# GPT vs RETRO: Exploring the Intersection of Retrieval and Parameter-
Efficient Fine-Tuning

Aleksander Ficek, Jiaqi Zeng11footnotemark: 1, Oleksii Kuchaiev
NVIDIA
<EMAIL_ADDRESS>
Equal contribution.

###### Abstract

Parameter-Efficient Fine-Tuning (PEFT) and Retrieval-Augmented Generation
(RAG) have become popular methods for adapting large language models while
minimizing compute requirements. In this paper, we apply PEFT methods
(P-tuning, Adapters, and LoRA) to a modified Retrieval-Enhanced Transformer
(RETRO) and a baseline GPT model across several sizes, ranging from 823
million to 48 billion parameters. We show that RETRO models outperform GPT
models in zero-shot settings due to their unique pre-training process but GPT
models have higher performance potential with PEFT. Additionally, our study
indicates that 8B parameter models strike an optimal balance between cost and
performance and P-tuning lags behind other PEFT techniques. We further provide
a comparative analysis of between applying PEFT to an Instruction-tuned RETRO
model and base RETRO model. This work presents the first comprehensive
comparison of various PEFT methods integrated with RAG, applied to both GPT
and RETRO models, highlighting their relative performance.

GPT vs RETRO: Exploring the Intersection of Retrieval and Parameter-Efficient
Fine-Tuning

Aleksander Ficek††thanks: Equal contribution., Jiaqi Zeng11footnotemark: 1,
Oleksii Kuchaiev NVIDIA<EMAIL_ADDRESS>

##  1 Introduction

Pre-trained large language models have made a demonstrable impact across
applications in academia and industry. Many use cases, however, require LLMs
adapted to specific tasks and unique information but lack the resources for
extensive retraining. To address this, Parameter-Efficient Fine-Tuning (PEFT)
(Han et al., 2024) and Retrieval-Augmented Generation (RAG) (Gao et al., 2023)
have become popular methods due to their effectiveness and efficiency,
inspiring new lines of research.

Figure 1: Average GPT vs RETRO scores of six datasets across model sizes of
823M to 48B parameters.

PEFT has been proven to be a comparable substitute to Supervised Fine-Tuning
(SFT) by achieving competitive performance at a fraction of the number of
updated parameters (Han et al., 2024). In this paper we select P-tuning (Liu
et al., 2023), Adapter modules (Houlsby et al., 2019) and Low-Rank Adaptation
(LoRA) (Hu et al., 2021) as representative PEFT methods. P-tuning involves
training continuous prompt embeddings to guide output for specific tasks
without modifying base model parameters. Adapters operate by training fully
connected layers inserted throughout the base model while keeping the
remaining parameters frozen. LoRA employs a similar strategy but further
decomposes the inserted layers into low-rank matrices, enhancing efficiency.

Retrieval-augmented generation (RAG) improves model quality by incorporating
external knowledge through mechanisms like BM-25 or TF-IDF (Robertson et al.,
2009), online web search (Page et al., 1999), or trained dense retriever
models (Karpukhin et al., 2020). Any LLM can be transformed into a retrieval-
augmented model by concatenating retrieved sources with the input query,
provided it fits within the model’s context window. Xu et al. (2023) found
that retrieval significantly improves GPT model quality on long context tasks,
reducing the "lost in the middle" effect (Liu et al., 2024) and offering
inherent efficiency benefits.

Alternatively, there exist multiple works (Borgeaud et al., 2022; Guu et al.,
2020; Izacard et al., 2023; Nakano et al., 2021) that have integrated
retrieval as part of model pretraining or fine-tuning to notable success when
compared to typical GPT models despite being a much lesser explored domain.
RETRO (Borgeaud et al., 2022) is of particular interest due to its unique
approach of incorporating a retrieval module directly into the transformer
architecture via a chunked-cross attention mechanism and ability to scale to
trillions of tokens resulting in reduced perplexity. Subsequently, Wang et al.
(2023b) showed that RETRO at sizes up to 9.5 billion parameters largely
outperforms GPT on specific knowledge-intensive tasks. Furthermore, Wang et
al. (2023a) illustrated that when scaled up to 48 billion parameters and
instruction-tuned, RETRO performed better than equivalent GPT models on
several question answering, reading comprehension and summarization tasks.

In this paper we continue the exploration of RETRO versus GPT through the lens
of parameter efficient finetuning. We apply P-tuning, Adapter modules and LoRA
to multiple tasks with retrieval for both RETRO and GPT models. To our
knowledge, this paper provides the first in-depth comparison of various
Parameter Efficient Fine-Tuning integrated with Retrieval-Augmented
Generation, uniquely applied to both GPT and RETRO models.

##  2 Related Work

In this section we focus on recent work that combine finetuning with
retrieval. A comprehensive survey Gao et al. (2023) synthetized multiple
comparative studies on PEFT and RAG, underscoring the potential benefits of
combining these approaches as a promising direction for future investigation.
There are multiple works that provide methods to combine RAG with fine-tuning
to improve accuracy (Zhang et al., 2024a, b; Rangan and Yin, 2024). Multiple
studies have explored the comparison between fine-tuning and retrieval.
Lakatos et al. (2024) and Ovadia et al. (2023) reported improved accuracy
using RAG over fine-tuning GPT models, while also noting suboptimal results
when combining the two methods. Gupta et al. (2024) demonstrated improved
outcomes by integrating both approaches for specific agriculture and geography
tasks. Additionally, Soudani et al. (2024) compared the efficacy of these
methods, including full and QLoRA fine-tuning (Dettmers et al., 2024), in low-
frequency entity question-answering tasks. These studies collectively suggest
the need for comprehensive investigation into multiple PEFT techniques
combined with RAG and maintain retrieval pretrained LLMs with PEFT to be
unexplored, thereby motivating our research.

|  | NQ | TQA | NQA | QASPER | QUALITY | QMSUM | AVERAGE
---|---|---|---|---|---|---|---|---
|  | GPT | RETRO | GPT | RETRO | GPT | RETRO | GPT | RETRO | GPT | RETRO | GPT | RETRO | GPT | RETRO
Extra Small | Zero-shot | 2.95 | 8.28 | 9.99 | 19.26 | 7.07 | 4.87 | 9.00 | 10.79 | 0.38 | 0.48 | 9.83 | 7.78 | 6.54 | 8.58
P-tuning | 24.74 | 7.60 | 63.63 | 24.61 | 16.74 | 6.69 | 24.54 | 11.94 | 24.59 | 17.45 | 18.25 | 13.63 | 28.75 | 13.65
Adapter | 38.48 | 23.69 | 67.99 | 59.60 | 17.76 | 15.42 | 23.52 | 20.96 | 24.26 | 25.93 | 19.74 | 14.42 | 31.96 | 26.67
LoRA | 37.09 | 22.13 | 67.31 | 59.02 | 18.08 | 15.81 | 23.54 | 19.85 | 24.93 | 25.65 | 19.27 | 13.79 | 31.70 | 26.04
Small | Zero-shot | 11.65 | 18.77 | 29.88 | 38.42 | 7.07 | 7.12 | 12.31 | 12.42 | 0.00 | 1.01 | 12.35 | 9.25 | 12.21 | 14.50
P-tuning | 39.27 | 18.58 | 70.31 | 61.13 | 19.98 | 15.13 | 24.75 | 20.34 | 22.77 | 24.11 | 18.76 | 14.61 | 32.64 | 25.65
Adapter | 42.29 | 23.68 | 73.21 | 64.91 | 21.40 | 18.10 | 27.29 | 20.55 | 24.93 | 25.07 | 20.17 | 15.03 | 34.88 | 27.89
LoRA | 39.27 | 28.06 | 72.34 | 64.59 | 20.98 | 17.90 | 24.83 | 21.28 | 25.79 | 24.69 | 20.31 | 14.46 | 33.92 | 28.50
Medium | Zero-shot | 23.67 | 24.11 | 51.00 | 52.17 | 8.90 | 6.39 | 9.01 | 10.04 | 1.44 | 0.14 | 11.28 | 9.15 | 17.55 | 17.00
P-tuning | 45.52 | 24.18 | 77.00 | 67.94 | 24.50 | 19.02 | 33.31 | 24.20 | 32.74 | 31.93 | 20.37 | 15.40 | 38.91 | 30.44
Adapter | 46.71 | 43.01 | 78.05 | 71.35 | 24.30 | 20.51 | 32.53 | 25.90 | 40.84 | 31.98 | 20.03 | 15.61 | 40.41 | 34.65
LoRA | 46.81 | 42.11 | 78.26 | 70.75 | 25.17 | 20.42 | 31.84 | 24.48 | 41.56 | 32.41 | 21.47 | 15.30 | 40.85 | 34.24
Fine-tuning | 41.34 | 29.79 | 79.82 | 68.84 | 22.33 | 19.37 | 49.67 | 23.53 | 37.01 | 33.56 | 21.95 | 15.29 | 42.02 | 31.73
Large | Zero-shot | 25.37 | 31.43 | 48.68 | 60.30 | 13.92 | 7.98 | 8.73 | 10.52 | 2.97 | 1.87 | 6.30 | 9.33 | 17.66 | 20.24
P-tuning | 45.20 | 15.78 | 78.33 | 73.22 | 25.21 | 21.58 | 34.24 | 24.50 | 47.65 | 39.93 | 20.07 | 15.00 | 41.78 | 31.67
Adapter | 47.48 | 44.43 | 79.68 | 73.57 | 26.37 | 22.03 | 32.12 | 26.09 | 46.74 | 38.06 | 20.81 | 15.22 | 42.20 | 36.57
LoRA | 47.33 | 44.48 | 79.79 | 73.63 | 25.85 | 21.49 | 32.25 | 25.21 | 42.62 | 39.31 | 21.67 | 15.02 | 41.58 | 36.53
Extra Large | Zero-shot | 26.97 | 33.49 | 44.71 | 62.87 | 11.89 | 10.07 | 11.58 | 13.38 | 3.07 | 0.96 | 7.65 | 9.99 | 17.65 | 21.79
P-tuning | 47.27 | 24.53 | 80.27 | 74.38 | 27.09 | 22.48 | 34.08 | 24.93 | 57.19 | 38.06 | 21.17 | 15.53 | 44.51 | 33.32
Adapter | 49.68 | 46.41 | 81.64 | 75.10 | 26.94 | 22.24 | 33.94 | 26.38 | 54.65 | 42.62 | 21.19 | 15.71 | 46.82 | 38.08
LoRA | 49.21 | 44.53 | 81.87 | 74.92 | 27.31 | 22.16 | 31.98 | 27.49 | 49.19 | 39.65 | 22.77 | 15.73 | 43.72 | 37.41

Table 1: A comprehensive comparison between GPT vs RETRO on six datasets. Bold
indicates the better result in each head-to-head comparison.

##  3 Experimental Setup

###  3.1 Datasets

To cover several task categories, we use six datasets suited to benefit from
retrieval and finetuning. We select Natural Questions (NQ) (Kwiatkowski et
al., 2019), TriviaQA (TQA) (Joshi et al., 2017), NarrativeQA (NQA) (Kočiskỳ et
al., 2018) and Qasper (Dasigi et al., 2021) for document question answering,
QuALITY (Pang et al., 2021) for multiple-choice question answering, and QMSum
(Zhong et al., 2021) for query-based summarization (see detail statistics in
Appendix B). Each of these datasets contain necessary external knowledge that
must be filtered via retrieval and response behaviour that encourages
finetuning. Following the official metrics, we use F1 score for evaluating
document QA, exact match for mutliple-choice QA and the geometric mean of
ROUGE-1/2/L (Lin, 2004) for summarization.

###  3.2 Models

In order to undertand the effect of model scales, we use base GPT models of
sizes 823M (Extra Small), 2.25B (Small), 8.5B (Medium), 22B (Large), and 43B
(Extra Large), as introduced in Wang et al. (2023a), which were pretrained on
a massive dataset of 1.2 trillion tokens. We employ the corresponding RETRO
models from the same work as the foundation for our retrieval pretrained LLM
experiments. Notably, the RETRO architecture features an encoder that extracts
neighbors from an external database, which increases the total model size to
877M, 2.47B, 9.5B, 24B, and 48B, respectively. Wang et al. (2023a) found
ablating the encoder after pretraining led to comparable results. In our paper
we include it so that adapter modules and LoRA layers are added throughout
decoder and encoder components. For more on the base models, we refer readers
to the original work.

###  3.3 Retrieval

We follow Wang et al. (2023a); Xu et al. (2023) to use Dragon+ (Lin et al.,
2023) as a retriever. Dragon+ is a dual encoder model that consists of a query
encoder and a context encoder. We first chunk each context document with 100
words, and then encode both the questions and all chunks independently with
corresponding encoders. The most relevant 5 chunks, ranked by the dot product
of the question embedding and chunk embedding, are retrieved as neighbors. For
GPT models, they are concatenated together (following the left to right order
from the most relevant to least relevant) as the context of the prompt for
generation. For RETRO models, they interact with the question during
generation through chunked cross-attention.

###  3.4 Parameter Efficient Fine-Tuning

We implement P-tuning in RETRO akin to GPT. Virtual tokens are added to the
beginning of the decoder. Based on the design of chunked cross-attention (Wang
et al., 2023a), left padding is added to ensure the length of input (virtual
tokens + context + question) is a multiple of chunk size. Adapter and LoRA
layers are inserted within the transformer blocks in both RETRO and GPT at the
feedforward and attention layers. In RETRO they are also inserted in the
transformer encoder ingesting the retrieved neighbors. We provide additional
hyperparameter tuning, resource utilization and prompt template details in
Appendix A.

##  4 Results

###  4.1 Main Results

Table 1 shows the comprehensive comparison between GPT and RETRO models across
five sizes and six datasets. From this table we observe:

(1) RETRO is generally better than GPT at zero-shot settings. This superiority
stems from its unique pre-training approach. By learning to extract salient
information from retrieved text and integrate it into its generation process,
RETRO develops the capability to harness relevant contextual knowledge,
ultimately leading to its good zero-shot performance. In contrast, GPT relies
on an auto-regressive loss during pre-training, focusing on accurately
predicting next tokens without the benefit of external retrievals. As a
result, GPT’s ability to learn context-aware question-answering is limited to
the presence of relevant data within the pre-training corpus, resulting in
significantly less targeted training compared to RETRO.

(2) Both RETRO and GPT models exhibit saturation points on these datasets
around the 8B mark, with a similar pattern emerging between the two models,
albeit with RETRO performing less well. This can be seen in Figure 1 and
suggests that, for a specific task, a medium-sized PEFT model strikes the
optimal balance between cost and performance, making it a sweet spot for many
applications.

(3) For GPT models, P-tuning performs worse at smaller models (<=8B), but it
performs as good or better than LoRA and Adapter at the large model sizes.
This difference is visualized in Figure 2 and Figure 3 (Appendix C). However,
for RETRO models, P-tuning generally under performs the other PEFT methods
across all model sizes.

(4) The performance ceiling for PEFT-tuned models is significantly higher for
GPT compared to RETRO as seen in Figure 4 (Appendix C). For example, using
medium-sized models, the average score of LoRA with GPT is 40.85, while with
RETRO it is 34.24. This disparity suggests that GPT has more room for
improvement with PEFT tuning. This phenomenon can also be possibly explained
by the two different pre-training strategies. Since GPT has been less targeted
on retrieval-augmented generation during pre-training, it opens a larger room
for improvement during fine-tuning.

(5) We also conduct full fine-tuning on medium-sized GPT and RETRO models. For
GPT model, we find that full fine-tuning achieves slightly better performance
than PEFT on average, with a notable exception on the QASPER dataset, where
the difference is more pronounced and full fine-tuning yields significantly
better results. In contrast, for RETRO model, full fine-tuning generally
performs worse than PEFT, demonstrating the effectiveness of PEFT. These
findings are consistent with the observations of Hu et al. (2021).

Figure 2: Comparison of Extra Large GPT and RETRO results averaged across 6
datasets.

###  4.2 Comparing to Instruction-tuned RETRO

Instruction tuning post retrieval-augmented pre-training (Wang et al., 2023a)
has been demonstrated to improve zero-shot performance on RETRO models. A
natural thought is that whether Instruction-tuned RETRO (I-RETRO) serve as a
better foundation for applying PEFT compared to the base RETRO. To investigate
this, we additionally apply PEFT to a medium-sized I-RETRO model and show
overall results in Table 2 and more granular results in Table 4 (Appendix C).
Our findings reveal that while I-RETRO exhibits improved performance in the
zero-shot setting, it has limited scope for further improvement using PEFT.
Even with substantial hyperparameter tuning, the average scores across six
datasets, using each of the three PEFT methods, demonstrate an approximately
10% gap between I-RETRO and base RETRO. We hypothesize that conceptually both
models should be tunable to similar performance but will leave that
exploration to future work.

| Average QA | QUALITY | QMSUM | Average
---|---|---|---|---
| I-RETRO | RETRO | I-RETRO | RETRO | I-RETRO | RETRO | I-RETRO | RETRO
Zero-shot | 27.65 | 23.79 | 3.35 | 0.14 | 11.04 | 9.15 | 20.83 | 17.00
P-tuning | 23.25 | 47.18 | 16.68 | 31.93 | 15.88 | 15.40 | 20.75 | 30.44
Adapter | 22.64 | 52.75 | 29.87 | 31.98 | 15.06 | 15.16 | 22.58 | 34.65
LoRA | 26.53 | 52.80 | 24.21 | 32.41 | 15.40 | 15.30 | 24.29 | 34.24

Table 2: Instruction-tuned RETRO evaluation results.

##  5 Conclusion

This study explores Parameter-Efficient Fine-Tuning (PEFT) methods applied to
Retrieval-Augmented Generation (RAG) models, comparing GPT and RETRO
architectures. RETRO generally outperforms GPT in zero-shot settings due to
their pre-training process that integrates external retrieval, enhancing
contextual understanding. However, GPT models show a higher performance
potential with PEFT, indicating more room for improvement during fine-tuning.
Both RETRO and GPT models perform optimally around the 8B parameter mark,
balancing cost and performance. While P-tuning is effective in larger models,
it lags behind other methods in smaller models, particularly for RETRO.
Applying PEFT to Instruction-tuned RETRO yields limited improvement compared
to base RETRO, suggesting a saturation point in leveraging pre-training and
fine-tuning benefits. Our comprehensive analysis offers valuable insights for
optimizing large language models with PEFT and RAG to the community.

## Limitations

Due to the breadth of experiments covered in this work we had to prioritze
certain experiments over others. This resulted in us using only the medium
sized GPT and RETRO models for additional finetuning and Instruction tuning
experiments. We believe these results generalize to the other model sizes but
leave that to be validated in future work.

## Potential Risks

The environmental impact associated with training and fine-tuning large models
is not negligable as it involves substantial computational resources and
energy consumption. While PEFT aims to alleviate this by reducing the number
of tunable parameters, works like ours still require significant compute to
distinguish which methods are more promising.

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##  Appendix A Details on Experimental Setup

###  A.1 Hyperparameter Tuning

We conducted hyperparemter search by selectively modifying number of virtual
tokens among the values of 40, 50, 90 and 100, Adapter/LoRA dimensions 16, 32
or 64 and learning rates 1e-4 to 1e-6.

###  A.2 Resource Utilization

In our experiments, we used up to 16 compute nodes, each with 8 A100-80GB SXM
GPUs. When model is smaller, we increased the data parallelism size, using
tools in NeMo framework.

###  A.3 Prompt Template

The template we used to present context to GPT models is as follows.

    title: {title}
    source: {source}
    title: {title}
    source: {source}
    title: {title}
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    Question: {question} Answer: The answer is

##  Appendix B Dataset Statistics

| NQ | TQA | NQA | QASPER | QUALITY | QMSUM
---|---|---|---|---|---|---
Train | 79168 | 78785 | 44002 | 2053 | 2018 | 1005
Validation | 8757 | 8837 | 11001 | 514 | 505 | 252
Test | 3610 | 11313 | 5859 | 1726 | 2086 | 272

Table 3: Number of samples in train/validation/test split for each dataset.

##  Appendix C Supplementary Figures and Tables

(a)

(b)

Figure 3: GPT vs RETRO comparisons on Extra Small and Medium sized models.

(a)

(b)

(c)

(d)

Figure 4: GPT vs RETRO seperate method comparisons.

| NQ | TQA | NQA | QASPER | QUALITY | QMSUM | Average
---|---|---|---|---|---|---|---
| I-RETRO | RETRO | I-RETRO | RETRO | I-RETRO | RETRO | I-RETRO | RETRO | I-RETRO | RETRO | I-RETRO | RETRO | I-RETRO | RETRO
Zero-shot | 30.39 | 24.11 | 53.25 | 52.17 | 12.23 | 6.39 | 14.72 | 10.04 | 3.35 | 0.14 | 11.04 | 9.15 | 20.83 | 17.00
P-tuning | 19.55 | 24.18 | 41.95 | 67.94 | 20.17 | 19.02 | 11.34 | 24.20 | 16.68 | 31.93 | 15.88 | 15.40 | 20.75 | 30.44
Adapter | 18.81 | 43.01 | 38.83 | 71.35 | 20.30 | 20.51 | 12.64 | 25.90 | 29.87 | 31.98 | 15.06 | 15.16 | 22.58 | 34.65
LoRA | 21.56 | 42.11 | 47.89 | 70.75 | 19.23 | 20.42 | 17.45 | 24.48 | 24.21 | 32.41 | 15.40 | 15.30 | 24.29 | 34.24

Table 4: Full results with Instruction-tuned RETRO. Bold indicates the better
result in each head-to-head comparison.


Eikonal quasinormal modes and shadow of string-corrected $d$-dimensional black
holes

Filipe Moura† and João Rodrigues‡

_† Departamento de Matemática, Escola de Tecnologias e Arquitetura and
Instituto de Telecomunicações,
ISCTE - Instituto Universitário de Lisboa,
Av. das Forças Armadas, 1649-026 Lisboa, Portugal_ _‡ Centro de Física da
Universidade de Coimbra,
Rua Larga, 3004-516 Coimbra, Portugal_

<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We compute the quasinormal frequencies of $d$-dimensional spherically
> symmetric black holes with leading string $\alpha^{\prime}$ corrections in
> the eikonal limit for tensorial gravitational perturbations and scalar test
> fields. We find that, differently than in Einstein gravity, the real parts
> of the frequency are no longer equal for these two cases. The corresponding
> imaginary parts remain equal to the principal Lyapunov exponent
> corresponding to circular null geodesics, to first order in
> $\alpha^{\prime}$. We also compute the radius of the shadow cast by these
> black holes.

##  1 Introduction

Black holes in the ringdown phase resulting from a black hole collision form a
dissipative system which can be described by black hole perturbation theory:
they loose energy by emitting gravitational radiation. Those decaying
oscillations are called quasinormal modes, and they are given in terms of
complex frequencies. The spectrum of black hole quasinormal modes (QNMs) does
not depend on what drives the perturbations: it is completely determined in
terms of intrinsic physical quantities of the black hole such as mass, charge
or spin, and eventually (beyond Einstein gravity) some other parameters of the
theory. This feature turns QNMs into preferential probes for testing theories
of gravity beyond Einstein, since the ringing frequencies can be directly
measured by the gravitational wave detectors. In spite of their classical
origin, QNMs may provide a glimpse into the eventual quantum nature of the
black hole. With the advent of gravitational wave astronomy, therefore,
interest in the study of black hole QNMs has raised. Some recent works
computing QNMs of black holes with higher derivative corrections coming from
different theories are [1, 2, 4, 3, 5].

Most of the times, QNMs have to be computed numerically. Nonetheless,
different analytical methods have been developed in order to compute QNMs in
some limiting cases. One of such cases is the eikonal limit. Experimental data
related to black holes coming either from the detection of gravitational waves
or from observation of the electromagnetic spectrum in their vicinity, does
not fix the near-horizon geometry; more information for that purpose can be
obtained from the shadow cast by a black hole. As we will see in section 4,
for spherically symmetric asymptotically flat black holes the radius of such
shadow can be related with the real part of QNM frequencies in the eikonal
limit.

In this article we will consider $d$-dimensional spherically symmetric black
holes in string theory and compute their quasinormal frequencies in the
eikonal limit and the radius of the black hole shadow. The article is
organized as follows: in section 2 we will review the string-corrected black
hole solution we will consider, the master equation and the corresponding
potentials for minimally coupled test scalar fields and tensorial
gravitational perturbations; in section 3 we analyze how to compute the
corresponding quasinormal frequencies in the eikonal limit in this context,
and we obtain such frequencies; in section 4 we review how to obtain the black
hole shadow from the quasinormal modes and we compute it for the black hole we
consider.

##  2 String-corrected spherically symmetric black holes and their tensorial
perturbations

A general static spherically symmetric metric in $d$ dimensions can always be
cast in the form

$ds^{2}=-f(r)\ dt^{2}+f^{-1}(r)\ dr^{2}+r^{2}d\Omega^{2}_{d-2}.$ (1)

A minimally coupled scalar field propagating in the background of a black hole
of the form (1) can be expanded as

$\Phi(t,r,\theta)=e^{i\omega
t}\sum_{\ell}\Phi_{\ell,\omega}(r)Y_{\ell}(\theta)\,,$ (2)

where $\omega$ is the wave frequency, $\ell$ is the angular quantum number
associated with the polar angle $\theta$ and $Y_{\ell}(\theta)$ are the usual
spherical harmonics defined over the $(d-2)$ unit sphere $\mathbb{S}^{d-2}$,
with the azimuthal angles in this case set to constants. In terms of the
tortoise coordinate $x$ defined by

$dx=\frac{dr}{f(r)},$ (3)

the field equation satisfied by each component $\Phi_{\ell,\omega}(r)$ (which
we denote by $\psi$) is given by

$\frac{d^{2}\psi}{d\,x^{2}}+\omega^{2}\psi=V\left[f(r)\right]\psi$ (4)

with the potential [6]

$V_{\textsf{0}}[f(r)]=f(r)\left(\frac{\ell\left(\ell+d-3\right)}{r^{2}}+\frac{\left(d-2\right)\left(d-4\right)f(r)}{4r^{2}}+\frac{\left(d-2\right)f^{\prime}(r)}{2r}\right).$
(5)

Perturbations of a $d-$dimensional spherically symmetric metric can be
decomposed according to their tensorial behavior on $\mathbb{S}^{d-2}$ as
scalar, vectorial or (for $d>4$) tensorial. Each type of perturbation is
described in terms of a master variable, which we also denote generically by
$\psi$. In Einstein gravity, this master variable obeys a second order
differential equation (“master equation”) analogous to (4), but with a
potential that also depends on the kind of perturbation one considers [7].
Specifically, tensorial perturbations obey an equation (4) with the same
potential (5) of scalar test fields.

In the presence of higher order corrections in the lagrangian, one can still
have spherically symmetric black holes of the form (1), but the master
equation obeyed by each perturbation variable is expected to change. Tensorial
metric perturbations in the presence of leading $\alpha^{\prime}$ corrections
have been studied in [8, 9], $\alpha^{\prime}$ being the inverse string
tension. Concretely, the following $d$–dimensional effective action with
purely gravitational $\alpha^{\prime}$ corrections has been considered:

$\frac{1}{16\pi
G}\int\sqrt{-g}\left(\mathcal{R}-\frac{4}{d-2}\left(\partial^{\mu}\phi\right)\partial_{\mu}\phi+\mbox{e}^{\frac{4}{d-2}\phi}\frac{\lambda}{2}\
\mathcal{R}^{\mu\nu\rho\sigma}\mathcal{R}_{\mu\nu\rho\sigma}\right)\mbox{d}^{d}x.$
(6)

This is the effective action of bosonic and heterotic string theories, to
first order in $\alpha^{\prime}$, with
$\lambda=\frac{\alpha^{\prime}}{2},\frac{\alpha^{\prime}}{4}$, respectively.
111Type II superstring theories do not have $\alpha^{\prime}$ corrections to
this order. In both cases one can consistently set all other bosonic and
fermionic fields present in the string spectrum to zero except for the dilaton
field $\phi$. We are also not considering compactification effects: action (6)
results from compactifying on a flat torus, leaving $d$ dimensions
uncompactified.

In [8, 9] it has been shown that, perturbing the field equations resulting
from this action, for tensorial perturbations of a spherically symmetric
metric (1) one also obtains a second order master equation (4). The
corresponding potential is given by

$\displaystyle V_{\textsf{T}}[f(r)]$ $\displaystyle=$ $\displaystyle\lambda\
\frac{f(r)}{r^{2}}\left[\left(\frac{2\ell\left(\ell+d-3\right)}{r}+\frac{\left(d-4\right)\left(d-5\right)f(r)}{r}+\left(d-4\right)f^{\prime}(r)\right)\left(2\frac{1-f(r)}{r}+f^{\prime}(r)\right)\right.$
(7) $\displaystyle+$
$\displaystyle\left.\Big{(}4(d-3)-(5d-16)f(r)\Big{)}\frac{f^{\prime}(r)}{r}-4\left(f^{\prime}(r)\right)^{2}+\left(d-4\right)f(r)f^{\prime\prime}(r)\right]+V_{\textsf{0}}[f(r)],$

with $V_{\textsf{0}}[f(r)]$ given by (5).

Spherically symmetric $d-$dimensional black hole solutions with leading
$\alpha^{\prime}$ corrections have been obtained in [10, 11]. Specifically
concerning the action (6), a solution of the respective field equations is of
the form (1), with

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
f_{0}(r)\left(1+\frac{\lambda}{R_{H}^{2}}\delta f(r)\right),$ (8)
$\displaystyle f_{0}(r)$ $\displaystyle=$ $\displaystyle
1-\frac{R_{H}^{d-3}}{r^{d-3}},$ (9) $\displaystyle\delta f(r)$
$\displaystyle=$ $\displaystyle-\frac{(d-3)(d-4)}{2}\
\frac{R^{d-3}_{H}}{r^{d-3}}\
\frac{1-\frac{R_{H}^{d-1}}{r^{d-1}}}{1-\frac{R^{d-3}_{H}}{r^{d-3}}}.$ (10)

When $\lambda=0$ we get the Tangherlini solution with $f(r)=f_{0}(r)$. This is
then clearly an $\alpha^{\prime}-$corrected Tangherlini solution, which has
been obtained and studied by Callan, Myers and Perry in [10]. Its only horizon
occurs at the same radius $r=R_{H}$ of the Tangherlini solution. For our
purposes, it is enough to just quote here the explicit expression for the
temperature of this black hole, given by

$T=\frac{d-3}{4\pi R_{H}}\left(1-\frac{\left(d-1\right)\left(d-4\right)}{2}\
\frac{\lambda}{R_{H}^{2}}\right).$ (11)

##  3 Quasinormal modes in the eikonal limit

In this section, we compute quasinormal modes in the eikonal limit for the
cases we have been considering: minimally coupled (test) scalar fields and
tensorial gravitational perturbations. The eikonal limit of quasinormal modes
corresponds to having a large multipole number $\ell$, or equivalently a large
real part of the QNM frequency. In this limit one can apply the WKB
approximation.

In the eikonal limit ($\ell\rightarrow+\infty$), we are left with only one
term from the potential (5):

$V_{0}^{\textsf{eik}}[f(r)]=\ell^{2}\frac{f(r)}{r^{2}}.$ (12)

In Einstein gravity (i.e. taking $\lambda=0$), this limit is the same for the
potentials corresponding not only to test scalar fields but also to all kinds
of gravitational perturbations: tensorial, vectorial and scalar [7], and so
are the corresponding eikonal quasinormal modes. We now briefly review how to
compute these quasinormal modes.

In order to apply the WKB approximation, we assume that the potential we are
dealing with vanishes at the horizon and at infinity, and that it has a single
maximum for some value $r>R_{H}$. Concretely for $V_{0}^{\textsf{eik}}[f(r)]$,
such maximization corresponds to having

$2f(r)=rf^{\prime}(r),$ (13)

which is precisely the equation defining a circular null geodesic. Indeed, in
[12] it was shown that for $d$-dimensional asymptotically flat spherically
symmetric black holes in Einstein gravity the eikonal quasinormal frequencies
are given in terms of parameters of the unstable circular null geodesics
around such black holes. Concretely, the quasinormal frequency corresponding
to the overtone mode $n$ is given by

$\omega_{n}=\ell\Omega-i\left(n+\frac{1}{2}\right)\Lambda,$ (14)

where $\Omega$ is the angular velocity at such geodesics and $\Lambda$ is the
principal Lyapunov exponent corresponding to such orbit. These quantities are
given by

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{\sqrt{f(r_{c})}}{r_{c}},$ (15) $\displaystyle\Lambda$
$\displaystyle=$
$\displaystyle\sqrt{-\frac{r_{c}^{2}}{2f(r_{c})}\frac{d^{2}}{dx^{2}}\left(\frac{f}{r^{2}}\right)_{r=r_{c}}}$
(16) $\displaystyle=$
$\displaystyle\sqrt{-\frac{r_{c}^{2}}{2}\left(f^{\prime}\left(\frac{f}{r^{2}}\right)^{\prime}+f\left(\frac{f}{r^{2}}\right)^{\prime\prime}\right)_{r=r_{c}}},$

with $r_{c}$ being the value of $r$ maximizing $V_{0}^{\textsf{eik}}[f(r)]$,
i.e. the radius of the circular null geodesic, a solution of (13). $x$ is the
tortoise coordinate (3).

In Einstein gravity we have the Tangherlini solution with $f(r)=f_{0}(r)$, as
we saw. The corresponding solution to (13) is given by

$r_{0}=R_{H}\sqrt[d-3]{\frac{d-1}{2}}.$ (17)

The quasinormal frequencies are obtained by simply replacing $r_{c}=r_{0}$ in
(15) and (16).

We wish to generalize the study of quasinormal modes in the eikonal limit for
the string–corrected Callan-Myers-Perry black hole given by (8). For
simplicity, we start by QNMs of test scalar fields, after which we study QNMs
of tensorial gravitational perturbations.

###  3.1 The eikonal limit for test scalar fields

The procedure for obtaining QNMs in the eikonal limit for test scalar fields
is essentially identical to the one we just saw in Einstein gravity. We want
to maximize the same potential, given by (12). Such maximization corresponds
to solving equation (13). The only change comes from the function $f(r)$,
which is now given by (8), including a $\lambda$ correction, corresponding to
the solution we have been considering. For such choice of $f(r)$, the value of
$r$ which is a solution of (13) is given, to first order in $\lambda$, by

$r_{c}=R_{H}\sqrt[d-3]{\frac{d-1}{2}}\left(1+\frac{\lambda}{R_{H}^{2}}\left(\frac{d-4}{2}-(d-4)\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right).$
(18)

Replacing this value in (15) and (16), respectively, we obtain

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\sqrt{\frac{d-3}{2}}\frac{1}{R_{H}}\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\left[1+\frac{\lambda}{R_{H}^{2}}\frac{d-4}{2}\left(\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}-1\right)\right],$
(19) $\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle\frac{d-3}{\sqrt{2}}\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\frac{1}{R_{H}}\left[1-\frac{\lambda}{R_{H}^{2}}\frac{d-4}{2}\left(1+(d-2)\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right].$
(20)

The $\lambda=0$ limits of (18), (19) and (20) correspond to the known results
in Einstein gravity.

We now express the results in terms of the black hole temperature. We simply
invert, to first order in $\lambda$, the expression $T(R_{H})$ given by (11);
we then replace $R_{H}$ by the result $R_{H}(T)$ in (19) and (20), obtaining

$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle 2\pi
T\sqrt{\frac{2}{d-3}}\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\left[1+8\pi^{2}T^{2}\lambda\frac{d-2}{(d-3)^{2}}\left(d-4+3\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right];$
(21) $\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle\frac{4\pi
T}{\sqrt{2}}\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\left[1+8\pi^{2}T^{2}\lambda\frac{(d-2)(d-4)}{(d-3)^{2}}\left(1-\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right].$
(22)

We can check that, when expressed in terms of a physical variable (in this
case the temperature), $\alpha^{\prime}$ corrections are positive for both
real and imaginary parts of the QNM frequencies.

###  3.2 The eikonal limit for tensorial perturbations

We now turn to the computation of QNM frequencies in the eikonal limit for
gravitational tensorial perturbations, described to first order in
$\alpha^{\prime}$ by the potential (7). In order to distinguish from the
scalar field case we have just seen, we denote these frequencies by

$\omega^{\textsf{T}}_{n}=\ell\Omega_{\textsf{T}}-i\left(n+\frac{1}{2}\right)\Lambda_{\textsf{T}}.$
(23)

The eikonal limit ($\ell\rightarrow+\infty$) of the potential (7), compared to
the one of (5), naturally has a $\lambda$ correction. This limit can be
expressed analogously to (12), but with $f(r)$ replaced by $g(r)$:

$\displaystyle V_{\textsf{T}}^{\textsf{eik}}[f(r)]$ $\displaystyle=$
$\displaystyle\ell^{2}\frac{g(r)}{r^{2}},$ (24) $\displaystyle g(r)$
$\displaystyle\equiv$ $\displaystyle
f(r)\left(1+\frac{2\lambda}{r^{2}}\left(2\left(1-f(r)\right)+rf^{\prime}(r)\right)\right).$
(25)

Because of the $\lambda$ correction in $g(r)$, the equation for the
maximization of $V_{\textsf{T}}^{\textsf{eik}}$ is no longer the equation
defining a circular null geodesic, as pointed out in [3]; in our case, it is
rather given by

$2g(r)=rg^{\prime}(r).$ (26)

Its solution is given, to first order in $\lambda$, by

$r_{t}=R_{H}\sqrt[d-3]{\frac{d-1}{2}}\left(1+\frac{\lambda}{R_{H}^{2}}\left(\frac{d-4}{2}-(2d-5)\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right).$
(27)

The procedure for the computation of
$\Omega_{\textsf{T}},\Lambda_{\textsf{T}}$ in (23) is similar to the one
described in the previous section for $\Omega,\Lambda$. We simply have to take
$g(r)$ instead of $f(r)$ and $r_{t}$ instead of $r_{c}$ in (15) and (16):

$\displaystyle\Omega_{\textsf{T}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{g(r_{t})}}{r_{t}},$ (28)
$\displaystyle\Lambda_{\textsf{T}}$ $\displaystyle=$
$\displaystyle\sqrt{-\frac{r_{t}^{2}}{2g(r_{t})}\frac{d^{2}}{dx^{2}}\left(\frac{g}{r^{2}}\right)_{r=r_{t}}}$
(29) $\displaystyle=$
$\displaystyle\sqrt{-\frac{r_{t}^{2}}{2}\frac{f(r_{t})}{g(r_{t})}\left(f^{\prime}\left(\frac{g}{r^{2}}\right)^{\prime}+f\left(\frac{g}{r^{2}}\right)^{\prime\prime}\right)_{r=r_{t}}}.$

We obtained the following results, to first order in $\lambda$:

$\displaystyle\Omega_{\textsf{T}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{d-3}{2}}\frac{1}{R_{H}}\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\left[1+\frac{\lambda}{R_{H}^{2}}\frac{1}{2}\left(3(d-2)\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}-(d-4)\right)\right],$
(30) $\displaystyle\Lambda_{\textsf{T}}$ $\displaystyle=$
$\displaystyle(d-3)\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\frac{1}{R_{H}}\left[1-\frac{\lambda}{R_{H}^{2}}\frac{d-4}{2}\left(1+(d-2)\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right].$
(31)

The $\lambda=0$ limits of (27), (30) and (31) correspond to the known results
in Einstein gravity.

Like we did for the case of test scalar fields, we now express the results in
terms of the black hole temperature, given by (11):

$\displaystyle\Omega_{\textsf{T}}$ $\displaystyle=$ $\displaystyle 2\pi
T\sqrt{\frac{2}{d-3}}\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\left[1+8\pi^{2}T^{2}\lambda\frac{d-4}{(d-3)^{2}}\left(d-2+\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right],$
(32) $\displaystyle\Lambda_{\textsf{T}}$ $\displaystyle=$ $\displaystyle 4\pi
T\left(\frac{2}{d-1}\right)^{\frac{d-1}{2(d-3)}}\left[1+8\pi^{2}T^{2}\lambda\frac{(d-2)(d-4)}{(d-3)^{2}}\left(1-\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right].$
(33)

Like for the case of test scalar fields these $\alpha^{\prime}$ corrections,
expressed in terms of the temperature, are positive both for the real and for
the imaginary parts of the QNM frequencies.

###  3.3 The equality of tensorial and test imaginary parts

We must point out that, because the equation for the maximization of
$V_{\textsf{T}}^{\textsf{eik}}$ does not coincide with the one defining a
circular null geodesic, the identifications of $\Omega_{\textsf{T}}$ with the
angular velocity at such geodesic and $\Lambda_{\textsf{T}}$ as the principal
Lyapunov exponent corresponding to such orbit are not expected to be valid for
these perturbations. That is the case for $\Omega_{\textsf{T}}$: by comparing
(19) to (30), we see that $\Omega$ and $\Omega_{\textsf{T}}$ are indeed
different. But that is not the case for $\Lambda_{\textsf{T}}$.

By comparing (20) to (31), we notice that the imaginary components of the
eikonal quasinormal frequencies, associated with scalar test fields and tensor
type gravitational perturbations, are equal _to first order in $\lambda$_:

$\Lambda=\Lambda_{\textsf{T}}+\mathcal{O}(\lambda^{2}).$ (34)

This property is surprising and remarkable. We saw that, in the absence of
higher order corrections, quasinormal modes of tensorial perturbations and
test fields were identical; in the eikonal limit, test fields and all kinds of
gravitational perturbations had the same QNM spectra. In all these cases, the
equality of the QNM spectra had its origin in the equality of the
corresponding perturbation potentials in the eikonal limit: the same potential
naturally gave rise to the same QNM frequencies. In the presence of leading
string corrections, in the cases we considered222The potentials for black
holes in the presence of string $\alpha^{\prime}$ corrections are known only
in the cases we considered: tensorial gravitational perturbations and test
fields. the perturbation potentials are different but the imaginary parts of
QNM frequencies remain the same (not the real parts, though).

We now provide an analytical proof of this remarkable equality, under the
assumptions we have been considering in this article. In order to simplify the
notation for this section, we define
$\lambda^{\prime}=\frac{\lambda}{R_{H}^{2}}.$ We will always discard
contributions of order $\mathcal{O}(\lambda^{\prime 2})$; every equation in
this section should thus have an extra “$+\mathcal{O}(\lambda^{\prime 2})$”
term, which we omit also to simplify the notation (but it is important to keep
this in mind).

We recall that $r_{c}$ and $r_{t}$ are the solutions to (13) and (26),
respectively:

$\displaystyle 2f(r_{c})=r_{c}f^{\prime}(r_{c}),$ (35) $\displaystyle
2g(r_{t})=r_{t}g^{\prime}(r_{t}).$ (36)

Here $f(r),\,g(r)$ correspond to the leading terms of the potentials
$V_{0}^{\textsf{eik}}[f(r)],\,V_{\textsf{T}}^{\textsf{eik}}[f(r)]$. We will
assume for a moment that $f(r),\,g(r)$ are generic functions, defined only by
conditions (12) and (24) respectively, in the presence of higher order
corrections proportional to a constant $\lambda^{\prime}\frac{}{}$, which for
a while we also take as generic. The only restriction we will consider is
that, in the absence of the higher order corrections, both $f(r),\,g(r)$
reduce to $f_{0}(r)$ given by (9). This means that both these functions can be
written in the form

$\displaystyle f(r)=f_{0}(r)+\lambda^{\prime}\delta f,$ $\displaystyle
g(r)=f_{0}(r)+\lambda^{\prime}\delta g.$ (37)

Similarly, $r_{t},\,r_{c}$ can also both be written in the form

$\displaystyle r_{t}=r_{0}+\lambda^{\prime}\delta r_{t},$ $\displaystyle
r_{c}=r_{0}+\lambda^{\prime}\delta r_{c}.$ (38)

$r_{0}$, given by (17), is the solution of

$2f_{0}(r_{0})=r_{0}f_{0}^{\prime}(r_{0}).$ (39)

From (16) and (29), and using (35) and (36), proving (34) amounts to prove
that

$\frac{f^{2}(r_{c})}{r_{c}^{2}}\left[2-\frac{f^{\prime\prime}(r_{c})}{f(r_{c})}r_{c}^{2}\right]=\frac{f^{2}(r_{t})}{r_{t}^{2}}\left[2-\frac{g^{\prime\prime}(r_{t})}{g(r_{t})}r_{t}^{2}\right].$
(40)

We can define the function

$\Delta f(r):=g(r)-f(r).$ (41)

From (37) we see that $\Delta f(r)$ is necessarily of order
$\lambda^{\prime}$. Similarly from (38) we can conclude that $r_{c}-r_{t}$ is
also of order $\lambda^{\prime}$. Knowing these facts, we may consider first
order in $\lambda^{\prime}$ Taylor expansions of
$f(r_{c}),f^{\prime\prime}(r_{c})$ around $r_{t}$. Using such expansions and
(37) we can write, always to first order in $\lambda^{\prime}$,

$\displaystyle f(r_{c})f^{\prime\prime}(r_{c})$ $\displaystyle=$
$\displaystyle\left[f(r_{t})+f^{\prime}(r_{t})(r_{c}-r_{t})\right]\left[f^{\prime\prime}(r_{t})+f^{\prime\prime\prime}(r_{t})(r_{c}-r_{t})\right]$
(42) $\displaystyle=$ $\displaystyle
f(r_{t})f^{\prime\prime}(r_{t})+(r_{c}-r_{t})\left[f_{0}(r_{0})f_{0}^{\prime\prime\prime}(r_{0})+f_{0}^{\prime}(r_{0})f_{0}^{\prime\prime}(r_{0})\right]$

and

$\begin{split}\frac{f^{2}(r_{t})}{g(r_{t})}g^{\prime\prime}(r_{t})=f^{2}(r_{t})\left(\frac{1}{f(r_{t})}-\frac{\Delta
f(r_{t})}{f^{2}(r_{t})}\right)g^{\prime\prime}(r_{t})=\left(f(r_{t})-\Delta
f(r_{t})\right)\left(f^{\prime\prime}(r_{t})+\Delta
f^{\prime\prime}(r_{t})\right)\\\ =f(r_{t})f^{\prime\prime}(r_{t})-\Delta
f(r_{0})f^{\prime\prime}_{0}(r_{0})+f_{0}(r_{0})\Delta
f^{\prime\prime}(r_{0}).\end{split}$ (43)

Knowing that
$\frac{1}{r_{c}^{2}}-\frac{1}{r_{t}^{2}}=-\frac{2}{r_{0}^{3}}(r_{c}-r_{t})$,
always to first order in $\lambda^{\prime}$, and also using (39), we have

$2\frac{f^{2}(r_{c})}{r_{c}^{2}}-2\frac{f^{2}(r_{t})}{r_{t}^{2}}=2f^{2}(r_{t})\left(\frac{1}{r_{c}^{2}}-\frac{1}{r_{t}^{2}}\right)+\frac{4}{r_{0}^{2}}f_{0}(r_{0})f_{0}^{\prime}(r_{0})(r_{c}-r_{t})=(r_{c}-r_{t})\frac{4}{r_{0}^{3}}f_{0}^{2}(r_{0}).$
(44)

From (42), (43) and (44) we can rewrite equation (40) as

$(r_{c}-r_{t})\left[\frac{4}{r_{0}^{3}}f_{0}^{2}(r_{0})-f_{0}(r_{0})f_{0}^{\prime\prime\prime}(r_{0})-f_{0}^{\prime}(r_{0})f_{0}^{\prime\prime}(r_{0})\right]+f_{0}(r_{0})\Delta
f^{\prime\prime}(r_{0})-\Delta f(r_{0})f^{\prime\prime}_{0}(r_{0})=0.$ (45)

Up to now, in this section we have kept the functions $f(r),g(r)$ generic. The
only assumption made was about $f_{0}$, given by (9), i.e., we are considering
a solution which is a higher order correction to the Tangherlini black hole.
The most general condition that such solution must follow in order to verify
the equality (34) is given by (45).

We now verify that (45) is indeed fulfilled by the solution and potential for
tensor-type perturbations we considered in this article, with $f(r),\,g(r)$
given respectively by (8) and (25). Taking these functions, the values of
$r_{c},\,r_{t}$ are those given in (18) and (27), from which we see that

$r_{c}-r_{t}=2\lambda^{\prime}\frac{R_{H}^{2}}{r_{0}}.$ (46)

From (25) we may rewrite the definition (41) as

$\Delta
f(r)=2\lambda^{\prime}f(r)\left(\frac{R_{H}}{r}\right)^{2}\left[2(1-f(r))+rf^{\prime}(r)\right].$
(47)

Differentiating twice $\Delta f(r)$ and also using (39), after some algebra we
get

$\Delta
f^{\prime\prime}(r_{0})=4\lambda^{\prime}R_{H}^{2}\left[\frac{f_{0}^{\prime\prime}(r_{0})}{r_{0}^{2}}-2\frac{f_{0}(r_{0})}{r_{0}^{4}}+\frac{1}{2}\frac{f_{0}(r_{0})}{r_{0}}f_{0}^{\prime\prime\prime}(r_{0})\right]$
(48)

which, combined with (47), gives

$f_{0}(r_{0})\Delta f^{\prime\prime}(r_{0})-\Delta
f(r_{0})f_{0}^{\prime\prime}(r_{0})=\lambda^{\prime}R_{H}^{2}\left[2\frac{f_{0}^{2}(r_{0})}{r_{0}}f_{0}^{\prime\prime\prime}(r_{0})-8\frac{f_{0}^{2}(r_{0})}{r_{0}^{4}}\right].$
(49)

Using (46) and plugging the equation above in (45) yields

$\lambda^{\prime}\frac{2R_{H}^{2}}{r_{0}}\left[f_{0}^{2}(r_{0})f^{\prime\prime\prime}_{0}(r_{0})-f_{0}(r_{0})f_{0}^{\prime\prime\prime}(r_{0})-f_{0}^{\prime}(r_{0})f_{0}^{\prime\prime}(r_{0})\right]=0.$
(50)

We found a condition which is equivalent to (40), but much simpler!
Considering (39), equation (50) holds true if and only if $r_{0}$ is a root of

$\mathcal{F}(r):=f_{0}^{\prime\prime\prime}(r)\left(f_{0}(r)-1\right)-\frac{2}{r}f_{0}^{\prime\prime}(r).$
(51)

Using (8), we can explicitly write

$\mathcal{F}(r)=\left(\frac{R_{H}}{r}\right)^{d-3}(d-3)(d-2)\left[\frac{2}{r^{3}}-(d-1)\frac{R_{H}^{d-3}}{r^{d}}\right].$
(52)

It is easy to see that the only real positive root of $\mathcal{F}$ is

$r=\left(\frac{d-1}{2}\right)^{\frac{1}{d-3}}R_{H}=r_{0}.$ (53)

This completes the proof of the equality (34) between the imaginary components
of test and tensor gravitational eikonal quasinormal frequencies of solution
(8) to first order in $\lambda^{\prime}$, as we wanted.

##  4 Black hole shadow and quasinormal modes

In this section, we provide an analytical expression for the radius of the
shadow cast by the Callan-Myers-Perry black hole (8).

In a $d$-dimensional asymptotically flat black hole space time, an observer at
spatial infinity can set up an Euclidean spatial coordinate system. In this
framework, the $(d-2)$-dimensional hyperplane that simultaneously contains the
black hole and is normal to the line connecting it to the observer is called
the observer’s sky. Now, we consider a light source behind the black hole,
from the observer’s perspective. As the photons, emitted by the source, travel
through space time, some will get trapped inside the black hole photon sphere,
while some will eventually reach the observer. The absence of photons that
ended trapped by the black hole will make the observer perceive some sort of
shadow in the observer’s sky. For the case of a spherical symmetric black hole
like (1), this shadow forms a disk with radius given by [13]

$R_{S}=\frac{r_{c}}{\sqrt{f(r_{c})}}$ (54)

where $r_{c}$ is the solution of (13): the radius of the unstable null
circular geodesic, the photon sphere, given, as we saw, by (18).

The asymptotic flatness of the black hole is crucial for the result (54) to be
valid, in Einstein gravity and also in the presence of higher derivative
corrections [5]. For the radius of the shadow of AdS black holes there is a
correction term, even in the absence of higher derivatives [14].

Comparing (15) with (54), we can immediately notice a relation between the
shadow radius $R_{S}$ and the real part of the eikonal quasinormal frequencies
associated with test scalar fields, as noticed in [15]:

$R_{S}=\frac{1}{\Omega}.$ (55)

In the absence of higher derivative corrections, this relation extends to all
quasinormal frequencies, since their eikonal limits are all identical.

Using (19) and (55), we can relate the shadow radius $R_{S}$ of the Callan-
Myers-Perry black hole to the horizon radius $R_{H}$:

$R_{S}=\sqrt{\frac{2}{d-3}}\left(\frac{d-1}{2}\right)^{\frac{d-1}{2(d-3)}}\left[1+\frac{\lambda}{R_{H}^{2}}\frac{d-4}{2}\left(1-\left(\frac{2}{d-1}\right)^{\frac{d-1}{d-3}}\right)\right]R_{H}.$
(56)

Equation (56) can be inverted, in order to obtain $R_{H}$ expressed in terms
of $R_{S}$. This way, as we mentioned, we can obtain information about the
near-horizon geometry from the black hole shadow.

Physical quantities such as the black hole mass and temperature, which are
written as functions of $R_{H}$, can then be expressed in terms of $R_{S}$ and
be estimated, if the shadow radius of a black hole that can be described by
the metric (8) turns to be measured. Other physical quantities such as the
quasinormal frequencies can also be expressed as functions of mass or
temperature and can, therefore, be estimated knowing $R_{S}$. Since
quasinormal frequencies can also be independently measured, at least in some
limits, these estimates can be confronted with experimental results.

##  5 Conclusions and future directions

In this work we computed the quasinormal frequencies in the eikonal limit
corresponding to test fields and tensorial gravitational perturbations for the
simplest case of a $d-$dimensional spherically symmetric black hole solution
with string corrections given by the Callan-Myers-Perry black hole (8). We
expressed the results in terms of the black hole temperature and we concluded
that $\alpha^{\prime}$ corrections are positive for both real and imaginary
parts of the QNM frequencies. Knowing the real part of the frequency
corresponding to test fields, we obtained the radius of the black hole shadow.

The most surprising result we obtained was the equality (34) between the
imaginary parts of the two different quasinormal frequencies. The proof of the
equality (34) we provided in section 3.3 is valid for the solution (8) we
worked with, and for the potential (7) for tensorial perturbations. Both the
solution and the potential result from the specific string-theoretical action
(6). One may wonder if a property similar to (34) is valid in general for
other theories with higher derivative corrections (such as Lovelock theories),
or if it is an intrinsic property of string-theoretical solutions, or
eventually just of this solution. We obtained a condition (45) that a generic
spherically symmetric black hole solution with higher derivative corrections
should obey in order to verify (34).

In Lovelock theories, differently than in string theory, the higher-derivative
corrections are not seen as a perturbative expansion. This way, the parameter
controlling e.g. the Gauss-Bonnet term may be taken to arbitrary order in the
solution to the (nonlinear higher order) field equations, even if it appears
only to first order in the lagrangian. In [3] quasinormal modes in the eikonal
limit of Gauss-Bonnet $d-$dimensional black holes have been computed for
tensorial, vectorial and scalar gravitational perturbations, up to second
order in the Gauss-Bonnet parameter. Similarly to our case, no agreement was
found for the real parts of the quasinormal modes even to first order in the
Gauss-Bonnet parameter. The agreement between the imaginary parts was observed
for all types of perturbations (also for vectorial and scalar gravitational
perturbations), but only to first order in the Gauss-Bonnet parameter – not to
second order, where it failed. That is probably the reason the agreement just
to first order was overlooked in [3].

It is tempting to question if the agreement we found could be extended, for
the same solution, to vectorial and scalar gravitational perturbations. In
order to verify such agreement, one would need to determine the potentials
corresponding to such perturbations for solutions corresponding to the stringy
lagrangian (6). The determination of such potentials, the study of the
stability of the Callan-Myers-Perry black hole (8) under such perturbations
and the calculation of the corresponding quasinormal modes are by themselves
relevant problems; the eventual verification of an identity like (34) is an
interesting extra motivation. Since in string theory $\alpha^{\prime}$
corrections come in a perturbative expansion, all these studies we propose
here could also be extended to black holes with higher order (namely
$\alpha^{\prime 3}$) corrections. One could then figure out if the identity
(34) is a more general property in string theory of if is just a special
feature of quadratic curvature corrections.

If one could conclude that the identity (34) was indeed a general property,
that would mean that the imaginary part of quasinormal frequencies in the
eikonal limit would be equal to the principal Lyapunov exponent corresponding
to circular null geodesics. If that statement is true, it could help to
clarify the question of the stability of these black holes under these
perturbations in different dimensions, a topic with many open problems. This
is why we think this subject deserves further study.

#### Acknowledgements

This work has been supported by Fundação para a Ciência e a Tecnologia under
contracts IT (UIDB/50008/2020 and UIDP/50008/2020) and project CERN/FIS-
PAR/0023/2019.

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