# UP-Diff: Latent Diffusion Model for Remote Sensing Urban Prediction

Zeyu Wang, , Zecheng Hao, Jingyu Lin, Yuchao Feng, and Yufei Guo This work is
supported by grants from the National Natural Science Foundation of China
under contracts No.12202412 and No.12202413.Zeyu Wang, Jingyu Lin, and Yufei
Guo are with Intelligent Science & Technology Academy of CASIC, China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>yfguo@pku.edu.cn). Zeyu Wang
is also with College of Information Science & Electronic Engineering, Zhejiang
University, China.Zecheng Hao is with School of Computer Science, Peking
University, China (e-mail: haozecheng@pku.edu.cn).Yuchao Feng is with College
of Engineering, Westlake University, China (e-mail:
fengyuchao@wioe.westlake.edu.cn)._Corresponding author: Yufei Guo_.

###### Abstract

This study introduces a novel Remote Sensing (RS) Urban Prediction (UP) task
focused on future urban planning, which aims to forecast urban layouts by
utilizing information from existing urban layouts and planned change maps. To
address the proposed RS UP task, we propose UP-Diff, which leverages a Latent
Diffusion Model (LDM) to capture position-aware embeddings of pre-change urban
layouts and planned change maps. In specific, the trainable cross-attention
layers within UP-Diff’s iterative diffusion modules enable the model to
dynamically highlight crucial regions for targeted modifications. By utilizing
our UP-Diff, designers can effectively refine and adjust future urban city
plans by making modifications to the change maps in a dynamic and adaptive
manner. Compared with conventional RS Change Detection (CD) methods, the
proposed UP-Diff for the RS UP task avoids the requirement of paired pre-
change and post-change images, which enhances the practical usage in city
development. Experimental results on LEVIR-CD and SYSU-CD datasets show UP-
Diff’s ability to accurately predict future urban layouts with high fidelity,
demonstrating its potential for urban planning. Code and model weights are
available at https://github.com/zeyuwang-zju/UP-Diff.

###### Index Terms:

Remote Sensing (RS), Urban Prediction (UP), UP-Diff, Latent Diffusion Model
(LDM), cross-attention.

##  I Introduction

Urban Prediction (UP) is a crucial field in city development, which forecasts
and analyzes future trends in urban growth [40, 32, 1]. It helps researchers
address the challenges arising from population growth [5], transportation
congestion [37], and climate changes [21, 18]. Currently, Remote Sensing (RS)
technology plays a critical role in city development by capturing RS images
about detailed urban landscapes [28, 6], which are essential for monitoring
the urban growth. By leveraging the RS technology, researchers can
significantly improve the accuracy and efficiency of urban layout prediction,
enabling them to make well-informed decisions to optimize the urban
development strategies.

Despite the advancements in RS technology for city development, current works
normally focus on the RS Change Detection (CD) task. As illustrated in Fig. 1
(a), RS CD aims to identify the differences between two images captured at
different time points of the same geographic area from the aerial view [30,
3]. However, in practical urban planning, our goal is to forecast future urban
layouts based on existing urban layouts and proposed modifications [19].
Current methods for RS CD primarily focus on detecting the layout changes
rather than predicting future urban developments. In addition, the requirement
for paired pre-change and post-change images poses a challenge for RS data
collection. Collection of paired RS images must be guaranteed at long
intervals, and the position and orientation of the probe must be accurate.
Therefore, it not only increases the cost of data acquisition and image
alignment but also impedes effective urban planning.

Figure 1: Illustration of the difference between (a) conventional RS Change
Detection (CD) and (b) our proposed RS Urban Prediction (UP).

In this paper, we propose a novel RS-based task focusing on urban planning,
called RS Urban Prediction (UP). As illustrated in Fig. 1 (b), the main
objective of the proposed RS UP task is to forecast future urban layouts by
leveraging information from existing urban layouts and planned change maps. To
address the challenge of the proposed task, we present UP-Diff, which is a
novel Latent Diffusion Model (LDM) building upon the Stable Diffusion (SD)
[25]. Firstly, UP-Diff incorporates a trainable ConvNeXt model [20] to encode
the pre-change urban layouts and planned change maps into position-aware
embeddings. Subsequently, these embeddings are fed into the trainable cross-
attention layers within the iterative diffusion modules of UP-Diff. It enables
the model to dynamically focus on the specific regions crucial for urban
planning. Meanwhile, we transfer the pre-trained SD model weights to mitigate
the challenge of limited annotated RS image pairs. Our main contributions are
as follows:

  * •

We propose the RS Urban Prediction (UP) task to forecast future urban layouts
based on existing urban RS layouts and planned change maps. To our knowledge,
this is the first work on RS-based urban prediction.

  * •

We also propose UP-Diff, which is a latent diffusion model with iterative
layout-aware attention mechanism to dynamically enhance the critical urban
regions.

  * •

Extensive experiments on LEVIR-CD [8] and SYSU-CD [29] datasets demonstrate
that UP-Diff can accurately predict the future urban layouts with high
fidelity.

##  II Related Works

###  II-A Remote Sensing Change Detection

Current research on RS CD normally adopts deep learning-based models.
Convolutional Neural Networks (CNNs) have been widely applied on this task
[38, 9, 35]. Later, Transformer-based models have also shown the ability to
capture long-range context and relationships between different positions [7,
11]. Recently, the advanced diffusion models, known for their effectiveness in
capturing complex patterns in RS images, have also shown promise in RS CD [4,
36].

###  II-B Diffusion Models

Diffusion models have emerged as the leading choice for image generation,
which originate from Denosing Diffusion Probabilistic Models (DDPMs) [31, 14].
Building upon this foundation, the Latent Diffusion Model (LDM) [25] has
pushed the boundaries by using a low-dimensional latent space [10]. Current
works on diffusion models normally guide the generation given text conditional
information [24, 27, 25]. Moreover, recent advancements have been made in the
field of layout-to-image diffusion models, where the spatial controls serve as
the input conditions [2, 41, 16, 17].

##  III Methods

Figure 2: Illustration of our proposed UP-Diff for Remote Sensing Urban
Prediction. (a) Training of the autoencoder for reconstruction. (b) Training
of UP-UNet for latent diffusion and denoising. C denotes the concatenation.

###  III-A Overview

Our proposed UP-Diff is illustrated in Fig. 2. To enhance the modeling
efficiency, UP-Diff follows LDM [25], which adopts an autoencoder to encode
the input image into the latent space $\boldsymbol{z}$ and learn the
representation through a reconstruction process. The autoencoder comprises the
encoder $\mathcal{E}_{\varphi}$ and the decoder $\mathcal{D}_{\phi}$. In order
to augment the model’s comprehension of the RS layout and target alterations,
we incorporate the embedded conditions through iterative attention-based
blocks within the UP-UNet in the latent space. It enables our UP-Diff model to
dynamically prioritize specific target urban regions. To our knowledge, UP-
Diff is the first diffusion model for the proposed RS UP task, where the
iterative diffusion process generates high-fidelity post-change RS images.

Figure 3: Qualitative results of the baseline methods and our UP-Diff on
LEVIR-CD and SYSU-CD datasets for the proposed RS UP task.

###  III-B Condition Embedding

Condition embedding is critical in our proposed UP-Diff. Firstly, the text
condition $\boldsymbol{c}$ is obtained by embedding “A remote sensing photo.”
via the CLIP text encoder [23]. Utilizing the cross-attention layer from the
SD model by incorporating $\boldsymbol{c}$ greatly enhances the awareness of
RS features.

Meanwhile, the layout condition $\boldsymbol{l}$ is obtained by embedding the
concatenated pre-change image $\boldsymbol{I}_{pre}$ and change map
$\boldsymbol{m}$ by a trainable ConvNeXt model [20] with parameters
$\boldsymbol{\psi}$:

$\boldsymbol{l}=\text{ConvNeXt}_{\boldsymbol{\psi}}(\text{Concat}[\boldsymbol{I}_{pre},\boldsymbol{m}]),$
(1)

where the semantic layout feature $\boldsymbol{l}$ is added after the
iterative attention-based blocks. The ConvNeXt model has been pre-trained on
large-scale natural image datasets. In addition, the tokenized layout
condition $\boldsymbol{l}^{*}$ is derived by flattening $\boldsymbol{l}$:

$\boldsymbol{l}^{*}=\text{Flatten}(\boldsymbol{l})+\text{PE},$ (2)

where PE denotes a trainable position embedding to enhance the position-aware
information. The tokenized $\boldsymbol{l}^{*}$ serves as a suitable form for
the gated cross-attention mechanism.

###  III-C UP-UNet

Our proposed UP-UNet, denoted as
$\boldsymbol{\epsilon}_{\boldsymbol{\theta}}$, serves as the core component
for the latent denoising iterations. UP-UNet inherits the spatial structure of
UNet [26], where the basic module consists of the residual convolutional
blocks [12] and iterative attention-based layers [34]. We keep the original
layers from SD model [25] and frozen the model weights to fully utilize the
generative model pre-trained on large-scale natural images. As illustrated in
Fig. 2, the UP-UNet is mainly composed of cascaded self-attention and cross-
attention layers, which can be formulated as below:

$\text{Attn}(Q,K,V)=\text{softmax}\left(\frac{Q\cdot
K^{T}}{\sqrt{d_{k}}}\right)\cdot V,$ (3)

where the matrices $Q$, $K$, and $V$ are derived from linear projection of the
input features. The token dimension of the matrix $K$ is denoted by $d_{k}$.
In self-attention mechanism, the matrices $Q$, $K$, and $V$ are derived from
the same input feature. In cross-attention mechanism, $Q$ is derived from one
modality, while $K$ and $V$ are obtained from another modality. The cross-
attention layer is followed by the feed-forward layer to increase the
representation ability, which is omitted in Fig. 2.

In specific, the trainable cross-attention layer in the middle of UP-UNet
block incorporates a gated mechanism, which focuses on the target RS layout
regions. The gated cross-attention layer can be formulated as:

$Q_{c}=Q_{l}+\lambda\cdot\text{tanh}(\gamma)(\text{Attn}(Q_{l},K_{l},V_{l})),$
(4)

where $\lambda$ is a hyperparameter and $\gamma$ is a learnable parameter
initialized as 0. The tokens $K_{l}$ and $V_{l}$ are obtained by the linear
projection from $\boldsymbol{l}^{*}$. Inside the iterative diffusion blocks of
UP-UNet, a trainable cross-attention layer is introduced between the fixed
self-attention and cross-attention layers, enabling the model to anticipate
the layout of change area. Through the implementation of a gated mechanism,
the model can dynamically adjust the attention weights to enhance its
perception on the target urban layout.

###  III-D Training & Sampling Formulation

We first fine-tune the autoencoder for reconstruction on the RS image
datasets. During the diffusion training process, the model weights of the
autoencoder are freezed, and the latent $\boldsymbol{z}_{0}$ is obtained by
encoding the post-change image $\boldsymbol{I}_{post}$ through
$\mathcal{E}_{\varphi}$. After that, the noise-contaminated latent features
$\boldsymbol{z}_{1},\boldsymbol{z}_{2},...,\boldsymbol{z}_{T}$ are achieved
via a Markov chain:

$q\left(\boldsymbol{z}_{t}\mid\boldsymbol{z}_{t-1}\right)=\mathcal{N}\left(\boldsymbol{z}_{t};\sqrt{1-\beta_{t}}\boldsymbol{z}_{t-1},\beta_{t}\mathbf{I}\right),$
(5)

where the hyperparameter $\beta_{t}$ corresponds to the variance of the
Gaussian distribution, and it undergoes a linear increase as $t$. Meanwhile,
we can directly obtain $\boldsymbol{z}_{t}$ based on $\boldsymbol{z}_{0}$ and
$\beta_{t}$ via the reparameterization [15]:

$\boldsymbol{z}_{t}=\sqrt{\overline{\alpha}_{t}}\boldsymbol{z}_{0}+\sqrt{1-\overline{\alpha}_{t}}\boldsymbol{\epsilon},$
(6)

where $\boldsymbol{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ denotes
the randomly sampled Gaussian noise. Meanwhile, $\alpha_{t}=1-\beta_{t}$ and
$\overline{\alpha}_{t}=\prod_{i=1}^{t}\alpha_{i}$. Therefore,
$\boldsymbol{z}_{T}$ converges to standard normal distribution
$\mathcal{N}(\mathbf{0},\mathbf{I})$ when $T\rightarrow\infty$. The training
objective of our proposed UP-UNet, denoted as
$\boldsymbol{\epsilon}_{\boldsymbol{\theta}}$, can be represented as:

$\min_{\boldsymbol{\theta}}\mathcal{L}_{\text{UP-
Diff}}=\mathbb{E}_{\boldsymbol{z}_{t},\boldsymbol{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I}),t,\boldsymbol{c}}\left[\left\|\boldsymbol{\epsilon}-\boldsymbol{\epsilon}_{\boldsymbol{\theta}}\left(\boldsymbol{z}_{t},t,\boldsymbol{l},\boldsymbol{c}\right)\right\|_{2}^{2}\right],$
(7)

where $t\sim\text{Uniform}(\\{1,...,T\\})$ is a random time step.
$\boldsymbol{l}$ and $\boldsymbol{c}$ denote the layout condition and text
condition, respectively.

During the inference process, we can reverse the diffusion process into the
denoising process. Starting from a Gaussian noise $\boldsymbol{z}_{T}$, our
proposed UP-UNet gradually generates less noisy
$\boldsymbol{z}_{T-1},\boldsymbol{z}_{T-2},...,\boldsymbol{z}_{0}$ by the
following iteration:

$\boldsymbol{z}_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}\left(\boldsymbol{z}_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\boldsymbol{\epsilon}_{\theta}\left(\boldsymbol{z}_{t},t,\boldsymbol{l},\boldsymbol{c}\right)\right)+\beta_{t}\boldsymbol{\epsilon}.$
(8)

Finally, we obtain the denoised $\boldsymbol{z}_{0}$, which is then decoded by
$\mathcal{D}_{\phi}$ into the generated post-change image
$\widehat{\boldsymbol{I}}_{post}$.

##  IV Experiments

###  IV-A Implementations

We implement UP-Diff using PyTorch 1.13.1 on Intel(R) Xeon(R) Platinum 8255C
<EMAIL_ADDRESS>and two NVIDIA Tesla V100 GPUs with CUDA 12.2. AdamW optimizer is
adopted with the learning rate of 5e-5 and 10,000 warmup training iterations.
The batch sizes per GPU are set to 2 and 4 for training the autoencoder and
the UP-Diff, respectively. Additionally, we employ random flips and crops on
the RS images for data augmentation.

###  IV-B Datasets

We conduct experiments on two datasets with different RS scenes and
resolutions to demonstrate the generalizability:

  * •

LEarning, VIsion, and Remote sensing Dataset (LEVIR-CD) [8] contains 445, 64,
and 128 RS image pairs for training, validation, and testing, respectively,
captured in Texas, USA. We conduct the experiments on the dataset with the
large resolution of 512 $\times$ 512.

  * •

Sun Yat-Sen University Dataset (SYSU-CD) [29] is a large-scale RS image
dataset gathered in Hong Kong, China, containing 12,000/4,000/4,000 samples
for training/validation/testing, respectively. We adopt the official
resolution of 256 $\times$ 256 for experiments.

###  IV-C Baselines

We compare our UP-Diff with the following baselines listed in Table I. For the
conventional deep learning models, we concatenate $\boldsymbol{I}_{pre}$ and
$\boldsymbol{m}$ as input for end-to-end training. For the diffusion-based
models, we implement them by replacing the original $\boldsymbol{I}_{post}$
with $\boldsymbol{m}$ as the input conditions.

###  IV-D Qualitative Comparison

The qualitative results of the baseline methods and our UP-Diff on LEVIR-CD
and SYSU-CD datasets are shown in Fig. 3. It can be seen that pre-change
images $\boldsymbol{I}_{pre}$ (the first column) are typically captured on
barren land, while the post-change images $\boldsymbol{I}_{post}$ (the last
column) display neatly arranged buildings. The baseline models employing end-
to-end training, including UNet, UNet++, SPADE, and VQVAE, can generate the
post-change images with general building outlines but lack fidelity and
diversity. The advanced VQGAN, DDPM, and DDPM-CD models produce high-fidelity
images, but some texture details do not accurately align with the actual
transformations depicted in the change maps. In contrast, our proposed UP-Diff
model can produce high-quality post-change images that faithfully capture
target urban modifications, such as the specific buildings, main roads, and
vessels on the water.

###  IV-E Quantitative Comparison

TABLE I: Quantitative comparison on the generated image quality.

Model | LEVIR-CD | SYSU-CD
---|---|---
LPIPS$\downarrow$ |  FID$\downarrow$ |  LPIPS$\downarrow$ |  FID$\downarrow$
UNet [26] | 0.504 | 196.28 | 0.521 | 139.59
UNet++ [42] | 0.497 | 185.48 | 0.523 | 120.05
SPADE [22] | 0.423 | 142.58 | 0.436 | 66.28
VQVAE [33] | 0.406 | 185.45 | 0.477 | 73.20
VQGAN [10] | 0.402 | 179.22 | 0.478 | 60.56
DDPM [14] | 0.420 | 134.63 | 0.430 | 61.03
DDPM-CD [4] | 0.418 | 144.38 | 0.414 | 49.80
UP-Diff | 0.342 | 117.79 | 0.400 | 34.57

  * •

$\downarrow$ denotes that lower values are better.

We evaluate the models from two quantitative aspects. On one hand, we evaluate
the quality of the generated images using Learned Perceptual Image Patch
Similarity (LPIPS) [39] and Fréchet Inception Distance (FID) [13]. On the
other hand, inspired by the assessment in semantic image synthesis [22], we
utilize a pre-trained RS CD model (DMINet [11]) to test the RS CD metrics
based on original input $\boldsymbol{I}_{pre}$ and output
$\widehat{\boldsymbol{I}}_{post}$. The metrics include Precision (Pre.),
Recall (Rec.), F1-score (F1), and Intersection over Union (IoU).

The results in Table I demonstrate that our UP-Diff model achieves the best
LPIPS and FID scores across both datasets, with scores of 0.342 LPIPS and
117.79 FID on the LEVIR-CD dataset, and 0.400 LPIPS and 34.57 FID on the SYSU-
CD dataset, respectively. The diffusion-based model DDPM-CD also exhibits
commendable performance, achieving LPIPS scores of 0.418 and 0.414 on the two
datasets, respectively. Meanwhile, from Table II, we can see that when we use
the generated post-change images $\widehat{\boldsymbol{I}}_{post}$ for
evaluation on the RS CD task, the evaluation scores all demonstrate our
excellent performance. For example, UP-Diff achieves the Pre. score of 92.23%
and F1 score of 82.41% on LEVIR-CD dataset, respectively. However, these
baseline methods exhibit obvious inferior performance when contrasted with our
method. We also show an example of the generated change maps through DDPM,
DDPM-CD, and our UP-Diff in Fig. 4. It can be seen that the result of our UP-
Diff aligns well with the ground-truth $\boldsymbol{I}_{post}$, while the
results from baseline methods exhibit differing degrees of deviation from
$\boldsymbol{I}_{post}$.

TABLE II: Quantitative comparison on the generated image for RS CD task. The
results are recorded in (%).

| Model |  Pre.$\uparrow$ |  Rec.$\uparrow$ |  F1$\uparrow$ |  IoU$\uparrow$
---|---|---|---|---|---
LEVIR-CD |  UNet [26] | 55.54 | 50.41 | 49.70 | 47.90
UNet++ [42] | 64.39 | 55.79 | 59.19 | 53.51
SPADE [22] | 80.33 | 64.52 | 71.46 | 62.32
VQVAE [33] | 70.00 | 59.82 | 64.96 | 57.43
VQGAN [10] | 84.56 | 69.63 | 74.79 | 65.43
DDPM [14] | 74.58 | 68.13 | 77.23 | 69.74
DDPM-CD [4] | 86.54 | 72.15 | 77.37 | 67.93
UP-Diff | 92.23 | 76.55 | 82.41 | 73.35
SYSU-CD |  UNet [26] | 77.23 | 84.63 | 79.10 | 66.26
UNet++ [42] | 77.71 | 85.29 | 79.62 | 66.94
SPADE [22] | 79.58 | 84.52 | 80.33 | 68.71
VQVAE [33] | 77.43 | 84.97 | 78.62 | 66.89
VQGAN [10] | 81.02 | 85.26 | 80.05 | 67.48
DDPM [14] | 80.14 | 83.29 | 81.48 | 69.74
DDPM-CD [4] | 84.11 | 85.43 | 84.74 | 74.33
UP-Diff | 86.82 | 86.10 | 86.45 | 76.86

  * •

$\uparrow$ denotes that higher values are better.

Figure 4: Qualitative comparison on the generated images for RS CD task.

##  V Conclusion

In this work, we introduce a novel task called Remote Sensing (RS) Urban
Prediction (UP) for forecasting future urban layouts. Meanwhile, we propose
the latent UP-Diff model tailored for the RS UP task, utilizing iterative
layout-based gated cross-attention layers to focus on critical regions. Our
qualitative and quantitative results on two datasets illustrate the
superiority of UP-Diff over baseline methods in the RS UP task, including
conventional deep learning and diffusion-based approaches. The post-change
images generated by UP-Diff align well with ground-truth images during
migration to the RS Change Detection (CD) task with a pre-trained model. In
the future, we aim to develop more efficient methods with mobility for the
proposed RS UP task. Furthermore, we plan to introduce a novel pluralistic RS
UP task, where the generated post-change images can exhibit great diversity,
contributing to flexible urban planning initiatives.

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August 2024 IPMU24-0033

Starobinsky inflation beyond the leading order

Shunsuke Toyama a,& and Sergei V. Ketov a,b,c,#,111The corresponding author

a Department of Physics, Tokyo Metropolitan University,

1-1 Minami-ohsawa, Hachioji-shi, Tokyo 192-0397, Japan

b Research School of High-Energy Physics, Tomsk Polytechnic University,

Tomsk 634028, Russian Federation

c Kavli Institute for the Physics and Mathematics of the Universe (WPI),

The University of Tokyo Institutes for Advanced Study, Kashiwa 277-8583, Japan

&<EMAIL_ADDRESS>#<EMAIL_ADDRESS>
Abstract

The Starobinsky model of cosmological inflation in four spacetime dimensions
is reviewed with the emphasis on impact of quantum gravity corrections. As a
specific example of the quantum corrections, the Grisaru-Zanon quartic
curvature terms in the gravitational effective action of closed superstrings
are chosen. Those quartic curvature terms are compared to the Bel-Robinson
tensor squared in a flat Friedman universe, and the upper bound on the
effective string coupling constant is found by demanding unitarity (causality)
and the absence of ghosts. It is found that the quantum corrections to the
observables (tilts) of the cosmic microwave background radiation in the
Starobinsky inflation may be of the same order of magnitude as the next-to-
next-to-next classical contributions in the Starobinsky model with respect to
the inverse powers of the e-folding number at the horizon crossing.

##  1 Introduction

Ultra-violet (UV) completion of phenomenologically viable field-theoretical
models of cosmological inflation is important because inflation is sensitive
to high energy physics in the early universe and, hence, quantum corrections
to classical description of inflation may be important. The existence of the
UV completion to a particular inflation model allows one to treat it as the
effective field theory (EFT) originating from quantum gravity. The UV
completion is even more relevant to large-field inflation models, while the
Starobinsky inflation model [1] is one of them. Unfortunately, little is known
about quantum gravity, so UV-completion is often discussed in the framework of
Swampland conjectures [2], see also Refs. [3, 4, 5] for the applications of
the Swampland conjectures to the Starobinsky inflation.

String theory is a good candidate for the theory of quantum gravity, so it is
natural to seek an UV completion of the Starobinsky model in string theory,
see e.g., Ref. [6] for the earlier attempts. This task turned out to be
difficult because the Starobinsky model is defined in four space-time
dimensions, whereas string theory needs to be compactified from higher
dimensions to four dimensions, while the gravitational low-energy EFT in
string theory is subject to large ambiguities related to field redefinitions
of spacetime metric, see e.g., Ref. [7] for more.

The Starobinsky model of inflation is based on the $R^{2}$ gravity and can be
destabilized by the higher-order curvature terms in the gravitational EFT if
those terms have large coefficients. Inflation provides the mechanism for
generation of cosmological perturbations, while the Starobinsky model is in
excellent agreement with current measurements of the cosmic microwave
background (CMB) radiation, so the higher order curvature terms should be sub-
leading during inflation. Nevertheless, it makes sense to investigate
robustness of the Starobinsky inflation against specific quantum gravity
corrections derived from superstring theory. In the case of closed (type II)
superstrings, the leading $(\alpha^{\prime})^{3}$-correction beyond the
Einstein-Hilbert term is given by the terms quartic in the space-time
curvature, which were first derived by Grisaru and Zanon in 1985 from the
vanishing four-loop renormalization group beta-function of the supersymmetric
non-linear sigma model in two dimensions, describing propagation of a test
superstring in the gravitational background [8]. The same quartic curvature
terms arise from M-theory [9, 10] after dimensional reduction down to four
dimensions. The dependence of the gravitational EFT in string theory upon the
Ricci scalar curvature (and the Ricci tensor also) is known to be ambiguous,
see Ref. [7] for a review, because the perturbative string theory is
consistently defined only on Ricci-flat backgrounds.

In this paper, we combined the Starobinsky model with the Grisaru-Zanon
quantum gravity (superstring) correction, called the Starobinsky-Grisaru-Zanon
(SGZ) gravity, that is the generalization of the Einstein-Grisaru-Zanon
gravity introduced in Ref. [11]. It allowed us not only derive the
restrictions on the effective superstring coupling constant in front of the
quartic curvature terms but also compare the contributions of those quantum
corrections to the sub-leading terms (beyond the leading order with respect to
the e-folding number) in the inflationary (CMB) observables such as
cosmological tilts and their running, as well as derive the leading quantum
gravity (superstring) corrections to the Starobinsky solution.

The paper is organized as follows. In Sec. 2 the Starobinsky model of
inflation is reviewed both in the original (Jordan) frame and in the Einstein
(quintessence) frame. We do not follow historical developments but introduce
the Starobinsky model from the modern perspective. In Sec. 3 we define the SGZ
gravity in the perturbative setup with respect to the GZ term, and compare it
to the similar but different Starobinsky-Bel-Robinson (SBR) gravity [12, 13,
14] also having the quartic curvature terms in its action. In Sec. 4 we derive
the leading quantum corrections to the inflationary dynamics due to the GZ
term in the Jordan frame, in the first order with respect to its effective
(string) coupling constant. In Sec. 5 we find the upper limits on the
effective string coupling constant by demanding the absence of ghosts,
unitarity and causality. The leading quantum corrections to the CMB
observables are derived and compared against the classical subleading
contributions in Sec. 6. We conclude in Sec. 7.

##  2 Review of Starobinsky inflation

The Starobinsky model of inflation is the generally covariant and
nonperturbative extension of the Einstein-Hilbert (EH) gravity theory by the
term quadratic in the Ricci scalar curvature $R$. All the curvature-dependent
terms beyond the EH one are irrelevant in the Solar system, while they may
also be negligible during reheating after inflation in the weak-gravity
regime. However, it is not the case during inflation in the high curvature
regime where the $R^{2}$ term is the leading contribution (see below).

The Starobinsky model is the particular case of modified $F(R)$ gravity, and
it is geometrical because only gravitational interactions are used. A modified
gravity action has the higher derivatives and generically suffers from
Ostrogradsky instabilities and ghosts. However, in the most general modified
gravity action, whose Lagrangian is quadratic in the spacetime curvature, the
only ghost-free term is just given by $R^{2}$ with a positive coefficient,
which leads to the Starobinsky model with the action

$S_{\rm Star.}=\alpha\int\mathrm{d}^{4}x\sqrt{-g}R^{2}+{{M^{2}_{\rm
Pl}}\over{2}}\int\mathrm{d}^{4}x\sqrt{-g}R~{},\quad\alpha\equiv{{M^{2}_{\rm
Pl}}\over{12M^{2}}}~{},$ (1)

having the only parameter $\alpha$ or $M$, where $M_{\rm Pl}=1/\sqrt{8\pi
G_{\rm N}}\approx 2.4\times 10^{18}$ GeV, the spacetime signature is
$(-,+,+,+,)$ and the natural units are used, $\hbar=c=1$. The first term in
this action is scale invariant with the dimensionless parameter $\alpha$.

The origin of the $R^{2}$ term was originally proposed due to contributions of
quantized matter fields in the EH gravity [1]. However, because the EH term is
subleading during inflation, we adopt the opposite interpretation, namely,
with the EH term being originated from the scale-invariant gravity. For
instance, when starting from the scale-invariant action for gravity and a
scalar field $\phi$ as [15, 16]

$S[g_{\mu\nu},\phi]=\int\mathrm{d}^{4}x\sqrt{-g}\left[\alpha
R^{2}+\xi\phi^{2}R-{{{1}\over{2}}}(\partial\phi)^{2}-\lambda\phi^{4}\right]~{},$
(2)

one finds that it can undergo a phase transition (called dimensional
transmutation) due to quantum corrections, known as the Coleman-Weinberg
mechanism of spontaneous symmetry breaking [17]. It leads to the massive
scalar field $\phi$ that may be identified with dilaton or Higgs field having
a non-vanishing vacuum expectation value (VEV) in the effective action, as can
be demonstrated in the one-loop perturbation theory [15, 16]. As a result,
both the Planck mass and the EH term are generated with

${{{1}\over{2}}}M^{2}_{\rm Pl}=\xi\left\langle\phi\right\rangle^{2}~{},$ (3)

though this cannot be considered as the UV-completion of the Starobinsky
gravity.

The metric of a flat Friedman universe is given by

$ds^{2}=-dt^{2}+a^{2}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)~{}.$ (4)

Then the action (2) leads to equations of motion in the form

$2H\ddot{H}-\left(\dot{H}\right)^{2}+H^{2}\left(6\dot{H}+M^{2}\right)=0~{},\quad
H=\dot{a}/a~{},$ (5)

known as the Starobinsky equation in the literature, where the dots stand for
the time derivatives and $H(t)$ is Hubble function.

When searching for a solution to the Starobinsky equation in the form of left
Painlevé series, $H(t)=\sum^{k=p}_{k=-\infty}c_{k}(t_{0}-t)^{k}$, one finds
the Hubble function (see e.g., Ref. [14])

$\begin{split}{{H(t)}\over{M}}&={{M}\over{6}}(t_{0}-t)+{{1}\over{6M(t_{0}-t)}}-{{4}\over{9M^{3}(t_{0}-t)^{3}}}+{{146}\over{45M^{5}(t_{0}-t)^{5}}}\\\
&{}-{{11752}\over{315M^{7}(t_{0}-t)^{7}}}+{\cal
O}\left(M^{-9}(t_{0}-t)^{-9}\right)\end{split}$ (6)

valid for $M(t_{0}-t)>1$. This special solution is an attractor, while
$R=12H^{2}+6\dot{H}$.

In the slow-roll (SR) approximation defined by
$\left|\ddot{H}\right|\ll\left|H\dot{H}\right|$ and $\left|\dot{H}\right|\ll
H^{2}$, one gets the leading term in the Starobinsky solution as

$H(t)\approx\left({{M^{2}}\over{6}}\right)(t_{0}-t)$ (7)

that is entirely due to the $R^{2}$-term in the action. The attractor solution
spontaneously breaks the scale invariance of the $R^{2}$-gravity and,
therefore, implies the existence of the Nambu-Goldstone boson (called
scalaron) that is the physical excitation of the higher-derivative gravity. It
can be made manifest by rewriting the Starobinsky action into the quintessence
form after the field redefinition (or Legendre-Weyl transform) [18]

$\varphi=\sqrt{{{3}\over{2}}}M_{\rm Pl}\ln F^{\prime}(\chi)\quad{\rm and}\quad
g_{\mu\nu}\to{{2}\over{M^{2}_{\rm
Pl}}}F^{\prime}(\chi)g_{\mu\nu}~{},\quad\chi=R~{}.$ (8)

It yields

$S[g_{\mu\nu},\varphi]={{M^{2}_{\rm
Pl}}\over{2}}\int\mathrm{d}^{4}x\sqrt{-g}R-\int\mathrm{d}^{4}x\sqrt{-g}\left[{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+V(\varphi)\right]~{},$
(9)

in terms of the canonical inflaton $\varphi$ with the scalar potential

$V(\varphi)={{3}\over{4}}M^{2}_{\rm
Pl}M^{2}\left[1-\exp\left(-\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}\varphi/M_{\rm Pl}\right)\right]^{2}~{}.$
(10)

This potential has the infinite plateau (for the large $\phi$-field values of
the order $M_{\rm Pl}$ and beyond) that implies the approximate shift symmetry
of the inflaton field as the consequence of the scale invariance of the
$R^{2}$ gravity or due to the approximate scale invariance of the action (2)
in the large-curvature regime. The potential (10) also has the positive
”cosmological constant” given by the first term in the square brackets,
induced by the $R^{2}$ term in the action (1), which can be physically
interpreted as the energy driving inflation. The scale of inflation is
determined by the parameter $M$ that is identified with the inflaton mass. The
universality class of inflationary models is determined by the critical
parameter $\sqrt{2/3}$ in the exponential term [19].

The equivalent actions (1) and (9) are usually referred to the Jordan frame
and the Einstein frame, respectively. The approximate shift symmetry of the
potential (10) is the consequence of the approximate scale invariance of the
$R^{2}$ gravity, which requires the presence of the $R^{2}$ term in any viable
model of inflation based on modified $F(R)$-gravity. It becomes even more
transparent by using the inverse transformation from the Einstein frame to the
Jordan frame, having the parametric form [20]

$R=\left({{\sqrt{6}}\over{M_{\rm
Pl}}}{{dV}\over{d\varphi}}+{{4V}\over{M^{2}_{\rm
Pl}}}\right)e^{\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}\varphi/M_{\rm Pl}}~{},\quad
F=\left({{\sqrt{6}}\over{M_{\rm
Pl}}}{{dV}\over{d\varphi}}+{{2V}\over{M^{2}_{\rm
Pl}}}\right)e^{2\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}\varphi/M_{\rm Pl}}~{}.$ (11)

As is clear from these equations, in the SR approximation (chaotic inflation)
the first term in the brackets is much less than the second term, which
immediately implies $F(R)\sim R^{2}$.

The gravitational EFT during inflation does not have to be limited to the
terms given in Eq. (1) but should also include the higher-order curvature
terms. Those terms eliminate the infinite plateau in the inflaton potential
(10). The fact that the Starobinsky model of inflation is in excellent
agreement with the current CMB measurements (see below) implies that those
terms do not destabilize the Starobinsky inflation, which put restrictions on
their contributions.

It is convenient to use the e-foldings number $N$ instead of time $t$, which
are related by

$N=\int^{t_{0}}_{t}H(\tilde{t})d\tilde{t}~{},$ (12)

and the co-moving wavenumber $k=2\pi/\lambda$ related to $N(t)$ by the
equation $d\ln k=-dN$ .

The SR (running) parameters in the Einstein frame are defined by

$\varepsilon_{\rm sr}(\varphi)={{M^{2}_{\rm
Pl}}\over{2}}\left({{V^{\prime}}\over{V}}\right)^{2}\quad{\rm
and}\quad\eta_{\rm sr}(\varphi)=M^{2}_{\rm
Pl}\left({{V^{\prime\prime}}\over{V}}\right)$ (13)

in terms of the quintessence scalar potential $V$, where the primes denote the
derivatives with respect to $\varphi$. In the Jordan frame, one uses the
Hubble flow functions,

$\epsilon_{H}=-{{\dot{H}}\over{H^{2}}}~{},\quad\eta_{H}=\epsilon_{H}-{{\dot{\epsilon}_{H}}\over{2\epsilon_{H}H}}~{}~{}.$
(14)

The amplitude of scalar perturbations at the horizon crossing with the pivot
scale $k_{*}=0.05~{}{\rm Mpc}^{-1}$ is known from CMB measurements (called
WMAP normalization) as

$A_{s}={{V_{*}^{3}}\over{12\pi^{2}M^{6}_{\rm
Pl}({V_{*}}^{\prime})^{2}}}={{3M^{2}}\over{8\pi^{2}M^{2}_{\rm
Pl}}}\sinh^{4}\left({{\varphi_{*}}\over{\sqrt{6}M_{\rm Pl}}}\right)\approx
2\cdot 10^{-9}~{},$ (15)

where subscript (*) refers to the CMB pivot scale, in the case of Starobinsky
inflation. This allows us to fix the only parameter $M$ (or $\alpha$) and the
scale of inflation, $H_{\rm inf.}$, in the Starobinsky model as

${{M}\over{M_{\rm Pl}}}\approx{\cal O}(10^{-5}),\quad\alpha\approx{\cal
O}(10^{9}),\quad H\approx{\cal O}(10^{14})~{}{\rm
GeV},\quad{{R}\over{M^{2}_{\rm Pl}}}\approx{{12H^{2}}\over{M^{2}_{\rm
Pl}}}\approx 10^{-7}.$ (16)

It is worth mentioning here that the higher-order curvature terms in the
gravitational EFT beyond the Starobinsky model are given by power series with
respect to $H^{2}/M^{2}_{\rm Pl}\sim 10^{-8}$, so they must be sub-leading
during inflation unless they have very large coefficients. It is also worth
noticing that the large value of $\alpha$ required by CMB does not speak in
favor of generating the $R^{2}$-term by quantum matter contributions because a
single quantized matter field contributes in the 1-loop approximation about
$10^{-3}$ to the $\alpha$-parameter, so one needs about $10^{12}$ quantized
matter fields in order to achieve the desired result.

The primordial spectrum $P_{\zeta}(k)$ of 3-dimensional scalar (density)
perturbations $\zeta(x)$ in a flat Friedman universe is defined by the 2-point
correlation function

$\left\langle{{\delta\zeta(x)}\over{\zeta}}{{\delta\zeta(y)}\over{\zeta}}\right\rangle=\int{{d^{3}k}\over{k^{3}}}e^{ik\cdot(x-y)}{{P_{\zeta}(k)}\over{P_{0}}}~{}~{},$
(17)

where $k=2\pi/\lambda$ is the co-moving number. Similarly, one defines the
primordial spectrum $P_{t}(k)$ of tensor perturbations, see e.g., Ref. [19]
for more details The scale $k$ is simply related to the e-folds number $N$ via
$N=-\int^{k}d\tilde{k}/\tilde{k}$. The power spectra coincide with the
corresponding amplitudes $A_{s}(k)$ and $A_{t}(k)$, respectively.

Given the power spectra $P_{\zeta}(k)$ and $P_{t}(k)$, one defines the scalar
tilt $n_{s}(k)$, its running parameter $\alpha_{s}(k)$, the tensor tilt
$n_{t}(k)$ and its running parameter $\alpha_{t}(k)$ (all dimensionless) as

$n_{s}=1+{{d\ln P_{\zeta}(k)}\over{d\ln k}}~{},\quad\alpha_{s}={{d^{2}\ln
P_{\zeta}(k)}\over{(d\ln k)^{2}}}~{},\quad n_{t}={{d\ln P_{t}(k)}\over{d\ln
k}}~{},\quad\alpha_{t}={{d^{2}\ln P_{t}(k)}\over{(d\ln k)^{2}}}~{}~{},$ (18)

as well as the tensor-to-scalar ratio

$r(k)={{P_{t}}\over{P_{\zeta}}}=8\left|n_{t}\right|~{}.$ (19)

The Starobinsky model gives simple predictions for the cosmological tilts of
the scalar and tensor power spectra in the leading order with respect to the
e-folds $N_{*}$ evaluated when perturbations left the horizon (at the horizon
crossing) as [21, 22]

$n_{s}\approx
1-{{2}\over{N_{*}}}~{}~{},\quad\alpha_{s}\approx-{{2}\over{N^{2}_{*}}}~{}~{},\quad\alpha_{t}\approx-{{3}\over{N^{3}_{*}}}~{},\quad
r\approx{{12}\over{N^{2}_{*}}}~{}~{}.$ (20)

Therefore, tensor perturbations are suppressed with respect to scalar
perturbations by the extra factor of $N^{-1}_{*}$, whose value can be
estimated by comparing those predictions with CMB measurements [23, 24] ,

$n_{s}\approx 0.9649\pm 0.0042~{}(68\%{\rm CL})\quad{\rm and}\quad
r<0.032~{}(95\%{\rm CL})~{},$ (21)

that fit the Starobinsky model predictions for

$N_{*}=56\pm 8~{}.$ (22)

This prediction for the duration of inflation agrees with our calculations in
the Jordan frame, based on the solution (6). The corresponding times for the
end and the beginning of inflation are $M(t_{0}-t_{\rm end})\approx 2.5$ and
$M(t_{0}-t_{\rm start})\approx 27.7$, respectively.

In particular, excluding $N_{*}$ from Eqs. (20) yields the sharp prediction of
the Starobinsky model for the tensor-to-scalar ratio as

$r\approx 3(1-n_{s})^{2}~{}.$ (23)

The Starobinsky inflation does not exclude the higher-order curvature terms in
the action (1), though it implies that those terms should be subleading during
inflation, being suppressed by the powers of $H^{2}/M^{2}_{\rm Pl}\sim
10^{-8}$. The Starobinsky model is sensitive to quantum (UV) corrections
because of its high inflation scale and the inflaton field values near the
Planck scale during inflation. Hence, it is important to determine its UV-
cutoff $\Lambda_{\rm UV}$ of the Starobinsky model by studying scaling of
scattering amplitudes with respect to energy, $E/\Lambda_{\rm UV}$. A careful
calculation yields [25]

$\Lambda_{\rm UV}=M_{\rm Pl}~{}.$ (24)

Therefore, the predictions of the Starobinsky model for inflation and CMB make
sense and the model itself can be considered as a trustable effective field
theory after decoupling of heavy modes expected at the Planck scale [5].

##  3 Starobinsky-Grisaru-Zanon (SGZ) gravity

The SGZ action is defined by

$S_{\rm SGZ}[g]={{M_{\text{Pl}}^{2}}\over{2}}\int
d^{4}x\sqrt{-g}\,\left(R+{{1}\over{6M^{2}}}R^{2}-{{72\gamma}\over{M^{6}}}Z\right),$
(25)

where we have added the GZ (quantum) superstring correction [8]

$72Z=\left(R^{\mu\rho\sigma\nu}R_{\lambda\rho\sigma\tau}+{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}R^{\mu\nu\rho\sigma}R_{\lambda\tau\rho\sigma}\right)R_{\mu}^{\,\,\,\alpha\beta\lambda}R^{\tau}_{\,\,\,\alpha\beta\nu}$
(26)

to the Starobinsky action (1) with the new dimensionless coupling constant
$\gamma>0$ 222The parameter $\gamma$ introduced in Ref. [11] was rescaled here
by the factor $(M_{\rm Pl}/M)^{6}$..

The value of $\gamma$ cannot be calculated from string theory because the
action (25) is in four space-time dimensions, so $\gamma$ depends upon
compactification from ten to four dimensions and the unknown vacuum
expectation value of the string dilaton.

The SGZ gravity (25) is different from the SBR gravity defined by the action
[12, 14]

$S_{\rm SBR}={{M_{\rm Pl}^{2}}\over{2}}\int
d^{4}x\,\sqrt{-g}\left(R+{{1}\over{6M^{2}}}R^{2}-{{\beta}\over{8M^{6}}}T^{2}\right)~{},$
(27)

where $\beta$ is another dimensionless coupling constant, the $T^{2}$ stands
for the Bel-Robinson (BR) tensor squared, $T^{2}\equiv
T_{\rho\nu\lambda\mu}T^{\rho\nu\lambda\mu}$, with

$T^{\rho\nu\lambda\mu}\equiv
R^{\rho\sigma\eta\lambda}R^{\nu}{}_{\sigma\eta}{}^{\mu}+{}^{*}R^{\rho\sigma\eta\lambda}{}^{*}R^{\nu}{}_{\sigma\eta}{}^{\mu}=R^{\rho\sigma\eta\lambda}R^{\nu}{}_{\sigma\eta}{}^{\mu}+R^{\rho\sigma\eta\mu}R^{\nu}{}_{\sigma\eta}{}^{\lambda}-{{{1}\over{2}}}g^{\rho\nu}R^{\sigma\eta\xi\lambda}R_{\sigma\eta\xi}{}^{\mu}~{},$
(28)

and the star denoting the Hodge dual tensor in four dimensions.

The BR tensor was introduced by Bel and Robinson [26, 27] by analogy with the
energy-momentum tensor of Maxwell theory of electromagnetism,

$T^{\rm
Maxwell}_{\mu\nu}=F_{\mu\rho}F_{\nu}{}^{\rho}+{}^{*}F_{\mu\rho}{}^{*}F_{\nu}{}^{\rho}~{}~{},\quad
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}~{}~{}.$ (29)

There is an identity [28, 29]

$T^{2}=-{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}({}^{*}\\!R_{\mu\nu\lambda\rho}{}^{*}\\!R^{\mu\nu\lambda\rho})^{2}+{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}({}^{*}R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho})^{2}={\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}(P_{4}^{2}-E^{2}_{4})={\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}(P_{4}+E_{4})(P_{4}-E_{4})$ (30)

relating the BR tensor squared to the Euler and Pontryagin topological
densities in four dimensions, $E_{4}$ and $P_{4}$, respectively. There is
another well-known identity

$E_{4}={\cal G}_{\rm GB}\equiv
R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}-4R^{\mu\nu}R_{\mu\nu}+R^{2}$ (31)

that relates the Euler density to the Gauss-Bonnet invariant ${\cal G}_{\rm
GB}$ in four dimensions.

We consider the first two terms in the EGZ and EBR actions nonperturbatively,
but the last (quartic) curvature (BR or GZ) terms only perturbatively, in the
first order with respect to the coupling constants. Therefore, no Ostrogradski
ghosts arise. The difference between the GZ and BR terms in four dimensions,
in the context of superstrings/M-theory, was first noticed in Ref. [30].

In a flat Friedman universe (4) we find

$Z=H^{8}+2H^{6}\dot{H}+\hbox{\large{${{\textstyle{{11}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{6}}$}}}}}$}}H^{4}\dot{H}^{2}+\hbox{\large{${{\textstyle{{2}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{3}}$}}}}}$}}H^{2}\dot{H}^{3}+\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{12}}$}}}}}$}}\dot{H}^{4}~{},$ (32)

whereas

$\hbox{\large{${{\textstyle{{1}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{144}}$}}}}}$}}T^{2}=H^{8}+2H^{6}\dot{H}+H^{4}\dot{H}^{2}~{},$
(33)

which makes the difference manifest, though the first two leading terms
(relevant to the SR approximation) are the same.

##  4 GZ quantum corrections to Starobinsky inflation

The SGZ gravity equation of motion in a flat Friedman universe is given by

$\displaystyle{{}m^{6}H^{2}+6m^{4}H^{2}\dot{H}+2m^{4}H\ddot{H}-m^{4}\dot{H}^{2}}$
---
$\displaystyle{{}-12\gamma H^{8}+132\gamma H^{6}\dot{H}+44\gamma
H^{5}\ddot{H}+138\gamma H^{4}\dot{H}^{2}}$
$\displaystyle{{}+48\gamma H^{3}\dot{H}\ddot{H}+28\gamma
H^{2}\dot{H}^{3}+12\gamma H\dot{H}^{2}\ddot{H}-3\gamma\dot{H}^{4}=0}$
(34)

that extends the Starobinsky equation (5) by the $\gamma$-dependent terms. A
solution to this equation in the first order with respect to the (small)
$\gamma$-parameter, similarly to Eq. (6) reads

$H(t)=H_{0}(t)+\gamma H_{1}(t)~{},$ (35)

where

$\displaystyle{{{H_{1}(t)}\over{M}}=}$ | $\displaystyle{{}-{{1}\over{163296}}M^{7}(t_{0}-t)^{7}-{{2}\over{2835}}M^{5}(t_{0}-t)^{5}-{{391}\over{90720}}M^{3}(t_{0}-t)^{3}-{{9061}\over{306180}}M(t_{0}-t)}$
---|---
|
$\displaystyle{{}-{{127}\over{7776M(t_{0}-t)}}-{{1931203}\over{5358150M^{3}(t_{0}-t)^{3}}}+{\cal
O}(M^{-5}(t_{0}-t)^{-5})~{},}$
(36)

and the leading contribution during inflation comes from the 2nd term above.

Accordingly, the scalar curvature is given by

$\displaystyle{{{R}\over{M^{2}}}=}$ | $\displaystyle{{}~{}~{}{{M^{2}(t_{0}-t)^{2}}\over{3}}-{{1}\over{3}}-{{4}\over{9M^{2}(t_{0}-t)^{2}}}+{{16}\over{5M^{4}(t_{0}-t)^{4}}}-{{6908}\over{189M^{6}(t_{0}-t)^{6}}}}$
---|---
|
$\displaystyle{{}+\gamma\left[-{{M^{8}(t_{0}-t)^{8}}\over{40824}}-{{151M^{6}(t_{0}-t)^{6}}\over{58320}}+{{143M^{4}(t_{0}-t)^{4}}\over{122472}}-{{4163M^{2}(t_{0}-t)^{2}}\over{81648}}\right.}$
|
$\displaystyle{{}-{{68713}\over{7144200}}-{{5109281}\over{5143824M^{2}(t_{0}-t)^{2}}}+{{17584432631}\over{964467000M^{4}(t_{0}-t)^{4}}}}$
|
$\displaystyle{{}\left.-{{75802186291}\over{300056400M^{6}(t_{0}-t)^{6}}}+{\cal
O}(M^{-8}(t_{0}-t)^{-8})\right]~{},}$
(37)

where the leading $\gamma$-dependent contribution during inflation is due to
the 2nd term in the square brackets also.

The Hubble flow functions (14) are given by

$\displaystyle{\epsilon_{H}=}$ | $\displaystyle{{}~{}~{}{{6}\over{M^{2}(t_{0}-t)^{2}}}-{{6}\over{M^{4}(t_{0}-t)^{4}}}+{{48}\over{M^{6}(t_{0}-t)^{6}}}}$
---|---
|
$\displaystyle{{}+\gamma\left[-{{5M^{4}(t_{0}-t)^{4}}\over{4536}}-{{869M^{2}(t_{0}-t)^{2}}\over{11340}}-{{4591}\over{22680}}+{{38051}\over{34020M^{2}(t_{0}-t)^{2}}}\right.}$
|
$\displaystyle{{}\left.-{{1385}\over{648M^{4}(t_{0}-t)^{4}}}+{{51649511}\over{396900M^{6}(t_{0}-t)^{6}}}\right]+{\cal
O}(M^{-8}(t_{0}-t)^{-8})}$
(38)

and

$\displaystyle{\eta_{H}=}$ | $\displaystyle{{}~{}~{}{{6}\over{M^{4}(t_{0}-t)^{4}}}-{{96}\over{M^{6}(t_{0}-t)^{6}}}}$
---|---
|
$\displaystyle{{}+\gamma\left[-{{5M^{4}(t_{0}-t)^{4}}\over{1512}}-{{869M^{2}(t_{0}-t)^{2}}\over{5670}}-{{4591}\over{22680}}\right.}$
|
$\displaystyle{{}\left.+{{1385}\over{648M^{4}(t_{0}-t)^{4}}}-{{51649511}\over{198450M^{6}(t_{0}-t)^{6}}}\right]+{\cal
O}(M^{-8}(t_{0}-t)^{-8})~{},}$
(39)

where the leading $\gamma$-dependent contributions during inflation are due to
the 2nd terms in the square brackets too.

##  5 The upper bounds on $\gamma$

Some modified gravity models of inflation can be described by the effective
function $F(H^{2})$ entering equations of motion in a flat Friedman universe
[31]. Our equations of motion (34) are not of that type because they include
the higher time derivatives of the Hubble function, but they fall into that
type in the SR approximation, see Ref. [14] for applications to the SBR
gravity theory. In the case of SGZ gravity, we find

$R\left({{R}\over{12}}-H^{2}\right)-H\dot{R}=3M^{2}\left(H^{2}-{{12\gamma
H^{8}}\over{M^{6}}}+{{22\gamma H^{6}}\over{M^{4}}}\right)\equiv
3M^{2}F(H^{2})~{},$ (40)

so we have

$F(H^{2})=H^{2}-{{22\gamma}\over{M^{4}}}\left(H^{2}\right){}^{3}-{{12\gamma}\over{m^{6}}}\left(H^{2}\right){}^{4}~{}.$
(41)

The derivatives of $F(H^{2})$ with respect to $H^{2}$ are as follows:

$F^{\prime}(H^{2})\equiv{{dF}\over{d(H^{2})}}=1-66\gamma\left({{H}\over{M}}\right)^{4}-48\gamma\left({{H}\over{M}}\right)^{6}$
(42)

and

$F^{\prime\prime}(H^{2})=-{{132\gamma}\over{M^{4}}}H^{2}-{{144\gamma}\over{M^{6}}}H^{4}~{}.$
(43)

The effective Newton constant in the higher-derivative gravity theories
studied in Ref. [31] must obey the condition

$G_{\mathrm{eff.}}={{1}\over{8\pi
M_{\mathrm{Pl}}^{2}[F^{\prime}(H^{2})+4(H^{2}/M^{2})]}}>0~{},$ (44)

that implies

$F^{\prime}(H^{2})+4{{H^{2}}\over{M^{2}}}>0\quad{\rm or}\quad 1-66\gamma
h^{4}-48\gamma h^{6}+4h^{2}>0~{},$ (45)

where $h=H/M$. The maximal value of $h$ in the Starobinsky inflation is
$h_{\rm max.}\approx 4.6$, which implies

$\gamma<1.74\times 10^{-4}~{}.$ (46)

Another condition proposed in Ref. [31] from demanding the absence of negative
energy fluxes (or unitarity and causality constraints) reads

$-4\leq{{210H^{2}F^{\prime\prime}(H^{2})}\over{F^{\prime}(H^{2})+4(H^{2}/M^{2})}}\leq
4~{},$ (47)

that in our case is given by

$-1\leq{{210(-33\gamma h^{4}-38\gamma h^{6})}\over{1-66\gamma h^{4}-48\gamma
h^{6}+4h^{2}}}\leq 1~{}.$ (48)

It implies

$\gamma\leq{{1+4h^{2}}\over{12h^{4}(634h^{2}+583)}}~{}~{},$ (49)

and, therefore, the upper bound

$\gamma\leq 1.12\times 10^{-6}~{}.$ (50)

Both bounds (46) and (50) are slightly stronger than those found in Ref. [14]

##  6 Quantum corrections versus classical corrections to CMB observables
beyond the leading order

It is instructive to compare contributions of the GZ quantum correction to the
CMB observables against the classical contributions beyond the leading order
given by Eq. (20) within the possible range of $N_{*}$ in Eq. (22).

The subleading terms for the predicted CMB observables in the Starobinsky
inflation are given by [32, 33]

$\displaystyle{n_{s}=}$ | $\displaystyle{{}~{}~{}1-{{2}\over{N_{*}}}+{{2.4}\over{N^{2}_{*}}}-{{\ln(2N_{*})}\over{6N^{2}_{*}}}+{\cal O}\left({{\ln 2N_{*}}\over{N^{3}_{*}}}\right)~{},}$
---|---
$\displaystyle{\alpha_{s}=}$ | $\displaystyle{{}-{{2}\over{N^{2}_{*}}}+{\cal O}\left({{\ln 2N_{*}}\over{N^{3}_{*}}}\right)~{},}$
$\displaystyle{r=}$ | $\displaystyle{{}~{}~{}{{12}\over{N^{2}_{*}}}+{{2\ln(2N_{*})}\over{N^{3}_{*}}}-{{56.76}\over{N^{3}_{*}}}+{\cal O}\left(N^{-4}_{*}\right)~{},}$
$\displaystyle{\alpha_{t}=}$ | $\displaystyle{{}-{{3}\over{N^{3}_{*}}}+{\cal O}\left(N^{-4}_{*}\right)~{},}$
(51)

where $\ln 2N_{*}/N^{3}_{*}<4.1\cdot 10^{-5}$ and $N^{-4}_{*}<2\cdot 10^{-7}$.
The subleading contribution to the scalar tilt $n_{s}$ in the 3rd term above
can increase the $n_{s}$-value by $1.0\cdot 10^{-3}$, while the subleading
contribution to the tensor-to-scalar-ratio $r$, given by the 3rd term above,
can decrease the $r$-value by $5\cdot 10^{-4}$. All the subleading
contributions are within the observational errors given in Eq. (21).

On the other hand, when using Eqs. (38) and (39) in the first order with
respect to the string parameter $\gamma$ and taking their first-order
contributions to the scalar tilt $n_{s}$ and the tensor-to-scalar-ratio $r$ at
the horizon crossing,

$n_{s}\approx 1-6\epsilon_{H}+2\eta_{H}~{},\quad r=16\epsilon_{H}~{},$ (52)

with the maximal value of $\gamma$ from Eq. (50), we get the quantum
contributions to the $n_{s}$ and $r$ up to $+2.5\cdot 10^{-4}$ and $-4.9\cdot
10^{-5}$, respectively.

Therefore, the quantum contributions to the CMB tilts are smaller than the
subleading terms proportional to $N^{-2}_{*}$ by one order of magnitude but
may be of the same order of magnitude as the classical $N^{-3}_{*}$
contributions. The same conclusion also applies to the running parameters
$\alpha_{s}$ and $\alpha_{t}$.

##  7 Conclusion

The upcoming CMB measurements by CORE Collaboration [34], S4 Collaboration
[35], LiteBIRD Collaboration [36], NASA PICO Collaboration [37], the Simons
Observatory survey [38] and EUCLID Collaboration [39] are expected to probe
the tensor-to-scalar ratio $r$ in the range of $10^{-3}$ and improve the
precision value of $n_{s}$.

Should those measurements confirm Eq. (23), it would be a triumph of the
Starobinsky model of inflation. Should the predicted relation (23) be ruled
out, the question would arise about the origin of disagreement. If the
disagreement will be significant, it would rule out the Starobinsky inflation.
If, however, the disagreement will be small (say, within one order of
magnitude), one may expect that due to the sub-leading corrections to the
leading order predictions in Eq. (20). Then another question about the origin
of those small corrections would arise. The latter may be due to the sub-
leading terms in the classical Starobinsky model or due to quantum gravity
corrections to the gravitational EFT, or they have a very different origin,
say, due to reheating or new physics, see e.g., Ref. [40].

The main new result of this paper is about possible superstring (as quantum
gravity) corrections that may be of the same size as the next-to-next-to-next
classical corrections (in the $N^{-3}_{*}$-terms). Of course, the value of the
effective string coupling $\gamma$ may be much lower than the estimate found
in Eq. (50), which may reduce the size of quantum corrections even further.

## Acknowledgements

One of the authors (SVK) is grateful to Ignatios Antoniadis, Eugenio Bianchi,
Norma Borstnik, Gia Dvali, Maxim Khlopov, Elias Kiritsis and Holger Nielsen
for discussions. SVK was supported by Tokyo Metropolitan University, the
Japanese Society for Promotion of Science under the grant No. 22K03624, the
World Premier International Research Center Initiative (MEXT, Japan), and the
Tomsk Polytechnic University development program
Priority-2030-NIP/EB-004-375-2024.

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