# Densified pupil spectrograph as high-precision radial velocimetry: From
direct measurement of the Universe’s expansion history to characterization of
nearby habitable planet candidates

Taro Matsuo Department of Particle and Astrophysics, Graduate School of
Science, Nagoya University
Furocho, Chikusa-ku, Nagoya, Aichi 466-8601, Japan  NASA Ames Research Center,
Moffett Field, CA 94035, USA  Thomas P. Greene NASA Ames Research Center,
Space Science and Astrobiology Division, M.S. 245-6, Moffett Field, CA 94035,
USA  Mahdi Qezlou Department of Physics & Astronomy, University of California
Riverside, CA 92521, USA  Simeon Bird Department of Physics & Astronomy,
University of California Riverside, CA 92521, USA  Kiyotomo Ichiki  Department
of Particle and Astrophysics, Graduate School of Science, Nagoya University
Furocho, Chikusa-ku, Nagoya, Aichi 466-8601, Japan  Kobayashi-Maskawa
Institute for the Origin of Particles and the Universe, Nagoya University,
Chikusa-ku, Nagoya 464-8602, Japan  Yuka Fujii National Astronomical
Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan  Earth-Life
Science Institute, Tokyo Institute of Technology, Japan  Tomoyasu Yamamuro
Optcraft, Sagamihara, Kanagawa 252-0144, Japan

###### Abstract

The direct measurement of the Universe’s expansion history and the search for
terrestrial planets in habitable zones around solar-type stars require
extremely high-precision radial velocity measures over a decade. This study
proposes an approach for enabling high-precision radial velocity measurements
from space. The concept presents a combination of a high-dispersion densified
pupil spectrograph and a novel telescope line-of-sight monitor. The precision
of the radial velocity measurements is determined by combining the
spectrophotometric accuracy and the quality of the absorption lines in the
recorded spectrum. Therefore, a highly dispersive densified pupil spectrograph
proposed to perform stable spectroscopy can be utilized for high-precision
radial velocity measures. A concept involving the telescope line-of-sight
monitor is developed to minimize the change of the telescope line-of-sight
over a decade. This monitor allows the precise measurement of a long-term
telescope drift without any significant impact on the Airy disk when the
densified pupil spectra are recorded. We analytically derive the uncertainty
of the radial velocity measurements, which is caused by the residual offset of
the line-of-sights at two epochs. We find that the error could be reduced down
to approximately 1 $cm/s$, and the precision will be limited by another factor
(e.g., wavelength calibration uncertainty). A combination of the high
precision spectrophotometry and the high spectral resolving power could open a
new path toward the characterization of nearby non-transiting habitable planet
candidates orbiting late-type stars. We present two simple and compact high-
dispersed densified pupil spectrograph designs for the cosmology and exoplanet
sciences.

techniques: radial velocities — techniques: spectroscopic — cosmology:
cosmological parameters — planets and satellites: terrestrial planets

††journal: AJ

##  1 Introduction

Struve (1952) proposed the performance of high-dispersion spectroscopy to
search for unseen companions (i.e., exoplanets) by detecting the periodic
change in the radial velocity of a star due to the Doppler effect. After
improving the precision of the radial velocity measurements for over 20 years,
a Jupiter-mass companion with an orbital period of approximately 3.5 days
around Pegasi 51 was finally successfully discovered with a high-dispersion
ELODIE spectrograph at the Haute-Provence Observatory in 1995 (Mayor & Queloz,
1995). About a quarter of 4000 exoplanets confirmed so far have been detected
by the radial velocity measurements based on an exoplanet
database111http://exoplanet.eu/. This has rapidly expanded the research
field’s focus to search for life activity on exoplanets by characterizing the
terrestrial planet atmospheres in the habitable zones around various types of
stars. Several candidates found around nearby late-type stars, such as Proxima
Centauri, Teegarden, and Trappist-1, have been confirmed by the radial
velocity and transit photometry measurements (e.g., Anglada-Escudé et al.,
2016; Zechmeister et al., 2019; Gillon et al., 2016), and transit spectroscopy
and high-contrast imaging will characterize the atmospheres in the next two
decades. Thanks to the low contrast ratio between a planet and its host star
in the mid-infrared regime, the multi-band photometry and spectroscopy
characterize the atmospheres of non-transiting terrestrial planets in the
habitable zones around nearby late-type stars (Kreidberg & Loeb, 2016; Snellen
et al., 2017; Fujii & Matsuo, 2021). In contrast, no candidates around Sun-
like stars have been detected because of stricter requirements. The detection
of candidates requires maintaining the precision of an order of 10 $cm/s$ over
a few years. The precision of the current radial velocity planet-search
instruments, including HARPS (Pepe et al., 2002) and HIRES (Vogt et al.,
1994), is limited to approximately 1 $m/s$ over a few years before 2016
(Fischer et al., 2016). However, the next-generation radial velocity programs,
such as the Echelle Spectrograph for Rocky Exoplanets and Stable Spectroscopic
Observations(ESPRESSO: Pepe et al., 2014; González Hernández et al., 2018) and
the Extreme Precision Spectrograph (EXPRES: Jurgenson et al., 2016), could
achieve the precision of a few tens of $cm/s$ on the sky over a timescale of a
few hours to a few days (e.g., Petersburg et al., 2020; Pepe et al., 2021).
Thanks to advanced modal noise reduction techniques, advanced radial velocity
instruments mostly overcame the continuous change in the atmospheric seeing
conditions (Mahadevan et al., 2014; Halverson et al., 2015; Sirk et al., 2018;
Petersburg et al., 2018). Note that the modal noise in a multi-mode fiber
indicates coherent speckles at the output end of the fiber induced by optical
interference (e.g., Lemke et al., 2011; Oliva et al., 2019). In addition, a
precise wavelength calibration using the laser frequency comb (Wilken et al.,
2012; Molaro et al., 2013) compensates for a shift of the relative position
between the spectrograph and the detector. In addition to the ground-based
planet-survey programs, a simpler approach for space-based observatories has
been investigated as a probe mission concept study of the Astro2020 Decadal
Survey (Plavchan & Vasisht, 2019). This approach can avoid a fundamental
limitation caused by the telluric atmosphere, corresponding to $\sim$ 10 - 30
$cm/s$ in visible, and mitigate the stellar jitter with a wide observing
bandwidth ranging from UV to near-infrared. A compact high-dispersion
spectrograph suitable for space observatories could be realized by combining a
Virtually Imaged Phased Array (VIPA) originally proposed for telecommunication
(Shirasaki, 1996) as an Echelle grating with a general grating as a cross-
disperser (Bourdarot et al., 2018; Zhu et al., 2020). The large dispersion
angle of the VIPA can keep the size of the spectrograph with a resolving power
of more than 100,000 below 1 $m$ long. However, the radial velocity
measurements on both the ground- and space-based observatories will be
eventually limited by the stellar jitter caused by an inhomogeneous surface
with spots and plagues, which makes difficult to detect habitable planet
candidates around Sun-like stars. Based on the long-term monitor of the Sun
over three years (Dumusque et al., 2015, 2021), the Sun’s radial velocity
changes at the level of 1 $m/s$ from day to day.

Loeb (1998) proposed the application of existing high-precision radial-
velocity instruments developed for exoplanet search to constraints of the
cosmological parameters, such as the density parameters of the universe and
the Hubble constant, developing an idea on the direct measurement of its
global dynamics (Sandage, 1962). Two spectroscopic observations with a time
separation of 10 years would possibly allow us to detect the small redshift
drifts of extragalactic objects if the signal-to-noise ratio of the cosmic
signal is statistically proportional to the square root of the number of the
observed astronomical objects. Neutral hydrogen atoms (HI) between the object
and the Earth make a number of the Lyman alpha absorption lines in the shorter
wavelength range than the Lyman alpha break (i.e., Lyman alpha forest); hence,
quasars (hereafter referred to as QSOs) as one of the brightest extragalactic
objects are a suitable background source for this purpose. Many absorption
lines in the Lyman alpha forest improve the radial velocity precision. This
idea was first applied to the HI 21 $cm$ absorption line in the continuum
radio spectrum of a QSO, namely 3C286 (Davis & May, 1978). The redshift drift
of the HI 21 $cm$ again became the focus, thanks to the emergence of the Super
Kilometer Array (SKA) (Darling, 2012). The advantage of these redshift drift
measurements is looking over the dynamical evolution of the universe without
the cold dark matter model as the standard cosmological scenario. In contrast,
the other measurements, including the cosmic microwave background (CMB: Sachs
& Wolfe, 1967; Smoot et al., 1992), the baryon acoustic oscillation (BAO:
Sunyaev & Zeldovich, 1970; Peebles & Yu, 1970), and the Type Ia supernova
surveys (Goobar & Perlmutter, 1995), require the standard cosmological model.
Although the standard cosmological model has been widely accepted by
successfully describing the universe structure at different cosmic epochs, a
significant ($>4\sigma$) discrepancy between the Hubble constants derived from
the measurements of the early and late universe, called the Hubble tension,
was recently revealed (e.g., Riess et al., 2016; Verde et al., 2019). The
standard cosmological model may require modification and new physics (e.g., Di
Valentino et al., 2021). The initial condition of the CMB could not be
extrapolated to its entire history; thus, even though the other methods
successfully constrained the cosmological parameters and the Hubble constant
(e.g., Perlmutter et al., 1999; Spergel et al., 2003, 2007; Eisenstein et al.,
2005), it is valuable to directly measure them from the redshift drift of an
object without the $\Lambda$CDM model. Focusing on the fact that the Lyman
alpha forest is produced by nutral hydrogen medium between the QSO and the
observer, we can trace the expansion history of the Universe (i.e., redshift
drifts at various redshifts) from an observation of a QSO, which may lead to
solve the Hubble tension and discover new physics.

For this purpose, the COsmic Dynamic and EXo-earth experiment (CODEX) to be
mounted on the European Extremely Large Telescope (E-ELT) aims to reach an
order of a few $cm/s$ over a decade in the era of extremely large telescopes
(Pasquini et al., 2008). Note that the project name of CODEX was changed to
HIRES after the two science communities for the optical and near-infrared
high-resolution spectrographs were merged. The HIRES instrument could restrict
the cosmological parameters if the photon noise mainly limits the measurement
accuracy of the redshift drift (Liske et al., 2008; Maiolino et al., 2013).
However, the redshift drifts of QSOs at $z=2-5$ over a decade are expected to
be only a few $cm/s$ based on the latest results from the Planck satellite and
the Sloan Digital Sky Survey (Planck Collaboration et al., 2020; eBOSS
Collaboration et al., 2020). In other words, the systematic error of the
radial-velocity measurements should be reduced down to the same level, as
discussed in Section 2.

The uncertainty of the radial velocity measurements was first studied under
the assumption that the spectrograph can achieve a photon-noise-limited
performance (Connes, 1985; Bouchy et al., 2001; Figueira, 2018). In parallel,
radial velocity planet-search programs revealed that not only the systematic
noises, but also the random noises, such as the photon and detector noises,
limit the precision of the radial velocity measurements (Tuomi, 2013). While
the measurement errors of the radial velocity instruments are reduced in
proportion to the signal-to-noise ratio in its low regime, these errors are
almost constant in its high regime (Fischer et al., 2016). Following these
studies, Plavchan et al. (2015) revealed what types of systematic noises on
fiber-fed high-dispersion spectrographs placed in a temperature-controlled
room (e.g., Mayor et al., 2003) affect the radial velocity measurements. Each
instrumental noise has been quantitatively evaluated (e.g., Halverson et al.,
2016; Blackman et al., 2020). Next-generation radial velocity instruments
could reduce the noise down to an order of 10 $cm/s$, thanks to the relevant
technological advancements (e.g., Wilken et al., 2012; Mahadevan et al., 2014;
Halverson et al., 2015). While terrestrial planets in the habitable zones of
nearby G-type stars could be found in the next decade, whether the Doppler
drifts of the QSOs at $z=2-5$, corresponding to a few $cm/s$ over a decade,
are measured is still an open question. Even if the next-generation
instruments can achieve a photon-noise-limited performance, different
observatories should independently validate the measured cosmological
parameters. HIRES is the only instrument designed for this purpose.

Here, we notice that the radial velocity measurements have an important
similarity to transit spectroscopy. Two types of methods obtain the difference
between the spectra recorded at two epochs. Considering that transit
spectroscopy is directly limited by the photometric precision of the
spectrograph, there is also a relation between the precision of the radial
velocity measurements and the photometric precision. A densified pupil
spectrograph concept proposed for transit spectroscopy of exoplanets from
space (Matsuo et al., 2016) could be applied to the radial velocity
measurements. The densified pupil spectrograph achieves the high precision
spectrophotometry by producing the spectra of multiple sub-pupils on the
detector plane, to which the primary mirror is optically conjugated. Low-order
aberrations, such as the telescope line-of-sight jitter (i.e., pointing
stability) and the primary mirror deformation, do not, in principle, change
the spectral positions and sizes on the detector. On the other hand, the main
difference between the radial velocity measurements and transit spectroscopy
is the resolving power imposed on the spectrograph. While transit spectroscopy
reduces the resolving power due to a decrease in the shot noise for each
spectral element, the radial velocity measurements require a high resolving
power for resolving a number of the absorption lines recorded in the spectrum.
Therefore, we propose a high-dispersed densified pupil concept for enabling
the high-precision radial velocity measurements. Furthermore, we notice that a
combination of the high photometric precision and the high spectral resolving
power could measure the Doppler shift of the planet spectrum embedded in the
bright stellar light. As the systematic error is smaller, the stellar light
gives a smaller impact on the planet spectrum. The high photometric precision
allows us to observe planetary systems with high planet-to-star contrast
ratios. It means that a new approach toward characterizing the reflected light
of nearby non-transiting potential habitable planets can be developed without
a high angular and high contrast technique. Thus, the high-dispersion
densified pupil spectrograph concept may be useful not only for enabling the
high-precision radial velocity measurements but also for characterizing the
atmospheres of nearby habitable planet candidates.

However, we must consider the new impact of the telescope line-of-sight drift
on the photometric stability in visible and near-infrared regimes. While the
surface figure errors of the dispersive element placed on the focal plane
could be regarded as negligible at the mid-infrared wavelengths, the speckles
caused by the wavefront errors on the detector plane should be considered at
shorter wavelengths. The line-of-sight jitter changes the image position on
the dispersive element; thus, various speckle patterns are formed during
observations. Finally, the drift of the telescope line-of-sight during one
observation and its difference between two observations with a period of a
decade will limit the accuracy of the radial velocity measurements. We
introduce herein a new sensor concept for measuring the image position on the
dispersive element. This sensor could precisely work without any significant
impact on the point spread function (PSF) core at the same time when the
densified pupil spectra are recorded. This sensor is different from general
slit viewers placed on the spectrograph entrance (e.g., Rayner et al., 2003;
Iseki et al., 2008). The photon-noise-limited precision of the radial velocity
measurements may be possible by combining the densified pupil spectrograph
with this type of sensor.

This study proposes a new approach for achieving high-precision radial
velocity measurements over a decade and characterizing the atmospheres of
nearby habitable planet candidates on future space observatories. This concept
is mainly composed of two sub-systems: 1) a high-dispersion densified pupil
spectrograph; and 2) a telescope line-of-sight monitor. In Section 2, we
derive the precision of the radial velocity measurements and summarize the
requirements for enabling the photon-noise-limited performance. Section 3
presents an overview and the limitation based on an analysis of the wavefront
propagation through the spectrograph. Sections 4 and 5 show the optical
designs of the high-dispersion spectrographs optimized for direct measurement
of the universe’s expansion history and characterization of the atmospheres of
nearby habitable planet candidates orbiting late-type stars, respectively.
While the former spectrograph has the resolving power of approximately 10,000
in the optical wavelength regime, the resolving power of the latter one could
reach to 70,000 over the wavelength range of 750 to 980 $nm$. We investigate
the possibilities of whether the two science cases are realized with this
approach. Finally, we discuss why this proposed approach could be beneficial
for space telescopes, comparing this concept with ground-based fiber-fed
spectrographs for the Doppler planet search in Section 6.

##  2 Basics of radial-velocity measurements

Before presenting the concept overview, we will explain why the systematic
noise floor should be considered for enabling high-precision radial velocity
measurements. In this section, we derive the uncertainty of the radial
velocity measurements caused by the shot noise and the systematic instrumental
noise. We also discuss the requirements for enabling the photon-noise-limited
performance and compare them with those of transit spectroscopy.

###  2.1 Ideal limitation

According to the previous studies (Connes, 1985; Bouchy et al., 2001;
Figueira, 2018), the radial velocity signal for the $i$-th spectral element is

$\frac{\delta V}{c}(i)=\frac{N_{2}(i)-N_{1}(i)}{\lambda(i)\left(\frac{\partial
N_{true}}{\partial\lambda}(i)\right)},$ (1)

where $\lambda$ is the observation wavelength; $N_{1}(i)$ and $N_{2}(i)$ are
the numbers of the observed photons for the $i$-th spectral element at the two
epochs 1 and 2, respectively; and $N_{true}(i)$ is the true spectrum of a
target for the $i$-th spectral element. Each spectral element has a different
error, the estimation of the radial velocity is written as an weighted average
of the radial-velocity signals over the entire wavelength range:

$\left(\frac{\delta V}{c}\right)_{wav}=\frac{\sum_{i}(\frac{\delta
V}{c}(i)W(i))}{\sum_{i}W(i)},$ (2)

where $W(i)$ shows the optimum weight function for the $i$-th spectral
element. The weighted average is equal to the maximum likelihood estimate
under a condition that the optimum weight function is proportional to the
inverse square of the standard deviation of each measurement value:

$W(i)=\frac{1}{\sigma^{2}(i)},$ (3)

where $\sigma(i)$ is the random noise attached to the radial velocity signal
of the $i$-th spectral element. Given that there is no systematic effect
originated from the telescope and instrument, the signal variation is only
caused by the shot noise, $\sigma_{shot}$; this assumption corresponds to the
fundamental limitation case. Under this assumption, the variance of the
weighted average, $\sigma^{2}_{wav}$, which represents the square of the
uncertainty of the radial-velocity measurements, is presented as

$\displaystyle\sigma^{2}_{wav}$ $\displaystyle=$
$\displaystyle\frac{1}{\sum_{i}W(i)}$ (4) $\displaystyle=$
$\displaystyle\frac{1}{\sum_{i}\frac{1}{\sigma^{2}_{shot}(i)}},$

where we assumed that the spectrum slope is not changed during the two epochs.
Given that the shot noise is independently generated at each epoch, the root-
mean-square of the difference between the observed data at the two epochs,
$[N_{2}(i)-N_{1}(i)]_{RMS}$, is $\sqrt{2N_{true}(i)}$; hence, $\sigma^{2}(i)$
is written as follows:

$\sigma^{2}(i)=\frac{2N_{true}(i)}{\lambda(i)\left(\frac{\partial
N_{true}}{\partial\lambda}(i)\right)}.$ (5)

The weighted average variance under the fundamental limitation case is
rewritten as

$\displaystyle\sigma^{2}_{wav}$ $\displaystyle\simeq$
$\displaystyle\frac{2}{\sum_{i}\frac{\lambda^{2}(i)}{N_{true}(i)}\left(\frac{\partial
N_{true}}{\partial\lambda}(i)\right)^{2}}$ (6) $\displaystyle=$
$\displaystyle\frac{2}{Q_{photon}^{2}\sum_{i}N_{true}(i)},$

where $Q_{photon}$ is a dimensionless quantity that represents how much the
obtained spectrum is suitable for the radial velocity measurements in terms of
the spectral shape. The uncertainty of the radial velocity measurements is
decreased by the square root of the number of the photo-electrons collected
over the entire wavelength range, $\sqrt{\sum_{i}N_{true}(i)}$. The
$Q_{photon}$ factor is defined as follows:

$Q_{photon}\equiv\frac{\sqrt{\sum_{i}\frac{\lambda^{2}(i)}{N_{true}(i)}\left(\frac{\partial
N_{true}}{\partial\lambda}(i)\right)^{2}}}{\sqrt{\sum_{i}N_{true}(i)}}.$ (7)

The $Q_{photon}$ factor defined above is equal to the original $Q$ specified
by the previous studies (Connes et al., 1996; Bouchy et al., 2001). Note that,
if a change in the spectrum slope at the two epochs is not negligible, the
change contributes to the weighted average variance and the $Q_{photon}$
factor (Liske et al., 2008). The $Q_{photon}$ factor strongly depends on the
resolving power and the intrinsic spectrum of the target. A higher resolving
power resolves the spectral feature more; hence, the $Q_{photon}$ factor
generally increases as the resolving power is higher. The uncertainty of the
radial velocity measurements for the photon-noise-limited case is determined
by the $Q_{photon}$ factor and the number of the photons collected over the
observation bandwidth, $\sum_{i}N_{true}(i)$. The uncertainty is inversely
proportional to the square root of the integration time, which is the same as
the relationship between the signal-to-noise ratio and the integration time
for general observations.

###  2.2 Impact of systematic noise on the radial velocity measurements

In reality, additional random signal variations caused by the detector noise
and the random line-of-sight jitter exist. These signal variations increase
the variance of each spectral element, $\sigma^{2}$; thus, the integration
time required for reduction to the same variance level is longer than that for
the fundamental limitation case. Furthermore, some systematic signal
variations originate from unknown systematic effects, such as the change of
the detector gain and the telescope line-of-sight drift (i.e., low-frequency
pointing error). The static error cannot be smoothed out, even by averaging
the obtained data over a long term; therefore, the measurement precision would
be finally limited by the static systematic noises. Given that the systematic
noise distribution is a Gaussian function centered at 0 along the spectral
element, the variance of each spectral element, $\sigma^{2}$, can be written
as the sum squared of the random and systematic noises:

$\sigma^{2}(i,t_{integ})=\sigma^{2}_{shot}(i,t_{integ})+\sigma^{2}_{detector}(t_{integ})+\sigma^{2}_{jitter}(i,t_{integ})+\sigma^{2}_{static}(i),$
(8)

where $\sigma_{detector}$ and $\sigma_{jitter}$ are the signal variations
caused by the random detector noises and the telescope line-of-sight jitter,
respectively, $\sigma_{static}$ is a static noise caused by the telescope
pointing drift and the systematic change in the detector gain, and $t_{integ}$
is the integration time. All noises, except for the detector noises, have
different wavelength dependencies. As will be discussed in Section 3, the
signal variations induced by the line-of-sight jitter and drift originate from
a partial loss of the amplitude spread function (ASF) on the entrance slit of
the spectrograph (hearafter referred as to the ”motion loss”).

All noises, except for the static noises, could be roughly reduced by the
squared root of the integration time, $t_{integ}$; hence, the standard
deviation of each spectral element becomes the static systematic noise under
the condition that the integration time is infinite. Given that the systematic
noise floor of the $i$-th spectral element is $\epsilon(i)$, the systematic
error of the difference between the observed data at two epochs limited by the
static noise is approximately written as $\epsilon(i)N_{true}(i)$. When the
integration time, $t_{integ}$, goes to infinity, the weighted average variance
is reduced to

$\lim_{t_{integ}\to\infty}\sigma^{2}_{wav}\simeq\frac{1}{\sum_{i}\frac{1}{\sigma^{2}_{static}(i)}}.$
(9)

The static noise for the $i$-th spectral element is derived as follows through
its error propagation:

$\sigma_{static}(i)=\frac{\partial\left(\frac{\delta V}{c}(i)\right)}{\partial
u(i)}(u(i)-\overline{u}),$ (10)

where $u(i)$ shows the difference between the numbers of the photons collected
at two epochs for the $i$-th spectral element, $N_{2}(i)-N_{1}(i)$, and
$\overline{u}$ is its average. The radial velocity signal variance can be
rewritten with Equation 1 as follows because the systematic error of
$u(i)-\overline{u}$ is defined as $\epsilon(i)N_{true}(i)$:

$\lim_{t_{integ}\to\infty}\sigma^{2}_{wav}=\frac{1}{\sum_{i}\left(\frac{1}{\epsilon(i)}\frac{\lambda(i)}{N_{true}(i)}\left(\frac{\partial
N_{true}}{\partial\lambda}(i)\right)\right)^{2}}.$ (11)

The statistics of the static systematic noise is independent of that of the
object spectrum. $\left(\frac{\lambda(i)}{N_{true}(i)}\frac{\partial
N_{true}}{\partial\lambda}(i)\right)^{2}$ is roughly close to a log-normal
distribution. When the static noise is Gaussian distribution along the
wavelength, the right-hand side of Equation 11 is more than a few times
smaller than a simple multiplication of
$\frac{1}{\sum_{i}\left(\frac{\lambda(i)}{N_{true}(i)}\left(\frac{\partial
N_{true}}{\partial\lambda}(i)\right)\right)^{2}}$ and $\bar{\epsilon}$, where
$\bar{\epsilon}$ is the average of the systematic noise over the entire
wavelength range. This is because $\epsilon(i)$ close to 0 largely increases
the denominator in the right-hand side of Equation 11. We numerically
confirmed it, using the simulation spectra of the Lyaman alpha forest and the
Sun-like star, which will be introduced later in this subsection. In contrast,
if a strong correlation between the static noises of the different spectral
elements exists, the uncertainty of the measurements could be larger than for
the no-correlation case. The static noise tends to originate from the
telescope line-of-sight drift and the change in the detector gain; therefore,
the systematic noise errors are correlated to some extent. When the static
noise has the same value of $\bar{\epsilon}$ over the whole wavelength range,
the radial velocity signal variance is maximized:

$\lim_{t_{integ}\to\infty}\sigma^{2}_{wav}\leq\frac{\bar{\epsilon}^{2}}{Q_{sys}^{2}},$
(12)

where $Q_{sys}$ is

$Q_{sys}=\sqrt{\sum_{i}\left(\frac{\lambda(i)}{N_{true}(i)}\frac{\partial
N_{true}}{\partial\lambda}(i)\right)^{2}}.$ (13)

The systematic noises for all of the spectral elements contribute to the
variance of $\sigma_{wav}^{2}$ in the above equation; hence, the above
assumption corresponds to the worst scenario for the radial velocity
measurements.

$Q_{sys}$ is also a dimensionless quantity that almost looks the same as the
original $Q_{photon}$ shown in Equation 7. However, the two values are greatly
different. Figure 1 shows the $Q_{photon}$ and $Q_{sys}$ factors as a function
of the resolving power for two spectra: a Lyman-alpha forest of a QSO at $z=3$
in optical and a spectrum of a Sun-like star in optical. The Lyman alpha
forest spectra used in this study were generated from the Illustris-TNG
simulation222https://www.tng-project.org/ (Springel et al., 2018; Marinacci et
al., 2018; Naiman et al., 2018; Nelson et al., 2018; Pillepich et al., 2018a)
using the $fake\\_spectra$ 333https://github.com/sbird/fake_spectra package
(Bird et al., 2015). Appendix A describes the details of the simulation
parameters. The spectrum of a Sun-like star with the surface gravity of 4.5
$log(cm/s^{2})$, effective temperature of 5800 $K$, and solar metallicity was
generated by the $BT-settle$ model (Allard et al., 2012) in the theoretical
spectra web server444http://svo2.cab.inta-csic.es/theory/newov2/. The
difference of the two values became larger as the resolving power increased.
The number of the spectral elements was almost proportional to the resolving
power; thus, the ratio of $Q_{sys}$ to $Q_{photon}$ almost increased with the
square root of the resolving power (Figure 1).

Figure 2 shows the uncertainty of the radial velocity measures caused by the
systematic errors for the worst case. The measurement uncertainty was
determined by a combination of the resolving power and the systematic noise
error. The $Q_{sys}$ factor became larger as the resolving power became higher
(Figure 1); therefore, the instrumental noise floor requirement was mitigated
more for the higher resolving power. To achive 1 $cm/s$ precision, the noise
floor should be reduced down to a few and a few tens of parts-per-million
($ppm$) for the resolving powers of 10,000 and 100,000, respectively. Although
the precision of actual radial velocity measurements would lie above the worst
scenario in reality, the systematic noise floor requirement should be equal to
that derived based on the worst sceanrio for reliable measurements because the
distribution of the systematic errors along the wavelength is unknown. In this
paper, we applied Equation 12 as the radial velocity measurement unceratinty
caused by the systematic noise errors.

A change in the intrinsic spectrum of a target at two epochs could become
another systematic effect. In this study, we neglect the change in the
intrinsic spectrum over a decade. While spots and faculae on the surface limit
the precision of the radial velocity measurements in optical (e.g., Makarov et
al., 2009), the systematic effects would be largely reduced at longer
wavelengths (Marchwinski et al., 2015). The assumption is also appropriate for
the direct measurement of the cosmological parameters because a number of
objects are included in each QSO (Loeb, 1998).

Figure 1: $Q_{photon}$ and $Q_{sys}$ factors as a function of the resolving
power for (a) the Lyman alpha forest of a QSO at $z=3$ and (b) the spectrum of
the Sun-like star in optical, respectively. The blue and red lines show the
$Q_{photon}$ and $Q_{sys}$ factors, respectively. The wavelength ranges of the
two cases are approximately (a) 450 - 480 $nm$ and (b) 450 - 550 $nm$,
respectively. Section A describes in detail the simulation of the Lyman alpha
forest spectra.  Figure 2: Uncertainty of the radial-velocity measurements
caused by the systematic noise error for (a) the Lyman alpha forest of a QSO
at $z=3$ and (b) the spectrum of the Sun-like star in optical as a function of
the resolving power, respectively. The red, purple, and blue lines represent
the precisions of $1cm/s$, $10cm/s$, and $1m/s$, respectively.

###  2.3 Similarities and differences between radial velocity and transit
spectroscopy

The signal of the radial velocity measures is a change in spectra recorded at
two epochs along the wavelength direction; hence, high-precision radial
velocity measurements require not an absolute, but a relative photometric
precision. A spectrograph will reach the photon-noise-limited performance in
terms of the radial velocity measures under the assumption that the
observation and the instrument conditions at the two epochs are perfectly the
same. Conversely, when the conditions are different, the precision of the
radial velocity measurements would be limited by unknown systematic effects.
The transit spectroscopy signal is the difference between the spectra at in-
and out-of-transits; thus, both the radial velocity measurements and transit
spectroscopy need a similar instrument concept to improve their performances.
The key parameter of the instrument concept is ”stability,” which is defined
herein as the ability of keeping an instrument condition at two epochs. A high
stability indicates that the conversion factor of the incident photons (to the
primary mirror) to the digitalized data is constant at two epochs.

Based on the considerations in the previous subsection, the following two
conditions are required to achieve the photon-noise-limited performance. 1.
The random signal variation should be mainly determined by the photon noise:

$\sigma_{shot}(i,t_{integ})>\sqrt{\sigma^{2}_{detector}(t_{integ})+\sigma^{2}_{jitter}(i,t_{integ})}.$
(14)

2\. The uncertainty of the radial velocity measurements induced by the
systematic effects should be smaller than the signal variation due to the shot
noise:

$\frac{\bar{\epsilon}}{Q_{sys}}<\frac{1}{Q_{photon}\sqrt{\sum_{i}N_{true}(i)}}.$
(15)

These conditions are similar to those imposed for transit spectroscopy for
characterizing the exoplanet atmospheres.

Another key parameter for the two observing methods is the resolving power of
the spectrograph. The radial velocity and transit spectroscopy require
different resolving power. While transit spectroscopy demands relatively low
resolving power to reduce the shot noise for each spectral element, the radial
velocity measurement requires a higher resolving power to enhance the
$Q_{photon}$ and the newly defined $Q_{sys}$ factors. The higher $Q_{sys}$
could mitigate the requirement on the systematic noise floor. In addition, the
time-spans for maintaining the photon-noise-limited performance (i.e.,
photometric stability) are different for the two cases. Transit spectroscopy,
including the phase curve measurements, typically spans from a few hours to a
few days, while the radial velocity measurements for the search for Earth-
twins orbiting Sun-like stars and the direct measurement of the Universe’s
expansion history require the time-spans of a few years to a decade. Thus,
high-precision radial velocity measurements must simultaneously achieve both
high-dispersion spectroscopy and photon-noise-limited performance over a time
span of a few years to a decade. The requirements imposed on the radial
velocity measurements are tougher than those for transit spectroscopy.

##  3 Instrument concept

In this section, we propose an instrument concept for enabling high-precision
radial velocity measurements based on the requirements discussed in Section 2.
This concept is a combination of a high-dispersion densified pupil
spectrograph and a novel telescope line-of-sight monitor.

###  3.1 Overview

The direct measurement of the Universe’s expansion history and the search for
habitable planet candidates around Sun-like stars require a spectrograph that
maintains the photon-noise-limited performance over a decade. We present a new
possibility for realizing these science cases by developing an instrument
concept that is complementary to widely common high-dispersion spectrographs.
This instrument concept is mainly composed of two sub-systems: 1) a high-
dispersion densified pupil spectrograph; and 2) a telescope line-of-sight
monitor suitable for its spectrograph. Figure 3 shows an overview of this
instrument concept. The light coming from the primary telescope is divided
into two beams by a field stop, whose surface is coated by a reflective
material. While the light inside the field stop is introduced to the densified
pupil spectrograph, that outside the field stop is reflected by the surface
and is introduced to the telescope line-of-sight monitor. The densified pupil
spectrograph can minimize the impacts of the line-of-sight jitter and drift on
the photometric precision, while the line-of-sight monitor reduces the change
in the telescope line-of-sight over a long period. Although this line-of-sight
monitor looks like a conventional slit viewer (e.g., Rayner et al., 2003;
Iseki et al., 2008) and a coronagraphic low-order wavefront sensor (CLOWFS;
Guyon et al., 2009) in terms of dividing the incident beam into two on the
focal plane, the wavefront propagations through the two telescope pointing
sensors are totally different, as discussed in Section 3.3. In addition, the
position of each spectral element is stable on the detector; thus, the
spectrum shift caused by the Doppler effect may be able to be measured along
the wavelength without any wavelength calibrators, such as a gas absorption
cell and a Laser Frequency Comb (LFC), once the relation between the detector
position and the wavelength is precisely determined. Thus, a combination of
the densified pupil spectrograph and telescope line-of-sight monitor allows us
to perform highly stable spectroscopy against the telescope pointing drift and
its change over a decade.

We will briefly introduce how the densified pupil spectrograph works and
explain why the telescope line-of-sight monitor could improve the precision of
the radial velocity measurements. The light introduced to the densified pupil
spectrograph is first divided into several with a slice mirror composed of a
few rectangular mirrors at the entrance pupil, to which the primary mirror is
optically conjugated. Each divided beam is densified with two concave mirrors.
Densified sub-pupils are then formed at the spectrograph entrance,
corresponding to the pupil plane. The size of each densified sub-pupil is
close to the wavelength; hence, each beam is diffracted as a point source from
the spectrograph entrance. Each diffracted beam is collimated by a collimate
mirror, and the ASF is formed on a dispersive element of the focal plane.
Finally, the spectra of the densified sub-pupils are formed on the detector 1.
Figure 4 shows the conceptual diagram of the densified sub-pupil spectra. We
define the densified sub-pupil as one spectral element. Each spectral element
is sampled by a higher number of detector pixels compared to that of a general
spectrograph because of the division of the entrance pupil into several.
Thanks to this feature, each spectral element is less affected by the bad
pixels and the cosmic rays. The impact of the pixel-to-pixel quantum variation
on the wavelength calibration with a calibration source can be largely
mitigated, thanks to the larger detector samplings (Goda & Matsuo, 2018;
Matsuo et al., 2020). The long-term stability of the calibration source will
be a fundamental limitation on the wavelength calibration (see Section 4).

The detector plane is optically conjugated to the primary mirror; therefore,
any wavefront errors on the pupil plane, in principle, do not affect the
spectra of the densified sub-pupils. However, the telescope line-of-sight
jitter and drift introduce intensity variations even under the assumption that
the optical system is ideal because a motion loss (i.e., a partial loss of the
ASF) is generated on the field stop and the dispersive element, which are
placed on the focal plane. According to the previous study of Itoh et al.
(2017), the photometric variation caused by the motion loss on the focal plane
for a circular isotropic aperture is written as

$\sigma_{jitter}=\frac{\pi a}{2}\left|\frac{\partial}{\partial
r}PSF(r)|_{r=a}\right|(\Delta\theta)^{2},$ (16)

where $r$ is the radial distance from the center of the field (i.e.,
$r=\sqrt{\xi^{2}+\eta^{2}}$); $PSF$ is the point spread function formed on the
focal plane; $a$ is the field-stop radius in the radian unit; and
$\Delta\theta$ represents a change of the telescope line-of-sight at two
epochs normalized by the spatial resolution of a telescope. The photometric
variation caused by the motion loss was mainly determined by a combination of
the telescope line-of-sight jitter and the field-stop radius. The left panel
of Figure 5 shows the averaged photometric variation caused by the motion loss
in a bandwidth equivalent to the central wavelength. The line-of-sight jitter
requirement can be more mitigated as the field-stop radius is larger. To
achieve the photometric precision of 10 $ppm$, the jitter amplitude should be
reduced to 0.01 and 0.1 times the spatial resolution for the radii of 1 and 10
times its resolution, respectively. In Section 4, we will derive the optimum
field-stop radius, such that the photometric variation is smaller than 10
$ppm$.

However, even if the signal variation caused by the motion loss can be
negligible, the telescope line-of-sight jitter and drift degrade the
photometric precision via another intensity variation. The densified pupil
spectrograph stabilizes the spectra of the densified sub-pupils on the
detector plane because its detector is optically conjugated to the primary
mirror. A dispersive element is inevitably placed on the focal plane, and the
tilt in the phase of each densified sub-pupil on the P3 plane moves its Airy
disk formed on the dispersive element. The wavefront aberration introduced on
the dispersion element of the focal plane forms speckles on the detector
plane. The speckles are brighter because the wavelength is shorter (i.e., in
the visible and near-infrared regimes). As a result, a combination of the
newly added wavefront aberration with the change in the telescope line-of-
sight produces various speckle patterns on the detector during the
observation. Focusing on the fact that the line-of-sight jitter randomly
changes, the speckle patterns induced by the jitter are smoothed out for a
long-term observation. Conversely, the intensity variation caused by an offset
of the telescope line-of-sight at two epochs remains as a static noise
corresponding to the systematic noise floor of the photometry, $\epsilon(i)$.
Thus, the photometric variations induced by the telescope pointing drift could
inevitably degrade the radial velocity measurements at the visible and near-
infrared wavelengths.

However, it is hard to measure the telescope line-of-sight drift on the
densified pupil spectrograph because the detector plane is optically
conjugated to the primary mirror. The telescope line-of-sight drift could not
be precisely measured on general imaging spectrographs because the light
coming from an astronomical object is dispersed and is blurred along the
wavelength direction on the detector. In addition, a general slit viewer
cannot measure the telescope pointing drift at the same time when the spectrum
of an astronomical object is recorded because the ASF core is introduced to
the densified pupil spectrograph. Based on this background, we employed a
sensor for measuring the telescope line-of-sight, using the light blocked by
the field stop (i.e., the ASF base). Thanks to the simultaneous measurements
of the telescope pointing drift and the densified pupil spectra, we could
reduce the impact of the long-term drift on the photometric precision by
compensating it during the observation. This sensor would be suitable for
measuring the slow change in the telescope line-of-sight because the amount of
the photons introduced to the sensor is limited (right panel of Figure 5). In
contrast, if a bright object is observed with a large telescope, this monitor
possibly works as a real-time sensor, such as the wavefront sensors used for
adaptive optics (e.g., Hardy, 1998; Guyon, 2005) and the low-order wavefront
sensors for a coronagraph (Guyon et al., 2009).

In the following subsections, we analytically investigate how the wavefront
aberrations on the dispersive element affect the photometric precision based
on an analysis of the wavefront propagation through the densified pupil
spectrograph. We also introduce the telescope line-of-sight monitor,
describing how the monitor precisely measures the telescope line-of-sight.

Figure 3: Conceptual diagram of a densified pupil spectrograph combined with a
telescope line-of-sight monitor. The red and purple lines show the beams
introduced to the densified pupil spectrograph and the line-of-sight monitor,
respectively. The terms of ”P” and ”F” in parentheses represent the pupil and
focal planes, respectively. The red dashed line in the spectrograph shows the
beam propagation without a wave diffraction. The coordination system of the
pupil plane is $(x,y)$.  Figure 4: Conceptual diagram of the densified pupil
spectra on the detector 1. The left panel shows a pupil image on the pupil
slicer with five rectangular mirrors (P2). The red circle represents the
entrance pupil on the P2 plane. The pupil is divided into five with the pupil
slicer. The middle panel shows five densified pupil spectra on the detector 2
(P4). The five dotted rectangulars, named ”Slice 1, 2, …, 5,” represent the
five spectra formed on the P4 plane. The red arcs are spectral elements at a
specific wavelength. The right panel shows an enlarged view of a spectral
element in Slice 3. The difference of the wavelength from the edge to edge of
the spectral element corresponds to $\Delta\lambda$. Each spectral element is
sampled by $n_{s}$ detector pixels along the wavelength direction (i.e., $y$
direction). $n_{s}$ is determined such that the detector noise is smaller than
the photon noise from an astronomical object. $n_{s}$ is more reduced as the
resolving power is higher. Figure 5: (Left) Averaged photometric variation
caused by the motion loss over the observing bandwidth equivalent to the
central wavelength, $\lambda_{c}$, as the functions of the amplitude of the
telescope line-of-sight jitter and the field-stop radius, which are normalized
by $\frac{\lambda_{c}}{D}$. The red, purple, and blue solid lines represent
the photometric variation of 1, 10, and 100 $(ppm)$, respectively. (Right) The
peak intensity of the image formed on the telescope line-of-sight monitor,
which is normalized by that of the original PSF, as a function of the field-
stop radius in the unit of $\frac{\lambda_{c}}{D}$.

###  3.2 Impact of long-term drift on radial velocity measurements

In this subsection, we analytically derive how the difference of the line-of-
sights at two epochs affects the radial velocity measurements. No optical
element exists on the focal plane, except for the field stop before the
spectrograph entrance (i.e., the P3 plane); thus, the wavefront propagation
from the entrance of the spectrograph (P3) to the detector 1 (P4) is focused.

The coordinate systems of the pupil and the focal planes were set to $(x,y)$
and $(\xi,\eta)$, respectively. $\eta$ and $y$ were parallel to the dispersion
direction (i.e., wavelength one). When the wavefront aberration of each
densified sub-pupil on the P3 plane, including the telescope pointing error,
is $\phi_{P3}(x,y,t_{k})$ at the $k$-th epoch, the complex amplitude of its
densified sub-pupil at the $i$-th epoch is written as
$E_{P3}(x,y,\lambda(i),t_{k})=E_{0}(x,y,\lambda(i))\exp\left(i\frac{2\pi}{\lambda(i)}\phi_{P3}(x,y,t_{k})\right)$,
where $E_{0}(x,y,\lambda(i))$ is the electric field of the entrance pupil from
an observed astronomical object at the wavelength of $\lambda(i)$. The
electric field of each densified sub-pupil on the $i$-th dispersive element at
the $k$-th epoch is

$E_{F2}(\xi,\eta,\lambda(i),t_{k})=FT\left\\{E_{0}(x,y,\lambda(i))\exp\left(i\frac{2\pi}{\lambda(i)}\phi_{P3}(x,y,t_{k})\right)\right\\}E_{g}(\xi,\eta)\exp\left(i\frac{2\pi}{\lambda(i)}\phi_{g}(\xi,\eta,t_{k})\right),$
(17)

where $E_{g}(\xi,\eta)$ is the function of a perfect dispersive element;
$\phi_{g}(\xi,\eta,t_{k})$ is the wavefront aberration formed by passing
through the dispersive element at the $i$-th epoch; and $FT\\{o\\}$ represents
the Fourier transform of the function of $o$. The function of the dispersive
element, $E_{g}(\xi,\eta)$, is a periodic function; hence, the Fourier
transform of the function gives different tilts depending on the wavelength to
the electric field on the F2 plane. The light of each wavelength is focused to
a different detector position. Given that the center position of the densified
spectrum at the wavelength of $\lambda(i)$ is $(x_{\lambda},y_{\lambda})$, the
complex amplitude of each densified pupil spectrograph on the detector 1 at
the $k$-th epoch is presented as

$E_{P4}(x,y,t_{k})=E_{0}(x-x_{\lambda},y-y_{\lambda},\lambda(i))\exp\left(i\frac{2\pi}{\lambda(i)}\phi_{P3}(x-x_{\lambda},y-y_{\lambda},t_{k})\right)*FT\left\\{\exp\left(i\frac{2\pi}{\lambda(i)}\phi_{g}(\xi,\eta,t_{k})\right)\right\\}.$
(18)

If the wavefront aberration is not generated after passing through the
dispersive element (i.e., $\phi_{g}(\xi,\eta,t_{k})=0$), the intensity
distribution on detector 1 will be completely independent of the phase error
of the incident light;
$|E_{P4}(x,y,t_{k})|^{2}=|E_{0}(x-x_{\lambda},y-y_{\lambda},\lambda(i))|^{2}$.
Thus, the densified pupil spectrograph can stabilize the spectra of the
densified sub-pupils against the incident light phase if the dispersive
element is an ideal one.

However, no dispersive element generates the ideal diffracted wavefront (i.e.,
$\phi_{g}(\xi,\eta,t_{k})\neq 0$). When the wavefront error of
$\phi_{g}(\xi,\eta,t_{k})$ is smaller than the wavelength of $\lambda(i)$,
$\exp\left(i\frac{2\pi}{\lambda(i)}\phi_{g}(\xi,\eta,t_{k})\right)\simeq
1+i\frac{2\pi}{\lambda(i)}\phi_{g}(\xi,\eta,t_{k})$. The wavefront error of
$\phi_{g}(\xi,\eta,t_{k})$ is written as the sum of ripples with various
periods; therefore, the Fourier transform of
$\exp\left(i\frac{2\pi}{\lambda(i)}\phi_{g}(\xi,\eta,t_{k})\right)$ does not
generate only the main densified sub-pupils at the expected position of
$(x=x_{\lambda},y=y_{\lambda})$ but also faint sub-pupils around the expected
position (Traub & Oppenheimer, 2010). The copied faint sub-pupils are called
”speckles” in the field of the high-contrast science. When the ripple period
is $\xi_{0}$, the faint speckle is positioned at
$\pm\frac{\lambda(i)}{\xi_{0}}$ away from the expected position of
$(x=x_{\lambda},y=y_{\lambda})$. Here, a tilt error in the phase of
$\phi_{P3}(x-x_{\lambda},y-y_{\lambda},t_{k})$ moves the Airy pattern position
on the dispersive element, and the displacement changes the wavefront error
generated after passing through the dispersive element. When the Airy disk
displacement on the dispersive element caused by the tilt error is small
compared to the period of the ripples, the diffracted wavefronts at two
epochs, $t=t_{1}$ and $t_{2}$, are related as follows:

$\phi_{g}(\xi,\eta,t_{2})\simeq\phi_{g1}+\frac{\partial}{\partial\xi}\phi_{g1}\Delta\xi+\frac{\partial}{\partial\eta}\phi_{g1}\Delta\eta,$
(19)

where $\Delta\xi$ and $\Delta\eta$ are the Airy disk displacements on the
dispersive element between the two epochs along the $\xi$ and $\eta$
directions, respectively, and $\phi_{g1}$ represents the diffracted wavefront
error at $t=t_{1}$; $\phi_{g1}\equiv\phi_{g}(\xi,\eta,t=t_{1})$. The observed
spectra at $t=t_{1}$ and $t_{2}$ are written as follows based on the
abovementioned considerations:

$\displaystyle I_{P4}(x,y,\lambda(i),t_{1})$ $\displaystyle=$
$\displaystyle\left|E_{0}(x-x_{\lambda},y-y_{\lambda},\lambda(i))*FT\left\\{1+i\frac{2\pi}{\lambda(i)}\phi_{g1}\right\\}\right|^{2}$
$\displaystyle I_{P4}(x,y,t_{2})$ $\displaystyle\simeq$
$\displaystyle\left|E_{0}(x-x_{\lambda},y-y_{\lambda},\lambda(i))*FT\left\\{1+i\frac{2\pi}{\lambda(i)}\left(\phi_{g1}+\frac{\partial}{\partial\xi}\phi_{g1}\Delta\xi+\frac{\partial}{\partial\eta}\phi_{g1}\Delta\eta\right)\right\\}\right|^{2}.$
(20)

The difference of the spectra at $t=t_{1}$ and $t_{2}$ is

$\displaystyle I_{P4}(x,y,\lambda(i),t_{2})$ $\displaystyle-$ $\displaystyle
I_{P4}(x,y,\lambda(i),t_{1})$ (21) $\displaystyle\simeq$
$\displaystyle\left|E_{0}(x-x_{\lambda},y-y_{\lambda},\lambda(i))*FT\left\\{i\frac{2\pi}{\lambda(i)}\left(\frac{\partial}{\partial\xi}\phi_{g1}\Delta\xi+\frac{\partial}{\partial\eta}\phi_{g1}\Delta\eta\right)\right\\}\right|^{2}$
$\displaystyle+$
$\displaystyle\left(E_{0}(x-x_{\lambda},y-y_{\lambda},\lambda(i))*FT\left\\{i\frac{2\pi}{\lambda(i)}\left(\frac{\partial}{\partial\xi}\phi_{g1}\Delta\xi+\frac{\partial}{\partial\eta}\phi_{g1}\Delta\eta\right)\right\\}\right)+c.c.,$

where $c.c.$ stands for the complex conjugate of the second term in the right-
hand side of the equation. Thus, the telescope pointing error induced the
change of the recorded intensity under the condition that the dispersive
element generates the diffracted wavefront error, and the change of the
intensity between the two epochs limits the precision of the radial velocity
measurements.

The speckles at a wavelength affect the main densified sub-pupils at the
different wavelengths; thus, the ripples along the spectral direction on the
dispersive element have a more significant impact on the precision compared to
those along the direction perpendicular to that. For simplicity, we considered
how one ripple with a particular period forms the speckle on the detector and
limits the precision of the radial velocity measurements. The diffracted
wavefront error, $\phi_{g1}$, is written as the sum of the phase and amplitude
ripples with a spatial period of $\eta_{0}$ parallel to the dispersion
direction:

$\phi_{g1}=a\cos\left(2\pi\frac{\eta}{\eta_{0}}+\alpha\right)+ib\cos\left(2\pi\frac{\eta}{\eta_{0}}+\beta\right),$
(22)

where $a$ and $b$ are the amplitudes of the ripples in terms of the phase and
the amplitude, respectively, and $\alpha$ and $\beta$ are their spatial phases
on the dispersive element. From Equations 21 and 22, the difference of the
spectra at two epochs can be rewritten as

$I_{P4}(x,y,\lambda(i),t_{2})-I_{P4}(x,y,\lambda(i),t_{1})\simeq I_{+}+I_{-},$
(23)

where

$I_{\pm}=\Sigma_{i}\left(\frac{2\pi}{\lambda(i)}\right)^{2}\left(\frac{\pi\Delta\eta}{\eta_{0}}\right)\left(\frac{\pi\Delta\eta}{\eta_{0}}\mp
1\right)\left(a^{2}+b^{2}\pm
2ab\sin(\alpha-\beta)\right)I_{0}\left(x,y-y_{\lambda}\pm\frac{\lambda(i)}{\eta_{0}},\lambda(i)\right),$
(24)

where $I_{0}(x,y-y_{\lambda},\lambda(i))$ is the intensity distribution of the
ideal densified sub-pupil at the wavelength of $\lambda(i)$. The diffracted
wavefront error produces two asymmetric speckles at
$(x,y=y_{\lambda}\pm\frac{\lambda(i)}{\eta_{0}})$, whose intensities are
characterized by a combination of $a$, $b$, and $\Delta\eta$. When the spatial
frequency of a ripple on the diffraction grating is the diffraction limit
along the spectral direction, the shift of the speckle from the original
position is equal to the size of each spectral element. As shown in Equation
24, the single spectral element is affected by the speckles over the observing
wavelength range. Moreover, no pinned speckles formed by the interferences
between the rings of the main Airy disk and the speckles exists because the
densified sub-pupil is not spread on the detector plane like the diffraction
image.

The non-uniformity of the reflectance on the dispersive element is expected to
be much smaller than the wavefront error amplitude. While the reflectance non-
uniformity could be reduced down to 0.5 $\%$ in visible thanks to a precise
control of the coating thickness (e.g., Cohen & Hull, 2004; Lightsey et al.,
2012), the phase error of the diffracted wavefront by a reflection grating was
approximately 20 - 100 $nm$ peak-to-valley on the scale of 100 $mm$ in that
wavelength regime (Lee et al., 2012; Vannoni & Freijo-Martin, 2017). Under a
condition that $a>>b$, Equation 24 approximately becomes

$I_{\pm}\sim\Sigma_{i}\left(\frac{2\pi
a}{\lambda(i)}\right)^{2}\left(\frac{\pi\Delta\eta}{\eta_{0}}\right)\left(\frac{\pi\Delta\eta}{\eta_{0}}\mp
1\right)I_{0}\left(x,y-y_{\lambda}\pm\frac{\lambda(i)}{\eta_{0}},\lambda(i)\right).$
(25)

Equation 25 was derived under a simple assumption that there are only the
phase and amplitude ripples with a single period of $\eta_{0}$. However the
noise floor of the photometry for each spectral element, $\epsilon(i)$, should
be determined by the sum of the speckles produced by ripples with various
spatial periods. The amplitude distribution of the phase error is determined
by the power spectral density (PSD) of the diffracted wavefront error. In this
study, the PSD was set as follows based on a simple assumption that the PSD is
the same as the structure function of an astronomical telescope mirror:

$PSD(k)=\frac{A}{1+\left(\frac{k}{k_{0}}\right)^{3}},$ (26)

where $k$ is the spatial frequency. $A$ and $k_{0}$ were set to 400
$nm^{2}cm^{2}$ and 0.2 $cycles/cm$ as a fiducial value, respectively. When the
diffraction grating size is 10 $cm$, the root mean square (RMS) of the
diffracted wavefront error is approximately 10 $nm$, and the peak-to-valley
error at the spatial frequency of 0.1 $cm^{-1}$ is approximately 20 $nm$,
almost corresponding to those of the previous study (Vannoni & Freijo-Martin,
2017). The left panel of Figure 6 shows the photometric variation at 500 $nm$
due to the difference of the monochromatic speckle patterns at two epochs
induced by a change in the telescope line-of-sight. The monochromatic speckles
are fainter as separating from the original image position; thus, the noise
floor is mainly determined by only the speckles of the surrounding spectral
elements.

When the period of the phase ripple is matched to the full width half maximum
(FWHM) of the Airy disk formed on the dispersive element, the position of its
speckle is shifted to the spectral element next to the original one. The noise
floor of each spectral element can be easily calculated by normalizing the
spatial period of the ripples by the FWHM of the Airy disk. Given that the
phase ripple amplitude with the normalized period of $\eta_{m}$ is $a_{m}$,
Equation 25 can be rewritten as follows:

$I_{\pm}=\Sigma_{i}\Sigma_{m}\left(\frac{2\pi
a_{m}}{\lambda(i)}\right)^{2}\left(\frac{\pi\Delta\eta}{\eta_{m}}\right)\left(\frac{\pi\Delta\eta}{\eta_{m}}\mp
1\right)I_{0}\left(x,y-y_{\lambda}\pm\frac{\lambda(i)}{\eta_{m}},\lambda(i)\right).$
(27)

The right panel of Figure 6 shows the photometric precision limited by the
difference between the speckle patterns of 100 adjacent spectral elements at
two epochs. To achieve the photometric precision of 1 $ppm$, the offset of the
telescope line-of-sights at two epochs should be reduced to 0.03
$\frac{\lambda}{D}$ in the fiducial case that the RMS of the wavefront error
is 10 $nm$. Thus, a combination of the quality of the diffracted wavefront and
the offset of the telescope line-of-sight at two epochs determines the
difference of the spectra at two epochs,
$I_{P4}(x,y,\lambda(i),t_{2})-I_{P4}(x,y,\lambda(i),t_{1})$, corresponding to
the noise floor of the photometry for each spectral element, $\epsilon(i)$.
The limitation on the precision of the radial velocity measurements was
estimated through Equation 12. In Section 4, we estimate the precision of the
radial velocity measurements under an appropriate observing condition.

Figure 6: (a) Photometric variation at the wavelength of 500 $nm$ caused by a
change in the monochromatic speckle patterns induced by the difference between
the telescope line-of-sights at two epochs as a function of the distance along
the spectral direction normalized by the spectral element. The root mean
square of the diffracted wavefront error was set to approximately 10 $nm$. (b)
Photometric variation at the wavelength of 500 $nm$ caused by a change in the
speckle patterns of 100 spectral elements around its wavelength as a function
of the RMS of the diffracted wavefront error.

###  3.3 Monitor of the long-term drift

The precision of the radial velocity measurements is ultimately limited by the
difference of the telescope line-of-sights at two epochs. In exchange for the
high stability against the telescope pointing errors, the tilt error cannot be
estimated from the spectra recorded by the densified pupil spectrograph
because the spectra are insensitive to the phase of the pupil plane. It is
difficult even for conventional spectrographs to precisely measure the
telescope line-of-sight because the light from an astronomical object is
dispersed along the wavelength direction on the detector. We propose herein a
sensor working not only for the densified pupil spectrograph but also for
conventional ones. Figure 7 shows the conceptual design of the telescope line-
of-sight monitor. After the light outside the field stop is introduced to the
telescope line-of-sight monitor, the light is collimated and passed through a
pupil stop, and the light is then focused on a detector. The key parameter of
this monitor is the pupil stop diameter. When the diameter exactly matches
that of the entrance pupil (i.e., the primary mirror), this monitor can work
as a telescope line-of-sight monitor. The intensity distribution formed on
detector 2 includes an information on the telescope line-of-sight.

We will mathematically explain the wavefront propagation through the line-of-
sight monitor. The complex amplitude formed on the field stop (F1) is the
Fourier transform of the electric field on the entrance pupil (P1);
$E_{F1}(\xi,\eta,\lambda(i))=FT\\{E_{P1}(x,y,\lambda(i))\\}$. Based on
Babinet’s principle, the complex amplitude after the field stop can be written
as

$E_{F1}(\xi,\eta,\lambda(i))=FT\\{E_{P1}(x,y,\lambda(i))\\}\left(1-A_{field-
stop}(\xi,\eta)\right),$ (28)

where $A_{field-stop}(\xi,\eta)$ is the aperture function of the field stop
assumed to be a clear circular aperture. The complex field on the pupil stop
(P5) is

$E_{P5}(x,y,\lambda(i))=E_{P1}(x,y,\lambda(i))-E_{P1}(x,y,\lambda(i))*FT\\{A_{field-
stop}(\xi,\eta)\\},$ (29)

where $FT\\{A_{field-stop}(\xi,\eta)\\}$ is the Bessel function of the first
kind of order one. Most of the light is canceled by the destructive
interference, and the remaining faint light is concentrated at the edge of the
pupil stop. After passing through the pupil stop with the aperture function of
$A_{pupil-stop}(x,y)$, the complex amplitude on detector 2,
$E_{F3}(\xi,\eta)$, is mathematically derived as follows:

$E_{F3}(\xi,\eta,\lambda(i))=\left[FT\\{E_{P1}(x,y,\lambda(i))\\}-FT\\{E_{P1}(x,y,\lambda(i))\\}A_{field-
stop}(\xi,\eta)\right]*FT\\{A_{pupil-stop}(x,y)\\}.$ (30)

When the pupil stop diameter is equal to that of the entrance pupil,
$FT\\{A_{pupil-stop}(x,y)\\}$ becomes the ideal ASF of the telescope. The
electric field on detector 2, $E_{F3}(\xi,\eta,\lambda(i))$, is formed through
a convolution of the image on the field stop with the Fourier transformation
of the pupil stop. Concentric circle and ring patterns with the same period as
that of the ASF base on the field stop are formed on the detector. The
simulated image on detector 2 (Figure 7) is different from the original PSF.
The amplitude of $E_{F3}(\xi,\eta)$ is maximized when the center position of
the ASF is matched to that of the complex amplitude in the parentheses of
Equation 30. The telescope line-of-sight was reflected in the center position
of the amplitude in the parentheses of Equation 30; thus, this monitor allowed
us to measure, in principle, the pointing error from the intensity
distribution formed on the detector.

This sensor works in the broadband light because there are no chromatic
optical elements in the sensor system. Figure 8 shows the intensity images
formed on detector 2 for three different bandwidths. The image gradually
blurred as the bandwidth increased because the core size was proportional to
the wavelength. However, the center core of the image measured for estimating
the telescope line-of-sight was still clear even under the condition of the
bandwidth being equal to the observing wavelength. Note that the collimator
and the camera systems should be designed such that they do not generate a
chromatic aberration.

Even though the bandwidth can be increased, most of the light from an
astronomical object is required to be introduced to the densified pupil
spectrograph. The pointing error must be measured with the remaining faint
light. As shown in the right panel of Figure 5, the number of photons
introduced to the monitor was more limited as the field-stop radius increased.
For example, the peak intensity of the core formed on detector 2 was reduced
by factors of approximately $1.5\times 10^{-3}$ and $4\times 10^{-4}$ for the
field-stop radii of 5 and 10 $\frac{\lambda}{D}$, respectively. According to
the previous study of Thomas (2004), the measurement accuracy of the centroid
position was equal to the ratio of its FWHM to the square root of the number
of the photons for the photon-noise-limited case. The FWHM of the centroid
corresponded to the angular resolution of the telescope; therefore, the
precision of the telescope line-of-sight monitor is written as

$\sigma_{sensor}=\frac{\theta_{angular}}{\sqrt{N_{sensor}}},$ (31)

where $\theta_{angular}$ is the angular resolution of the telescope, and
$N_{sensor}$ is the number of photons introduced to this monitor, which is
determined by a combination of the number of the photons in the incident beam
to the telescope and the field-stop size.

Figure 9 shows the achievable precision of the telescope line-of-sight monitor
for a Lyman alpha forest of a QSO and a nearby Sun-like star when observing
using 15 $m$\- and 2 $m$-diameter telescopes, respectively. When observing a
Sun-like star at 30 $pc$ with the 2 $m$ class telescope, this monitor collects
a sufficient number of photons for only a few milli-second under the condition
of the field-stop radius being less than 5 $\frac{\lambda}{D}$. The line-of-
sight jitter is possibly monitored with an accuracy of 0.01
$\frac{\lambda}{D}$ in the same manner as the wavefront sensors for adaptive
optics (e.g., Hardy, 1998; Guyon, 2005) and the low-order wavefront sensors
for a coronagraph (Guyon et al., 2009). In contrast, the slow telescope line-
of-sight with a timescale of a few tens of seconds can be measured with this
monitor for the Lyman alpha forest because the QSO is much fainter than the
nearby stars. The required exposure time strongly depends on the telescope
diameter and field-stop diameters. The telescope diameter is smaller; thus,
this monitor measures only the slower change in the telescope line-of-sight.
However, the high-frequency line-of-sight jitter is randomly changed and has
no significant impact on the radial velocity measurements. The signal
variation caused by the high-frequency jitter could be smoothed out by
averaging the data over a long integration time. In other words, the long-term
drift will increase the difference of the telescope line-of-sights at two
epochs and limit the precision of the radial velocity measurements. As shown
in Figure 9, this monitor can fully measure the long-term drift; hence, the
precision of the radial velocity measurements could be improved by working at
the same time when the densified pupil spectra are recorded. The telescope
line-of-sight monitor is suitable for the radial velocity measurements of
distant QSOs and nearby stars. The appropriate field-stop diameter should be
determined by the balance between the signal fluctuations caused by the motion
loss and the shot noise of an astronomical object. If the telescope pointing
jiitter is small, the field-stop size could be reduced, thanks to the smaller
motion loss. We will determine the appropriate field-stop size under an
appropriate condition in Section 4.

Finally, we mention the difference between this monitor and the Lyot
coronagraph (Lyot, 1939). The monitor configuration is the same as that of the
Lyot coronagraph. The complex amplitude shown in Equation 30 is equal to that
formed on the detector plane of the Lyot coronagraph. The only difference
between the two systems is the pupil/Lyot stop diameter. The Lyot stop
diameter should be smaller than that of the entrance pupil because the
unwanted diffracted light from the edge of the occulting mask/field stop on
the focal plane is blocked. In contrast, the pupil stop of this monitor
transmits half of the light coming from the field stop. The simulated images
before and after the pupil stop shown in Figure 7 depict the outer half of the
bright ring in the image before P5 is blocked by the pupil stop.

Figure 7: Conceptual diagram of a telescope line-of-sight monitor. The light
was divided into two by the field stop. The light inside the field stop is
introduced to the densified pupil spectrograph indicated by the red line. The
remaining light was reflected by the field stop and introduced to the monitor,
as indicated by the purple line. The upper and lower images show the intensity
distributions on each plane on linear and log scales, respectively. The
simulation images were generated under the following conditions; 1) a clear
circular aperture with a tilt error of 1 $\frac{\lambda}{D}$ along the
horizontal direction prepared as the entrance pupil; 2) a field stop radius
set to 10 $\frac{\lambda}{D}$; 3) a pupil stop diameter equal to that of the
entrance pupil. Figure 8: Images formed on the telescope line-of-sight monitor
for (a) the monochromatic light, (b) the bandwidth of half the central
wavelength, and (c) the bandwidth equivalent to the central wavelength. The
upper panels show the two-dimensional intensities on the linear scale. The
lower panels show their intensity distributions along the horizontal
direction. The values of the vertical axis represent the average ratio of the
intensity on the sensor to the peak intensity of the original PSF. The tilt of
the incident light was set to 0. Figure 9: Precision of the telescope line-of-
sight monitor for the (a) Lyman alpha forest of a QSO and (b) the spectrum of
a Sun-like star at 30 $pc$ as a function of the field stop radius and exposure
time. The blue, purple, and red lines represent the precisions of $10^{-1}$,
$10^{-2}$, and $10^{-3}$ $\frac{\lambda_{c}}{D}$, respectively. The monitor
bandwidth was simply assumed to be equal to the central wavelength of 0.5
$\micron$. The telescope diameters for (a) and (b) were set to 15 $m$ and 2
$m$, respectively. The optical throughput, except for the losses caused by the
field stop and the pupil stop, was set to 0.9. The flux of the Lyman alpha
spectrum was simply set to 0.23 $m$Jy, corresponding to 18.0 mag in the $V$
band. The flux of the Sun-like star was calculated based on the parameters
listed in Table 2.

##  4 Direct measurement of the universe’s expansion history

Section 3 described an approach for enabling high-precision radial velocity
measurements through developing the densified pupil spectrograph originally
proposed for the mid-infrared transit spectroscopy of exoplanets. This
instrument concept was composed of a high-dispersion densified pupil
spectrograph and a monitor for measuring the telescope line-of-sight, working
at the same time when the densified pupil spectra are recorded. The present
section presents a new compact and simple optical design developed from the
existing densified pupil spectrograph. This new design allows us to
simultaneously acquire the spectra of approximately 10 astronomical objects at
different field-of-views. We evaluate how much the expansion history of the
Universe at high redshits could be directly measured with this instrument
design under an appropriate assumption.

###  4.1 Spectrograph design for direct measurement of the Universe’s
expansion history

This subsection shows an instrument design optimized for the direct
measurement of the Universe’s expansion history. We first determine the main
optical parameters of the spectrograph based on the performance of the radial
velocity instrument analytically derived in the previous sections. We next
present the optical design of a spectrograph satisfying the main optical
parameters.

####  4.1.1 Specification

We determined the main parameters of the spectrograph, such that the two main
sciences can be performed. The optical system introduced herein is a tentative
solution. The optical design should be finally optimized under a balance
between its spectrograph volume and its specification for enabling the science
case. As discussed in Section 2, the resolving power of the spectrograph
enhanced high $Q_{photon}$ and $Q_{sys}$ factors and reduced an imapct of the
systematic instrumental noise on the precision. As shown in Figure 1, both the
$Q_{photon}$ and $Q_{sys}$ factors for the Lyman alpha forest were almost
constant in the regime of the resolving power higher than 10,000. Considering
that a higher resolving power limits the wavelength range more and increases
the spectrograph volume, we set the resolving power to 10,000.

We focused on obtaining the spectra of the QSOs at z $>$ 2 because it was
difficult to present a compact optical design in the ultraviolet wavelength
regime. The number of lens materials available for that regime was very
limited, and a chromatic aberration remained. The longer wavelength edge of
the Lyman alpha forest is 365 $nm$ for the QSOs at $z=2$; thus, it was
prefarable to set the shorter edge of the wavelength range to that number. The
Lyman alpha forests of the high-redshift QSOs at z $\sim$ 5 can be fully
obtained when the longer edge of the wavelength range is 670 $nm$. Thus, we
set the wavelength range of the spectrograph to $<$ 365 to 670 $nm$.

We determined the optimum angular size of the field stop for the field-of-view
of the spectrograph. As discussed in Section 3, the field-stop radius should
be determined by the balance between the photometric variation caused by the
motion loss and the precision of the telescope line-of-sight monitor. The
photometric variation due to the motion loss induced by the telescope line-of-
sight jitter decreased as the field-stop radius became larger (Figures 5). In
other words, the line-of-sight jitter requirement was stricter for the smaller
field-stop radius. For example, to reduce the photometric variation caused by
the motion loss down to 10 $ppm$, the field-of-view should be larger than 4
$\frac{\lambda}{D}$ under the condition of the line-of-sight jitter being 0.05
$\frac{\lambda}{D}$. In contrast, the amount of the light introduced to the
telescope line-of-sight monitor decreased for the larger field stop radius,
and the monitor precision was limited. As shown in Figure 9, a long exposure
time was required for achieving a higher monitor precision because the Lyman
alpha forest stamped on the spectrum of a high-redshift QSO was very faint.
The instrumental noise could be less than 1 $ppm$ when the difference of the
line-of-sights at two epochs is reduced to 0.001 $\frac{\lambda}{D}$ (Figure
6). The field stop radius should be less than 4 $\frac{\lambda}{D}$ to achieve
the measurement precision of 1 $ppm$ with a modest exposure time of less than
1,000 s. Thus, the field-of-view was set to 4 $\frac{\lambda}{D}$.

####  4.1.2 Optical design

We constructed a new simple and compact optical design based on the determined
spectrograph specification. This new design was developed from the existing
densified pupil design for the Origins Space Telescope (Matsuo et al., 2018).
Figure 10 shows two optical designs optimized for different wavelength ranges.
After passing through the field stop with a radius of 4 $\frac{\lambda}{D}$,
the beam was first divided into two by a dichroic mirror in terms of the
wavelength range: 1) 305 - 500 $nm$ for the short channel; and 2) 463 - 660
$nm$ for the long one. The two designs were almost identical, except for the
optical coating, echelle grating, and lens design. After passing through the
dichroic mirror, the incident beam with 3 $mm$ diameter was collimated by a
lens with a 42.9 $mm$ focal length and divided into five with a compact beam
slicer (Bowen-Walraven type image slicer) (Bowen, 1938; Walraven & Walraven,
1972). Although this type of image slicer is widely applied to high-resolution
spectrographs (e.g., Schwab et al., 2011; Tajitsu et al., 2012), this was the
first time it was used as a pupil slicer. The upper panel of Figure 11 shows
the footprints on the plane after the Bowen-Walraven type image slicer was
used. A focus alignment plate was added to the Bowen-Walraven type image
slicer. All sub-pupils were focused on the reflection grating without defocus.
This new optical design did not require a pupil densification with two concave
mirrors because of the small diameter of the entrance beam reduced by the
image slicer. The divided beams were collimated by a Cassegrain system
composed of two mirrors with an effective focal length of 1800 $mm$. An
Echelle grating with a pitch interval of 200 (110) $lines/mm$ for the short
(long) channel dispersed the collimated five beams. A cross-disperser of a
reflection grating separated the overlapping spectral bands with diffraction
orders of 18 to 29 (25 to 35) in the short (long) channel. The Litrrow
configuration was chosen. The incident beam angle to the Echelle grating was
equal to its brazed angle. The camera lens system was designed under the
condition of each spectral element being sampled by approximately 2.5 pixels
along the spectral direction at the central wavelength of each order. In this
case, the number of each spectral element was approximately 122, which will
reduce the impacts of an uncorrected flat-field error and a finite pixel
sampling of the PSF on the precision of the radial velocity measurements.
Considering that a photon counting Electron Multiplying CCD (EMCCD) camera
with a large format of 4k x 4k has now been tested (Daigle et al., 2018), the
format and the pixel pitch of the detector for this spectrograph were set to
2k x 2k and 12 $\micron$, respectively. The left and right panels of Figure 11
show the Echelle format of the shorter and longer channels, respectively. The
field-of-view of the spectrograph was set to 4 $\frac{\lambda}{D}$ at the
longest wavelength of 660 $nm$. The wavelength range of each unit is limited;
thus, the anti-reflection coating can be optimized. The optical throughput of
the spectrograph was approximately 0.63 assuming that the throughput of each
anti-reflection coating was 0.995, the diffraction efficiency was 0.8, and the
efficiency of the pupil slicer was 0.9. Considering that the telescope
throughput was 0.9 and the quantum efficiency of the detector was 0.8, we set
the total throughput to 0.45. The spectrograph size optimized for each band
was approximately 550 (L) x 300 (W) x 100 (H) $mm$. Table 1 compiles the
optical design parameters.

Figure 10: High-dispersion spectrograph design optimized for the (a)(b) short (305 - 500 $nm$) and (c)(d) long (463 - 660 $nm$) channels. A dichroic mirror between the field stop and the collimator lens, which is not shown in this diagram, divides a beam into two in terms of the wavelength. The color of the ray indicates a different grating order: 29, 23, and 18 grating orders for the blue, green, and orange rays in the short channel and 35, 29, and 25 grating orders in the long one, respectively. Each beam has a field of view of 4 $\frac{\lambda}{D}$ in radius. The entrance beam with 3 $mm$ diameter was divided into five without any densification. The footprint on the plane 1 after the Bowen-Walraven type image slicer was used is shown in the left upper panel of (b). Figure 11: (a)(c) Echelle formats of the short and long channels. The vertical lines indicate the different diffraction orders of 18 to 29 and 25 to 35 for the short and long channels, respectively. The detector size of each channel was assumed to be 30.7 x 30.7 $mm$. The panels (b) and (d) show the wavelength ranges of each order for the short and long channels, respectively. Table 1: Parameters of the optical designs optimized for the short and long channels | Short channel | Long channel
---|---|---
Wavelength range | 305 - 500 $nm$ | 463 - 660 $nm$
Resolving power at the central wavelength of each order | 10,800 | 10,800
Field-of-view | 4 $\frac{\lambda}{D}$ at 500 $nm$ | 4 $\frac{\lambda}{D}$ at 660 $nm$
Incident beam diameter | 3 $mm$ | 3 $mm$
Focal length of the collimator lens | 42.9$mm$ | 42.9 $mm$
Number of sub-pupils | 5 | 5
Focal length of the Cassegrain system | 1800 $mm$ | 1800 $mm$
Type of cross-disperser | Reflection grating | Reflection grating
Pitch interval of the Echelle grating | 200 $lines/mm$ | 110 $lines/mm$
Brazed angle | 65.5 $degree$ | 65.5 $degree$
Diffraction order | 18 - 29 | 25 - 35
Focal length of the camera lens system | 100 $mm$ | 100 $mm$
Number of samplings along spectral direction | 2.5 | 2.5
Detector format | 2 k x 2 k | 2 k x 2 k
Pixel size of the detector | 15 $\micron$ | 15 $\micron$

###  4.2 Performance

Based on the specification of the designed spectrograph unit, we evaluated the
precision of the radial velocity measurements for direct measurement of the
Universe’s expansion history. The spectrum of the Lyman alpha forest was
generated through the $fake\\_spectra$ package (Bird et al., 2015) (Appendix
A). For comparison, the two redshifts of $z=3$ and 4 were focused in this
evaluation. Based on the Million Quasars (MILLIQUAS) catalog (Flesch, 2021),
the apparent $B$ band mangnitudes of the brightest targets at approximately
$z$ = 3 and 4 are 16.01 and 17.29, respectively. We note that both of them are
confirmed as QSOs in the catalog. The applied values almost correspond to
those of the objects introduced in the previous study of Liske et al. (2008).

The telescope for the direct measurement of the Universe’s expansion history
was assumed to be the LUVOIR-A concept (The LUVOIR Team, 2019). The telescope
line-of-sight jitter of the LUVOIR-A concept was set to 0.3 $milli-arcsecond$
(Sacks et al., 2018). The jitter value almost corresponded to 0.05
$\frac{\lambda}{D}$ at the longest wavelength of 660 $nm$. Considering that
the field-stop radius was $4\frac{\lambda}{D}$ at that wavelength, the motion
loss on the field stop caused by the line-of-sight jitter will cause the
photometric variation of approximately 10 $ppm$ (Figure 5). However, the
photometric variation is randomly distributed; thus, the impact of the motion
loss on the photometry can be canceled out by adding a number of exposures.
The photometric degradation caused by the motion loss was not considered in
this evaluation. When we observed a Lyman alpha forest spectrum with an
apparent magnitude of 17.3 $mag$ in the $B$ band with a LUVOIR-A concept
telescope, the number of the obtained electrons per second for each spectral
element was approximately 62 $e/s$ on average. Thanks to the advanced
technology for the low-flux detectors in visible (e.g., Wilkins et al., 2014),
we simply neglected the detector noise (e.g., read noise and dark current).

The precision of the radial velocity measurements was mainly limited by the
instrumental systematic noise, wavelength calibration source, and shot noise.
The systematic noise floor caused by the difference between the line-of-sights
at two epochs can be reduced down to 1 $ppm$, thanks to the line-of-sight
sensor. However, the photometric precision was not only limited by the optical
wavefront errors, but also by the detector system. For example, the previous
studies showed that the photometric precision was an order of 10 $ppm$ based
on the characterization of only the detector system in the experiment room
(Matsuo et al., 2019; Schlawin et al., 2021). In this calculation, the
systematic noise was assumed to be determined by the root sum square of the
two independent noises. The wavelength calibration error was also considered,
which is mainly caused by a long-term change in the LFC spectrum, an
imperfection of the flat-field light source, and a variation in the signal-to-
noise ratio of the calibration source along the wavelength (e.g., Blackman et
al., 2020). The long-term stability of the source limited the precision of the
radial velocity measurements to $\sim$ 2 $cm/s$ (Halverson et al., 2016). In
contrast, the number of the detector samplings for each spectral element in
this spectrograph system was approximately 120, which is much larger than
those of other high-precision radial velocity instruments. The impacts of the
uncorrected flat-field error and the finite pixel samplings of the PSF will be
largely mitigated. In this calculation, the uncertainty of the wavelength
calibration was simply set to 2 $cm/s$ considering that only the calibration
source stability limits the precision. Finally, the precision will
fundamentally be limited by the shot noise associated with the spectrum
recorded by the densified pupil spectrograph. Table 2 presents all of the
parameters used for this evaluation. Note that the impact of a change in the
spectrograph temperature on the precision was not considered because the
thermal stability of the LUVOIR telescope concept could be achieved to 1 $mK$
(Yang et al., 2019), which was much smaller than those of the ground-based
observatories (e.g., Blackman et al., 2020).

Figure 12 shows the uncertainty of the radial velocity measurements for the
direct measurement of the Hubble parameters at various redshifts. This
proposed instrument concept could reach precision of $\sim$ 3 and 4 $cm/s$ for
the Lyman alpha forests of QSOs at $z$ = 3 and 4, respectively, when the
integration time at each epoch was 200 days. To measure the expansion history
at various redshifts, the wavelength range was equally divided into 3 for both
the QSOs at $z$ = 3 and 4. The divided wavelength ranges for the QSOs at $z$ =
3 and 4 were fixed to 603 and 1008 $nm$, respectively. The calibration error
will principally limit the precision of the radial velocity measurements when
the integration time is infinite. Finally, the systematic error of an order of
10 $ppm$ did not affect the precision for this science case.

Table 2: Parameters of the targets and the observatories for the direct measurement of the Hubble parameters Model spectrum | $fake\\_spectra$
---|---
Flux densities of the brightest QSOs | 0.90 $m$Jy at $z$=3, 0.36 $m$Jy at $z=$4
Flux densities of the faint QSOs | 0.23 $m$Jy at $z=$3, 0.091 $m$Jy at $z$=4
Telescope diameter | 1500 $cm$
Line-of-sight jitter | 0.3 $mas$
Number of the field-of-views | 10
Systematic noise of the optical system | 1 $ppm$
Systematic noise of the detector system | 10 $ppm$
Wavefront calibration error | 2 $cm/s$
Figure 12: Precision of the radial velocity measurements for the direct
measurement of the Universe’s expansion history, as a function of the
integration time at each epoch. The Lyman alpha forests of QSOs at $z=3$ and 4
were used (Appendix A). The parameters used for this evaluation are compiled
in Table 3. The wavelength ranges of the Lyman alpha forests were equally
divided into 3 to trace the Universe’s expansion history at high redshifts.
The divided wavelength intervals for the QSOs at $z$=3 and 4 were 603 and 1008
$nm$, respectively.

###  4.3 Expansion history of the Universe

The Lyman alpha forest of the QSO is formed by the absorptions of neutral
hydrogen atmos between the QSO and the observer; hence, the redshift drifts of
the hydrogen medium at various redshifts can be directly measured from the
difference between the radial velocity signals at two epochs with a time span
of more than a few years (Sandage, 1962; Loeb, 1998). In this subsection, we
introduce how the instrument concept could constrain the Universe’s expansion
history based on the performance derived in the previous subsection.

The two Universe scale factors at the astrnomical object and the observer are
related as follows:

$\frac{a(t_{obs})}{a(t_{obj})}=1+z_{obj},$ (32)

where $z_{obj}$ is the redshift of the object, $a$ is the Universe scale
factor, and $t_{obj}$ and $t_{obs}$ are the time at the object and the
observer, respectively. The change in the redshifts of the source at $z_{obj}$
at two epochs with a time span of $\Delta t_{obj}$ is

$\displaystyle\Delta z$ $\displaystyle=$ $\displaystyle
1+z_{obj}|_{t=t_{obj}+\Delta t_{obj}}-\left(1+z_{obj}|_{t=t_{obj}}\right)$
(33) $\displaystyle=$ $\displaystyle\frac{a(t_{obs}+\Delta
t_{obs})}{a(t_{obj}+\Delta t_{obj})}-\frac{a(t_{obs})}{a_{obj}(t_{obj})}$
$\displaystyle\simeq$
$\displaystyle\frac{\dot{a}(t_{obs})}{a(t_{obs})}\left(1+z_{obj}\right)\Delta
t_{obs}-\frac{\dot{a}(t_{obj})}{a(t_{obj})}\frac{a(t_{obs})}{a(t_{obj})}\Delta
t_{obj},$

where $\dot{a}$ is the time differential of the Universe scale factor. The
Taylor expansion was used because the time intervals of $\Delta t_{obj}$ and
$\Delta t_{obs}$ are much shorter than $t_{obj}$ and $t_{obs}$. Considering
that the comoving distance between the object and the observer is constant,
the light emitted from the same astronomical objects at the two time of
$t_{obj}$ and $t_{obj}+\Delta t_{obj}$ is observed by the observer at the two
time of $t_{obs}$ and $t_{obs}+\Delta t_{obs}$, respectively:

$\int_{t_{obj}}^{t_{obs}}\frac{1}{a(t)}dt=\int_{t_{obj}+\Delta
t_{obj}}^{t_{obs}+\Delta t_{obs}}\frac{1}{a(t)}dt.$ (34)

Here, the time intervals of $\Delta t_{obj}$ and $\Delta t_{obs}$ are very
small; hence, $\frac{\Delta t_{obj}}{a(t_{obj})}=\frac{\Delta
t_{obs}}{a(t_{obs})}$. The change of the redshift with a time interval of
$\Delta t_{obj}$ shown in Equation 33 is

$\displaystyle\Delta z$ $\displaystyle\simeq$
$\displaystyle\frac{\dot{a}(t_{obs})}{a(t_{obs})}\left(1+z_{obj}\right)\Delta
t_{obs}-\frac{\dot{a}(t_{obj})}{a(t_{obj})}\Delta t_{obs},$ (35)
$\displaystyle=$ $\displaystyle H_{0}(1+z_{obj})\Delta
t_{obs}-H(z_{obj})\Delta t_{obs},$

where $H(z)$ is the Hubble parameter, and $H_{0}$ is the Hubble constant at
the observer. The Hubble parameter can be directly measured from the redshift
drift (e.g., Liske et al., 2008):

$H(z)=H_{0}(1+z_{obj})-\frac{\Delta z}{\Delta t_{obs}}.$ (36)

Figure 13 compares the Hubble parameter based on the cosmological model with
the redshift drifts derived from the expected performance for the Lyman alpha
forests of the QSOs at $z=3$ and 4. Compared to the other methods for
measuring the Hubble parameter, such as the differential age (DA) and the BAO,
the redshift drifts could constrain the Hubble parameter at high redshifts
($z>2$). The measurement values derived from the other methods have the
uncertainty on the age estimation of galaxies or a cosmological model-
dependence. In contrast, this method directly measures the Hubble parameters
without any uncertainty and model depndence; hence, the comparison of the
measurement values from the other methods with the radshift drift ones is
valuable. This direct measurement of the expansion history of the Universe may
solve the Hubble tension and discover new physics.

Figure 13: Direct measurement of the Hubble parameter at high redshifts with
the proposed instrument concept. The red and purple dots show the redshift
drift (RD) measurements for the Lyman alpha forests of the QSOs at $z=3$ and
4, respectively. The solid line represents $H(z)$ with the parameters of
$\Omega_{\lambda}=0.7$, $\Omega_{m}=0.3$, $H_{0}=67.6km/s/Mpc$. For the
comparison, the measurement values from the differential age (DA; red points)
and the baryon acoustic oscillation (BAO; blue points) were also plotted
(Zhang & Xia, 2016). The observing wavelength range was equally divided into 3
for the Lyman alpha forests of the QSOs at $z=3$ and 4. The wavelength
intervals for the QSOs at $z$ = 3 and 4 were equal to 603 and 1008 $nm$,
respectively. The measurement values were affected by both the shot noise and
the systematic noise. The error bar represents the 1-sigma precision limited
by the shot noise.

##  5 Characterization of the reflected and thermal light from habitable
planet candidates

We introduce this evaluation method to characterize the reflected and thermal
light from the habitable planet candidates orbiting the late-type stars and
investigate optimum optical parameters of the spectrograph for this science
case. Based on this investigation, we present an optical design of the high-
dispersed densified pupil spectrograph. We also discuss the detectability of
the reflected light with the proposed instrument concept.

###  5.1 Measurement of the reflected light in visible

Habitable planet candidates orbiting nearby late-type stars were recently
discovered (e.g., Anglada-Escudé et al., 2016; Gillon et al., 2016;
Zechmeister et al., 2019). The reflected light from these planets is embedded
in the bright light of their host stars; thus, a high-contrast instrument that
spatially resolves the planetary system and mainly suprresses the light of
only the host star is required to detect the reflected light. By contrast,
because the planetary orbital motion imprints the Doppler shift information on
the reflected light, the high-resolution spectroscopy of the reflected light
motivates us to estimate the orbital inclination and the true planet mass. A
number of the absorption lines in the visible host stellar spectrum could be
utilized for this purpose. This approach directly measures the stellar
spectrum during the observation; hence, an exact stellar spectrum model is not
required. The high-resolution spectrograph could also resolve a number of
absorption lines produced by the molecules in the planetary atmosphere. This
approach helps us estimate the atmospheric compositions of nearby non-
transiting planets.

We investigated herein the detectability of the reflected light from the
nearby temperate Earth-sized planets with a high-precision radial velocity
instrument instead of a high-contrast equipment. All of the nearby habitable
planet candidates orbit late-type stars; therefore, the target systems were
set to the virtual Proxima Centauri and Trappist 1 systems at 5 $pc$ as
examples of the middle and late M type stars. The spectra of the host stars
were generated through the $BT-settle$ model (Allard et al., 2012). The
contrast ratio between the planet and its host star produced for the Proxima
Centauri and Trappist 1 systems was used (Lin & Kaltenegger, 2020). The
spectrum of each planet light was derived by multiplying the host stellar
spectrum and the contrast ratio. Table 3 compiles the target systems
parameters.

Next, we modified the $Q_{photon}$ and $Q_{sys}$ factors for the case where
the object light was embedded in the bright background. It was very difficult
to spatially resolve the planet and its host star light, even with a very
large space telescope, because of the very small angular separations between
the nearby late-type stars and their habitable zones. Note that some proposed
coronagraphs had very small inner working angles ($\sim 1\frac{\lambda}{D}$)
(e.g., Guyon et al., 2014; Itoh & Matsuo, 2020; Matsuo et al., 2021) and
encouraged us to directly detect the reflected light from the habitable planet
candidates orbiting nearby late-type stars. The radial velocity signal of the
faint reflected light was embedded in the bright stellar light. In this case,
the $i$-th pixel signal for the planetary velocity of $V_{p}$ can be
approximated as follows:

$\frac{\delta
V_{p}}{c}(i)\approx\frac{N_{s,2}(i)-N_{s,1}(i)}{\lambda(i)\left(\frac{\partial
N_{p}}{\partial\lambda}(i)\right)},$ (37)

where $N_{s,j}(i)$ is the number of the photons from the host stellar light at
$j$-th epoch, and $N_{p}(i)$ is the number of the photons from the reflected
light. The uncertainty is rewritten as

$\sigma^{2}_{wav}=\frac{2}{Q_{photon,p}^{2}\sum_{i}N_{s}(i)}.$ (38)

The modified $Q_{photon,p}$ factor for this science case is

$\displaystyle Q_{photon,p}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\sum_{i}\frac{\lambda^{2}(i)}{N_{s}(i)}\left(\frac{\partial
N_{p}}{\partial\lambda}(i)\right)^{2}}}{\sqrt{\sum_{i}N_{s}(i)}}$ (39)
$\displaystyle=$ $\displaystyle
C_{p/s}\frac{\sqrt{\sum_{i}\frac{\lambda^{2}(i)}{f_{s}(i)}\left(\frac{\partial
f_{p}}{\partial\lambda}(i)\right)^{2}}}{\sqrt{\sum_{i}f_{s}(i)}},$

where $C_{p/s}$ is defined as the contrast ratio between the maximum values of
$N_{p}$ and $N_{s}$ over the wavelength range, and $f_{p}$ and $f_{s}$ are the
normalized values by the maximum values. $f_{p}$ and $f_{s}$ are the same
order of magnitude; thus,
$\frac{\sqrt{\sum_{i}\frac{\lambda^{2}(i)}{f_{s}(i)}\left(\frac{\partial
f_{p}}{\partial\lambda}(i)\right)^{2}}}{\sqrt{\sum_{i}f_{s}(i)}}$ was almost
equal to the original $Q_{photon}$ factor. Therefore, the uncertainty of the
radial velocity measurements was determined by a combination of the contrast
ratio, the square root of the number of the host-star photons acquired over
the wavelength range, and the original $Q_{photon}$ factor. The $Q_{sys}$
factor for detecting the reflected light embedded in the stellar light can be
also rewritten as follows:

$\displaystyle Q_{sys,p}$ $\displaystyle=$
$\displaystyle\sqrt{\sum_{i}\left(\frac{\lambda(i)}{N_{s}(i)}\right)^{2}\left(\frac{\partial
N_{p}}{\partial\lambda}(i)\right)^{2}}$ (40) $\displaystyle=$ $\displaystyle
C_{p/s}\sqrt{\sum_{i}\left(\frac{\lambda(i)}{f_{s}(i)}\right)^{2}\left(\frac{\partial
f_{p}}{\partial\lambda}(i)\right)^{2}}$ $\displaystyle\sim$ $\displaystyle
C_{p/s}Q_{sys}.$

As with the $Q_{photon,p}$ factor, the $Q_{sys,p}$ factor was approximately
written as a multiplication of the contrast ratio of the planet to its host
stellar light, $C_{p/s}$, and the original $Q_{sys}$ factor.

Based on the above consideration, we calculated the $Q_{photon}$ and $Q_{sys}$
factors for each target system, using the planet and its host star spectra
simulated by the previous studies. We set the wavelength range to 750 to 980
$nm$ as the broadband light of the reflected light and extracted the
wavelength range of 758 to 770 $nm$ to evaluate whether or not the Oxygen A
line could be detected as a potential biosignature molecule. Although the
Oxygen A line was out of the wavelength range of the instrument concept
proposed in Section 4, the wavelength range can be changed by a small
modification of the optical design. Figure 14 shows the $Q_{photon}$ and
$Q_{sys}$ factors as a function of the resolving power. Compared to the
original $Q_{photon}$ and $Q_{sys}$ factors shown in Figure 1, these factors
were reduced by a factor of $10^{6.5}$, almost corresponding to the contrast
ratio between the planet and its host star. This result was consistent with
what we analytically showed in Equations 39 and 40. The $Q_{photon}$ and
$Q_{sys}$ factors for the Trappist 1-like system were enhanced because of the
more mitigated contrast ratio than the Proxima Cen-like system.

We calculated the uncertainty of the radial velocity measurements from the
$Q_{photon}$ factor. Figure 15 shows the uncertainty for the broadband light
and the Oxygen A line in each target system as a function of the integration
time. The orbital velocities of the virtual planets with the same orbital
parameters as Proxima Cen b and Trappist 1 e were estimated to be
approximately 50 $km/s$ under the parameters compiled in Table 3; thus, the
planetary orbital motion and the molecules in the atmosphere can be detected
at 1 sigma level when the uncertainty of the radial velocity measurements is
less than 0.00017, corresponding to an orbital velocity of 50 $km/s$.
Accordingly, 0.8 and 5 days were required as an integration time for detecting
the orbital motion and the molecule around the Proxima Cen system with 10,000
resolving power. However, the required integration times could be reduced by a
factor of 30 if the resolving power is 7 times higher than that of the
instrument concept in Section 3. While the $Q_{photon}$ and $Q_{sys}$ factors
for the Trappist 1-like system were larger than those for the Proxima Cen-like
system, the required integration time was longer due to the limited number of
the incident photons.

Figure 16 shows the dependency of the noise floor and the resolving power on
the uncertainty of the radial velocity measurements. The uncertainty was
calculated based on the $Q_{sys}$ factors. The noise floor of a few tens of
$ppm$ was acceptable for detecting the planetary orbital motion in each target
system under 10,000 resolving power. In contrast, the noise floor should be
reduced down to a few $ppm$ to detect the oxygen molecule. The noise floor was
also more mitigated for a higher resolving power. When the resolving power is
70,000, the systematic noise floor of 10 $ppm$ level is acceptable. This noise
level has been achieved in the room experiments (Matsuo et al., 2019; Schlawin
et al., 2021). In distinction from the science case of the direct measurement
of the Universe’s expansion history, a resolving power higher than 70,000 is
desierable for detecting the planetary orbital motion and the molecules from
the reflected light. When the resolving power of this proposed concept is
enhanced up to 70,000, this instrument concept could provide a new promising
approach for characterizing the reflected light from the non-transiting
planets orbiting nearby late-type stars. The observable targets by this
approach within a feasible integration time would be middle and late M-type
stars within 10 and 5 $pc$, respectively. However, this evaluation ignored the
impact of the stellar variability on the photometric precision. How the
stellar activity affects the detectability should be investigated as a future
work.

Based on these considerations, the increase in the resolving power of the
spectrograph to 70,000 cannot only reduce the required integration time for
detecting the atmospheres of exoplanets but also mitigate the condition on the
systematic noise error. In the next subsection, we present an optical design
of a seven times higher dispersed spectrograph than that for the direct
measurement of the Hubble parameters.

The abovementioned results were in the same order of magnitude as those
roughly estimated in terms of the signal-to-noise ratio. The signal-to-noise
ratio for detecting the planet signal embedded in the stellar light could be
improved by the square root of the number of the absorption lines resolved by
a high-resolution spectrograph (Snellen et al., 2015):

$\displaystyle\left(\frac{S}{N}\right)_{detection}$ $\displaystyle\approx$
$\displaystyle\frac{S_{p}}{\sqrt{S_{s}}}\sqrt{N_{lines}}$ (41)
$\displaystyle=$ $\displaystyle\frac{C_{p/s}\sqrt{N_{lines}}}{\sqrt{S_{s}}},$

where $S_{p}$ and $S_{s}$ are the numbers of the planet and star photons in
each spectral element, respectively, and $N_{lines}$ is the number of
absorption lines included in the wavelength range. We assumed that the shot
noise caused by the stellar light dominated the detector noise. When the
signal-to-noise ratio was equal to 1, the uncertainty of the radial velocity
measurements was approximately $\frac{c}{R}$, where $R$ is the resolving
power. For example, when the resolving power was 10,000, the number of the
resolved lines was approximately 3200 and 100 for the broadband light and the
Oxygen A line, respectively. The integration times required for detecting an
orbital motion with $\delta V_{p}$ of 30 $km/s$ in the Proxima Cen-like system
are 11.4 and 239 h for the broadband light and Oxygen A line cases,
respectively. As shown in Figure 15, these values were in a good agreement
with those required for reducing the uncertainty of the radial velocity
measurements to 0.0001, corresponding to the orbital motion of 30 $km/s$.

Table 3: Parameters of the two target systems | Proxima Centauri-like system | Trappist 1-like system
---|---|---
Distance | 5 $pc$ | 5 $pc$
Temperature of host star | 3000 $K$ | 2500 $K$
Stellar mass | 0.12 $M_{\odot}$ | 0.09 $M_{\odot}$
Stellar radius | 0.16 $R_{\odot}$ | 0.12 $R_{\odot}$
Planet radius | 1 $R_{\oplus}$ | 1 $R_{\oplus}$
Semi-major axis | 0.049 $AU$ | 0.029 $AU$
Composition of atmosphere | Earth-like composition | Earth-like composition
Surface pressure of planet | 1 $bar$ | 1 $bar$
Figure 14: $Q_{photon}$ (blue lines) and $Q_{sys}$ (red lines) factors of the
broadband light and the Oxygen A line as a function of the resolving power for
the (a) Proxima Cen-like and (b) Trappist 1-like systems. The solid and dashed
lines indicate the broadband and Oxygen A line cases, respectively. The
broadband light and the Oxygen A line range from 750 to 980 $nm$ and 758 to
770 $nm$, respectively. Figure 15: Uncertainty of the radial velocity
measurements in two different resolving powers: 10,000 (green lines) and
70,000 (purple lines) as a function of the integration time for the (a)
Proxima Cen-like and (b) Trappist 1-like systems. The solid and dashed lines
indicate the broadband and Oxygen A line cases, respectively. The horizontal
line represents the uncertainty of 50 $km/s$, almost corresponding to the
orbital velocities of Proxima Cen b and Trappist 1e.  Figure 16: Uncertainty
of the radial velocity measurements caused by the systematic noise error in
the broadband light and the Oxygen A line for the Proxima Cen-like (upper
panels) and Trappist 1-like systems (lower panels) as a function of the
resolving power. The red line in each panel represents the 50 $km/s$
uncertainty almost corresponding to the orbital velocities of Proxima Cen b
and Trappist 1e.

###  5.2 Spectrograph design for charcterization of the atmospheres of
habitable planet candidates

We present an optical design of the high-dispersed densified pupil
spectrograph for characterizing the atmospheres of habitable planet
candidates. The resolving power was set to approximately 70,000, instead of
that of 10,000. The wavelength of the spectrograph ranges from 745 to 984 $nm$
to detect the Oxygen A line at 0.76 $\micron$ as a biosignature and a strong
water vapor line around 0.95 $\micron$ as an indicator of the habitability.
Figure 17 shows the optical design of the spectrograph. Although the optical
configuration was the same as that for the direct measurement of the Hubble
parameters, only the optical parameters were changed to increase the resolving
power by a factor of seven. Table 4 compiles the parameters of the optical
design.

The volume of the spectrograph was 750 (L) x 750 (W) x 250 (H) $mm$. This
small spectrograph design could be utilized for space telescopes. This
proposed approach could realize a combination of the high presicion
spectrophotometry with the high resolving power, which opens a new path toward
characterization of the atmospheres of nearby habitable planet candidates. We
discuss the reason why this proposed concept could more reduce the volume of
the spectrograph compared to conventional high-dispersion spectrographs with
the same field of view in Section 6.

The increase in the resolving power of the spectrograph to 100,000 is
achievable by optimizing the optical parameters. The required integration time
could be more reduced than that shown in Figure 15. However, because the
higher resolving power more limits the wavelength range with the same detector
format, another spectral channel is required to keep the same wavelength
range, which widens the size of the spectrograph. The resolving power should
be determined such that the resource of a space telescope and the science
merit are balanced.

Figure 17: High-dispersion spectrograph design optimized for characterization of the atmospheres of habitable planet candidates. The blue, green, and orange rays indicate the diffraction orders of 83, 72, and 64, respectively. Each ray has a field-of-view of 4 $\frac{\lambda}{D}$ in radius. The entrance beam with 3 $mm$ diameter was divided into five without any densification. Figure 18: (a) Echelle format on the detector and (b) the wavelength range of each order. The 20 vertical lines show the different diffraction orders of from 64 to 83 indicated by the numbers shown above the lines. The detector size was assumed to be 48.3 x 48.3 $mm$. Table 4: Parameters of the optical designs for characterization of habitable planet candiates Wavelength range | 745 - 982 $nm$
---|---
Resolving power at the central wavelength of each order | 71,000
Field-of-view | 4 $\frac{\lambda}{D}$ at 900 $nm$
Incident beam diameter | 3 $mm$
Focal length of the collimator lens | 42.9 $mm$
Number of sub-pupils | 5
Focal length of the Cassegrain system | 8400 $mm$
Type of cross-disperser | Reflection grating
Pitch interval of the Echelle grating | 30 $lines/mm$
Brazed angle | 70 $degree$
Diffraction order | 64 - 83
Focal length of the camera lens system | 420 $mm$
Number of samplings along spectral direction | 2.5
Detector format | 4 k x 4 k
Pixel size of the detector | 12 $\micron$

###  5.3 Measurement of the thermal emissions at mid-infrared wavelengths

This instrument concept could be also applied to a longer wavelength regime.
The densified pupil spectrograph concept was originally proposed for transit
spectroscopy at the mid-infrared wavelengths (Matsuo et al., 2016) and applied
to the Origins Space Telescope concept that was proposed as one of the four
large space mission concepts. This telescope concept provided a very wide
wavelength range of approximately 3 to 600 $\micron$, thanks to the telescope
and the instruments that cooled down to the cryogenic temperature (Leisawitz
et al., 2018). As one of the baseline instruments, the mid-infrared
spectrograph employed the densified pupil spectrograph to search for
biosignature molecules through a highly stable transit spectroscopy of Earth-
sized planets orbiting late-type stars (Meixner et al., 2019). Recently,
focusing on the fact that the contrast ratio between the temperate planet and
its host star is mitigated to $10^{-4}$ beyond 10 $\micron$, Fujii & Matsuo
(2021) showed that the thermal emissions from the non-transiting temperate
Earth-sized planets could be detected with a high-dispersion spectrograph
mounted on a large cryogenic telescope, such as the Origins Space Telescope.
However, the thermal emission is still embedded in the host star in that
wavelength regime; therefore, the instrumental systematic noise could obscure
the absorption lines formed by the molecules in the planetary atmosphere.
Considering that the shot noise caused by the bright host star was in the
order of 100 $ppm$ in the numerical simulation in the previous study, the
noise floor of the spectrograph should be reduced down to a few tens of $ppm$
to detect the molecules in the atmospheres. This proposed instrument concept
could reduce the systematic noise down to a few $ppm$ and increase the
resolving power at the same time; thus, this science case could be realized by
this instrument concept.

##  6 high-dispersion densified pupil spectrograph for space telescopes

We discuss why high-dispersion densified pupil spectrograph designs are
suitable for space observatories. These designs have two major advantages over
general high-dispersion spectrographs that are widely used for the Doppler
planet search on ground-based telescopes: 1. Utilization of a high quality PSF
provided by a space telescope and 2. Miniaturization of a highly-disperesed
densified pupil spectrograph.

###  6.1 Utilization of a high quality PSF

Space telescopes provide stable, high quality PSFs, and the densified pupil
spectrograph concept takes advantage of this to minimize light losses and
variations due to pointing jitter. We now compare this to modern fiber-fed
spectrographs that are traditionally used for precision radial velocity
measurements on ground-based telescopes (e.g., ESPRESSO, EXPRES, KPF). These
modern spectrographs are typically located in a temperature- and vibration-
controlled room, that are fed by a long optical fiber that provides mechanical
isolation from the telescope.

There are two main differences between the two types of spectrographs. One is
the amount of light from a target that is largely lost from the observation.
Only part of the PSF passes through a signle-mode fiber that feeds a
spectrograph, the remaining light is rejected at the edge of the fiber. As
shown in Figure 5, the photometric variation caused by the motion loss is a
few hundred ppm even under a good line-of-sight jitter of 0.05
$\frac{\lambda}{D}$ when the fiber acts as a small field stop on order of the
PSF size. In contrast, the field of view (field stop) of the densified pupil
spectrograph concept can be larger than 4 $\frac{\lambda}{D}$, depending on
the amount of the incident light to the line-of-sight sensor; as the incident
light to the sensor is brighter, the field of view could be larger. The larger
field of view significantly reduces the photometric variation due to the
motion loss as shown in Figure 5.

Another difference is whether the angle of the incident light to the
spectrograph is precisely monitored or not. Because the slit viewer mounted on
a conventional spectrograph produces images of the slit field on the focal
plane (fiber-fed or not), the slit view is not designed to precisely measure
the incident angle of the light to the spectrograph. The main role of the slit
viewer is to confirm whether the incident light is introduced to the
spectrograph. In contrast, the proposed line-of-sight sensor for the densified
pupil spectrograph concept can re-form a PSF-like core from the remaining
light on the detector. Because the incident angle of the light directly
reflects in the position of the core, a positional precision of 0.001
$\frac{\lambda}{D}$ could be achievable, which leads to reduce the systematic
error due to the difference of the line-of-sights at two epochs (less movement
means less error). Based on these considerations, this proposed densified
pupil approach can utilize high-quality PSFs provided by space telescopes to
significantly improve photometric precision over modern fiber-fed
spectrographs. We note that a multi-mode fiber for wider field of view cannot
deliver the high quality PSF to the detector, which will induce an impact on
the photometric precision.

###  6.2 Miniaturization

The size of the spectrograph shown in Section 5 (summarized in Table 4) could
be fit onto an approximately 750 x 750 $mm$ bench, achieving the resolving
power of more than 70,000. We discuss the reason why the volume of the
densified pupil spectrograph can be reduced compared to that of conventional
high-dispersion spectrographs with the same field of view. As introduced in
Section 3, the densified pupil spectrograph concept divides a pupil into five
slices, and the beam size along the direction orthogonal to the spectral
direction can be reduced by the number of the divided subpupils. Although
additional small space (80 x 50 $mm$) is needed for a Bowen-Walraven type
image slicer working as a pupil slicer, this pupil division allows reducing
the size of the spectrograph in two ways. First, the angle between the
incident and output beams to the Echelle grating can be smaller. The distance
between the Echelle grating and the cross-disperser is inversely proportional
to the number of the subpupils; the distance can be reduced by a factor of
five compared to conventional high-dispersion spectrographs. Second, when the
distance from the Echelle grating to the cross-disperser is reduced, the
diameter of the incident beam to the camera lens system can be also reduced by
almost the same factor. Thus, this proposed approach can miniaturize the
spectrograph, which is useful for space telescopes.

##  7 Conclusion

The high-precision radial velocity measurements allowed us to open new science
frontiers from the direct measurement of the Universe’s expansion history to
the characterization of the habitable planet candidates orbiting nearby late-
type stars. We introduced herein two important indications, $Q_{photon}$ and
$Q_{sys}$, to evaluate how suitable the obtained spectrum is for the radial
velocity measurements. While the $Q_{photon}$ factor was used for evaluating
the principal limitation on the measurements, the $Q_{sys}$ factor bridged the
relation between the systematic noise floor and the precision of the radial
velocity measurements. We analytically described how the photometric precision
could affect their precision using the $Q_{photon}$ and $Q_{sys}$ factors. The
impact of the photometric error on the precision decreased as the resolving
power increased. However, the two factors were saturated in the resolving
power higher than 10,000 for the Lyman alpha forests of the high-redshift
QSOs. The resolving power and the photometric precision limited by the
systematic instrumental noise must be simultaneously enhanced.

Based on these considerations, we proposed a new approach to enable high-
precision radial velocity measurements over a decade. This instrument concept
was composed of a high-dispersion densified pupil spectrograph and a novel
line-of-sight sensor minimizing the impact of a telescope pointing error on
the spectro-photometric performance. The densified pupil spectrograph produced
the spectra of the densified sub-pupils on the detector plane, to which the
primary mirror was optically conjugated, and largely reduced the photometric
variation caused by the PSF movement on the detector plane due to the
telescope line-of-sight error. This was because the position of each spectrum
is, in principle, independent of the low-order wavefront errors.

However, the difference between the line-of-sights at two epochs with a time
baseline of a decade still affected the photometric accuracy, even though the
densified pupil spectrograph was applied to the radial velocimetry. The light
passing through a diffraction grating was aberrated, and speckles were formed
on the science detector, on which the densified pupil spectra were recorded. A
change in the line-of-sights moved the position of the ASF on the dispersion
grating; thus, the speckle pattern changed, leading to the degradation of the
photometric precision. Stabilizing the speckle patterns on the detector is
very important for the precision improvement. Therefore, the role of the line-
of-sight sensor was to accurately measure the telescope line-of-sight and give
the measurement value to a tilt compensation device (e.g., a real-time tip-
tilt mirror). In contrast, most of the incident light was introduced to the
densified pupil spectrograph as the radial velocity measurements to improve
the signal-to-noise ratio and reduce the systematic noise caused by the motion
loss on the field stop. Therefore, this new line-of-sight sensor precisely
measured the line-of-sight from the remaining light without any significant
impact on the main lobe of the ASF. In other words, the line-of-sight sensor
could perform at the same time when the densified pupil spectrograph is
recorded. Note that the remaining light had little information on the line-of-
sight; hence, a general slit viewer does not work under the configuration.

This line-of-sight sensor produces a PSF-like image on its detector from the
outer rings of the ASF on the field stop. This can be realized by putting a
pupil stop with a diamter that is excatly similar with that of the entrance
pupil, which is put after the outer part of the ASF was introduced to the
optical sensor system. The PSF-like image was not equal to the original PSF
normally formed on the focal plane. The image was formed through a convolution
of the Fourier transform of the pupil stop with the ASF formed on the field
stop. The PSF-like image position reflected the telescope line-of-sight;
therefore, the difference between the line-of-sights at two epochs can be
measured. However, the precision of the tilt measurements will be limited by
the number of photons available for this sensor type because the remaining
light was used. We confirmed that the sensor could fully measure the line-of-
sight with 0.001 $\frac{\lambda}{D}$ accuracy for an integration time of a few
minutes under an appropriate assumption. We quantitatively derived the impact
of the systematic error on the photometric precision based on an analytical
wavefront propagation from the diffraction grating to the detector. A
photometric precision of 1 $ppm$ is achievable in optical when the RMS of the
diffracted wavefront passing through the diffraction grating is a few tens of
$nm$.

We presented herein two compact densified pupil spectrograph designs with
resolving powers of 10,000 and 70,000 by advancing the design originally
proposed for mid-infrared transit spectroscopy. The former and latter are
optimized for the direct measurment of the Universe’s expansion history and
the characterization of nearby habitable planet candidates, respectively.
While the Lyman alpha forest could be fully resolved by the resolving power of
10,000, a seven times higher resolving power is preferable for the
characterization of the planet atmospheres to reduce the integration time and
mitigate the requirement on the systematic noise floor.

For the measurement of the Hubble parameters at various redshifts, two
channels could cover a wide wavelength range of 305 to 660 $nm$ with an
optical throughput of approximately 63 % (not including the qauntum efficiency
and the telescope throughput). Thanks to the line-of-sight monitor, the
systematic error due to the difference of the incident angles to the
spectrograph at two epochs could be reduced down to 1 $ppm$. Even though the
noise floor is limited to the leve of 10 $ppm$ by the detector system, the
precision of the radial velocity measurements could be improved to $\sim$ 2
$cm/s$, which is a fundamental limitation caused by the stability of a
calibration source. The Hubble parameters could be directly measured at
various redshifts ranging of 1.5 to 4 from measurement of a QSO over a decade
at the same precision as the indirect methods, such as the baryon acoustic
oscillation.

To detect the Oxygen A line as a biosignature and the water vapor lines as an
indicator of habitability, we set the wavelength range to 745 - 982 $nm$ and
the resolving power to 71,000, which increases $Q_{photon}$. We successfully
present a compact optical design of the high-dispersed densified pupil
spectrograph, increasing the resolving power by a factor of seven. The volume
of the spectrograph could be within 750 (L) x 750 (W) x 250 (H) $mm$. We
investigated the possibility of whether or not the reflected light from
virtual habitable planet candidates orbiting nearby non-transiting Proxima
Cen- and Trappist 1-like stars can be detected with this instrument concept.
Assuming that the telescope diameter is 15 $m$, the Oxygen A line could be
detected with a feasible integration time for the Proxima Cen- and Trappist
1-like stars within 10 and 5 $pc$, respectively. Because the high resolving
power enhances the $Q_{sys}$ factor, the required condition on the systematic
noise floor is also mitigated to the level of 10 $ppm$, which was already
achieved by a prototype low-dispersion densified pupil spectrograph in an
experiment room. Thus, a combination of the high spectral resolving power and
the high precision spectrophotometry could open a new path toward
characterizing the atmospheres of nearby non-transiting habitable planet
candidates without spatially separating the faint planet light from the bright
stellar light.

Furthermore, focusing on the fact that the visible stellar spectrum imprinted
on the reflected light has a number of the absorption lines over the wide
wavelength range, we could easily collect a sufficient number of photons to
detect the reflected light. This approach directly measures the stellar
spectrum as well as the reflected light; hence, an exact stellar spectrum
model is not required. Consequently, we found that the reflected light,
including a number of the stellar absorption lines, could enhance the
$Q_{photon}$ and $Q_{sys}$ factors and allow us to measure the planetary
orbital motions with only a few hours integration. The instrument design
specification was fully satisfied with the detection of the broadband
reflected light. Thus, before the characterization of the planet atmospheres
is performed, the true planet mass could be determined for the first a few
hours. We note that this evaluation ignored the impact of the host stellar
activity on the detectability of the planetary orbital motions and
atmospheres. A detailed investigation on the detectability of the reflected
light with a stable high-dispersion spectrograph is required in the next step.

Finally, we discussed why the highly-dispersed densified pupil spectrograph is
suitable for space applications, comparing this concept with conventional
fiber-fed high-dispersion spectrographs. This concept utilizes a stable and
highly qualified PSF provided by a space telescope. Because the field of view
of the spectrograph is more than 4 $\frac{\lambda}{D}$, the qualified PSF
could be delivered to the detector without any significant light loss. The
modest field of view could reduce the photometric variation due to the motion
loss on the focal plane. In addition, the qualified PSF allows us to precisely
measure the incident angle of the light to the spectrograph from the remaining
light out of the field of view. Furthermore, thanks to the divided sub-pupils,
this concept could more reduce the volume of the spectrograph compared to the
general high-dispersion spectrographs. This miniaturization will enable to
increase the resolving power under a limited resource of the space telescope.

##  Appendix A Lyman alpha forest spectra

This section briefly describes the simulations and the post-processing steps
used to generate the Lyman-$\alpha$ forest spectra used in this proposal. We
used the TNG300-1 simulation 555https://www.tng-project.org/ (Springel et al.,
2018; Marinacci et al., 2018; Naiman et al., 2018; Nelson et al., 2018;
Pillepich et al., 2018a). TNG300-1 has a simulation box of $205h^{-1}Mpc$
(comoving) and contains $2500^{3}$ dark matter particles and an equal number
of gas cells on average. The dark matter particle mass was $3.98\times
10^{7}h^{-1}M_{\odot}$ and the average target gas cell mass was $7.44\times
10^{6}\ h^{-1}M_{\odot}$. The gravitational softening length, which sets the
minimum gravitational force resolution, is $1\ h^{-1}kpc$. The cosmological
parameters were $\Omega_{\Lambda}=0.6911$, $\Omega_{m}=0.3089$,
$\Omega_{b}=0.0486$, $\sigma_{8}=0.8159$, $n_{s}=0.9667$ and $h=0.6774$,
consistent with the Planck 2015 results (Planck Collaboration et al., 2016).
The simulation was run using the cosmological simulation code AREPO (Springel,
2010) and includes models for gravitational dynamics, hydrodynamics, star
formation, supernova winds and black hole accretion and heating. The
gravitational dynamics were computed using the TreePM algorithm, while the gas
dynmics used a moving mesh on a dynamically constructed Voronoi mesh. The
supernova and the black hole wind models were tuned to reproduce several
important properties of the galaxy stellar mass function at $z=0$ and
reproduced numerous independent observations (Nelson et al., 2018). For
further details on the simulations we refer the reader to Pillepich et al.
(2018b) and Weinberger et al. (2017).