SciPost Physics Submission Cornering Extended Starobinsky Ination with CMB and SKA Tanmoy Modak1 Lennart R over1 Bj orn Malte Sch afer2

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SciPost Physics Submission

Cornering Extended Starobinsky Inﬂation with CMB and SKA

Tanmoy Modak1, Lennart R¨over1, Bj¨orn Malte Sch¨afer2,

Benedikt Schosser1, and Tilman Plehn1

1Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Germany

2Astronomisches Recheninstitut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg,

Germany

April 14, 2023

Abstract

Starobinsky inﬂation is an attractive, fundamental model to explain the Planck measure-

ments, and its higher-order extension may allow us to probe quantum gravity eﬀects. We

show that future CMB data combined with the 21cm intensity map from SKA will mean-

ingfully probe such an extended Starobinsky model. A combined analysis will provide a

precise measurement and intriguing insight into inﬂationary dynamics, even accounting

for correlations with astrophysical parameters.

Content

1 Introduction 2

2 Extended Starobinsky model 3

3 Future CMB data 6

3.1 LiteBIRD and CMB-S4 likelihoods 6

3.2 Combined CMB projections 7

4 SKA data 10

4.1 SKA likelihood 10

4.2 Modeling the redshift dependence 13

4.3 Combined SKA and CMB projections 14

5 Outlook 16

A HSR projections 17

References 17

arXiv:2210.05698v2 [astro-ph.CO] 13 Apr 2023

SciPost Physics Submission

1 Introduction

Inﬂation [1–3] provides a simple and elegant solution to the observed ﬂatness and horizon

problems and naturally explains the absence of exotic relics. It also seeds primordial

density ﬂuctuations, from which the cosmic large-scale structure evolves. These structures

can be observed in the cosmic microwave background (CMB) anisotropies [4,5] and in the

large-scale distribution of galaxies.

Among inﬂationary models, Starobinsky or R2-inﬂation [1,6–9] is one of the best-ﬁtting

models to data [5, 10, 11] of the early Universe. It simply extends the action of general

relativity (GR) by a quadratic term in the Ricci-scalar. For the near-scale invariant power

spectrum, deviations from GR manifest themselves primarily in a weak running of the

spectral index. The value of the scalar amplitude and the spectral index reported by

Planck [5, 10, 11] can be accounted for by adjusting the coeﬃcient of the R2-term. The

extended Starobinsky model with higher-order curvature modiﬁcations is motivated by

quantum gravity, but also from a purely phenomenological point of view [12–22] and it

may shed light on the UV-completion of Einstein gravity. In this paper we extend the

Starobinsky model by an R3-term and study the constraining power of future cosmological

data.

Planck’s observations of the cosmic microwave background (CMB) temperature and

polarisation anisotropies have advanced our understanding of inﬂation tremendously [5].

The next generation of CMB experiments will further develop this legacy. We focus on two

future CMB experiments, LiteBIRD [23–25] and CMB-S4 [26–29]. The LiteBIRD satellite

mission will detect primordial B-mode polarisation with moderate resolution, but excellent

sensitivity. CMB-S4 stands for the next generation of ground-based detectors, which are

going to be installed over the next decade, with excellent sensitivity and resolution, but

limited sky coverage [26–29].

We supplement the CMB measurements with the 21cm intensity mapping by the

Square Kilometre Array (SKA) [30–42], as a second window to primordial structures. We

are primarily interested in the redshift range z= 8 ... 10 and k= 0.01 ... 0.2 Mpc−1[43].

The combined datasets well pick up variations in the spectral index to probe the extended

Starobinsky model over a large range of scales. Structure formation at these scales is

described well by linear physics with Gaussian statistics [44–47]. The low astrophysical

systematics due to X-ray, UV-sources [48–52] or baryonic feedback processes [53–55] al-

low us to extract inﬂationary parameters from 21cm tomography. While we will use some

simplifying assumptions, the modelling of the reionisation process at high redshift has

reached a high degree of sophistication [56–61] and takes care of astrophysical processes,

which are likewise modelled in machine learning approaches [62, 63].

In Sec. 2 we ﬁrst discuss the details of the inﬂationary dynamics, deriving the required

equivalent inﬂationary potential for extended Starobinsky models using the Einstein-

Jordan duality. We then start with future CMB data and discuss the expected likelihoods

for LiteBIRD and CMB-S4 in Sec. 3.1 and results in Sec. 3.2. In Sec. 4 we study the

21cm intensity mapping by SKA, again detailing the likelihood in Sec. 4.1, followed by a

discussion of the modelling of the neutral hydrogen fraction as a function of redshift as the

most important astrophysical parameter in Sec. 4.2. The results on probing the extended

Starobinsky model with SKA and the next generation of CMB experiments are discussed

in Sec. 4.3. We summarize our results in Sec. 5 and update our results on the slow-roll

parametrization in the Appendix.

SciPost Physics Submission

2 Extended Starobinsky model

The Starobinsky model [1,6] is one of the simplest inﬂationary models, yet best-ﬁtting to

Planck data [5]. It is deﬁned in the Jordan frame as

SJ=1

2Zd4x√−gJf(R),(1)

where gJdenotes the determinant of space-time metric gµν Jwith signature convention

(−,+,+,+), MP= (8πG)−1/2, and

f(R) = M2

PR+1

6M2R2,(2)

with M2>0. The original Starobinsky model approximates general f(R) gravity models

with an attractor behavior in the large-ﬁeld regime, where a single mass parameter M

accounts for the observed nearly-scale invariant power spectrum and spectral index [5].

Probing an actual inﬂationary potential complements results based on an eﬀective recon-

struction of inﬂationary potentials in the slow-roll approximation [43, 64]. We extend the

original Starobinsky model by a R3-curvature term,

f(R) = M2

PR+1

6M2R2+c

36M4R3,(3)

where cis a dimensionless coeﬃcient, which can be generated by quantum corrections.

Higher-order terms involving derivatives, Ricci tensors and Riemann tensors typically

involve ghosts [65], and we neglect them in favor of the R3-term as a phenomenological

window to physics beyond the simple Starobinsky model.

The corresponding scalar-tensor theory can be found by a Legendre transformation of

Eq.(1),

SJ=1

2Zd4x√−gJf(s) + f0(s)(R−s)

SJ≡Zd4x√−gJM2

2Ω2R−V(s)

with Ω2=f0(s)

= 1 + 1

3M2s+c

12M4s2

and V(s) = 1

2sf0(s)−f(s).(4)

The Legendre transform is well deﬁned as long as f(R) is convex, for Eq.(3) translating

into s > −2M2/c. The action in Eq.(1) can be expressed in the Einstein frame through

the conformal transformation gµν E= Ω2gµν J,

SE=Zd4x√−gEM2

2RE−1

2gµν E(∇µϕ∇νϕ)−VE(ϕ),(5)

with the canonical ﬁeld ϕand

ϕ=r3

2MPln Ω2,(6)

VE(ϕ) = V(s)

Ω(s)4s=s(ϕ)

,(7)

R= Ω2RE+ 3Eln Ω2−3

2gµν

E∂µln Ω2∂νln Ω2.(8)

SciPost Physics Submission

Here, E=gµν

E∂µ∂νis the d’Alembert operator. This way, modiﬁcations of the gravi-

tational law are mapped onto an additional ﬁeld ϕsubjected to dynamics in a potential

V(ϕ). This has the tremendous advantage that the standard inﬂationary formalism can

be applied for computing the ﬁeld dynamics and the associated generation of structures.

In the potential one has to use s(ϕ), as found by inverting Ω2in Eq.(6) and solving for

s(ϕ). We ﬁnd

s(ϕ) = 









2M2

c"r1+3c(eq2

MP−1) −1#for c6= 0

−3M21−eq2

MPfor c= 0 .

(9)

The potential can be expressed as

VE(ϕ) =

Pcs(ϕ)3

M2+ 3s(ϕ)2

36M21 + s(ϕ)

3M2+cs(ϕ)2

12M42.(10)

For c= 0 it can be put into the standard R2or Starobinsky form

VE(ϕ) = 3M2

PM2

41−e−q2

MP2

.(11)

Here s(ϕ) has two solutions, but from Eq.(9) we know that we need to satisfy the convexity

condition s > −M2/(2c) and c > 0, while the potential VE(ϕ) has to remain positive at

large ﬁeld values. While the secondary solution can fulﬁll the convexity condition for

c < 0, the potential becomes unbounded from below for large ﬁeld values. In Fig 1 we

illustrate VE(ϕ) for some sample parameter choices.

To study the inﬂationary dynamics we split ϕinto a classical background ¯ϕand a

perturbation δϕ,

ϕ(xµ) = ¯ϕ(t) + δϕ(xµ).(12)

The perturbed spatially ﬂat Friedmann-Robertson-Walker (FRW) metric can be expanded

as [66–68]

ds2=−(1 + 2A)dt2+ 2a(t)(∂iB)dxidt+a(t)2[(1 −2ψ)δij + 2hij] dxidxj,(13)

Figure 1: The shape of the inﬂationary potential for few reference choices of Mand c.

SciPost Physics Submission

where a(t) is scale factor and tis the cosmic time. The A, B, ψ deﬁne scalar and hij tensor

metric perturbations.

With the above deﬁnitions the background ﬁeld equation can be written as

¯ϕ+ 3H˙

¯ϕ+VE, ¯ϕ= 0,(14)

where H= d(ln a)/dtis the Hubble function fulﬁlling

H2=1

3M2

P1

2˙

¯ϕ2+VEand ˙

H=−1

2M2

¯ϕ2,(15)

The slow-roll parameter can then be deﬁned as

≡ − ˙

H2.(16)

Inﬂation ends when = 1.

Splitting ϕ(xµ) into a background ﬁeld ¯ϕ(t) and gauge-dependent ﬁeld ﬂuctuations

δϕ(xµ) motivates the gauge-independent Mukhanov-Sasaki variables for the ﬂuctuations [67,

69–71],

Q=Q+˙

¯ϕ

Hψwith Q=Dκϕ|κ=0 =dϕ

dκ|κ=0 ,(17)

where κis the trajectory in ﬁeld space. The gauge-invariant ﬁeld ﬂuctuations Qfulﬁll

Q+ 3H˙

Q+k2

a2+VE, ¯ϕ¯ϕ−1

Pa3

dta3

H˙

¯ϕ2Q= 0 ,(18)

where VE, ¯ϕ¯ϕis the double derivative of the potential VE( ¯ϕ) with respect to ¯ϕ. The gauge-

invariant curvature perturbation Ris deﬁned as [67, 68]

R=H

¯ϕQ , (19)

and we are interested in the power spectrum of the gauge-invariant curvature perturba-

tion [67, 72]

hR(k1)R(k2)i= (2π)3δ(3)

D(k1+k2)PR(k1) with PR(k) = |R|2.(20)

The dimensionless power spectrum for the curvature perturbation is given by

PR(t;k) = k3

2π2PR(k).(21)

The spectral index nsof the power spectrum of the adiabatic ﬂuctuations is deﬁned as

ns= 1 + d ln PR(k)

d ln k.(22)

On the other hand, the mode equation for the tensor amplitude is

v00

k+k2−a00

avk= 0 ,(23)

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SciPostPhysicsSubmissionCorneringExtendedStarobinskyInationwithCMBandSKATanmoyModak1,LennartRover1,BjornMalteSchafer2,BenediktSchosser1,andTilmanPlehn11InstitutfurTheoretischePhysik,UniversitatHeidelberg,Germany2AstronomischesRecheninstitut,ZentrumfurAstronomiederUniversitatHeidelberg,Germany...

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