# Constraining Cosmic Inflation with Observations: Prospects for 2030

Benedict Bahr-Kalus,1 David Parkinson,1,2 and Richard Easther,3
1Korea Astronomy and Space Science Institute, Yuseong-gu, Daedeok-daero 776,
Daejeon 34055, Korea
2University of Science and Technology, Daejeon 34113, Korea
3Department of Physics, University of Auckland, Private Bag 92019, Auckland,
New Zealand E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The ability to test and constrain theories of cosmic inflation will advance
substantially over the next decade. Key data sources include cosmic microwave
background (CMB) measurements and observations of the distribution of matter
at low-redshift from optical, near-infrared, and 21cm intensity surveys. A
positive detection of a CMB B-mode consistent with a primordial stochastic
gravitational wave background (SGWB) is widely viewed as a smoking gun for an
inflationary phase. Still, a null result does not exclude inflation. However,
in a significant class of inflationary scenarios, a low SGWB amplitude is
correlated with a more significant running, $\alpha_{s}$, in the primordial
density perturbations than is seen with the simplest inflationary potentials.
With this motivation, we forecast the precision with which the spectral index
$n_{\rm{s}}$ and $\alpha_{\rm{s}}$ can be constrained by currently envisaged
observations, including CMB (Simons Observatory, CMB-S4 and LiteBIRD),
optical/near infra-red (DESI and SPHEREx), and 21cm intensity mapping (Tianlai
and CHIME) surveys. We identify optimal combinations of datasets for
constraining the running and show that they may yield additional and
informative constraints on the overall inflationary parameter space if the
SGWB remains undetected.

###### keywords:

cosmology: inflation – large-scale structure of the Universe – cosmic
background radiation

††pubyear: 2022††pagerange: Constraining Cosmic Inflation with Observations:
Prospects for 2030–Constraining Cosmic Inflation with Observations: Prospects
for 2030

##  1 Introduction

Cosmic inflation is a compelling explanation for the origin of the initial
perturbations that seed large-scale structure formation. It generically
predicts an initial spectrum of adiabatic, Gaussian, super-horizon density
perturbations, which grow under gravity to generate the distribution of matter
and radiation observed in present times (Starobinsky, 1979; Guth, 1981; Linde,
1982; Guth & Pi, 1982; Hawking, 1982; Starobinsky, 1983). These predictions
are tested by and are consistent with observations of the cosmic microwave
background (CMB; Liddle & Lyth, 1992; Spergel et al., 2003; Akrami et al.,
2020) anisotropies at recombination.

Inflation makes an additional prediction: the existence of a scale-free
primordial stochastic gravitational wave background (SGWB), sourced by quantum
fluctuations of spacetime during the inflationary epoch itself (Spergel et
al., 2003; Ade et al., 2018; Aghanim et al., 2020; Tristram et al., 2021; Ade
et al., 2021). These tensor modes distort the linear polarisation angle of the
CMB, generating a spectrum of B-modes in the polarisation distribution on the
sky. Gravitational waves undergo minimal reprocessing, and detecting an
inflationary SGWB would provide a remarkable window into the early universe.
Moreover, in the simplest inflationary models (single-field, slow-roll), there
is even a direct consistency relation between the scalar (density) and tensor
(gravitational wave) power spectra (Liddle & Lyth, 1992).

Detecting tensor modes in the CMB would be a strong indication that inflation
did happen in the very early universe and would permit decisive tests of
inflationary models motivated by specific ultrahigh-energy physics scenarios.
The SGWB is commonly parametrised via the ratio of amplitudes of the scalar
(density) and tensor spectra, denoted $A_{s}$ and $A_{T}$, respectively. By
convention, this is measured at the comoving wavenumber
$k=0.05\mathrm{Mpc}^{-1}$ and denoted

$r=\frac{A_{T}}{A_{s}}\,.$ (1)

However, there is no clear expectation regarding the likely amplitude of the
inflationary SGWB. In fact, in many scenarios, the SGWB is effectively
undetectable by any conceivable experiment.

Up until now, much of the focus in both the observational and theoretical
communities has been on the value of $r$, given that a scale-free, primordial
stochastic gravitational wave background would provide strong evidence for the
existence of an inflationary phase in the very early universe. In 2021 the
best constraints on $r$ moved from $r<0.06$ (BICEP2 Collaboration et al.,
2018) to $r<0.036$ at 95% confidence (Ade et al., 2021), via a combination of
data from the BICEP3/Keck array and Planck. Several CMB polarisation
experiments are in active development, including the Simons Observatory (Ade
et al., 2019), CMB-S4 (The CMB-S4 Collaboration et al., 2020) and LiteBIRD
(Hazumi et al., 2020), the last with the potential to measure $r$ to a
precision of $\sigma(r)\sim 10^{-4}$.

Algebraically simple potentials favour larger values of $r$ and such scenarios
can be successfully embedded in candidate theories of high-energy physics
(e.g. Silverstein & Westphal (2008)). However, there is no requirement
prevents $r$ from being vanishingly small. A failure to detect primordial
gravitational waves will not falsify the overall scenario but will only rule
out ‘large-field’ slow-roll inflation, a category which includes many once-
canonical models. Ironically, the more tightly $r$ is constrained, the more
plausible a very low value becomes, since the theoretical lower bounds on its
value lie many orders of magnitude below the reach of any plausible
experiment.

In the slow-roll framework, observables yield information about the shape of
the inflaton potential. This connection underpins the inflationary inverse
problem, or “reconstruction” (Lidsey et al., 1997). In particular, specifying
values of $n_{s}$ and $r$ sets the first two terms in the Hubble Slow Roll
expansion. This yields a prediction for the amount of accelerated expansion
that will transpire between the pivot scale leaving the horizon and inflation
coming to an end. Easther et al. (2022) show that current constraints on
$n_{s}$ and $r$ imply all such two-parameter models predict an inflationary
phase that lasts too long to be consistent with other cosmological
constraints.

There are several resolutions to this situation within the context of the
overall inflationary paradigm. For example, inflation may end with a sharp
discontinuity (a “cliff edge”), or the potential may be too complex to be
accurately represented by the analogue of a low-order Taylor expansion.
However, arguably the simplest resolution is working at next-to-leading order
in the systematic slow-roll expansion. The additional term this introduces
maps roughly to the third derivative of the potential, and, given an
appropriate value, it ends inflation without excessive expansion.

This term contributes to the running, or scale dependence of $n_{s}$,
$\alpha_{\rm{s}}=dn_{\rm{s}}/d\ln k$, where $k$ is the comoving wavenumber.
For most simple inflationary scenarios
$10^{-4}\lesssim|\alpha_{\rm{s}}|\lesssim 10^{-3}$ (Adshead et al., 2011).
However, if $\alpha_{\rm{s}}$ is chosen so that the duration of inflation is
consistent with current observational constraints, we typically need
$\alpha_{s}\lesssim-10^{-3}$. Moreover, if the bound on $r$ tightens the
likely magnitude of $\alpha_{s}$ will increase (Easther et al., 2022).

These values are far still smaller than those currently allowed by
observations, $\alpha_{\rm{s}}=-0.0045\pm 0.0067$ (at 68% confidence) (Planck
Collaboration et al., 2020). However, these constraints are set to tighten
substantially in the coming decade, thanks to measurements of the primordial
density power spectrum from large-scale galaxy surveys.111We focus on the two-
point function; the detection of primordial non-Gaussianity by any near-term
experiment would immediately rule out all the inflationary models considered
here.  These include the optical Dark Energy Spectroscopic Instrument (DESI)
galaxy redshift survey (DESI Collaboration et al., 2016) and the Spectro-
Photometer for the History of the Universe, Epoch of Reionization and Ices
Explorer (SPHEREx) galaxy survey (Doré et al., 2014), and the Canadian
Hydrogen Intensity Mapping Experiment (CHIME) (Bandura et al., 2014) and
Tianlai (Xu et al., 2015) 21cm intensity mapping experiments in radio.222These
are far from the only surveys. Other funded optical or 21cm intensity mapping
surveys that will produce cosmologically relevant results before 2030 include
the Baryon acoustic oscillations from integrated neutral gas observations
project (BINGO; Abdalla et al., 2022), Euclid (Amendola et al., 2018), the
Javalambre-Physics of the Accelerated Universe Astrophysical Survey (J-PAS;
Benitez et al., 2014), the Hydrogen Intensity and Real-time Analysis
eXperiment (HIRAX; Saliwanchik et al., 2020), the Legacy Survey of Space and
Time (LSST) at the Vera C. Rubin Observatory (Ivezić et al., 2019), the
MeerKAT Large Area Synoptic Survey (MeerKLASS; Santos et al., 2017), the
Widefield ASKAP L-band Legacy All-sky Blind surveY (WALLABY; Koribalski et
al., 2020), the WHT Enhanced Area Velocity Explorer (WEAVE; Pieri et al.,
2016), and the 4-metre Multi-Object Spectroscopic Telescope (4MOST; De Jong et
al., 2019). However, we expect DESI, SPHEREx and CHIME to be the best of their
classes for the purposes considered here. As explained in the text, we leave
consideration of Lyman-$\alpha$ forest surveys, such as the Hobby-Eberly
Telescope Dark Energy Experiment (HETDEX; Gebhardt et al., 2021) for future
work.

In what follows, we assemble a self-consistent compendium of forecasts for the
cosmological constraints on “inflationary observables” expected from
experiments currently deployed or under development. Given the genuine
possibility that the inflationary SGWB is vanishingly tiny, we focus on the
corner of parameter space in which $n_{\rm{s}}$ and $\alpha_{\rm{s}}$ are the
principal inflationary observables. We shall see that the larger values of
$|\alpha_{\rm{s}}|$ that are associated with very small $r$ may begin to be
accessible in the coming decade, even if they will not be rigorously testable
on this timescale.

The dynamical system that governs single-field slow-roll inflation and its
associated observables is well studied. However, the quality of present-day
observational data significantly changes our priors regarding the possible
form of the potential. The broader goal of this paper is to assess the ability
of future CMB and large-scale structure surveys to constrain inflationary
models in light of our current knowledge of the primordial perturbations. As
already noted, the tension between scenarios described by only the first two
slow-roll parameters and the data can be resolved in several ways. However,
the existence of a well-delineated set of scenarios with low $r$ and large
$\alpha_{s}$ provides an additional target for cosmological experiments.

The paper is structured as follows. In Section 2, we sketch the connections
between the power spectra, the slow-roll expansion and the expectations for
“inflationary” observables. Section 3 summarises the Fisher forecast approach
and the details of the surveys we will be simulating. In Section 4 we forecast
constraints from the different future CMB experiments and large-scale
structure surveys for $r$, $n_{s}$ and $\alpha_{\rm{s}}$, including joint
constraints, with a focus on the running $\alpha_{\textrm{s}}$. We conclude in
Section 5.

##  2 Inflation and the Running Index

Single-field, slow-roll inflation is the simplest implementation of the
broader inflationary paradigm. Its dynamics are governed by a field $\phi$
with potential $V(\phi)$ minimally coupled to Einstein’s gravity, which obeys
the equations of motion

$\displaystyle
H^{2}=\frac{1}{3M_{\rm{P}}^{2}}\left(\frac{\dot{\phi}^{2}}{2}+V(\phi)\right)\,,$
(2) $\displaystyle\dot{H}=-\frac{1}{2M_{\rm{P}}^{2}}\dot{\phi}^{2}\,,$ (3)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}=0\,.$ (4)

As usual, $a$ is the scale factor, $H$ is the Hubble parameter, and
$M_{\rm{P}}$ is the reduced Planck mass which, in natural units, is related to
Newton’s constant via $G=1/M_{\rm{P}}^{2}$.

The primordial observables are expressed in terms of the slow-roll parameters,
which can be defined in terms of the potential,

$\epsilon=\frac{M_{\rm{P}}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\,,\quad\eta=M_{\rm{P}}^{2}\frac{V^{\prime\prime}}{V}\,,\quad\xi=M_{\rm{P}}^{4}\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}}\,,$
(5)

where a dash denotes a derivative with respect to $\phi$. To lowest order the
key observables are

$n_{s}=1-6\epsilon_{0}+2\eta_{0}\,,\quad
r=16\epsilon_{0}\,,\quad\alpha_{s}=-2\xi_{0}\,,$ (6)

where the index $0$ denotes the value of the parameter when the pivot scale
leaves the horizon. A consequence of these relationships is that any set of
inflationary observables can be mapped to an inflationary trajectory: given
values for $n_{s}$, $\alpha_{s}$ and $r$, one can predict the amount of
accelerated growth that will occur after the pivot scale leaves the horizon on
the assumption that higher order terms are negligible. The dynamical system
that describes single-field inflation has long been understood (Lidsey et al.,
1997), and the possible inflationary mechanisms it supports are conveniently
summarized in the so-called zoo-plot (Dodelson et al., 1997).

Today, observational constraints significantly sharpen priors on $n_{s}$ and
$r$ and the corresponding slow-roll parameters. Recalling that
$r\propto\epsilon$ and $\epsilon\propto(V^{\prime}/V)^{2}$ we can divide the
parameter space into three distinct regimes,
$|V^{\prime}|\gtrsim|V^{\prime\prime}|$, $|V^{\prime}|\sim|V^{\prime\prime}|$
and $|V^{\prime}|\lesssim|V^{\prime\prime}|$, with $M_{\rm{P}}=1$ (Easther et
al., 2022). The first possibility is increasingly close to being
observationally untenable. Conversely, if the CMB experiments now under
construction do not detect an SGWB, it will follow that
$|V^{\prime\prime}|\gtrsim 5|V^{\prime}|$, leaving only the third alternative.
Consequently, even if the coming decade does not see a detection of an SGWB
constraints on the overall inflationary parameter space will still advance
both quantitatively and qualitatively.

The relationship between the spectral index and the derivatives of the
potential in Eq. 6 can be viewed as a differential equation, which can be
solved to yield a potential which delivers any pair of $n_{s}$ and $r$ at the
pivot. This leads to the so-called universality classes (Mukhanov, 2013;
Roest, 2014; Creminelli et al., 2015; Abazajian et al., 2016) of inflation and
is one basis of a widely repeated claim that $n_{s}\sim 1/N$ and
$\alpha_{s}\sim-1/N^{2}$, where $N$ is the remaining number of e-folds of
inflation.

The correlation between the running and the duration of inflation has been
explored a number of times (Malquarti et al., 2004; Makarov, 2005; Easther &
Peiris, 2006; Adshead & Easther, 2008; Adshead et al., 2011). In the potential
expansion, the scale-dependence of the $\eta$ parameter is given by

$\frac{d\eta}{dN}\approx{M_{P}^{4}}\left[\frac{V^{\prime}}{V}\frac{V^{\prime\prime\prime}}{V}-\left(\frac{V^{\prime}}{V}\right)^{2}\frac{V^{\prime\prime}}{V}\right]\,,$
(7)

where we have made use of the generic single-field slow-roll result

$\frac{dN}{d\phi}=\frac{1}{M_{\rm{P}}}\frac{1}{\sqrt{2\epsilon}}\,.$ (8)

If $\epsilon$ and thus $V^{\prime}/V$ is small, $(V^{\prime}/V)^{2}$ is very
small, while the measured value of $n_{s}$ fixes $V^{\prime\prime}/V$. In
these circumstances, the $V^{\prime}V^{\prime\prime\prime}/V^{2}$ term cannot
be considered irrelevant without an implicit tuning. In this case, the
dynamical system underlying the inflationary phase is more effectively treated
in the Hubble slow roll hierarchy (Liddle et al., 1994; Hoffman & Turner,
2001; Kinney, 2002), which can be explored analytically in this low-$\epsilon$
limit (Adshead & Easther, 2008).

The gradual tightening of constraints on $r$ and $n_{s}$ has reached the point
that any inflationary models described by only the first two slow roll
parameters lead to an inflationary phase that is inconsistent with limits on
the amount of accelerated expansion that occurs after astrophysical
perturbations leave the horizon.333This quantity is not precisely defined
unless the post-inflationary thermalisation mechanism is specified (Dodelson &
Hui, 2003; Liddle & Leach, 2003; Adshead et al., 2011; Muñoz & Kamionkowski,
2015). However, an upper bound follows from assuming immediate thermalisation
as inflation ends. For these cases, we can compute the value of
$V^{\prime\prime\prime}$ that will ensure that inflation terminates
appropriately, which also sets the value of $\alpha_{s}$ at the pivot (Easther
et al., 2022). Given these assumptions, $|\alpha_{s}|$ is larger than the
estimate of $1/(1-n_{s})^{2}$ derived from universality class arguments.

In doing so, we implicitly assume that $V^{\prime\prime\prime}/V$ is the “last
relevant term” in the Taylor expansion of the potential or, more accurately,
its Hubble Slow Roll analogue. We then constrain $\alpha_{s}$ by specifying an
overall duration of the inflationary phase, assuming that inflation ends via
the breakdown of slow-roll (Easther et al., 2022; Stein & Kinney, 2022). If
either of these conditions is violated, the arguments here do not constrain
$\alpha_{s}$. However, given current observational constraints on the
primordial spectra, this scenario represents a simple, distinctive and
testable class of inflationary model and is thus of significant interest.

Figure 1: Contours of constant $\alpha_{\mathrm{s}}$ in the $n_{\mathrm{s}}$ −
$\ln r$ plane with a three-term slow-roll hierarchy and $\xi$ set to give
$N=55$ when the pivot leaves the horizon. The vertical lines mark the current
$66$ and $95\%$ contours for $n_{\mathrm{s}}$ from Planck Collaboration et al.
(2020).

Fig. 1 illustrates the relationship between $n_{s}$, $r$ and $\alpha_{s}$ if
we assume that 55 e-folds of inflation occur after the pivot scale leaves the
horizon. This implies rapid but not instantaneous thermalisation, and in
scenarios with a long phase in which the post-inflationary universe is
effectively matter-dominated (which is a distinct possibility), the
corresponding value of $\alpha_{s}$ would increase. While much attention is
paid to the tightening bounds on $r$, the measured value of $n_{s}$ is also
critical – the closer this parameter is to unity, the larger
$V^{\prime\prime\prime}$ and thus $\alpha_{s}$ must be to match constraints on
the duration of inflation. Current bounds on $n_{s}$ are overlaid on Fig. 1.
If the permitted values of $n_{s}$ contract toward the centre of its current
range as further data become available, expectations for the magnitude of
$\alpha_{s}$ will likewise increase.

##  3 Fisher forecasting

###  3.1 Fisher matrix

To make meaningful comparisons between multiple possible combinations of
survey data we first need to create a mutually consistent set of forecasts. A
forecast assumes some likelihood function $\mathcal{L}(\theta)$, where the
probability of the data ($D$) can be evaluated with regards to assumed (fixed)
values of the parameters of interest ($\theta$),

$\mathcal{L}(\theta)=P(D|\theta)\,.$ (9)

When faced with real data, this likelihood function can be used to evaluate
the probability distribution of these parameters. But even before the data is
taken, a forecast can be made to estimate the effectiveness of the planned
survey or experiment. A common approach is the Fisher forecast (Kendall &
Stuart, 1977; Tegmark et al., 1997; Hobson et al., 2009), where the data is
simulated assuming some true values of the parameters $\hat{\theta}$, and the
log-likelihood function is Taylor-expanded around the extremum. For a given
realisation of the future data, the expansion can be modelled as

$\ln\mathcal{L}(\theta)\approx\ln\mathcal{L}(\theta^{*})+\frac{1}{2}\sum_{ij}(\theta_{i}-\theta^{*}_{j})^{T}H_{ij}(\theta_{i}-\theta^{*}_{j})\,,$
(10)

where $\theta^{*}$ is the maximum likelihood value of the parameters, and $H$
is the Hessian of the log-likelihood,

$H_{ij}=\left.\frac{\partial^{2}\ln\mathcal{L}}{\partial\theta_{i}\partial\theta_{j}}\right|_{\theta^{*}}\,.$
(11)

Averaging over all possible realisations of the data, the maximum likelihood
value $\theta^{*}$ should approach the fiducial value $\hat{\theta}$, and the
Fisher matrix is the expectation of the Hessian

$F_{ij}=\langle H_{ij}\rangle\,.$ (12)

Thus the Fisher matrix encodes the curvature of the likelihood surface around
the peak, and in the case of a Gaussian likelihood distribution, can be
related to the parameter probability covariance matrix $\mathbf{C}$,

$\mathbf{F}=\mathbf{C}^{-1}\,.$ (13)

This is the limiting best case of the Cramér–Rao bound.

A forecast 68% constraint on a parameter can therefore be obtained by
computing the curvature of the likelihood with the respect to the parameters.
In some cases, this can be performed analytically, but many likelihoods (and
almost all of those we encounter in cosmology) have a non-linear relationship
between input parameters and output value. Therefore the partial derivative
will need to be computed through some finite differencing method.

Since the values of the observable data and errors on the data are assumed
fixed given the fiducial values, the Fisher matrix depends on the sensitivity
of the theoretical prediction for the observable with respect to the
parameter(s) of interest. For the next generation of cosmological surveys,
accurate predictions of observables become difficult, due to the presence of
systematic effects associated with how the data is taken. This is already the
case with CMB power spectrum measurements, where foreground dust or
synchrotron emission can be confused with primordial CMB anisotropies. These
foregrounds can be modelled using nuisance parameters $\theta_{\mathbf{n}}$.
Since we do not know these noise parameters to infinite precision we have
included them in our Fisher analysis, alongside the standard set of
cosmological parameters $\theta_{c}$. The nuisance parameters will differ
between different types of observations (i.e. CMB vs optical galaxy redshift
surveys), so confusion between a cosmological parameter and a nuisance
parameter for a single experiment may be alleviated in combination.

###  3.2 Fiducial parameters

Our primary goal is to assess the sensitivity of future surveys to parameters
relating to the power spectra of primordial scalar and tensor perturbations.
These parameters usually are independent of (and thus relatively uncorrelated
with) the cosmological parameters that control the evolution of the late-time
universe. We assume a concordance Flat $\Lambda$CDM cosmology, where the dark
energy is a cosmological constant with no time evolution, and the total
density is equal to the critical density, i.e.

$\Omega_{\Lambda}=1-\Omega_{\mathrm{CDM}}-\Omega_{b}\,.$ (14)

The primordial power spectrum for the scalar perturbations is parameterised as

$P_{s}(k)=A_{s}\left(\frac{k}{k_{0}}\right)^{n_{s}-1+\frac{1}{2}\alpha_{\rm{s}}\ln(k/k_{0})+\ldots}\,.$
(15)

We assume that the running of the running $\beta_{s}=d\ln\alpha_{\rm{s}}/d\ln
k=0$, and that higher order derivatives are also absent. We also assume the
absence of any features (e.g. Adams et al. (2001); Hazra et al. (2014)) or
cut-off (e.g. Abazajian et al. (2014)) in the power spectrum.

For the spectrum of tensor perturbations, the amplitude is expressed in terms
of the tensor-to-scalar ratio (Eq. 1). We work with an inflationary prior so
that the slow-roll consistency relation

$n_{T}=-\frac{r}{8}\,.$ (16)

fixes the index of the tensor power spectrum. The fiducial values of the
cosmological parameters are chosen to be the Planck best fit cosmological
parameters and are listed in Tab. 1.

Name | Symbol | Value |
---|---|---|---
Physical baryon density | $\Omega_{b}h^{2}$ | 0.02237 | varying
Physical CDM density | $\Omega_{\mathrm{CDM}}h^{2}$ | 0.12 | varying
Hubble parameter | $H_{0}$ | 67.36 | varying
Matter density | $\Omega_{\rm m}$ | $\Omega_{b}+\Omega_{\mathrm{CDM}}$ | derived
Cosmological constant density | $\Omega_{\Lambda}$ | $1-\Omega_{\rm m}$ | derived
Optical depth to reionization | $\tau$ | 0.0925 | varying
Physical neutrino density† | $\Omega_{\nu}h^{2}$ | $4.28\times 10^{-4}$ | fixed
Curvature parameter | $\Omega_{K}$ | $0$ | fixed
Dark energy equation of state | $w$ | $-1$ | fixed
Primordial spectral index | $n_{\rm s}$ | 0.9649 | varying
of scalar fluctuations |  |  |
Amplitude of scalar fluctuations | $A_{\rm s}$ | $2.215\times 10^{-9}$ | varying
Running of primordial | $\alpha_{\rm s}$ | 0 | varying
spectral index |  |  |
Tensor to scalar ratio | $r$ | 0 | varying
Table 1: Fiducial values assumed in our Fisher forecasts.

###  3.3 Future surveys

A number of cosmological surveys are currently in different phases of design
and construction. To forecast how constraints on the primordial power spectrum
will improve over the period 2022-2030, we focus on those that are currently
deployed or an advanced state of preparation, namely

  * •

SO: Simons Observatory (Ade et al., 2019) \- a ground-based CMB experiment
being built at the same site as the Simons Array and the Atacama Cosmology
Telescope. SO consists of four telescopes; three small aperture telescopes
(SATs) and one large aperture telescope (LAT). It will cover the wavelength
range 27-280 GHz.

  * •

CMB-S4 (Lawrence et al., 2017): a ground-based CMB experiment being
constructed at both the South Pole and Atacama desert sites.

  * •

LiteBIRD: Lite (Light) satellite for the studies of B-mode polarization and
inflation from cosmic background Radiation Detection (Hazumi et al., 2020) \-
a satellite CMB observatory covering the wavelength range 40 to 400 GHz.

  * •

DESI: Dark Energy Spectroscopic Instrument, (DESI Collaboration et al., 2016)
\- a ground-based optical spectroscopic galaxy redshift survey using the
Mayall 4m telescope at Kitt Peak. It will measure the redshifts of Luminous
Red Galaxies (LRGs) up to $z=1$, Emission-Line Galaxies (ELGs) up to $z=1.7$,
and Quasars (QSOs) in the range $2.1<z<3.5$. The survey will cover $\approx
14,000~{}\mathrm{deg}^{2}$.

  * •

SPHEREx: Spectro-Photometer for the History of the Universe, Epoch of
Reionization, and Ices Explorer (Doré et al., 2014) is a space-based
observatory that will conduct an all-sky survey in the near-infrared, with
photometry over the range 0.75–5.0 microns. Redshifts are determined
photometrically by fitting templates to the NIR spectra, in particular, the
1.6 $\mu$m bump. Its launch date is planned for April 2025.

  * •

CHIME: Canadian Hydrogen Intensity Mapping Experiment (Bandura et al., 2014)
is a ground-based radio observatory that will detect neutral hydrogen from
total 21-cm emission over some relatively de-localised angular region in the
sky. It will cover 60% of the sky area over the redshift range $0.8<z<2.5$.

  * •

Tianlai (Xu et al., 2015): a ground-based radio observatory, which will survey
the distribution of neutral hydrogen in the universe, in a similar manner to
CHIME.

###  3.4 CMB Fisher forecasting

Field | $f_{\mathrm{sky}}$ | $\ell_{\mathrm{min}}$ | $\ell_{\mathrm{max}}$
---|---|---|---
Simons Observatory Large Aperture Telescope
\- T, E | 0.4 | 30 | 8000
\- B | 0.4 | 30 | 1000
Simons Observatory Small Aperture Telescope
\- E | 0.1 | 30 | 8000
\- B | 0.1 | 30 | 300
CMB-S4 Large Aperture Telescope
\- T | 0.4 | 30 | 3000
\- E | 0.4 | 30 | 5000
CMB-S4 Small Aperture Telescope
\- E, B | 0.03 | 21 | 335
LiteBIRD
\- T, E, B | 0.9 | 2 | 200
Table 2: Observed sky fractions $f_{\mathrm{sky}}$ and multipole ranges
expected for the CMB experiments considered in this work.

CMB experiments observe large areas of the (spherical) sky so it is convenient
to analyse their data using spherical harmonics $Y_{\ell}^{m}(\theta,\phi)$
such that the fluctuations $\delta X$ of the field $X\in\\{T,E,B\\}$ in
direction $(\theta,\phi)$ can be written as

$\delta X(\theta,\phi)=\sum_{\ell m}a_{\ell m}^{X}Y_{\ell}^{m}(\theta,\phi).$
(17)

Assuming statistical isotropy and Gaussian fluctuations, we can fully describe
the statistics of the fluctuation fields by their angular power spectra

$C^{XY}_{\ell}=\left\langle\left|a_{\ell m}^{X\ast}a_{\ell
m}^{Y}\right|\right\rangle.$ (18)

Describing the likelihood function $\mathcal{L}$ in terms of angular power
spectra $C^{XY}_{\ell}$, the Fisher matrix reads (e.g. Bermejo-Climent et al.,
2021)

$F_{ij}=\frac{1}{2}\operatorname{tr}\left\\{\frac{\partial\mathbf{C}}{\partial\theta_{i}}\mathbf{C}^{-1}\frac{\partial\mathbf{C}}{\partial\theta_{j}}\mathbf{C}^{-1}\right\\},$
(19)

where $\mathbf{C}^{XY}\equiv C^{XY}_{\ell}+N_{\ell}\delta_{\mathrm{K}}^{XY}$
and, to take the trace, one has to take into account that the number of modes
corresponding to each multipole is given by $(2\ell+1)f_{\mathrm{sky}}$. The
observed sky fractions $f_{\mathrm{sky}}$ and the $\ell$-ranges considered
when taking the trace are listed in Table 2. We adopt the noise model

$N_{\ell}=N_{\mathrm{red}}\left(\frac{\ell}{\ell_{\mathrm{knee}}}\right)^{\alpha_{\mathrm{knee}}}+N_{\mathrm{white}}$
(20)

from Ade et al. (2019) with the parameters listed in Table 3. The values of
these parameters vary between frequency bands. By observing in multiple
frequency bands, CMB experiments can distinguish cosmological signals from
foreground contamination, whose spectral dependence is typically different
from the blackbody curve of the CMB itself. A conservative assumption for the
noise of the combined CMB maps is that it is equal to the minimum noise of
each frequency band. We, therefore, choose effective noise parameters in a way
that, for any given multipole $\ell$, Eq. (20) approximates the minimum
$N_{\ell}$ among all frequency bands. The parameter values for the Simons
Observatory are modelled after Ade et al. (2019), the ones for CMB-S4 after
Lawrence et al. (2017), and we take $N_{\mathrm{white}}$ predictions for
LiteBIRD from Sailer et al. (2021). The remaining LiteBIRD noise parameters
are taken from Nerval et al. (2019). The noise models are plotted in Figure 2.

Instrument |  $N_{\mathrm{white}}$ [$\mathrm{\mu K^{2}}$] |  $N_{\mathrm{red}}$ [$\mathrm{\mu K^{2}}$] | $\ell_{\mathrm{knee}}$ | $\alpha_{\mathrm{knee}}$
---|---|---|---|---
Simons Observatory Large Aperture Telescope
\- temperature | $2.8\times 10^{-6}$ | $1.8\times 10^{-4}$ | 1000 | -3.5
\- polarisation | $2.8\times 10^{-6}$ | $2.8\times 10^{-6}$ | 700 | -1.4
Simons Observatory Small Aperture Telescope
\- polarisation | $3.1\times 10^{-7}$ | $3.1\times 10^{-7}$ | 25 | -2.5
CMB-S4 Large Aperture Telescope
\- temperature | $1.5\times 10^{-7}$ | $1.5\times 10^{-7}$ | 3400 | -4.7
\- polarisation | $1.5\times 10^{-7}$ | $1.5\times 10^{-7}$ | 340 | -4.7
CMB-S4 Small Aperture Telescope
\- polarisation | $8.5\times 10^{-8}$ | $8.5\times 10^{-8}$ | 60 | -2.5
LiteBIRD
\- polarisation | $1.7\times 10^{-7}$ | $1.7\times 10^{-7}$ | 12 | -1.5
Table 3: Effective noise-model parameters used in CMB forecasts. Figure 2:
Angular power spectra (in black) and noise spectra (in colour) used in our CMB
forecasts. The EE and BB noise spectra are slightly shifted towards the bottom
in order to make them visible where they coincide with the temperature noise.

A primordial B-mode signal is expected to be dominated by secondary B-modes
generated by gravitational lensing. Consequently, ‘delensing’ – reconstructing
the lensing potential and subtracting the lensing contribution to the B-modes
– will be key to the analysis of this data. In practice, this cancellation is
likely to be incomplete, and we introduce a nuisance parameter
$A_{\mathrm{lens}}$ to marginalise over the residual. Sherwin & Schmittfull
(2015) predict that half of the lensing contribution can be subtracted from
the Simons Observatory B-mode signal. For CMB-S4, we use the conservative
estimate $A_{\mathrm{lens}}=0.27$ where delensing is supported by the large
area telescopes based in Chile (The CMB-S4 Collaboration, 2018).

Galactic thermal dust and synchrotron radiation add significantly to the
polarisation signal. The spectra of these two effects are consistent with
power laws whose exponents we take from Adam et al. (2016) for the thermal
dust and from Choi & Page (2015) for the synchrotron power spectrum. Including
these two terms, our complete $C_{\ell}$ model reads

$\displaystyle C_{\ell}^{XY}=$ $\displaystyle
A_{\mathrm{lens}}C_{\ell}^{XY,\mathrm{lensed}}+\left(1-A_{\mathrm{lens}}\right)C_{\ell}^{XY,\mathrm{raw}}$
$\displaystyle+A_{\mathrm{dust}}\ell^{-2.42}+A_{\mathrm{synch}}\ell^{-2.3}\text{(this
line not for TT)},$ (21)

where we obtain $C_{\ell}^{XY,\mathrm{lensed}}$ and
$C_{\ell}^{XY,\mathrm{raw}}$ from the Cosmic Linear Anisotropy Solving System
(Blas et al., 2011; Di Dio et al., 2013, CLASS), and $A_{\mathrm{dust}}$ and
$A_{\mathrm{synch}}$ are free parameters with zero-centred Gaussian priors
with variance $5\times 10^{-4}$. This choice reproduces the forecasts
presented in Ade et al. (2019).

###  3.5 LSS Fisher forecasting

The most common summary statistic, when redshift information is available, is
the Cartesian power spectrum

$P_{x}(\mathbf{k})=\frac{1}{(2\pi)^{3}}\left\langle\delta_{x}^{\ast}(\mathbf{k})\delta_{x}(\mathbf{k})\right\rangle$
(22)

of the over-density field

$\delta_{x}(\mathbf{k})=\int\delta_{x}(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}}\operatorname{d}^{3}\mathbf{r}$
(23)

of tracer $x$ in Fourier basis $\mathbf{k}$. However, this is not a natural
basis for dealing with wide-angle effects. For instance, ignoring the presence
of a large magnification bias parameter $s_{\mathrm{g}}$ can render forecasts
on $\sigma(n_{\mathrm{s}})$ too optimistic by a factor of 1.25 for Stage IV
spectroscopic galaxy survey (Bellomo et al., 2020). Alternative approaches,
such as the angular power spectrum in tomographic redshift bins (e.g. Asorey
et al., 2012; Loureiro et al., 2019; Tanidis & Camera, 2021) or the 3D power
spectrum in a spherical Fourier-Bessel basis (e.g. Heavens, 2003; Leistedt et
al., 2012; Wang et al., 2019; Gebhardt & Doré, 2021), can naturally account
for wide angle effects. However, the former loses some of the information
contained in radial modes, while the latter complicates the cosmological
interpretation at scales much smaller than the survey size. On the other hand,
the 3D power spectrum has been widely used in the last decades, and many
analysis tools exist. We, therefore, expect that the key science analyses in
upcoming collaborations will still be done using Cartesian 3D power spectra,
despite their shortcomings, and make our forecasts accordingly.

Figure 3: Assumed values for the number densities $n(z)$ used in DESI and
SPHEREx forecasts.

Assuming a statistically isotropic universe, the angle-averaged power spectrum
$P_{\mathrm{g}}(k)$ contains all information on inflation, even in redshift
space. In this case, the Fisher matrix reads (Tegmark, 1997)

$F_{ij}=\frac{f_{\mathrm{sky}}}{2}\int_{k_{\mathrm{min}}}^{k_{\mathrm{max}}}\frac{\partial
P_{x}(k,z)}{\partial\theta_{i}}\frac{\partial
P_{x}(k,z)}{\partial\theta_{j}}\frac{V_{k}V_{\mathrm{eff}}(k,z)}{\left[P_{x}(k,z)+P_{\mathrm{N}}(z)\right]^{2}}\operatorname{d}k,$
(24)

where $V_{k}=\frac{k^{2}}{(2\pi)^{2}}$ and $P_{\mathrm{N}}(z)$ is the noise
power spectrum. If we use galaxies as tracers, the effective survey volume is

$V_{\mathrm{eff}}(k,z)=V_{\mathrm{survey}}\left(\frac{\bar{n}(z)P(k,z)}{1+\bar{n}(z)P(k,z)}\right)^{2}$
(25)

for a survey with varying average density $\bar{n}(z)$ (such as galaxy
surveys) or it is equal to the survey volume $V_{\mathrm{survey}}$ (e.g. for
HI intensity mapping). We plot the values of $\bar{n}(z)$ that we assume for
our forecasts in Figure 3. We downloaded the SPHEREx $n(z)$ and $b(z)$ tables
from https://github.com/SPHEREx/Public-products. The DESI values are inspired
by preliminary target selection papers (Raichoor et al., 2020a; Yèche et al.,
2020; Zhou et al., 2020).

Survey | $f_{\mathrm{sky}}$ | $k_{\mathrm{min}}$ | $k_{\mathrm{max}}$ | $b$
---|---|---|---|---
DESI
\- QSO | 0.33 | 0.0016 | 0.24 | 2.4
\- ELG | 0.33 | 0.0018 | 0.21 | 1.5
\- LRG | 0.33 | 0.0025 | 0.16 | 2.4
SPHEREx
\- $\sigma_{z}<0.003(1+z)$ | 0.7 | 0.001 | 0.2 | 1.4
\- $0.003(1+z)<\sigma_{z}<0.01(1+z)$ | 0.7 | 0.001 | 0.2 | 1.2
\- $0.01(1+z)<\sigma_{z}<0.03(1+z)$ | 0.7 | 0.001 | 0.2 | 1.5
\- $0.03(1+z)<\sigma_{z}<0.1(1+z)$ | 0.7 | 0.001 | 0.2 | 1.4
\- $0.1(1+z)<\sigma_{z}<0.2(1+z)$ | 0.7 | 0.001 | 0.2 | 1.3
CHIME | 0.6 | 0.0012 | 0.26 | 1.0
Table 4: Observed sky fractions $f_{\mathrm{sky}}$, $k$ ranges, and bias
values $b$ expected for the LSS experiments considered in this work.

Another crucial ingredient for Equation (24) is the range of scales between
$k_{\mathrm{min}}$ and $k_{\mathrm{max}}$ that each survey is going to be able
to probe. For SPHEREx, we adopted $k_{\mathrm{min}}=0.001$ and
$k_{\mathrm{max}}=0.2$ from Doré et al. (2014). For DESI and CHIME, we use
$k_{\mathrm{min}}=2\pi/\sqrt[3]{V_{\mathrm{survey}}}$ which we list in Table
4. This corresponds to the maximum scale attainable for a spherical survey
with volume $V_{\mathrm{survey}}$. As neither DESI nor CHIME observe perfect
spherical volumes, they will be able to measure longer modes in certain
directions than those we consider here. However, these will be noisy and prone
to direction-dependent systematic effects, and we do not take them into
consideration. Since we only operate with a linear clustering model, we also
do not consider scales above $k_{\mathrm{max}}$ where non-linear effects
become important. We obtain $k_{\mathrm{max}}$ for DESI and CHIME by
interpolating Table 1 of Seo & Eisenstein (2003) for the mean redshift of the
probe in question. With this choice of $k_{\mathrm{max}}$, we are only
marginally affected by small-scale corrections from, e.g., massive neutrinos
or alternative dark energy models.

####  3.5.1 Galaxy Surveys

Given a model matter power spectrum $P_{\mathrm{m}}(k)$ (that we can compute
again using CLASS), we use the Kaiser (1987) formula to compute the model
galaxy power spectrum

$P_{\mathrm{g}}(k,z)=b^{2}P_{\mathrm{m}}(k,z)\left[1+\frac{2\beta}{3}+\frac{\beta^{2}}{5}\right].$
(26)

Here $b$ is the large-scale galaxy bias which we treat as a free parameter;
$\beta$ is the linear growth rate of matter perturbations (which CLASS
computes for a given set of cosmological parameters) divided by the galaxy
bias $b$. We use this simple formula in our DESI forecasts where the
spectroscopic redshift errors are negligibly small. For SPHEREx forecasts,
however, redshift errors will be significant. The uncertainty in the redshift
estimation of tracers washes out some of the information from radial modes.

Assuming Gaussian distributed redshifts with variance $\sigma_{z}$ and
following the notation of Asorey et al. (2012), the 3D galaxy power spectrum
reads

$P_{\mathrm{g}}(k,\mu,z)=b^{2}\left(1+\beta\mu^{2}\right)^{2}P_{\mathrm{m}}(k,z)e^{-k^{2}\sigma_{\mathrm{t}}^{2}(z)\mu^{2}},$
(27)

where $\mu$ is the cosine of the angle between the mode and the line-of-sight
direction and $\sigma_{\mathrm{t}}(z)={c\sigma_{z}(z)}/{H_{\mathrm{fid}}(z)}$.
Note that we use the fiducial value $H_{\mathrm{fid}}(z)$ of the Hubble-
Lemaître parameter since using the $H(z)$ of the varied cosmology would mean
that the redshift uncertainty of the power spectrum would provide us with
information on the expansion rate. Averaging over all angles $\mu$, we obtain
the monopole power spectrum

$\displaystyle P_{\mathrm{g}}(k,z)=$ $\displaystyle
b^{2}P_{\mathrm{m}}(k,z)\left[\sqrt{\pi}\left(4k^{4}\sigma_{\mathrm{t}}^{4}+4k^{2}\sigma_{\mathrm{t}}^{2}\beta+3\beta^{2}\right)\operatorname{erf}(k\sigma_{\mathrm{t}})\right.$
$\displaystyle\left.-2k\sigma_{\mathrm{t}}e^{-k^{2}\sigma_{\mathrm{t}}^{2}}\beta\left(2k^{2}\sigma_{\mathrm{t}}^{2}(\beta+2)+3\beta\right)\right]/\left(8k^{5}\sigma_{\mathrm{t}}^{5}\right).$
(28)

At scales where $k\ll\frac{1}{\sigma_{\mathrm{t}}}$, we can expand equation
(3.5.1) to find the well-known Kaiser formula (cf. Eq. 26) with a quadratic
correction in $k\sigma_{\mathrm{t}}$:

$P_{\mathrm{g}}(k,z)=b^{2}P_{\mathrm{m}}(k,z)\left[1+\frac{2\beta}{3}+\frac{\beta^{2}}{5}-k^{2}\sigma_{\mathrm{t}}^{2}\left(\frac{1}{3}+\frac{2\beta}{5}+\frac{\beta^{2}}{7}\right)\right].$
(29)

The shot noise of galaxies is already accounted for in
$V_{\mathrm{eff}}(k,z)$. Thus, $P_{\mathrm{N}}(z)=0$ in Equation 24.

####  3.5.2 HI Intensity Mapping

When making forecasts for HI intensity mapping experiments, we proceed
similarly as for spectroscopic galaxy surveys. Since intensity mapping cannot
resolve single objects we model the power spectrum of the line intensity

$P_{\mathrm{HI}}(k,z)=\bar{T}_{\mathrm{b}}^{2}(z)b_{\mathrm{HI}}^{2}(z)P_{\mathrm{m}}(k,z)$
(30)

in terms of the temperature (Santos et al., 2015)

$T_{\mathrm{b}}(z)\approx
180h\frac{H_{0}}{H(z)}\Omega_{\mathrm{HI}}(z)(1+z)^{2}\mathrm{mK},$ (31)

where we approximate the density of HI tracers

$\Omega_{\mathrm{HI}}(z)=0.00048+0.00039z-0.000065z^{2}$ (32)

with the fit of Cunnington et al. (2019). As in the spectroscopic case, we
treat the bias $b_{\mathrm{HI}}$ a free parameter. Following Xu, Wang & Chen
(2015), we model the noise power spectrum as

$P_{\mathrm{N}}(k,z)=\frac{4\pi
f_{\mathrm{sky}}\lambda^{2}T_{\mathrm{sys}}^{2}y(z)r_{\mathrm{a}}^{2}(z)}{A_{\mathrm{e}}\Omega_{\mathrm{FoV}}t_{\mathrm{tot}}},$
(33)

where $A_{\mathrm{e}}$ is the effective collecting area of each element,
$\lambda$ is the redshifted wavelength of HI emission, $T_{\mathrm{sys}}$
designates the system temperature, $t_{\mathrm{tot}}$ the total integration
time, $\Omega_{\mathrm{FOV}}$ symbolises the field of view of a single
pointing, $y(z)=\frac{\lambda(1+z)}{H(z)}$ and $r_{\mathrm{a}}$ denotes the
angular diameter distance.

Figure 4: Forecast error contours on $n_{\mathrm{s}}$ and
$\alpha_{\mathrm{s}}$ from LSS surveys. An asterisk in the label denotes that
a Planck18 prior on the 6 Planck-$\Lambda$CDM-model parameters has been
applied. We do not apply any prior on $\alpha_{\mathrm{s}}$. Figure 5: 95%
confidence intervals for $\alpha_{\mathrm{s}}$ (left) and $n_{\mathrm{s}}$
(right) from WMAP and Planck (in black) along with forecast constraints for
upcoming surveys (in blue). The vertical area shaded in grey marks the
$\alpha_{\mathrm{s}}$-range between $-10^{-3}$ and $-10^{-4}$. DESI∗ refers to
DESI forecasts with a Planck 2018 prior on the 6 Planck-$\Lambda$CDM-model
parameters without a prior on $\alpha_{\mathrm{s}}$.

##  4 Results

###  4.1 CMB Forecasts

Figure 6: Forecast posterior contours of the spectral index $n_{\mathrm{s}}$, its running $\alpha_{\mathrm{s}}$ and the tensor-to-scalar ratio $r$ of the three CMB experiments considered in this work. survey | $10^{3}\sigma_{n_{\mathrm{s}}}$ | $10^{3}\sigma_{\alpha_{\mathrm{s}}}$ | $10^{4}r<$
---|---|---|---
DESI, Planck18444The Planck18 prior here is just a prior on the 6-parameter $\Lambda$CDM model, i.e., there is no prior on $\alpha_{\mathrm{s}}$ and no measurement on $r$ presented here. | 7.2 | 17 | -
CHIME | 7.4 | 5.1 | -
SPHEREx | 26 | 4.6 | -
SO | 5.6 | ${}^{+6.5}_{-6.6}$ | 99
\+ DESI | 4.5 | 6.2 | 99
\+ CHIME | ${}^{+2.9}_{-2.8}$ | 3.5 | 99
\+ SPHEREx | 1.6 | 2.4 | 98
S4 | 5.9 | ${}^{+6.8}_{-6.9}$ | 34
\+ DESI | 4.8 | 6.3 | 34
\+ CHIME | 3.0 | 3.7 | 34
\+ SPHEREx | 1.7 | 2.4 | 33
\+ CHIME, DESI, SPHEREx | 1.5 | 1.8 | 33
LiteBIRD | 140 | 67 | 0.98
LiteBIRD, S4(ℓ>200) | 3.7 | 4.9 | 0.98
\+ CHIME, SPHEREx | 1.2 | 1.7 | 0.97
\+ CHIME, DESI, SPHEREx | 1.2 | 1.7 | 0.97
Table 5: Forecast 95% confidence intervals of the spectral index
$n_{\mathrm{s}}$ and its running $\alpha_{\mathrm{s}}$, as well as forecast
upper limits on $r$ at 95% confidence level.

We show forecasts of $n_{\mathrm{s}}$, $\alpha_{\mathrm{s}}$ and $r$ in Figure
6 and list our expected constraints in Table 5. The three CMB experiments
under consideration, the Simons Observatory, CMB-S4 and LiteBIRD, are
complementary to each other. To measure the running $\alpha_{\mathrm{s}}$, we
need a long lever arm measuring the spectral index on many different scales.
From the CMB, we get those with the Large Aperture Telescopes (LAT) of the
partial-sky, high-resolution experiments, Simons Observatory and CMB-S4.
Interestingly, we shall see tighter constraints on $n_{\mathrm{s}}$ and
$\alpha_{\mathrm{s}}$ from the Simons Observatory even though it is expected
to run before CMB-S4. This behaviour is because the Simons LAT is less noisy
at large scales than CMB-S4 LAT. However, CMB-S4 will greatly improve
constraints on $r$ compared to the Simons Observatory, so the two experiments
are complementary to each other.

While the partial-sky, high-resolution experiments will significantly tighten
constraints on the running $\alpha_{\mathrm{s}}$, LiteBIRD will provide strong
constraints on $r$. Note that our forecasts depend, to some degree, on the
choice of the pivot scale $k_{0}$ (cf. Eq. 15). We adopt
$k_{0}=0.05\;\mathrm{Mpc}^{-1}$ throughout this work for internal consistency.
LiteBIRD is sensitive up to $\ell_{\mathrm{max}}=200$ only, which roughly
corresponds to $k\approx 0.0014\;\mathrm{Mpc}^{-1}$ in our fiducial cosmology.
This scale is far below the pivot scale chosen here, and other authors
commonly use $k_{0}=0.002\;\mathrm{Mpc}^{-1}$ in their LiteBIRD forecasts.
Using a larger pivot scale (i.e. smaller value of $k_{0}$), the LiteBIRD
constraints on $n_{\mathrm{s}}$ become tighter, though not competitive with
the Simons Observatory or CMB-S4.

We find that the best we can do in terms of $n_{\mathrm{s}}$,
$\alpha_{\mathrm{s}}$ and $r$ with CMB experiments in 2030 is to combine
LiteBIRD data at large scales ($\ell<200$) and CMB-S4 at small scales
($\ell>200$). Even though the Simons Observatory alone constrains
$n_{\mathrm{s}}$ and $\alpha_{\mathrm{s}}$ better than CMB-S4 when combined
with LiteBIRD, LiteBIRD beats both the Simons and CMB-S4 large-scale noise. At
small scales, CMB-S4 outshines the Simons Observatory. Our forecast errors are
$\sigma_{n_{\mathrm{s}}}=0.0037$, $\sigma_{\alpha_{\mathrm{s}}}=0.0049$ and
$r<9.8\times 10^{-5}$, assuming non-detection of positive $r$.

###  4.2 LSS Forecasts

Present constraints on $n_{\mathrm{s}}$ and $\alpha_{\mathrm{s}}$ have
primarily come from the CMB since large-scale structure surveys have not
covered large enough volumes yet to measure the galaxy power spectrum at
wavenumbers $k$ larger than the matter-radiation equality scale. However, as
galaxy surveys probe larger volumes it will soon be feasible to measure
$\alpha_{\mathrm{s}}$ and $n_{\mathrm{s}}$ from LSS surveys alone. Figure 4
shows our forecast probability contours. As the position of the radiation
equality scale depends on the Hubble-Lemaître rate $H_{0}$ and the density of
cold dark matter $\omega_{\mathrm{cdm}}$, $n_{\mathrm{s}}$ and
$\alpha_{\mathrm{s}}$ constraints from LSS are also dependent on these two
parameters.

The baryon acoustic oscillation (BAO) feature provides stringent constraints
on $H(z)r_{\mathrm{d}}$ (Bautista et al., 2020; De Mattia et al., 2021; Du Mas
des Bourboux et al., 2020; Gil-Marin et al., 2020; Hou et al., 2020; Neveux et
al., 2020; Raichoor et al., 2020b). Thus, to measure the Hubble-Lemaître rate
$H_{0}$, one needs constraints on the sound horizon scale at the end of the
Compton drag epoch $r_{\mathrm{d}}$ from CMB experiments. It is, therefore,
customary to present LSS forecasts on cosmological parameters with a CMB
prior. We show DESI contours with a Planck18 prior (Aghanim et al., 2020) and
refer to it as DESI∗ in the remainder of this work. We summarise all single
survey constraints on $\alpha_{\mathrm{s}}$ in Figure 5 along with the 95%
confidence ranges from Planck and WMAP legacy data. Even though DESI alone
will not provide tight constraints on $\alpha_{\mathrm{s}}$, with a Planck18
prior, it will provide an important semi-independent consistency check of the
Planck18 measurement of $\alpha_{\rm{s}}=-0.0045\pm 0.0067$ (Planck
Collaboration et al., 2020). This is because the Planck18 prior does not put
any limits on $\alpha_{\mathrm{s}}$ but only on the parameters
$\omega_{\mathrm{b}}$, $\omega_{\mathrm{cdm}}$, $h$, $n_{\mathrm{s}}$,
$A_{\mathrm{s}}$ and $\tau$. Here, we consider only direct-tracer measurements
of DESI. However, DESI will also infer the matter distribution through
Lyman-$\alpha$ (Ly-$\alpha$) forest features in the spectra of high redshift
($2.1<z<3.5$) quasars (Abareshi et al., 2022). As the linear regime spans a
wide range of $k$, Ly-$\alpha$ forest observations are, in principle, an
excellent probe of $\alpha_{\mathrm{s}}$. DESI Collaboration et al. (2016)
forecast that constraints on $\alpha_{\mathrm{s}}$ from direct tracers will
improve by almost a factor of three when combining them with Ly-$\alpha$
measurements. Even though this is an impressive improvement, we keep
Ly-$\alpha$ forecasts for future work as this requires excellent control of
broad-band systematics whose modelling requires more careful analysis (for
instance, along the lines of Cuceu et al., 2022), and CHIME and SPHEREx are
expected to provide similar $\alpha_{\mathrm{s}}$ constraints.

In Figure 5, we show all single survey constraints on $\alpha_{\mathrm{s}}$
and $n_{\mathrm{s}}$. One can see that DESI* will tighten the Planck18
$n_{\textrm{s}}$ constraints on $n_{\mathrm{s}}$. However, these will be
beaten by the Simons Observatory. CHIME and SPHEREx are expected to provide
tighter constraints on $\alpha_{\mathrm{s}}$ than CMB experiments but all the
CMB experiments will measure $n_{\mathrm{s}}$ more accurately than LSS
surveys. While DESI does not quite cover enough volume to provide competitive
$n_{\mathrm{s}}$ constraints, SPHEREx will not resolve small scales well
enough to measure $\omega_{\mathrm{cdm}}$ with high accuracy, which is
somewhat degenerate with $n_{\mathrm{s}}$ at large scales.

###  4.3 Combining CMB and LSS measurements

Figure 7: Forecast CMB-S4 and SPHEREx posterior contours as an example of how
combining CMB data with large-scale structure data improves constraints on
inflation. Figure 8: Similar to Figure 5 showing only forecasts including
forecasts on data combinations. The vertical area shaded in grey in the left
plot, again, marks the $\alpha_{\mathrm{s}}$-range between $-10^{-3}$ and
$-10^{-4}$.

As already discussed, CMB experiments generally have a better handle on
$n_{\mathrm{s}}$ than their LSS counterparts. On the other hand, by providing
3D information, LSS surveys probe $n_{\mathrm{s}}$ on different scales, giving
them more leverage on the running of the spectral index $\alpha_{\mathrm{s}}$.
Assuming there are no tensions between CMB and LSS results we can use this
synergy to obtain tighter constraints on inflationary parameters. We
illustrate this in Figure 7 which shows the forecast probability contours in
the $n_{\mathrm{s}}$-$\alpha_{\mathrm{s}}$ plane for CMB-S4 and SPHEREx.
CMB-S4 will yield tight error bars on $n_{\mathrm{s}}$ but will leave
$\alpha_{\mathrm{s}}$ relatively unconstrained. With SPHEREx $n_{\mathrm{s}}$
and $\alpha_{\mathrm{s}}$ are perfectly degenerate, and we will not get any
competitive constraints on either parameter. When combined with CMB-S4 data
this degeneracy is broken, almost halving the CMB-S4 $n_{\mathrm{s}}$ and
SPHEREx $\alpha_{\mathrm{s}}$ credible intervals.

Table 5 lists forecasts for all combinations of CMB and LSS probes that we
considered. Even though DESI will add to cosmological constraints when
combined with Planck18, it will not significantly add to our knowledge about
inflation once we have data from the Simons Observatory and CMB-S4. However,
in the real world, where tensions can arise, DESI will provide key consistency
checks on the new CMB data.

Combining CMB data with CHIME, SPHEREx, or both, we can learn substantially
more about inflation than we can from these experiments on their own, as
illustrated in the whisker plots of Figure 8. Merging CMB-S4 with data from
DESI, CHIME and SPHEREx, we can hope to attain
$\sigma_{\alpha_{\mathrm{s}}}=0.0018$ at 95% C.L. by 2030 or shortly
thereafter. Consequently, the class of scenarios with low $r$ and running
$\alpha_{\mathrm{s}}\lesssim-10^{-3}$ identified in Easther et al. (2022) will
be at the threshold of testability within the coming decade.

##  5 Conclusions and Discussion

The simplest implementation of inflation, i.e. two-parameter single-field
slow-roll, has come under stress thanks to recent data from the BICEP3/Keck
array and Planck. The resulting bounds on $n_{\mathrm{s}}$ and $r$ suggest
that inflation would last too long (Ade et al., 2021; Easther et al., 2022) in
these models to be consistent with broader consistency conditions on the
duration of accelerated expansion. Including a non-trivial third derivative in
the inflaton potential introduces an additional slow-roll parameter $\xi$ and,
hence, a running spectral index $\alpha_{\mathrm{s}}$. By mapping the slow-
roll parameters $\epsilon$, $\eta$ and $\xi$ to an inflationary trajectory, we
can choose $\xi$ so that inflation has a physically reasonable duration. To
end inflation at $N=55$ in this scenario requires
$-0.0023<\alpha_{\mathrm{s}}<-0.0004$ for any value of $n_{\mathrm{s}}$ within
the Planck-95% range.

Taking a broader view, this analysis underlines the progress that has been
made in observational cosmology since the first observation of primordial
anisotropies in the CMB (Smoot et al., 1992). In 1997, the inflationary
parameter space included models with spectral indices on either side of unity
(Dodelson et al., 1997) and large values of $r$. By the end of the 9-year WMAP
campaign (Spergel et al., 2003; Hinshaw et al., 2013) there was 5-$\sigma$
evidence that $n_{\mathrm{s}}$ was less than unity. These bounds tightened
further with the advent of Planck, and the fraction of the $n_{s}$-$r$ plane
in the original “zoo plot” that is consistent with observations has shrunk
dramatically.

Looking to the future, we forecast the ability of the Simons Observatory,
CMB-S4, LiteBIRD, DESI, SPHEREx, and CHIME to constrain $r$, $n_{\mathrm{s}}$
and $\alpha_{\mathrm{s}}$ using the Fisher approach, assuming a Planck six-
parameter $\Lambda$CDM cosmology and a negligible primordial stochastic
gravitational wave background. The CMB experiments, Simons Observatory, CMB-S4
and LiteBIRD, are expected to tighten constraints on $n_{\mathrm{s}}$ and $r$
and, if inflation does not produce sizeable tensor perturbations, LiteBIRD can
determine that $r<10^{-4}$ (at 95% C.L.). Such a bound on $r$ would provide
further evidence for the claim that any inflaton potential contains higher
derivatives. On the LSS side, due to its longer lever arms, we see tighter
constraints on $\alpha_{\mathrm{s}}$, although these are degenerate with
$n_{\mathrm{s}}$ within single surveys. For instance, we expect
$\sigma_{\alpha_{\mathrm{s}}}\sim 0.005$ (again at 95% C.L.) from CHIME and
SPHEREx. Constraints from DESI are expected to be more than three times wider
when inferring the matter density field from direct tracers only.

Combining these constraints with estimates from the Lyman-$\alpha$ forest data
from DESI may bring the error down to a similar level as with CHIME and
SPHEREx. However, this would require a more careful analysis which we leave
for future work. CMB experiments break the degeneracy in the LSS constraints,
and the error on $\alpha_{\mathrm{s}}$ shrinks to 0.0017 using a combination
of LiteBIRD, CMB-S4 at $\ell>200$, CHIME and SPHEREx. Thus, if inflation is
sourced by a potential with a nontrivial third derivative, evidence for the
running of the spectral index $\alpha_{\mathrm{s}}$ may be available by the
end of this decade.

We have seen that we can expect two orders of magnitude improvement in
measurements of $r$ and an order of magnitude improvement in measurements of
$\alpha_{\mathrm{s}}$ over the coming decade. However, it is entirely possible
that the number of non-trivial parameters in the “inflationary sector” of the
concordance cosmology (Baumann et al., 2009) need not change at all between
1992 and 2032.

While there is no compelling argument as to why $r$ should be in the
observable range, $\alpha_{\mathrm{s}}$ is non-zero in most-inflationary
models. Adshead et al. (2011) catalogued $\alpha_{\mathrm{s}}$ in a variety of
simple models, finding values that are typically ${\cal{O}}(10^{-4})$, and
thus several times smaller than the precision we expect from the coming
generation of experiments.

The scenario explored by Easther et al. (2022) provides a clear class of
models in which the magnitude of $\alpha_{\mathrm{s}}$ exceeds this generic
expectation, and is close to the expected sensitivity of forthcoming
experiments. This provides an additional route by which next-generation
experiments may usefully constrain a significant class of inflationary models.
As previously explained, a more considerable running is not required given
that other mechanisms within the overall inflationary paradigm can limit the
duration of inflation. However, it is a largely unrecognised correlation and
one that has become more interesting in the light of present data.

To our knowledge, this work gives the first “unified” forecast with like-for-
like comparisons for a wide variety of forthcoming experiments and their
possible combinations. We plan to extend this framework to include more
speculative scenarios with the longer-term goal of assisting the design of the
successor generation of surveys by understanding how these proposals can
interact to gain the tightest possible constraints on the overall inflationary
parameter space.

## Acknowledgements

The authors would like to thank Cullan Howlett, Changbom Park, and Arman
Shafieloo for valuable discussions around the project. BBK and DP are
supported by the project 우주거대구조를 이용한 암흑우주 연구 (“Understanding Dark Universe
Using Large Scale Structure of the Universe”), funded by the Ministry of
Science. RE acknowledges support from the Marsden Fund of the Royal Society of
New Zealand.

This work was supported by the high-performance computing clusters Seondeok at
the Korea Astronomy and Space Science Institute. This research made
substantial use of class (Blas et al., 2011), the ChainConsumer package
(Hinton et al., 2020), the Numpy package (Oliphant, 2006) and Matplotlib
(Hunter, 2007).

## Data Availability

No new primary data sets were generated in the production of this article. The
Fisher matrix forecasts are available from the author upon request.

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