11institutetext: Grupo de Astrofísica Molecular, Instituto de Física
Fundamental (IFF-CSIC), C/ Serrano 121, 28006 Madrid, Spain
11email<EMAIL_ADDRESS>22institutetext: Centro de Desarrollos
Tecnológicos, Observatorio de Yebes (IGN), 19141 Yebes, Guadalajara, Spain
33institutetext: Observatorio Astronómico Nacional (OAN, IGN), Madrid, Spain

# Pure hydrocarbon cycles in TMC-1: Discovery of ethynyl cyclopropenylidene,
cyclopentadiene and indene. ††thanks: Based on observations carried out with
the Yebes 40m telescope (projects 19A003, 20A014, 20D023, and 21A011) and the
Institut de Radioastronomie Millimétrique (IRAM) 30m telescope. The 40m
radiotelescope at Yebes Observatory is operated by the Spanish Geographic
Institute (IGN, Ministerio de Transportes, Movilidad y Agenda Urbana). IRAM is
supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).

J. Cernicharo 11 M. Agúndez 11 C. Cabezas 11 B. Tercero 2233 N. Marcelino 11
J. R. Pardo 11 P. de Vicente 22

(Received; accepted)

We report the detection for the first time in space of three new pure
hydrocarbon cycles in TMC-1: $c$-C3HCCH (ethynyl cyclopropenylidene), $c$-C5H6
(cyclopentadiene) and $c$-C9H8 (indene). We derive a column density of 3.1
$\times$1011 cm-2 for the former cycle and similar values, in the range (1-2)
$\times$1013 cm-2, for the two latter molecules. This means that
cyclopentadiene and indene, in spite of their large size, are exceptionally
abundant, only a factor of five less abundant than the ubiquitous cyclic
hydrocarbon $c$-C3H2. The high abundance found for these two hydrocarbon
cycles, together with the high abundance previously found for the propargyl
radical (CH2CCH) and other hydrocarbons like vinyl and allenyl acetylene
(Agúndez et al. 2021; Cernicharo et al. 2021a, b), start to allow us to
quantify the abundant content of hydrocarbon rings in cold dark clouds and to
identify the intermediate species that are probably behind the in situ bottom-
up synthesis of aromatic cycles in these environments. While $c$-C3HCCH is
most likely formed through the reaction between the radical CCH and $c$-C3H2,
the high observed abundances of cyclopentadiene and indene are difficult to
explain through currently proposed chemical mechanisms. Further studies are
needed to identify how are five- and six-membered rings formed under the cold
conditions of clouds like TMC-1.

###### Key Words.:

molecular data – line: identification – ISM: molecules – ISM: individual
(TMC-1) – astrochemistry

##  1 Introduction

Since the hypothesis that polycyclic aromatic hidrocarbons (PAHs) are the
carriers of the unidentified infrared bands (Léger & Puget 1984; Allamandola
et al. 1985), many efforts have been devoted to understand the chemical
processes leading to the formation of these molecular species (see, e.g.,
Joblin & Cernicharo  2018). Circumstellar envelopes around carbon-rich
Asymptotic Giant Branch (AGB) stars have been suggested as the factories of
PAHs (Cherchneff et al. 1992). The detection of benzene in the carbon-rich
protoplanetary nebula CRL 618 (Cernicharo et al. 2001) suggests a bottom-up
approach in which the small hydrocarbons formed during the AGB phase, such as
C2H2 and C2H4, interact with the ultraviolet (UV) radiation produced by the
star in its evolution to the white dwarf phase (Woods et al.  2002; Cernicharo
2004). Other hypotheses involve the proccesing of dust grains around evolved
stars, either through UV photons (Pilleri et al. 2015) or by chemical
processes (Martínez et al. 2020). Hence, it has been surprising to see that
cyanide derivatives of PAHs have been found in a cold prestellar core such as
TMC-1, which is well protected against UV radiation (McGuire et al. 2018,
2021). It is unlikely that these PAH-cyanides arise from a reservoir of PAHs
existing since the early stages of the cloud, since these relatively small
PAHs would not have survived the diffuse cloud stage. Although McGuire et al.
(2021) propose a reasonable chemical network starting with the phenyl radical
that could explain the observed abundances of cyanonaphthalene, the chemical
routes leading to benzene itself are still unclear. An in situ formation
mechanism for benzene must involve abundant hydrocarbons containing from 2 to
4 carbon atoms. Moreover, some of these species have to permit an easy
cyclization in 2-3 steps to have an efficient yield of benzene or phenyl
radical. The propargyl radical, CH2CCH, has been found recently in TMC-1 by
Agúndez et al. (2021) with an abundance close to 10-8 relative to H2. In
addition, complex hydrocarbons such as vinyl and allenyl acetylene have been
also observed with very large abundances (Cernicharo et al. 2021a, b). These
hydrocarbons may hold the key to form the first aromatic ring, from which
larger PAHs can grow.

In this Letter we report the discovery of three pure hydrocarbon cycles (see
Fig. 1): $c$-C3HCCH (ethynyl cyclopropenylidene), $c$-C5H6 (cyclopentadiene),
and $c$-C9H8 (indene)111We have learned in a virtual seminar during the final
phase of preparation of this work that a paper has been submitted (Burkhardt
et al., 2021) claiming the detection of this species in TMC-1. We do not know
neither its content nor the quality of the detection..

Figure 1: Structures of the three species detected in this work.  Figure 2:
Selected transitions of $c$-C3HCCH in TMC-1. The abscissa corresponds to the
rest frequency of the lines assuming a local standard of rest velocity of the
source of 5.83 km s-1. Frequencies and intensities for the observed lines are
given in Table LABEL:obs_line_parameters. The ordinate is the antenna
temperature, corrected for atmospheric and telescope losses, in milli Kelvin.
The quantum numbers for each transition are indicated in the upper left corner
of the corresponding panel. The red lines show the computed synthetic spectrum
for this species for Tr=10 K and a column density of 3.1$\times$1011 cm-2.
Cyan labels indicated the multiplicative factor applied to the model to match
the intensity of the observed lines.

##  2 Observations

New receivers, built within the Nanocosmos
project222https://nanocosmos.iff.csic.es/, and installed at the Yebes 40m
radiotelescope, were used for the observations of TMC-1. The Q-band receiver
consists of two HEMT cold amplifiers covering the 31.0-50.3 GHz band with
horizontal and vertical polarizations. Receiver temperatures vary from 22 K at
32 GHz to 42 K at 50 GHz. The backends are $2\times 8\times 2.5$ GHz fast
Fourier transform spectrometers with a spectral resolution of 38.15 kHz
providing the whole coverage of the Q-band in both polarisations. The main
beam efficiency varies from 0.6 at 32 GHz to 0.43 at 50 GHz. A detailed
description of the system is given by Tercero et al. (2021).

The line survey of TMC-1 ($\alpha_{J2000}=4^{\rm h}41^{\rm m}41.9^{\rm s}$ and
$\delta_{J2000}=+25^{\circ}41^{\prime}27.0^{\prime\prime}$) in the Q-band was
performed in several sessions. Previous results on the detection of C3N- and
C5N- (Cernicharo et al. 2020b), HC5NH+ (Marcelino et al. 2020), HC4NC
(Cernicharo et al. 2020c), and HC3O+ (Cernicharo et al. 2020a) were based on
two observing runs performed in November 2019 and February 2020. Two different
frequency coverages were used, 31.08-49.52 GHz and 31.98-50.42 GHz, in order
to check that no spurious spectral ghosts are produced in the down-conversion
chain. Additional data were taken in October, December 2020 and January-April
2021. The observing procedure was frequency-switching with a frequency throw
of 10 MHz for the two first runs and of 8 MHz for all the others.

All data were analyzed using the GILDAS
package333http://www.iram.fr/IRAMFR/GILDAS.

##  3 Results

The sensitivity of our TMC-1 data is better than previously published line
surveys of this source at the same frequencies (Kaifu et al. 2004) by a factor
10-20. In fact, it has been possible to detect many individual lines from
molecules that were reported previously only by stacking techniques (Marcelino
et al. 2021). The recent discovery of some molecules containing the ethynyl
group (CCH), such as vinyl and allenyl acetylene (Cernicharo et al. 2021a, b),
and of the propargyl radical (Agúndez et al. 2021), prompted us to search for
other chemically related hydrocarbons. Line identification in this work was
done using the catalogues MADEX (Cernicharo 2012), CDMS (Müller et al. 2005),
and JPL (Pickett et al. 1998).

###  3.1 Ethynyl cyclopropenylidene, $c$-C3HCCH

Cyclopropenylidene ($c$-C3H2) is an abundant species in cold dark clouds.
Several lines of the para species of this molecule have been detected in our
line survey. They are analyzed in Appendix D. The reaction of this molecule
with CCH and CN could lead to the formation of $c$-C3HCCH and c-C3HCN. Ethynyl
cyclopropenylidene ($c$-C3HCCH; see Fig. 1) is one of the various C5H2 isomers
(see Appendix B) studied by microwave spectroscopy in the laboratory (Travers
et al.  1997; McCarthy et al. 1997; Gottlieb et al. 1998). In addition to
$c$-C3HCCH, these isomers are $l$-H2C5, the bent HCC(CH)CC, and the cycle
$c$-H2C3CC. All these species are implemented in the MADEX code (Cernicharo
2012). In the case of $c$-C3HCCH, the dipole moment components are
$\mu_{a}$=2.04 D, and $\mu_{b}$=2.89 D (Travers et al.  1997).

A search for $c$-C3HCCH resulted in the detection of 13 rotational lines,
which gives a high degree of confidence in the detection of this species.
Nevertheless, the lines are weak, with intensities around 1 mK. Some of them
are shown in Fig. 2. The derived line parameters (see Appendix A) are given in
Table LABEL:obs_line_parameters. A few lines that could have been detected in
our survey appear blended with lines from other species or are affected by
negative features produced in the folding of frequency switching procedure.
These cases are indicated in Table LABEL:obs_line_parameters. An analysis of
the observed intensities through a rotational diagram provides a rotational
temperature of 8 $\pm$ 3 K and a column density for $c$-C3HCCH of (3.1 $\pm$
0.8) $\times$ 1011 cm-2. We assumed a linewidth of 1.0 km s-1, which
corresponds to the observed average value, and a source of uniform brightness
temperature with a diameter of 80′′ (Fossé et al. 2001). We performed a model
fitting directly the observed line profiles as described by Cernicharo et al.
(2021c), with the result that the best match between the computed synthetic
spectrum and the observations corresponds to Tr = 10 K and a column density
similar to that derived from the rotation diagram. Figure 2 shows, in red, the
computed synthetic spectrum. The correction factors applied to each line to
match the observations are indicated in the Figure. Taken into account the
signal-to-noise ratio of the data, these factors are within the expected
uncertainties of the fit.

Adopting a column density of H2 of 1022 cm-2 for TMC-1 (Cernicharo & Guélin
1987), the abundance of $c$-C3HCCH relative to H2 is 3.1 $\times$ 10-11, which
is significantly below those of $c$-C3H2 and $c$-C3H (5.9 $\times$ 10-9 and
6.2 $\times$ 10-10, respectively; see Appendices D and E), ethynyl derivatives
such as vinyl, methyl, and allenyl acetylene (Cernicharo et al. 2021a, b;
Cabezas et al. 2021), and propylene (Marcelino et al. 2007). We derive the
following abundance ratios: $c$-C3H2/$c$-C3HCCH $\sim$ 190 (see Appendix D),
$c$-C3H/$c$-C3HCCH $\sim$ 20, and $c$-C3H2/$c$-C3H $\sim$ 10\. Upper limits to
the column densities of the isomers of $c$-C3HCCH are analyzed in Appendix B.
Improved rotational and distortion constants for $c$-C3HCCH are given in
Appendix C.

Figure 3: Same as Fig. 2 but for the observed transitions of $c$-C5H6 towards
TMC-1. The red line shows the computed synthetic spectrum for cyclopentandiene
assuming $T_{r}$ = 10 K and $N$($c$-C5H6) = 1.2 $\times$ 1013 cm-2. Cyan
labels, when present, indicate the multiplicative factor applied to the best
fit model to match the observations. Figure 4: Same as Fig. 2 but for the
selected transitions of $c$-C9H8 observed towards TMC-1. The red line shows
the computed synthetic spectrum for indene assuming $T_{r}$ = 10 K and
$N$($c$-C9H8) = 1.6 $\times$ 1013 cm-2. Cyan labels, when present, indicate
the multiplicative factor applied to the best fit model to match the
observations.

###  3.2 Cyclopentadiene, $c$-C5H6

The recent detection by stacking techniques of two cyanide derivatives of
cyclopentadiene in TMC-1 (Lee et al. 2021) makes very likely that
cyclopentadiene itself is also present in this source. Cyclopentadiene (see
Fig. 1) has been observed in the laboratory up to 386.6 GHz by different
authors (Laurie  1956; Scharpen & Laurie 1965; Benson & Flygare  1970; Bogey
et al. 1988). The molecule has a low dipole moment of 0.416 D along its $b$
axis (Laurie  1956). Because of the C2v symmetry of the molecule, two spin
species, ortho and para, with $K_{a}+K_{c}$ odd and even, and statistical
weights 9 and 7, respectively, have to be considered. The lowest ortho level
is the 11,1 and it is 0.6 K above the lowest para level (00,0). We searched
for this species in our TMC-1 data and we detected all the strong lines in the
31-50 GHz frequency range. They are shown in Fig. 3 and the line parameters
are given in Table LABEL:obs_line_parameters. Only one para line, the
$4_{1,3}-3_{2,2}$, which is affected by a negative frequency-switching
artifact, is missing. Another para line, the $4_{0,4}-3_{1,3}$ is strongly
blended with a line from $c$-C3HD (see third panel from the top in Fig. 3).
Taking into account the large geometrical section of the molecule and the low
dipole moment, we could expect to have its rotational levels thermalized at
the kinetic temperature of the cloud, 10 K. Assuming the same source
parameters than for $c$-C3HCCH we derive a total column density for
cyclopentadiene of (1.2 $\pm$ 0.3) $\times$ 1013 cm-2, with identical
contributions from the ortho and para species. The fractional abundance of
$c$-C5H6 relative to H2 is thus 1.2 $\times$ 10-9. This abundance is identical
to that of CH2CHCCH (Cernicharo et al. 2021b) and H2CCHCCH (Cernicharo et al.
2021c), and lower than that of the propargyl radical by a factor of $\sim$7
(Agúndez et al. 2021). Compared with its cyano-derivatives (Lee et al. 2021),
cyclopentadiene is 15 and 63 times more abundant that 1- and
2-cyanocyclopentadiene, respectively.

###  3.3 Indene, $c$-C9H8

Indene, $c$-C9H8, has a bicyclic structure with a six-membered ring fused to a
five-membered ring (see Fig. 1). It has been studied in the microwave
laboratory by Li et al. (1979) and Caminati (1993). The accuracy of these
measurements is moderate, with typical uncertainties for the measured
frequencies of $\sim$100 kHz. Its dipole moment is also relatively low, with
$\mu_{a}$ = 0.50 D and $\mu_{b}$ = 0.37 D (Caminati 1993). Frequency
predictions in the 31-50 GHz range have uncertainties between 10 and 80 kHz.
We searched for indene in our data and found a large number of lines above the
3$\sigma$ level. Nineteen of them are shown in Fig. 4. Line parameters for all
the 47 observed lines of indene detected in TMC-1 are given in Table
LABEL:obs_line_parameters. Improved rotational and distortion constants for
indene are provided in Appendix C and Table 3.

In order to derive a column density for indene we have assumed that the
rotational temperature is 10 K. Taking into account the large geometrical
section of the molecule and its low dipole moment, this assumption seems
reasonable. The model fit procedure, consisting in a fit to the observed line
profiles, provides a column density for indene of (1.6 $\pm$ 0.3) $\times$
1013 cm-2, which is very similar to that of cyclopentadiene, vinyl and allenyl
acetylene, and only a factor of five below that of $c$-C3H2. The fractional
abundance of indene relative to H2 is thus 1.6 $\times$ 10-9. The discovery of
this large hydrocarbon through the old fashion line-by-line detection
procedure is solid and robust. For this species, the observed averaged value
of the linewidth is close to 1.0 km s-1, such as for c-CH3CCH. Most lines of
polar species have linewidths of 0.6 km s-1. In the case of indene, its low
dipole moment and its easy to be thermalized conditions, could favour the
emission from the less dense regions of the cloud. This could suggest that
this species prevails in the molecular envelope of TMC-1. However, c-CH3CCH
has a larger dipole moment and requires a high density to be close to
thermalization as derived in section 3.1. Further observations with higher
spectral resolution are needed to discriminate between excitation and possible
contribution from different cloud components.

##  4 Discussion

###  4.1 $c$-C3HCCH

The most obvious route to $c$-C3HCCH is the reaction between CCH and $c$-C3H2.
To our knowledge, this reaction has not been studied either experimentally or
theoretically, but it is likely that it occurs fast at low temperatures with
the H loss channel being the major one, based on the reactivity of CCH with
other unsaturated hydrocarbons (Vakhtin et al. 2001). In addition, the two
reactants are quite abundant in TMC-1 (e.g., Agúndez & Wakelam 2013). If we
implement this reaction with a rate coefficient of 10-10 cm3 s-1 in a chemical
model similar to that presented by Agúndez et al. (2021) and we assume that
$c$-C3HCCH is mostly destroyed by reactions with C atoms and C+ and H+ ions,
the peak abundance calculated for $c$-C3HCCH is $\sim$ 5 $\times$10-11
relative to H2, which is in very good agreement with the value observed in
TMC-1. Although the subset of reactions involving $c$-C3HCCH is clearly
incomplete, this exercise shows that the reaction CCH \+ $c$-C3H2 could be a
plausible formation route for this cyclic hydrocarbon in TMC-1. Other
reactions that could also yield $c$-C3HCCH are $c$-C3H + C2H2 and C2 \+
CH2CCH, although little is known on their chemical kinetics.

Similarly to CCH, the radical CN has been also found to react rapidly with
unsaturated hydrocarbons at low temperatures (e.g., Sims et al. 1993).
Therefore, the reaction between CN and $c$-C3H2 is a very likely source of
$c$-C3HCN, which is the cyanide derivative analogue of $c$-C3HCCH. It is
interesting to note that ethynyl derivatives are more abundant than cyanide
derivatives in TMC-1, e.g., CH3CCCCH/CH3CCCN = 7.6 and CH2CCHCCH/CH2CCHCN =
4.4 (Marcelino et al. 2021; Cernicharo et al. 2021b), which is thought to
reflect the CCH/CN abundance ratio of 10 observed in this source (Pratap et
al. 1997). Therefore, we could expect $c$-C3HCN to be a few times less
abundant than $c$-C3HCCH. We do not detect $c$-C3HCN in our sensitive TMC-1
data, which means that it must have a column density below 2.5 $\times$ 1011
cm-2 (3$\sigma$ upper limit; see Appendix B). Therefore, the abundance ratio
$c$-C3HCCH/$c$-C3HCN in TMC-1 must be $>$ 1.2. It is therefore likely that a
reduced noise level from deeper observations may lead to the detection of
$c$-C3HCN in TMC-1.

###  4.2 Cyclopentadiene, $c$-C5H6

Given the high abundance derived for cyclopentadiene in TMC-1, 1.2 $\times$
10-9 relative to H2, it is not straightforward to find an efficient formation
route. Several reactions that could lead to $c$-C5H6 are not efficient under
the cold conditions of TMC-1. For example, the reaction between CH2CCH + C2H4
has an activation barrier (Saeys et al. 2003), while the reactions of C2H3
with either CH3CCH or CH2CCH2 probably have also barriers, as occurs in other
reactions of C2H3 with unsaturated hydrocarbons (Miller et al. 2000; Ismail et
al. 2007; Goldsmith et al. 2009). The reaction between C3H5 and C2H2 yields
cyclopentadiene, but it has a barrier (Bouwman et al. 2015). The reaction C2H
+ CH2CHCH3 is fast at low temperatures and produces several C5H6 isomers,
although the experimental data seems to be inconsistent with cyclopentadiene
being one of them (Bouwman et al. 2012).

The reaction between CH and butadiene (CH2CHCHCH2) is calculated to produce
$c$-C5H6 with no barrier (McCarthy et al. 2021). Adopting a rate coefficient
of 10-10 cm3 s-1 for this reaction in our chemical model, based on Agúndez et
al. (2021), and assuming that c-C5H6 is mostly removed through reactions with
C atoms and C+ and H+ ions, we calculate a peak abundance of $\sim$ 3 $\times$
10-13 relative to H2 for $c$-C5H6. This is more than three orders of magnitude
below the observed value. In the chemical model, butadiene is essentially
formed by the reaction CH + CH2CHCH3 (Daugey et al. 2005; Loison & Bergeat
2009; Ribeiro & Mebel 2016) with a peak abundance of $\sim$ 3 $\times$10-10
relative to H2. Butadiene is an interesting potential precursor of $c$-C5H6,
and also of benzene (Jones et al. 2011), but to play such a role it needs to
have an abundance much higher than calculated by current gas-phase chemical
models. Unfortunately, it is a non-polar molecule and thus it cannot be
detected through radio techniques.

Given the high abundance derived for the radical CH2CCH (Agúndez et al. 2021),
reactions between radicals, such as CH2CCH + C2H5 and C2H3 \+ CH2CHCH2, are
also potential sources of $c$-C5H6. Routes involving ions, such as
condensation rections between cationic and neutral hydrocarbons (Herbst &
Leung  1989), may also lead to $c$-C5H6. Theoretical studies on these
reactions would help to evaluate their role in the synthesis of
cyclopentadiene. Whatever the formation route to $c$-C5H6 is, this hydrocarbon
cycle is the most obvious precursor of the two cyanide derivatives $c$-C5H5CN
detected in TMC-1 (McCarthy et al. 2021; Lee et al. 2021).

###  4.3 Indene, $c$-C9H8

The high abundance derived for this aromatic molecule, in fact the first pure
polycyclic aromatic hydrocarbon found in space, also challenges to find out an
efficient formation route in TMC-1. Such a route should involve a fast
reaction between two abundant species. Potential formation reactions like
$c$-C6H6 \+ CH2CCH and $c$-C6H5 \+ C3H4 have activation barriers (Kislov &
Mebel 2007; Vereecken & Peeters 2003; Mebel et al. 2017). Reactions such as
$c$-C5H6 \+ C4H3 or $c$-C5H5 \+ C4H4 could be efficient producing indene.
Doddipatla et al. (2021) suggest that indene can be formed through the
barrierless reaction between the radical CH and styrene ($c$-C6H5C2H3).
However, it is unknown whether the precursor styrene is abundant enough in
TMC-1, and it is not obvious which formation pathway can lead to it.
Doddipatla et al. (2021) present a tentative synthetic route, from which they
calculate abundances of up to 10-12 for indene, well below the value derived
from observations here. It is clear that further research is needed to explain
the presence of indene in TMC-1 with an abundance as high as 1.6 $\times$10-9
relative to H2.

##  5 Conclusions

We report the detection of $c$-C3HCCH, cyclopentadiene, and indene in TMC-1.
While the observed abundance of $c$-C3HCCH, a few 10-11 relative to H2, can be
accounted for by the reaction between CCH and $c$-C3H2, the fairly high
abundances found for cyclopentadiene and indene, above 10-9, are difficult to
explain by currently proposed formation routes. The detection of very abundant
cyclopentadiene and indene should promote further research to elucidate which
plausible bottom-up mechanisms of formation of such complex aromatic
hydrocarbons can be at work in cold dense clouds like TMC-1.

###### Acknowledgements.

We thank ERC for funding through grant ERC-2013-Syg-610256-NANOCOSMOS. We also
thank Ministerio de Ciencia e Innovación of Spain (MICIU) for funding support
through projects AYA2016-75066-C2-1-P, PID2019-106110GB-I00,
PID2019-107115GB-C21 / AEI / 10.13039/501100011033, and PID2019-106235GB-I00.
M.A. thanks MICIU for grant RyC-2014-16277. We would like to thank our
referee, S. Yamamoto, for his useful comments and suggestions.

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##  Appendix A Line parameters of $c$-C3HCCH, $c$-C5H6, and $c$-C9H8

Line parameters for the different molecules studied in this work were obtained
by fitting a Gaussian line profile to the observed data. A window of $\pm$ 15
km s-1 around the vLSR of the source has been considered for each transition.
The derived line parameters for the three molecular species discovered in this
work are given in Table LABEL:obs_line_parameters.

Table 1: Observed line parameters for c-C3HCCH in TMC-1 $J_{K_{a},K_{c}}$ |  $\nu_{obs}$ a |  $\int T_{A}^{*}dv$ b |  $\Delta v$ c |  $T_{A}^{*}$d |
---|---|---|---|---|---
| (MHz) | (mK km s-1) | (km s-1) | (mK) |
$c$-C3HCCH |  |  |  |  |
$1_{1,0}-1_{0,1}$ | 31525.012$\pm$0.03 | 1.30$\pm$0.23 | 1.35$\pm$0.28 | 0.90$\pm$0.20 |
$2_{1,1}-2_{0,2}$ | 31838.475$\pm$0.03 | 1.98$\pm$0.34 | 1.34$\pm$0.20 | 1.38$\pm$0.24 |
$5_{1,5}-4_{1,4}$ | 31904.045$\pm$0.03 | 0.53$\pm$0.18 | 0.76$\pm$0.35 | 0.65$\pm$0.24 | A
$3_{1,2}-3_{0,3}$ | 32312.929$\pm$0.03 | 0.87$\pm$0.16 | 0.87$\pm$0.17 | 0.93$\pm$0.20 | A
$9_{0,9}-8_{1,8}$ | 32429.470$\pm$0.03 | 0.88$\pm$0.20 | 1.13$\pm$0.28 | 0.73$\pm$0.21 |
$5_{0,5}-4_{0,4}$ | 32646.191$\pm$0.03 | 1.34$\pm$0.19 | 1.32$\pm$0.24 | 0.95$\pm$0.18 |
$4_{1,3}-4_{0,4}$ | 32953.663$\pm$0.03 | 1.07$\pm$0.22 | 1.16$\pm$0.33 | 0.87$\pm$0.22 |
$5_{1,4}-4_{1,3}$ | 33460.002$\pm$0.03 | 0.70$\pm$0.20 | 0.40$\pm$0.23 | 1.35$\pm$0.22 | B
$5_{1,4}-5_{0,5}$ | 33767.455$\pm$0.03 | 1.24$\pm$0.23 | 1.18$\pm$0.24 | 0.88$\pm$0.21 |
$6_{1,5}-6_{0,6}$ | 34762.885$\pm$0.03 | 1.09$\pm$0.25 | 1.37$\pm$0.38 | 0.75$\pm$0.21 |
$7_{1,6}-7_{0,7}$ | 35950.042$\pm$0.02 |  |  |  | C
$8_{1,7}-8_{0,8}$ | 37340.311$\pm$0.02 |  |  |  | D
$1_{1,1}-0_{0,0}$ | 37752.281$\pm$0.01 |  |  |  | E
$6_{1,6}-5_{1,5}$ | 38278.543$\pm$0.01 |  |  |  | C
$9_{1,8}-9_{0,9}$ | 38946.462$\pm$0.03 |  |  |  | E
$6_{0,6}-5_{0,5}$ | 39149.938$\pm$0.03 | 0.75$\pm$0.15 | 0.87$\pm$0.16 | 0.80$\pm$0.21 |
$10_{0,10}-9_{1,9}$ | 40056.165$\pm$0.03 | 1.06$\pm$0.18 | 0.81$\pm$0.20 | 1.23$\pm$0.21 |
$2_{1,2}-1_{0,1}$ | 43979.464$\pm$0.03 | 0.98$\pm$0.18 | 0.96$\pm$0.19 | 0.95$\pm$0.26 |
$c$-C5H6f |  |  |  |  |
$3_{1,2}-2_{2,1}$ | 37454.017$\pm$0.02 | 1.18$\pm$0.22 | 0.79$\pm$0.16 | 1.40$\pm$0.30 |
$3_{2,2}-2_{1,1}$ | 38092.525$\pm$0.02 | 0.85$\pm$0.17 | 0.68$\pm$0.15 | 1.17$\pm$0.24 |
$4_{0,4}-3_{1,3}$ | 38224.588e |  |  |  | F
$4_{1,4}-3_{0,3}$ | 38224.787$\pm$0.02 | 0.75$\pm$0.14 | 0.68$\pm$0.15 | 1.03$\pm$0.20 |
$4_{1,3}-3_{2,2}$ | 46314.423e |  |  |  $\leq$1.5 |
$4_{2,3}-3_{1,2}$ | 46352.738$\pm$0.02 | 1.37$\pm$0.26 | 0.56$\pm$0.12 | 2.33$\pm$0.47 |
$3_{3,1}-2_{2,0}$ | 46499.116$\pm$0.02 | 0.91$\pm$0.18 | 0.48$\pm$0.09 | 1.78$\pm$0.36 |
$5_{0,5}-4_{1,4}$ | 46767.415$\pm$0.02 | 3.49$\pm$0.23 | 0.62$\pm$0.05 | 5.33$\pm$0.39 | G
$c$-C9H8f |  |  |  |  |
$11_{4,7}-10_{4,6}$ | 31364.868$\pm$0.02 | 0.47$\pm$0.20 | 1.12$\pm$0.39 | 0.40$\pm$0.20 |
$12_{3,10}-11_{3,9}$ | 32083.322$\pm$0.01 | 0.80$\pm$0.18 | 1.11$\pm$0.32 | 0.68$\pm$0.18 |
$11_{3,8}-10_{3,7}$ | 32171.624$\pm$0.01 | 0.83$\pm$0.28 | 0.89$\pm$0.33 | 0.88$\pm$0.20 |
$13_{2,12}-12_{2,11}$ | 32481.036$\pm$0.01 | 0.74$\pm$0.16 | 0.83$\pm$0.18 | 0.84$\pm$0.21 |
$14_{0,14}-13_{1,13}$ | 32526.303$\pm$0.01 | 0.94$\pm$0.20 | 1.01$\pm$0.24 | 0.87$\pm$0.18 |
$14_{1,14}-13_{1,13}$ | 32527.242$\pm$0.01 | 0.78$\pm$0.21 | 1.05$\pm$0.33 | 0.70$\pm$0.18 |
$14_{0,14}-13_{0,13}$ | 32528.322$\pm$0.01 | 1.14$\pm$0.21 | 1.65$\pm$0.15 | 0.65$\pm$0.18 |
$14_{1,14}-13_{0,13}$ | 32529.269$\pm$0.01 | 0.61$\pm$0.19 | 0.60$\pm$0.24 | 0.97$\pm$0.20 |
$13_{1,12}-12_{1,11}$ | 32555.319$\pm$0.03 | 1.15$\pm$0.26 | 1.87$\pm$0.45 | 0.57$\pm$0.20 | A
$12_{2,10}-11_{2,9}$ | 33087.091$\pm$0.02 | 0.21$\pm$0.11 | 0.48$\pm$0.20 | 0.40$\pm$0.16 |
$12_{6,7}-11_{6,6}$ | 33104.538$\pm$0.01 |  |  |  $\leq$0.48 |
$12_{4,9}-11_{4,8}$ | 33105.876$\pm$0.02 | 0.30$\pm$0.16 | 0.40$\pm$0.20 | 0.45$\pm$0.16 |
$12_{6,6}-11_{6,5}$ | 33124.468$\pm$0.02 | 0.56$\pm$0.17 | 0.84$\pm$0.19 | 0.70$\pm$0.19 | A
$12_{5,8}-11_{5,7}$ | 33262.106$\pm$0.01 | 0.91$\pm$0.18 | 0.81$\pm$0.20 | 1.05$\pm$0.22 |
$12_{5,7}-11_{5,6}$ | 33524.184$\pm$0.02 | 0.26$\pm$0.10 | 0.34$\pm$0.20 | 0.72$\pm$0.18 |
$13_{3,11}-12_{3,10}$ | 34484.248$\pm$0.01 | 0.72$\pm$0.16 | 0.96$\pm$0.26 | 0.70$\pm$0.18 |
$12_{4,8}-11_{4,7}$ | 34608.352$\pm$0.01 | 0.88$\pm$0.16 | 1.11$\pm$0.22 | 0.75$\pm$0.18 |
$14_{2,13}-13_{2,12}$ | 34736.637$\pm$0.01 | 0.60$\pm$0.19 | 1.02$\pm$0.36 | 0.55$\pm$0.21 |
$15_{0,15}-14_{1,14}$ | 34771.228$\pm$0.02 |  |  |  $\leq$ 0.45 |
$15_{1,15}-14_{1,14}$ | 34771.684$\pm$0.02 | 0.30$\pm$0.08 | 0.54$\pm$0.17 | 0.53$\pm$0.15 |
$15_{0,15}-14_{0,14}$ | 34772.190$\pm$0.02 | 0.49$\pm$0.32 | 0.94$\pm$0.33 | 0.49$\pm$0.15 | A
$15_{1,15}-14_{0,14}$ | 34772.612$\pm$0.02 |  |  |  $\leq$0.45 |
$14_{1,13}-13_{1,12}$ | 34776.320$\pm$0.02 | 0.40$\pm$0.08 | 0.33$\pm$0.15 | 1.13$\pm$0.17 |
$12_{3,9}-11_{3,8}$ | 35011.580$\pm$0.01 | 0.57$\pm$0.13 | 0.63$\pm$0.15 | 0.85$\pm$0.19 |
$13_{2,11}-12_{2,10}$ | 35211.250$\pm$0.01 | 0.43$\pm$0.08 | 0.49$\pm$0.22 | 0.82$\pm$0.17 |
$13_{4,10}-12_{4,9}$ | 35766.651$\pm$0.02 | 0.68$\pm$0.17 | 1.12$\pm$0.26 | 0.57$\pm$0.19 |
$13_{6,8}-12_{6,7}$ | 35959.203$\pm$0.01 |  |  |  $\leq$ 0.60 |
$13_{6,7}-12_{6,6}$ | 36008.349$\pm$0.02 | 0.64$\pm$0.15 | 1.00$\pm$0.34 | 0.60$\pm$0.20 | A
$13_{5,9}-12_{5,8}$ | 36101.117$\pm$0.02 | 0.64$\pm$0.17 | 1.10$\pm$0.40 | 0.47$\pm$0.16 |
$13_{5,8}-12_{5,7}$ | 36605.949$\pm$0.01 | 0.71$\pm$0.15 | 0.71$\pm$0.15 | 0.94$\pm$0.21 |
$14_{3,12}-13_{3,11}$ | 36835.520$\pm$0.01 | 0.80$\pm$0.16 | 1.02$\pm$0.25 | 0.74$\pm$0.18 |
$15_{2,14}-14_{2,13}$ | 36986.145$\pm$0.02 | 0.32$\pm$0.11 | 0.45$\pm$0.16 | 0.59$\pm$0.19 |
$15_{1,14}-14_{1,13}$ | 37006.756$\pm$0.01 | 0.60$\pm$0.14 | 0.65$\pm$0.16 | 0.87$\pm$0.19 |
$16_{1,16}-15_{1,15}$ | 37016.001$\pm$0.02 | 0.39$\pm$0.15 | 0.77$\pm$0.27 | 0.47$\pm$0.20 |
$16_{0,16}-15_{0,15}$ | 37016.269$\pm$0.02 | 0.68$\pm$0.21 | 1.05$\pm$0.35 | 0.57$\pm$0.20 | A
$14_{2,12}-13_{2,11}$ | 37318.440$\pm$0.01 | 0.30$\pm$0.12 | 0.60$\pm$0.20 | 0.47$\pm$0.20 |
$13_{3,10}-12_{3,9}$ | 37665.778$\pm$0.01 | 0.84$\pm$0.16 | 0.78$\pm$0.17 | 1.01$\pm$0.21 |
$13_{4,9}-12_{4,8}$ | 37842.548$\pm$0.01 | 0.41$\pm$0.16 | 0.64$\pm$0.21 | 0.70$\pm$0.19 |
$14_{4,11}-13_{4,10}$ | 38361.360$\pm$0.01 |  |  |  | C
$14_{6,9}-13_{6,8}$ | 38829.224$\pm$0.01 | 0.87$\pm$0.17 | 0.83$\pm$0.16 | 1.00$\pm$0.23 |
$14_{5,10}-13_{5,9}$ | 38918.551$\pm$0.01 |  |  |  $\leq$0.69 |
$14_{6,8}-13_{6,7}$ | 38939.641$\pm$0.02 |  |  |  $\leq$0.69 |
$15_{3,13}-14_{3,12}$ | 39147.950$\pm$0.02 | 0.38$\pm$0.13 | 0.48$\pm$0.20 | 0.75$\pm$0.23 |
$16_{2,15}-15_{2,14}$ | 39232.293$\pm$0.02 | 1.02$\pm$0.27 | 0.68$\pm$0.19 | 1.46$\pm$0.24 |
$16_{1,15}-15_{1,14}$ | 39242.748$\pm$0.02 |  |  |  $\leq$1.6 |
$17_{1,17}-16_{1,16}$ | 39260.308$\pm$0.02 | 0.71$\pm$0.23 | 0.54$\pm$0.18 | 1.21$\pm$0.40 |
$17_{0,17}-16_{0,16}$ | 39260.513$\pm$0.02 | 1.04$\pm$0.26 | 0.61$\pm$0.15 | 1.61$\pm$0.40 |
$15_{2,13}-14_{2,12}$ | 39447.608$\pm$0.03 | 0.54$\pm$0.18 | 0.82$\pm$0.29 | 0.62$\pm$0.22 |
$14_{5,9}-13_{5,8}$ | 39799.463$\pm$0.03 | 0.51$\pm$0.18 | 1.10$\pm$0.33 | 0.43$\pm$0.20 |
$14_{3,11}-13_{3,10}$ | 40118.461$\pm$0.02 |  |  |  $\leq$0.65 |
$15_{4,12}-14_{4,11}$ | 40886.724$\pm$0.03 | 0.58$\pm$0.20 | 0.95$\pm$0.35 | 0.57$\pm$0.26 |
$14_{4,10}-13_{4,9}$ | 40983.754$\pm$0.03 | 0.37$\pm$0.17 | 0.60$\pm$0.29 | 0.58$\pm$0.24 |
$16_{3,14}-15_{3,13}$ | 41432.310$\pm$0.03 |  |  |  $\leq$0.90 |
$17_{2,16}-16_{2,15}$ | 41476.748$\pm$0.03 |  |  |  | B
$17_{1,16}-16_{1,15}$ | 41482.002$\pm$0.03 |  |  |  $\leq$0.90 |
$18_{1,18}-17_{1,17}$ | 41504.704$\pm$0.01 | 0.95$\pm$0.20 | 0.74$\pm$0.18 | 1.21$\pm$0.29 | H
$18_{0,18}-17_{0,17}$ | 41504.704$\pm$0.01 |  |  |  | H
$16_{2,14}-15_{2,13}$ | 41608.793$\pm$0.02 | 0.63$\pm$0.27 | 0.49$\pm$0.17 | 1.20$\pm$0.33 |
$15_{5,11}-14_{5,10}$ | 41698.205$\pm$0.02 |  |  |  $\leq$0.90 |
$19_{1,19}-18_{1,18}$ | 43748.996$\pm$0.02 | 1.00$\pm$0.30 | 0.92$\pm$0.38 | 1.09$\pm$0.35 | H
$19_{0,19}-18_{0,18}$ | 43748.996$\pm$0.02 |  |  |  | H
$20_{1,20}-19_{1,19}$ | 45993.359$\pm$0.02 | 1.24$\pm$0.31 | 0.64$\pm$0.23 | 1.83$\pm$0.42 | H
$20_{0,20}-19_{0,19}$ | 45993.369$\pm$0.02 |  |  |  | H
Table 1: continued. 444
$a$$a$footnotetext: Observed frequency assuming a vLSR of 5.83 km s-1.
$b$$b$footnotetext: Integrated line intensity in mK km s-1.
$c$$c$footnotetext: Linewidth at half intensity derived by fitting a Gaussian
function to the observed line profile (in km s-1).
$d$$d$footnotetext: Antenna temperature in milli Kelvin.
$e$$e$footnotetext: Predicted frequency.
$f$$f$footnotetext: Measured frequencies assume a vLSR of 5.83 km s-1. The
difference between observed and predicted frequencies are $\pm$15 kHz.
$A$$A$footnotetext: Partially blended with another feature. Fit still possible
providing reasonable results.
$B$$B$footnotetext: Blended with another feature. Fit uncertain.
$C$$C$footnotetext: Fully blended with a negative feature produced in the
frequency switching folding. Fit unreliable. Frequency corresponds to the
predicted value from the rotational constants of Table 1.
$D$$D$footnotetext: Fully blended with a line from HCCCH2CN. Frequency
corresponds to the predicted value from the rotational constants of Table 2.
$E$$E$footnotetext: A narrow line is clearly visible but it is affected by a
negative feature. Frequency corresponds to the predicted value from the
rotational constants of Table 1.
$F$$F$footnotetext: Fully blended with a line from $c$-C3HD. Frequency
corresponds to the predicted value.
$G$$G$footnotetext: The $5_{1,5}-4_{0,4}$ line, which belongs to the para
species, is fully blended with the $5_{0,5}-4_{1,4}$ transition of the ortho
species. Both lines have similar line strength.
$H$$H$footnotetext: Unresolved doublet.

##  Appendix B The isomers of $c$-C3HCCH

The isomeric family with formula H2C5 is composed by five molecular species.
The nonpolar pentadiynylidene (HCCCCH) and ethynyl cyclopropenylidene
($c$-C3HCCH) are the most stable isomers, whose energy separation is very
small $\thicksim$0.043 eV. H2CCCCC is a high energy isomer and lies 0.564 eV
above the ground. Other two species are part of this isomeric family,
HCC(CH)CC and $c$-C3H2CC, placed at 0.737 and 0.910 eV respectively, above the
most stable forms, respectively (Seburg et al. 1997).

The rotational transitions in the 31-50 GHz of HCC(CH)CC, one of the C5H2
isomers, can be predicted with high accuracy from the laboratory data of
Gottlieb et al. (1998). We obtain a 3$\sigma$ upper limit to its column
density of 2$\times$1010 cm-2. Note that the dipole moment of this species is
4.52 D (Gottlieb et al. 1998). An even larger dipole moment has been
calculated by the same authors for the isomer $c$-C3H2CC (8.2 D). For this
species we derive a 3$\sigma$ upper limit to its column density of
9$\times$109 cm-2. Finally, the cumulenic species H2C5 has been detected
recently from our data and the results will be published elsewhere (Cabezas et
al. 2021, in preparation).

An additional derivate of $c$-C3H2 is cyano propenylidene, $c$-C3HCN, which
has been observed in the laboratory by McCarthy et al. (1999). We searched for
it in our data and derive a 3$\sigma$ upper limit to its column density of
2.5$\times$1011 cm-2.

##  Appendix C Improved rotational constants for $c$-C3HCCH and c-C9H8

The frequencies we have measured in TMC-1 can be used to improve the
rotational and distortion constants of $c$-C3HCCH and indene. We have used the
fitting code FITWAT described in Cernicharo et al. (2018).

Table 2 provides the results obtained by fitting the laboratory data of
Travers et al.  (1997) for c-C3HCCH alone, and those obtained from a fit to
the merged laboratory plus the TMC-1 frequencies. An improvement of the
uncertainty in the rotational and distortion constants is obtained. The fit to
the laboratory data alone results in exactly the same constants than those
obtained by Travers et al.  (1997).

Table 3 provide the same information but for indene. The 50 lines measured in
TMC-1 (three of them are unresolved doublets) provide a significant
improvement of the rotational and distortion constants for this species.

The merged fits are recommended to predict the frequency of the rotational
transitions of both species with uncertainties between 10 and 200 kHz up to
115 GHz.

Table 2: Rotational and distortion constants of $c$-C3HCCH

Constant |  Laboratorya | This work
---|---|---
$A$ (MHz) | 34638.7012(27) | 34638.7023(26)
$B$ (MHz) | 3424.87685(43) | 3424.87678(42)
$C$ (MHz) | 3113.63856(50) | 3113.63900(44)
$\Delta_{J}$ (kHz) | 0.2877(80) | 0.2918(74)
$\Delta_{JK}$ (kHz) | 29.53(24) | 29.72(21)
Number of lines | 13 | 26
$\sigma$(kHz) | 2.1 | 26
$J_{max}$, $K_{max}$ | 7, 1 | 10,1
$\nu_{max}$ (GHz) | 26.130 | 43.979
555
Values between parenthesis correspond to the uncertainties of the parameters
in units of the least significant digits.
$a$$a$footnotetext: Laboratory frequencies from Travers et al.  (1997).

Table 3: Rotational and distortion constants of $c$-C9H8

Constant |  Laboratorya | This work
---|---|---
$A$ (MHz) | 3775.048(15) | 3775.0469(80)
$B$ (MHz) | 1580.8656(21) | 1580.86511(79)
$C$ (MHz) | 1122.2460(18) | 1122.24773(51)
$\Delta_{J}$ (kHz) | 0.0326(38) | 0.03349(97)
$\Delta_{JK}$ (kHz) | 0.050(11) | 0.0517(73)
$\Delta_{K}$ (kHz) | 0.37(12) | 0.329(82)
$\delta_{J}$ (kHz) | 0.0117(12) | 0.01068(51)
$\delta_{K}$ (kHz) | 0.068(19) | 0.0775(94)
Number of lines | 78 | 128
$\sigma$(kHz) | 67 | 54
$J_{max}$, $K_{max}$ | 30,13 | 30,13
$\nu_{max}$ (GHz) | 39.3 | 45.99
666
Values between parenthesis correspond to the uncertainties of the parameters
in units of the least significant digits.
$a$$a$footnotetext: Laboratory frequencies from Li et al. (1979); Caminati
(1993).

##  Appendix D Column density of $c$-C3H2

The polar carbene ring molecule $c$-C3H2 is widespread in interstellar and
circumstellar clouds (Matthews & Irvine  1985). For the physical conditions of
TMC-1 we expect only a few lines of this molecule to be strong enough within
the frequency coverage of our survey. In fact, only one ortho line, the
$3_{21}-3_{12}$ at 44104.777 MHz has an upper energy level below 20 K. This
line appears in our data in absorption as shown in Fig. 5. We have checked
that the absorption is real and not produced by a negative feature created
during the folding of the frequency switching data by analyzing separately the
two frequency throws used during the observations. This line was predicted to
be in absorption by Avery & Green (1989) for the typical densities of cold
dark clouds, but to the best of our knowledge it has not been reported
previously. The other two lines of the ortho species in our survey have upper
level energies around 45 K and are not detected. Absorption from $c$-C3H2 in
cold dark clouds has been previously reported for its para line
$2_{20}-2_{11}$ at 21.6 GHz (Matthews et al. 1986). For the para species three
lines are detected in our data as shown in Fig. 5. They are the
$4_{40}-4_{31}$ at 35.36 GHz, the $4_{31}-4_{22}$ at 42.231 GHz, and the
$2_{11}-2_{02}$ at 46.755 GHz. The first one appears also in absorption. We
have checked, as for $3_{21}-3_{12}$ ortho transition, that the absortion is
real. The other two para lines are in emission.

Collisional rates between $c$-C3H2 and He adapted to the low temperatures of
cold dark clouds are available from Avery & Green (1989) and Khalifa et al.
(2019). We performed Large Velocity Gradient (LVG) calculations for ortho and
para cyclopropenylidene species by varying the density and the column density
and by adopting the most recent set of collisional rates. The radius of the
source is fixed to 40′′ (Fossé et al. 2001) and the linewidth to 0.5 km s-1.
The best fit to the three observed para lines is obtained for $n$(H2) =(
4.0$\pm$0.5)$\times$104 cm-2, and $N$($p$-C3H2)=(1.4 $\pm$ 0.3) $\times$ 1013
cm-2. Adopting the derived density for the para species, then the ortho
absorption line can be reproduced for a column density $N$($o$-C3H2) = 4.5
$\times$ 1013 cm-2. Hence, the ortho to para abundance ratio of $c$-C3H2 is
$\sim$ 3, which is the expected value from the spin degeneracy of the two
species. The total column density of cyclopropenylidene is 5.9 $\times$ 1013
cm-2, in very good agreement with the value derived by Fossé et al. (2001) of
5.8 $\times$ 1013 cm-2. Nevertheless, the derived abundances rely on the
accuracy of the collisional rate coefficients. The synthetic spectrum computed
from these parameters is shown by the red (ortho) and blue (para) lines in
Fig. 5, and reproduces nicely the ortho and para absorption lines, and the two
para emission lines. If we use the collisional rates calculated by Avery &
Green (1989), a reasonable fit can be obtained reproducing the two absorption
lines. In this case the derived volume density is $n$(H2) = (2.8 $\pm$ 0.5)
$\times$ 104 cm-3 and the best fit to the intensities correspond to
$N$($o$-C3H2) = (2.8 $\pm$ 0.5) $\times$ 1013 cm-2 and $N$($p$-C3H2) = (3.2
$\pm$ 0.5) $\times$ 1013 cm-2. With these less accurate collisional rates the
ortho/para ratio is $\sim$1\. Moreover, deviations of up to 20% between
calculated and observed intensities are observed for the best model fit. These
deviations are not observed when we use the most recent calculations of
Khalifa et al. (2019), which are based in a more precise determination of the
potential energy surface of the $c$-C3H2/He system. For both set of rates the
derived total column density of $c$-C3H2 is practically identical, $\sim$ 6
$\times$ 1013 cm-2. In view of the complex system of absorption and emission
lines shown by this molecular species, collisional rates using H2 as collider
are highly desirable.

Figure 5: Observed transitions of $c$-C3H2 in TMC-1. The abscissa corresponds
to the rest frequency of the lines assuming a local standard of rest velocity
of the source of 5.77 km s-1. The ordinate is the antenna temperature,
corrected for atmospheric and telescope losses, in mK. The quantum numbers for
each transition are indicated in the upper or lower right corner of the
corresponding panel. The red line shows the computed synthetic spectrum for
this species (see text). Blue labels indicate the multiplicative factor
applied, when needed, to the model to match the intensity of the observed
lines.

##  Appendix E Column density of $c$-C3H

Cyclopropanediylidenyl, $c$-C3H, has been detected towards several
astrophysical environments including cold dark clouds, evolved stars, and
translucent molecular clouds (Yamamoto et al.  1987; Mangum & Wooten 1990;
Turner et al.  2000; Cernicharo et al. 2000). Several lines pertinining to the
fine and hyperfine structure of the $2_{1,1}-2_{1,2}$ rotational transitions
of this species are within our line survey of TMC-1. The data for all the
strongest components, and some of the weak ones, are shown in Fig. 6. Several
of the panels of this Figure show a significant number of unknown features.
This is an important drawback for any stacking procedure used to detect
molecules producing intensities below 1 mK for their rotational transitions.
At this level of sensitivity TMC-1 cannot be considered as a line-poor source
and detections have to be performed using the standard procedure of line-by-
line detection.

With only one transition we have to assume a rotational temperature in order
to derive a column density for the molecule. The energy of the upper level of
the different fine and hyperfine components of the $2_{1,1}-2_{1,2}$
transition is $\sim$ 6.5 K. Hence, the column density will depend slightly on
the adopted value of $T_{r}$ (see, e.g., the error analysis carried out by
Cernicharo et al. 2021c for HC3S+). Assuming $T_{r}$ = 10 K, a source diameter
of 80′′ and a linewidth of 0.6 km s-1, we obtain a column density for $c$-C3H
of (1.2 $\pm$ 0.2) $\times$ 1012 cm-2. This value is a factor five smaller
than the one derived by Yamamoto et al.  (1987), who observed the
$2_{1,2}-1_{1,1}$ transition at $\sim$ 91.5 GHz with the Nobeyama
radiotelescope. They assumed a rotational temperature of 5 K. In order to
understand this large difference we have used the data gathered with the IRAM
30m telescope during the 3 mm line survey of TMC-1 (Marcelino et al. 2007). We
obtain intensities for the $2_{1,2}-1_{1,1}$ hyperfine components that are
similar to those of Yamamoto et al.  (1987). The 3 mm data are shown in Fig.
7. Assuming an identical rotational temperature for the two transitions it is
not possible to get a reliable fit to the observed intensities. In fact, the 3
mm lines are well explained with a column density of (6.2 $\pm$ 0.4) $\times$
1012 cm-2 for $T_{r}$ = 5 K, a very similar value to that of Yamamoto et al.
(1987). However, the rotational temperature has to be decreased to 3.7 K to
explain with the same column density the observed intensities for the
$2_{1,1}-2_{1,2}$. The two transitions share the same lower level (the
fundamental one), but they could have significant different excitation
temperatures if the collisional rates from the ground to the two upper levels
are different. Unfortunately no collisional rates for $c$-C3H are available in
the literature.

Figure 6: Fine and hyperfine components of the $2_{1,1}-1_{1,1}$ transition of
$c$-C3H in TMC-1 as observed with the Yebes 40m radiotelescope. The abscissa
corresponds to the rest frequency of the lines assuming a local standard of
rest velocity of the source of 5.83 km s-1. The ordinate is the antenna
temperature, corrected for atmospheric and telescope losses, in mK. The
quantum numbers for each transition are indicated in the upper or lower right
corner of the corresponding panel. The red line shows the computed synthetic
spectrum for this species for a column density of 6.2 $\times$ 1012 cm-2 and
$T_{r}$ = 3.7 K (see text).  Figure 7: Fine and hyperfine components of the
$2_{1,2}-1_{1,1}$ transition of $c$-C3H in TMC-1 as observed with the IRAM 30m
radiotelescope (data from Marcelino et al. 2007). The abscissa corresponds to
the rest frequency of the lines assuming a local standard of rest velocity of
the source of 5.83 km s-1. The ordinate is the antenna temperature, corrected
for atmospheric and telescope losses, in mK. The quantum numbers for each
transition are indicated in the upper or lower right corner of the
corresponding panel. The red line shows the computed synthetic spectrum for
this species for a column density of 6.2 $\times$ 1012 cm-2 and $T_{r}$ = 5 K
(see text). Violet labels indicate the observed minus calculated frequencies
when larger than 5 kHz.


UFIFT-QG-23-05

Remembrance of Things Past

R. P. Woodard† and B. Yesilyurt∗

Department of Physics, University of Florida,
Gainesville, FL 32611, UNITED STATES

ABSTRACT

Nonlinear sigma models on de Sitter background have proved a useful prototype
for quantum gravity in summing the large logarithms which arise from loop
corrections. We consider a model whose evolution is described, at leading
logarithm order, by the trace of the coincident, doubly differentiated scalar
propagator. An analytic approximation for this quantity on an arbitrary
expansion history is applied to generalize the resummed de Sitter result to
any cosmological background which has experienced primordial inflation. In
addition to analytic expressions, we present explicit numerical results for
the evolution in a plausible expansion history. The large scales of primordial
inflation are transmitted to late times.

PACS numbers: 04.50.Kd, 95.35.+d, 98.62.-g

† e-mail<EMAIL_ADDRESS>
∗ e-mail<EMAIL_ADDRESS>
##  1 Introduction

Many things change when quantum field theory is applied to a cosmological
background,

$ds^{2}=-dt^{2}+a^{2}(t)d\vec{x}\\!\cdot\\!d\vec{x}\qquad\Longrightarrow\qquad
H(t)\equiv\frac{\dot{a}}{a}\quad,\quad\epsilon(t)\equiv-\frac{\dot{H}}{H^{2}}\;.$
(1)

One of these is the way the accelerated expansion ($\epsilon<1$) of primordial
inflation causes the energy-time uncertainty principle to predict the
production of cosmological wavelength particles which are both massless and
not conformally invariant [1]. This is the basis for the primordial spectra of
gravitons [2] and scalars [3].

Of course these particles must interact with themselves and with other quanta,
which can change kinematics, long range forces, and even the expansion rate.
Effects mediated by scalars are easier to compute, but model-dependent. Even
on the simplest background (de Sitter, with $\epsilon=0$) graviton effects
require heroic computations [4, 5, 6, 7] and suffer from potential gauge
dependence [8, 9], but they can do things which scalars cannot, and no one
doubts that they can be described by general relativity as a low energy
effective field [10, 11, 12, 13, 14].

An important feature of loop corrections from inflationary gravitons is that
they sometimes grow in time as more and more quanta are ripped from the
vacuum. For example, a single loop of gravitons on de Sitter background
changes the mode function of plane wave gravitons [15] and the Newtonian
potential [16] to,

$\displaystyle u(t,k)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
u_{0}(t,k)\Biggl{\\{}1+\frac{16GH^{2}}{3\pi}\ln^{2}(a)+O(G^{2})\Biggr{\\}}\;,\qquad$
(2) $\displaystyle\Psi(t,r)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{GM}{ar}\Biggl{\\{}1+\frac{103G}{15\pi
a^{2}r^{2}}-\frac{8GH^{2}}{\pi}\ln^{3}(a)+O(G^{2})\Biggr{\\}}\;.\qquad$ (3)

Similar results have been reported for 1-loop graviton corrections to fermions
[17, 18], to massless, minimally coupled scalars [19], and to electromagnetism
[20, 21].

It is fascinating to wonder what happens when the factors of $\ln[a(t)]$ grow
large enough to overwhelm the inflationary loop counting parameter of
$GH^{2}$. One would also like to generalize the background from de Sitter to
an arbitrary cosmological background (1) in order to search for effects which
persist long after inflation has ended. Nonlinear sigma models provide a good
theoretical framework for such studies because they possess the same
derivative interactions as quantum gravity, without the complicated index
structure and the gauge problem [22, 23, 24, 25]. A recent de Sitter
background study [26, 27] of this model,

$\mathcal{L}=-\frac{1}{2}\partial_{\mu}A\partial_{\nu}Ag^{\mu\nu}\sqrt{-g}-\frac{1}{2}\Bigl{(}1\\!+\\!\frac{1}{2}\lambda
A\Bigr{)}^{2}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;.$ (4)

showed that the large logarithms can be summed by combining a variant of
Starobinksy’s stochastic formalism [28, 29] with a variant of the
renormalization group. For example, the leading logarithms in the expectation
value of the field $A(x)$ sum to the form,

$\Bigl{\langle}\Omega\Bigl{|}A(x)\Bigr{|}\Omega\Bigr{\rangle}\longrightarrow\frac{2}{\lambda}\Biggl{[}\sqrt{1+\frac{\lambda^{2}H^{2}\ln(a)}{16\pi^{2}}}-1\Biggr{]}+\Bigl{(}{\rm
Stochastic\ acceleration}\Bigr{)}\;,$ (5)

where the final term speeds the field down its effective potential $V_{\rm
eff}(A)=-\frac{3H^{4}}{8\pi^{2}}\ln|1+\frac{1}{2}\lambda A|$.

The evolution of the scalar background (5) is entirely a stochastic effect.
One derives it by integrating out the differentiated $B$ fields in the
presence of a constant $A$ background,

$\displaystyle\frac{\delta S[A,B]}{\delta A}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{]}-\frac{\lambda}{2}\Bigl{(}1\\!+\\!\frac{\lambda}{2}A\Bigr{)}\partial_{\mu}B\partial_{\nu}Bg^{\mu\nu}\sqrt{-g}\;,\qquad$
(6) $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle\partial_{\mu}\Bigl{[}\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}A\Bigr{]}-\frac{\lambda\sqrt{-g}\,g^{\mu\nu}\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})|_{x^{\prime}=x}}{2(1+\frac{\lambda}{2}A)}=0\;.\qquad$
(7)

Because (7) is the equation of motion for a scalar potential model, one can
sum its leading logarithms using Starobinsky’s stochastic formalism [28, 29].
The stochastic random field $\mathcal{A}(t,\vec{x})$ of this formalism obeys
the Langevin equation,

$-3H(t)\Bigl{[}\dot{\mathcal{A}}(t,\vec{x})-\dot{\mathcal{A}}_{0}(t,\vec{x})\Bigr{]}=\frac{\frac{\lambda}{2}\mathcal{T}[a](t)}{1+\frac{\lambda}{2}\mathcal{A}(t,\vec{x})}\;,$
(8)

where the stochastic “jitter” $\mathcal{A}_{0}(t,\vec{x})$ is the infrared-
truncated, free field mode sum,

$\mathcal{A}_{0}(t,\vec{x})\equiv\int_{a_{i}H_{i}}^{aH}\\!\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\sqrt{\frac{H^{2}(t_{k})C(\epsilon(t_{k}))}{2k^{3}}}\Biggl{\\{}e^{-i\vec{k}\cdot\vec{x}}\alpha(\vec{k})+e^{i\vec{k}\cdot\vec{x}}\alpha^{\dagger}(\vec{k})\Biggr{\\}}\quad,\quad
a(t_{k})H(t_{k})\equiv k\;,$ (9)

where inflation begins at time $t_{i}$ and
$C(\epsilon)\equiv\frac{1}{\pi}\Gamma^{2}(\frac{1}{2}+\frac{1}{1-\epsilon})[2(1-\epsilon)]^{\frac{2}{1-\epsilon}}$.
Here $\mathcal{T}[a](t)$ stands for the trace of the coincident,
differentiated propagator which depends functionally on the expansion history
$a(t)$,

$\mathcal{T}[a](t)\equiv
g^{\mu\nu}\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}\;.$
(10)

Neglecting the stochastic jitter (which merely accelerates the rolling of
$\mathcal{A}(t,\vec{x})$ down its potential) gives a first order equation
which can be integrated (from $\mathcal{A}(t_{i},\vec{x})=0$) to give,

$\mathcal{A}(t,\vec{x})=\frac{2}{\lambda}\Biggl{[}\sqrt{1-\frac{\lambda^{2}}{6}\int_{t_{i}}^{t}\\!\\!dt^{\prime}\frac{\mathcal{T}[a](t^{\prime})}{H(t^{\prime})}}\\!-1\Biggr{]}+\Bigl{(}{\rm
Stochastic\ acceleration}\Bigr{)}\;.$ (11)

The de Sitter result (5) was obtained by using dimensional regularization on
de Sitter background to evaluate the trace of the coincident, differentiated
propagator [30, 31],

$\mathcal{T}\Bigl{[}{\rm de\
Sitter}\Bigr{]}(t)=-\frac{H^{D}\Gamma(D)}{(4\pi)^{\frac{D}{2}}\Gamma(\frac{D}{2})}\longrightarrow-\frac{3H^{4}}{8\pi^{2}}\qquad\Bigl{(}D=4\Bigr{)}\;.$
(12)

The purpose of this paper is to evaluate (11) using a recently developed
analytic approximation of $\mathcal{T}[a](t)$ for a general expansion history
which has undergone primordial inflation [32].111 Note also the related study
of scalar perturbations amplified during inflation to explain dark energy and
the Hubble tension [33, 34, 35, 36, 37, 38]. In section 2 we review these
results. In particular, we specify the renormalization condition, which was
not needed for the de Sitter result (12). Section 3 devises a plausible
expansion history and presents numerical results for (11) in this expansion
history. Of course this requires us to make explicit choices for the coupling
constant $\lambda$ and for the renormalization scale. Our conclusions comprise
section 4.

##  2 Generalizing from de Sitter

The purpose of this section is to review the analytic approximation developed
in [32] for the trace of the coincident, doubly differentiated scalar
propagator (10) on a general cosmological geometry (1) which has undergone
primordial inflation. We first give the primitive results, both during
inflation and afterwards. Then we implement renormalization.

###  2.1 General Primitive Results

Dolgov and Pelliccia have shown that $\mathcal{T}[a](t)$ can be inferred from
the coincidence limit of the scalar propagator $\mathcal{V}[a](t)\equiv
i\Delta(x;x)$ [39],

$\mathcal{T}[a](t)=-\frac{1}{2}\left(\frac{d}{dt}+(D-1)H\right)\dot{\mathcal{V}}[a](t)\
,$ (13)

where a dot indicates differentiation with respect to co-moving time $t$. One
obtains $\mathcal{V}[a](t)$ for a general expansion history (which has
undergone primordial inflation) by first expressing it as a dimensionally
regulated, spatial Fourier mode sum of the norm-square of the mode function
$\mathcal{\tilde{V}}(t,k)=u(t,k)u^{*}(t,k)$,

$\mathcal{V}(t)=\int\\!\\!\frac{d^{D-1}k}{(2\pi)^{D-1}}\,\mathcal{\tilde{V}}(t,k)=\frac{2}{\Gamma(\frac{D-1}{2})(4\pi)^{\frac{D-1}{2}}}\int_{a_{i}H_{i}}^{\infty}\\!\\!\\!\\!dk\
k^{D-2}\mathcal{\tilde{V}}(t,k)\ .$ (14)

We then employ appropriate analytic approximations for
$\mathcal{\tilde{V}}(t,k)$ depending upon the relation between the physical
wave number $k/a(t)$ and the Hubble parameter $H(t)$. During inflation
$a(t)H(t)$ grows and there are two such regions:

  * •

Ultraviolet — for $a(t)H(t)<k<\infty$; and

  * •

Infrared — for $a(t_{i})H(t_{i})<k<a(t)H(t)$.

Only the ultraviolet modes require dimensional regularization. When the
appropriate approximations are inserted in the mode sum the result during
inflation consists of a divergent part and a finite part [32],

$\displaystyle\mathcal{V}_{\mathrm{div}}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{1}{8}\left(\frac{D-2}{D-4}\right)\frac{[D-2\epsilon(t)]H^{D-2}(t)}{\Gamma\left(\frac{D-1}{2}\right)(4\pi)^{\frac{D-1}{2}}}\
,\qquad$ (15) $\displaystyle\mathcal{V}_{\mathrm{fin}}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{H^{2}(t)}{8\pi^{2}}+\frac{1}{4\pi^{2}}\int_{t_{i}}^{t}\\!\\!dt^{\prime}H^{3}\left(t^{\prime}\right)\left[1-\epsilon\left(t^{\prime}\right)\right]C\left(\epsilon\left(t^{\prime}\right)\right)\
.\qquad$ (16)

Note that the horizon crossing relation $k=a(t_{k})H(t_{k})$ has been used to
change variables from $\frac{dk}{k}$ to $[1-\epsilon(t)]H(t)dt$. Recall that
inflation begins at time $t_{i}$, and the function $C(\epsilon)$ was defined
after equation (9). Finally, note that the key derivative for (13) is,

$\dot{\mathcal{V}}_{\mathrm{fin}}=\frac{H^{3}}{4\pi^{2}}\Bigl{[}1-(1-\epsilon)[1-C(\epsilon)]\Bigr{]}\
.$ (17)

Inflation ends at co-moving time $t_{e}$, after which $a(t)H(t)$ falls and
some of the infrared modes experience 2nd horizon crossing. This partitions
the mode sum into three regions:

  * •

Ultraviolet — for $a(t_{e})H(t_{e})<k<\infty$;

  * •

Near Infrared — for $a(t)H(t)<k<a(t_{e})H(t_{e})$; and

  * •

Far Infrared — for $a(t_{i})H(t_{i})<k<a(t)H(t)$.

If a mode experiences 1st horizon crossing at time $t$, then $t_{2}(t)$
represents the time at which it experiences 2nd crossing. Similarly, if a mode
experiences 2nd crossing at time $t$ then $t_{1}(t)$ represents the time at
which it experiences 1st horizon crossing. Of course the ultraviolet
divergence does not change but the finite part after the end of inflation
becomes [32],

$\displaystyle\mathcal{V}_{\mathrm{fin}}=\frac{(2-\epsilon)H^{2}}{8\pi^{2}}\ln\left(\frac{k_{e}}{aH}\right)-\frac{H^{2}}{8\pi^{2}}\left(\frac{k_{e}}{aH}\right)^{2}$
(18)
$\displaystyle+\frac{1}{4\pi^{2}a^{2}(t)}\int_{t_{e}}^{t}\\!\\!\\!dt^{\prime}\left[\epsilon\left(t^{\prime}\right)-1\right]H\left(t^{\prime}\right)a^{2}\left(t^{\prime}\right)\times\cos^{2}\left[\int_{t_{2}\left(t_{k}\right)}^{t}\\!\\!\\!\\!dt^{\prime\prime}\frac{k}{a\left(t^{\prime\prime}\right)}\right]\times
H^{2}\left(t_{1}\right)C\left(\epsilon\left(t_{1}\right)\right)$
$\displaystyle+\frac{1}{4\pi^{2}}\int_{t}^{t_{2}\left(t_{i}\right)}\\!\\!\\!\\!dt^{\prime}\left[\epsilon\left(t^{\prime}\right)-1\right]H\left(t^{\prime}\right)\times
H^{2}\left(t_{1}\right)C\left(\epsilon\left(t_{1}\right)\right)\ .$

The factor
$\cos^{2}\left[\int_{t_{2}\left(t_{k}\right)}^{t}dt^{\prime\prime}\frac{k}{a\left(t^{\prime\prime}\right)}\right]$
oscillates wildly and averages to 1/2 inside the integral. The key derivative
for (13) after inflation is,

$\displaystyle\begin{aligned}
&\dot{\mathcal{V}}_{\mathrm{fin}}=-\frac{\left[2\epsilon(2-\epsilon)+\frac{1}{H}\dot{\epsilon}\right]H^{3}}{8\pi^{2}}\ln\left(\frac{k_{e}}{aH}\right)-\frac{(1-\epsilon)(2-\epsilon)H^{3}}{8\pi^{2}}+\frac{H^{3}}{4\pi^{2}}\left(\frac{k_{e}}{aH}\right)^{2}\\\
&-\frac{H(t)}{4\pi^{2}a^{2}(t)}\int_{t_{e}}^{t}\\!\\!dt^{\prime}\left[\epsilon\left(t^{\prime}\right)-1\right]H\left(t^{\prime}\right)a^{2}\left(t^{\prime}\right)\times
H^{2}\left(t_{1}\right)C\left(\epsilon\left(t_{1}\right)\right)\ .\
\end{aligned}$ (19)

###  2.2 Renormalization

The coincident field $A^{2}(x)$ is a composite operator and we must use
composite operator renormalization to remove the divergence (15) which
afflicts it [40, 41]. The only local, dimension two operator with which
$A^{2}$ can mix is the Ricci scalar $R=(D-1)[D-2\epsilon]H^{2}$. Hence the
appropriate counterterm is,

$\Delta\mathcal{V}[a](t)=\Bigl{(}\frac{D\\!-\\!2}{8}\Bigr{)}\frac{[D\\!-\\!2\epsilon(t)]H^{2}(t)}{\Gamma\left(\frac{D-1}{2}\right)(4\pi)^{\frac{D-1}{2}}}\Bigl{(}\frac{\mu^{D-4}}{D-4}\Bigr{)}\
,$ (20)

where $\mu$ is the scale of dimensional regularization. Adding (20) to (15)
and taking the unregulated limit gives a finite residual,

$\lim_{D\rightarrow 4}\Bigl{[}\mathcal{V}_{\rm
div}[a](t)-\Delta\mathcal{V}[a](t)\Bigr{]}=\frac{R(t)}{48\pi^{2}}\ln\Bigl{[}\frac{\mu}{H(t)}\Bigr{]}\equiv\mathcal{V}_{\rm
res}[a](t)\;.$ (21)

It is useful to make the running dimensionless with the parameterization
$\mu=e^{\alpha}\times H(t_{e})$,

$\mathcal{V}_{\rm
res}[a](t)=\frac{R(t)}{48\pi^{2}}\ln\Bigl{[}\frac{H(t_{e})}{H(t)}\Bigr{]}+\frac{\alpha
R(t)}{48\pi^{2}}\equiv\frac{R(t)}{48\pi^{2}}\ln\Bigl{[}\frac{H(t_{e})}{H(t)}\Bigr{]}+\mathcal{V}_{\rm
run}[a](t)\;.$ (22)

Combining the first term with $\mathcal{V}_{\rm fin}$ defines the renormalized
result,

$\mathcal{V}_{\rm ren}(t)\equiv\mathcal{V}_{\rm
fin}[a](t)+\frac{R(t)}{48\pi^{2}}\ln\Bigl{[}\frac{H(t_{e})}{H(t)}\Bigr{]}\;.$
(23)

##  3 A Plausible Expansion History

The purpose of this section is to develop an explicit expansion history so
that equation (11) can be numerically evolved. We begin by grafting a simple
model of scalar-driven inflation onto the $\Lambda$CDM model of post-
inflationary cosmology. Then a convenient dimensionless formulation is
introduced for the various independent and dependent variables. Finally,
explicit results are presented for the scalar background.

###  3.1 Attaching Inflation to a Hot Big Bang

Our cosmology begins with inflation supported by the potential $V(\varphi)$ of
a scalar inflaton $\varphi$,

$\ddot{\varphi}+3H\dot{\varphi}+V^{\prime}(\varphi)=0\qquad\Longrightarrow\qquad\left\\{\begin{matrix}H^{2}=\frac{8\pi
G}{3}[\frac{1}{2}\dot{\varphi}^{2}+V(\varphi)]\\\
\epsilon_{\varphi}=\frac{3\dot{\varphi}^{2}}{\dot{\varphi}^{2}+2V(\varphi)}\end{matrix}\right\\}\;.$
(24)

In order to exploit simple slow roll expressions for the purposes of
estimating parameters we chose the quadratic model,

$V(\varphi)=\frac{c^{2}\varphi^{2}}{16\pi G}\qquad,\qquad c=7.1\times
10^{-6}\;.$ (25)

With initial value $\varphi(t_{i})=15/\sqrt{8\pi G}$ one gets about 56.8
e-foldings of inflation. This model agrees with the observed scalar amplitude
and spectral index [42] but it badly violates the increasingly tight bounds on
the tensor-to-scalar ratio [43]. However, for the illustrative purposes of
this study that should not matter. Indeed, a more realistic plateau potential
[44] would increase the accuracy of our analytic approximation (16) [32].

Inflation ends when $\epsilon_{\varphi}(t_{e})=1$, after which
$\epsilon_{\varphi}$ oscillates between 0 and 3 with constant amplitude and
increasing frequency. We want to graft this expansion history onto that of the
$\Lambda$CDM model,

$\epsilon_{\Lambda}(t)=\frac{2\Omega_{r}(1\\!+\\!z)^{4}+\frac{3}{2}\Omega_{m}(1\\!+\\!z)^{3}}{\Omega_{r}(1\\!+\\!z)^{4}+\Omega_{m}(1\\!+\\!z)^{3}+\Omega_{\Lambda}}\qquad,\qquad
1\\!+\\!z\equiv\frac{a(t_{0})}{a(t)}\;.$ (26)

where $t_{0}$ is the current time. The $\Lambda$CDM parameters are [45],

$\Omega_{m}\approx 0.315\qquad,\qquad\Omega_{\Lambda}\approx
0.685\qquad,\qquad\Omega_{r}\approx\frac{\Omega_{m}}{3390}\;.$ (27)

Of course the $\Lambda$CDM model is not accurate at very early times when the
number of relativistic particles grows, but it will serve for our purpose of
providing a numerical framework to illustrate our analytic approximation (18).
Rather than devise an elaborate theory of reheating, we simply interpolate
$\epsilon_{\varphi}(t)$ into $\epsilon_{\Lambda}(t)$,

$\epsilon(t)\equiv\frac{1}{2}\Bigl{[}1-\tanh(n\\!-\\!n_{\rm
eq})\Bigr{]}\\!\times\\!\epsilon_{\varphi}(t)+\frac{1}{2}\Bigl{[}1+\tanh(n\\!-\\!n_{\rm
eq})\Bigr{]}\\!\times\\!\epsilon_{\Lambda}(t)\qquad,\qquad
n\equiv\ln\Bigl{[}\frac{a(t)}{a(t_{i})}\Bigr{]}\;,$ (28)

where $n_{\rm eq}=59.1$. Figure 1 shows the resulting first slow roll
parameter.

Figure 1: Evolution of the composite slow roll parameter (28) over cosmic
history.

###  3.2 Dimensionless Formulation

It is desirable to convert the evolution variable from the co-moving time $t$
to the dimensionless number of e-foldings since the beginning of inflation,
$n\equiv\ln[\frac{a(t)}{a(t_{i})}]$. The resulting change in calculus is,

$\frac{d}{dt}=H\frac{d}{dn}\qquad,\qquad\frac{d^{2}}{dt^{2}}=H^{2}\Bigl{[}\frac{d^{2}}{dn^{2}}-\epsilon\frac{d}{dn}\Bigr{]}\qquad,\qquad\int\\!\\!dt=\int\\!\\!\frac{dn}{H}\;.$
(29)

Hence the relation (13) between $\mathcal{T}[a](t)$ and $\mathcal{V}[a](t)$
becomes,

$\mathcal{T}_{\rm
ren}=-\frac{H^{2}}{2}\Bigl{[}\frac{d}{dn}+(3\\!-\\!\epsilon)\Bigr{]}\frac{d\mathcal{V}_{\rm
ren}}{dn}\;.$ (30)

And the key integral in expression (11) can be written as,

$\int_{t_{i}}^{t}\\!\\!dt^{\prime}\frac{\mathcal{T}_{\rm
ren}}{H(t^{\prime})}=-\frac{1}{2}\\!\\!\int_{0}^{n}\\!\\!dn^{\prime}\Bigl{[}\frac{d}{dn^{\prime}}+(3\\!-\\!\epsilon)\Bigr{]}\frac{d\mathcal{V}_{\rm
ren}}{dn^{\prime}}\equiv-\frac{\mathcal{I}(n)}{16\pi G}\;.$ (31)

###  3.3 Numerical Results

As Figure 2 shows, the running term (22) makes only a small contribution
during inflation, and virtually nothing afterwards.

Figure 2: Evolution of ($8\pi G$ times) the running contribution (22) to the
coincident propagator for different values of $\alpha$. Note that
$\mathcal{V}_{\rm run}$ vanishes exactly during radiation domination, and is
too small to show up afterwards.

Figure 3 shows the much larger contribution from the dimensionless integral
$\mathcal{I}(n)$ which was defined in expression (31). Note that it grows
during inflation, and then falls off afterwards. The expectation value of the
dimensionless field can be expressed in terms of $\mathcal{I}(n)$ as,

$\Bigl{\langle}\Omega\Bigl{|}\sqrt{8\pi
G}\,A(x)\Bigr{|}\Omega\Bigr{\rangle}=\frac{2\sqrt{8\pi
G}}{\lambda}\left[\sqrt{1\\!+\\!\frac{1}{12}\left(\frac{\lambda}{\sqrt{8\pi
G}}\right)^{2}\mathcal{I}(n)}\\!-\\!1\right]+\Bigl{(}{\rm Stochastic\
acceleration}\Bigr{)}\;.$ (32)

Figure 3: Evolution of $\mathcal{I}(n)$ from the beginning of primordial
inflation to today.

Of course the right hand side of (32) depends on our choice of the
dimensionless coupling constant $\lambda/\sqrt{8\pi G}$. Figure 4 shows the
analytic part for a convenient choice. We have ignored the “stochastic
acceleration” term which increases the growth during inflation, and disappears
afterwards. Even at the present day one can see that the dimensionless field
retains more than half the value it built up during inflation. Had we employed
a model with a larger period of primordial inflation this effect would have
been correspondingly greater.

Figure 4: Evolution of the expectation value of $\sqrt{8\pi G}\,A$ for
$\lambda=\frac{1}{10}\times\sqrt{8\pi G}$.

One can see from Figure 3 that $\mathcal{I}(n)$ is never larger than about
$4\times 10^{-9}$. This is because the inflationary model of section 2
persists for only the minimum of 50-60 e-foldings required to explain the
Horizon Problem. One can envisage much longer periods of inflation; extending
the number of e-foldings by a factor increases the peak value of
$\mathcal{I}(n)$ by roughly that same factor. Because the dimensionless
coupling $\lambda/\sqrt{8\pi G}=\frac{1}{10}$ chosen for Figure 4 is of order
one, the square root in equation (32) is near unity and perturbation theory
never breaks down,

${\rm
Perturbative}\qquad\Longrightarrow\qquad\Bigl{\langle}\Omega\Bigl{|}\sqrt{8\pi
G}\,A(x)\Bigr{|}\Omega\Bigr{\rangle}\simeq\frac{\lambda\mathcal{I}(n)}{12\sqrt{8\pi
G}}\;.$ (33)

However, expression (32) is nonpertutbative, and we can make perturbation
theory break down, either by increasing the duration of primordial inflation
or else by increasing the dimensionless coupling. The result in that case
becomes,

${\rm
Nonperturbative}\qquad\Longrightarrow\qquad\Bigl{\langle}\Omega\Bigl{|}\sqrt{8\pi
G}\,A(x)\Bigr{|}\Omega\Bigr{\rangle}\simeq\sqrt{\frac{\mathcal{I}(n)}{3}}\;.$
(34)

Figure 5 illustrates this regime for $\lambda=10^{6}\times\sqrt{8\pi G}$.

Figure 5: Evolution of the expectation value of $\sqrt{8\pi G}\,A$ for
$\lambda=10^{6}\times\sqrt{8\pi G}$.

The alert reader will note from Figures 3 through 5 that the decrease due to
modes experiencing 2nd horizon crossing ceases at the onset of the current
phase of cosmic acceleration. Indeed, one might expect to see the decline
reversed because modes again begin exiting the horizon. However, it must be
recalled that the strength of the amplitude scales like the square of the
Hubble parameter at 1st crossing, as per equation (62) in Ref. [32]. Because
the current Hubble parameter is some 55 orders of magnitude smaller than its
value during inflation, the predicted late time growth would not be visible on
our plots, even if they were extended beyond the present time.

##  4 Conclusions

We have analyzed the evolution of the expectation value of the scalar field
$A$ in a non-linear sigma model (4) on a general cosmological background (1)
that experiences primordial inflation. We did this by solving the Starobinsky
Langevin equation (7) for this theory [26] which involves $\mathcal{T}[a](t)$,
the trace of the doubly differentiated, coincident propagator (10). Expression
(13) gives $\mathcal{T}[a](t)$ in terms of the coincident propagator
$\mathcal{V}[a](t)$. A good analytic approximation [32] for the dimensionally
regulated, primitive form of $\mathcal{V}[a](t)$ was reviewed in section 2.1,
and subjected to composite operator renormalization in section 2.2. The final
result takes the form of a small “running” term proportional to the local
Ricci scalar, plus a much larger contribution (23), with (16) and (17)
pertaining during inflation and (18) and (19) afterwards.

Although our analytic approximations should apply for any cosmology which
which has experienced primordial inflation, we devised an plausible expansion
history in section 3.1 to provide explicit numerical results. Figure 1 shows
the first slow roll parameter of this cosmology, from which the Hubble
parameter and the scale factor can be constructed by integration. The small
running contribution (22) to $\mathcal{V}[a](t)$ is shown in Figure 2. Figure
3 gives the key integral (31) which determines the expectation value of the
field shown in Figures 4 and 5.

A major goal of this study was to quantify the extent to which the high scales
of primordial inflation are transmitted to late times. In this regard it is
worth noting that the expectation value of $A$ would vanish if the Universe
was eternally radiation dominated (that is, $\epsilon(t)=2$), or if it was
flat. The fact that we get nonzero results, even during radiation domination,
is due to the initial phase of primordial inflation. Note also from Figures 4
and 5 that, even at very late times, the field retains a substantial amount of
the amplitude it built up during primordial inflation. Note finally that an
arbitrarily large amplitude could be built up by extending the duration of
primordial inflation, whereas there can be no extension of the post-
inflationary diminution.

This study demonstrates that it is not only possible to sum the series of
leading logarithms of nonlinear sigma models on de Sitter, but also on an
arbitrary cosmological background (1) which has undergone primordial
inflation. And we have seen two crucial things:

  * •

That evolution never stops; and

  * •

That an arbitrarily large effect from primordial inflation can be transmitted
to late times.

The next step is extending this formalism to quantum gravity in order to
discover what becomes of graviton loop corrections such as (2-3) at late times
and after perturbation theory has broken down.

Acknowledgements

This work was partially supported by NSF grant PHY-2207514 and by the
Institute for Fundamental Theory at the University of Florida.

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