power spectra have been used in the analyses of CMB
anisotropy and other data, (deCruzetal2022, ). It is interesting that there
appears to be some additional ambiguity in the form of non-flat inflation
model power spectra, caused by the ambiguity in the form of the assumed non-
flat inflation initial conditions, compared to what happens in the flat
inflation case.

In Section II we summarize the background geometry of the closed and open
models and the Einstein-scalar-field model equations of motion. For more
detailed descriptions see Refs. (RatraPeebles1995, ; Ratra2017, ). In Section
III we determine the inflaton potential energy densities we use and solve the
spatially homogeneous background equations of motion in the inflation epoch of
the closed and open models. We solve the linear perturbation equations in
Section IV, where we compute the late-time primordial power spectra during
inflation in the closed and open models. We conclude in Section V. Appendix A
list some results in the spatially-flat tilted inflation model, that we use
for comparison to some of our results on smaller scales during inflation in
the non-flat models when spatial curvature is unimportant. Appendices B and C
describe primordial power spectra definitions and conventions in the flat,
closed, and open models.

##  II Technical Preliminaries

###  II.1 Spatially homogeneous background geometries

The positive spatial curvature (closed) FLRW model has line element

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dt^{2}-a^{2}(t)H_{ij}(\vec{x})dx^{i}\,dx^{j}$ $\displaystyle=$ $\displaystyle
dt^{2}-a^{2}(t)\left[d\chi^{2}+{\rm sin}^{2}\\!(\chi)\left\\{d\theta^{2}+{\rm
sin}^{2}\\!(\theta)\,d\phi^{2}\right\\}\right],$

where $a(t)$ is the cosmological scale factor, $H_{ij}(\vec{x})$ is the metric
on the closed spatial hypersurfaces, the ‘radial’ coordinate $0\leq\chi<\pi$,
and $\theta,\phi$ are the usual angular coordinates on the two-sphere. The
square of the distance between two points, $(t,\chi,\theta,\phi)$ and
$(t,\chi^{\prime},\theta^{\prime},\phi^{\prime})$, is

$\displaystyle\sigma^{2}=2a^{2}(t)\left[-1+{\rm cos}(\gamma_{3})\right],$ (2)
$\displaystyle{\rm cos}(\gamma_{3})={\rm cos}(\chi){\rm
cos}(\chi^{\prime})+{\rm sin}(\chi){\rm sin}(\chi^{\prime}){\rm
cos}(\gamma_{2}),$ (3)

where $\gamma_{2}$ is the usual angle between the two points $(\theta,\phi)$
and $(\theta^{\prime},\phi^{\prime})$ on the two-sphere,

$\displaystyle{\rm cos}(\gamma_{2})={\rm cos}(\theta){\rm
cos}(\theta^{\prime})+{\rm sin}(\theta){\rm sin}(\theta^{\prime}){\rm
cos}(\phi-\phi^{\prime}).$ (5)

The negative spatial curvature (open) FLRW model has line element

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
dt^{2}-a^{2}(t)H_{ij}(\vec{x})dx^{i}\,dx^{j}$ $\displaystyle=$ $\displaystyle
dt^{2}-a^{2}(t)\left[d\chi^{2}+{\rm sinh}^{2}\\!(\chi)\left\\{d\theta^{2}+{\rm
sin}^{2}\\!(\theta)\,d\phi^{2}\right\\}\right],$

where $H_{ij}(\vec{x})$ is now the metric on the open spatial hypersurfaces,
and $\chi\ (0\leq\chi<\infty),\theta,$ and $\phi$ are defined above. The
square of the distance between two points, $(t,\chi,\theta,\phi)$ and
$(t,\chi^{\prime},\theta^{\prime},\phi^{\prime})$, is

$\displaystyle\sigma^{2}=2a^{2}(t)\left[1-{\rm cosh}(\gamma_{3})\right],$ (7)
$\displaystyle{\rm cosh}(\gamma_{3})={\rm cosh}(\chi){\rm
cosh}(\chi^{\prime})-{\rm sinh}(\chi){\rm sinh}(\chi^{\prime}){\rm
cos}(\gamma_{2}),$ (8)

and $\gamma_{2}$ is defined in Eq. (5).

###  II.2 Einstein-scalar-field model conventions

The Einstein-scalar-field action, for metric tensor $g_{\mu\nu}$ and inflaton
$\Phi$, is

$\displaystyle S=$ (10) $\displaystyle{m_{p}{}^{2}\over 16\pi}\int
dt\,d^{3}\\!x\sqrt{-g}\left[-R+{1\over
2}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-{1\over 2}V(\Phi)\right].$

Here $m_{p}=G^{-1/2}$ is the Planck mass and $V$ is the scalar field potential
energy density. Varying, we find the inflaton and gravitation equations of
motion,

$\displaystyle{1\over\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)+{1\over
2}V^{\prime}(\Phi)=0,$ (11) $\displaystyle R_{\mu\nu}={8\pi\over
m_{p}{}^{2}}\left(T_{\mu\nu}-{1\over 2}g_{\mu\nu}T\right),$ (12)

where a prime denotes a derivative with respect to $\Phi$ and $T$ is the trace
of the stress-energy tensor

$\displaystyle T_{\mu\nu}=$ $\displaystyle{m_{p}{}^{2}\over
16\pi}\left[\partial_{\mu}\Phi\partial_{\nu}\Phi-{1\over
2}g_{\mu\nu}\left\\{g^{\lambda\rho}\partial_{\lambda}\Phi\partial_{\rho}\Phi-V(\Phi)\right\\}\right].$
(13)

To derive the equations of motion for the spatially homogeneous background
fields and for the spatial inhomogeneities, we linearize Eqs. (11) – (II.2)
about an open or closed FLRW model and a spatially homogeneous scalar field.
We work in synchronous gauge, with line element

$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left[H_{ij}(\vec{x})-h_{ij}(t,\vec{x})\right]dx^{i}dx^{j},$
(14)

where the background metric on the closed [open] spatial hypersurfaces,
$H_{ij}$, is given in Eq. (II.1) [Eq. (II.1)], and the metric perturbations
are denoted by $h_{ij}$. The expansion for the scalar field is

$\displaystyle\Phi(t,\vec{x})=\Phi_{b}(t)+\phi(t,\vec{x}),$ (15)

where $\Phi_{b}$ and $\phi$ are the spatially homogeneous and inhomogeneous
parts of the inflaton field (the inflaton perturbation $\phi$ should not be
confused with the angular variable $\phi$ of Sec. II.1).

##  III Tilted Closed and Open Inflation Models and Spatially Homogeneous
Background Solutions

The Einstein-scalar-field model equations of motion for the spatially
homogeneous background fields, derived in Sec. III.A of Ref.
(RatraPeebles1995, ) for the open case and in Sec. III.A of Ref. (Ratra2017, )
for the closed case, are

$\displaystyle\ddot{\Phi}_{b}+3{\dot{a}\over a}\dot{\Phi}_{b}+{1\over
2}V^{\prime}(\Phi_{b})=0,$ (16) $\displaystyle\left({\dot{a}\over
a}\right)^{2}={1\over
12}\left[\dot{\Phi}_{b}{}^{2}+V(\Phi_{b})\right]+{\kappa^{2}\over a^{2}},$
(17) $\displaystyle{\ddot{a}\over a}=-{1\over 6}\dot{\Phi}_{b}{}^{2}+{1\over
12}V(\Phi_{b}),$ (18)

where an overdot denotes a derivative with respect to time and
$\kappa^{2}=+1(-1)$ for open (closed) spatial hypersurfaces.

Motivated by the power-law expansion inflation model in the spatially-flat
case, (LucchinMatarrese1985, ; Ratra1989, ; Ratra1992, ), for suitable scalar
field potential energy densities, discussed below, the background Friedmann
equation (17) during closed or open inflation becomes

$\displaystyle\left({\dot{a}\over a}\right)^{2}={Q\over
a^{q}}+{\kappa^{2}\over a^{2}},$ (19)

where $Q$ and $q$ are constants and $0<q<2$ for inflation. In the closed case
where $\kappa^{2}=-1$, we require, at the waist at $t=t_{i}$, the initial
condition $\dot{a}(t_{i})=0$, so the right hand side of Eq. (19) must also
vanish at the waist, which results in $Qa_{i}^{p}=1$ where $a_{i}=a(t_{i})$
and $p=2-q$. In the closed case the inflation model includes only the $t\geq
t_{i}$ part of the spacetime.

The integral of Eq. (19) is

$\displaystyle\sqrt{\kappa^{2}}(t-t_{0})$ $\displaystyle=$ $\displaystyle
a\,{}_{2}F_{1}(1/2,1/p;1+1/p;-Qa^{p}/\kappa^{2})$ $\displaystyle-
a_{0}\,{}_{2}F_{1}(1/2,1/p;1+1/p;-Qa_{0}^{p}/\kappa^{2}),$

where ${}_{2}F_{1}$ is the Gauss hypergeometric function, see Ch. 15 of Ref.
(AS, ), and $a_{0}=a(t=t_{0})$ is the constant of integration.

In the open case, this is

$\displaystyle t=a\,{}_{2}F_{1}(1/2,1/p;1+1/p;-Qa^{p})+{\rm constant},$ (21)

and in the closed case, where $t_{i}$ is at the waist,

$\displaystyle t-t_{i}$ $\displaystyle=$ $\displaystyle
ia\,{}_{2}F_{1}(1/2,1/p;1+1/p;Qa^{p})$ (22) $\displaystyle-
iQ^{-1/p}\sqrt{\pi}{\Gamma(1+1/p)\over\Gamma(1/2+1/p)},$

where $\Gamma$ is the Euler Gamma function.

In the flat limit where $\kappa^{2}\rightarrow 0$, or
$Qa^{p}/\kappa^{2}\rightarrow\infty$, Eq. (III) becomes
$a\propto(t-t_{0})^{2/q}$, the usual flat-space tilted power-law inflation
result, see Eq. (2.5) of Ref. (Ratra1992, ). In the $q\rightarrow 0$ limit
Eqs. (21) and (22) reduce to the correct open and closed slow-roll untilted de
Sitter inflation relations, $a(t)={{\rm sinh}(\sqrt{Q}(t-t_{0}))}/\sqrt{Q}$
and $a(t)={{\rm cosh}(\sqrt{Q}(t-t_{0}))}/\sqrt{Q}$, see Eq. (4.10) of Ref.
(RatraPeebles1995, ) and Eq. (105) of Ref. (Ratra2017, ).

In the open case, conformal time

$\displaystyle{\tilde{t}}={1\over p\sqrt{\kappa^{2}}}{\rm
ln}\left[{\sqrt{Qa^{p}/\kappa^{2}+1}-1\over\sqrt{Qa^{p}/\kappa^{2}+1}+1}\right],$
(23)

where $-\infty\leq{\tilde{t}}\leq 0$, while in the closed case, conformal time

$\displaystyle{\tilde{t}}={2\over p\sqrt{-\kappa^{2}}}{\rm
tan}^{-1}\left[\sqrt{Qa^{p}/(-\kappa^{2})-1}\right],$ (24)

where $0\leq{\tilde{t}}\leq\pi/p$.

With the scalar field potential energy density in the open case,

$\displaystyle V(\Phi)={2(6-q)Q\over(\kappa^{2}/Q)^{q/p}\left[{\rm
sinh}\left\\{p\Phi/{\sqrt{8q}}\right\\}\right]^{2q/p}}$ (25)

and in the closed case,

$\displaystyle V(\Phi)={2(6-q)Q\over(-\kappa^{2}/Q)^{q/p}\left[{\rm
cosh}\left\\{p\Phi/{\sqrt{8q}}\right\\}\right]^{2q/p}},$ (26)

the homogeneous part of the scalar field equation of motion (16) and the
Friedmann equation (17)999More precisely, Eqs. (27) and (28) reduce the
Friedmann equation (17) to Eq. (19) which is solved by Eqs. (21) and (22). are
satisfied by, in the open case,

$\displaystyle a=(\kappa^{2}/Q)^{1/p}\left[{\rm
sinh}\left\\{p\Phi_{b}/{\sqrt{8q}}\right\\}\right]^{2/p},$ (27)

and in the closed case,

$\displaystyle a=(-\kappa^{2}/Q)^{1/p}\left[{\rm
cosh}\left\\{p\Phi_{b}/{\sqrt{8q}}\right\\}\right]^{2/p}.$ (28)

In the closed case we have used the initial condition $\Phi_{b}(t_{i})=0$.

In the untilted, slow-roll, small $q$ limit, both scalar field potential
energy densities, Eqs. (25) and (26), become $\propto(1-\epsilon\Phi)$, where
$\epsilon=\sqrt{q/2}$, which are the potential energy densities used in
untilted very-slow-roll open (RatraPeebles1995, ) and closed (Ratra2017, )
inflation model computations. However. the $a(\Phi_{b})$ equations (27) and
(28) appear to not behave sensibly at $q=0$ and so it appears that these
models do not make sense at $q=0$. For small $q$ the potential energy
densities, Eqs. (25) and (26), change only slowly with $\Phi$ and at $q=0$
they are flat. At $q=0$ the scalar field will not move if is initially at
rest, but $a(t)$ grows, resulting in a breakdown of Eqs. (27) and (28) at
$q=0$.

In the limit that $\Phi$ is large, both scalar field potential energy
densities, Eqs. (25) and (26), become $\propto{\rm exp}(-\sqrt{q/2}\Phi)$,
which is the potential energy density used in the standard flat-space tilted
inflation model (LucchinMatarrese1985, ; Ratra1989, ; Ratra1992, ).

##  IV Linear Scalar Perturbations During Inflation

###  IV.1 Synchronous gauge linear scalar perturbation equations

The scalar parts of the synchronous gauge inflation-epoch linear perturbation
equations in spatial momentum space are derived in Secs. II and III.A of Refs.
(RatraPeebles1995, ; Ratra2017, ) for the open and closed cases.

With $-A(A+2)$, integer $A=0,1,2\cdots$ ($A\geq 2$ modes are physical), being
the closed model spatial Laplacian eigenvalue, and $-(A^{2}+1)$, $A>0$, being
the open model spatial Laplacian eigenvalue, we define $k^{2}=A^{2}+1$ and
$\bar{k}^{2}=A^{2}+4$ for the open case and $k^{2}=A(A+2)$ and
$\bar{k}^{2}=(A-1)(A+3)$ for the closed case.

The linear scalar perturbation equations for the spatial momentum space scalar
field $\phi(A,t)$, trace of the metric perturbation $h(A,t)$ (the perturbation
to the size of the proper volume element), and the trace-free part of the
metric perturbation ${\cal H}(A,t)$ (the shearing perturbation of the volume
element) modes are

$\displaystyle\ddot{\phi}+3{\dot{a}\over a}\dot{\phi}+{k^{2}\over
a^{2}}\phi+{1\over 2}V^{\prime\prime}(\Phi_{b})\phi={1\over
2}\dot{h}\dot{\Phi}_{b},$ (29) $\displaystyle\ddot{h}+2{\dot{a}\over
a}\dot{h}=2\dot{\Phi}_{b}\dot{\phi}-{1\over 2}V^{\prime}(\Phi_{b})\phi,$ (30)
$\displaystyle\dot{\cal H}={k^{2}\over\bar{k}^{2}}\left[{3\over
2}\dot{\Phi}_{b}\phi-\dot{h}\right],$ (31)
$\displaystyle\ddot{h}+6{\dot{a}\over a}\dot{h}$ $\displaystyle+$
$\displaystyle{(\kappa^{2}+\bar{k}^{2})\over a^{2}}h+\ddot{\cal
H}+3{\dot{a}\over a}\dot{\cal H}$ (32) $\displaystyle+$
$\displaystyle{(\kappa^{2}+\bar{k}^{2})\over a^{2}}{\cal H}+{3\over
2}V^{\prime}(\Phi_{b})\phi=0,$ $\displaystyle\ddot{\cal H}+3{\dot{a}\over
a}\dot{\cal H}-{k^{2}\over 3a^{2}}{\cal H}-{k^{2}\over 3a^{2}}h=0.$ (33)

###  IV.2 Synchronous gauge linear scalar perturbation solutions

Using Eqs. (19) and (25)—(28), Eqs. (29) and (30) can be re-expressed as

$\displaystyle[Qa^{p}+\kappa^{2}]{d^{2}\\!\phi\over da^{2}}+{1\over
2a}[(8-q)Qa^{p}+6\kappa^{2}]{d\phi\over da}+{k^{2}\over a^{2}}\phi+$ (34)
$\displaystyle\ {(6-q)\over
4a^{2}}[2qQa^{p}+(2+q)\kappa^{2}]\phi={\sqrt{2qQ}\over
2a^{q/2}}[Qa^{p}+\kappa^{2}]^{1/2}{dh\over da},$

and

$\displaystyle[Qa^{p}+\kappa^{2}]{d^{2}\\!h\over da^{2}}+{1\over
2a}[(6-q)Qa^{p}+4\kappa^{2}]{dh\over da}=$ (35) $\displaystyle\
{2\sqrt{2qQ}\over a^{q/2}}[Qa^{p}+\kappa^{2}]^{1/2}{d\phi\over
da}+{(6-q)\sqrt{2qQ}\over 2a^{(2+q)/2}}[Qa^{p}+\kappa^{2}]^{1/2}\phi.$

These can be combined to give a third order equation for $\phi$

$\displaystyle[Qa^{p}+\kappa^{2}]{d^{3}\\!\phi\over da^{3}}+{1\over
2a}[2(8-q)Qa^{p}+(10+q)\kappa^{2}]{d^{2}\\!\phi\over da^{2}}$ $\displaystyle\
+{1\over 4a^{2}}[(48-10q-q^{2})Qa^{p}+(12-q)(2+q)\kappa^{2}+4k^{2}]{d\phi\over
da}$ $\displaystyle\ +{q\over
8a^{3}}[2p(6-q)Qa^{p}+(6-q)(2+q)\kappa^{2}+4k^{2}]\phi$ $\displaystyle\ =0.$
(36)

Changing variables from $\phi$ and $a$ to $f$ and $x$ where $\phi=f/a^{q/2}$
and $x=Qa^{p}/\kappa^{2}$, this equation becomes

$\displaystyle x^{2}(x+1){d^{2}\\!Z\over dx^{2}}+\left[{11\over
2}x^{2}+4x\right]{dZ\over dx}+$ (37)
$\displaystyle\left[{(40-41q+10q^{2})\over
2p^{2}}x+{(11-8q+2q^{2}+k^{2}/\kappa^{2})\over p^{2}}\right]Z=0,$

where $Z(x)=df/dx$. The general solution $Z(x)$ of Eq. (37) can be expressed
in terms of Gauss hypergeometric functions and this can be integrated once
with respect to $x$ to get $f(x)$ and so $\phi(a)$.

Defining $A_{\pm}$, $B$, $D$, and $G_{\pm}$,

$\displaystyle 4pA_{\pm}=3p-2W\pm(2+q),$ (38) $\displaystyle pB=p-W,$ (39)
$\displaystyle pD=p+W,$ (40) $\displaystyle 4pG_{\pm}=3p+2W\pm(2+q),$ (41)

where

$\displaystyle W=\sqrt{-8-4q+q^{2}-4k^{2}/\kappa^{2}},$ (42)

the scalar field solution of the linear perturbation equations is

$\displaystyle a^{q/2}\phi$ $\displaystyle=$ $\displaystyle\tilde{c}$
$\displaystyle+$
$\displaystyle\tilde{c}_{+}x^{(B-2)/2}{}_{3}F_{2}(A_{+},A_{-},B/2-1;B,B/2;-x)$
$\displaystyle+$
$\displaystyle\tilde{c}_{-}x^{(D-2)/2}{}_{3}F_{2}(G_{+},G_{-},D/2-1;D,D/2;-x).$

Here $\tilde{c}$ and $\tilde{c}_{\pm}$ are constants of integration, the
$\tilde{c}$ solution is a gauge solution corresponding to the remnants of time
translation invariance in synchronous gauge, and ${}_{3}F_{2}$ is a
generalized hypergeometric function, see Ch. IV of Ref. (EHTF, ) and Ch. 16 of
Ref. (OLBC, ).

In terms of $x$, Eq. (35) is

$\displaystyle x^{2}(x+1){d^{2}\\!h\over dx^{2}}+\left[{(8-3q)\over
2p}x^{2}+{(3-q)\over p}x\right]{dh\over dx}=$ (44) $\displaystyle\
{2\sqrt{2q}\over p}x^{3/2}(x+1)^{1/2}{d\phi\over dx}+{(6-q)\sqrt{2q}\over
2p^{2}}x^{1/2}(x+1)^{1/2}\phi.$

Using Eq. (IV.2) to evaluate the right hand side of this equation, it can be
solved to give the trace of the metric perturbation solution

$\displaystyle h$ $\displaystyle=$ $\displaystyle
c_{2}-2c_{1}(-1)^{1/p}\sqrt{x+1}\,{}_{2}F_{1}(1/2,1+1/p;3/2;x+1)$
$\displaystyle+$ $\displaystyle{3\sqrt{2q}\over
p}\left({Q\over\kappa^{2}}\right)^{q/2p}\tilde{c}(-1)^{1/p}\sqrt{x+1}$
$\displaystyle\ \ \ \ \times{}_{2}F_{1}(1/2,1/p;3/2;x+1)$ $\displaystyle+$
$\displaystyle{3\sqrt{2q}\over
2p}\left({Q\over\kappa^{2}}\right)^{q/2p}\tilde{c}_{+}\int
dx{x^{(B-2)/2-1/p}\over\sqrt{x+1}}$ $\displaystyle\ \
\times\big{\\{}{}_{3}F_{2}(A_{+},A_{-},B/2-1;B,B/2;-x)$ $\displaystyle\ \ \ \
\ +{(B-2)\over 3B}{}_{3}F_{2}(A_{+},A_{-},B/2;B,B/2+1;-x)\big{\\}}$
$\displaystyle+$ $\displaystyle{3\sqrt{2q}\over
2p}\left({Q\over\kappa^{2}}\right)^{q/2p}\tilde{c}_{-}\int
dx{x^{(D-2)/2-1/p}\over\sqrt{x+1}}$ $\displaystyle\ \
\times\big{\\{}{}_{3}F_{2}(G_{+},G_{-},D/2-1;D,D/2;-x)$ $\displaystyle\ \ \ \
\ +{(D-2)\over 3D}{}_{3}F_{2}(G_{+},G_{-},D/2;D,D/2+1;-x)\big{\\}},$

where the integrals can be done (and expressed as infinite series) but this is
not of use to us and so these series are not recorded here. In Eq. (IV.2)
$\tilde{c}$ and $\tilde{c}_{\pm}$ are the constants in Eq. (IV.2) and $c_{1}$
and $c_{2}$ are constants of integration, with $c_{2}$ corresponding to a
gauge mode.

In terms of $x$, Eq. (31) is

$\displaystyle{d{\cal H}\over
dx}={k^{2}\over\bar{k}^{2}}\left[{3\sqrt{2q}\over
2p}{\phi\over\sqrt{x(x+1)}}-{dh\over dx}\right].$ (46)

Using Eqs. (IV.2) and (IV.2) to evaluate the right hand side of this equation,
it can be solved to give the trace-free metric perturbation solution

$\displaystyle{\bar{k}^{2}\over k^{2}}{\cal H}=$ $\displaystyle
c_{3}+2c_{1}(-1)^{1/p}\sqrt{x+1}\,{}_{2}F_{1}(1/2,1+1/p;3/2;x+1)$
$\displaystyle-$ $\displaystyle{\sqrt{2q}\over
2p}\left({Q\over\kappa^{2}}\right)^{q/2p}\tilde{c}_{+}{(B-2)\over B}\int
dx{x^{(B-2)/2-1/p}\over\sqrt{x+1}}$ $\displaystyle\ \ \ \
\times{}_{3}F_{2}(A_{+},A_{-},B/2;B,B/2+1;-x)$ $\displaystyle-$
$\displaystyle{\sqrt{2q}\over
2p}\left({Q\over\kappa^{2}}\right)^{q/2p}\tilde{c}_{-}{(D-2)\over D}\int
dx{x^{(D-2)/2-1/p}\over\sqrt{x+1}}$ $\displaystyle\ \ \ \
\times{}_{3}F_{2}(G_{+},G_{-},D/2;D,D/2+1;-x),$

where the integrals can be done (and expressed as infinite series) but this
not of use to us and so these series are not recorded here. In Eq. (IV.2)
$\tilde{c}_{\pm}$ and $c_{1}$ are the constants in Eqs. (IV.2) and (IV.2), and
$c_{3}$ is a constant of integration corresponding to a gauge mode.

Using the solutions given in Eqs. (IV.2), (IV.2) and (IV.2), the left-hand
sides of Eqs. (32) and (33) are proportional. Requiring they vanish results in
the relation

$\displaystyle c_{1}=\sqrt{2\over q}{1\over
p}\left({Q\over\kappa^{2}}\right)^{q/2p}\tilde{c}{\bar{k}^{2}\over\kappa^{2}}.$
(48)

We are unable to analytically establish that the coefficients of
$\tilde{c}_{\pm}$ — the gauge-invariant contributions — vanish in these
equations, as they must. However, we are convinced that they indeed do vanish
since, as discussed below, the complete numerical solution (for given sets of
parameter values) of the linear perturbation equations results in power
spectra that are in very good agreement with the analytic power spectra we
have derived. Additionally, we show below that on smaller scales when spatial
curvature is unimportant, the primordial power spectra in these tilted non-
flat inflation models are identical to the primordial power spectrum in the
tilted flat model of Refs. (LucchinMatarrese1985, ; Ratra1992, ; Ratra1989, ),
where in the computation of Ref. (Ratra1992, ) it was shown that the
corresponding coefficients of $\tilde{c}_{\pm}$ vanished in the corresponding
equations.

###  IV.3 Gauge-invariant variables solutions

For scalar perturbations there are two independent gauge-invariant variables,
invariant under the remnants of general coordinate invariance in synchronous
gauge (Ratra1991, ). We choose these to be

$\displaystyle\Delta_{\Phi}=$ $\displaystyle\
{1\over\dot{\Phi}_{b}{}^{2}+V(\Phi_{b})}\left[2\dot{\Phi}_{b}\dot{\phi}+V^{\prime}(\Phi_{b})\phi+6{\dot{a}\over
a}\dot{\Phi}_{b}\phi\right],$ (49) $\displaystyle A_{\Phi}=$ $\displaystyle\
{1\over\dot{\Phi}_{b}{}^{2}+V(\Phi_{b})}\left[2\dot{\Phi}_{b}\dot{\phi}+V^{\prime}(\Phi_{b})\phi-\dot{\Phi}_{b}{}^{2}(h+{\cal
H})\right].$ (50)

Another useful gauge-invariant combination is

$\displaystyle{\mathcal{R}}_{\Phi}={\dot{\Phi}_{b}^{2}+V(\Phi_{b})\over
6\dot{\Phi}_{b}^{2}}\left[\Delta_{\Phi}-A_{\Phi}\right].$ (51)

During inflation, from the $\phi$, $h$, and ${\cal H}$ solutions of the
previous sub-section, the gauge-invariant variables are

$\displaystyle\Delta_{\Phi}$ $\displaystyle=$ $\displaystyle{p\sqrt{2q}\over
12}\left({Q\over\kappa^{2}}\right)^{q/2p}\sqrt{x+1}$ $\displaystyle\times$
$\displaystyle\Big{[}(B-2)\tilde{c}_{+}x^{(B-2)/2-1/p}{}_{2}F_{1}(A_{+},A_{-};B;-x)$
$\displaystyle+(D-2)\tilde{c}_{-}x^{(D-2)/2-1/p}{}_{2}F_{1}(G_{+},G_{-};D;-x)\Big{]},$

and

$\displaystyle A_{\Phi}$ $\displaystyle=$ $\displaystyle-{q\over
6}\left({Q\over\kappa^{2}}\right)^{q/2p}\sqrt{x+1}$ $\displaystyle\times$
$\displaystyle\Big{[}{6\over\sqrt{2q}}\tilde{c}_{+}x^{(B-2)/2-1/p}$
$\displaystyle\ \ \ \ \ \ \times{}_{3}F_{2}(A_{+},A_{-},B/2-1;B,B/2;-x)$
$\displaystyle+{6\over\sqrt{2q}}\tilde{c}_{-}x^{(D-2)/2-1/p}$ $\displaystyle\
\ \ \ \ \ \ \times{}_{3}F_{2}(G_{+},G_{-},D/2-1;D,D/2;-x)$
$\displaystyle-\left({3q\over
2}{\kappa^{2}\over\bar{k}^{2}}x+1\right){p\over\sqrt{2q}}(B-2)\tilde{c}_{+}x^{(B-2)/2-1/p}$
$\displaystyle\ \ \ \ \ \ \times{}_{2}F_{1}(A_{+},A_{-};B;-x)$
$\displaystyle-\left({3q\over
2}{\kappa^{2}\over\bar{k}^{2}}x+1\right){p\over\sqrt{2q}}(D-2)\tilde{c}_{-}x^{(D-2)/2-1/p}$
$\displaystyle\ \ \ \ \ \ \times{}_{2}F_{1}(G_{+},G_{-};D;-x)$
$\displaystyle-{3\over 2}{\kappa^{2}\over\bar{k}^{2}}\sqrt{2q}{(B-2)\over
B}\tilde{c}_{+}x^{B/2-1/p}$ $\displaystyle\ \ \ \ \ \
\times{}_{3}F_{2}(A_{+},A_{-},B/2;B,B/2+1;-x)$ $\displaystyle-{3\over
2}{\kappa^{2}\over\bar{k}^{2}}\sqrt{2q}{(D-2)\over D}\tilde{c}_{-}x^{D/2-1/p}$
$\displaystyle\ \ \ \ \ \
\times{}_{3}F_{2}(G_{+},G_{-},D/2;D,D/2+1;-x)\Big{]}.$

We assume that the gauge-invariant solutions, those proportional to
$\tilde{c}_{\pm}$, obey Eqs. (32) and (33), and use these and Eq. (48) to
simplify the expression for $A_{\Phi}$ to that given in Eq. (IV.3), see the
discussion around Eq. (48).

From the expressions for $\Delta_{\phi}$ and $A_{\phi}$ above,

$\displaystyle{\mathcal{R}}_{\Phi}$ $\displaystyle=$
$\displaystyle\left({Q\over\kappa^{2}}\right)^{q/2p}\sqrt{x+1}$
$\displaystyle\times$
$\displaystyle\Big{[}{1\over\sqrt{2q}}\tilde{c}_{+}x^{(B-2)/2-1/p}$
$\displaystyle\ \ \ \ \ \ \times{}_{3}F_{2}(A_{+},A_{-},B/2-1;B,B/2;-x)$
$\displaystyle+{1\over\sqrt{2q}}\tilde{c}_{-}x^{(D-2)/2-1/p}$ $\displaystyle\
\ \ \ \ \ \ \times{}_{3}F_{2}(G_{+},G_{-},D/2-1;D,D/2;-x)$
$\displaystyle-{\sqrt{2q}p\over
8}{\kappa^{2}\over\bar{k}^{2}}(B-2)\tilde{c}_{+}x^{B/2-1/p}$ $\displaystyle\ \
\ \ \ \ \times{}_{2}F_{1}(A_{+},A_{-};B;-x)$ $\displaystyle-{\sqrt{2q}p\over
8}{\kappa^{2}\over\bar{k}^{2}}(D-2)\tilde{c}_{-}x^{D/2-1/p}$ $\displaystyle\ \
\ \ \ \ \times{}_{2}F_{1}(G_{+},G_{-};D;-x)$ $\displaystyle-{\sqrt{2q}\over
4}{\kappa^{2}\over\bar{k}^{2}}{(B-2)\over B}\tilde{c}_{+}x^{B/2-1/p}$
$\displaystyle\ \ \ \ \ \ \times{}_{3}F_{2}(A_{+},A_{-},B/2;B,B/2+1;-x)$
$\displaystyle-{\sqrt{2q}\over 4}{\kappa^{2}\over\bar{k}^{2}}{(D-2)\over
D}\tilde{c}_{-}x^{D/2-1/p}$ $\displaystyle\ \ \ \ \ \
\times{}_{3}F_{2}(G_{+},G_{-},D/2;D,D/2+1;-x)\Big{]}.$

###  IV.4 Initial conditions and constants of integration

Defining $\hat{c}_{\pm}=\tilde{c}_{\pm}\sqrt{\kappa^{2}/Q}$, and ignoring the
first, gauge-dependent, term on the right hand side of Eq. (IV.2), a more
convenient form of the inflaton field perturbation solution is

$\displaystyle a\phi$ $\displaystyle=$
$\displaystyle\hat{c}_{+}x^{(B-1)/2}{}_{3}F_{2}(A_{+},A_{-},B/2-1;B,B/2;-x)$
$\displaystyle+$
$\displaystyle\hat{c}_{-}x^{(D-1)/2}{}_{3}F_{2}(G_{+},G_{-},D/2-1;D,D/2;-x).$

We assume as an initial condition that the conformally-rescaled scalar field
perturbation ($a\phi$) is in the conformal time harmonic oscillator ground
state, when non-flat inflation initiated, on small length scales. I.e., we
require

$\displaystyle{\rm lim}_{A\rightarrow\infty}a\phi(A,t)=\left({16\pi\over
m_{p}{}^{2}}\right)^{1/2}{e^{-iA\tilde{t}}\over\sqrt{2A}},$ (56)

where conformal time $\tilde{t}$ is defined, for the open and closed models,
in Eqs. (23) and (24). See Refs. (Ratra1985, ; Ratra1989, ; Ratra1992, ;
Ratra1994, ; RatraPeebles1995, ; Ratra2017, ) for discussions of such initial
conditions in a variety of flat and non-flat inflation models. These initial
conditions result in de Sitter invariant inflaton two-point correlation
functions in the very-slow-roll flat and closed de Sitter inflation models.
The non-flat open and closed not-necessarily very-slow-roll inflation models
which we apply them to here have less symmetry than the flat and closed de
Sitter inflation models and so it is possible that our assumption here of the
absence of an additional subdominant at large $A$ correction, that cannot be
determined from Eq. (56), and that might be important at small $A$, might not
be justified in the non-flat not-necessarily very-slow-roll inflation models
(such corrections do not contribute in the flat and closed de Sitter inflation
models).101010I thank A. Guth for emphasizing this point.

We have not been able to analytically compute the large $A$ asymptotic limit
of Eq. (IV.4).111111Except for the $\hat{c}_{-}$ term in the closed case where
the result agrees with the expression given in Eq. (57) below. However, if we
set $\hat{c}_{+}=0$ in both non-flat models, and choose for the closed model

$\displaystyle\hat{c}_{-}=\left({16\pi\over
m_{p}{}^{2}}\right)^{1/2}{2^{-2A/p}(-1)^{A/p}\over\sqrt{2A}},$ (57)

and for the open model

$\displaystyle\hat{c}_{-}=\left({16\pi\over
m_{p}{}^{2}}\right)^{1/2}{i2^{-i2A/p}\over\sqrt{2A}},$ (58)

it may numerically be shown in both non-flat models that the initial condition
of Eq. (56) is satisfied, (Namjoo2020, ). Additionally, we show below that on
smaller scales when spatial curvature is unimportant, the primordial power
spectra in these tilted non-flat inflation models (which depend on these
expressions for $\hat{c}_{\pm}$), are identical to the primordial power
spectrum in the tilted flat inflation model of Refs. (LucchinMatarrese1985, ;
Ratra1992, ; Ratra1989, ), where in Ref. (Ratra1992, ) the corresponding
asymptotic limits were computed.

###  IV.5 $P_{\mathcal{R}}$ at late time during inflation in the open and
closed models

Here we record expressions for the power spectra at late time during inflation
in the open and closed tilted inflation models. The power spectrum definition
and conventions are given in Appendices B and C.

Using Eqs. (IV.3) and (57), and setting $\hat{c}_{+}=0$, we find at late time
during inflation in the closed model

$\displaystyle\sqrt{|P_{\mathcal{R}}|}$ $\displaystyle=$
$\displaystyle\left({16\pi\over
m_{p}{}^{2}}\right)^{1/2}\\!\\!\\!Q^{1/p}{(2+q)p\over\sqrt{\pi
q}}\left|-1+{W\over p}\right|$ (59) $\displaystyle\times$
$\displaystyle\\!\\!\\!\\!\\!\\!{2^{-(6-4q+2A-W)/p}\over\sqrt{A}(A-1)(A+3)}\left|{\Gamma\left(1+W/p\right)\Gamma\left((2+q)/(2p)\right)\over\Gamma\left((2+W)/p\right)}\right|,$

where

$\displaystyle W=\sqrt{-8-4q+q^{2}+4A(A+2)},$ (60)

while Eqs. (IV.3) and (58), and setting $\hat{c}_{+}=0$, give at late time
during inflation in the open model

$\displaystyle\sqrt{|P_{\mathcal{R}}|}$ $\displaystyle=$
$\displaystyle\left({16\pi\over
m_{p}{}^{2}}\right)^{1/2}\\!\\!\\!Q^{1/p}{(2+q)p\over\sqrt{\pi
q}}\left|-1+{W\over p}\right|$ (61) $\displaystyle\times$
$\displaystyle\\!\\!\\!\\!\\!\\!{2^{-(6-4q)/p}\over\sqrt{A}(A^{2}+4)}\left|{\Gamma\left(1+W/p\right)\Gamma\left((2+q)/(2p)\right)\over\Gamma\left((2+W)/p\right)}\right|,$

where

$\displaystyle W=\sqrt{-12-4q+q^{2}-4A^{2}},$ (62)

and in both cases the scalar spectral power-law index $n=(2-3q)/(2-q)$. In the
closed case, for fixed parameter values, the power spectrum of Eq. (59) agrees
very well with power spectra computed numerically in the model described by
the potential energy density of Eq. (26), (Namjoo2020, ).

In the large $A$ limit where $W=2A$ ($2iA$) in the tilted closed (open) case,
the $P_{\mathcal{R}}$ expressions in Eqs. (59) and (61) reduce identically to
the flat-space tilted inflation model expression of Eq. (83). This shows that
on small scales at late times during tilted non-flat inflation, when spatial
curvature is not important, the tilted closed and open model primordial power
spectra are identical to the tilted flat model primordial power spectrum. This
is a useful consistency test of our analyses, include the initial conditions
we have used here, as there is somewhat less uncertainty about initial
conditions in the flat model compared to the closed, as well as possibly even
the open, model. Given that potentially only a tiny $\sim 1$% deviation from
flatness is what is of interest, it is reassuring that this initial conditions
consistency test is passed.

##  V Conclusion

We have extended from the very-slow-roll, untilted, linear inflaton potential
energy density open and closed inflation models of Refs. (RatraPeebles1995, ;
Ratra2017, ) to not-necessarily very-slowly-rolling, tilted, non-linear
inflaton potential energy density open and closed inflation models. We have
determined power spectra for quantum-mechanically produced spatial
inhomogeneities in these models. These power spectra can be used to
characterize spatial inhomogeneities in closed and open inflation models, and
have been used in analyses of CMB anisotropy and other data, (deCruzetal2022,
). They differ from those that have previously been used for this purpose,
(Planck2016, ; Planck2020, ). The power spectra of Refs. (Planck2016, ;
Planck2020, ) can also be generated by quantum fluctuations during (so far,
only in closed) inflation, assuming different initial conditions,
(Guthetal2022, ).

Recent hints of observational tension with a few predictions of the spatially-
flat $\Lambda$CDM model provides motivation for studying non-flat cosmological
models as well as other alternatives. Moreover, even if space is flat, to
properly establish this from CMB anisotropy data requires use of consistent
non-flat cosmological models — such as those constructed here — and the
primordial power spectra in these models, and in Ref. (Guthetal2022, ) for the
closed case. It might be significant that in the non-flat inflation model
there appears to be additional freedom in the form of the inflation generated
power spectrum, compared to the spatially-flat case.

###### Acknowledgements.

I acknowledge very valuable discussions with A. Guth, M. H. Namjoo, C.-G.
Park, and J. de Cruz. This work was supported in part by DOE grant DE-
SC0011840.

##  Appendix A ${\mathcal{R}}_{\Phi}$ in the spatially-flat tilted inflation
model

In the spatially-flat tilted inflation model of Refs. (LucchinMatarrese1985, ;
Ratra1992, ; Ratra1989, ), the scalar field potential energy density during
inflation is

$\displaystyle\\!\\!\\!\\!V(\Phi)=\left({6-q\over 3}\right){16\pi\over
m_{p}{}^{2}}\rho^{(0)}_{\Phi}{\rm exp}\left[-\sqrt{q\over
2}(\Phi-\Phi^{(0)})\right],$ (63)

where $\Phi^{(0)}$ and $\rho^{(0)}_{\Phi}$ are the scalar field and the scalar
field energy density during inflation at scale factor $a_{0}$. The scalar
field energy density during inflation is

$\displaystyle\rho_{\Phi}=\rho^{(0)}_{\Phi}\left({a_{0}\over a}\right)^{q}.$
(64)

During inflation, the gauge-invariant

$\displaystyle{\mathcal{R}}_{\Phi}$ $\displaystyle=$ $\displaystyle-{1\over
6}\left({q\over 2}\right)^{(5-q)/q}{aH^{(10-q)/(2q)}\over k}$
$\displaystyle\times$ $\displaystyle\Big{[}c_{+}H^{(1)}_{\nu+1}\left({2k\over
paH}\right)+c_{-}H^{(2)}_{\nu+1}\left({2k\over paH}\right)\Big{]},$

where $H$ is the Hubble parameter, $\nu=(2+q)/(2p)$, $H^{(i)}_{\nu+1}$ are
Hankel functions, and from the initial conditions the constant of integration
$c_{-}=0$ and

$\displaystyle c_{+}=$ (66) $\displaystyle\left({16\pi\over
m_{p}{}^{2}}\right)^{1/2}{k\over 2}\left({q\pi\over
p}\right)^{1/2}\left(a_{0}M^{2/q}\right)^{-5/2}e^{i(\nu-1/2)\pi/2},$

where

$\displaystyle M={q\over 2}\left({8\pi\over
3m_{p}{}^{2}}\rho_{\Phi}^{(0)}\right)^{1/2}.$ (67)

##  Appendix B Relation between $P$ and ${\mathcal{P}}$ in flat, open, and
closed models

In this Appendix we define two power spectra we use and relate them to the
two-point function in spatial momentum space. In this Appendix we do not
explicitly indicate the time dependence of the fields.

In the flat model, defining the Fourier expansion of a position space field

$\displaystyle\zeta(\vec{x})=\int{d^{3}k\over(2\pi)^{3}}e^{i\vec{k}\cdot\vec{x}}\zeta(\vec{k}),$
(68)

and

$\displaystyle\langle\zeta(\vec{k})\zeta^{*}(\vec{k^{\prime}})\rangle=P_{\zeta}(k)(2\pi)^{3}\delta^{(3)}(\vec{k}-\vec{k^{\prime}}),$
(69)

where $P_{\zeta}(k)$ is the power spectrum and $k=|\vec{k}|$, we have

$\displaystyle\langle\zeta(\vec{x})\zeta^{*}(\vec{x^{\prime}})\rangle=\int{d^{3}k\over(2\pi)^{3}}\int{d^{3}k^{\prime}\over(2\pi)^{3}}e^{i(\vec{k}\cdot\vec{x}-\vec{k^{\prime}}\cdot\vec{x^{\prime}})}\langle\zeta(\vec{k})\zeta^{*}(\vec{k^{\prime}})\rangle$
$\displaystyle\ \
=\int{d^{3}k\over(2\pi)^{3}}e^{i\vec{k}\cdot(\vec{x}-\vec{x^{\prime}})}P_{\zeta}(k).$
(70)

Setting $\vec{x}=\vec{x^{\prime}}$, we have

$\displaystyle\langle|\zeta(\vec{x})|^{2}\rangle=\int{d^{3}k\over(2\pi)^{3}}P_{\zeta}(k)=\int_{0}^{\infty}{dk\over
k}{k^{3}P_{\zeta}(k)\over 2\pi^{2}},$ (71)

and so define the power spectrum

$\displaystyle\mathcal{P}_{\zeta}(k)={k^{3}P_{\zeta}(k)\over 2\pi^{2}}$ (72)

which gives the power in a logarithmic wavenumber interval.

In the open model, defining

$\displaystyle\zeta(\vec{\Omega})=\int_{0}^{\infty}dA\sum_{BC}Z_{ABC}(\vec{\Omega})\zeta(A),$
(73)

where $B$ and $C$ are ‘magnetic’ integral indices, $Z_{ABC}$ is defined in Eq.
(2.9) of Ref. (RatraPeebles1995, ), and

$\displaystyle\langle\zeta(A)\zeta^{*}(A^{\prime})\rangle=P_{\zeta}(A)\delta(A-A^{\prime})\delta_{B,B^{\prime}}\delta_{C,C^{\prime}},$
(74)

we have

$\displaystyle\langle\zeta(\vec{\Omega})\zeta^{*}(\vec{\Omega^{\prime}})\rangle$
$\displaystyle\ \ =\int dA\int
dA^{\prime}\sum_{BC}\sum_{B^{\prime}C^{\prime}}Z_{ABC}(\vec{\Omega})Z^{*}_{A^{\prime}B^{\prime}C^{\prime}}(\vec{\Omega^{\prime}})\langle\zeta(A)\zeta^{*}(A^{\prime})\rangle$
$\displaystyle\ \ =\int
dA\sum_{BC}Z_{ABC}(\vec{\Omega})Z^{*}_{ABC}(\vec{\Omega^{\prime}})P_{\zeta}(A)$
$\displaystyle\ \ =\int_{0}^{\infty}{dA\over
A}{A^{3}P_{\zeta}(A)\over(2\pi)^{3/2}}{P^{-1/2}_{iA-1/2}({\rm
cosh}(\gamma_{3}))\over\sqrt{{\rm sinh}(\gamma_{3})}},$ (75)

where we have used Eq. (2.11) of Ref. (RatraPeebles1995, ), $P^{\mu}_{\nu}$ is
the associated Legendre function of the first kind, and ${\rm
cosh}(\gamma_{3})$ is defined in Eq. (8). Setting
$\vec{\Omega}=\vec{\Omega^{\prime}}$ and using Eq. (A.3) of Ref. (Ratra1994,
), we have

$\displaystyle\langle|\zeta(\vec{\Omega})|^{2}\rangle=\int_{0}^{\infty}{dA\over
A}{A^{3}P_{\zeta}(A)\over 2\pi^{2}},$ (76)

and so define

$\displaystyle\mathcal{P}_{\zeta}(A)={A^{3}P_{\zeta}(A)\over 2\pi^{2}}.$ (77)

In the closed model, defining

$\displaystyle\zeta(\vec{\Omega})=\sum_{A=2}^{\infty}\sum_{BC}Y_{ABC}(\vec{\Omega})\zeta(A),$
(78)

where $B$ and $C$ are ‘magnetic’ integral indices, $Y_{ABC}$ is defined in Eq.
(9) of Ref. (Ratra2017, ), and

$\displaystyle\langle\zeta(A)\zeta^{*}(A^{\prime})\rangle=P_{\zeta}(A)\delta_{A,A^{\prime}}\delta_{B,B^{\prime}}\delta_{C,C^{\prime}},$
(79)

we have

$\displaystyle\langle\zeta(\vec{\Omega})\zeta^{*}(\vec{\Omega^{\prime}})\rangle$
$\displaystyle\ \
=\sum_{A=2}^{\infty}\sum_{A^{\prime}=2}^{\infty}\sum_{BC}\sum_{B^{\prime}C^{\prime}}Y_{ABC}(\vec{\Omega})Y^{*}_{A^{\prime}B^{\prime}C^{\prime}}(\vec{\Omega^{\prime}})\langle\zeta(A)\zeta^{*}(A^{\prime})\rangle$
$\displaystyle\ \
=\sum_{A=2}^{\infty}\sum_{BC}Y_{ABC}(\vec{\Omega})Y^{*}_{ABC}(\vec{\Omega^{\prime}})P_{\zeta}(A)$
$\displaystyle\ \ =\sum_{A=2}^{\infty}{1\over
A+1}{(A+1)^{3}P_{\zeta}(A)\over(2\pi)^{3/2}}{P^{-1/2}_{A-1/2}({\rm
cos}(\gamma_{3}))\over\sqrt{{\rm sin}(\gamma_{3})}},$ (80)

where we have used Eq. (11) of Ref. (Ratra2017, ), and ${\rm cos}(\gamma_{3})$
is defined in Eq. (3). Setting $\vec{\Omega}=\vec{\Omega^{\prime}}$ and using
Eq. (3.9.2 (8)) on page 163 of Ref. (EHTF, ), we have

$\displaystyle\langle|\zeta(\vec{\Omega})|^{2}\rangle=\sum_{A=2}^{\infty}{1\over
A+1}{(A+1)^{3}P_{\zeta}(A)\over 2\pi^{2}},$ (81)

(the factor of $1/(A+1)=1/\nu$ is chosen for consistency with the definition
of Ref. (LesgourguesTram2014, ), see below) and so define

$\displaystyle\mathcal{P}_{\zeta}(A)={(A+1)^{3}P_{\zeta}(A)\over 2\pi^{2}}.$
(82)

##  Appendix C $P_{\mathcal{R}}$ and ${\mathcal{P}_{\mathcal{R}}}$ at late
time in the flat exponential potential and the open and closed linear
potential inflation models, and comparison to CAMB and CLASS input power
spectra

In the flat model of Refs. (LucchinMatarrese1985, ; Ratra1992, ; Ratra1989, ),
at late time during inflation, using the results of Appendix A and Eq. (9.1.9)
of Ref. (AS, ), we have

$\displaystyle P_{\mathcal{R}}=Ck^{n-4},$ (83)

where $n=(2-3q)/(2-q)$ and the proportionality constant

$\displaystyle C={4\over m_{p}{}^{2}q}\Gamma^{2}\\!\\!\left({6-q\over
2p}\right)p^{4/p}\left({8\pi\over
3m_{p}{}^{2}}\rho_{\Phi}^{(0)}a_{0}{}^{q}\right)^{2/p},$ (84)

so, from Eq. (72),

$\displaystyle\mathcal{P_{\mathcal{R}}}={C\over 2\pi^{2}}k^{n-1}.$ (85)

In the limit of small $q=2\epsilon^{2}$, Eq. (83) reduces to

$\displaystyle P_{\mathcal{R}}={2\pi h^{2}\over
m_{p}{}^{2}\epsilon^{2}}k^{-3},$ (86)

where $h^{2}=4\pi(6-q)\rho_{\Phi}^{(0)}/(9m_{p}{}^{2})$.

In the open linear scalar field potential energy density model of Ref.
(RatraPeebles1995, ), with $V(\Phi)=12h^{2}[1-\epsilon\Phi]$ where $h$ is a
constant, at late times during inflation

$\displaystyle P_{\mathcal{R}}={2\pi h^{2}\over
m_{p}{}^{2}\epsilon^{2}}{1\over A(A^{2}+1)},$ (87)

so

$\displaystyle\mathcal{P_{\mathcal{R}}}={h^{2}\over\pi
m_{p}{}^{2}\epsilon^{2}}{A^{2}\over A^{2}+1}.$ (88)

In the closed linear scalar field potential energy density model of Ref.
(Ratra2017, ), at late times during inflation

$\displaystyle P_{\mathcal{R}}={2\pi h^{2}\over
m_{p}{}^{2}\epsilon^{2}}{1\over A(A+1)(A+2)},$ (89)

so

$\displaystyle\mathcal{P_{\mathcal{R}}}={h^{2}\over\pi
m_{p}{}^{2}\epsilon^{2}}{(A+1)^{2}\over A(A+2)}.$ (90)

In the limit $A\gg 1$, Eqs. (87) and (89) become

$\displaystyle P_{\mathcal{R}}={2\pi h^{2}\over
m_{p}{}^{2}\epsilon^{2}}A^{-3},$ (91)

which is identical to the expression in Eq. (86). As expected, on small scales
at late times during non-flat inflation when spatial curvature is unimportant,
the very-slow-roll closed and open inflation model primordial power spectra
are identical to the very-slow-roll flat inflation model primordial power
spectrum.

The CAMB and CLASS input power spectra are defined in Ref.
(LesgourguesTram2014, ). The definition in their Eq. (3.23) is a little
unusual given the $(2\pi)^{3}$ in their Eq. (3.24), but the normalization of
the power spectrum is an adjustable parameter to be determined by fitting to
data.

Comparing Ref. (LesgourguesTram2014, ) Eqs. (3.25) and (3.26) in the flat
case, we see that their Eq. (3.26) is identical to Eq. (85) above.

Reference (TramLesgourgues2013, ) defines the negative of the eigenvalue of
the spatial Laplacian to be $k^{2}/|K|$ in their Eq. (1.11). Here
$K=-H_{0}^{2}\Omega_{k0}$ is negative (positive) for open (closed) spatial
hypersurfaces and $\Omega_{k0}$ is the current value of the spatial curvature
density parameter. In the second line below Eq. (3.4) of Ref.
(LesgourguesTram2014, ) (where they define $q$ in terms of $k$) they define
another wavenumber $\nu=q/\sqrt{|K|}$. It can be seen that in the open model
their $\nu$ is identical to the $A$ we use here while in the closed model
their $\nu$ is identical to the $A+1$ we use here.

Reference (LesgourguesTram2014, ) uses an unusual but now standard convention.
They define $\mathcal{P_{\mathcal{R}}}=A_{s}k^{n-1}$ to be the flat space
expression, see their Eq. (3.26). To make this clear, in what follows, we put
a superscript ${\rm FS}$ on this and write $\mathcal{P^{\rm
FS}_{\mathcal{R}}}=A_{s}k^{n-1}$. What we call $\mathcal{P_{\mathcal{R}}}$
they refer to as

$\displaystyle\widetilde{\mathcal{P}}_{\mathcal{R}}(\nu)={\nu^{2}\over\nu^{2}-\hat{K}}\mathcal{P^{\rm
FS}_{\mathcal{R}}}.$ (92)

Here $\hat{K}=K/|K|$ which is $-1(+1)$ for open (closed) hypersurfaces. The
above expression can be derived from their Eq. (3.29) by changing variables
from $q$ to $\nu$. That our $\mathcal{P_{\mathcal{R}}}$ is their
$\widetilde{\mathcal{P}}_{\mathcal{R}}$ follows from their Eq. (3.28) by
rewriting $d\nu\,\nu^{2}$ as $(d\nu/\nu)\,\nu^{3}$ and combining the $\nu^{3}$
with the last factor in the integrand and then comparing to the flat-space
expression in their Eq. (3.25).

Setting $n=1$ in the Ref. (LesgourguesTram2014, ) non-flat expressions, as we
want to compare to the linear scalar field potential energy density inflation
expressions above, we find in the open case the Ref. (LesgourguesTram2014, )
formula is

$\displaystyle\mathcal{P_{\mathcal{R}}}\propto{A^{2}\over A^{2}+1},$ (93)

that is in agreement with Eq. (88) above, and in the closed case the Ref.
(LesgourguesTram2014, ) formula is

$\displaystyle\mathcal{P_{\mathcal{R}}}\propto{(A+1)^{2}\over A(A+2)},$ (94)

which agrees with Eq. (90) above.

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