# Rogue planets and brown dwarfs: Predicting the populations of free-floating
planetary mass objects observable with JWST

Aleks Scholz SUPA, School of Physics & Astronomy, University of St Andrews,
North Haugh, St Andrews, KY16 9SS, United Kingdom  Koraljka Muzic CENTRA,
Faculdade de Ciências, Universidade de Lisboa, Ed. C8, Campo Grande, 1749-016
Lisboa, Portugal  Ray Jayawardhana Department of Astronomy, Cornell
University, Ithaca, NY 14853, USA  Lyra Quinlan  James Wurster SUPA, School of
Physics & Astronomy, University of St Andrews, North Haugh, St Andrews, KY16
9SS, United Kingdom

###### Abstract

Free-floating (or rogue) planets are planets that are liberated (or ejected)
from their host systems. Although simulations predict their existence in
substantial numbers, direct observational evidence for free-floating planets
with masses below $\sim 5\,M_{\mathrm{Jup}}$ is still lacking. Several cycle-1
observing programs with JWST aim to hunt for them in four different star-
forming clusters. These surveys are designed to be sensitive to masses of
$1-15\,M_{\mathrm{Jup}}$ (assuming a hot-start formation), which corresponds
to spectral types of early L to late T for the ages of these clusters. If the
existing simulations are not wide off the mark, we show here that the planned
programs are likely to find up to 10-20 giant rogue planets in moderate
density clusters like NGC1333 or IC348, and several dozen to $\sim 100$ in
high-density regions like NGC2024 and the Orion Nebula Cluster. These numbers
correspond to 1-5% of the total cluster population; they could be
substantially higher if stars form multiple giant planets at birth. In
contrast, the number of free-floating brown dwarfs, formed from core collapse
(‘like stars’) is expected to be significantly lower, only about 0.25% of the
number of stars, or 1-7 for the clusters considered here. Below
10$\,M_{\mathrm{Jup}}$ that number drops further by an order of magnitude. We
also show that the planned surveys are not at risk of being significantly
contaminated by field brown dwarfs in the foreground or background, after
spectroscopic confirmation. Taken together, our results imply that if a
population of L and T dwarfs were to be found in these JWST surveys, it is
expected to be predominantly made up of rogue planets.

##  1 Introduction

Free-floating (or rogue) planets are objects that are formed in a disk around
a young star, and subsequently ejected from their planetary system. According
to simulations, these rogue planets are expected to be a significant component
of the substellar population, in particular for masses below the Deuterium
burning limit (e.g. van Elteren et al., 2019; Parker & Quanz, 2012). So far we
have very limited knowledge of this population. Deep surveys of clusters,
star-forming regions, and the field have only been able to dip into the
planetary-mass domain (e.g. Scholz et al., 2012; Peña Ramírez et al., 2012;
Gagné et al., 2017; Lodieu et al., 2021; Miret-Roig et al., 2022; Bouy et al.,
2022), and found a few dozen of spectroscopically confirmed objects below the
Deuterium burning limit, which could have formed either from core collapse
(‘like stars’) or in a protoplanetary disk (‘like planets’). Meanwhile,
microlensing surveys tentatively indicate the existence of a population of
free-floating objects with planetary masses (Mróz et al., 2017; McDonald et
al., 2021). Our knowledge of the demographics of exoplanets will not be
complete without firm observational constraints on the number of those that
got ejected (or liberated) from their host planetary systems.

Table 1: JWST GTO programs approved for cycle 1, with the potential to find
free-floating planets.

Target | NGC1333 | IC348 | NGC2024 | ONC
---|---|---|---|---
Program ID | 1202 | 1229a | 1190 | 1256
Instrument | NIRISS | NIRCam | NIRCam | NIRCam
Mode | Slitless spectroscopy | Imaging | Imaging | Imaging
FOV (sqarcmin) | 30 | 20 | 10 | 80
On-source time (s) | 3100 | 300 | 6660 | 770
Depth (K-mag) | 21 | 23 | 24 | 23
Distance (pc)b | 296 | 324 | 414 | 403
Number of starsc | 200 | 500 | 800 | 2600
Scheduled ford | 8-10/2023 | 8-10/2022 | 2-3/2023 | 9-10/2022

a Program 1229 also includes a NIRSpec spectroscopy run at a later time, and
parallel observations with NIRISS. b From Kuhn et al. (2019), except for
NGC2024 where we use the distance cited in van Terwisga et al. (2020). c From
Luhman et al. (2016) for Perseus and Kuhn et al. (2015) for Orion. d The JWST
schedule is dynamic, that means, the actual observing dates may be different.

The advent of JWST offers a tremendous boost to this field of research. It is
the first telescope that has the capability to find and characterise directly
free-floating planets down to masses of approximately 1$\,M_{\mathrm{Jup}}$ or
even below. For the first time, we will be able to produce a robust census of
the sub-stellar population with masses between 1 and 15$\,M_{\mathrm{Jup}}$,
and probe directly the predictions from the simulations mentioned above. In
cycle 1, four guaranteed time observing (GTO) programs will be in position to
do exactly that, targeting four different young clusters of stars, two in
Perseus and two in Orion, all within 500 pc. Beyond cycle 1 a wide range of
programs to survey other regions and to characterise the first samples of
free-floating planets can be expected.

Any survey that aims to find free-floating planets in a specific star forming
region will also find objects that are forming like stars in the same region,
as well as contaminating brown dwarfs in the foreground and background of the
cluster that share spectral characteristics with free-floating planets. In
this paper, we aim to gauge likely outcomes of the planned JWST programs, by
estimating how many free-floating planets we expect to find in a given region,
and how that number compares to the total number of objects with similar
characteristics that will be detected in the same surveys.

In the following we will distinguish between brown dwarfs, as objects that are
formed alongside stars in the collapse and fragmentation of molecular clouds,
and free-floating or rogue planets, as objects formed in and then ejected from
a (proto-)planetary system. We acknowledge that at the time of writing there
is no observational test available to distinguish between formation scenarios
for individual objects, although research on exactly that topic will also
benefit greatly from JWST observations. Most of the observational evidence to
date supports the view that free-floating substellar objects down to the
Deuterium burning limit are formed predominantly ‘like stars’ (see review by
Luhman, 2012, and references therein). It is also plausible to assume that
this population will diminish in number per mass bin as we probe lower and
lower masses and approach the opacity limit for fragmentation (Bate, 2012).
The purpose of this paper is to find out what type of objects, from which
formation channel, we should expect to find below the Deuterium burning limit,
based on our current understanding. Therefore it makes sense to distinguish
populations based on formation scenarios.

We proceed as follows. In section 2 we summarise the GTO programs aiming to
find free-floating planets in young clusters. In section 3 we collate and
contextualise predictions from work on simulating the dynamical evolution of
planetary systems, to estimate the number of free-floating giant planets in
young clusters. In section 4, we estimate the expected number of brown dwarfs
(i.e. formed ‘like stars’) in the planetary mass domain. We devise a method to
estimate the number of field brown dwarfs that may contaminate surveys for
free-floating planets in section 5. Finally, we summarize and discuss these
estimates in section 6.

##  2 Summary of relevant JWST cycle 1 programs

In JWST’s cycle 1, four GTO programs will conduct deep surveys in nearby star
forming regions, and are in principle able to find young free-floating objects
down to masses comparable to Jupiter. The four target regions are all within
500 pc, two in Perseus (NGC1333 and IC348) and two in Orion (NGC2024 and the
Orion Nebula Cluster, short ONC). These four programs are summarised in Table
1, with their basic parameters estimated from the published survey design.
While the survey in NGC1333 with the WFSS mode of the instrument NIRISS
(Willott et al., 2022) is a spectroscopy campaign, the remaining three are
designed as multi-band imaging surveys with NIRCam, likely to be followed by
spectroscopy.111We note that another program with similar science goals is
GO2640, targeting the distant (6 kpc) cluster Westerlund 2 with NIRCam.

The four clusters offer a diverse set of initial conditions. NGC1333 and IC348
are of moderate density, host a few hundred stars and brown dwarfs each, but
are without any O stars (Luhman et al., 2016). NGC2024 and the ONC are rich
clusters, embedded in HII regions heated by massive stars (although in the
case of NGC2024 the source of the heating is unclear, see van Terwisga et al.
(2020) and references therein). NGC1333, NGC2024 and the ONC are very young,
with typical ages below or around 1 Myr, while IC348 is a few Myr old. In all
four clusters, deep surveys from the ground or with HST have revealed a rich
substellar population (Levine et al., 2006; Scholz et al., 2012; Da Rio et
al., 2012; Alves de Oliveira et al., 2013).

All four programs have been scheduled in the JWST Long Range Plan, with
anticipated observations in late 2022 for IC348 and the ONC, and 2023 for
NGC2024 and NGC1333. In Table 1 we estimate the survey footprint and the
K-band depth for the observations. The footprint (the field of view) is easy
to infer from the design of the mosaics as shown in the Astronomer’s Proposal
Tool (APT). The depth can be estimated from the given on-source exposure times
using the JWST Exposure Time Calculator (ETC, version 1.7.0.1). We adopt a
substellar source with a $T_{\mathrm{eff}}=1000$ K model spectrum, and change
the normalization in the K-band until sufficient signal-to-noise ratio (SNR)
is achieved. We assume a desired SNR of 10, reached in the wide- and medium
band filters used in these four campaigns, with central wavelength $>1.5\,\mu
m$. Images at shorter wavelength may not be sensitive to the same level. We
use the detector setup as specified in the APT descriptor, and medium
background for all regions.

The NIRISS observations in NGC1333 will provide immediate spectral information
in the H- and K-bands, with a slitless instrument, as a result these are
relatively shallow, with a projected depth at $H\sim 22$ or $K\sim 21$
(Willott et al., 2022) The three NIRCam imaging programs are deeper, by 2-3
mag, but will require spectroscopy follow-up. The observations in the ONC
cover the widest field, the ones in NGC2024 have the longest integration times
(but for a small field). The choice of filters and wavelength coverage, not
discussed here further, is also different from program to program. All
programs have their individual strengths, and are highly complementary to each
other.

How do the projected depths of the programs compare to the instrument
sensitivities as determined from JWST commissioning data (Rigby et al., 2022)?
For NIRISS in WFSS mode, the throughput is 30% better, resulting in on-sky
sensitivities that are 7-20% improved compared to the ETC. For NIRCam, the
throughput is comparable to pre-launch expectations, and the point source
sensitivity is slightly better: 7.3 compared to 10 nJy, for a SNR of 10 in
10000 s exposures. This estimated sensitivity corresponds to a K-band (Vega)
magnitude of about $K_{\mathrm{lim}}=27.3$, compared to $27.0$ for the value
taken from the ETC.

Infrared images of star forming regions are challenging for source extraction
and precise photometry, due to the strongly varying background. The early
released images of NGC3324, program ID 2731 (Pontoppidan et al., 2022), may
provide a good example of what to expect from NIRCam imaging of star forming
regions. The pipeline-provided photometry catalogue for the F200W image with a
central wavelength of $2.0\,\mu m$ and an on-source exposure time of 1610 s is
complete up to a Vegamag around 25 in this band. The F200W filter has
approximately the same zeropoint as the 2MASS K-band ($\pm 0.2$ mag). Scaling
to 10000 s, this is still 1.3 mag shallower than $K_{\mathrm{lim}}$ as given
in the commissioning report. The ETC estimates a SNR of $\sim 10$ for the
Carina imaging at the completeness limit achieved in the actual data. Thus,
while imaging of star forming regions unsurprisingly is not going to be as
sensitive as in typical fields, these cross-checks provide re-assurance that
the numbers from the ETC are realistic, albeit perhaps slightly pessimistic.

All programs listed in Table 1 aim to measure the mass function, therefore it
seems appropriate to give some indications about the depth in terms of mass. A
useful benchmark is a $1\,M_{\mathrm{Jup}}$ mass object at the age of around 1
Myr. According to ‘hot start’ models of planet formation, such an object
should have a brightness of $M_{K}=14$ (Spiegel & Burrows, 2012), which
translates to $K=21.3$ for Perseus or $K=22$ for Orion, all for $A_{V}=0$.
This is comparable to the predictions from Lyon isochrones, which give
$M_{K}=13.7$ (COND) or $M_{K}=14.6$ (DUSTY) for this type of object (Baraffe
et al., 2002).

All programs in Table 1 should reach this limit. In fact, the NIRISS survey in
NGC1333 was designed explicitly for this mass and age limit. The three imaging
surveys will be deeper, and depending on the depth of the follow-up
spectroscopy could reach objects with masses below $1\,M_{\mathrm{Jup}}$. The
spectroscopic survey for IC348, already planned for cycle 1, should
approximately match the depth of the imaging campaign. In this paper we will
focus on the mass range between 1 and 15 $\,M_{\mathrm{Jup}}$ for which all
surveys are sensitive.

We note that when making statements about the depth of the surveys, age and
extinction need to be specified. With $A_{V}=10$, the brightness would drop by
1 mag in the K-band. Also, with increasing age the brightness drops, to
$M_{K}=16-17$ for 5 Myr.

As another note of caution, with ‘cold start’ models, the predicted brightness
of planets would drop by several magnitudes, to $M_{K}=17-18$ (Spiegel &
Burrows, 2012). If these are realistic, the mass range of the JWST surveys
would be severely limited (for example, in the case of NGC1333, the survey
would only reach about $5\,M_{\mathrm{Jup}}$). Recent observational tests of
these model tracks (Flagg et al., 2019; Berardo & Cumming, 2017) as well as
theoretical considerations (Marleau & Cumming, 2014; Marleau et al., 2019)
make hot start scenarios far more plausible, hence, we ignore the cold start
scenario in the following.

##  3 Simulations of planet ejection

Several groups have simulated the dynamical fate of planetary systems in
clustered birth environments (see Parker, 2020, for a review). These
simulations, with a varied set of initial conditions and assumptions,
generally predict the presence of a population of free-floating planets in
star forming regions. The main mechanisms that are responsible for the
liberation of young planets from their host stars are encounters with other
stars in a cluster and planet-planet interactions. Most of the cited work
below comes from N-body simulations, using various approaches to reduce the
complexity of the numerical problem.

The predicted fraction of ejected planets in clusters from these simulations
typically ranges from about 5 to 25% (e.g., Parker & Quanz, 2012; Liu et al.,
2013; Zheng et al., 2015; Forgan et al., 2015). This rate depends on a wide
variety of parameters. It increases with higher cluster density and
substructure, and could perhaps reach 50% or more for very dense, structured
clusters (Daffern-Powell et al., 2022) or to 1-2% for low-density clusters
(Cai et al., 2017). For environments like NGC1333 with moderate initial
density (500-2000 $M_{\odot}$pc-3, Parker & Alves de Oliveira 2017), a
realistic estimate is probably at the lower end of the quoted range. In fact,
Adams et al. (2006) argued that for a cluster like NGC1333 the disruption of
planetary systems must be rare. For massive, dense clusters like the ONC, we
would expect higher ejection rates (14% in the simulations by van Elteren et
al. 2019, which mimic the ONC). On the other hand, disks may be disrupted more
often in high-density environments, either by dynamical encounters with stars
or by the radiation field from massive stars (Winter et al., 2022), inhibiting
planet formation, and thus reducing the number of free-floating planets in an
indirect way. The ejection rate may also depend on the properties of the host
stars, in particular, it may increase with stellar mass (Fujii & Hori, 2019).
Binary systems are more likely to eject planets than single stars, thus, the
binary fraction in a cluster matters as well (Wang et al., 2020).

The age of the cluster is critical: While the number of ejected planets in a
cluster rises strongly in the first few Myrs after planet formation, and then
plateaus, a significant portion of the liberated planets will escape from the
cluster, also on timescales of a few Myrs (van Elteren et al., 2019). This
means that free-floating planets are only expected to be found in young
clusters for a relatively narrow age range, conceivably between 1 and 5 Myr.
For extremely young clusters, planets are not formed yet or have not had time
to experience events that may lead to ejection. For slightly older clusters,
the free-floating planets will have escaped into the field population (see
also Smith & Bonnell 2001). As noted in Section 2, the brightness of these
objects drops steeply with age, making them much harder to detect in older
clusters.

So far, simulations have not explored the combination of all these effects
systematically. Taken together, we would expect significant variations in the
number of free-floating planets as a function of environmental parameters,
such as the space density of stars. However, it may be challenging to directly
measure these variations in the mass function, especially given the low
numbers that are expected; see the discussion in Mužić et al. (2019). We note
that during main-sequence and post-main-sequence lifetimes more planets will
be liberated (Veras & Moeckel, 2012; Veras et al., 2011), which would increase
the number of free-floating planets in the field, but not affect star forming
regions. Observations with JWST, no matter the outcome, will put strong
constraints on the quoted simulations.

In the following we use the simulations as guidance to estimate the number of
expected free-floating planets in young clusters with masses above
1$\,M_{\mathrm{Jup}}$, which are directly detectable with JWST (see Section
2). Given the dependencies outlined above, this is aimed to be an order of
magnitude estimate.

To start, we need to establish how many giant planets are likely to be formed
in the first place. Direct imaging surveys for giant planets typically detect
planets with masses $>1\,M_{\mathrm{Jup}}$ and orbits $>5$ au around a few
percent of their target stars (Biller et al., 2013; Vigan et al., 2017; Gaudi
et al., 2021). Specifically, for young systems Vigan et al. (2021) find that
$\sim 6$% host at least one such planet. In addition, long term radial
velocity studies constrain the occurrence rate for $>1\,M_{\mathrm{Jup}}$
planets on orbital separations larger than 1 au to about 5-15% (Wolthoff et
al., 2022; Wittenmyer et al., 2020; Fulton et al., 2021). About 1-3% of stars
will host a hot or warm Jupiter on orbits $<1$ au (Wright et al., 2012;
Wittenmyer et al., 2020). Microlensing studies indicate that 17% of stars have
planets of 0.3-10$\,M_{\mathrm{Jup}}$ and separations 0.5-10 au (Cassan et
al., 2012).

Note that all these studies focus on stars older than the star forming regions
we are interested in. They will therefore not include planets that have been
ejected or planets that have plunged into the star during the first Myrs.
Thus, the occurrence rate may be significantly higher for young stars.
Overall, it is plausible to assume that at least around 20% of very young
stars form a planet with a Jupiter mass or more, most of these will reside on
separations of 1 au or larger. With the ejection rates from simulations, as
quoted above, that means between 1 and 5% of stars eject a giant planet.

The approximate number of young stars in the four clusters in question is
cited in Table 1. For NGC1333 we would expect about 40 of them to have formed
a giant planet, for IC348 100, for NGC2024 160 and for the ONC 520. Again
using the ejection rates from simulations, the number of free-floating planets
with masses above 1$\,M_{\mathrm{Jup}}$ would be 2-10 in NGC1333, 5-25 in
IC348, 8-40 in NGC2024, and 26-130 in the ONC. These numbers are summarised in
Table 2.

Figure 1: Mass function in NGC1333 (black dots), matched by a log-normal mass
function (dashed line) with $M_{c}=0.3\,M_{\odot}$ and $\sigma=0.5$, and a
power-law mass function (dash-dotted line) with $\alpha=0.7$. The shaded
region show the variation when the parameters of the mass functions are
changed in the plausible range (see text). The red dotted line marks the upper
mass limit considered in our estimates.

The estimates come with a number of serious caveats. Just to name a few: As
mentioned above, some of these rogue planets will leave the cluster on
timescales comparable to the age of the stars, and some may not have had
sufficient time yet to get ejected. Also, the surveys introduced in Section 2
do not cover the entirety of the clusters. Thus, not all these planets will
reside in the cluster areas covered by JWST in cycle 1. Follow-up observations
to cover wider areas to complete the census may be warranted.

The entire architecture of the young planetary systems has not been fully
included yet in the available simulations of ejections. The ejection rate may
be a function of planet orbital separations, planet masses, and overall
numbers of planets. Inner planets may be better protected against ejection
(Cai et al., 2017), which may significantly change the numbers and
characteristics of rogue planets. At least for separations $>5$ au, the
simulations by van Elteren et al. (2019) do not show such dependencies in
single and multi-planet systems.

Maybe most importantly, most of the cluster simulations quoted above focus on
the disruptions caused by stellar encounters. The impact of processes internal
to the young planetary system, like planet-planet interactions and planet-disc
interactions (Baruteau et al., 2014), on the free-floating population has not
been fully explored yet. Planet-planet interactions in particular will cause
instabilities, drive up eccentricities, and lead to ejections (Carrera et al.,
2019; Li et al., 2021). Planet-planet interactions will especially affect
giant planets; they can act as ’catalysts for planetary system disruption’
(Cai et al., 2017). So far, however, in simulations attempting to quantify the
population of free-floating planets, the outcomes are largely determined by
encounters with stars, partly due to the simplified setup of planetary
systems. For example, in the simulations by van Elteren et al. (2019) the
majority of the ejected planets have experienced an encounter with another
star within 1 Myr prior to ejection (about 70%).

If stars typically produce several giant planets, from which most are ejected
by planet-planet interactions, the free-floating planet population could be
significantly larger than estimated above (Veras & Raymond, 2012; Li et al.,
2015). Emsenhuber et al. (2021) discusses giant planet formation as a ‘self-
limiting process’, meaning that more giant planets are likely to lead to more
dynamical instabilities within the system, with ejections as one possible
outcome. How many giant planets are typically formed at birth is still an open
question. Internal processes will produce free-floating planets irrespective
of cluster environment, therefore, the surveys of low or moderate density
clusters (NGC1333 and IC348) are particularly relevant to constrain those
mechanisms.

Table 2: Number of very low mass objects according to a log-normal or a power-
law mass function.

| Fractiona | NGC1333 | IC348 | NGC2024 | ONC
---|---|---|---|---|---
Free-floating planetsb | 1-5% | 2-10 | 5-25 | 8-40 | 26-130
Log-normal MF, 1-15$\,M_{\mathrm{Jup}}$ | 0.25% | 1 | 1 | 2 | 7
– range | 0.1-1.4% | 0-3 | 0-7 | 0-11 | 0-36
Power-law MF, 1-15$\,M_{\mathrm{Jup}}$ | 11% | 22 | 55 | 88 | 286
– range | 4-31% | 8-62 | 20-155 | 32-248 | 104-806
Contamination L/T dwarfsc |  | 1.4 | 15 | 0.2 | 1.0

a Fraction of objects relative to the total number of stars and brown dwarfs
in a cluster (see Table 1). b According to the estimates in Section 3. c
According to the estimates in Section 5.

##  4 Planetary-mass brown dwarfs

In addition to ejected planets, we expect that young clusters will harbor
brown dwarfs with masses comparable to giant planets that are formed by cloud
fragmentation and core collapse, i.e. in a way similar to stars. Here we
estimate how many such objects we expect to find in deep surveys.

Star formation simulations predict that the outcome is a population of stars
with masses in agreement with a log-normal mass function (Bate, 2012), with a
cutoff at the opacity limit for fragmentation. A commonly quoted explanation
for this is the central limit theorem (Offner et al., 2014). The lower mass
limit has not been determined yet empirically, but is expected to be in the
range between 1 and 10$\,M_{\mathrm{Jup}}$ (Bate, 2012). The statistics of the
stellar population predicted by simulations will depend on the specific
formulation and ingredients, e.g., turbulence (Padoan & Nordlund, 2004),
radiative feedback (Guszejnov et al., 2016), stellar winds (Krumholz et al.,
2012), magnetic fields (Price & Bate, 2007; Wurster et al., 2019) are all
expected to alter the shape of the IMF and the star formation rate
significantly.

Observationally, however, the log-normal mass function for local galactic star
forming regions is found to be universal, or near universal (Bastian et al.,
2010). The parameters of the log-normal distribution, the critical mass
$M_{c}$ and the width $\sigma$, have been empirically robustly determined for
a wide range of clusters. Damian et al. (2021) measured the mass function in 8
clusters with a range of environmental parameters, and find on average
$M_{c}=0.32\pm 0.02$ and $\sigma=0.47\pm 0.02$. All observational surveys to
date of galactic young clusters are consistent with $M_{c}=0.25-0.35$ and
$\sigma=0.4-0.6$. This is the range of parameters we adopt here. Instead of
the log-normal form, many observational surveys parameterise the mass function
as a power law $dN/dM\sim M^{-\alpha}$. Observed mass functions for brown
dwarfs are consistent with $\alpha=0.7$, with values ranging from 0.5 to 0.9,
for many clusters (Mužić et al., 2017) and also in the field (Kirkpatrick et
al., 2021). Again this points to a universal mass function. In the following
we therefore assume that all clusters share a consistent underlying mass
function for the objects that form from core collapse.

To determine the empirical scaling of the mass function, we use the sample of
100 members of the cluster NGC1333 with estimated masses from Scholz et al.
(2012), covering the domain of brown dwarfs and low-mass stars. The same
sample has also been used in Offner et al. (2014) and is shown to be
consistent with the low-mass mass function in many other clusters. The sample
comprises about half of the known members of NGC1333. Compared to the census
by Luhman et al. (2016), which comprises about 200 members, our IMF sample is
missing the stars around and above $0.7\,M_{\odot}$ as well as some objects in
the outskirts of the cluster.

The mass function for NGC1333 together with a log-normal parameterisation is
shown in Fig. 1. The log-normal mass function has been shifted to match the
datapoints for high-mass brown dwarfs and very low mass stars. We also show in
Figure 1 the standard power-law mass function, with a slope of $\alpha=0.7$
(for $dN/dM$), or $\Gamma=-0.3$ (for $dN/dlog(M)$). This has again been
shifted to match the datapoints in NGC1333. For about an order of magnitude in
masses, from 0.05 to 0.5 $M_{\odot}$, log-normal and power-law
parameterisation are in good agreement. They diverge strongly in the planetary
mass domain below the Deuterium burning limit.

We note that for the two lowest mass bins in NGC1333 the current data clearly
exceed the expected log-normal mass function. A similar excess has been
observed in a number of other clusters and associations, (Miret-Roig et al.,
2022; Gagné et al., 2017; Mužić et al., 2019), see also Figure 2 in Offner et
al. (2014). This could be an indication of a second population, potentially
ejected giant planets, but at this stage the data are not sufficiently robust
to be confident.

We can now add up the expected number of objects according to the assumed
underlying mass function, at first for NGC1333. To do that we use the mass
functions shown in Figure 1. With a log-normal mass function, as expected for
objects forming like stars, we expect 0.5 (between 0 and 3) objects with
masses between 1 and 15$\,M_{\mathrm{Jup}}$ – but since the mass function is
declining steeply, most of these would be near the top end of the relevant
mass range ($M>10M_{\mathrm{Jup}}$). This corresponds to 0.25% of the known
cluster population in NGC1333, including stars and brown dwarfs. The power-law
mass function on the other hand predicts 21 (7-62) objects in the same mass
range, or 11% of the cluster population. More than half of these have masses
below $10\,M_{\mathrm{Jup}}$.

These results can easily be scaled to other clusters, again assuming the
universality of the mass function, and using the number of stars and brown
dwarfs in the cluster as scaling factor. In Table 2 we summarise these
estimates for the four clusters in question here.

Figure 2: Comparison between spectra for young and old brown dwarfs, both at
spectral type L4. The young template is from Luhman et al. (2017), the old is
DENIS-P J170548.38-051645.7, published as template in Burgasser et al. (2010),
obtained from the SpeX Prism Spectral Libraries. All spectra are rebinned to a
resolution of 0.01$\,\mu m$ using Spectres (Carnall, 2017), and scaled to an
arbitrary flux level. On the left side, we show both template spectra at high
SNR, on the right side we added Gaussian noise (for a SNR of 20) to the young
template.

This exercise demonstrates that the population of objects below the Deuterium
burning limit that are formed ‘like stars’ from core collapse, and form the
continuation of the log-normal mass function, is expected to be very small. In
particular, it will be negligible for masses of $1-10\,M_{\mathrm{Jup}}$. The
numbers of expected objects from core collapse is far below the expected
numbers of ejected planets, especially in the 1-10 $M_{\mathrm{Jup}}$ mass
domain. In contrast, the power-law mass function predicts a substantial
population of substellar objects down to masses comparable to Jupiter,
exceeding the estimates for ejected planets we made in Section 3.

The analysis shows how valuable the JWST surveys will be compared to existing
ground-based observations: The current surveys are typically incomplete below
$10\,M_{\mathrm{Jup}}$. Given the uncertainties it is difficult to distinguish
between an extended power-law and a log-normal mass function. On the other
hand, JWST can probe sufficiently deep to make that distinction. If the
observed mass function of free-floating objects below the Deuterium burning
limit is found to be inconsistent with the log-normal mass function, we would
have clear evidence for the presence of free-floating, ejected planets in the
observed clusters.

##  5 Contamination by field brown dwarfs

Every cluster survey is sensitive to a range of objects in the foreground and
background of the target region. In surveys for young L and T dwarfs, the most
problematic type of contamination is from field brown dwarfs of the same
spectral type. While young L and T dwarfs have characteristic spectra, as well
as different colours (Cruz et al., 2009; Almendros-Abad et al., 2022), these
distinctions can be difficult to detect at low signal-to-noise ratios. In
Figure 2, we show a comparison of spectra for a young and an old template, for
spectral type L4, for a resolution comparable to what can be achieved with
NIRISS/WFSS or NIRSpec/PRISM. For the young template, we also add a version
with 5% noise of the signal (or SNR of 20). The differences in the shape of
the spectrum as well as the overall relative fluxes in different bands can
still be appreciated at this noise level. If more noise is present, the
differences between young and old become difficult to discern.

Additionally, in the absence of spectra, more exotic objects in the
background, particularly very late giants and red-shifted AGNs, could in
principle mimic the colour signature of young brown dwarfs. We will neglect
those exotic background contaminants here because we assume that any
substellar candidate will need a spectrum (or a complete spectral energy
distribution in the near/mid-infrared) for confirmation (see also Almendros-
Abad et al. 2022 for further discussion). We note that in regions with strong
extinction (ONC, NGC2024), the cloud acts as a screen and prevents background
contamination that way.

This leaves field brown dwarfs along the pencil beam towards the cluster as
the most important source of potential contamination. To estimate the number
of expected contaminating objects in a survey, we need to know the survey
depth in a given band, as well as the survey footprint. Those numbers are
summarised in Table 1 for the four cycle 1 projects investigated here,
obtained from the project description as currently published (see Section 2).
In addition, we obtain the space densities as a function of spectral type from
Best et al. (2021). We use the parametric form of the absolute J-band
magnitude as a function of spectral type provided by Filippazzo et al. (2015),
and convert to absolute K-band magnitude using the (J-K) colour vs. spectral
type relation (Figure 6 in Best et al., 2021).

We choose here to divide the relevant spectral range into four bins, L0-L5,
L5-T0, T0-T5, and T5-T9. The space densities for those bins are 2.22, 2.20,
1.12, 4.48 $\times 10^{-3}$ pc-3, with an error of 5-10% (Best et al., 2021).
The absolute magnitudes in the K-band are estimated to be 11.3, 13.0, 14.4,
16.9, with a considerable spread of about 0.5 mag. With those absolute
magnitudes, and the survey depth (an apparent magnitude), we can estimate the
limiting distance of the survey as a function of spectral type. For NGC2024
and the ONC we adopt the cluster distance as maximum limiting distance, due to
the screening effect of the cloud. With the survey footprint, we can then
define a survey volume for each project – a pyramid extending to the limiting
distance. Multiplying this volume with the space densities yields the expected
rate of contamination.

Results from this exercise are summarised in Figure 3, for the four surveys
and as a function of spectral type. The errorbars in this figure take into
account $\pm 0.5$ mag uncertainty in survey depth, as well as $\pm 0.25$ mag
in the absolute magnitudes. All other uncertainties are negligible. For
NGC1333, we expect little contamination by field dwarfs – in total, the
calculation yields 1.4 objects in this field (with a range 0.5-4.0), most of
them are going to be early L dwarfs. IC348 could in principle find
considerable numbers of contaminating brown dwarfs. Our calculation yields 15,
with a range from 5 to 42. Similar to NGC1333, most are expected to be early L
dwarfs. For the ONC the total number is 0.9, with a relatively small errorbar
(0.8-1.0), and almost uniformly distributed with spectral type. This is a
result of defining the cloud as a screen that eliminates background
contamination, and thus a portion of the bright L dwarfs. For NGC2024 the
expected numbers are very small, in total less than 0.2, due to the small
field of view. Again the contamination rate is flat vs. spectral type due to
the screening of the cloud. We note that qualitatively similar results were
obtained by Caballero et al. (2008) using a different method and a different
test case.

These numbers should be considered upper limits, for a variety of reasons. For
one, most of the contamination is expected for early L dwarfs, which are
visible to larger distances. Since young early L dwarfs are bright compared to
the survey depth, distinguishing them from the field brown dwarfs with
spectroscopy should be straightforward (see Figure 2). The WFSS survey in
NGC1333 will immediately give spectral information, and the one in IC348 is
associated with a follow-up spectroscopy campaign with NIRSpec. Similar
campaigns can be expected for the Orion observations.

Second, with spectroscopy included, the surveys will not reach the depth
expected for imaging. There will be a number of faint objects at the limits of
the imaging surveys for which spectroscopic follow-up is prohibitively
expensive in terms of integration time. Adopting a putative spectroscopic
depth of $K=22$, the number of contaminants drops to 3.8 for IC348, 0.8 for
the ONC, and 0.1 for NGC2024. As a reminder, the estimated absolute K-band
magnitude for a $1\,M_{\mathrm{Jup}}$, 1 Myr free-floating planet is
$M_{K}\sim 14$, which translates to $K\sim 21$ for Perseus and $K\sim 22$ for
Orion (at $A_{V}=0$). Finally, while the extinction in NGC1333 and IC348 is
moderate and will not completely block the background contamination, it will
screen out some fainter sources and reduce the overall survey volume.

We also note that extinction is going to alter the spectral energy
distributions for most young free-floating planets, which should help
distinguishing them from reddening-free field brown dwarfs. Taken all that
into account, contamination by field brown dwarfs is not going to be an issue
for these surveys. However, spectroscopy with decent signal-to-noise ratio is
going to be needed to confirm individual objects.

While the space densities of brown dwarfs are only known for the local volume
out to distances of 25 pc, we assume here that they are broadly the same when
probing larger distances in the foreground of the clusters investigated here.
This is a plausible assumption: Recent work based on Gaia data does not show
strong variations in stellar density out to distances of 500 pc (Miyachi et
al., 2019; Widmark et al., 2022) towards the galactic directions of Perseus
and Orion, relevant to this work. Similarly, the number counts of stars from
the Besancon Galaxy model (Robin et al., 2003) only change within 10% of the
total as a function of galactic sightline for the regions investigated here.
If anything, the number densities are expected to drop off for distances
$>100\,$pc, due to the galactic structure (see Fig. 1 in Caballero et al.,
2008), again confirming that our estimates should be seen as upper limits.

Figure 3: Number of contaminating field dwarfs in the four surveys discussed
in the text, as a function of spectral types. Spectral types are numerically
defined with L0 being 0.0 and T9 being 19.0. Errorbars take into account
$\pm$0.5 mag uncertainty in survey depth and $\pm$0.25 mag error in absolute
magnitudes. For NGC2024 and the ONC, we assumed that the cloud acts a screen
to eliminate background contamination.

##  6 Summary and discussion

In the preceding sections we have investigated the expected numbers of free-
floating planets in JWST surveys, plus the number of brown dwarfs formed by
core collapse, plus the expected number of brown dwarf contaminants. Our
estimates are tailored to four specific JWST cycle 1 programs, but can be
translated and scaled easily to other target clusters in future programs. In
this section, we first summarise our findings, and then put them into context.

In Section 3 we showed that based on our current understanding and judged by
the outcomes of existing simulations, we expect that nearby star forming
clusters within 500 pc host between a few to a hundred free-floating (or
rogue) planets that are formed in a disk and are later ejected, with masses
above $1\,M_{\mathrm{Jup}}$. These objects would be ejected either due to
encounters with other stars or due to planet-planet interactions. This
corresponds to a few percent of the number of stars in the cluster. These
estimates come with a number of caveats. In particular, the numbers could be
significantly higher if stars form typically several giant planets from which
most are ejected. Empirical measurements will be key in guiding the
development of simulations.

We also show, in Section 4, that the number of planetary-mass brown dwarfs
that are likely to be formed by core collapse like stars, is very low, about
0.25% of the number of stars for the mass range 1-15$\,M_{\mathrm{Jup}}$, and
an order of magnitude lower for 1-10$\,M_{\mathrm{Jup}}$. That means, the mass
function in this domain will be very different depending on which population
dominates. If ejected planets are very rare and significantly less common than
estimated above, we expect a steep dropoff following a log-normal mass
function below the Deuterium burning limit. If ejected planets are as abundant
as simulations predict, we should see a clear deviation from the log-normal
mass function below $10\,M_{\mathrm{Jup}}$. Or put in simple terms: If
observations find significant numbers of free-floating objects with masses
comparable to Jupiter, those are expected to be rogue planets, and not very
low mass brown dwarfs.

Our estimates can be contrasted with existing observational programs that also
target this mass range. In 2011, Sumi et al. published a contentious paper
claiming that the number of planets that are unbound or on wide orbits exceeds
the number of stars (Sumi et al., 2011). This claim triggered a lot of
interest, but did not hold up to further scrutiny. Mróz et al. (2017) find a
95% upper limit of 0.25 free-floating/wide orbit Jupiter-mass planets per
star, estimated from microlensing detections, superseding earlier estimates
from the same method. Some or most of these could be explained by wide (10 au
or wider) Jupiter-type planets (Clanton & Gaudi, 2017), i.e. the upper limit
for free-floating planets would be smaller. Further constraints are expected
from the Kepler/K2 microlensing campaign, but have not been published yet
(McDonald et al., 2021). In the terminology of our paper, this estimate would
include actual ejected planets and low-mass brown dwarfs formed by core
collapse. Microlensing probes the field population; as pointed out in Section
3, we expect a larger number of free-floating planets per star in the field
compared to young clusters, due to escape from the clusters and further
ejections from planetary systems during stellar evolution. As more stars turn
into white dwarfs or neutron stars, some of their planets are going to be
released into the field population. Over timescales of Gyrs, the number of
free-floating planets per star will therefore increase. Adding our estimates
from Section 3 and 4, we would expect in the range of $\sim 0.05$ free-
floating object (planet or brown dwarf) with Jupiter- or super-Jupiter masses
per star, consistent with the limit from microlensing.

Direct searches for ultra-low mass free-floating objects in clusters are
currently only sensitive to masses $>5\,M_{\mathrm{Jup}}$. These surveys tend
to find that objects with masses below the Deuterium burning limit account for
1-14% (Lucas et al., 2006), 2-5% (Scholz et al., 2012), 2-7% (Miret-Roig et
al., 2022), and $\sim 10$% (Peña Ramírez et al., 2012) of the number of stars
and brown dwarfs in the studied clusters. We caution that determining masses
for objects around the Deuterium burning limit is a difficult process, and
therefore these numbers come with significant errorbars. As already indicated
in Section 4, these existing estimates exceed what is expected from a log-
normal mass function, as pointed out by several groups. Although these surveys
are not as deep as the ongoing work with JWST, they are already finding
numbers of planetary mass objects that are comparable to the sum of the
numbers for free-floating planets and very low mass brown dwarfs estimated in
this paper. This may indicate that ejected planets are more abundant than
expected from the existing simulations, or that existing surveys overestimate
the number of objects in this mass domain. Probing the mass ranges below
$5\,M_{\mathrm{Jup}}$ with JWST will certainly provide clarification on this
issue.

Young rogue planets with Jupiter-like masses will have spectral types of L and
T. In Section 5 we also estimate the number of brown dwarfs in the foreground
and background of the young clusters that may be found among those objects
with L and T spectral types. For rich embedded clusters where the cloud
shields the background, like the ONC, the contamination by brown dwarfs is
orders of magnitude lower than the expected numbers of young objects residing
in the clusters with those spectral types. For sparsely populated or low
extinction environments, like NGC1333, the field brown dwarf contamination is
still very low, and dominated by early L type objects which are
straightforward to identify. Follow-up spectroscopy will be needed for a
detailed characterisation, but even with low signal-to-noise ratio and
resolution, the contamination by field brown dwarfs in these surveys will be
negligible.

##  7 Acknowledgements

We would like to thank the referee, Richard Parker, for a concise and
constructive report on our paper. AS, KM, and RJ are PI and Co-Is of the
JWST/GTO program 1202, ’The NIRISS Survey for Young Brown Dwarfs and Rogue
Planets’. KM acknowledges funding by the Science and Technology Foundation of
Portugal (FCT), grants No. PTDC/FIS-AST/28731/2017 and UIDB/00099/2020. LQ
contributed to this paper as part of a summer research project, for which
funding was kindly provided by Rita Tojeiro and Vivienne Wild. This research
has benefitted from the SpeX Prism Spectral Libraries, maintained by Adam
Burgasser at http://www.browndwarfs.org/spexprism.

## References

  * Adams et al. (2006) Adams, F. C., Proszkow, E. M., Fatuzzo, M., et al. 2006, ApJ, 641, 504. doi:10.1086/500393
  * Almendros-Abad et al. (2022) Almendros-Abad, V., Mužić, K., Moitinho, A., et al. 2022, A&A, 657, A129. doi:10.1051/0004-6361/202142050
  * Alves de Oliveira et al. (2013) Alves de Oliveira, C., Moraux, E., Bouvier, J., et al. 2013, A&A, 549, A123. doi:10.1051/0004-6361/201220229
  * Baraffe et al. (2002) Baraffe, I., Chabrier, G., Allard, F., et al. 2002, A&A, 382, 563. doi:10.1051/0004-6361:20011638
  * Baruteau et al. (2014) Baruteau, C., Crida, A., Paardekooper, S.-J., et al. 2014, Protostars and Planets VI, 667. doi:10.2458/azu_uapress_9780816531240-ch029
  * Bastian et al. (2010) Bastian, N., Covey, K. R., & Meyer, M. R. 2010, ARA&A, 48, 339. doi:10.1146/annurev-astro-082708-101642
  * Bate (2012) Bate, M. R. 2012, MNRAS, 419, 3115. doi:10.1111/j.1365-2966.2011.19955.x
  * Beleznay & Kunimoto (2022) Beleznay, M. & Kunimoto, M. 2022, arXiv:2207.12522
  * Berardo & Cumming (2017) Berardo, D. & Cumming, A. 2017, ApJ, 846, L17. doi:10.3847/2041-8213/aa81c0
  * Best et al. (2021) Best, W. M. J., Liu, M. C., Magnier, E. A., et al. 2021, AJ, 161, 42. doi:10.3847/1538-3881/abc893
  * Biller et al. (2013) Biller, B. A., Liu, M. C., Wahhaj, Z., et al. 2013, ApJ, 777, 160. doi:10.1088/0004-637X/777/2/160
  * Bouy et al. (2022) Bouy, H., Tamura, M., Barrado, D., et al. 2022, arXiv:2206.00916
  * Burgasser et al. (2010) Burgasser, A. J., Cruz, K. L., Cushing, M., et al. 2010, ApJ, 710, 1142. doi:10.1088/0004-637X/710/2/1142
  * Caballero et al. (2008) Caballero, J. A., Burgasser, A. J., & Klement, R. 2008, A&A, 488, 181. doi:10.1051/0004-6361:200809520
  * Cai et al. (2017) Cai, M. X., Kouwenhoven, M. B. N., Portegies Zwart, S. F., et al. 2017, MNRAS, 470, 4337. doi:10.1093/mnras/stx1464
  * Carnall (2017) Carnall, A. C. 2017, arXiv:1705.05165
  * Carrera et al. (2019) Carrera, D., Raymond, S. N., & Davies, M. B. 2019, A&A, 629, L7. doi:10.1051/0004-6361/201935744
  * Cassan et al. (2012) Cassan, A., Kubas, D., Beaulieu, J.-P., et al. 2012, Nature, 481, 167. doi:10.1038/nature10684
  * Clanton & Gaudi (2017) Clanton, C. & Gaudi, B. S. 2017, ApJ, 834, 46. doi:10.3847/1538-4357/834/1/46
  * Cruz et al. (2009) Cruz, K. L., Kirkpatrick, J. D., & Burgasser, A. J. 2009, AJ, 137, 3345. doi:10.1088/0004-6256/137/2/3345
  * Daffern-Powell et al. (2022) Daffern-Powell, E. C., Parker, R. J., & Quanz, S. P. 2022, MNRAS, 514, 920. doi:10.1093/mnras/stac1392
  * Damian et al. (2021) Damian, B., Jose, J., Samal, M. R., et al. 2021, MNRAS, 504, 2557. doi:10.1093/mnras/stab194
  * Da Rio et al. (2012) Da Rio, N., Robberto, M., Hillenbrand, L. A., et al. 2012, ApJ, 748, 14. doi:10.1088/0004-637X/748/1/14
  * Emsenhuber et al. (2021) Emsenhuber, A., Mordasini, C., Burn, R., et al. 2021, A&A, 656, A70. doi:10.1051/0004-6361/202038863
  * Filippazzo et al. (2015) Filippazzo, J. C., Rice, E. L., Faherty, J., et al. 2015, ApJ, 810, 158. doi:10.1088/0004-637X/810/2/158
  * Flagg et al. (2019) Flagg, L., Johns-Krull, C. M., Nofi, L., et al. 2019, ApJ, 878, L37. doi:10.3847/2041-8213/ab276d
  * Forgan et al. (2015) Forgan, D., Parker, R. J., & Rice, K. 2015, MNRAS, 447, 836. doi:10.1093/mnras/stu2504
  * Fulton et al. (2021) Fulton, B. J., Rosenthal, L. J., Hirsch, L. A., et al. 2021, ApJS, 255, 14. doi:10.3847/1538-4365/abfcc1
  * Fujii & Hori (2019) Fujii, M. S. & Hori, Y. 2019, A&A, 624, A110. doi:10.1051/0004-6361/201834677
  * Gagné et al. (2017) Gagné, J., Faherty, J. K., Mamajek, E. E., et al. 2017, ApJS, 228, 18. doi:10.3847/1538-4365/228/2/18
  * Gaudi et al. (2021) Gaudi, B. S., Meyer, M., & Christiansen, J. 2021, ExoFrontiers; Big Questions in Exoplanetary Science, 2. doi:10.1088/2514-3433/abfa8fch2
  * Guszejnov et al. (2016) Guszejnov, D., Krumholz, M. R., & Hopkins, P. F. 2016, MNRAS, 458, 673. doi:10.1093/mnras/stw315
  * Kirkpatrick et al. (2021) Kirkpatrick, J. D., Gelino, C. R., Faherty, J. K., et al. 2021, ApJS, 253, 7. doi:10.3847/1538-4365/abd107
  * Krumholz et al. (2012) Krumholz, M. R., Klein, R. I., & McKee, C. F. 2012, ApJ, 754, 71. doi:10.1088/0004-637X/754/1/71
  * Kuhn et al. (2015) Kuhn, M. A., Getman, K. V., & Feigelson, E. D. 2015, ApJ, 802, 60. doi:10.1088/0004-637X/802/1/60
  * Kuhn et al. (2019) Kuhn, M. A., Hillenbrand, L. A., Sills, A., et al. 2019, ApJ, 870, 32. doi:10.3847/1538-4357/aaef8c
  * Levine et al. (2006) Levine, J. L., Steinhauer, A., Elston, R. J., et al. 2006, ApJ, 646, 1215. doi:10.1086/504964
  * Li et al. (2015) Li, Y., Kouwenhoven, M. B. N., Stamatellos, D., et al. 2015, ApJ, 805, 116. doi:10.1088/0004-637X/805/2/116
  * Li et al. (2021) Li, J., Lai, D., Anderson, K. R., et al. 2021, MNRAS, 501, 1621. doi:10.1093/mnras/staa3779
  * Liu et al. (2013) Liu, H.-G., Zhang, H., & Zhou, J.-L. 2013, ApJ, 772, 142. doi:10.1088/0004-637X/772/2/142
  * Lodieu et al. (2021) Lodieu, N., Hambly, N. C., & Cross, N. J. G. 2021, MNRAS, 503, 2265. doi:10.1093/mnras/stab401
  * Lucas et al. (2006) Lucas, P. W., Weights, D. J., Roche, P. F., et al. 2006, MNRAS, 373, L60. doi:10.1111/j.1745-3933.2006.00244.x
  * Luhman (2012) Luhman, K. L. 2012, ARA&A, 50, 65. doi:10.1146/annurev-astro-081811-125528
  * Luhman et al. (2016) Luhman, K. L., Esplin, T. L., & Loutrel, N. P. 2016, ApJ, 827, 52. doi:10.3847/0004-637X/827/1/52
  * Luhman et al. (2017) Luhman, K. L., Mamajek, E. E., Shukla, S. J., et al. 2017, AJ, 153, 46. doi:10.3847/1538-3881/153/1/46
  * Marleau & Cumming (2014) Marleau, G.-D. & Cumming, A. 2014, MNRAS, 437, 1378. doi:10.1093/mnras/stt1967
  * Marleau et al. (2019) Marleau, G.-D., Mordasini, C., & Kuiper, R. 2019, ApJ, 881, 144. doi:10.3847/1538-4357/ab245b
  * McDonald et al. (2021) McDonald, I., Kerins, E., Poleski, R., et al. 2021, MNRAS, 505, 5584. doi:10.1093/mnras/stab1377
  * Miret-Roig et al. (2022) Miret-Roig, N., Bouy, H., Raymond, S. N., et al. 2022, Nature Astronomy, 6, 89. doi:10.1038/s41550-021-01513-x
  * Miyachi et al. (2019) Miyachi, Y., Sakai, N., Kawata, D., et al. 2019, ApJ, 882, 48. doi:10.3847/1538-4357/ab2f86
  * Mróz et al. (2017) Mróz, P., Udalski, A., Skowron, J., et al. 2017, Nature, 548, 183. doi:10.1038/nature23276
  * Mužić et al. (2017) Mužić, K., Schödel, R., Scholz, A., et al. 2017, MNRAS, 471, 3699. doi:10.1093/mnras/stx1906
  * Mužić et al. (2019) Mužić, K., Scholz, A., Peña Ramírez, K., et al. 2019, ApJ, 881, 79. doi:10.3847/1538-4357/ab2da4
  * Offner et al. (2014) Offner, S. S. R., Clark, P. C., Hennebelle, P., et al. 2014, Protostars and Planets VI, 53. doi:10.2458/azu_uapress_9780816531240-ch003
  * Padoan & Nordlund (2004) Padoan, P. & Nordlund, Å. 2004, ApJ, 617, 559. doi:10.1086/345413
  * Parker & Quanz (2012) Parker, R. J. & Quanz, S. P. 2012, MNRAS, 419, 2448. doi:10.1111/j.1365-2966.2011.19911.x
  * Parker & Alves de Oliveira (2017) Parker, R. J. & Alves de Oliveira, C. 2017, MNRAS, 468, 4340. doi:10.1093/mnras/stx739
  * Parker (2020) Parker, R. J. 2020, Royal Society Open Science, 7, 201271. doi:10.1098/rsos.201271
  * Peña Ramírez et al. (2012) Peña Ramírez, K., Béjar, V. J. S., Zapatero Osorio, M. R., et al. 2012, ApJ, 754, 30. doi:10.1088/0004-637X/754/1/30
  * Pontoppidan et al. (2022) Pontoppidan, K., Blome, C., Braun, H., et al. 2022, arXiv:2207.13067
  * Price & Bate (2007) Price, D. J. & Bate, M. R. 2007, MNRAS, 377, 77. doi:10.1111/j.1365-2966.2007.11621.x
  * Rigby et al. (2022) Rigby, J., Perrin, M., McElwain, M., et al. 2022, arXiv:2207.05632
  * Robin et al. (2003) Robin, A. C., Reylé, C., Derrière, S., et al. 2003, A&A, 409, 523. doi:10.1051/0004-6361:20031117
  * Scholz et al. (2012) Scholz, A., Jayawardhana, R., Muzic, K., et al. 2012, ApJ, 756, 24. doi:10.1088/0004-637X/756/1/24
  * Scholz et al. (2013) Scholz, A., Geers, V., Clark, P., et al. 2013, ApJ, 775, 138. doi:10.1088/0004-637X/775/2/138
  * Smith & Bonnell (2001) Smith, K. W. & Bonnell, I. A. 2001, MNRAS, 322, L1. doi:10.1046/j.1365-8711.2001.04321.x
  * Spiegel & Burrows (2012) Spiegel, D. S. & Burrows, A. 2012, ApJ, 745, 174. doi:10.1088/0004-637X/745/2/174
  * Sumi et al. (2011) Sumi, T., Kamiya, K., Bennett, D. P., et al. 2011, Nature, 473, 349. doi:10.1038/nature10092
  * van Elteren et al. (2019) van Elteren, A., Portegies Zwart, S., Pelupessy, I., et al. 2019, A&A, 624, A120. doi:10.1051/0004-6361/201834641
  * van Terwisga et al. (2020) van Terwisga, S. E., van Dishoeck, E. F., Mann, R. K., et al. 2020, A&A, 640, A27. doi:10.1051/0004-6361/201937403
  * Vigan et al. (2017) Vigan, A., Bonavita, M., Biller, B., et al. 2017, A&A, 603, A3. doi:10.1051/0004-6361/201630133
  * Vigan et al. (2021) Vigan, A., Fontanive, C., Meyer, M., et al. 2021, A&A, 651, A72. doi:10.1051/0004-6361/202038107
  * Veras et al. (2011) Veras, D., Wyatt, M. C., Mustill, A. J., et al. 2011, MNRAS, 417, 2104. doi:10.1111/j.1365-2966.2011.19393.x
  * Veras & Moeckel (2012) Veras, D. & Moeckel, N. 2012, MNRAS, 425, 680. doi:10.1111/j.1365-2966.2012.21552.x
  * Veras & Raymond (2012) Veras, D. & Raymond, S. N. 2012, MNRAS, 421, L117. doi:10.1111/j.1745-3933.2012.01218.x
  * Wang et al. (2020) Wang, Y.-H., Perna, R., & Leigh, N. W. C. 2020, MNRAS, 496, 1453. doi:10.1093/mnras/staa1627
  * Widmark et al. (2022) Widmark, A., Widrow, L. M., & Naik, A. 2022, arXiv:2207.03492
  * Willott et al. (2022) Willott, C. J., Doyon, R., Albert, L., et al. 2022, PASP, 134, 025002. doi:10.1088/1538-3873/ac5158
  * Winter et al. (2022) Winter, A. J., Haworth, T. J., Coleman, G. A. L., et al. 2022, MNRAS. doi:10.1093/mnras/stac1564
  * Wittenmyer et al. (2020) Wittenmyer, R. A., Wang, S., Horner, J., et al. 2020, MNRAS, 492, 377. doi:10.1093/mnras/stz3436
  * Wolthoff et al. (2022) Wolthoff, V., Reffert, S., Quirrenbach, A., et al. 2022, A&A, 661, A63. doi:10.1051/0004-6361/202142501
  * Wright et al. (2012) Wright, J. T., Marcy, G. W., Howard, A. W., et al. 2012, ApJ, 753, 160. doi:10.1088/0004-637X/753/2/160
  * Wurster et al. (2019) Wurster, J., Bate, M. R., & Price, D. J. 2019, MNRAS, 489, 1719. doi:10.1093/mnras/stz2215
  * Zheng et al. (2015) Zheng, X., Kouwenhoven, M. B. N., & Wang, L. 2015, MNRAS, 453, 2759. doi:10.1093/mnras/stv1832


# Extrapolation of Hubble Curves

A. E. Pavlov Institute of Mechanics and Power Engineering, Russian State
Agrarian University – Moscow Timiryazev Agricultural Academy, Timiryazevskaya
str. 49, Moscow 127550, Russia
<EMAIL_ADDRESS>
###### Abstract

We present exact solutions to the Friedmann equation in coordinate time and
conformal time in elliptic Legendre integrals. Both approaches describe the
modern Hubble diagram with the same accuracy. Hubble curves are extrapolated
for large redshift values.

###### pacs:

04.20.Cv, 98.80.Es

##  I Introduction

In recent years, two independent collaborations “High Supernova” and
“Supernova Cosmology Project” obtained new unexpected data about cosmological
evolution at very large distances – hundreds and thousands megaparsecs
expressed in redshift values $z=1\div 1.7$ RiessNobel . Surprisingly, it was
found that the decrease of brightness with distance happen noticeably faster
than it was expected according to the standard cosmological model with the
matter dominance. Supernovae are situated at distances further than it was
predicted. Therefore, according to the standard cosmological model, in the
last period, the cosmological expansion proceeds with acceleration. Dynamics,
by unknown reasons passes from the deceleration stage to an acceleration one
of expansion. Observable data testify that the Universe is filled mainly not
with massive dust that can not provide accelerating expansion but with
unspecified enigmatic substance of other nature – “dark energy”.

Theory of elementary particles fails to identify a theory for dark matter and
dark energy. There are some free parameters that must be consistent with the
experimental data. The important purpose of testing a cosmological theory is
vitiated Ram . Recently, a large analysis of observational data on the
expansion rate of the Universe at small $z$ has been carried out Denzel ;
Soltis ; RiessApJL . It should be noted that the observational data for the
Hubble constant differ.

Einstein’s General Relativity was formulated in redundant variables. It is a
theory with constraints, so it has been studied by methods of Hamiltonian
dynamics. Friedmann equation that utilized for interpretations of the modern
cosmological data is the Hamiltonian constraint by its meaning. In conformal
metric, the Universe is not expanding but the elementary particle’s masses are
depended on time Narlikar . Both approaches describe the modern Hubble diagram
with the same accuracy Zakharov:2010nf . The theoretical curves interpolating
the Hubble diagram are expressed in analytical form. The functions belong to
the class of meromorphic Weierstrass functions PavlovMIPh . Of interest is the
extrapolation of the Hubble curves for large redshifts.

##  II Friedmann equation in classical cosmology

In the standard cosmology the Friedmann equation is used for fitting SNe Ia
data. The closed cosmological model is set by space interval

$\gamma_{ij}{\rm d}x^{i}{\rm d}x^{j}=a^{2}(t)\left({\rm
d}\chi^{2}+\sin^{2}\chi({\rm d}\theta^{2}+\sin^{2}\theta{\rm
d}\phi^{2})\right),$ (1)

where $a(t)$ is the radius of the Universe as a function of the coordinate
time $t$, $(\chi,\theta,\phi)$ are dimensionless angle variables. According to
the Hamiltonian theory of gravity, the Friedmann equation is a Hamiltonian
constraint Pavlov2017 :

${\cal H}_{\bot}:=\sqrt{\gamma}\left(K_{ij}K^{ij}-K^{2}-R+16\pi G\rho\right).$
(2)

Here, $\gamma$ is the determinant of the 3-sphere metric, $R=6/a^{2}$ is the
Ricci scalar of the 3-sphere, $K_{ij}$ are components of the extrinsic
curvature tensor

$K_{ij}:=-\frac{1}{2}\dot{\gamma}_{ij},$ (3)

the dot denotes the derivative with respect to the coordinate time $t$, $G$ is
the Newton constant, $\rho$ is the matter density, the trace of the extrinsic
curvature tensor (3) is

$K:=\gamma^{ij}K_{ij}=-3\left(\frac{\dot{a}}{a}\right)\equiv 3H(t).$ (4)

The quantity $H\equiv\dot{a}/a$ is the Hubble parameter defining the expansion
rate of the Universe. Its present value $H_{0}$ calls the Hubble constant.
Hence, the Hamiltonian constraint (2), after substitutions (3), (4) takes the
well-known form of the Friedmann equation:

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{1}{a^{2}},$ (5)

The energy continuity equation

$\dot{\rho}=-3(\rho+P)\left(\frac{\dot{a}}{a}\right),$ (6)

with an equation of state $P=w\rho$, which connects the pressure $P$ and the
energy density $\rho$, yields the dependence of the density of the scale
factor. So,

$\bullet$ for interstellar dust $P=0:\rho\sim a^{-3};$

$\bullet$ for radiation $P=\rho/3:\rho\sim a^{-4};$

$\bullet$ for vacuum $P=-\rho:\rho\sim\Lambda;$

$\bullet$ for rigid state matter $P=\rho:\rho\sim a^{-6}.$

Introducing for the vacuum density, non-relativistic, relativistic, and rigid
state matters the cosmological parameters $\Omega_{\rm\Lambda}$, $\Omega_{\rm
M}$, $\Omega_{\rm rad},$ $\Omega_{\rm rigid},$ we present the energy density
as the sum

$\displaystyle\rho=\frac{3H_{0}^{2}}{8\pi G}\times$ $\displaystyle\times$
$\displaystyle\left[\Omega_{\rm\Lambda}+\Omega_{\rm
M}\left(\frac{a_{0}}{a}\right)^{3}+\Omega_{\rm
rad}\left(\frac{a_{0}}{a}\right)^{4}+\Omega_{\rm
rigid}\left(\frac{a_{0}}{a}\right)^{6}\right].$

The cosmological parameters are constrained

$\Omega_{\rm\Lambda}+\Omega_{\rm M}+\Omega_{\rm rad}+\Omega_{\rm
rigid}+\Omega_{\rm curv}=1,$ (8)

where for curvature term is

$\Omega_{\rm curv}\equiv-\frac{1}{a_{0}^{2}H_{0}^{2}}.$ (9)

Substituting (II), (8), (9) into the right side of the Friedmann equation (5),
we obtain

$\displaystyle\frac{1}{x^{2}}\left(\frac{{\rm d}x}{{\rm d}t}\right)^{2}=$
$\displaystyle=$ $\displaystyle
H_{0}^{2}\left[\Omega_{\rm\Lambda}+\frac{\Omega_{\rm
curv}}{x^{2}}+\frac{\Omega_{\rm M}}{x^{3}}+\frac{\Omega_{\rm
rad}}{x^{4}}+\frac{\Omega_{\rm rigid}}{x^{6}}\right].$

Here the variable $x$ is given as a ratio of a scale $a(t)$ to a modern value
$a_{0}=1$:

$x\equiv\frac{a(t)}{a_{0}}=\frac{1}{1+z},$ (11)

$z$ is a redshift of spectral lines,

$H_{0}=h\cdot 10^{5}m/s/Mpc,\quad h=0.72\pm 0.08$ (12)

is the Hubble constant. The data of modern astronomical observations are
fitted using cosmological parameters RiessNobel :

$\Omega_{\Lambda}=0.72,\quad\Omega_{\rm M}=0.28.$ (13)

The contributions of vacuum and baryonic matter prevail. The contributions of
the remaining terms in the Friedmann equation (II) are negligible.

Since the spacetime interval for a photon is zero, we have a relation between
the intervals of time and space $c{\rm d}t=-a(t){\rm d}r$. Using the above
notation (11), we find the formula for the distance

$-a_{0}r=c\int\frac{{\rm d}t}{x}=c\int\frac{{\rm d}x}{x}\frac{1}{{\rm d}x/{\rm
d}t}.$ (14)

Substituting the Friedmann equation (II) with significant cosmological
parameters into the integrand (14), we obtain the formula for calculating the
distance to a luminous object

$r(z)=\frac{c}{H_{0}\sqrt{\Omega_{\Lambda}}}\int\limits_{1/(1+z)}^{1}\frac{{\rm
d}x}{\sqrt{x^{4}+p^{3}x}},$ (15)

where we denoted the ratio of partial contributions as
$p\equiv\sqrt[3]{{\Omega_{\rm M}}/{\Omega_{\Lambda}}}.$ The integral

$I=\int\frac{{\rm d}x}{\sqrt{x^{4}+p^{3}x}},$ (16)

can be expressed with using elliptic integrals Whittaker . To do this, we
represent the radical expression in the form of a product of second-order
polynomials

$x^{4}+p^{3}x=(x^{2}-px+p^{2})(x^{2}+px).$

Next, we apply the linear fractional substitution $x\mapsto t$

$x=\frac{(1+\sqrt{3})t+(1-\sqrt{3})}{2(t+1)}p.$

The differential will be replaced as follows

${\rm d}x=-\frac{\sqrt{3}p}{(t+1)^{2}}{\rm d}t,$

and the integral (16) becomes

$I=-\frac{2}{\sqrt{(2\sqrt{3}-3)}p}\int\frac{{\rm
d}t}{\sqrt{(1+t^{2})(b^{2}t^{2}-1)}},$ (17)

where $b\equiv(2+\sqrt{3}).$ With the trigonometric substitution
$t\mapsto\varphi$

$bt=\sec\varphi\equiv\frac{1}{\cos\varphi}$

the integrand is reduced to the standard form

$\frac{{\rm d}t}{\sqrt{(1+t^{2})(b^{2}t^{2}-1)}}=\sqrt{1-k^{2}}\frac{{\rm
d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}},$

where $k^{2}\equiv b^{2}/(1+b^{2})$. As a result, the definite integral (15)
is expressed in terms of an elliptic integral of the first kind according to
Legendre:

$F(\varphi,k):=\int\limits_{0}^{\varphi}\frac{{\rm
d}\phi}{\sqrt{1-k^{2}\sin^{2}\phi}}$

Hence, we obtain the formula desired

$r(z)=\frac{2c}{\sqrt[4]{3}p\sqrt{\Omega_{\Lambda}}H_{0}}\left[F(\varphi(z),k)-F(\varphi(0),k)\right].$
(18)

Here $k^{2}$ is the modulus of the elliptic integral of the first kind
according to Legendre

$k\equiv\frac{2+\sqrt{3}}{\sqrt{1+(2+\sqrt{3})^{2}}},$

and the angle $\varphi(z)$ is defined by the formula:

$\varphi(z)={\arccos}\left(\frac{(\sqrt{3}+1)p(1+z)-2}{(\sqrt{3}+2)[(\sqrt{3}-1)p(1+z)+2]}\right).$

In standard cosmology, the luminosity distance $d_{L}(z)_{SC}$ is related to
the coordinate distance $r$:

$d_{L}(z)_{SC}=(1+z)a_{0}r(z).$ (19)

Modern observational cosmology is based on the Hubble diagram. Effective
stellar magnitude – redshift relationship

$m(z)-M=5{\rm lg}[d_{L}(z)_{SC}]+{\cal M},$ (20)

is used to test cosmological theories ($d_{L}$ in megaparsecs) RiessNobel .
Here $m(z)$ is the apparent magnitude, $M$ is its absolute magnitude, and
${\cal M}=25$ is a constant.

##  III Conformal General Relativity

The Einstein equations are not invariant under conformal transformations. This
raises the questions, whether it is suitable to work with such equations
globally on cosmological distances and times Fock . Conformal transformations
shrink or stretch the spacetime intervals between two points described by the
same coordinate system

${\rm d}s^{2}=\Omega^{2}{\rm d}\bar{s}^{2}$

on the conformal to each other manifolds ${\cal M}$ and ${\cal\bar{M}}$. Under
the conformal transformation the components of the spacetime metric and
corresponding metric determinant transform as

$g_{\mu\nu}=\Omega^{2}\bar{g}_{\mu\nu},\qquad\sqrt{-g}=\Omega^{4}\sqrt{-\bar{g}}.$
(21)

The Ricci scalar of the spacetime transforms according to

$R=\Omega^{-2}\left(\bar{R}-6\frac{\bar{\Box}\Omega}{\Omega}\right),$ (22)

where the d’Alembertian operator ${\bar{\Box}}$ is taken with respect to the
conformally rescaled metric. In order to understand the problem of conformal
invariance let us start our discussion with the Hilbert action ProblemsGauge

$S_{\rm H}=\frac{1}{2}\int\,{\rm d}^{4}x\sqrt{-g}\frac{1}{6}R,$ (23)

where one puts

$\kappa^{2}\equiv 8\pi G=6.$

The conformal transformation (21) together with (22) yields the Hilbert action
transformed

$\bar{S}_{\rm H}=\frac{1}{2}\int\,{\rm
d}^{4}x\sqrt{-\bar{g}}\Omega^{2}\left(\frac{1}{6}\bar{R}-\frac{\bar{\Box}\Omega}{\Omega}\right).$
(24)

The action (24) is not conformal invariant. The additional term in (24) looks
as a kinetic term of an arbitrary scalar field. The idea is to couple a
massless scalar field $\Phi$ in order to absorb the conformal factor

$S_{\rm H\Phi}=\frac{1}{2}\int\,{\rm
d}^{4}x\sqrt{-g}\left(\frac{1}{6}\right)R\Phi^{2}.$ (25)

After the conformal transformation one yields

$\bar{S}_{\rm H\Phi}=\frac{1}{2}\int\,{\rm
d}^{4}x\sqrt{-\bar{g}}\left(\frac{1}{6}\bar{R}\bar{\Phi}^{2}-\frac{\bar{\Box}\Omega}{\Omega}\bar{\Phi}^{2}\right),$
(26)

where we take into account the scalar field rescaled

$\Phi=\Omega^{-1}\bar{\Phi},$ (27)

so the conformal factor in the action (26) is absorbed.

Then, let us consider the action for a massless scalar field

$S_{\Phi}=-\frac{1}{2}\int\,{\rm d}^{4}x\sqrt{-g}\Phi\Box\Phi.$ (28)

After the conformal transformation (32) is applied, with taking into account
the field transformation (27), one yields the transformed field action

$\bar{S}_{\Phi}=-\frac{1}{2}\int\,{\rm
d}^{4}x\sqrt{-\bar{g}}\left(\bar{\Phi}\bar{\Box}\bar{\Phi}-\frac{\bar{\Box}\Omega}{\Omega}\bar{\Phi}^{2}\right),$
(29)

where we have used the relation

$\Box\Phi=\frac{1}{\Omega^{3}}\left(\bar{\Box}\bar{\Phi}-\frac{\bar{\Box}\Omega}{\Omega}\bar{\Phi}^{2}\right).$

As we see the scalar field action (29) is not conformal invariant.

Now, if we add the action of the massless scalar field (29) with its action
coupled with gravitation field (26), we get the conformally invariant action

$S=\frac{1}{2}\int\,{\rm
d}^{4}x\sqrt{-{g}}\Phi\left(\frac{1}{6}R\Phi-\Box\Phi\right).$ (30)

The action (30) can be presented in a general form including the massive
scalar field and after discarding the boundary term

$S=\frac{1}{2}\int\,{\rm
d}^{4}x\sqrt{-{g}}\left(\frac{1}{6}R\Phi^{2}+g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-m^{2}\Phi^{2}\right).$
(31)

Here we used the expression for the covariant d’Alembertian

$\Box\Phi\equiv\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{-g}g^{\mu\nu}\frac{\partial}{\partial
x^{\nu}}\Phi\right).$

The action with the mass term (31) is conformally invariant if we take into
account the mass scaling

$m=\Omega^{-1}\bar{m}.$

With the particular $1/6$ coefficient of $R$ (31), this action is equivalent
to that of General Relativity Deser . Dilaton theories are successfully
developed (see, e.g. ProblemsGauge ; Deser ; Feza ).

In this paper we consider the spatial conformal transformations Pavlov2017 .
Under the conformal transformation the components of the space metric and
corresponding metric determinant transform as

$\gamma_{ij}=\Psi^{4}\bar{\gamma}_{ij},\qquad\sqrt{\gamma}=\Psi^{6}\sqrt{\bar{\gamma}}.$
(32)

The components of extrinsic curvature transform as

$K_{ij}=\Psi^{-2}A_{ij}+\frac{1}{3}\Psi^{4}\bar{\gamma}_{ij}K,$ (33)

where $A_{ij}$ are components of the traceless part of the tensor $K_{ij}$,
$K$ is its trace.

The definitions (32), (33) are redundant. These definitions are invariant
under the conformal transformation Brown

$\displaystyle\bar{\gamma}_{ij}$ $\displaystyle\longrightarrow$
$\displaystyle\xi^{4}\bar{\gamma}_{ij},$ $\displaystyle\Psi$
$\displaystyle\longrightarrow$ $\displaystyle\xi^{-1}\Psi,$ $\displaystyle
A_{ij}$ $\displaystyle\longrightarrow$ $\displaystyle\xi^{-2}A_{ij},$
$\displaystyle K$ $\displaystyle\longrightarrow$ $\displaystyle K,$

for any field $\xi$. The determinant of the metric is unspecified. The metric
tensor represents a conformal equivalent class of metrics Brown .

The Ricci scalar of the space transforms according to

$R=\Psi^{-4}\left(\bar{R}-8\frac{\bar{\triangle}\Psi}{\Psi}\right),$ (34)

where the Laplacian ${\bar{\triangle}}$ is taken with respect to the
conformally rescaled metric. The conformal Hamiltonian constraint

$\displaystyle{\cal H}_{\bot}=$
$\displaystyle=\frac{1}{2}\bar{\gamma}A^{ij}A_{ij}-\frac{1}{3}\bar{\gamma}\Psi^{12}K^{2}-\frac{1}{2}\bar{\gamma}\Psi^{8}\bar{R}+4\bar{\gamma}\Psi^{7}\bar{\triangle}\Psi$
$\displaystyle+\bar{\gamma}\Psi^{12}\rho=0$ (35)

is invariant under the conformal transformation (32).

##  IV Conformal Friedmann equation

For studying dynamics of gravitational field, the transition to the conformal
metric with components $\tilde{\gamma}_{ij}$ is implemented (32). The factor
$\Psi$ is defined as a power of the ratio of the spatial metric (1)
determinant $\gamma$

$\Psi:=\left(\frac{\gamma}{f}\right)^{1/12}={\sqrt{\frac{a(t)}{a_{0}}}}$

and a background metric determinant $f$. For the background space the present
day sphere with the modern radius $a_{0}$ is taken:

$f_{ij}{\rm d}x^{i}{\rm d}x^{j}=a_{0}^{2}\left({\rm
d}\chi^{2}+\sin^{2}\chi({\rm d}\theta^{2}+\sin^{2}\theta{\rm
d}\phi^{2})\right).$ (36)

Hence, the conformal metric with components $\tilde{\gamma}_{ij}$ is the
static metric of the modern Universe with radius $a_{0}$. Under the conformal
transformation (32) the Ricci scalar $R$ is transformed as

$R=\Psi^{-4}\tilde{R}=\left(\frac{a_{0}}{a(t)}\right)^{2}\tilde{R}.$ (37)

The conformal Hamiltonian constraint takes the form of the Lichnerowicz – York
equation

$-\frac{1}{9}K^{2}-\frac{1}{6}\Psi^{-4}\tilde{R}+\left(\frac{8\pi
G}{3}\right)\rho=0.$ (38)

The conformal Friedmann equation is obtained after the substitution in (38)
the interval of the coordinate time by the interval of conformal time: $c{\rm
d}t=a{\rm d}\eta$:

$\left(\frac{{a}^{\prime}}{a}\right)^{2}=\frac{8\pi G}{3}a^{2}\rho-{k},$ (39)

where the prime denotes the derivative with respect to the conformal time.
Since we will describe the physics in the conformal frame, we have to use the
conformal time. The conformal Friedmann equation becomes

$\displaystyle\frac{1}{x^{2}}\left(\frac{{\rm d}x}{{\rm d}\eta}\right)^{2}=$
$\displaystyle=$ $\displaystyle
H_{0}^{2}\left[\Omega_{\rm\Lambda}x^{2}+\Omega_{\rm curv}+\frac{\Omega_{\rm
M}}{x}+\frac{\Omega_{\rm rad}}{x^{2}}+\frac{\Omega_{\rm
rigid}}{x^{4}}\right].$

The energy continuity equation in conformal time takes the form:

${\rho}^{\prime}=-3(\rho+p)\left(\frac{{a}^{\prime}}{a}\right).$ (41)

The Hubble parameter in classical cosmology $H(t)$ and the conformal Hubble
parameter ${\cal H}(\eta)$ are connected:

$H(t)\equiv\frac{\dot{a}}{a}=\frac{a^{\prime}}{a^{2}}\equiv\frac{1}{a}{\cal
H}(\eta).$

Leaving significant cosmological parameters for fitting the Hubble diagram, we
get

$\frac{1}{x^{2}}\left(\frac{{\rm d}x}{{\rm
d}\eta}\right)^{2}=\left(\frac{{\cal
H}_{0}}{c}\right)^{2}\left[\frac{\Omega_{\rm M}}{x}+\frac{\Omega_{\rm
rigid}}{x^{4}}\right],$ (42)

where ${\cal H}_{0}=H_{0}$ is a conformal Hubble constant. Interpretation of
the Hubble diagram, based on conformal Friedmann equation with parameters

$\Omega_{\rm rigid}=0.755,\qquad\Omega_{\rm M}=0.245,$ (43)

yields the same qualitative approximation as the standard cosmological model
with parameters $\Omega_{\Lambda}=0.72,$ $\Omega_{\rm M}=0.28$ Zakharov:2010nf
. A parameter $\Omega_{\rm rigid}$ corresponds to a rigid state matter, when
the energy density is equal to the pressure, that happens under a
nucleosynthesis regime in stars Narlikar . This unusual equation of state
arises when describing a quantum vacuum. In quantum physics, the equation
$p=\epsilon$ describes the Casimir vacuum of virtual massive boson and fermion
particles EoS . This confirms the manifestation of vacuum in the Hubble
diagram.

If we identify the observed quantities with conformal quantities, then the
evolution of lengths in cosmology is replaced by the evolution of masses
Narlikar . The choice of the length scale is equivalent to the choice of the
mass scale when describing physical processes in Nature. The requirement of
conformal symmetry leads to scaling of the masses of objects, which, in turn,
leads to a new (non-Doppler) interpretation of the cosmological redshift of
atomic spectra. The introduction of a relative standard means the
identification of observable quantities with conformal fields and coordinates,
which leads to a conformal mass cosmology instead of the standard distance
evolution cosmology. Photons emitted by atoms on distant stars billions of
years ago ”remember“ the characteristics of the atoms. The wavelength of such
a photon changes with the inverse electron mass, which is responsible for the
transition. Atoms were determined by their masses even at that distant time.
Astronomers are currently comparing the spectrum with spectrum of the same
atoms on Earth, but with an increased mass over the past time
$m(\eta)=m_{I}a(\eta),$ where $m_{I}$ is an initial mass of an atom, $a(\eta)$
is the conformal factor as a function of photon time $\eta$. As a result, a
redshift $z$ of the Fraunhofer spectral lines is observed

$1+z=\frac{\lambda m_{I}}{[\lambda a(t)]m_{I}}=\frac{\lambda
m_{I}}{\lambda[a(t)m_{I}]}.$

The wavelength of the photon $\lambda$ does not change during all the time it
moves in space, since its rest mass is equal to zero. The cost of such an
interpretation is negligible – the masses of elementary particles turn out to
be time-dependent, which is not comparable to the acceptance of the collapse
of the space in which we live, so called Big Bang.

It is appropriate to remind the correct statement of Steven Weinberg about
interpretation of experimental data on redshifts WeinbergThree :

> “I do not want to give the impression that everyone agrees with this
> interpretation of the red shift. We do not actually observe galaxies rushing
> away from us; all we are sure of is that the lines in their spectra are
> shifted to the red, i.e. towards longer wavelengths. There are eminent
> astronomers who doubt that the red shifts have anything to do with Doppler
> shifts or with expansion of the universe”.

From the conformal Friedmann equation (42) with significant cosmological
parameters we obtain the formula for calculating the conformal time to a
luminous object. In conformal coordinates, the behavior of photons is exactly
the same as in the Minkowski space. The time intervals between the emission of
two photons and between their absorption are the same. The coordinate time
interval ${\rm d}t=-a{\rm d}r$ and the conformal time interval ${\rm
d}\eta=-{\rm d}r$ are different GRub .

The analytical formula for calculating the distance to a luminous object can
be obtained after integrating (42). From the conformal Friedmann equation (42)
with significant cosmological parameters we obtain the formula for calculating
the conformal time to a luminous object

$\eta(z)=\frac{c}{H_{0}\sqrt{\Omega_{\rm
M}}}\int\limits_{1/(1+z)}^{1}\frac{x{\rm d}x}{\sqrt{P(x)}},$ (44)

where the polynomial $P(x)$ can be represented in a factorized form:

$P(x)=x^{3}+q^{3}\equiv(x+q)(x^{2}-qx+q^{2}),$

where $q\equiv\sqrt[3]{\Omega_{\rm rigid}/\Omega_{\rm M}}.$

The formula for calculating the distance to a luminous object includes the
integral of the rational function

$I=\int\frac{x{\rm d}x}{\sqrt{P(x)}}.$ (45)

Let us show how it can be expressed in terms of the elliptic integrals
Whittaker . We use the trigonometric substitution $x\mapsto\varphi$:

$x=-q+\sqrt{3}q\tan^{2}\frac{\varphi}{2}.$

After changing the variable, we get

$P(\varphi)=\frac{3\sqrt{3}q^{3}\tan^{2}(\varphi/2)}{\cos^{4}(\varphi/2)}[1-k^{2}\sin^{2}\varphi],$

where $k^{2}\equiv(2+\sqrt{3})/4.$ The differential is transformed according
to the formula:

${\rm d}x=\sqrt{3}q\frac{\tan(\varphi/2)}{\cos^{2}(\varphi/2)}{\rm d}\varphi.$

As a result of changing the variable, the integral (45) will take the form:

$\displaystyle I=$ $\displaystyle-$
$\displaystyle\sqrt{\frac{q}{\sqrt{3}}}\int\frac{{\rm
d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}}$ (46) $\displaystyle+$
$\displaystyle\sqrt{\sqrt{3}q}\int\frac{\tan^{2}(\varphi/2)\,{\rm
d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}}.$

We transform the second integral using the equality

$\displaystyle\int\frac{\tan^{2}(\varphi/2)\,{\rm
d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}}=2\tan(\varphi/2)\sqrt{1-k^{2}\sin^{2}\varphi}$
$\displaystyle-2\int\sqrt{1-k^{2}\sin^{2}\varphi}\,{\rm
d}\varphi+\int\frac{{\rm d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}}.$

Then the integral (46) takes the form

$\displaystyle I$ $\displaystyle=$ $\displaystyle
2\sqrt{\sqrt{3}q}\tan(\varphi/2)\sqrt{1-k^{2}\sin^{2}\varphi}$
$\displaystyle-$ $\displaystyle
2\sqrt{\sqrt{3}q}\int\sqrt{1-k^{2}\sin^{2}\varphi}$ $\displaystyle+$
$\displaystyle(\sqrt{3}-1)\sqrt{\frac{q}{\sqrt{3}}}\int\frac{{\rm
d}\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}}.$

Substituting the limits of integration, we find the required distance

$\displaystyle r(z)=\frac{2c}{H_{0}\sqrt{\Omega_{\rm
M}}}\left[\sqrt{(1+q)(1-k^{2}\sin^{2}\varphi(0))}\right.$
$\displaystyle\left.-\sqrt{\frac{1+q(1+z)}{(1+z)}(1-k^{2}\sin^{2}\varphi(z))}\right.$
$\displaystyle\left.+\frac{(\sqrt{3}-1)}{2}\sqrt{\frac{q}{\sqrt{3}}}\left[F(k,\varphi(0))-F(k,\varphi(z))\right]\right.$
$\displaystyle\left.-\sqrt{\sqrt{3}q}\left[E(k,\varphi(0))-E(k,\varphi(z))\right]\right].$
(47)

Here, beside of the elliptic integral of the first kind $F(k,\varphi)$, the
elliptic integral of the second kind according to Legendre $E(k,\varphi)$ is
appeared

$E(k,\varphi):=\int\limits_{0}^{\varphi}\sqrt{1-k^{2}\sin^{2}\phi}\,{\rm
d}\phi,\qquad k^{2}\equiv\frac{2+\sqrt{3}}{4}$

with modulus $k^{2}$. The angle $\varphi$ is determined by the formula

$\varphi(z)=2\arctan\sqrt{\frac{1+q(1+z)}{\sqrt{3}q(1+z)}}.$

The conformal luminosity distance $d_{L}(z)_{CC}$ is related to the standard
luminosity distance $d_{L}(z)_{SC}$ as GRub

$d_{L}(z)_{CC}=(1+z)d_{L}(z)_{SC}=(1+z)^{2}r(z).$

The effective magnitude – redshift relationship in the conformal cosmology has
the form

$m(z)-M=5{\rm lg}[d_{L}(z)_{CC}]+{\cal M}.$ (48)

##  V Comparative analysis

The analytical formulas (20), (48) are applicable to extrapolate the Hubble
diagram for redshifts exceeding modern observational data on supernovae. For
large values of $z$, the conformal model curve rises significantly above the
standard model curve (see Fig. 1). The deviation exceeds the errors of
astronomical observations. This discrepancy has already been noted in Behnke ,
where the problem been solved in the dilaton theory of gravity.

Figure 1: Extrapolation of modern Hubble curves of two cosmological models:
effective magnitude - redshift ratio. For large values of $z$, the conformal
model curve rises significantly above the standard model curve. The deviation
exceeds the errors of astronomical observations. Future refined astronomical
observations may suggest which model should be preferred.

In the standard cosmology, there are characteristics of the Hubble diagram:
Hubble parameter, a deceleration and a jerk RiessNobel

$\displaystyle H(t)$ $\displaystyle\equiv$
$\displaystyle+\left(\frac{\dot{a}}{a}\right)=H_{0}\sqrt{\frac{\Omega_{\rm
M}}{a^{3}}+\Omega_{\Lambda}},$ (49) $\displaystyle q(t)$ $\displaystyle\equiv$
$\displaystyle-\left(\frac{\ddot{a}}{a}\right)\left(\frac{\dot{a}}{a}\right)^{-2}=\frac{\Omega_{\rm
M}/2-\Omega_{\Lambda}a^{3}}{\Omega_{\rm M}+\Omega_{\Lambda}a^{3}},$ (50)
$\displaystyle j(t)$ $\displaystyle\equiv$
$\displaystyle+\left(\frac{\dot{\ddot{a}}}{a}\right)\left(\frac{\dot{a}}{a}\right)^{-3}=1.$
(51)

The deceleration parameter $q$ changes its sign during the evolution of the
Universe at the inflection point

$a^{*}=\sqrt[3]{\frac{\Omega_{\rm M}}{2\Omega_{\Lambda}}},$

$j$-parameter remains constant. It should be noted that the value of the
Hubble constant (12) varies quite a lot depending on the research methods:
analysis of angular fluctuations of the microwave background or the use of
traditional astronomical methods Dolgov . But, the existence of the inflection
point does not depend on the value $H_{0}$.

We define analogous parameters for the conformal time

$\displaystyle{\cal H}(\eta)$ $\displaystyle\equiv$
$\displaystyle+\left(\frac{{a}^{\prime}}{a}\right),$ (52)
$\displaystyle{q}(\eta)$ $\displaystyle\equiv$
$\displaystyle-\left(\frac{{a}^{\prime\prime}}{a}\right)\left(\frac{{a}^{\prime}}{a}\right)^{-2},$
(53) $\displaystyle{j}(\eta)$ $\displaystyle\equiv$
$\displaystyle+\left(\frac{{a}^{\prime\prime\prime}}{a}\right)\left(\frac{{a}^{\prime}}{a}\right)^{-3}.$
(54)

We calculate the conformal parameters using the conformal Friedmann equation
(42). Hubble parameter

${\cal H}(\eta)=\frac{{\cal H}_{0}}{a^{2}}\sqrt{\Omega_{\rm rigid}+\Omega_{\rm
M}a^{3}}>0,$

deceleration parameter

$q(\eta)=\left(\frac{\Omega_{\rm rigid}-(\Omega_{\rm M}/2)a^{3}}{\Omega_{\rm
rigid}+\Omega_{\rm M}a^{3}}\right)>0,$

hence, the scale factor grows with deceleration; the jerk parameter

$j(\eta)=\frac{3\Omega_{\rm rigid}}{\Omega_{\rm rigid}+\Omega_{\rm M}a^{3}}>0$

changes from 3 to $3\Omega_{\rm rigid}$. Dimensionless parameters $q(\eta)$
and $j(\eta)$ remain positive during the evolution. The Universe does not
undergo the so-called jerk which is an artifact of approach of the standard
cosmological model. Future refined astronomical observations will suggest
which model should be preferred.

##  VI Discussion

The conformal gravity theory has been advanced as a candidate to standard
Einstein gravity PhMannheim . There are hopes that an alternative to dark
matter and dark energy is conformal gravity Mann . In papers tHooft1 ; tHooft2
; tHooft3  a nontrivial connection between Einstein gravity and Weyl conformal
gravity is found.

## Acknowledgement

I am grateful to Professor A. B. Arbuzov for fruitful discussions on modern
cosmological problems and the Laboratory of Theoretical Physics named after N.
N. Bogoliubov, Joint Institute for Nuclear Research (Dubna) for hospitality. I
obliged Professor J. David Brown for acquaintance with his paper on conformal
invariance in General Relativity. I express my gratitude for the interest in
the work to the participants of ${\rm XVI}$-th International Conference
”Finsler Extension of Relativity Theory”, and of the seminar of the People’s
Friendship University of Russia (Moscow).

## References

  * (1) A. G. Riess, Rev. Mod. Phys. 84, 1165 (2012).
  * (2) R. G. Vishwakarma, J. V. Narlikar, Research in Astron. Astrophys. 10, 1195 (2010).
  * (3) Ph. Denzel, et al., Monthly Notices of the Royal Astronomical Society. 501, 784 (2020).
  * (4) J. Soltis, et al., The Astrophysical Journal Letters. 908, L5 (2021).
  * (5) A. G. Riess, et al., The Astrophysical Journal Letters. 908, L6 (2021).
  * (6) J. V. Narlikar, Violent Phenomena in the Universe (Oxford University Press, 1984).
  * (7) A. F. Zakharov, V. N. Pervushin, Int. J. Mod. Phys. D 19, 1875 (2010).
  * (8) A. E. Pavlov, RUDN J. Math. Inform. Sc. Phys. 25, 390 (2017).
  * (9) V. A. Fock, Theory of Space, Time and Gravitation (Pergamon Press, 1959).
  * (10) M. P. Da̧browski, D. Behnke, D. Blaschke. In: Problems of Gauge Theories: On occasion of 60th birthday of V. N. Pervushin. Eds.: B. M. Barbashov, V. V. Nesterenko. D2-2004-66 (JINR, Dubna, 2004).
  * (11) S. Deser, Annals of Physics, 59, 248 (1970).
  * (12) F. G$\ddot{\rm u}$rsey, Annals of Physcis, 24, 211 (1963).
  * (13) A. E. Pavlov, Grav. Cosmol. 23, 208 (2017).
  * (14) J. D. Brown, Phys. Rev. D 71, 104011 (2005).
  * (15) E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1927).
  * (16) A. E. Pavlov, Mod. Phys. Lett. A 35, 2050271 (2020).
  * (17) S. Weinberg, The First Three Minutes: A Modern View of the Origin of the Universe (Basic Books, 1977).
  * (18) D. S. Gorbunov, V. A. Rubakov, Introduction to the Theory of the Early Universe: Hot Big Bang Theory (World Scientific, 2011).
  * (19) D. Behnke, et al., Phys. Lett. B 530, 20 (2002).
  * (20) S. I. Blinnikov, A. D. Dolgov, Physics – Uspekhi 62, 529 (2019).
  * (21) M. Shaposhnikov, D. Zenh$\ddot{\rm a}$usern, Phys. Lett. B 671, 187 (2009).
  * (22) Ph. D. Mannheim, Prog. Part. Nucl. Phys. 56, 340 (2006).
  * (23) Ph. D. Mannheim, Found. Phys. 42, 388 (2012).
  * (24) G. ’t Hooft, Probing the small distance structure of canonical quantum gravity using the conformal group, arXiv:1009.0669v2 [gr-qc].
  * (25) G. ’t Hooft, The conformal constraint in canonical quantum gravity, arXiv:1011.0061v1 [gr-qc].
  * (26) G. ’t Hooft, A class of elementary particle models without any adjustable real parameters, arXiv:1104.4543v1 [gr-qc].