# First JVLA Radio Observation on PDS 70

Hauyu Baobab Liu Department of Physics, National Sun Yat-Sen University, No.
70, Lien-Hai Road, Kaohsiung City 80424, Taiwan, R.O.C.  Center of Astronomy
and Gravitation, National Taiwan Normal University, Taipei 116, Taiwan  Simon
Casassus Departamento de Astronomía, Universidad de Chile, Casilla 36-D,
Santiago, Chile  Data Observatory Foundation, to Data Observatory Foundation,
Eliodoro Yańẽz 2990, Providencia, Santiago, Chile  Millennium Nucleus on Young
Exoplanets and Their Moons (YEMS), Santiago, Chile  Ruobing Dong Department of
Physics & Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada
Kiyoaki Doi Department of Astronomical Science, School of Physical Sciences,
Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa, Mitaka,
Tokyo 181-8588, Japan  National Astronomical Observatory of Japan, 2-21-1
Osawa, Mitaka, Tokyo 181-8588, Japan  Jun Hashimoto Astrobiology Center,
National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588,
Japan  Subaru Telescope, National Astronomical Observatory of Japan, Mitaka,
Tokyo 181-8588, Japan  Department of Astronomical Science, School of Physical
Sciences, Graduate University for Advanced Studies (SOKENDAI), 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Takayuki Muto  Division of Liberal Arts,
Kogakuin University, 1-24-2, Nishi-Shinjuku, Shinjuku-ku, Tokyo 163-8677,
Japan

###### Abstract

PDS 70 is a protoplanetary system that hosts two actively accreting gas
giants, namely PDS 70b and PDS 70c. The system has a $\sim$60–100 au dusty
ring that has been resolved by the Atacama Large Millimeter/Submillimeter
Array (ALMA), along with circumplanetary disks around the two gas giants. Here
we report the first Karl G. Jansky Very Large Array (JVLA) Q (40–48 GHz), Ka
(29–37 GHz), K (18–26 GHz), and X (8–12 GHz) bands continuum observations, and
the complementary ALMA Bands 3 ($\sim$98 GHz) and 4 ($\sim$145 GHz)
observations towards PDS 70. The dusty ring appears azimuthally asymmetric in
our ALMA images. We obtained firm detections at Ka and K bands without
spatially resolving the source; we obtained a marginal detection at Q band,
and no detection at X band. The spectral indices ($\alpha$) are 5$\pm$1 at
33–44 GHz and 0.6$\pm$0.2 at 22–33 GHz. At 10–22 GHz, the conservative lower
limit of $\alpha$ is 1.7. The 33–44 GHz flux density is likely dominated by
the optically thin thermal emission of grown dust with $\gtrsim$1 mm maximum
grain sizes, which may be associated with the azimuthally asymmetric
substructure induced by planet-disk interaction. Since PDS 70 was not detected
at X band, we found it hard to explain the low spectral index at 22–33 GHz
only with free-free emission. Hence, we attribute the dominant emission at
22–33 GHz to the emission of spinning nanometer-sized dust particles, while
free-free emission may partly contribute to emission at this frequency range.
In some protoplanetary disks, the emission of spinning nanometer-sized dust
particles may resemble the 20–50 GHz excess in the spectra of millimeter-sized
dust. The finding of strong continuum emission of spinning nanometer-sized
particles can complicate the procedure of constraining the properties of grown
dust. Future high resolution, multi-frequency JVLA/ngVLA and SKA observations
may shed light on this issue.

Circumstellar dust (236) — Protoplanetary disks (1300) — Pre-main sequence
(1289) — Planet formation (1241)

††facilities: JVLA††software:  astropy (Astropy Collaboration et al., 2013,
2018, 2022), Numpy (van der Walt et al., 2011), CASA (McMullin et al., 2007),

##  1 Introduction

Table 1: Observations Date | Band | API rmsaaThe RMS phase at the JVLA site measured with the Atmospheric Phase Interferometer (API; for more details see https://science.nrao.edu/facilities/vla/docs/manuals/oss2013A/performance/gaincal/api). The API a 2-element interferometer separated by 300 meters, observing an 11.7 GHz beacon from a geostationary satellite. The default API rms upper limits for the X, Ku, K, Ka, and Q band observations are 30∘, 15∘, 10∘, 7∘, and 5∘. |  uv-range | Freq. range | Flux/Passband/Gain calibrators | Synthesized beambbMeasured from multi-frequency synthesis images generated using aggregated continuum bandwidths and Natural (i.e., Briggs Robust$=$2) weighting. | rms noisebbMeasured from multi-frequency synthesis images generated using aggregated continuum bandwidths and Natural (i.e., Briggs Robust$=$2) weighting.
---|---|---|---|---|---|---|---
(UTC) |  | (∘) | (meters) | (GHz) |  | ($\theta_{\mbox{\scriptsize maj}}\times\theta_{\mbox{\scriptsize min}}$; ∘) | ($\mu$Jy beam-1)
Project: JVLA (19B-023; PI: H. B. Liu)
2019-Nov-08 | X | 4–7 | 36-19070 | 8–12 | 3C286/3C286/J1427-4206 | 16$\times$4$\farcs$9; $-$0.61∘ | 10
2019-Nov-15 | K | 4–6 | 33-1031 | 18–26 | 3C286/J1427-4206/J1427-4206 | 32$\times$2$\farcs$4; $-$0.43∘ | 17
2019-Nov-17 | K |  $\sim$7 | 32-1030 | 18–26 | 3C286/J1427-4206/J1427-4206 | 35$\times$2$\farcs$7; $-$0.19∘ | 16
2020-Jan-04 | Q |  $\sim$3 | 34-1031 | 40–48 | 3C286/3C286/J1427-4206 | 9$\farcs$6$\times$1$\farcs$2; $-$0.35∘ | 100
2020-Jan-20 | Ka |  $\sim$4 | 36–3381 | 29–37 | 3C286/3C286/J1427-4206 | 7$\farcs$0$\times$3$\farcs$0; $-$14∘ | 23
Project: ALMA (2022.1.01477.S; PI: H. B. Liu)
2023-Mar-26 | 3 | $\cdots$ | 14–1241 | 91.5–95.2; 101.6–105.3 | J1427-4206/J1427-4206/J1407-4302 | 0$\farcs$49$\times$0$\farcs$45; $-$30∘ | 18
2023-Mar-25 | 4 | $\cdots$ | 14–1246 | 140.0–142.7; 149.0–152.8 | J1427-4206/J1427-4206/J1407-4302 | 0$\farcs$43$\times$0$\farcs$33; 44∘ | 44

Planet-disk interaction plays an important role in trapping grown dust, which
could potentially aid in dust growth thereby facilitate the formation of
second-generation planetesimals and planets (for a recent review see Birnstiel
2023 and references therein). Constraining dust distributions and dust growth
in planet-hosting protoplanetary disks is crucial for understanding the
related physical processes. Limited by the sensitivity and image fidelity of
the present optical/infrared observing facilities, the candidates of planet-
hosting protoplanetary disks remain rare; most of those candidates still
require confirmation (e.g., Kraus & Ireland 2012; Currie et al. 2022).

The pre-main sequence star PDS 70 is a K7-type (0.76 $M_{\odot}$), weak-line T
Tauri type star (Keppler et al. 2018; Müller et al. 2018) at $\sim$113 pc
distance (Gaia Collaboration et al. 2023). The previous Subaru-HiCIAO
coronagraphic polarimetric imaging observations at 1.6 $\mu$m wavelength
resolved that it is associated with an extended disk of $\sim$100 au radius,
which presents a gap extending from $\sim$17–60 au radii (0′′.15–0′′.53;
Hashimoto et al. 2012). In addition, Hashimoto et al. (2015) compared the
Subaru-HiCIAO near infrared images with the 1.3 mm image taken with the
Submillimeter Array (SMA). Based detailed Monte Carlo radiative transfer
models (also see Dong et al. 2012), they suggested that there are at least two
distinct dust components in the disk which have large and small grain sizes.
It appeared that the gap edges of the large and small dust components are at
different radii, which are 80 au and 65 au, respectively. This may be a
signpost of dust filtration, which may be due to the presence of
(proto)planets.

Newer optical and infrared imaging observations equipped with extreme adaptive
optics have discovered the 2–17 Jupiter masses ($M_{\mbox{\scriptsize Jup}}$)
planet orbiting PDS 70, known as PDS 70b (Keppler et al. 2018; Müller et al.
2018; Hashimoto et al. 2020; Wagner et al. 2018), followed by the detection of
another 4–12 $M_{\mbox{\scriptsize Jup}}$ planet, PDS 70c (Haffert et al.
2019; Hashimoto et al. 2020). These two planets orbit the host pre-main
sequence star at 23 au (0$\farcs$2) and 35 au (0$\farcs$31) radii,
respectively. The resolved massive planets at known locations make PDS70 a
unique and ideal laboratory for investigating dust processing in a planet-
hosting disk using (sub)millimeter and centimeter wavelength observations.

Intensive Atacama Large Millimeter/Submillimeter Array (ALMA) dust continuum
observations at 225–345 GHz frequencies ($\lambda\sim$1.3–0.87 mm) have been
carried out towards PDS 70 (Long et al. 2018; Isella et al. 2019; Benisty et
al. 2021; Facchini et al. 2021; Casassus & Cárcamo 2022). They resolved the
gap/ring structures in the PDS 70 disk. Moreover, they detected the
circumplanetary disks around PDS 70b and c (Isella et al. 2019; Benisty et al.
2021; Casassus & Cárcamo 2022). These observations have provided the
unprecedentedly clear picture of the structures of the bulk of the dust
reservoirs in the PDS 70 system. However, the recent surveys and population
synthesis studies have indicated that the 230–345 GHz optical depths of most
of the Class II protoplantary disks in the Taurus, Ophiuchus, and Lupus star-
forming regions are $\gtrsim$5–10 (Chung et al. 2024; Delussu et al. 2024, and
references therein). In addition, some case studies have shown that a good
amount of grown dust may be hidden in localized substructures that can only be
discerned at $\lesssim$100 GHz frequencies (e.g., Hashimoto et al. 2022, 2023;
Liu et al. 2024). Continuum observations at yet lower frequencies, reaching
into the microwaves, will reach an optically thinner regime, and will provide
constraints on the dust mass and on the properties of the largest grains
observables. Such observations of grain growth are crucial to understand the
planet-formation process.

|  |  |
---|---|---|---
|  |  |
Figure 1:  JVLA continuum images (upper row, color images and contours) and
the synthesized dirty beams (bottom row, color images). Upper row :– From left
to right are the Q band (40–48 GHz) image (contours are (1-$\sigma$)
$\times$[-2, 2, 3]); Ka band (29–37 GHz) image (contours are (1-$\sigma$)
$\times$[-3, -2, 2, 3, 4]); K band (18–26 GHz) image (contours are
(1-$\sigma$) $\times$[-2, 2, 4, 6, 8]) that were generated by jointly imaging
the two epochs of K band observations (Table 1); X band (8–12 GHz) image
(contours are (1-$\sigma$) $\times$[-2, 2]). The synthesized clean beams are
shown in the lower left. We note that the two panels on the left and the two
panels on the right are presented on different angular scales due to the very
elongated synthesized (clean) beams of the latter cases. The magenta ellipse
in each panel shows the location and size of the PDS 70 millimeter ring (c.f.
Long et al. 2018; Isella et al. 2019; Benisty et al. 2021; Facchini et al.
2021; Casassus & Cárcamo 2022). Bottom row :– Color images show the
synthesized dirty beams of the observations presented in the upper row.

Here we report our JVLA (18–48 GHz) observations and the ALMA Bands 3 (97.5
GHz) and 4 (145 GHz) observations towards PDS 70. The technical details of our
observations are provided in Section 2. The results are introduced in Section
3. Section 4 discusses our hypotheses of the dominant emission mechanisms at
18–48 GHz. Our conclusion is given in Section 5. Appendix A briefly discusses
various strategies of measuring flux densities. Appendix B described how we
produced models of dust spectral energy distributions (SEDs) to compare with
the JVLA observations.

##  2 Observations and data reduction

###  2.1 JVLA

We have carried out the JVLA observations towards PDS 70 at Q (40–48 GHz), Ka
(29–37 GHz), K (18–26 GHz), and X (8–12 GHz) bands in 2019 (Project code:
19B-023, PI: Hauyu Baobab Liu), which are summarized in Table 1. The pointing
and phase referencing centers were R.A. (J2000)$=$14${}^{\mbox{\scriptsize
s}}$08${}^{\mbox{\scriptsize m}}$10$\farcs$150, Decl.
(J2000)$=-$41∘23′52$\farcs$5\. We adopted the standard 3-bit continuum setup
of the WIDAR correlator, which provided full RR, RL, LR, and LL correlations.

The K and Q bands observations were carried out in the most compact (D) array
configuration. The Ka band observations were carried out in the D-north-C
(DnC) array configuration that the western and eastern arms of the array were
in the D configuration while the northern arm of the array was in the C array
configuration. The X band observations were carried out in a moving array
configuration that most of the antennae were on the pads of the D array
configuration while three antennae were on the pad of the A array
configuration. The target source is in the far south which transited at a
$\sim$14.6∘ elevation. In all of the observations, the antennae in the
northern arm of the array were severely shadowed and thus mostly had to be
flagged. Effectively, we were observing with an array with only the western
and eastern arm, which poorly determined the location of the target source in
the north-south direction. After flagging the shadowed and malfunctioned
antennae, in general, we were left with $\sim$15 available antennae.

|
---|---
Figure 2:  ALMA continuum images at Bands 3 (left) and 4 (right). Synthesized
beams of these images are shown in the lower left. Magenta ellipse are the
same as those in Figure 1.

We manually calibrated the JVLA data following the standard strategy (see
below) using the Common Astronomy Software Applications (CASA 6.4.1; McMullin
et al. 2007) software package. For all the JVLA observations, after
implementing the antenna position corrections, weather information, gain-
elevation curve, and opacity model, we bootstrapped delay fitting and passband
calibrations, and then performed complex gain calibration. We applied the
absolute flux reference to our complex gain solutions, and then applied all
derived solution tables to the target source.

We performed multi-frequency synthesis (mfs; nterms$=$1) imaging (Rau &
Cornwell 2011) using the CASA tclean task. We adopted the Briggs Robust$=$2.0
weighting to minimize the root-mean-square (RMS) noises in Jy beam-1 units. In
the imaging procedure, the dirty images were first obtained from the inverse
Fourier transform of the Robust$=$2.0 weighted visibilities; the synthesized
dirty beams were the inverse Fourier transforms of the sampling functions in
the visibility domain. The tclean task used the synthesized dirty beams to
deconvolve the dirty images and at the end outputed the deconvolution model
(i.e., the clean components) and the clean residual image. The tclean task
defined the synthesized clean beam by performing two-dimensional (2D) Gaussian
fittings to the synthesized dirty beam. Finally, the tclean task produced the
clean images by convolving the deconvolution model with the synthesized clean
beams, and then superimposing the clean residual image. Table 1 summarizes the
achieved synthesized clean beams and the root-mean-square (RMS) noises that
were measured from the clean residual images made for the individual epochs of
observations. We jointly imaged the two epochs of K band observations, which
yielded a $\theta_{\mbox{\scriptsize maj}}\times\theta_{\mbox{\scriptsize
min}}$=32${}^{\prime\prime}\times$2$\farcs$4 (P.A.=$-$0.43∘) synthesized beam
and a RMS noise of 12 $\mu$Jy beam-1.

The synthesized beam of the X band image appears grainy due to the three
antennae on the A array configuration pads (Figure 1). The synthesized beam of
the Ka band image appeared highly non-Gaussian in the central region and thus
cannot be well represented by the 2D Gaussian synthesized clean beams (Figure
1). The clean algorithm can still deconvolve the dirty image in this case,
although the interpretation of the restored image and the Jy beam-1 intensity
unit are less trivial (Appendix A).

###  2.2 ALMA

We have carried out the ALMA Bands 3 (89.6–93.4 GHz/101.6–105.4 GHz) and 4
(137.0–138.9 GHz/149.0–152.9 GHz) standard FDM continuum observations in the
C43-5 array configuration (Project code: 2022.1.01477.S, PI: Hauyu Baobab Liu;
Table 1). The pointing and phase referencing centers were R.A.
(ICRS)$=$14${}^{\mbox{\scriptsize s}}$08${}^{\mbox{\scriptsize
m}}$10$\farcs$1067, Decl. (ICRS)$=-$41∘23′53$\farcs$059.

We manually calibrated the ALMA data following the standard strategy that is
similar to that for the JVLA data calibration (Section 2.1) using CASA 6.4.1
(McMullin et al. 2007). In addition, we performed gain-phase self-calibration.
We produced continuum data by binning the line-free channels. We performed mfs
(nterms$=$1) imaging (Rau & Cornwell 2011) for the continuum data using the
CASA tclean task. We adopted the Briggs Robust$=$2.0 weighting to minimize the
RMS noises in Jy beam-1 units. The achieved image qualities are summarized in
Table 1.

---
Figure 3:  The 8–48 GHz spectral profile of PDS 70. Upper panel :– Flux
densities measured based on fitting 2D Gaussians in the visibility domain (red
dots) and reading peak intensities in the image domain (cyan dots); triangles
show the 3-$\sigma$ upper limits at 10 GHz. The sizes of some symbols are
larger than the error bars. Gray solid and dash-dotted lines show the spectral
indices of 2.0 and 4.0, respectively. Orange and blue lines show the SED
models for a grown ($a_{\mbox{\scriptsize max}}=$1.5 mm) dust component and a
spinning nanometer-sized dust component, respectively (Appendix B); black line
shows the integrated SED of these two dust components. Bottom panel :– The
22–33 GHz and 33–44 GHz spectral indices and the lower limit of the 10–22 GHz
spectral index derived based on the measurements presented in the upper panel.
They are compared with the spectral indices of optically thin dust slabs at 20
K temperature with various $a_{\mbox{\scriptsize max}}$ values, which were
evaluated based on the DSHARP-default dust opacities (Birnstiel et al. 2018).

##  3 Results

Upper row of Figure 1 and Figure 2 show the clean images obtained from our
JVLA and ALMA observations, respectively. The dirty beams of the JVLA
observations are shown in the bottom row of Figure 1.

The ALMA Band 3 and 4 observations marginally resolved the PDS 70 ring (Figure
2). Intriguingly, the PDS 70 ring appears azimuthal asymmetric in these ALMA
images, showing a bright crescent in the northwest (c.f. Benisty et al. 2021).

We detected PDS 70 at Ka (33 GHz) and K bands (22 GHz) with a significance of
at least 4-$\sigma$ and 8-$\sigma$, respectively. At Q band (44 GHz), the
$\sim$4-$\sigma$ intensity peak is 3$\farcs$2 offset from the PDS 70 ring,
which is smaller than the $\sim$10′′ FWHM of a clean beam. The offset of the Q
band intensity peak is attributed to the residual phase and delay errors that
are inevitable in high-frequency observations at low elevations. We did not
detect PDS 70 at $\gtrsim$2-$\sigma$ significance at X band (10 GHz).

We measured the flux densities from the ALMA images by integrating the pixel
values over the inner 4′′ square region. Thanks to the high signal-to-noise
ratios achieved by the ALMA observations, varying the location and shape of
the square region does not change the obtained flux density by more than the
nominal 5% absolute flux errors we quoted.

We compared two methods for measuring flux densities from the JVLA images: (i)
reading the peak intensities in the image domain, and (ii) performing 2D
Gaussian fittings in the visibility domain. We considered the peak intensities
as lower limits of flux densities because they are less immune to imaging
artifacts. We considered the flux densities obtained from the 2D Gaussian
fittings as upper limits because they are potentially confused by unidentified
background/foreground sources. We provide further discussion about flux
density measurements in Appendix A.

The flux density measurements are summarized in Table 2. As we are not aware
of any strong confusion source within the field of view of our JVLA
observations, we think that the flux densities measured using visibility
fittings are closer to the actual flux densities. We have inspected the 4.85
GHz and 0.834 GHz images taken with the Parkes-MIT-NRAO (PMN) radio continuum
surveys (Griffith & Wright 1993) and the Sydney University Molonglo Sky Survey
(SUMSS; Mauch et al. 2003) and did not find strongly confusing radio sources,
although the rms noises of these surveys were relatively high ($\sim$1 mJy
beam-1). Moreover, the adjacent infrared sources detected by the Wide-field
Infrared Survey Explorer (WISE)111Obtained from the NASA/IPAC Extragalactic
Database (NED). The NASA/IPAC Extragalactic Database (NED) is funded by the
National Aeronautics and Space Administration and operated by the California
Institute of Technology. are $\gtrsim$1′ away, whose radio emission unlikely
can confuse our JVLA Q and Ka bands observations. Our discussion (Section 4)
will mainly focus on measurements made by visibility fittings, although the
methods of measuring flux density do not lead to qualitative differences in
our discussion and conclusion.

Upper panel of Figure 3 summarizes the flux density measurements, while the
bottom panel shows the spectral indices derived from these measurements. The
spectral indices ($\alpha$) are 5$\pm$1 and 0.6$\pm$0.2 at 33–44 GHz and 22–33
GHz, respectively. We were surprised by the bright 22 GHz emission, the low
22–33 GHz spectral index, and the 10 GHz non-detection (Figure 3). Based on
the 22 GHz flux density and the 3-$\sigma$ upper limit of the 10 GHz flux
density (Table 1), we derived a lower limit of the 10–22 GHz spectral index to
be 1.7. Our working hypothesis to interpret this SED is provided in Section
4.1.

##  4 Discussion

###  4.1 Interpreting the SED

Table 2: Flux density measurements Observations | Flux density ($\mu$Jy)
---|---
| Reading peakaaFlux densities obtained from reading the peak intensities in the clean images. | Gaussian fittingsbbFlux densities obtained from performing 2D Gaussian fittings in the visibility domain.
JVLA
18–26 GHz (Nov.15) | 114$\pm$17 | 114$\pm$17
18–26 GHz (Nov.17) | 110$\pm$16 | 122$\pm$16
29–37 GHz | 105$\pm$23 | 148$\pm$23
40–48 GHz | 393$\pm$100 | 665$\pm$100
ALMA
97.5 GHz | 4.1$\pm$0.2ccObtained from integrating over the inner 4′′ square region, nominally assuming a 5% absolute flux error.
145 GHz | 15$\pm$0.75ccObtained from integrating over the inner 4′′ square region, nominally assuming a 5% absolute flux error.

####  4.1.1 Emission at $\gtrsim$30 GHz

The $\gtrsim$30 GHz SED of PDS 70 (Figure 3) can be interpreted by dust
thermal emission that is optically thick at $\gtrsim$50 GHz and optically thin
at lower frequencies. For example, we compare the observed SED with the SED
model that was evaluated based on the DSHARP-default dust opacity table,
taking both absorption and scattering opacities into account (for more details
see Appendix B.1; c.f. Birnstiel et al. 2018). We assumed a 20 K mean dust
temperature, which may be consistent with that in the PDS 70 dust ring
(Hashimoto et al. 2012, 2015).

To gauge the maximum dust grain size ($a_{\mbox{\scriptsize max}}$), in the
bottom panel of Figure 3, we compared the observed spectral indices with the
spectral indices of the dust slabs that have $\Sigma_{\mbox{\scriptsize
dust}}=$0.01 g cm-2 dust column densities and $a_{\mbox{\scriptsize max}}=$0.3
mm, 1.0 mm, 2.0 mm, 3.0 mm, 10.0 mm maximum grain sizes. At $\sim$40 GHz
frequency, which is the optically thin part of the spectrum, the spectral
indices of the dust slab models are maximized when the $a_{\mbox{\scriptsize
max}}$ value is in between 1 mm and 2 mm, which makes a good comparison with
the observed high spectral index at 33–44 GHz (Figure 3, bottom). Using the
other $a_{\mbox{\scriptsize max}}$ values which make the spectral index of
this grown dust component smaller will yield a too low spectral index in the
integrated SED model, after mixing with free-free emission or the emission of
the spinning nanometer-sized dust component that has a very low spectral index
(more in Section 4.1.2; Figure 3). Therefore, in our fiducial model which is
presented in the upper panel of Figure 3, we adopted $a_{\mbox{\scriptsize
max}}=$1.5 mm as a working hypothesis. The $a_{\mbox{\scriptsize max}}$ in our
fiducial model should be regarded as a lower limit. It is plausible to assume
a slightly larger $a_{\mbox{\scriptsize max}}$ value. However, when
$a_{\mbox{\scriptsize max}}$ is comparably greater than 1 cm, the lower
absorption opacity spectral index ($\beta$) will start to make the 33–44 GHz
spectral index lower, making it harder to explain the 33–44 GHz spectral
index.

To match the observed $\gtrsim$30 GHz SED, in the fiducial model, the dust
column density ($\Sigma_{\mbox{\scriptsize dust}}$) and solid angle ($\Omega$)
were chosen to be 5.0 g cm-2 and 2$\cdot$10-12 Sr, respectively. The overall
dust mass in this grown dust component is $\sim$180 $M_{\oplus}$, which is
uncertain due to the assumptions of dust opacities.

The values of $\Sigma_{\mbox{\scriptsize dust}}$ and $\Omega$ are degenerated
and thus were loosely constrained. Nevertheless, the $\Omega$ given in our
fiducial model should be regarded as an upper limits, and thus the
$\Sigma_{\mbox{\scriptsize dust}}$ should be a lower limits. We found that if
we increase $\Omega$ and decrease $\Sigma_{\mbox{\scriptsize dust}}$ by the
same factor, the flux densities at 97.5 GHz and 145 GHz will immediately
exceed what have been measured in the ALMA observations (Table 2). On the
other hand, for a model with a 2 times larger $\Sigma_{\mbox{\scriptsize
dust}}$ and a 2 times smaller $\Omega$, the emission at 33 and 44 GHz will be
too optically thick to reproduce the observed, 5$\pm$1 spectral index (Figure
3).

The solid angle $\Omega$ is considerably smaller than that of the millimeter
ring resolved at $>$200 GHz frequencies (Benisty et al. 2021; Isella et al.
2019; Casassus et al. 2019). It is comparable to that of a circular dust slab
that has a $\sim$0$\farcs$18 angular radius, while the geometry of the grown
dust component was not resolved in our JVLA observations. For example, the
grown dust component in our model may represent either a few spatially
unresolved clumps, or a crescent that has $\sim$0$\farcs$1 characteristic
width and $\sim$0$\farcs$3 characteristic length, or a combination of those
two cases. This is consistent with the shape of the crescent in the northwest
(Figure 2; see also Figure 8 of Benisty et al. 2021). This crescent may be a
local concentration of grown dust, which is similar to what has been recently
spatially resolved in the DM Tau disk (Liu et al. 2024).

Finally, we remark that if we extrapolate $F_{\mbox{\scriptsize 97.5 GHz}}$ to
44 GHz by assuming a spectral index of 2.0, the extrapolated 44 GHz flux
density will be 840 $\mu$Jy, which is only 26% higher than our measurement
listed in Table 2. This extrapolation implies that our 44 GHz flux density
measurement was unlikely to be significantly underestimated, otherwise, the
44–97.5 GHz spectral index will become lower than 2.0, which seems unlikely.
It is less likely that our 44 GHz flux density measurement was overestimated,
since the calibration errors tend to yield decoherent signal. Usually, the
JVLA Q band (40–48 GHz) observations are the most difficult to calibrate. Our
measurements at lower frequencies were likely less affected by observational
issues or calibration errors. Since our two epochs of 22 GHz measurements
achieved good consistency, they are unlikely to have been biased by serious
calibration errors (Table 2). We do not have any evidence that the 33 GHz flux
density was seriously biased (i.e., underestimated) due to calibration issues.
Even if the 33 GHz flux density was underestimated, the interpretation for the
observed SED (Figure 3) will be qualitatively unchanged. In that case, we will
have to use a larger $\Sigma_{\mbox{\scriptsize dust}}$ value for the grown
dust component (Appendix B.4).

Figure 4:  Similar to the top panel of Figure 3, but use free-free emission
(Appendix B.2) instead of the emission of spinning nano-meter sized dust to
model the flux densities at low frequencies.

####  4.1.2 Emission at $\lesssim$30 GHz

With an assumed $\sim$20 K dust temperature, the emission at 22–33 GHz has a
too low spectral index (0.6$\pm$0.2; Figure 3) to be interpreted with the
continuum emission of an isothermal dust slab that has a $>$0.1 $\mu$m
$a_{\mbox{\scriptsize max}}$, even with the consideration of the anomalous
reddening effect (Liu 2019). In some previous studies on the relatively
massive disks, such low spectral indices were interpreted by the emission of
optically thick dust sources that has a temperature gradient along the line-
of-sight (e.g., Li et al. 2017; Galván-Madrid et al. 2018; Liu et al. 2019;
Wright et al. 2022). This cannot be the case of PDS 70 since the high spectral
index at 33–44 GHz (Figure 3; Table 2) needs to be interpreted with optically
thin dust emission in the Rayleigh-Jeans limit (Hildebrand 1983).

To interpret the SED of PDS 70, besides the thermal emission of the normally
considered dust ($a_{\mbox{\scriptsize max}}>$0.1 $\mu$m), another emission
mechanism is needed. We first discuss the possibility of interpreting the SED
with normal grown dust thermal emission (Appendix B.1) and free-free emission
(Appendix B.2). The spectrum of the grown dust component is the same as that
of the grown dust model introduced in Section 4.1.1 (Figure 3).

The spectral index of optically thin free-free emission is $\sim-$0.1. In
principle, the spectral index of optically thick free-free emission can be as
high as 2.0 (Rybicki & Lightman 1986). However, an entirely optically thick
free-free emission source with a $\sim$2.0 spectral index is not necessarily
realistic. The realistic ionized wind or jet naturally has an optically thick
core and the relatively optically thin halo (Reynolds 1986). Therefore, the
spectral index of unresolved, relatively optically thick ionized wind or jet
is theoretically expected to be $\sim$0.6 (Wright & Barlow 1975; Anglada et
al. 1998), which is considerably lower than our constrained lower limit at
10–22 GHz. If the actual 10–22 GHz spectral index is constrained to be $>$2,
it will become impossible to interpret the 22 GHz emission with free-free
emission. Moreover, since the optical depths and spectral indices of free-free
emission do not vary with frequency abruptly (Appendix B.2), when using
optically thick free-free emission to interpret the X and K bands non-
detection/observations with a $>$1.7 spectral index, we found that the SED
model will inevitably show large excess at Ka band (33 GHz; Figure 4; see also
the discussion in Appendix B.4). Due to these reasons, we disfavor
interpreting the $\lesssim$30 GHz emission only with free-free emission.

It is not impossible that the source PDS 70 presented radio variability
although the previous radio monitoring surveys have indicated that the chance
is low (e.g. Liu et al. 2014; Dzib et al. 2015; Coutens et al. 2019; Getman et
al. 2022). We note that the hosts of many known disks that presented radio
flares are close binaries (Salter et al. 2008; Curiel et al. 2019; Chung et
al. 2024) while PDS 70 is not a binary system. If PDS 70 underwent a flare of
free-free emission during the two epochs of our K band (22 GHz) observations
and was quiescent during our X band (10 GHz) observations, the apparent 10–22
GHz spectral index could be biased high. Nevertheless, we do not think this is
particularly likely, due to the proximity of our X band and K band
observations in time (Table 1). Moreover, we obtained very consistent 22 GHz
flux densities in the two epochs of K band observations (Table 3), which may
disfavor the interpretation with significant and short duration radio
variability.

The 22–44 GHz spectral profile in Figure 3 appears similar to a combination of
the Rayleigh-Jeans tail of the standard thermal emission of grown dust (c.f.
Hildebrand 1983) and the anomalous microwave emission (AME; for a review see
Dickinson et al. 2018). In the following discussion, we consider the
possibility of attributing the origin of AME to spinning nanometer-sized dust
(Draine & Hensley 2012; Hoang et al. 2018).

In Figure 3, we compare our JVLA observations with the emission of a spinning
nanometer-sized dust component and the emission of the grown dust component
(the same as the grown dust component introduced in Section 4.1.1). We
approximated the SED of the spinning nanometer-sized dust component by quoting
the empirical formula in Hoang et al. (2018) (see Appendix B.3). This SED
model mimics the spectral shape of the observational data, however, has a
tension with the 3-$\sigma$ upper limit at X band (10 GHz). This tension is
merely due to that the empirical formula has a too low spectral index in the
low frequency tail. This tension can be largely alleviated if the spectral
profile of the spinning nano-meter sized dust is narrower. If we adopt the
more realistic SED model of spinning nanometer-sized dust particles (c.f.
Rafikov 2006; Hoang et al. 2018), the narrower spectral profile can be
achieved by using a smaller characteristic grain size (e.g. $\lesssim$2Å) and
a small characteristic width of grain size distribution (e.g., $\lesssim$0.2
in the log space) The detailed model for spinning nano-meter sized dust is
beyond the scope of the present observational study. We remark that if we take
our JVLA Q band measurement of the PDS70 flux density as a spurious detection
and consider the Q band flux density quoted in Table 2 as an upper limit
(Section 3; Figure 1), then the spinning nanometer-sized dust would be the
most prominent emission source at 22 and 33 GHz (see also the discussion in
Appendix B.4).

###  4.2 Physical implication

In a protoplanetary disk, individual nanometer-sized dust particles may carry
net electric charges. Nanometer-sized dust particles thus play an important
role in controlling how the disk is coupled with the magnetic field. Forming a
Keplerian disk may require depleting the nanometer-sized dust particles,
otherwise, the magnetic braking can be too efficient (Zhao et al. 2016). In a
protoplantary disk, nanometer-sized dust particles may be replenished by
destructing larger-sized dust particles (e.g., Hoang et al. 2019).

Infrared observations can detect nanometer-sized dust particles at the disk
surfaces (e.g., Geers et al. 2009; Kokoulina et al. 2021 and references
therein), however, do not sample the disk mid-plane. Whether or not nanometer-
sized dust particles are present in the disk mid-plane has been controversial.
Some previous observations on low-mass (Greaves et al. 2018; Hoang et al.
2018; Curone et al. 2023) and massive disks (Wright et al. 2023) claimed that
the microwave emission of spinning nanometer-sized dust can explained the
centimeter SEDs of their target sources. Nevertheless, it is not necessarily
the only plausible interpretation for their data. Wright et al. (2023) also
considered the possibility of interpreting the same observations with $e^{-}$
free-free emission which is relevant to massive disks, while Curone et al.
(2023) preferred an interpretation of strong free-free variability.

The present work reinforces that it is realistic to consider the emission of
spinning nanometer-sized dust particles in the modeling of $\lesssim$50 GHz
SEDs, although it is not yet clear whether or not this component is prominent
in every protoplanetary disk. The emission of the spinning nanometer-sized
dust particles may resemble the 30–50 GHz excess of millimeter-sized dust
emission. The possibility of detecting the emission of spinning nanometer-
sized dust particles thus makes it harder to confirm the presence of grown
dust if nanometer-sized dust particles commonly present in protoplanetary
disks (Greaves & Mason 2022). In this case, simultaneously detecting the
anomalously reddened and bluened features in the well-sampled SED (Liu 2019;
Liu et al. 2021) and spatially resolving the polarization of dust self-
scattering (Kataoka et al. 2015) may be the most robust approach of confirming
dust growth in the millimeter-sized regime. The wide bandwidth survey (e.g.,
1–50 GHz) on a large number of protoplanetary disks with the JVLA, or in the
future with the Next Generation Very Large Array (ngVLA) and/or the Square
Kilometer Array (SKA), will shed light on these issues.

##  5 Conclusion

We present the first Karl G. Jansky Very Large Array (JVLA) observations on
the PDS 70 disk at Q (40–48 GHz), Ka (29–37 GHz), K (18–26 GHz), and X (8–12
GHz) bands that were taken from late 2019 and early 2020, and the
complementary ALMA Bands 3 (97.5 GHz) and 4 (145 GHz) observations. We
detected PDS 70 at modest significance at Ka and K bands and at a marginal
significance at Q band; we did not detect PDS 70 at X band. The PDS 70 ring
appears lopsided in the ALMA images. The lopsidedness likely signifies the
presence of a dust crescent in the northwest. Based on the resolved spectral
profile and the X band non-detection, we suggest that the flux density at
$\gtrsim$33–44 GHz is dominated by the optically thin thermal emission of a
grown dust component that may have a $\gtrsim 1$ mm maximum grain size. The
grown dust component may represent substructures (e.g., the dust crescent) in
the PDS 70 ring, given that is solid angle is smaller than that of the bulk of
the PDS 70 ring. At lower frequencies the flux densities are not consistent
with the standard dust emission, and the most plausible explanation is that
they are dominated by the emission of spinning nanometer-sized dust particles
The future JVLA, ngVLA, and SKA surveys are important to address how common is
the emission feature of the nanometer-sized dust particles in the
protoplanetary disks.

We thank the referee for the useful suggestions. This paper makes use of the
following ALMA data: ADS/JAO.ALMA#2022.1.01477.S. ALMA is a partnership of ESO
(representing its member states), NSF (USA) and NINS (Japan), together with
NRC (Canada), NSTC and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. H.B.L. is supported by the National
Science and Technology Council (NSTC) of Taiwan (Grant Nos.
111-2112-M-110-022-MY3). T.M. is supported by JSPS KAKENHI Grant Number
JP23K03463. K.D. is supported by JSPS KAKENHI Grant Number JP22KJ1435.

##  Appendix A Discussion on the impact of flux extraction systematics

It is common to measure the overall flux densities by performing 2D Gaussian
fittings in the (clean) image domain or by integrating the flux densities over
the regions in the clean image where the target source(s) is detected above a
certain significance level (e.g., 2-$\sigma$). These methods work well when
the intensity distribution of the target source is resolved with high
significance, such that the sum of the flux density in the clean components is
close to the overall flux density of the target source.

However, when the intensity distribution is resolved at modest significance
(e.g., 2–4-$\sigma$), it is difficult to recover a high fraction of the
overall flux densities in the clean components. Nevertheless, as long as the
innermost region of the synthesized dirty beam is approximately Gaussian, the
sum of the flux densities in the clean components and the flux densities in
the clean residual image is still representative of the overall flux density
of the target source (c.f. Appendix of Casassus & Cárcamo 2022). Therefore,
the two methods of measuring flux densities in the image domain mentioned
above may still work reasonably well.

When the synthesized dirty beam is highly non-Gaussian, the clean components
and the clean residual image are represented by very different beams (i.e.,
point spread functions), leading to an ambiguous Jy beam-1 intensity unit in
the finally restored clean images. In such cases, integrating flux densities
in the image domain will yield biased overall flux densities. In the case that
the target source is spatially unresolved, and when the residual phase and
delay errors are negligible, one can simply measure the overall flux density
by reading the peak intensity value (in Jy beam-1 units) in the image domain,
since the Jy beam-1 unit peak intensity does not depend on the shape of the
synthesized dirty and clean beams. However, if there are some residual phase
and delay errors, the peak intensity should be regarded as a lower limit of
the overall flux density since the phase or delay errors can lead to angular
dispersion and potential decoherence of the flux densities.

Fitting visibility amplitudes is rather immune to phase errors. However, in
the visibility domain, it is hard to distinguish any potential confusing
sources from the target source. Therefore, the flux densities obtained from
visibility fittings should be regarded as upper limits.

In each of the two epochs of K band observations, we verified that the
following four approaches, (i) performing 2D Gaussian fittings in the clean
image domain, (ii) performing 2D Gaussian fittings in the visibility domain,
(iii) reading the peak intensities, and (iv) integrating the flux densities
over the regions that are above 2-$\sigma$ significance, yielded consistent
flux density measurements. We found that reading peak intensities in the image
domain and performing 2D Gaussian fittings in the visibility domain yield
reasonably consistent flux density measurements at Ka and Q bands (Table 2).
Making integration or performing 2D Gaussian fittings in the image domain led
to significant underestimates of flux densities at Ka and Q bands.

##  Appendix B Models of dust emission

###  B.1 Grown dust

To produce the SED models for the grown dust component (Figure 3), we quoted
the DSHARP-default opacity and the Equations (10)–(20) presented in Birnstiel
et al. (2018). For a dust emission source that the angle between the surface
normal and the line-of-sight is $\theta$, those equations evaluate the
observed intensity $I^{\mbox{\scriptsize out}}_{\nu}$ at frequency $\nu$ using
the Eddington-Barbier approximation

$I^{\mbox{\scriptsize
out}}_{\nu}\simeq(1-e^{-\Delta\tau/\mu})S_{\nu}\left((\frac{1}{2}\Delta\tau-\tau_{\nu})/\mu=2/3\right),$
(B1)

where $\Delta\tau$ is the total optical depth evaluated based on the effective
total extinction, $\tau_{\nu}$ is the optical depth evaluated at a specific
scale height, and $\mu=\cos\theta$. The source function is expressed by

$S_{\nu}(\tau_{\nu})=\epsilon^{\mbox{\scriptsize
eff}}_{\nu}B_{\nu}(T_{\mbox{\scriptsize dust}})+(1-\epsilon^{\mbox{\scriptsize
eff}}_{\nu})J_{\nu}(\tau_{\nu}),$ (B2)

where $\epsilon^{\mbox{\scriptsize eff}}_{\nu}$ is the effective total
extinction at frequency $\nu$, $B_{\nu}(T_{\mbox{\scriptsize dust}})$ is the
Planck function at dust temperature $T_{\mbox{\scriptsize dust}}$, and
$J_{\nu}(\tau_{\nu})$ is the mean intensity that can be obtained by
analytically solving the equations of radiative transfer (Miyake & Nakagawa
1993).

The dust grains were assumed to have compact morphology and is composed of
water ice (Warren, 1984), astronomical silicates (Draine, 2003), troilite, and
refractory organics (Henning & Stognienko 1996). We assumed a power-law grain
size distribution function (i.e., $n(a)\propto a^{-q}$) in between the minimum
and maximum grain sizes ($a_{\mbox{\scriptsize min}}$, $a_{\mbox{\scriptsize
max}}$), where the values of $q$ and $a_{\mbox{\scriptsize min}}$ were assumed
to be 3.5 and $10^{-4}$ mm, respectively. The size-averaged opacities derived
based on these assumptions are not sensitive to the assumed
$a_{\mbox{\scriptsize min}}$ values.

Figure 5:  Similar to the top panel of Figure 3, but only use the thermal
emission of a grown ($a_{\mbox{\scriptsize max}}=$1.5 mm) dust component to
model the flux densities. There are eight curves, which present the models
with $\Sigma_{\mbox{\scriptsize dust}}=$1, 5, 10, 15, 20, 25, 30, 35 g cm-2,
respectively. The solid angle and temperature are the same as the assumptions
for the grown dust component in Figure 3.

###  B.2 Free-free emission

Free-free emission is the thermal emission of ionized gas that can be
described by

$F_{\nu}=B_{\nu}(T)(1-{e}^{-\tau_{\nu}^{ff}})\Omega,$ (B3)

where $\Omega$ is the observed solid angle, $\tau_{\nu}^{ff}$ is the optical
depth at frequency $\nu$, and
$B_{\nu}(T)$$=$$(2h\nu^{3}/c^{2})$$(e^{h\nu/k_{B}T}-1)^{-1}$ is the Planck
function at temperature $T$, $h$ and $k_{B}$ are Planck and Boltzmann
constants.

Following Keto (2003) and Mezger & Henderson (1967), we approximate
$\tau_{\nu}^{ff}$ by:

$\tau_{\nu}^{ff}=8.235\times
10^{-2}\left(\frac{T_{e}}{\mbox{K}}\right)^{-1.35}\left(\frac{\nu}{\mbox{GHz}}\right)^{-2.1}\left(\frac{\mbox{EM}}{\mbox{pc\,cm${}^{-6}$}}\right),$
(B4)

where $T_{e}$ is the electron temperature, and EM is the emission measure
defined as EM$=$$\int n_{e}^{2}d\ell$, $n_{e}$ is the electron number volume
density.

For the model present in Figure 4, we chose $T=8000$ K, EM$=$109 pc cm-6, and
$\Omega=1.3\times 10^{-15}$ Sr. This choice is to make the free-free emission
optically thick/thin at below and above $\sim$22 GHz. These parameters are
degenerate due to the lack of observational constraints at $<$22 GHz.

We note that Equations B3 and B4 describe the thermal emission of an ionized
gas slab that has uniform temperature and surface density. When EM is
arbitrarily high, such uniform ionized gas slabs can be optically thick and
thus show $\sim$2.0 spectral indices at a specific frequency. However, in
spite that an optically thick ionized gas slab can reproduce the $>$1.7
spectral index in PDS 70 at 10–22 GHz, such ionized gas slab may not represent
the realistic ionized gas structures in young stellar objects.

In young stellar objects, the ionized gas structures that are spatially
extented enough to contribute to the observed flux densities, are ionized
wind/jet (Reynolds 1986). The realistic ionized gas wind/jet will likely have
a surface density gradient. In that case, the observations at any frequency
will always detect a mixture of optically thick and thin emission which makes
the spectral index lower than 2.0 (Wright & Barlow 1975; Reynolds 1986;
Anglada et al. 1998).

###  B.3 Spinning nanometer-sized dust

We quote Equation (15) of Hoang et al. (2018) to represent the SED of the
spinning nanometer-sized dust particles. The equation is reproduced as
follows:

$F_{\nu}=F_{sd,0}\left(\frac{\nu}{\nu_{pk}}\right)^{2}e^{1-(\frac{\nu}{\nu_{pk}})^{2}},$
(B5)

where $\nu_{pk}$ and $F_{sd,0}$ are the frequency and the flux density at the
peak of the spinning dust spectrum, respectively. The same equation was also
utilized in the studies on the SEDs of external galaxies made by Draine &
Hensley (2012).

We adopted $\nu_{pk}=$21 GHz and $F_{sd,0}=$0.07 mJy. We note that this
equation is an empirical formulation rather than the full analytical
expression for the spectrum of the spinning nano-meter sized dust. We argue
that quoting this Equation is appropriate for our present purpose of comparing
with the observations that were not very well sampled in the frequency domain.
The more sophisticated SED model of spinning nanometer-sized dust particles,
which involves a large number of free parameters (Rafikov 2006; Hoang et al.
2018), may be tested after the detailed spectral profile at $\sim$5–50 GHz is
resolved by future observations.

###  B.4 Consideration of measurement uncertainties

In Section 4, we discussed the SED model that fits the JVLA flux density
measurements made by fitting 2D Gaussian in the visibility domain (Table 2).
Here we briefly discuss the dust emission model that fits the JVLA flux
density measurements made by reading the intensity peaks (i.e., the cyan
points in Figure 3), which we consider have larger uncertainties (Section 3,
Appendix A). The two methods of measuring flux densities yielded very
consistent measurements at K band (22 GHz) while reading intensity peaks
yielded considerably lower flux density measurements at Ka (33 GHz) and Q (44
GHz) bands.

There needs at least one emission mechanism that contributes significantly at
K band (22 GHz), which may be either the emission of spinning nanometer-sized
dust or free-free emission. In either case, fractionally, this emission
mechanism contributes more at Ka band if we adopt the intensity peak instead
of the results of 2D Gaussian fits. This makes it harder to explain the
observed steep spectral slope at 33–44 GHz (Figure 3).

When we adopt the intensity peak instead of the results of 2D Gaussian fits,
free-free emission cannot be a prominent emission mechanism at K band as it
would contribute too much emission at low spectral index at 33–44 GHz
frequencies. The emission of spinning nanometer-sized dust may be the only
plausible dominant emission mechanism at K band while there needs some
adjustments in the grain size distributions to make the flux density drops
rapidly at $>$26 GHz frequencies (c.f. Hoang et al. 2018).

Finally, if we ignore the 33 GHz measurement (e.g., considering that it is
subject to a large systematic calibration error), it would be possible to fit
the 22 GHz and 44 GHz measurements by only considering a grown dust component
($a_{\mbox{\scriptsize max}}\gtrsim$1 mm), without considering the emission of
spinning nanometer-sized dust (Appendix B.3). However, in this case, the close
to $\sim$2.0 spectral index between 22 GHz and 44 GHz (Figure 3) will imply
that the dust emission is not optically thin even at $\sim$22 GHz. To achieve
such a high optical depth, the lower limit of dust mass will be $\sim$5 times
higher than the dust mass given by our fiducial model, which appears
unrealistic to us (Figure 5). In this sense, it is realistic to consider that
at least part of the 22 GHz flux density is contributed by the emission of
spinning nanometer-sized dust.

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Measuring Unruh radiation from accelerated electrons

G. Gregori1a∗, G. Marocco1b,2, S. Sarkar1b, R. Bingham3,4, C. Wang5

1a Atomic & Laser Physics, Parks Road, Oxford OX1 3PU, UK

1b Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU,
UK

2 Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, Berkeley CA 94720,
USA

3 STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX, UK

4 Department of Physics, University of Strathclyde SUPA, Glasgow G4 0NG, UK

5 Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, UK

(* Corresponding author<EMAIL_ADDRESS>
## Abstract

Detecting thermal Unruh radiation from accelerated electrons has presented a
formidable challenge due not only to technical difficulties but also for lack
of conceptual clarity about what is actually seen by a laboratory observer. We
give a summary of the current interpretations along with a simpler heuristic
description that draws on the analogy between the Unruh effect and radiation
from a two-level atomic system. We propose an experiment to uniquely test
whether there is emission of thermal photons from an accelerated electron.

## Introduction

An era of accelerated expansion or ‘inflation’ soon after the Big Bang is
invoked to address several fundamental issues of the Friedmann-Lemaître
cosmology, viz. the horizon and flatness problems [1]. Whereas these problems
are not well-posed (as they require unjustifiable extrapolation back to
$t=0$), it is widely accepted that the generation of a nearly scale-invariant
spectrum of scalar density perturbations during inflation is crucial for
seeding the growth of the observed large-scale structure in the Universe. The
underlying physical mechanism is gravitationally driven particle production
through quantum processes [2, 3, 4], which elegantly brings together general
relativity and quantum field theory. The challenge of physically modelling
such processes is however formidable. We consider whether it may be possible
to _experimentally_ explore gravitational particle production using ultra-high
intensity lasers, thus furthering our theoretical understanding of the very
early Universe.

It is well known that particle-production phenomena can occur in a curved or
dynamic spacetime [5, 6, 7]. Famously, thermal radiation arises from particle
production near the event horizon of a black hole, an effect known as Hawking
radiation [8, 9]. It had been noted earlier that particle production also
occurs due to the varying gravitational field in an expanding Universe [10, 2,
3, 4]. (Although Schrödinger [10] had remarked on this ‘alarming phenomenon’
even earlier, his argument was not correct in detail [11].) Of particular
interest is the inflationary epoch in the early Universe when the Hubble
expansion rate was very high and spacetime was very curved and dynamic.
Understanding particle production during inflation should help to embed it in
a physical framework [12] and also elucidate non-thermal production of
supermassive dark matter in the early universe [13].

Simply put, an expanding space-time causes creation operators to evolve into
superpositions of creation and annihilation operators. This so-called
Bogoliubov transformation implies the creation of particle-antiparticle pairs
— a consequence of the mappings between the Fock (particle number) states
associated with different reference frames. This can happen in a curved
spacetime or in an accelerating frame in flat spacetime [14, 15], both of
which may be interpreted via the equivalence principle as due to gravitational
particle production .

Gravitational particle production has not been directly seen. Whereas
primordial density perturbations are indeed observed via their imprint on the
cosmic microwave background (CMB), to establish that these were generated
during inflation requires detection of the concommitant primordial
gravitational waves via their B-mode polarisation signature in the CMB; such a
claim was made by the BICEP collaboration but subsequently shown to be
unjustified [16, 17]. However, soon to be commissioned multi-petawatt laser
facilities [18] will enable us to access and control the unprecedented large
acceleration they can impart to e.g. electrons which may lead to the
observation for the first time of particle production in a non-inertial
reference frame. This would constitute the experimental confirmation of the
Fulling-Davies-Unruh effect [14, 15, 19] which is considered to be equivalent
to Hawking radiation as a consequence of black hole thermodynamics, and a
pointer to quantum gravity.

Our purpose here is to clarify what can be measured in such experiments and to
guide experimentalists with simple models that follow from basic principles.
We do not attempt rigorous derivations but rather aim to provide physically
meaningful insights. This paper is an expanded version of what we have
presented at the Multi Petawatt Prioritization workshop [20].

## State of the art of our theoretical understanding

As mentioned above, recent developments in ultra-high intensity lasers [21]
have revived interest in the possibility of detecting both the Schwinger
effect and testing non-perturbative QED effects [22, 23, 24, 25]. Projects
such as the European Extreme Light Infrastructure [18] will provide radiation
beams of intensities exceeding $10^{23}\,\mathrm{\>W/cm^{2}}$. An electron
placed at the focus of such beams should then experience an acceleration
comparable to what it would feel at the event horizon of a $6\times 10^{18}$
kg ($3\times 10^{-12}\,M_{\rm Sun}$) black hole. Indeed for such low mass
black holes the surface gravity is strong enough that pairs of entangled
photons can be produced from the vacuum, with one of the pair escaping to
infinity. The black hole radiates and the spectrum has the blackbody form at
the Hawking temperature [8]. However astrophysical black holes are usually
shrouded by the matter they are accreting. Given the difficulties of directly
observing Hawking radiation, Unruh proposed, by appealing to the Equivalence
Principle (of gravitational and inertial accelerations), that a similar effect
(hereafter called just the ‘Unruh effect’) can be measured by an accelerated
observer [19].

While it is generally accepted that the semi-classical derivation of Hawking
radiation is sound, it nevertheless makes use of several approximations that
have not been tested. Similarly, while an accelerated observer is believed to
experience the equivalent Unruh radiation, it has been unclear what a detector
in the laboratory frame will actually measure. Views are split, with some
researchers believing that the answer is contained in ordinary quantum field
theory, while others point out that additional effects due to the acceleration
must be included. A possible reason for such differing views is that defining
what constitutes an experimental proof of Hawking and Unruh radiation requires
a coherent understanding of rather disparate fields: viz. gravity, non-
equilibrium quantum field theory in curved space-time, and high-intensity
lasers.

Figure 1: Heuristic interpretation of the Unruh effect for a two-level atomic
system. When the atom is at rest (top row), high order (virtual) processes
involving the simultaneous emission and absorption of photons are permitted.
However, when the atom is accelerated (bottom row), emission and absorption
can become decorrelated leaving the atom in an excited state and emitting a
photon to a distant observer.

The exact derivation of the Unruh effect is not simple, but physical intuition
can be gained by considering an idealized two-level atom (with energy
separation $\Delta E=h\nu$) that is subject to a constant acceleration $a$
[26, 27]. In the accelerated state, higher-order processes by which a photon
is simultaneously emitted and absorbed can also be considered (see Figure 1).
Since the acceleration incrementally changes the velocity of the atom, the
frequency at which the photon is emitted can be slightly different than the
frequency at which it is absorbed. However, if this change in frequency is
larger than the linewidth of the atomic transition, the emitted photon
disentangles from the absorbed one, and we are left with an atom in the
excited state and a real photon being radiated away as shown in Figure 1. By
Heisenberg’s principle this occurs when $c^{2}/a\lesssim\ell$, where
$\ell=c/\nu$ is the photon correlation length (i.e., wavelength), and we have
assumed the atom moves at close to the speed of light with an acceleration
time $t_{\text{acc}}\sim c/a$. This is the essence of the Unruh effect.
Contrary to what is often said in the literature, the Unruh effect does _not_
require the presence of an event horizon — it is a very general process which
is associated with all accelerated bodies.

If the acceleration continues for a sufficiently long time (we will define
what we mean by this later), the process of emission and absorption of photons
by the atom reaches a steady state. The Unruh effect requires that the
duration of the acceleration be at least comparable to the inverse of
$A_{21}$, the Einstein coefficient for spontaneous emission (for the case of
naturally broadened lineshape). The frequency uncertainty due to the
acceleration is

$\Delta\nu\gtrsim\frac{1}{2\pi t_{acc}}=\frac{a}{2\pi c},$ (1)

so we can set $A_{21}\sim a/2\pi c$. The spectral flux of emitted photons is
then given by

$F_{\nu}\sim h\nu A_{21}\phi_{\nu}\ell^{-2}\sim\frac{ha\nu^{2}}{2\pi c^{3}},$
(2)

where we have taken the photon correlation length to be characteristic size of
the emission volume and $\phi_{\nu}=1/\nu$. In the final step of this
heuristic derivation of the Unruh effect, let us assume that the flux of
emitted photons can be represented in terms of a blackbody function at some
temperature $T_{\text{U}}$. In the low-energy regime, this is simply given by
the Raleigh-Jeans formula:

$F_{\nu}=\frac{2\pi\nu^{2}}{c^{2}}k_{B}T_{\text{U}}.$ (3)

Equating the last two expressions, we get

$T_{\text{U}}=\frac{\hbar a}{2\pi k_{B}c},$ (4)

which is just the Unruh temperature [19].

From the considerations above we see that the key element in the realization
of the Unruh effect is the atom having discrete energy levels, i.e. a system
that can change its state or its internal energy. We call such a system a
detector. Any particle with internal structure, e.g. a proton or a neutron, is
a detector according to this definition. If we describe the two level detector
as a classical harmonic oscillator, the damping rate is obtained by comparing
the correlation length of the system to that of the transition, i.e.,
$\gamma_{cl}=(32\pi^{3}/3)\,\nu(\ell/\lambda)^{2}\sim\nu^{3}(\ell/c)^{2}$
[28], where $\ell\sim c/\nu$ for an atomic transition. Using this the
coefficient of spontaneous emission is [29, 27]:

$A_{21}\sim\frac{g^{2}}{4\pi}\gamma_{cl}\sim\frac{g^{2}}{4\pi}\left(\frac{a}{2\pi
c}\right),$ (5)

where, for generality, we have introduced a coupling constant, $g^{2}/4\pi$
(for discrete atomic transitions involving the emission and absorption of
photons, $g^{2}/4\pi=1$). The arguments here can be easily generalised. in
Figure 1 we see that the accelerated atom can, in principle, couple to the
emission and absorption of other particles e.g. gravitons and neutrinos, and
possibly even those beyond the Standard Model such as axions, dark photons or
millicharged particles [30, 31].

## Larmor power and radiation reaction

A direct observation of the Unruh effect as outlined above would require the
acceleration of atoms. However, because of their larger mass compared to that
of electrons, the acceleration of atoms or ions is challenging. Using
radiation pressure acceleration [32], even at intensities $I_{\text{L}}\sim
10^{25}\,\mathrm{\>W/cm^{2}}$, we expect only $T_{\text{U}}\sim 10^{-3}$ eV.
This temperature would indeed be high enough to affect the ionization state of
the bound levels in the accelerated atom, hence the measurement of the
ionization balance or line emission spectral changes from the atom as function
of the acceleration (particularly for rotational/vibrational spectra of
molecules) would provide possible ways for the detection of Unruh radiation.
However the required laser intensity is well above what is achievable at any
current laser facility. In addition, at $I_{\text{L}}\sim
10^{25}\,\mathrm{\>W/cm^{2}}$ QED effects such as copious production of
electron-positron pairs [33] would already make any detection of changes in
the ionization balance very challenging. As discussed in Ref. [34], for small
accelerations, the Unruh signal from an atom inside a cavity can resonate with
its normal modes, enhancing the overall emission. This may be a promising
avenue for a future experiment; however, the experimental realization of such
a detector is still far from being realised. Recently the possibility of
accelerating an electron in a magnetic field has been considered [35]. While
such a system exhibits finite transitions set by the Landau levels and thus
its behaviour resembles that of an atom, its realisation would still push the
limits of the currently available laser and magnetic field technology.

For all of these reasons, Unruh and others have instead adopted an alternative
approach to exploit the mathematical analogy between trans-sonic flowing water
[36], Bose-Einstein condensates [37] and other analogue systems [38] and the
behaviour of quantum fields in the vicinity of a black hole horizon. While
these experiments have successfully demonstrated the mathematical soundness of
Hawking’s solution, they may have fallen short in proving that the radiation
is actually emitted by non-inertial bodies and that the underlying theory is
indeed physically correct [39].

Electrons, instead, are much easier to accelerate. They would be the ideal
detector to test the Unruh predictions. Large accelerations can already be
realised in the laboratory using current high intensity lasers. For example,
lasers with intensity $I_{\text{L}}\sim 10^{19}$ W/cm2 can accelerate
electrons to an Unruh temperature of about 1 eV [40]. While, in principle,
this can be measured in the laboratory, there is a fundamental issue that has
seldom been discussed in the literature: the electron is a point particle with
no internal structure. This implies that what we mean as Unruh radiation for
an accelerated electrons must be carefully defined. Because of this Unruh
radiation has been a controversial subject in the literature [41],
particularly concerning how to distinguish it from other classical and quantum
radiation processes involving acceleration of charged particles.

Here we address these issues in a more unified approach, with the aim of
clarifying different interpretations of what is meant by radiation from an
accelerated electron. While the electron has no internal energy states, its
interaction with the accelerated vacuum (the Unruh photons) results in
continuous changes of its momentum via electromagnetic scattering [42, 43].
First, for electromagnetic scattering, we take $g^{2}/4\pi=\alpha$ in Eq. (5),
where $\alpha$ is the fine structure constant. These energy changes can be
infinitely small, so $\nu\rightarrow 0$ and $A_{21}\rightarrow 0$. This means
that those small continuous transitions require infinitely long times to
equilibrate and consequently the power radiated by the Unruh effect at those
frequencies is very small. Eq.(5) still remains valid if $\gamma_{cl}\lesssim
a/2\pi c$, implying that the uniform acceleration must continue at least for a
time

$t\gtrsim t_{eq}=\left(\frac{64\pi^{4}h^{2}}{3m^{2}c^{3}a}\right)^{1/3},$ (6)

where $t_{eq}\sim 1/\nu$ is the thermalization time and we have taken
$\ell=\lambda_{C}$, where $\lambda_{C}=h/mc$ is the Compton wavelength, i.e.
the correlation length of the structure-less detector.

We conclude this section by noticing that the power emitted in Unruh radiation
is

$P_{\nu}=\frac{F_{\nu}\ell^{2}}{t_{acc}}\sim{\cal
B}\left(\frac{\hbar}{c^{2}}\right)\frac{g^{2}a^{2}}{4\pi},$ (7)

where the constant ${\cal B}\sim 2/3$ can be understood from the fact that
both transitions where the detector undergoes either an energy gain or an
energy loss are possible (hence the factor of 2) and because of the isotropy
of space in three dimensions (giving the factor 1/3). For the case of an
electron, assuming that it had sufficient time to thermalize, we indeed
reproduce the classical Larmor formula for electromagnetic radiation, which
can be seen as the limit power achieved by accelerated detectors with no
internal structure [44]. The same conclusion was recently also reached in Ref.
[45].

This stands if we consider only the lower order (classical) limit implied by
the spectral flux in Eq. 2. Going beyond the Rayleigh-Jean regime, Eq. 3
becomes

$F_{\nu}=\frac{2\pi h\nu^{3}}{c^{2}}\frac{1}{e^{h\nu/k_{B}T_{\text{U}}}-1}.$
(8)

The above expression assumes that the detector experiences no recoil. However,
as photons scatter with the detector, the latter does experience a finite
recoil $\mu$. The occupation of states described by the Planck function must
then be changed to [46, 45]

$F_{\nu}=\frac{2\pi
h\nu^{3}}{c^{2}}\frac{1}{e^{(h\nu+\mu)/k_{B}T_{\text{U}}}-1},$ (9)

where $\mu=\hbar^{2}k^{2}/2m$ plays the role of a chemical potential. Here $m$
is the mass of the detector and $\hbar k$ represents the recoil momentum.
Since the average photon energy is of order of $k_{B}T_{\text{U}}$, then
$\hbar kc\sim k_{B}T_{\text{U}}$, and we can approximate
$\mu\sim(k_{B}T_{\text{U}})^{2}/2mc^{2}$. Regarding $\mu$ as a small
correction, we have

$F_{\nu}=\frac{2\pi\nu^{2}}{c^{2}}k_{B}T_{\text{U}}\left[1-\frac{(k_{B}T_{\text{U}})^{2}}{2mc^{2}h\nu}\right]\sim\frac{ha\nu^{2}}{2\pi
c^{3}}\left(1-\frac{k_{B}T_{\text{U}}}{2mc^{2}}\right),$ (10)

where in the last step inside the parenthesis we have taken $h\nu\sim
k_{B}T_{U}$. Using the same reasoning as above, the power emitted by Unruh
radiation is

$P_{\nu}\sim{\cal
B}\left(\frac{\hbar}{c^{2}}\right)\frac{g^{2}a^{2}}{4\pi}\left(1-\eta\frac{k_{B}T_{\text{U}}}{2mc^{2}}\right),$
(11)

where again ${\cal B}\sim 2/3$ and $\eta$ is a coefficient that depends on the
details of the detector. For electrons, $\eta=24$ [45]. The above formula
shows that recoil (quantum) corrections to the Larmor formula contain terms of
order $a^{2}k_{B}T_{\text{U}}/mc^{2}$ [47, 45]. This is what is usually
referred to as the Unruh effect for accelerated electrons.

If we now use Eq.(11) and repeat the same calculation used to derive the
Abraham–Lorentz formula for the radiation reaction force, we obtain:

${\bf F}_{RR}\sim{\cal
B}\left(\frac{\hbar}{c^{2}}\right)\frac{g^{2}}{4\pi}\left(1-\eta\frac{k_{B}T_{\text{U}}}{mc^{2}}\right)\mathbfit{\dot{a}}.$
(12)

This provides a first order quantum correction to the non-relativistic
Abraham–Lorentz radiation reaction force. Recently, it has been argued [45]
that the observation of quantum corrections to radiation reaction are an
experimental proof of the Unruh effect. Indeed, quantum radiation reaction
effects have been seen in aligned crystals [48] as well as with high power
lasers [49]. However, since quantum corrections to the Abraham–Lorentz formula
(and its relativistic counterpart) can also be derived directly from QED [25]
without appealing to the Unruh effect, the claim [45] must be reconsidered.

## Avenues to the detection of the Unruh effect

Here we propose a variation of the experimental scheme suggestedin Ref. [50]
as a possible way to directly measure the Unruh effect from accelerated
electrons. This is based on the observation that the quantum recoil correction
to the radiation reaction formula seems to renormalize the fine structure
constant [45]

$\alpha\longrightarrow\left(1-\eta\frac{k_{B}T_{\text{U}}}{mc^{2}}\right)\alpha,$
(13)

which is a consequence of the accelerated electron being dressed by a thermal
photon field. We will discuss this in more detail below. Following directly
our previous paper [50], let us now consider the case of an electron
oscillating in the field produced by a linearly polarized high-intensity laser
beam. We can consider the acceleration to be constant if the laser frequency
is such that (see Eq. 6):

$\nu_{\text{L}}\lesssim\left(\frac{3m^{2}c^{3}a}{64\pi^{4}h^{2}}\right)^{1/3}\approx
4.9\times 10^{17}\left(\frac{I_{\text{L}}}{10^{20}\,\,\rm
W/cm^{2}}\right)^{1/6}\,\,\rm Hz,$ (14)

where $I_{\text{L}}$ is the laser intensity. The above relation is well
satisfied for all current high-intensity optical lasers. Once the electron is
accelerated the second step is to probe its motion using another laser beam.
If the probe laser vector’s potential is denoted as $\bf A$, the effective
interaction Hamiltonian (including the quantum recoil correction seen above)
describing the Thomson scattering process is [50]

${\cal H}_{\rm int}({\bf
r})=\frac{e^{2}}{2m}\left(1-\eta\frac{k_{B}T_{\text{U}}}{mc^{2}}\right)^{2}{\bf
A}\cdot{\bf A}\sim\frac{e^{2}}{2m}{\rm e}^{-\mathbfit{q}\cdot\mathbfit{r}}{\bf
A}\cdot{\bf A},$ (15)

where we have used ${\mathbfit{a}}\sim{\mathbfit{r}}/t_{acc}^{2}$ (with
$\mathbfit{r}$ the position vector of the electron), such that
${\mathbfit{q}}=2\eta k_{B}T_{\text{U}}\mathbfit{a}/mc^{4}$. The probe
frequency must be high enough to sample time scales that are much shorter than
the acceleration equilibration time, that is $c/\lambda_{p}\gtrsim
t_{eq}^{-1}$ (where $\lambda_{p}$ is the probe laser wavelength). In practice,
this implies that the probe must be operated at X-ray wavelengths, and X-ray
free electron lasers [51] are thus ideally suited for this.

We now take an ensemble of uncorrelated electrons at a temperature $T$, all
placed inside the high-power laser spot where they are accelerated as seen in
Figure 2 showing the schematic of the suggested experimental setup. Because of
the modified Hamiltonian, the differential cross section for Thomson
scattering which is the quantity which is measured in the experiment, is
proportional to the dynamic structure factor, $S({\bf
k}+i\mathbfit{q},\omega)$. This is given by [50, 52]

$S({\bf k}+i\mathbfit{q},\omega)=\int S_{0}({\bf k},\omega^{\prime})F({\bf
k}+i\mathbfit{q},\omega-\omega^{\prime})d\omega^{\prime},$ (16)

where ${\bf k}={\bf k_{i}}-{\bf k_{f}}$ is the scattering wavenumber, that is,
the difference between the wavenumber of the incoming x-ray photons and the
wavenumber of the scattered ones as shown in Figure 2, and [50]

$F({\bf k}+i\mathbfit{q},\omega)=\sqrt{\frac{a}{\pi
q}}\exp\left[\frac{(\omega-{\bf k}\cdot{\bf v})^{2}}{qa}\right].$ (17)

Here, $\bf v$ is the velocity of the electrons. For a classical gas in
equilibrium at a temperature $T$ we have [50],

$\displaystyle S_{0}({\bf k},\omega)=$ $\displaystyle\left(\frac{m}{2\pi
k^{2}k_{B}T}\right)^{1/2}\exp\left[-\frac{m}{2k^{2}k_{B}T}\left(\omega-\frac{\hbar
k^{2}}{2m}\right)^{2}\right].$ (18)

The electron motion induced by the high-intensity optical laser remains along
the acceleration direction, which also corresponds to the laser polarization
axis. As shown in Figure 2, we can choose the geometry of the scattering probe
such that ${\bf k}\cdot{\bf v}=0$, and the total structure factor probed by
the X-ray probe reduces to

$\displaystyle S({\bf k}+i\mathbfit{q},\omega)=$
$\displaystyle\left(\frac{m}{2\pi k^{2}k_{B}T_{\rm
eff}}\right)^{1/2}\exp\left[-\frac{m}{2k^{2}k_{B}T_{\rm
eff}}\left(\omega-\frac{\hbar k^{2}}{2m}\right)^{2}\right],$ (19)

implying that the modified Hamiltonian (15) results in the electrons being
probed by the Thomson scattering beam exhibit an effective temperature given
by [50]

$T_{\rm
eff}=T\left(1+\frac{maq}{2k_{B}k^{2}}\right)=T\left[1+4\eta\left(\frac{\pi
k_{B}}{\hbar kc}\right)^{2}T_{\text{U}}^{3}\right].$ (20)

Thus the probe measures a broadened electron velocity distribution

$\displaystyle\frac{\Delta T}{T}=\frac{T_{\rm eff}-T}{T}\approx
0.1\sin^{2}(\theta/2)\times$ (21)
$\displaystyle\left(\frac{I_{\text{L}}}{10^{20}\,\,\rm
W/cm^{2}}\right)^{2/3}\left(\frac{\lambda_{p}}{0.1\,\,\rm nm}\right)^{2},$

where $\theta$ is the scattering angle. The expected broadening is within
current experimental capabilities of detection [53].

Figure 2: Schematic of the setup for the proposed experiment to detect Unruh
radiation. The black arrows indicate the polarization direction of the optical
laser and FEL beams (both vertical in the figure), and ${\bf k}_{\rm OL}$ is
the waveumber of the high-intensity optical laser.

We note that this uncertainty in the outgoing photon energy can also be
interpreted as an uncertainty in the rest mass of the electron [52], that is,
by setting $k_{B}\Delta T=k_{B}(T_{\rm eff}-T)=(\Delta m)c^{2}$, we obtain

$\Delta m=\frac{maq}{2k^{2}c^{2}}=4\eta\left(\frac{\pi}{\hbar
kc^{2}}\right)^{2}(k_{B}T_{\text{U}})^{3}.$ (22)

The above expression is a restatement of the fact that an electron immersed in
an intense plane-wave field exhibits an intensity-dependent alteration of its
mass shell condition [54, 55]. Eq. (22) can be written in the more common mass
shift form by noticing that $a\sim eE_{\text{L}}/m$ and $\omega_{p}\sim kc$
(where $E_{\text{L}}$ and $\omega_{p}$ are the high-intensity optical laser
electric field and probe frequency, respectively):

$\Delta
m^{2}\sim\frac{q^{2}}{4k^{2}}\frac{e^{2}E_{\text{L}}^{2}}{\omega_{p}^{2}c^{2}}.$
(23)

Classically, the exchanged momentum is just $q\sim 2k$, and for the case of a
single laser ($\omega_{p}=\omega_{\text{L}}$, with $\omega_{\text{L}}$ the
drive laser frequency) the above reduces to the usual mass shift expression
[56].

## Relation to the fine structure constant

While the fine structure constant is one of the best measured quantities in
physics, there is no theory that can predict its value from fundamental
principles. It has been speculated that if the electron mass equals the
electromagnetic self-mass calculated with a cut-off set by the Planck length
$\Lambda=(\hbar G/c^{3})^{1/2}$, then it would eliminates the logarithmic
divergences of QED and the fine structure constant should then scale as [57,
58]

$\alpha=\alpha_{0}\left[\ln\left(\frac{\lambda_{C}}{\Lambda}\right)^{2}\right]^{-1},$
(24)

where $\alpha_{0}$ is some undetermined coefficient. If we assume that the
electron mass changes by $m\rightarrow m+\delta m$, then the previous scaling
implies that

$\alpha\rightarrow\alpha\left(1+\frac{2\alpha}{\alpha_{0}}\frac{\delta
m}{m}\right),$ (25)

which is compatible with Eq. (13) if $\delta m\sim T_{\text{U}}$. We also note
that, based on the discussions in the previous section, taking $\hbar kc\sim
k_{B}T_{\text{U}}$, we can rewrite Eq. (22) as:

$\delta m\equiv\Delta m\sim\frac{4\pi^{2}\eta}{c^{2}}k_{B}T_{\text{U}}\sim
T_{\text{U}},$ (26)

which is also consistent with the above renormalization of the fine structure
constant.

Thus if the measurement reveals a broadening as in Eq. (Avenues to the
detection of the Unruh effect) it would provide direct evidence of the fact
that the probe photons are scattering off electrons that are dressed by an
excited vacuum. However, the detection of thermal radiation from an
accelerated oscillator is not, by itself, sufficient to prove the Unruh
effect. What is needed is a general theoretical framework to enable
construction of a robust experimental test. Such a framework must be able to
address e.g. that in the laboratory acceleration is not constant (as assumed
in Unruh’s derivation) [19], and compare Unruh radiation against conventional
QED processes [59].

## Conclusions

The significance of experimentally detecting the Unruh effect can hardly be
overstated. It will be a major scientific milestone marking a new era of
experimental tests of quantum gravity as it will illuminate fundamental issues
related to the (im)possibility of information loss and the smallest possible
scales in Nature.

We note that such experiments at future high-intensity laser facilities may
have impact on diverse technology applications such as quantum computing (as a
quantum process under unscreenable gravitational fluctuations).

## Acknowledgements

GG would like to thank Dr Antonino di Piazza (MPIK) for several discussions
regarding this topic. CW is grateful to the Cruickshank Trust for support.

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