# Palm: Predicting Actions through Language Models
@ Ego4D Long-Term Action Anticipation Challenge 2023

Daoji Huang Otmar Hilliges Luc Van Gool Xi Wang
ETH Zürich

###### Abstract

We present Palm, a solution to the Long-Term Action Anticipation (LTA) task
utilizing vision-language and large language models. Given an input video with
annotated action periods, the LTA task aims to predict possible future
actions. We hypothesize that an optimal solution should capture the
interdependency between past and future actions, and be able to infer future
actions based on the structure and dependency encoded in the past actions.
Large language models have demonstrated remarkable commonsense-based reasoning
ability. Inspired by that, Palm chains an image captioning model and a large
language model. It predicts future actions based on frame descriptions and
action labels extracted from the input videos. Our method outperforms other
participants in the EGO4D LTA challenge and achieves the best performance in
terms of action prediction. Our code is available at
https://github.com/DanDoge/Palm.

##  1 Introduction

Predicting future actions from egocentric videos is inherently a challenging
task given the uncertainty of the future, and very often there exist multiple
plausible action candidates and executive orders. Previous solutions [3, 2]
rely on Transformer-based feature extraction followed by multiple
classification heads which predict verbs and nouns independently. ICVAE[6]
proposes to use a predicted scenario, which summarizes the activity depicted
in the video, as a condition for action prediction, but such data-driven
models are inherently limited by the dataset and data distribution it is
trained on. We argue that a shared feature is not enough to predict future
actions as it cannot model the complex dependency of verbs and nouns within
the same action and the dependency across past and future actions. We
hypothesize that leveraging commonsense knowledge embedded in large language
models can help us discover the underlying structure and dependency of
different activities and make better predictions of future actions.

We propose Palm to predict actions through large language models (LLMs). We
first extract essential information from the input video through an image
captioning module and an action recognition module. The captions and
recognized actions are then used to create a prompt which is fed into a large
language model for future action anticipation. The variability of the
predictions is achieved by sampling multiple solutions from LLMs. We present
our approach in detail in Section 2, followed by experimental analysis in
Section 3.

Figure 1: Palm: Predicting Actions through Language Models. Given an input
video, Palm uses an image captioner and an action recognizer to describe its
content. The obtained image captions and action labels are combined to create
a prompt, which is inputted into a large language model. A sequence of future
actions is then predicted by the large language model.

##  2 Method

Our idea is to use natural language to describe past events and perform
reasoning and prediction in the semantic space, leveraging the commonsense
knowledge embedded in large language models. Figure 1 shows an overview of the
proposed method Palm.

###  2.1 Task Definition

The task of long-term action anticipation (LTA) involves predicting a sequence
of $20$ future actions in chronological order from an input video clip that
spans approximately 5 minutes. In addition, the segment boundaries of each
action in the videos are also provided as part of the input. Each action is
described by a (verb, noun) pair.

###  2.2 Prompt Design

We formulate the action anticipation task as a sentence completion task and
predict future actions by prompting large language models (LLMs) with past
action descriptions. Out-of-the-box large language models do not work well on
the LTA task in a zero-shot setting. Following the common practice, we design
the prompt with a few examples to maximize the reference given to the model.
Figure 2 shows the template of our prompts, which consists of an instruction,
a few examples, followed by a query. A prefix instruction is to adapt the
model to the LTA task. Each example contains $N$ full-sentence narrations and
corresponding $N$ action labels of (verb, noun) pairs. We also give the future
20 actions in the examples. Similarly, a query is composed of narrations and
action labels of the past $N^{\prime}$ actions. We use a larger number of
action observations $N^{\prime}>N$ in the query.

⬇ You are going to complete an action sequence, an action is one (verb, noun)
pair. A complete sequence consists of 28 actions. You will also be given a
text description of the past actions for reference. Narrations: [caption] x N
Actions: [action] x (N+Z) ... Narrations: [caption] x N Actions: [action] x
(N+Z) Narrations: [caption] x N’ Actions: [action] x N’
---
Figure 2: Prompt template we used to query large language models. The prompt
consists of an instruction paragraph, a few examples, and a query.

#### Past actions.

Accurate action descriptions are crucial as they implicitly provide the
corresponding context of the video input to large language models and give
possible action candidates. We adopt the Transformer model used in the Ego4D
[3] baseline to recognize the actions from the observed video clips. We use
video features extracted from EgoVLP [5], a vision-language model trained on
the Ego4D dataset and widely used in various Ego4D challenges. A Transformer
is used to extract a single feature vector from the input video features,
followed by two classification heads to predict the verbs and nouns.

#### Narrations.

The past action labels do not capture all the visual information and context,
e.g. the location of the actions or the background objects. Our idea is to use
an image captioning model to capture the action context, providing more
information related to the visual input. To do so, we use the given action
segments and generate a caption from each past action. More specifically, we
use the middle frame of each action segment and generate a caption starting
with the prefix ”A person is”.

#### Prompt selection.

Inspired by a maximal-marginal-relevance-based (MMR) exemplar selection
strategy [10], we aim to select a set of examples that are semantically
relevant to the query while being diverse enough to provide nonrepetitive
information. Specifically, we iteratively select a set of examples $p_{i}\in
T$ from the training set $D$ that are semantically close to the query prompt
$q$ but also diverse enough to provide additional information.

$p_{j}={\operatorname*{arg\,max}}_{p_{j}\in
D/T}\lambda\mathcal{S}(q,p_{j})-(1-\lambda)\max_{p_{i}\in
T}\mathcal{S}(p_{i},p_{j}),$ (1)

where $S$ measures the semantic similarity between two descriptions and the
parameter $\lambda$ is to balance between similarity and diversity. We use
MPNet [9] to extract textural embeddings and cosine similarity to measure
semantic similarity.

###  2.3 Language Model Inference

Large language models will output comma-separated verb-noun pairs following
the format in the prompt. We then extract the verb-noun pairs and append them
to our prediction if both verb and noun fall in the Ego4D label space. For
predictions that have less than 20 actions, we pad it with the last action. We
sample LLMs multiple times to obtain $K$ predictions.

##  3 Experiments

We train the action recognition model on version 2 of the Ego4D dataset and
use the EgoVLP [5] features of four past actions as input (dim=3072). Training
configurations are otherwise the same as in the Ego4D baseline. We use GPT-
Neo-1.3B [1] as the large language model and BLIP2-opt-2.7B [4] as the image
captioning model. The prompt includes 8 examples with $N=8$ past actions and
$Z=20$ future actions. The query has $N^{\prime}=\min(12,\mbox{action\\_idx})$
previous actions, where action_idx is the index of last observed action. In
other words, action_idx corresponds to the total number of actions contained
in the input video. Following the requirement of Ego4D, we predict an action
sequence starting from the action with $\mbox{action\\_idx}=7$ and generate
five sequences for each video.

#### Evaluation metric.

We evaluate our method using the metric of edit distance. Given $K$ predicted
sequences with each consisting $Z$ future actions
$\\{\\{(\hat{n}_{z,k},\hat{v}_{z,k})\\}_{z=1}^{Z}\\}_{k=1}^{K}$, the action
edit distance is measured as the best individual edit distance with the ground
truth action sequence $\\{(n_{z},v_{z})\\}_{z=1}^{Z})$,

$\Delta_{E_{tot}}=\min\limits_{k=1..K}\Delta_{E}(\\{(\hat{n}_{z,k},\hat{v}_{z,k})\\}_{z=1}^{Z},\\{(n_{z},v_{z})\\}_{z=1}^{Z}).$
(2)

Similar metrics can be defined for verb and noun predictions respectively.

###  3.1 Leaderboard results

Tabel 1 shows the comparison of our method to the Ego4D baseline on the test
set. We achieve first place in the leaderboard of the CVPR 2023 competitions.
Figure 3 shows one example. Notice the similarity between the query captions
and the retrieved examples and how the output captures the variability of
future actions.

Figure 3: Qualitative result of our framework with the input images, the prompt and one prediction. Ground truth labels are shown in the last row. Table 1: Results on the Ego4D test set. Method |  Verb ED $\downarrow$ |  Noun ED $\downarrow$ |  Action ED $\downarrow$
---|---|---|---
Ego4D baseline | 0.7169 | 0.7359 | 0.9253
Brunos | 0.7004 | 0.7092 | 0.9142
PaMsEgoAI | 0.6838 | 0.6785 | 0.8933
Palm | 0.6956 | 0.6506 | 0.8856

###  3.2 Ablation

We examine the effectiveness of our main contributions in this section. Here
the version 2 validation set is used. We compare two different caption models
vit-gpt2 [7] and BLIP2 [4] in Table 2. For overall actions, BLIP2 yields
better results, especially in terms of noun prediction. Further adding MMR
prompt selection yields the best result over all metrics.

Table 2: Ablation of captioning models. Model |  Prompt Selection |  Verb ED $\downarrow$ |  Noun ED $\downarrow$ |  Action ED $\downarrow$
---|---|---|---|---
vit-gpt2 [7] | random | 0.7168 | 0.7026 | 0.9046
BLIP2 [4] | random | 0.7235 | 0.6908 | 0.9025
BLIP2 [4] | MMR | 0.7165 | 0.6767 | 0.8934

Table 3 shows the importance of using narrations and actions in the prompts.
While both parts are beneficial, past actions seem to be more important in
providing object-related context, resulting in lower edit distance in terms of
noun and action predictions.

Table 3: Results of ablating the prompt content. Narrations | Actions |  Verb ED $\downarrow$ |  Noun ED $\downarrow$ |  Action ED $\downarrow$
---|---|---|---|---
✗ | ✓ | 0.7275 | 0.6798 | 0.9006
✓ | ✗ | 0.7152 | 0.7338 | 0.9206
✓ | ✓ | 0.7165 | 0.6767 | 0.8934

Table 4 shows the benefit of using a larger language model which allows for a
larger prompt size. We compare GPT-Neo-1.3B [1] with all variants in the GPT-2
[8] family where the number of examples is maximized within the token limit.
The increasing model size consistently lowers verb and noun errors and thus
has better action edit distance. With more model parameters and larger prompt
size, GPT-Neo-1.3B outperforms all GPT-2 variants. An even larger language
model would likely lead to further improvements.

Table 4: Results of ablating the language models. Model | # Example |  Verb ED $\downarrow$ |  Noun ED $\downarrow$ |  Action ED $\downarrow$
---|---|---|---|---
GPT-2-small | 3 | 0.7507 | 0.6851 | 0.9090
GPT-2-medium | 3 | 0.7369 | 0.6906 | 0.9074
GPT-2-large | 3 | 0.7228 | 0.6814 | 0.9027
GPT-2-xl | 3 | 0.7296 | 0.6795 | 0.8983
GPT-Neo-1.3B | 8 | 0.7165 | 0.6767 | 0.8934

###  3.3 Limitations

Our model is limited by the quality of recognized past actions. More
specifically, edit distance gets larger when the accuracy of recognized
actions is lower. Table 5 shows a strong negative correlation between the
accuracy of past action recognition and edit distance. We measure the accuracy
as the mean accuracy of the actions, i.e. the percentage of recognized actions
being equal to ground truth actions. Figure 4 shows typical failure cases
where a wrongly recognized noun leads to incorrect prediction, resulting in
edit-distance being 1, i.e. all predicted actions are incorrect.

Figure 4:  Failure case examples when action recognition fails.  Table 5: Linear regression coefficient of action recognition accuracy and edit distance. The standard deviations of coefficients are shown in brackets. | Verb Accuracy | Noun Accuracy
---|---|---
Verb ED | -0.1298 (0.003) | 0.0160 (0.002)
Noun ED | -0.1012 (0.005) | -0.1835 (0.004)
Action ED | -0.1057 (0.002) | -0.0725 (0.002)

##  4 Conclusion

We present Palm, a framework that leverages pretrained vision-language and
large language models for long-term action anticipation. We demonstrate the
effectiveness of performing the LTA task in the language domain and the
benefits of using large vision-language and language models. However, a pre-
trained large language model might not be able to predict future action
sequences when past action descriptions are incorrect.

## References

  * [1] Sid Black, Leo Gao, Phil Wang, Connor Leahy, and Stella Biderman.  GPT-Neo: Large Scale Autoregressive Language Modeling with Mesh-Tensorflow, Mar. 2021.
  * [2] Srijan Das and Michael S. Ryoo.  Video + clip baseline for ego4d long-term action anticipation, 2022.
  * [3] Kristen Grauman, Andrew Westbury, Eugene Byrne, Zachary Chavis, Antonino Furnari, Rohit Girdhar, Jackson Hamburger, Hao Jiang, Miao Liu, Xingyu Liu, et al.  Ego4d: Around the world in 3,000 hours of egocentric video.  arXiv preprint arXiv:2110.07058, 2021.
  * [4] Junnan Li, Dongxu Li, Silvio Savarese, and Steven Hoi.  BLIP-2: bootstrapping language-image pre-training with frozen image encoders and large language models.  In ICML, 2023.
  * [5] Kevin Qinghong Lin, Alex Jinpeng Wang, Mattia Soldan, Michael Wray, Rui Yan, Eric Zhongcong Xu, Difei Gao, Rongcheng Tu, Wenzhe Zhao, Weijie Kong, et al.  Egocentric video-language pretraining.  arXiv preprint arXiv:2206.01670, 2022.
  * [6] Esteve Valls Mascaro, Hyemin Ahn, and Dongheui Lee.  Intention-conditioned long-term human egocentric action anticipation.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pages 6048–6057, January 2023.
  * [7] NLP Connect.  vit-gpt2-image-captioning (revision 0e334c7), 2022.
  * [8] Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever.  Language models are unsupervised multitask learners.  2019\.
  * [9] Kaitao Song, Xu Tan, Tao Qin, Jianfeng Lu, and Tie-Yan Liu.  Mpnet: Masked and permuted pre-training for language understanding, 2020\.
  * [10] Xi Ye, Srini Iyer, Asli Celikyilmaz, Ves Stoyanov, Greg Durrett, and Ramakanth Pasunuru.  Complementary explanations for effective in-context learning.  ArXiv, abs/2211.13892, 2022.


# The Role of Tensor Force in Heavy-Ion Fusion Dynamics

Lu Guo1<EMAIL_ADDRESS>Cédric Simenel2<EMAIL_ADDRESS>Long Shi1
Chong Yu1 1 School of Physics, University of Chinese Academy of Sciences,
Beijing 100049, China  2 Department of Nuclear Physics, RSPE, Australian
National University, Canberra, Australia

###### Abstract

The tensor force is implemented into the time-dependent Hartree-Fock (TDHF)
theory so that both exotic and stable collision partners, as well as their
dynamics in heavy-ion fusion, can be described microscopically. The role of
tensor force on fusion dynamics is systematically investigated for
${}^{40}\mathrm{Ca}+\mathrm{{}^{40}Ca}$,
${}^{40}\mathrm{Ca}+\mathrm{{}^{48}Ca}$,
${}^{48}\mathrm{Ca}+\mathrm{{}^{48}Ca}$,
${}^{48}\mathrm{Ca}+\mathrm{{}^{56}Ni}$, and
${}^{56}\mathrm{Ni}+\mathrm{{}^{56}Ni}$ reactions which vary by the total
number of spin-unsaturated magic numbers in target and projectile. A notable
effect on fusion barriers and cross sections is observed by the inclusion of
tensor force. The origin of this effect is analyzed. The influence of
isoscalar and isovector tensor terms is investigated with the T$IJ$ forces.
These effects of tensor force in fusion dynamics are essentially attributed to
the shift of low-lying vibration states of colliding partners and nucleon
transfer in the asymmetric reactions. Our calculations of above-barrier fusion
cross sections also show that tensor force does not significantly affect the
dynamical dissipation at near-barrier energies.

Nuclear fusion dynamics, Tensor force, Low-lying vibrations, Nucleon transfer,
Time-dependent Hartree-Fock theory

The tensor interaction is of great interests in nuclear physics. It is indeed
crucial to explain the properties of the deuteron. It plays a significant role
in nuclear structure, in particular in exotic nuclei far from stability with
extreme ratios between proton number and neutron number. The fact that exotic
nuclei may show distinct characteristics from those seen in usual stable
nuclei could in part be attributed to tensor interaction. In particular, in
nuclear dynamics, the tensor force changes not only the spin-orbit splitting,
but also the intrinsic excitations which may give rise to dynamical effects
and are more complicated than those arising from simple shell evolution. The
introduction of tensor force improved the systematic agreement between model
predictions and experimental data in the shell evolution of exotic nuclei
Otsuka  _et al._ (2006), spin-orbit splitting Colò  _et al._ (2007), Gamow-
Teller and charge exchange spin-dipole excitations Bai  _et al._ (2010). In
spite of these indications in nuclear structure, most calculations for
reaction dynamics ignored tensor force for decades both in macroscopic-
microscopic approaches and self-consistent mean-field methods.

The microscopic mechanisms involved in low-energy heavy-ion collisions
originate from the effective interaction between the nucleons, the
understanding of which remains one of the main challenges in nuclear physics.
Microscopic models of heavy-ion collisions Negele (1982); Wen  _et al._ (2013,
2014) can then be used to test some properties of the nuclear force. In such
collisions, various couplings between the relative motion and the internal
degrees of freedom of colliding partners complicate the description of the
reaction mechanisms. Through coupling, these internal degrees of freedom,
which include low-lying vibrations Morton  _et al._ (1994); Stefanini  _et
al._ (1995); Simenel  _et al._ (2013a), nucleon transfer Washiyama  _et al._
(2009); Simenel (2010); Sekizawa (2017), rotations Leigh  _et al._ (1995), and
high-lying giant resonances Diaz-Torres  _et al._ (2008) have been shown to
modify the reaction dynamics and, subsequently, the outcome of the reaction
itself Dasso  _et al._ (1985).

The effect of tensor force on these couplings in heavy-ion fusion dynamics is
an open question which we address in the present study. In particular, low-
lying collective states and nucleon transfer, which have the strongest impact
on near-barrier fusion, are very sensitive to the underlying shell structure,
which is in turn affected by the tensor force. Another possible effect of
tensor force is to modify dissipation in heavy-ion collisions. The latter has
been the subject of theoretical studies at energies well above the Coulomb
barrier Dai  _et al._ (2014a); Stevenson  _et al._ (2016); Shi and Guo (2017).
To our knowledge, however, the interplay between the tensor force and
dissipation at near-barrier energies has never been investigated. Our work is
the first attempt at a study of the effects induced by tensor force on the
low-energy fusion dynamics.

In order to study the role of tensor force in fusion dynamics, it is desirable
to develop a theoretical framework in which the dynamical effects, such as
couplings and dissipation, are all treated on the same footing. To this end,
we incorporated the full tensor terms of Skyrme energy density functional
(EDF) into the microscopic time-dependent Hartree-Fock (TDHF) approach, which
automatically includes one-body dissipation as well as the couplings between
collective motion and internal degrees of freedom at the mean-field level.
TDHF theory is a well-defined microscopic framework and extensively applied to
the study of heavy-ion collisions Umar and Oberacker (2006); Guo  _et al._
(2007, 2008); Simenel (2011); Guo and Nakatsukasa (2012); Sekizawa and Yabana
(2013); Umar  _et al._ (2014); Vo-Phuoc  _et al._ (2016); Wang and Guo (2016);
Umar  _et al._ (2017); Yu and Guo (2017); Williams  _et al._ (2018) and
vibration dynamics Lacroix  _et al._ (2004); Maruhn  _et al._ (2005);
Nakatsukasa and Yabana (2005); Umar and Oberacker (2005); Reinhard  _et al._
(2007); Simenel and Chomaz (2009); Fracasso  _et al._ (2012); Avez and Simenel
(2013). For recent reviews, see Refs. Simenel (2012); Nakatsukasa  _et al._
(2016).

Most TDHF calculations employ Skyrme effective interaction Skyrme (1956), in
which the two-body tensor force was proposed in its original form as

$\displaystyle\begin{split}v_{T}&=\dfrac{t_{\mathrm{e}}}{2}\bigg{\\{}\big{[}3({\sigma}_{\mathrm{1}}\cdot\mathbf{k}^{\prime})({\sigma}_{\mathrm{2}}\cdot\mathbf{k}^{\prime})-({\sigma}_{\mathrm{1}}\cdot{\sigma}_{\mathrm{2}})\mathbf{k}^{\prime\mathrm{2}}\big{]}\delta(\mathbf{r}_{\mathrm{1}}-\mathbf{r}_{\mathrm{2}})\\\
&+\delta(\mathbf{r}_{\mathrm{1}}-\mathbf{r}_{\mathrm{2}})\big{[}3({\sigma}_{\mathrm{1}}\cdot\mathbf{k})({\sigma}_{\mathrm{2}}\cdot\mathbf{k})-({\sigma}_{\mathrm{1}}\cdot{\sigma}_{\mathrm{2}})\mathbf{k}^{\mathrm{2}}\big{]}\bigg{\\}}\\\
&+t_{\mathrm{o}}\bigg{\\{}3({\sigma}_{\mathrm{1}}\cdot\mathbf{k}^{\prime})\delta(\mathbf{r}_{\mathrm{1}}-\mathbf{r}_{\mathrm{2}})({\sigma}_{\mathrm{2}}\cdot\mathbf{k})-({\sigma}_{\mathrm{1}}\cdot{\sigma}_{\mathrm{2}})\mathbf{k}^{\prime}\delta(\mathbf{r}_{\mathrm{1}}-\mathbf{r}_{\mathrm{2}})\mathbf{k}\bigg{\\}}.\end{split}$
(1)

The coupling constants $t_{\textrm{e}}$ and $t_{\textrm{o}}$ represent the
strengths of triplet-even and triplet-odd tensor interactions, respectively.
The operator $\mathbf{k}=\frac{1}{2i}(\nabla_{1}-\nabla_{2})$ acts on the
right, and
$\mathbf{k}^{\prime}=-\frac{1}{2i}(\nabla^{\prime}_{1}-\nabla^{\prime}_{2})$
acts on the left.

With the inclusion of central, spin-orbit and tensor forces, the full version
of Skyrme EDF is expressed as

$\displaystyle\begin{split}\mathcal{H}&=\mathcal{H}_{0}+\sum_{\rm{t=0,1}}\Big{\\{}C_{\rm{t}}^{\rm{s}}\mathbf{s}_{\rm{t}}^{2}+C_{\rm{t}}^{\Delta{s}}\mathbf{s}_{\rm{t}}\cdot\Delta\mathbf{s}_{\rm{t}}+C_{\rm{t}}^{\nabla
s}(\nabla\cdot\mathbf{s}_{\rm{t}})^{2}\\\
&+C_{\rm{t}}^{F}\big{(}\mathbf{s}_{\rm{t}}\cdot\mathbf{F}_{\rm{t}}-\frac{1}{2}\big{(}\sum_{\mu=x}^{z}J_{\rm{t},\mu\mu}\big{)}^{2}-\frac{1}{2}\sum_{\mu,\nu=x}^{z}J_{\rm{t},\mu\nu}J_{\rm{t},\nu\mu}\big{)}\\\
&+C_{\rm{t}}^{\rm{T}}\big{(}\mathbf{s}_{\rm{t}}\cdot\mathbf{T}_{\rm{t}}-\sum_{\mu,\nu=x}^{z}J_{\rm{t},\mu\nu}J_{\rm{t},\mu\nu}\big{)}\Big{\\}},\end{split}$
(2)

where $\rm{t}$ denotes the isospin and $\mathcal{H}_{0}$ is the simplified
functional without tensor terms as used in Sky3D code Maruhn  _et al._ (2014)
and most TDHF calculations. The coupling constants $C$ have been defined in
Refs. Lesinski  _et al._ (2007); Davesne  _et al._ (2009). Due to the
computational complexity, various approximations to nuclear force have been
employed, which restrict the number of degrees of freedom accessible during a
collision, and hence the nature and degree of dissipation dynamics. For
instance, the inclusion of spin-orbit interaction solved the conflict between
TDHF predictions and experiments Umar  _et al._ (1986), and turned out to play
an important role in fusion and dissipation dynamics Maruhn  _et al._ (2006);
Dai  _et al._ (2014b). In our code, we incorporated all the time-even and
time-odd density dependence of the tensor terms shown in Eq. (2). Our TDHF
calculations have been performed in a fully three-dimensional Cartesian space
and without any symmetry restriction. As pointed out in Refs. Lesinski  _et
al._ (2007); Stevenson  _et al._ (2016), the terms containing the gradient of
spin density may cause the spin instability in both nuclear structure and
reaction studies, so we set $C_{\rm{t}}^{\Delta{s}}=C_{\rm{t}}^{\nabla s}=0$
in our calculations.

The Skyrme tensor force has been constructed in two ways. One is to add
perturbatively to the existing standard interactions, for instance, the
existing Skyrme parameter SLy5 Chabanat  _et al._ (1998) plus tensor force,
denoted as SLy5t Colò  _et al._ (2007). The comparison between calculations
with SLy5 and SLy5t addresses the question on how much of the changes is
caused by tensor force itself. Another approach is to readjust the full set of
Skyrme parameters self-consistently. This strategy has been adopted in Ref.
Lesinski  _et al._ (2007) and led to the set of T$IJ$ parametrizations with a
wide range of isoscalar and isovector tensor couplings. Due to its fitting
strategy, the contributions from the tensor force and the rearrangement of all
other terms could be physically entangled.

The contribution of tensor force is expected to be nearly zero for the ground
state of spin-saturated nuclei, whereas it might be significant for the nuclei
with one level out of two spin-orbit partners filled. We have thus chosen five
representative reactions ${}^{40}\mathrm{Ca}+\mathrm{{}^{40}Ca}$,
${}^{40}\mathrm{Ca}+\mathrm{{}^{48}Ca}$,
${}^{48}\mathrm{Ca}+\mathrm{{}^{48}Ca}$,
${}^{48}\mathrm{Ca}+\mathrm{{}^{56}Ni}$, and
${}^{56}\mathrm{Ni}+\mathrm{{}^{56}Ni}$ quantified by $N_{s}=0,1,2,3,$ and 4,
respectively, where $N_{s}$ is the total number of spin-unsaturated magic
number in target and projectile. In these collisions, the reaction partners
are closed-shell corresponding to 20 (spin-saturated) and 28 (spin-
unsaturated) neutron or proton magic numbers.

Figure 1: (Color online.) Difference of fusion barrier heights with tensor
(SLy5t) and without tensor (SLy5) computed with the FHF static (open circles)
and TDHF dynamical (solid circles) methods.

The effect of tensor force on fusion could have static (e.g., modification of
ground-state density) and dynamical (e.g., modification of couplings and
dissipation) origins. In order to disentangle the contributions from static
and dynamical effects, both frozen Hartree-Fock (FHF), where the nuclei keep
their ground-state densities Brueckner  _et al._ (1968); Simenel and Avez
(2008); Washiyama and Lacroix (2008); Simenel  _et al._ (2013a); Shi and Guo
(2017); Simenel  _et al._ (2017), and TDHF calculations of fusion barriers
have been performed. Dynamical fusion barriers can be computed directly from
TDHF Simenel and Avez (2008); Washiyama and Lacroix (2008); Shi and Guo
(2017), or with the density-constrained TDHF technique Umar and Oberacker
(2006).

Figure 1 shows the variation of static (FHF) and dynamic (TDHF) fusion barrier
heights due to the tensor force in SLy5t as a function of $N_{s}$. For the
spin-saturated reaction ${}^{40}\mathrm{Ca}+\mathrm{{}^{40}Ca}$, the static
barrier obtained from FHF calculations with SLy5t is same as with SLy5, as
expected, due to the nearly zero contributions of tensor force to the ground
state of spin-saturated nucleus ${}^{40}\mathrm{Ca}$. For the other spin-
unsaturated reactions, the static (FHF) barrier with SLy5t is systematically
higher than SLy5. This indicates a fusion hindrance due to tensor force in
this mass region.

The variation of dynamical barriers due to tensor force presents different
behavior from the static results, as shown in Fig. 1. The dynamical barriers
appear staggering as a function of $N_{s}$. In particular, a distinguishing
characteristics is the lower dynamical barrier by the inclusion of tensor
force in ${}^{48}\mathrm{Ca}+\mathrm{{}^{56}Ni}$. This indicates the
importance of dynamical effects which can remarkably modify the barrier
height. These dynamical effects, which could be low-lying vibration states,
nucleon transfer, and dynamical dissipation, strongly affect the near-barrier
fusion. In the following, we will investigate these three dynamical effects
and their interplay in the role of tensor force on fusion dynamics. For
symmetric reactions, the role of nucleon transfer is minimized. Therefore, the
effect of tensor force on the vibrational modes will be first studied focusing
on symmetric reactions. In a second step, the role of tensor force in nucleon
transfer will be investigated in asymmetric reactions.

The effect of tensor terms on the isovector giant dipole resonance has been
studied in Ref. Fracasso  _et al._ (2012). Here, we focus on the low-lying
quadrupole and octupole modes which have a stronger impact on the near-barrier
fusion. The peak energies of low-lying vibration states, RPA results Cao  _et
al._ (2009), and experimental data Raman  _et al._ (1987); Kibédi and Spear
(2002) are listed in Tab 1. In the quadrupole excitation of
${}^{40}\mathrm{Ca}$, we did not find the low-lying collective states. This is
consistent with the fact that, in this nucleus, the $2_{1}^{+}$ state does not
exhibit a strong increase of collectivity as compared to the single-particle
picture Raman  _et al._ (1987). The $3_{1}^{-}$ state of octupole excitation
in ${}^{40}\mathrm{Ca}$ is pushed 0.13 MeV down by tensor force. This
indicates the tensor force has small effect on the vibration states of
${}^{40}\mathrm{Ca}$ and, subsequently, the fusion dynamics through the
couplings. This is consistent with the barrier in
${}^{40}\mathrm{Ca}$+${}^{40}\mathrm{Ca}$ as shown in Fig. 1. The other states
are at higher energies and have a smaller strength. This $3_{1}^{-}$ state is
then expected to have a much stronger effect on the near-barrier reaction
mechanisms than the other octupole states. We also observe the difference in
the peak energies of vibration states between TDHF and RPA approaches,
although the trend of energy variance induced by tensor force is the same.
This could possibly arise from the different treatment of center-of-mass
(c.m.) corrections. Since c.m. corrections lead to ambiguities in TDHF
calculations of heavy-ion collisions, they are neglected in present vibration
studies to allow a consistent treatment of structure and dynamics of colliding
partners. The inclusion of c.m. corrections in vibration states and a detailed
comparison between TDHF and RPA calculations will be the subject of future
works.

Table 1: Peak energies (in MeV) of low-lying vibration states in ${}^{40}\mathrm{Ca}$, $\mathrm{{}^{48}Ca}$, and $\mathrm{{}^{56}Ni}$ with and without tensor force. The RPA results Cao  _et al._ (2009) and experimental data Raman  _et al._ (1987); Kibédi and Spear (2002) are also shown. Nucleus | State | Method |  $E_{\nu}$(SLy5) |  $E_{\nu}$(SLy5t) | Expt.
---|---|---|---|---|---
${}^{40}\mathrm{Ca}$ | $3_{1}^{-}$ | This work | 3.64 | 3.51 | 3.74
|  | RPA | 3.78 | 3.02 |
${}^{48}\mathrm{Ca}$ | $2_{1}^{+}$ | This work | 2.98 | 4.03 | 3.83
|  | RPA | 3.05 | 3.88 |
| $3_{1}^{-}$ | This work | 5.13 | 6.46 | 4.51
|  | RPA | 4.78 | 6.16 |
${}^{56}\mathrm{Ni}$ | $2_{1}^{+}$ | This work | 2.30 | 2.76 | 2.70
| $3_{1}^{-}$ | This work | 9.53 | 9.68 |

The situation is quite different for the spin-unsaturated nucleus
${}^{48}\mathrm{Ca}$. As seen in Fig. LABEL:vibration-Ca48, the tensor force
shifts both the quadrupole and octupole strength distributions to higher
energies. In particular, the energy of $2_{1}^{+}$ ($3_{1}^{-}$) state is
pushed 1.05 MeV (1.33 MeV) up as shown in Tab. 1. Fusion is dynamically
hindered by tensor force in ${}^{48}\mathrm{Ca}$-involved reactions due to the
increase of the energy of the low-lying collective vibrations. This is a
plausible explanation for the increase of dynamical barrier due to tensor
force in the symmetric system ${}^{48}\mathrm{Ca}$+${}^{48}\mathrm{Ca}$ , as
shown in Fig. 1. We also observe that the $2_{1}^{+}$ state by the inclusion
of tensor force in SLy5t is reasonably close to the experimental value of 3.83
MeV Raman  _et al._ (1987). However, the $3_{1}^{-}$ state apparently
overestimates the experiment Kibédi and Spear (2002), which may be due to the
limitation of mean-field approximation and the choice of the interaction.

Table 2: Average number of transferred neutrons (n) and protons (p) with and without tensor force in asymmetric central collisions at 0.1 MeV below the TDHF barrier. The $+(-)$ sign indicates transfer from the light (heavy) fragment.  Reactions |  $\mathrm{N_{n}}$(SLy5) |  $\mathrm{N_{n}}$(SLy5t) |  $\mathrm{N_{p}}$(SLy5) |  $\mathrm{N_{p}}$(SLy5t)
---|---|---|---|---
${}^{40}\mathrm{Ca}$+${}^{48}\mathrm{Ca}$ | -0.85 | -0.25 | +0.95 | +0.94
${}^{48}\mathrm{Ca}$+${}^{56}\mathrm{Ni}$ | +1.37 | +1.88 | -0.29 | -1.06

The similar effect of tensor force appears in the exotic nucleus
${}^{56}\mathrm{Ni}$, but the effect is much weaker than in
${}^{48}\mathrm{Ca}$ due to the underlying shell structure. This weaker effect
results in a smaller increase of dynamical barrier in
${}^{56}\mathrm{Ni}$+${}^{56}\mathrm{Ni}$ compared with
${}^{48}\mathrm{Ca}$+${}^{48}\mathrm{Ca}$, as shown in Fig 1.

For the asymmetric reactions, both the low-lying vibrations and nucleon
transfer are crucial for the fusion dynamics. As the tensor force affects the
single-particle levels, this could clearly affect transfer and then the TDHF
fusion barrier. We calculate the transferred nucleon numbers with and without
tensor force at the sub-barrier collisions, as listed in Tab. 2. For
${}^{40}\mathrm{Ca}$+${}^{48}\mathrm{Ca}$, less neutrons are transferred from
${}^{48}\mathrm{Ca}$ to ${}^{40}\mathrm{Ca}$ in SLy5t, and the transferred
protons going from ${}^{40}\mathrm{Ca}$ to ${}^{48}\mathrm{Ca}$ are nearly
same in SLy5 and SLy5t. This small transfer (less than one neutron in average)
results in a negligible effect on the fusion barrier. As observed in Fig. 1,
the small variation of dynamical barrier in this reaction indicates the
vibrational couplings are dominated by the $3_{1}^{-}$ state in
${}^{40}\mathrm{Ca}$, which is slightly affected by tensor force. In addition,
the fact that there is only one spin-unsaturated magic number also accounts
for the small effect of tensor force both in static and dynamic barriers.

The nucleon transfer presents distinct behavior in
${}^{48}\mathrm{Ca}$+${}^{56}\mathrm{Ni}$. More neutrons are transferred from
${}^{48}\mathrm{Ca}$ to ${}^{56}\mathrm{Ni}$ in SLy5t because the energy gap
between the most high-lying occupied orbit $1f_{7/2}$ in ${}^{48}\mathrm{Ca}$
and the most low-lying unoccupied orbit $2p_{3/2}$ in ${}^{56}\mathrm{Ni}$ is
reduced by about 1 MeV by the tensor force. The large neutron transfer
decreases the dynamical barrier Jiang  _et al._ (2014). Also more protons are
observed to transfer from ${}^{56}\mathrm{Ni}$ to ${}^{48}\mathrm{Ca}$ in
SLy5t. The proton transfer is more subtle, as the nuclear part lowers the
barrier, while the Coulomb part increases it Godbey  _et al._ (2017). Another
effect, the couplings to the low-lying vibration states in
${}^{48}\mathrm{Ca}$ and ${}^{56}\mathrm{Ni}$, increases the dynamical
barrier. The two dynamical effects, including the modifications of vibration
states and nucleon transfer, are opposite in fusion dynamics, and hence cancel
each other to some extent. The lower dynamical barrier in SLy5t as seen in
Fig. 1 indicates the nucleon transfer plays a dominant role in
${}^{48}\mathrm{Ca}$+${}^{56}\mathrm{Ni}$.

Figure 3: (Color online.) Static FHF and dynamic TDHF fusion barriers with six
Skyrme forces as well as experimental value Stefanini  _et al._ (2009) for
${}^{48}\mathrm{Ca}+\mathrm{{}^{48}Ca}$ in the upper panel, and the peak
energies of $2_{1}^{+}$ and $3_{1}^{-}$ states in the lower panel.

We now proceed to a comparison among the results of various forces, for which
the coupling constants are listed in Tab. 3, for the reaction
${}^{48}\mathrm{Ca}+\mathrm{{}^{48}Ca}$ as an example. The upper panel of Fig.
3 shows that the static barrier systematically overestimates the experimental
data Stefanini  _et al._ (2009) and dynamical barrier, because the couplings
to the internal degrees of freedom have been neglected in static FHF
calculations. The dynamical barrier by the inclusion of tensor force in SLy5t
are in better agreement than SLy5. Note that the improvement is rather
remarkable given the fact that no free parameters are adjusted to reproduce
the dynamical properties, e.g., fusion barrier and cross section in TDHF
calculations. However, one should note that this improvement could be spurious
as it is partly due to the pushing up of $E_{3_{1}^{-}}$ in
${}^{48}\mathrm{Ca}$ which disagrees with experimental data. Nevertheless,
this shows that the tensor force plays a significant role on fusion, which is
the main focus of this work. A better reproduction of experimental data
requires an improvement of the fitting procedure of the tensor terms, which is
beyond the scope of this work. For T$22$ and T$44$ the dynamical barriers are
similar, indicating the isoscalar tensor coupling has negligible effect in
this reaction. By comparing the results with T$26$, T$44$, and T$62$, the
barrier height increases as the isovector tensor coupling decreases. This
clear dependence of isoscalar and isovector tensor coupling may be due to the
interplay between tensor terms and rearrangement of mean-field. Although the
isoscalar tensor with the proton and neutron single particle spectrum moving
in the same way could affect these vibration modes, the effect seems to be
canceled by the refitting of the parameters. However, the refitting does not
absorb the variations of isovector tensor in the same way. Another interesting
observation is, as shown in the lower panel of Fig. 3, the peak energies of
$2_{1}^{+}$ and $3_{1}^{-}$ states in ${}^{48}\mathrm{Ca}$ display a very
similar dependence on the tensor coupling as the dynamical TDHF barrier
behaves. This indicates the fusion dynamics in symmetric reactions is
dominated by the low-lying vibrations of colliding partners. Due to the subtle
interplay between nuclear structure and reaction dynamics, these T$IJ$ forces
may have distinct impact on the different heavy-ion collisions.

Table 3: Isoscalar and isovector spin-current coupling constants in units of MeVfm5. Force | $\mathrm{C}^{\mathrm{J}}_{0}$ | $\mathrm{C}^{\mathrm{J}}_{1}$
---|---|---
T22 | 0 | 0
T26 | 120 | 120
T44 | 120 | 0
T62 | 120 | -120
SLy5 | 15.65 | 64.55
SLy5t | -19.35 | -70.45

To investigate the importance of tensor terms on dynamical effects such as
dissipation, we computed above-barrier fusion cross sections in
$\mathrm{{}^{40}Ca}+\mathrm{{}^{40}Ca}$ and
${}^{48}\mathrm{Ca}+\mathrm{{}^{48}Ca}$. For the spin-saturated reaction
$\mathrm{{}^{40}Ca}+\mathrm{{}^{40}Ca}$, the fusion cross sections with and
without tensor force are quite close to each other, which is consistent with
the observation of the same fusion barriers between SLy5 and SLy5t as shown in
Fig. 1. We observe the calculated cross sections overestimate the experimental
data Montagnoli  _et al._ (2012) by about 20%. Similar overestimations have
been obtained for other systems like $\mathrm{{}^{16}O}+\mathrm{{}^{208}Pb}$
Simenel and Avez (2008); Simenel  _et al._ (2013b).

Figure 4: (Color online.) The fusion cross sections calculated with various
forces as well as the older (denoted as Exp1) Trotta  _et al._ (2001) and
recent experimental data (Exp2) Stefanini  _et al._ (2009) for
$\mathrm{{}^{48}Ca}+\mathrm{{}^{48}Ca}$ in the upper panel, and the deviation
of theoretical fusion cross section from the recent experiment Stefanini  _et
al._ (2009), defined as
$\mathrm{P_{\sigma}}=({\sigma_{\mathrm{th}}-\sigma_{\mathrm{exp}}})/{\sigma_{\mathrm{exp}}}$
in the lower panel.

The fusion cross sections in 48Ca$+^{48}$Ca with various forces together with
the experimental data Trotta  _et al._ (2001); Stefanini  _et al._ (2009) are
given in the upper panel of Fig. 4. The older experimental data (denoted as
Exp1) Trotta  _et al._ (2001) is systematically higher than the recent data
(Exp2) Stefanini  _et al._ (2009). The variation between the calculations is
globally within the fluctuations of experimental data. To discriminate the
variation between the calculations, the deviation $P_{\sigma}$ is shown in the
lower panel of Fig. 4. We see that some forces lead to results in better
agreement than others. In particular, the results in SLy5t lead to a much
better agreement than in SLy5. This observation can be traced back to the
increase of the dynamical barrier in SLy5t as seen in Fig. 3(a). The T$62$
interaction also leads to very similar cross sections as SLy5t, which is again
interpreted by the fact that the barriers with T$62$ and SLy5t are very close
(see Fig. 3(a)). Another interesting observation is that the theoretical cross
sections above barrier have all very similar slopes, whatever the forces. This
is a strong indication that dissipation is not significantly affected by the
tensor force at the near-barrier energies, and the effect on the fusion cross
section mainly comes from the change of fusion barrier height.

In summary, we incorporate the full tensor force into TDHF calculations so
that both exotic and stable collision partners, as well as their dynamics, can
be described microscopically. We systematically investigate the role of tensor
force on fusion dynamics, and a notable effect is observed in the spin-
unsaturated reactions. The isoscalar and isovector tensor terms in T$IJ$
forces play distinct role in the fusion dynamics due to the subtle interplay
between nuclear structure and reaction dynamics. Since the near-barrier fusion
can be strongly affected by the couplings to intrinsic degrees of freedom of
the collision partners, three dynamical origins, including the vibration
states, nucleon transfer, and dynamical dissipation, are investigated to
interpret the observed effects of tensor force in fusion dynamics. In the
symmetric collisions, the pushing-up of low-lying vibration states is
responsible for the observed fusion hindrance. In the asymmetric collisions,
the nucleon transfer and modification of low-lying vibration states may play a
similar or opposite effect on the fusion dynamics, depending on the underlying
shell structure of collision partners. The above-barrier fusion cross sections
do not show significant effect of the tensor force on the dynamical
dissipation.

##  I Acknowledgments

This work is partly supported by NSF of China (Grants No. 11175252 and
11575189), Presidential Fund of UCAS, NSFC-JSPS International Cooperation
Program (Grant No. 11711540016), and by the Australian Research Council Grants
No. FT120100760 and DP180100497. The computations in present work were
performed on the High-performance Computing Clusters of SKLTP/ITP-CAS and
Tianhe-1A supercomputer of the CNSC in Tianjin. Part of the calculations have
been performed on the NCI National Facility in Canberra, Australia, which is
supported by the Australian Commonwealth Government.

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# New constraints on primordial features from the galaxy two-point correlation
function

Mario Ballardini<EMAIL_ADDRESS>Dipartimento di Fisica e
Astronomia, Alma Mater Studiorum Università di Bologna, via Gobetti 93/2,
I-40129 Bologna, Italy  INAF/OAS Bologna, via Piero Gobetti 93/3, I-40129
Bologna, Italy  INFN, Sezione di Bologna, viale C. Berti Pichat 6/2, I-40127
Bologna, Italy  Department of Physics & Astronomy, University of the Western
Cape, Cape Town 7535, South Africa  Fabio Finelli  INAF/OAS Bologna, via Piero
Gobetti 93/3, I-40129 Bologna, Italy  INFN, Sezione di Bologna, viale C. Berti
Pichat 6/2, I-40127 Bologna, Italy  Federico Marulli  Dipartimento di Fisica e
Astronomia, Alma Mater Studiorum Università di Bologna, via Gobetti 93/2,
I-40129 Bologna, Italy  INAF/OAS Bologna, via Piero Gobetti 93/3, I-40129
Bologna, Italy  INFN, Sezione di Bologna, viale C. Berti Pichat 6/2, I-40127
Bologna, Italy  Lauro Moscardini  Dipartimento di Fisica e Astronomia, Alma
Mater Studiorum Università di Bologna, via Gobetti 93/2, I-40129 Bologna,
Italy  INAF/OAS Bologna, via Piero Gobetti 93/3, I-40129 Bologna, Italy  INFN,
Sezione di Bologna, viale C. Berti Pichat 6/2, I-40127 Bologna, Italy  Alfonso
Veropalumbo  Dipartimento di Fisica, Università degli Studi Roma Tre, via
della Vasca Navale 84, I-00146 Rome, Italy

###### Abstract

Features in the primordial power spectrum represent the imprinted signal in
the density perturbations of the physics and evolution of the early Universe.
A measurement of such signals will represents the need to go beyond the
minimal assumption made for the initial conditions of the cosmological
perturbations. For the first time, we study different templates with undamped
oscillations or a bump from the two-point correlation function measured from
BOSS DR12 galaxies constraining the amplitude of the features to be at most a
few percent. Constraints are competitive to the ones obtained with Planck DR3.

Introduction. Measurements of cosmic microwave background (CMB) anisotropies,
such as those from Planck DR3 Planck:2018vyg ; Planck:2018jri ; Planck:2019kim
, have contributed significantly to the characterization of the initial
conditions of the Universe. The tight CMB constraints to spatial curvature,
isocurvature fluctuations, and primordial non-Gaussianity, all agree with
predictions of the standard single field slow-roll models Starobinsky:1980te ;
Guth:1980zm ; Sato:1980yn ; Linde:1981mu ; Albrecht:1982wi ; Hawking:1982ga ;
Linde:1983gd .

While in these simplest realizations the nearly exponential expansion driven
by a scalar field $\phi$ slowly rolling down a sufficiently flat potential
$V(\phi)$ predicts a nearly scale-invariant primordial power spectrum (PPS),
several classes of inflationary models predict features in the PPS such as
scale-dependent oscillations.

Indeed, primordial features (PF) are expected to appear as deviations from a
power-law in the PPS due to local or non-local modifications of the inflaton
potential Starobinsky:1992ts ; Adams:2001vc ; Chen:2008wn ; Flauger:2009ab ,
to the presence of new interactions and heavy particles Chung:1999ve ;
Barnaby:2009mc ; Chen:2011tu ; Chen:2011zf , non Bunch-Davies initial
conditions Danielsson:2002kx ; Easther:2002xe ; Martin:2003kp , or also as
imprints of alternatives to cosmic inflation Chen:2011zf ; Chen:2011tu ;
Chen:2018cgg . Moreover, phenomenologically they can be connected to
persistent anomalies in the CMB temperature angular power spectrum measured by
WMAP WMAP:2003syu and by Planck observations Planck:2013jfk ; Planck:2015sxf ;
Planck:2018jri ; PF have been also proposed to explain anomalies such as the
CMB lensing anomaly, the $H_{0}$ and the $S_{8}$ tensions Planck:2018jri ;
Domenech:2020qay ; Keeley:2020rmo ; Hazra:2022rdl .

In order to go beyond the current $2\sigma-3\sigma$ evidence of some PF
patterns from CMB anisotropy measurements WMAP:2003syu ; Planck:2013jfk ;
Meerburg:2013dla ; Easther:2013kla ; Chen:2014cwa ; Planck:2015sxf ;
Planck:2018jri ; Zeng:2018ufm ; Braglia:2021ckn ; Braglia:2021sun ;
Braglia:2021rej , it has been shown how future large-scale structure
observations will be able to improve current constraints on these oscillatory-
feature models. This will give the opportunity to further investigate the
presence of any salient feature in the matter power spectrum, complementing
the constraints based on CMB anisotropy measurements to smaller scales, and
being very powerful to constrain high frequency oscillations appearing on the
PPS, see Huang:2012mr ; Chen:2016zuu ; Chen:2016vvw ; Ballardini:2016hpi ;
Slosar:2019gvt ; Beutler:2019ojk ; Ballardini:2019tuc . In this perspective,
an intensive effort has been put in place to study the nonlinear dynamic of
the cold dark matter (CDM) power spectrum for initial conditions containing
PF, see Refs. Vlah:2015zda ; Beutler:2019ojk ; Vasudevan:2019ewf ;
Ballardini:2019tuc ; Chen:2020ckc ; Li:2021jvz .

The goal of this Letter is to provide new constraints on the amplitude of PF
for models with superimposed undamped oscillations or a bump performing the
first analysis of PF models with the galaxy two-point correlation function
(2PCF) from the Sloan Digital Sky Survey III Baryon Oscillation Spectroscopic
Survey Data Release 12 (BOSS DR12).

Models of oscillatory primordial power spectrum.

Figure 1: PPS for the three templates analysed. For M1 (red), we fix ${\cal
A}_{\rm lin}=0.05$, $\omega_{\rm lin}=2$, and $\phi_{\rm lin}=0$. For M2
(blue), we fix ${\cal A}_{\rm log}=0.05$, $\omega_{\rm log}=5$, and $\phi_{\rm
log}=0$. For M3 (green), we fix $\tilde{{\cal A}}_{\rm IR}\equiv{\cal A}_{\rm
IR}/A_{\rm s}=0.1$ and $k_{\rm IR}=0.01$ Mpc-1.

Starting from the standard power-law PPS of the comoving curvature
perturbations $\cal R$ on superhorizon scales, which is given by

${\cal P}_{{\cal R},\,0}(k)=A_{\rm s}\left(\frac{k}{k_{*}}\right)^{n_{\rm
s}-1}$ (1)

where $A_{\rm s}$ and $n_{\rm s}$ are the amplitude and the spectral index of
the comoving curvature perturbations at a given pivot scale $k_{*}=0.05\ {\rm
Mpc}^{-1}$, we consider deviations as

${\cal P}_{\cal R}(k)={\cal P}_{{\cal R},\,0}(k)\left[1+{\cal P}_{\rm
PF}(k)\right]\,.$ (2)

There are many kinds of oscillatory patterns models such as features that
oscillates linearly or logarithmically in $k$ as well as oscillations with
constant or scale-dependent amplitude Starobinsky:1992ts ; Adams:2001vc ;
Danielsson:2002kx ; Easther:2002xe ; Martin:2003kp ; Wang:2002hf ; Bean:2008na
; Chen:2008wn ; Flauger:2009ab ; Chen:2010bka ; Chen:2011zf ; Chen:2011tu ;
Chen:2018cgg . We therefore consider the following templates with superimposed
oscillations on the PPS

${\cal P}_{\rm PF}(k)={\cal A}_{\rm X}\sin\left(\omega_{\rm X}\Xi_{\rm
X}+2\pi\phi_{\rm X}\right)\,,$ (3)

where ${\rm X}=\\{{\rm lin,\,log}\\}$ and $\Xi_{\rm
X}=\\{k/k_{*},\,\log(k/k_{*})\\}$. These two templates with undamped
oscillatory features (hereafter M1 and M2) are described by three extra
parameters: an amplitude ${\cal A}_{\rm X}$, a frequency $\omega_{\rm X}$, and
a phase $2\pi\phi_{\rm X}$ that we restrict to the range $[0,\,2\pi]$. As
third model (hereafter M3), we consider a localized bump Barnaby:2009dd

${\cal P}_{\rm PF}(k)=\tilde{\cal A}_{\rm IR}\left(\frac{\pi
e}{3}\right)^{3/2}\left(\frac{k}{k_{\rm
IR}}\right)^{3}e^{-\frac{\pi}{2}\left(\frac{k}{k_{\rm
IR}}\right)^{2}}\left(\frac{k}{k_{*}}\right)^{1-n_{\rm s}}\,,$ (4)

described by two extra parameters: an amplitude that we rescale with the
amplitude of comoving curvature perturbations $\tilde{{\cal A}}_{\rm
IR}\equiv{\cal A}_{\rm IR}/A_{\rm s}$ and a scale $k_{\rm IR}$ connected with
the position of the bump. We show the effect of the feature parameters on
${\cal P}_{\rm PF}(k)$ in Fig. 1.

Data. In order to constraint these models, we consider the full DR12 data
set111https://data.sdss.org/sas/dr12/boss/papers/clustering/ split into three
redshift bins $0.2<z<0.5$, $0.5<z<0.75$, and (overlapping) $0.4<z<0.6$,
following Ref. Ross:2016gvb . We study also the case combining the first two
non-overlapping redshift bins neglecting their correlation.

We analyse the DR12 data set with the publicly available library
CosmoBolognaLib222https://gitlab.com/federicomarulli/CosmoBolognaLib
Marulli:2015jil by performing a Markov Chain Monte Carlo (MCMC) analysis.

For all redshift bins, we use the first two even multipole moments, i.e., the
monopole and the quadrupole, of the anisotropic correlation function
$\xi(s,\mu)$ measured using the minimum-variance unbiased estimator
Landy:1993yu using density-field reconstruction Eisenstein:2006nk ;
Padmanabhan:2012hf

$\xi(s,\mu)=\frac{DD(s,\mu)-2DS(s,\mu)+SS(s,\mu)}{SS(s,\mu)}\,,$ (5)

where $DD$, $SS$, and $DS$ are the numbers of galaxy-galaxy, random-random,
and galaxy-random pairs in bins of $s$ and $\mu$.

Unless otherwise noted, we use a flat $\Lambda$CDM cosmology given by
$\Omega_{\rm m}=0.31$, $\Omega_{\rm b}h^{2}=0.0220$, $h=0.676$, $n_{\rm
s}=0.97$, $\sigma_{8}=0.8$, and three massless neutrinos. This is consistent
with Planck DR3 results Planck:2018vyg and is the same cosmological model as
used in the papers studying the BOSS DR12 combined sample Ross:2016gvb . We
consider the post-reconstrunced galaxy 2PCF on the range of scales $20<s\
[h^{-1}{\rm Mpc}]<180$ divided in $\Delta s=5\ h^{-1}{\rm Mpc}$ bins.

parameter | flat prior range
---|---
$\alpha_{\perp},\,\alpha_{\parallel}$ | $[0.7,1.3]$
$B_{0},\,B_{2}$ | $[0,5]$
$A^{0}_{0},\,A^{2}_{0},\,A^{0}_{1},\,A^{2}_{1}$ | $[-100,100]$
$A^{0}_{2},\,A^{2}_{2}$ | $[-1000,1000]$
${\cal A}_{\rm lin,\,log}$ | $[0,0.5]$
$\tilde{{\cal A}}_{\rm IR}$ | $[0,1]$
$\omega_{\rm lin}$ | $[1.5,13]$
$\omega_{\rm log}$ | $[1,50]$
$k_{\rm IR}$ [Mpc-1] | $[0.001,0.1]$
$\phi_{\rm lin,\,log}$ | $[0,1]$
Table 1: Priors for model parameters.

Anisotropic analysis of the BAO peak in the multipoles of the 2PCF. To build
the model for the anisotropic 2PCF, we start modelling the nonlinear CDM power
spectrum in redshift space Kaiser:1987qv ; Fisher:1993pz ; Xu:2012hg ;
Xu:2012fw :

$\displaystyle P^{\rm CDM}(k,\mu)=[1+\beta\mu^{2}R(k,\Sigma_{r})]^{2}F_{\rm
fog}(k,\Sigma_{s})P_{\rm nw}(k)$
$\displaystyle\qquad\qquad\qquad\times\left[1+P_{\rm
BAO}(k)e^{-k^{2}\Sigma_{\rm NL}^{2}(\mu)/2}\right]$
$\displaystyle\qquad\qquad\qquad\times\left[1+P_{\rm
PF}(k)e^{-k^{2}\Sigma_{\rm PF}^{2}/2}\right]\,,$ (6)

where

$\displaystyle F_{\rm
fog}(k,\mu,\Sigma_{s})=\frac{1}{\left(1+k^{2}\mu^{2}\Sigma_{s}^{2}/2\right)^{2}}\,$
(7) $\displaystyle P_{\rm BAO}(k)=\frac{P_{\rm lin}(k)}{P_{\rm nw}(k)}-1\,$
(8) $\displaystyle\Sigma_{\rm
NL}^{2}(\mu,\Sigma_{\parallel},\Sigma_{\perp})=\mu^{2}\Sigma_{\parallel}^{2}+(1-\mu^{2})\Sigma_{\perp}^{2}\,.$
(9)

The linear power spectrum $P_{\rm lin}(k)$ is obtained with
CAMB333https://camb.readthedocs.io/en/latest/ Lewis:1999bs ; Howlett:2012mh
and the no-wiggle $P_{\rm nw}(k)$ is obtained with fitting formulae from Ref.
Eisenstein:1997ik , in both cases using our fiducial cosmology. We fix the
streaming scale $\Sigma_{s}=4\ h^{-1}{\rm Mpc}$, respectively. The radial and
transverse components of the standard Gaussian damping of baryon acoustic
oscillations (BAO) are fixed at $\Sigma_{\parallel}=4\ h^{-1}{\rm Mpc}$ and
$\Sigma_{\perp}=2.5\ h^{-1}{\rm Mpc}$.
$R(k,\Sigma_{r})=1-e^{-k^{2}\Sigma_{r}^{2}/2}$ is the smoothing applied in
reconstruction and $\Sigma_{r}=15\ h^{-1}{\rm Mpc}$ is the smoothing scale
used when deriving the displacement field Seo:2015eyw . Motivated by the weak
dependence of current measurements of the 2PCF on the choice of damping scales
(see Ref. Ross:2016gvb ), we fix $\Sigma_{\rm NL}=\Sigma_{\rm PF}$.

Figure 2: Nonlinear galaxy 2PCF monopole (top panels) and quadrupole (bottom
panels) computed at $z=0.51$ for M1, M2, and M3 (from left to right). We fix
the standard cosmological and model parameters to the baseline and we vary one
feature parameter at a time. For M1 (left panel), we fix ${\cal A}_{\rm
lin}=0.05$ and we vary $\omega_{\rm lin}=\\{2,\,10\\}$, $\phi_{\rm
lin}=\\{0,\,0.5\\}$. For M2 (central panel), we fix ${\cal A}_{\rm log}=0.05$
and we vary $\omega_{\rm log}=\\{5,\,20\\}$, and $\phi_{\rm
log}=\\{0,\,0.5\\}$. For M3 (right panel), we fix $\tilde{{\cal A}}_{\rm
IR}\equiv{\cal A}_{\rm IR}/A_{\rm s}=0.1$ and we vary $k_{\rm
IR}=\\{0.0075,\,0.01,\,0.025,\,0.05\\}$ Mpc-1.

Following Refs. Ross:2016gvb , we fit to the data using the following model
for the monopole and quadrupole of the galaxy 2PCF:

$\displaystyle\xi_{0}(s)=B_{0}\xi_{0}^{\rm
CDM}(s,\alpha_{\perp},\alpha_{\parallel})+A_{0}^{0}+\frac{A_{0}^{1}}{s}+\frac{A_{0}^{2}}{s^{2}}\,,$
(10) $\displaystyle\xi_{2}(s)=\frac{5}{2}\left[B_{2}\xi_{2}^{\rm
CDM}(s,\alpha_{\perp},\alpha_{\parallel})-B_{0}\xi_{0}^{\rm
CDM}(s,\alpha_{\perp},\alpha_{\parallel})\right]$
$\displaystyle\qquad\qquad+A_{2}^{0}+\frac{A_{2}^{1}}{s}+\frac{A_{2}^{2}}{s^{2}}\,,$
(11)

where $\xi_{0}^{\rm DM}(s,\alpha_{\perp},\alpha_{\parallel})$ and
$\xi_{2}^{\rm DM}(s,\alpha_{\perp},\alpha_{\parallel})$ are the CDM 2PCF
monopole and quadrupole computed at the fiducial cosmology, and {$B_{0}$,
$B_{2}$, $A^{0}_{0}$ $A^{2}_{0}$, $A^{0}_{1}$, $A^{2}_{1}$, $A^{0}_{2}$,
$A^{2}_{2}$} are considered as nuisance parameters. We impose uniform flat
prior on all parameters (see Tab. 1 for the prior ranges).

The effect of each of the feature parameters on the nonlinear galaxy 2PCF
monopole and quadrupole is illustrated in Fig. 2 for a reference central
redshift $z=0.51$ of the BOSS DR12 sample. We see that M1 present a sharp peak
or deep, depending on the value of the phase $\phi_{\rm lin}$, corresponding
to the scale of the 2PCF located at $s\sim 20\cdot\omega_{\rm lin}\,$Mpc. The
signal is smoothed by nonlinear effects similarly to the case of the BAO
Beutler:2019ojk ; Ballardini:2019tuc , damping the primordial oscillations on
the small scales of the Fourier space matter power spectrum and leading to a
broad bump on the 2PCF. Note that for $\omega_{\rm lin}\sim 7.5$ and
$\phi_{\rm lin}=0$ the peak coincides with the position of the BAO one, i.e.,
$s_{\rm BAO}\sim 150\,$Mpc. M2 leaves a broad pattern on all scales in the
2PCF. Finally, we see, differently from what happened for the oscillatory
templates, M3 generates a smooth and broad change of the 2PCF monopole shape.
It is interesting to note that while the first two templates leave a clear
oscillatory pattern also on the quadrupole, for the third model there is a
negligible effect on the quadrupole.

Results for the anisotropic analysis.

|  $\Lambda$CDM | M1 | M2 | M3
---|---|---|---|---
sample | $\alpha_{\perp}$ | $\alpha_{\parallel}$ | $\alpha_{\perp}$ | $\alpha_{\parallel}$ | $\alpha_{\perp}$ | $\alpha_{\parallel}$ | $\alpha_{\perp}$ | $\alpha_{\parallel}$
$0.2<z<0.5$ | $0.979\pm 0.014$ | $1.017\pm 0.028$ | $0.982\pm 0.017$ | $1.004\pm 0.029$ | $0.976\pm 0.015$ | $1.022\pm 0.027$ | $0.980\pm 0.014$ | $1.015\pm 0.028$
$0.4<z<0.6$ | $0.991\pm 0.014$ | $0.988\pm 0.025$ | $0.994\pm 0.016$ | $0.989\pm 0.027$ | $0.992\pm 0.014$ | $0.986\pm 0.025$ | $0.992\pm 0.014$ | $0.988\pm 0.025$
$0.5<z<0.75$ | $0.994\pm 0.023$ | $0.953\pm 0.041$ | $0.998\pm 0.024$ | $0.948\pm 0.045$ | $0.995\pm 0.025$ | $0.950\pm 0.044$ | $0.992\pm 0.021$ | $0.957\pm 0.040$
combined | $0.976\pm 0.014$ | $1.031\pm 0.027$ | $0.985\pm 0.018$ | $1.017\pm 0.027$ | $0.982\pm 0.016$ | $1.016\pm 0.028$ | $0.979\pm 0.014$ | $1.018\pm 0.029$
Table 2: Marginalized mean values and 68% CL for the 2D BAO parameters
$\alpha_{\perp}$ and $\alpha_{\parallel}$ for the $\Lambda$CDM, M1, M2, and
M3.

We start performing the standard BAO analysis on BOSS DR12 data for
$\Lambda$CDM cosmology to validate our pipeline, using the measured post-
reconstructed monopole and quadrupole, for each redshift bin. Means and
uncertainties for $\alpha_{\perp}$ and $\alpha_{\parallel}$, marginalized over
{$B_{0}$, $B_{2}$, $A^{0}_{0}$ $A^{2}_{0}$, $A^{0}_{1}$, $A^{2}_{1}$,
$A^{0}_{2}$, $A^{2}_{2}$}, agree well with the ones found in Ref. Ross:2016gvb
; we collect the results in Tab. 2. We repeat the BAO analysis in the presence
of the template with linear oscillations (M1), logarithmic ones (M2), and a
bump (M3). The constraints on $\alpha_{\perp},\,\alpha_{\parallel}$ are
affected in some cases by $<20\%$ larger uncertainties, see Tab. 2.

Figure 3: Marginalized joint posterior distributions of the features
parameters for M1, M2 and M3 models (from top to bottom).

Figure 4: Marginalized joint posterior distribution of the value of the
feature parameters for M1 model (top panel) and M2 (bottom panel) obtained
with the BOSS DR12 combined redshift bin and for the Planck DR3.

For M1, the marginalized 95% Confidence Level (CL) upper bound on the
amplitude of the primordial oscillations analysing the 2PCF is ${\cal A}_{\rm
lin}<0.029$ at $z=0.38$, ${\cal A}_{\rm lin}<0.024$ at $z=0.51$, ${\cal
A}_{\rm lin}<0.023$ at $z=0.61$ on the frequency range $1.5<\omega_{\rm
lin}<13$; combining the two non-overlapping samples we obtain ${\cal A}_{\rm
lin}<0.025$ at 95% CL. For M2, the marginalized 95% CL upper bound on the
amplitude of the primordial oscillations is ${\cal A}_{\rm log}<0.048$ at
$z=0.38$, ${\cal A}_{\rm log}<0.023$ at $z=0.51$, ${\cal A}_{\rm log}<0.047$
at $z=0.61$ on the frequency range $1<\omega_{\rm log}<50$; combining the two
non-overlapping samples we obtain ${\cal A}_{\rm log}<0.027$ at 95% CL. For
M3, the marginalized 95% CL upper bound on the amplitude of the bump is
$\tilde{\cal A}_{\rm IR}<0.88$ at $z=0.38$, $\tilde{\cal A}_{\rm IR}<0.92$ at
$z=0.51$, $\tilde{\cal A}_{\rm IR}<0.86$ at $z=0.61$; combining the two non-
overlapping samples we obtain $\tilde{\cal A}_{\rm IR}<0.34$ at 95% CL. For a
simple model where the inflation field $\phi$ interacts with field $\chi$
through the interaction

${\cal L}=-\frac{g^{2}}{2}\left(\phi-\phi_{0}\right)^{2}\chi^{2}\,,$ (12)

we can derive constraints on the coupling $g$ thanks to the relation ${\cal
A}_{\rm IR}\simeq 1.01\times 10^{-6}g^{15/4}$ for a single burst of particle
production during inflation Barnaby:2009dd . We find for the combined sample
$g^{2}<0.021$ at 95% CL (for $A_{\rm s}=2.1\times 10^{-9}$), corresponding to
a reasonable coupling values for models with phase transition during
inflation.

The bounds on the first two templates with oscillatory features can be
compared with the results inferred from CMB anisotropy observations obtained
with Planck DR3 for the same templates Planck:2018jri . The frequency coverage
in the CMB analysis is wider for both templates, i.e.,
$1<\omega_{X}<10^{2.1}$. On the restricted frequency range covered by our
analysis, the constraints from the 2PCF are comparable if not slightly tighter
compared to the CMB ones. Planck DR3 constraints at 95 % CL correspond to
${\cal A}_{\rm lin}<0.029$ ($<0.028$) for TT (TTTEEE) for M1 and ${\cal
A}_{\rm log}<0.043$ ($<0.023$) for TT (TTTEEE) for M2. In Fig. 4, we overplot
the marginalised posterior distribution on the feature amplitude depending on
the frequency from our BOSS DR12 analysis with the one obtained from CMB
Planck DR3. We are able to put tight constraints on some of the best-fit
frequencies detected with CMB Planck data Planck:2018jri using BOSS DR12
measurements. In particular, the best-fit for M1 from TTTEEE data
corresponding to ${\cal A}_{\rm lin}=0.015$ and $\omega_{\rm lin}\simeq 10$,
and the best-fit for M2 from TT data corresponding to ${\cal A}_{\rm
log}=0.024$ and $\omega_{\rm lin}\simeq 32$ are at odds with current LSS data.

Our constraints for M1 and M2 are consistent with the ones obtained in
Beutler:2019ojk where the same two templates have been constrained with the
isotropic power spectrum monopole in Fourier space on the frequency ranges
$5<\omega_{\rm lin}<45$ and $10<\omega_{\rm log}<80$, finding bounds on the
features amplitude from BOSS DR12 up to a factor of 2.3 and 3.1 tighter than
the ones obtained from CMB for $\omega_{\rm lin}>10$ and $\omega_{\rm
log}>20$, respectively.

Conclusions. In this Letter, we have derived for the first time constraints on
primordial feature in the PPS through the galaxy two-point galaxy correlation
function. We have considered undamped superimposed oscillations and a
localized bump motivated by the production of particles during inflation. Our
results show that the BOSS DR12 2PCF has the potential to achieve
sensitivities competitive to current limits from Planck DR3 CMB anisotropies,
improving the bounds on the average amplitude of such features and putting at
odds some of the CMB best-fit frequencies found in Planck data. The
complementarity between 2PCF and CMB measurements of primordial features opens
exciting synergies, as verification of the primordial interpretation of
anomalies in the CMB temperature and polarization angular power spectrum, that
will lead the way to new studies of the physics of the early universe.

Future galaxy clustering measurements, such as those expected from DESI
DESI:2016fyo , Euclid EUCLID:2011zbd , SPHEREx Dore:2014cca , will
significantly improve the current bounds obtained here in different aspects.
Beyond a lower shot noise and higher redshift bins, a wider redshift volume
will allow to probe larger separation scales and therefore target even higher
frequencies for superimposed oscillations than those studied here.

## Acknowledgments

We all acknowledge financial support from the contract ASI/ INAF for the
Euclid mission n.2018-23-HH.0. FF acknowledge support by the agreement n.
2020-9-HH.0 ASI-UniRM2 “Partecipazione italiana alla fase A della missione
LiteBIRD”. LM acknowledges support from the grant PRIN-MIUR 2017 WSCC32.

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