# Synchrotron Spectral Curvature from 22 MHz to 23 GHz

A. Kogut11affiliation: Code 665, Goddard Space Flight Center, Greenbelt, MD
20771<EMAIL_ADDRESS>
###### Abstract

We combine surveys of the radio sky at frequencies 22 MHz to 1.4 GHz with data
from the ARCADE-2 instrument at frequencies 3 to 10 GHz to characterize the
frequency spectrum of diffuse synchrotron emission in the Galaxy. The radio
spectrum steepens with frequency from 22 MHz to 10 GHz. The projected spectral
index at 23 GHz derived from the low-frequency data agrees well with
independent measurements using only data at frequencies 23 GHz and above.
Comparing the spectral index at 23 GHz to the value from previously published
analyses allows extension of the model to higher frequencies. The combined
data are consistent with a power-law index $\beta=-2.64\pm 0.03$ at 0.31 GHz,
steepening by an amount $\Delta\beta=0.07$ every octave in frequency.
Comparison of the radio data to models including the cosmic ray energy
spectrum suggests that any break in the synchrotron spectrum must occur at
frequencies above 23 GHz.

###### Subject headings:

radio continuum: general, radiation mechanisms: non-thermal

††slugcomment: Accepted by The Astrophysical Journal

##  1\. Introduction

Synchrotron emission from relativistic cosmic ray electrons accelerated in the
Galactic magnetic field dominates the diffuse radio continuum at frequencies
below 1 GHz. It is an important foreground contaminant for measurements of the
cosmic microwave background radiation, and also serves to probe the Galactic
magnetic field and cosmic ray distributions. Measurements of the synchrotron
frequency spectrum are thus of interest to several areas in astrophysics.

An isotropic distribution of relativistic electrons at a single energy
$E=\gamma mc^{2}$ propagating in a uniform magnetic field $B$ has emissivity

$\epsilon(\nu)=\frac{\sqrt{3}e^{3}}{mc^{2}}B\sin\alpha F(x)~{},$ (1)

where $\alpha$ is the pitch angle between the magnetic field and the line of
sight, and

$F(x)=x\int^{\infty}_{x}K_{5/3}(x^{\prime})dx^{\prime}$ (2)

is defined in terms of the modified Bessel function of order $5/3$ with
variable $x=\nu/\nu_{c}$ and

$\nu_{c}=\frac{3}{4\pi}\frac{e}{mc}\gamma^{2}B\sin\alpha$ (3)

(Schwinger, 1949; Westfold et al., 1959; Oster, 1961). For a power-law
distribution of electron energies $N(E)~{}\propto~{}E^{p}$ propagating in a
uniform magnetic field, the synchrotron emission is also a power law,

$T_{A}(\nu)\propto\nu^{\beta}$ (4)

where $T_{A}$ is antenna temperature, $\nu$ is the radiation frequency, and

$\beta=\frac{p-3}{2}$ (5)

(Rybicki & Lightman, 1979).

Measurements of the synchrotron spectral index provide important input for
models of cosmic ray propagation. Solar modulation reduces the local cosmic
ray electron density for electron energies below a few GeV so that synchrotron
emission provides the most direct probe of low-energy cosmic rays.
Measurements of the cosmic ray spectrum above a few GeV in turn inform models
of the high-frequency synchrotron spectrum. Energy losses from cosmic ray
propagation steepen the cosmic ray spectrum, increasing $p$ toward higher
energies. The observed steepening from $p\sim-2.6$ at 5 GeV to $p\sim-3.2$ at
50 GeV predicts a corresponding steepening in the synchrotron spectrum from
$\beta\sim-2.8$ at 1 GHz to $\beta\sim-3.1$ at 100 GHz (Strong, Moskalenko, &
Ptuskin, 2007).

Comparison of the cosmic ray spectra to the predicted synchrotron spectrum is
complicated by confusion from competing radio emission sources. The diffuse
radio continuum is a superposition of the cosmic microwave background,
synchrotron emission, free-free emission from the warm ionized interstellar
medium, and emission from interstellar dust. A number of authors have
attempted to disentangle the various emission sources to determine the
synchrotron spectral index (for a recent review see Appendix A of Strong,
Orlando, & Jaffe (2011)). Despite some discrepant results, the general trend
shows a steepening of the synchrotron spectrum from $\beta\sim-2.5$ at 22 MHz
to $\beta\sim-3.0$ above 23 GHz, in rough agreement with the observed cosmic
ray spectra.

Several factors contribute to the observed scatter in estimates of the
synchrotron spectral index. Most estimates, particularly those below 23 GHz,
assume a power-law spectrum for synchrotron and do not explicitly model
spectral steepening. Comparisons between closely-separated frequencies more
accurately reflect the local synchrotron spectrum, but have larger
uncertainties from competing emission sources or measurement offsets. Analyses
with broader frequency coverage reduce foreground and offset uncertainties but
average over any spectral steepening.

Two additional effects are important for analyses including data from the
Wilkinson Microwave Anisotropy Probe (WMAP) at frequencies 23 to 94 GHz. A
growing body of evidence suggests that a substantial fraction of the diffuse
continuum near 23 GHz consists of electric dipole radiation from a population
of small, rapidly spinning dust grains (Kogut et al., 1996; de Oliveira-Costa
et al., 1997, 2004; Miville-Deschênes et al., 2008; Dobler & Finkbeiner, 2008;
Ysard, Miville-Deschênes, & Verstraete, 2010; Kogut et al., 2011; Gold et al.,
2011; Planck collaboration, 2011). Spinning dust emission is expected to peak
at frequencies near 20 GHz (Draine & Lazarian, 1998; Ali-Haïmoud et al., 2009;
Hoang, Draine, & Lazarian, 2010; Ysard & Verstraete, 2010). Analyses that
ignore this component to attribute the observed emission only to synchrotron
radiation tend to over-predict the synchrotron amplitude at frequencies near
23 GHz, biasing the derived spectral index to flatter values when comparing to
lower frequencies and steeper values when comparing to higher frequencies.

A second systematic error can result from improper treatment of offsets in the
data. Measurements from radio surveys at frequencies below 20 GHz generally
include the absolute intensity (zero level) of the sky. The WMAP differential
radiometers are insensitive to any constant (monopole) intensity on the sky;
the zero level of the WMAP sky maps is set so that the map intensity in the
Galactic polar caps matches a cosecant fit to the mid-latitude sky (Bennett et
al., 2003; Hinshaw et al., 2009). Analyses that directly compare low-frequency
radio surveys to the WMAP data without subtracting a monopole component from
the radio data will miss the corresponding emission in the WMAP bands, biasing
the derived spectral index to steeper values.

Figure 1.— Toy model showing effect of improper zero level subtraction on the
derived spectral index between 408 MHz and 23 GHz.

The zero level bias can be significant. The Haslam et al. (1981) survey at 408
MHz is commonly used to model synchrotron emission. The North Galactic pole
has measured temperature $19\pm 3$ K at 408 MHz, while a $\csc|b|$ fit to the
same 408 MHz map predicts a polar contribution of only $5.1\pm 0.6$ K. Similar
results apply to the Southern hemisphere, where the measured polar cap
temperature of $21\pm 3$ K significantly exceeds the value $4.0\pm 0.5$ K
obtained from a $\csc|b|$ fit. Only 2.7 K of the difference can be attributed
to emission from the cosmic microwave background, leaving a large residual.

Figure 1 illustrates the bias induced by including this residual at 408 MHz
but excluding it from the WMAP data. We take the 408 MHz map, remove the 2.7 K
CMB monopole, and scale the remaining (radio) emission to 23 GHz using a
power-law index $\beta=-2.7$ to produce a toy model of the radio sky at 23
GHz. Following the WMAP processing, we then remove a monopole from the scaled
map so that the map temperature in the south polar cap matches the $\csc|b|$
fit. We then compute the bias in apparent spectral index by comparing the 408
MHz map to the scaled 23 GHz map before and after removing the scaled
monopole. Figure 1 shows the bias in spectral index at 23 GHz, binned by
Galactic latitude. Dis-similar treatment of the map zero level creates a
spatially varying bias $\Delta\beta\approx 0.15$, comparable to the total
spectral steepening predicted by the measured cosmic ray spectra. The bias is
largest in regions where the sky brightness is faintest, at high latitudes or
away from the Galactic center.

Table 1Sky Surveys Used for Synchrotron Analysis Frequency | Calibration | Offset | Relative
---|---|---|---
(GHz) | Uncertainty | Uncertainty (K) | Uncertaintya
0.022 | 0.05 | 5000 | 0.15
0.045 | 0.10 | 250 | 0.11
0.408 | 0.10 | 3.0 | 0.17
1.420 | 0.05 | 0.5 | 0.63
3.20 | 0.001 | 0.011 | 0.10
3.41 | 0.001 | 0.006 | 0.07
7.98 | 0.001 | 0.036 | 0.89
8.33 | 0.001 | 0.042 | 2.64
9.72 | 0.001 | 0.003 | 0.34
10.49 | 0.001 | 0.002 | 0.27
aQuadrature sum of calibration and offset uncertainties,
divided by the mid-latitude sky temperature.

Measurement uncertainties in the absolute level of the sky brightness can also
introduce bias in estimates of the synchrotron spectral index. Many of the
low-frequency surveys have uncertainty in the measured zero level approaching
30% of the polar cap brightness. As with the toy model above, such measurement
errors introduce spatially dependent biases that are largest where the sky
brightness is faintest. Minimizing uncertainty in the derived synchrotron
spectrum requires a combination of measurements with good sky coverage and
good control of offset uncertainty at frequencies where competing emission
sources are faint. No such ideal data set yet exists. In this paper, we model
the synchrotron spectral index and curvature using low-frequency radio surveys
with high sky coverage but large offset uncertainty, combined with higher-
frequency measurements with limited sky coverage but still useful offset
uncertainty.

##  2\. Sky Maps

We model synchrotron emission using radio surveys at 22 MHz (Roger et al.,
1999), 45 MHz (Maeda et al., 1999; Alvarez et al., 1997), 408 MHz (Haslam et
al., 1981), and 1420 MHz (Reich, Testori, & Reich, 2001; Reich & Reich, 1986).
These surveys have full or nearly-full sky coverage at frequencies where
Galactic radio emission is significant, with gain and zero-level systematics
controlled at the 10–20% level. We supplement the radio surveys with sky maps
from the Absolute Radiometer for Cosmology, Astrophysics, and Diffuse Emission
(ARCADE 2) instrument111 The ARCADE data are available at the Legacy Archive
for Microwave Background Data Analysis, http://lambda.gsfc.nasa.gov at 3, 8,
and 10 GHz (Kogut et al., 2011). The ARCADE 2 data observe both the Galactic
plane and mid-latitude regions ($|b|<40\arcdeg$) with sufficient control of
zero-level uncertainty to constrain the synchrotron curvature relative to the
lower-frequency radio surveys.

Table 1 summarizes the input sky maps. The increase in the offset uncertainty
at low frequency is compensated by a corresponding increase in sky brightness.
The final column shows the relative measurement uncertainty for a mid-latitude
region, defined as the ratio of the combined offset and calibration
uncertainty to the measured brightness at $(l,b)=(17\arcdeg,-35\arcdeg)$ after
removing the CMB monopole. The selected maps provide roughly uniform relative
sensitivity to synchrotron emission over 2.5 decades of frequency.

We convert all maps to units of antenna temperature and subtract the CMB
monopole at (thermodynamic) temperature 2.725 K from the measured sky
temperatures. We then convolve each map to the 11$\fdg$6 angular resolution of
the ARCADE 2 instrument. At frequencies of 10 GHz and below, both thermal dust
emission and spinning dust emission are negligible. Free-free emission,
however, can still be appreciable. We correct the convolved maps by scaling
the WMAP 7-year maximum entropy free-free model (Gold et al., 2011) to each
frequency using spectral index $-2.15$, convolving the scaled model to
11$\fdg$6 angular resolution, and subtracting the convolved model from each
sky survey. The resulting maps are dominated by synchrotron emission.

Figure 2.— Sky coverage for this analysis. The plot shows the 408 MHz sky
survey convolved to 11$\fdg$6 angular resolution. Pixels common to all 10
radio surveys are shown in color. The sky coverage is limited by the ARCADE 2
observations but includes the Galactic plane, mid-latitude sky, and portions
of the North Galactic Spur (radio Loop I).

##  3\. Analysis

The input sky maps define a data set $T(\hat{n},\nu)$ sampled at discrete
pixel directions $\hat{n}$ and 10 discrete frequencies $\nu$ ranging from 22
MHz to 10 GHz. We restrict the analysis to the 8% of the sky observed at all
10 frequencies. Figure 2 shows the resulting sky coverage. Within the common
sky coverage, we model synchrotron emission as a modified power law

$T(\hat{n},\nu)=A(\hat{n})\left(\frac{\nu}{\nu_{0}}\right)^{\beta+C\ln(\nu/\nu_{0})}$
(6)

with spectral index $\beta$ and curvature $C$ defined with respect to
reference frequency $\nu_{0}$ = 310 MHz. The adopted value for $\nu_{0}$
minimizes covariance between the fitted amplitude $A$ and spectral index
$\beta$, simplifying extrapolation to other frequencies.

For each pixel $\hat{n}$ we define a 10 by 10 data covariance matrix $M$ with
diagonal elements determined by the instrument noise, calibration and offset
uncertainty. The input maps are not all linearly independent. Measurements at
22 MHz used the 408 MHz map to determine the declination dependence of the
gain. We estimate the resulting correlation of spatial structure in the two
maps at 50%. The ARCADE 2 maps have independent instrument noise but share a
fraction of the offset uncertainty related to absolute thermometry uncertainty
and ground glint (Singal et al., 2011). All 10 maps share a common model for
free-free emission. We conservatively estimate the uncertainty in the free-
free correction at 30% of the free-free amplitude; however, the results do not
change significantly as the model free-free amplitude is varied by as much as
50%. Off-diagonal elements in $M$ include these effects.

Figure 3.— Sky temperatures and best-fit model (solid line) for a 4$\arcdeg$
diameter patch on the Galactic plane centered on the the brightest pixels in
the ARCADE 2 sky coverage. For clarity, the data are plotted relative to the
best-fit power-law model ($\beta=-2.56$) with zero curvature (dashed line).
All fits include the non-trivial covariance between individual data points.
The data are well described by a model with spectral index $\beta=-2.60\pm
0.04$ and curvature $C=-0.081\pm 0.028$.  Figure 4.— Sky maps of best-fit
spectral parameters. Left to right: synchrotron amplitude, spectral index, and
curvature evaluated at reference frequency 310 MHz. The top panels show
results from the 10-frequency fit, while the bottom panels include the
spectral constraint at 23 GHz.

For each pixel, a least-squares minimization determines the best-fit
parameters $A$, $\beta$, and $C$. Figure 3 shows the measured temperature and
best-fit model for the brightest Galactic plane region
$(l,b)=(52\arcdeg,0\arcdeg)$ within the common sky coverage. The data show
evidence for spectral curvature, with best-fit values $\beta=-2.60\pm 0.04$
and $C=-0.081\pm 0.028$ evaluated at $\nu_{0}=310$ MHz. The spectral curvature
$C$ for this region is significant at approximately 3 standard deviations
compared to the baseline model with $C=0$.

We may extrapolate the spectral models to compare the results at frequencies
10 GHz and below to independent determinations of the spectral index using
WMAP data at frequencies 23 GHz and above. We compute the antenna temperature
of the modeled spectra to derive the effective power-law index for frequencies
near 23 GHz. Note that this is not equivalent to evaluating Eq. 6 at $\nu=23$
GHz, which would yield the scaling from 310 MHz to 23 GHz but not the power-
law index at 23 GHz. The mean for all 258 pixels in the common sky coverage is
$\beta_{23}=-3.02$ with standard deviation $0.22$.

The extrapolated value compares well with independent determinations of the
spectral index above 23 GHz. Kogut et al. (2004) analyze WMAP polarization
data to derive synchrotron spectral index $\beta=-3.2\pm 0.1$ averaged over
the full sky. Dunkley et al. (2009) use a Bayesian analysis of polarization
data and find the mean spectral index $\beta=-3.03\pm 0.04$ with pixel-to-
pixel standard deviation $0.25$ over the high-latitude sky. Gold et al. (2011)
use template fitting techniques to derive spectral index $\beta=-3.13$ between
23 and 33 GHz.

The mean spectral index derived from the 10 low-frequency radio surveys agrees
with the value derived from WMAP data at higher frequencies. Much of the
scatter in the extrapolated spectral indices results from pixels at high
latitude where the emission is faintest. The extrapolated index in these
pixels can reach unphysical values. We reduce the scatter in the fitted
spectral parameters by applying additional constraints using WMAP data at 23
GHz and above. The simplest such constraint, adding WMAP temperature data to
the multi-frequency fit, is problematic. Not only would the procedure need to
include additional free parameters to account for emission from thermal dust
or spinning dust (both negligible at lower frequencies), but each of the low-
frequency maps would require a correction to remove the monopole contribution
missing from the WMAP data. Although the WMAP zero level is clearly defined by
a $\csc|b|$ fit to mid-latitude data, the astrophysical interpretation of a
similar procedure applied to low-frequency radio surveys is less clear. The
coldest region of the radio sky is not at the Galactic poles, but at mid-
latitudes above the Galactic anti-center. Subtraction of too large a monopole
can leave unphysical negative residuals. Limited sky coverage exacerbates this
problem.

Figure 5.— Spectral index evaluated at 310 MHz. The dashed line shows the
distribution from the 10-frequency radio data while the solid line includes
the prior at 23 GHz.

We avoid these problems by using a constraint based on the spectral index
derived solely from WMAP data. For each pixel, we use the radio data (Table 1)
to fit the synchrotron amplitude $A(\hat{n})$ over a 2-dimensional grid in the
spectral parameters $\beta$ and $C$ (Eq. 6). At each grid point, we compute
the $\chi^{2}$ value $R^{T}M^{-1}R$ where $M^{-1}$ is the inverse covariance
matrix and $R$ is the difference vector between the measured and modeled
temperatures. We then use the spectral parameters $\beta$ and $C$ to evaluate
the power-law index at 23 GHz and compare the resulting value to a prior. We
use the difference between the extrapolated spectral index and the prior to
augment the $\chi^{2}$ at each grid point,

$\chi^{2}\rightarrow\chi^{2}+\left(\frac{\beta_{23}-\beta_{p}}{\sigma_{p}}\right)^{2},$
(7)

where $\beta_{23}$ is the model spectral index evaluated at 23 GHz, and
$\beta_{p}\pm\sigma_{p}~{}=~{}-3.1\pm 0.1$ is the prior at 23 GHz. The minimum
$\chi^{2}$ over the entire grid then defines the best-fit model at that pixel.
This allows inclusion of the spectral information derived from frequencies
above 23 GHz without confusion from either additional emission components
(thermal or spinning dust) above 23 GHz or the missing zero level in the WMAP
data.

Figure 6.— Spectral curvature at 310 MHz. The dashed line shows the
distribution from the 10-frequency radio data while the solid line includes
the prior at 23 GHz. The mean curvature $C=-0.052$ corresponds to a steepening
of the local spectral index by an amount $\Delta\beta=0.07$ every octave in
frequency.

Figure 4 shows the best-fit spectral parameter maps, while Figures 5 and 6
show the distribution of the best-fit values for the spectral index and
curvature. Including the constraint at 23 GHz, the best-fit spectral index has
mean value $-2.64$ and standard deviation $0.03$ at reference frequency
$\nu_{0}=310$ MHz. The best-fit curvature has mean value $-0.052$ with
standard deviation $0.005$. The corresponding spectral index at 23 GHz is
$\langle\beta_{23}\rangle=-3.09$ with standard deviation 0.05. Inclusion of
the prior at 23 GHz does not induce a significant shift in the mean for the
extrapolated spectral index, but does significantly reduce the pixel-to-pixel
scatter.

Comparison of the parameter distributions with and without the spectral
constraint at 23 GHz demonstrates that the addition of the spectral constraint
mainly affects the fitted curvature values (Figs. 5 and 6). The adopted value
for the spectral constraint comes from independent analysis of WMAP data and
is weighted toward regions of higher synchrotron intensity. We test whether
this creates a bias in the fitted curvature values by splitting the observed
sky coverage into two subsets of equal area, defined by the brightest and
faintest 50% of the fitted amplitudes at 310 MHz. Within each subset, we
compare the mean and standard deviation of the fitted curvature derived from
the 10-frequency fit or the enhanced fit including the constraint at 23 GHz.
The “bright” subset shows no shift in the mean spectral curvature when the 23
GHz constraint is added (although the scatter is significantly reduced). The
‘faint” subset shows both a reduction in scatter and a modest shift in the
mean value, with curvature parameter steepening from -0.034 to -0.049 when the
23 GHz constraint is included. This shift is less than one standard deviation:
given the limited sky coverage, the available radio data do not yet provide
significant evidence for spatial variation in the synchrotron curvature.

##  4\. Discussion

Radio data show statistically significant steepening of the synchrotron
spectrum from 22 MHz to 10 GHz. The nearly uniform relative uncertainty of the
selected data minimizes dependence of the fitted parameters on offset or
calibration errors at any one frequency. We test whether the best-fit
parameters are particularly sensitive to any one input map by repeating the
analysis after dropping one or two maps from the fit. We select either one
radio survey (22 MHz, 45 MHz, 408 MHz, or 1420 MHz) or a pair of ARCADE
frequency channels (3 GHz, 8 GHz, or 10 GHz) and repeat the fit after deleting
the corresponding elements from the data vector $T$ and covariance matrix $M$.
The resulting shift in either the spectral index or curvature parameters is
smaller than the pixel-to-pixel standard deviation using all 10 frequency
channels. Systematic errors in the offset or temperature calibration do not
appear to dominate the multi-frequency analysis.

The steepening of the synchrotron spectrum is broadly consistent with models
of cosmic ray propagation in the Galactic magnetic field. Jaffe et al. (2011)
combine radio observations with the galprop222http://galprop.stanford.edu
cosmic ray propagation code to model synchrotron emission on the Galactic
plane. They find a power-law index $-2.8<\beta<-2.74$ from 408 MHz to 2.3 GHz
and $-2.98<\beta<-2.91$ from 2.3 GHz to 23 GHz. The corresponding values for
the spectral steepening model presented here are $\beta=-2.76$ from 408 MHz to
2.3 GHz and $\beta=-2.97$ from 2.3 GHz to 23 GHz, in good agreement with the
cosmic ray model.

We may use the best-fit values in each pixel to predict the synchrotron
spectrum at higher frequencies where the emission is fainter and competing
sources stronger. Previous attempts to disentangle competing emission from
free-free, synchrotron, thermal dust, and spinning dust emission using WMAP
data have suffered from degeneracy between the synchrotron and spinning dust
emission, both of which are falling at frequencies above 33 GHz (see, e.g.,
the discussion in Gold et al. (2011)). Extending the synchrotron curvature
observed at lower frequencies into the millimeter band reduces confusion
between the spinning dust and synchrotron spectra and may facilitate
characterization of both the spatial distribution and frequency spectrum of
spinning dust emission in the interstellar medium.

Table 2Local Power-Law Spectral Index Frequency | Power-Law
---|---
(GHz) | Index
0.022 | -2.36
0.045 | -2.44
0.408 | -2.67
3.3 | -2.89
23 | -3.09
33 | -3.13
41 | -3.15
61 | -3.19
94 | -3.24

Table 2 shows the local power-law index $T\propto\nu^{\beta}$ at selected
frequency bands. The modeled spectrum steepens by $\Delta\beta=0.07$ every
octave in frequency, from $\beta=-2.67$ at 408 MHz to $\beta=-3.24$ at 94 GHz.
Note, however, that the spectral steepening observed at low frequencies can
not continue indefinitely. _Fermi_ measurements of the cosmic ray energy
spectrum are consistent with a single power law from energy 7 GeV to 1 TeV
(Abdo et al., 2009; Ackermann et al., 2010, 2012). If anything, the _Fermi_
data suggest a modest flattening of the cosmic ray energy spectrum at higher
energies, which would induce a positive curvature to the synchrotron spectrum
at frequencies above 23 GHz.

Figure 7.— Comparison of the synchrotron spectral index as a function of
frequency for different models. The break in the spectral index for the
Strong, Orlando, & Jaffe (2011) cosmic ray model at frequencies of a few GHz
is not reproduced by the ARCADE 2 observations at 3–10 GHz, which prefer
models with nearly constant spectral curvature.

Strong, Orlando, & Jaffe (2011) combine radio data at 22, 45, 150, 408, and
1420 MHz with WMAP data at 23 through 94 GHz and Fermi Large Area Telescope
cosmic ray measurements and the galprop code to estimate the magnetic field
intensity and synchrotron spectrum from 22 MHz to 94 GHz. Figure 7 compares
the resulting synchrotron spectral index (diffusion model with injection index
1.3) to the curvature model from this paper. The cosmic ray model analyzes a
limited latitude range $10\arcdeg<|b|<50\arcdeg$, which we follow in Figure 7
by excluding pixels at latitudes $|b|<10\arcdeg$. The two methods agree for
frequencies below 408 MHz (where they share common radio data) but differ at
higher frequencies. The cosmic ray model shows an increased synchrotron
curvature from 408 MHz to a few GHz, followed by a spectral break to near-
constant index $\beta\sim-3.1$ at higher frequencies. The ARCADE 2 data at
3–10 GHz do not reproduce these features, but are instead consistent with
constant spectral curvature from 22 MHz to 10 GHz (Fig 3).

Both models agree at frequencies near 23 GHz. The spectral index at 23 GHz
derived from the 10-frequency radio data without the 23 GHz prior is
consistent with independent measurements of the index above 23 GHz and with
the full radio fit including the 23 GHz prior. The radio data, taken alone, do
not support a break in the synchrotron spectrum at GHz frequencies. Comparison
of the radio fit to the cosmic ray model suggests that any spectral break in
the synchrotron spectrum must occur at frequencies above 23 GHz. Direct
confirmation of the synchrotron spectrum above 23 GHz remains a challenge.

##  5\. Conclusions

Radio data are consistent with a synchrotron spectrum that steepens with
frequency from 22 MHz to 10 GHz. Direct comparison of low-frequency radio
surveys with the WMAP data at 23 to 94 GHz is complicated both by the presence
of additional emission components at higher frequencies and by the subtraction
of a substantial monopole component of sky emission by the differential WMAP
instrument. The synchrotron spectral index at 23 GHz, derived using only
lower-frequency radio surveys, is consistent with the value derived
independently using only data at higher frequencies. We extend the radio data
by comparing the extrapolated index at 23 GHz to a prior based on higher-
frequency data. The combined data have mean spectral index $\beta=-2.64\pm
0.03$ and curvature $C=-0.052\pm 0.005$ at reference frequency 0.31 GHz. The
measured spectrum steepens by an amount $\Delta\beta=0.07$ every octave in
frequency. Comparison of the radio data to models including the cosmic ray
energy spectrum suggests that any break in the synchrotron spectrum must occur
at frequencies above 23 GHz.

This research is based upon work supported by the National Aeronautics and
Space Administration through the Science Mission Directorate under the
Astronomy and Physics Research and Analysis suborbital program.

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# Countable modular groups of infinite type surfaces

Rogelio Niño Hernández

###### Abstract

We prove that every infinite type surface $S$ without boundary and finite
genus has a Riemann surface structure such that its modular group of
quasiconformal homeomorphisms is countable.

##  1 Introduction

Consider a surface without boundary, denoted as $S$, and its topological
mapping class group, defined as the quotient of the group of orientation-
preserving homeomorphisms of $S$ by the subgroup of homeomorphisms isotopic to
the identity ($Map(S)=Homeo^{+}(S)/Homeo_{0}(S)$). Consider also a Riemann
surface structure $R$ on $S$ and the subgroup $Mod(R)$ of $Map(S)$ for which
there exists a quasiconformal representative with respect to $R$.

In the case where the surface $S$ is of finite topological type (meaning its
fundamental group is finitely generated), both $Map(S)$ and $Mod(R)$ are
always isomorphic and countable. However, when the surface $S$ is of infinite
topological type (with a non-finitely generated fundamental group), the
situation changes. In this context, the mapping class group $Map(S)$ is always
uncountable. Furthermore, certain homeomorphisms cannot be realized as
elements within $Mod(R)$, regardless of the chosen Riemann surface structure
$R$.

In his work [Mat05], Matsuzaki shows that for the surface
$S=\mathbb{S}^{2}\setminus K$, where $K$ represents the Cantor set, there
exists a Riemann surface structure $R$, wherein the associated group $Mod(R)$
is countable. Chandra, Patel and Vlamis [CPV] ask ”How does $Mod(R)$ sits on
$Map(S)$?”, we extend Matsuzaki’s results to give a partial answer to this
question. Our main result is:

######  Theorem 1.

For every infinite type surface $S$ of finite genus there exists a Riemann
surface structure $R$ such that $Mod(R)$ is countable.

The strategy we use follows closely that of Matsuzaki [Mat05], which can be
roughly summarized as follows: using the notion of a core tree defined by
Bavard and Walker [BW18] we give $S$ a convenient pants decomposition which
will be used to construct a Riemann surface structure $R$ on $S$. This
structure comes with an exhaustion sequence of Riemann subsurfaces of finite
type $\\{R_{n}\\}$. The exhaustion $\\{R_{n}\\}$ has the crucial property:
there exists an $N$ such that for every $n\geq K\geq N$ all K-q.c. elements in
Mod(R) leaves $R_{n}$ invariant modulo free homotopy.

This enables us to define a restriction map $\Psi_{n}:Mod(R)_{n}\to
Mod(R_{n})$ that establishes a link between the set of equivalence classes of
n-quasiconformal homeomorphisms of $R$ and the at most countable group
$Mod(R_{n})$, which comprises free equivalence classes of quasiconformal
homeomorphisms of $R_{n}$. We establish the countability of $Mod(R)$ by
showing the injectivity of $\Psi_{n}$.

Figure 1(()): Core tree. The root is blue colored and the ends are represented
on the circle.

Figure 1(()): Pair of pants decomposition

## Acknowledgements

We thank Ferrán Valdez for his valuable feedback that improved this
manuscript. Thanks also for Carlos Matheus and the CMLS, École Polytechnique
for providing the facilities where this research was done. Finally many thanks
to the Solomon Lefschetz Laboratory at UNAM, CONACYT Ph.d. grant, and UNAM
PAPIIT IN-101422 for providing funding during the visit at L’X.

##  2 Preliminaries

As previously mentioned, for the construction of the Riemann surface $R$ we
use a convenient pants decomposition of $S$. This is achieved using the notion
of core tree, denoted as $T=T(S)$, which we explain in what follows.

####  2.0.1 Core trees

For a detailed construction we refer to Lemma 2.3.1 of [BW18]. We recall here
briefly the properties of a core tree:

  * •

It is an infinite rooted tree where each vertex can only have degrees $1,2$ or
$3$.

  * •

The vertices are either marked or unmarked. The unmarked only have degree $1$
or $3$. The root if unmarked is of degree $3$.

  * •

The space $Ends(T)$ is homeomorphic to $Ends(S)$. Here $Ends(\cdot)$ is the
space of ends of a topological space. See [Ric63] for an exposition on ends.

See Fig. 1 for an example of a core tree. The core tree serves as a
scaffolding to construct the desired pairs of pants decomposition on a surface
$S(T)$ homeomorphic to $S$. We detail its contruction in what follows. This is
essentially section $2.3$ of [BW18].

  * •

Replacing vertices. Let $v\in V(T(S))$ be unmarked. If $v$ has degree $3$ then
replace $v$ with a pairs of pants. If $v$ has degree zero then replace it with
nothing.

If $v$ is marked then replace it with a torus with $deg(v)$ boundary
components. Each torus has its own pairs of pants decomposition as in the
Figure 2.

Figure 2: Tori with boundaries and with a pants decomposition. From left to
right, we have the torus with one, two or three boundary components.

  * •

Glueings. We glue pairs of boundary components of the tori and the pairs of
pants if there was an edge connecting the replaced vertices.

  * •

Removing boundaries Once glued we have a surface $S^{\prime}$. Since $S$ has
no boundary replace the boundary components of $S^{\prime}$ by punctures. Now
this new surface is $S(T)$ which we denote also as $S$ since it is
homeomorphic to $S$.

In this pants decomposition of $S$, we have pairs of pants which can have
$0,1$ or $2$ punctures. We refer to them as pairs of pants of type $0$,$1$ and
$2$, respectively. See Fig. 1 for an example of genus $0$ and Figs. 3.1 and
3.1 for positive genus.

######  Remark 1.

Every surface $S$ of infinite type is accompanied by a non-unique core tree,
since there is an element of choice.

####  2.0.2 Prefered core tree

For technical reasons that will be discussed later, it is convenient to work
with a particular class of core trees. These are core trees having at most one
_exterior tree_. An exterior tree is a subtree comprising three unmarked
vertices and two leafs. The exterior trees corresponds to pairs of pants of
type $2$. As the following Lemma shows, for the surfaces that interest us, we
can always find such a core tree.

######  Lemma 1.

Let $S$ be of infinite type with $g=0$ and $End(S)$ having infinitely many
countable isolated points. Then there is a core tree, $T(S)$ with at most one
exterior tree.

###### Proof.

Let $T(S)$ be a core tree of $S$. Since $T(S)$ has countable number of
vertices there is at most a countable number of exterior trees. Then there is
a enumeration of them, denoted by $\\{t_{i}\\}_{i\in\mathbb{N}}$. We will
remove them so that after the removal the new core tree $T^{\prime}(S)$ has
the property that $S(T^{\prime})$ is homeomorphic to $S$ with at most one
exterior tree. Let $p_{i}$ denote the geodesic path from the vertex of degree
2 of $t_{i}$ to the root. Here $p_{i}(j)$ denotes the vertices along the path
with $p_{i}(0)$ the initial vertex and $p_{i}(m_{i})$ the root,
$j\in\mathbb{N}$. If $p_{i}(0)$ is the root, we don’t do anything. Denote by
$T(p_{i}(j))$ the connected component of $T(S)\setminus p_{i}(j)$ which
contains the geodesic segment $[p_{i}(0),p_{i}(j-1)]$, for $0<j\leq m_{i}$.
Since the geodesic $p_{i}$ is finite there is a unique maximal finite subtree
$T(p_{i}(l))$ which has $t_{i}$ as the solely numbered exterior tree. Replace
$T(p_{i}(l))$ with one single leaf. Note that $p_{i}(l)$ does not change its
degree. If $p_{i}(l)\neq p_{i}(m_{i})$ then $p_{i}(l)$ is not the vertex of a
new exterior tree otherwise $T(p_{i}(l))$ would not have been maximal. The
only possibility after this procedure is that a new exterior tree occurs on
the root. Finally, since for every $t_{i}$ we replaced finitely many leafs
with one single leaf and $Ends(S)$ has infinitely many countable isolated
points, we have that $Ends(T^{\prime})$ is homeomorphic to $Ends(T)$. Hence
$S(T^{\prime})$ is homeomorphic to $S(T)$.

∎

##  3 Proof of the main result.

The proof begins in this section, which is divided in two parts. In the first
part we construct a complete and of the first kind Riemann surface structure
$R$ on $S$. In the second part we show countability of the group $Mod(R)$.

###  3.1 Constructing the Riemann surface structure

By prescribing lengths to the closed curves of a pants decomposition of $S$ we
obtain a hyperbolic structure on it. This is because each pair of pants of
type $0,1$ and $2$ becomes a hyperbolic pairs of pants with cusps. Each
hyperbolic pair of pants with $0,1$ or $2$ cusps will be called hyperbolic
pairs of pants of type $0,1$ and $2$. The resulting hyperbolic structure on
$S$ may not be complete and of the first kind. Specially since we are not
doing any twists along the curves. This is a problem since we would allow
boundaries at infinity which makes $Mod(R)$ uncountable as explained in the
proof of Proposition 1 of [Mat05]. Thanks to the topology given by the core
tree we can use the results of Basmajian and Šarić [BŠ19] to obtain a complete
and of the first kind Riemann surface structure.

Case 1. $End(S)$ does not have a finite number of isolated points, and genus
of $S$ is $0$.

By Lemma 1 we choose $T(S)$ having at most one exterior tree. We will only
label edges of $T(S)$ that do not incide a leaf as follows. Choose an edge
$E_{1}$ of the root vertex that does not meet a leaf and label it $1!$. We
assign the number $3!$ to the other incident edges of the endpoints $E_{1}$
(as long as they do not meet a leaf). For the edges that got a number we
repeat the process as with $E_{1}$. Their remaining unlabeled neighboring
edges will have the number $5!$. We proceed by induction so that the edges are
assigned the odd numbers. See Fig 3.1 for an example.

Figure 3(()): Core tree with at most one exterior tree. The root is in blue.

Figure 3(()): Surface with at most one hyperbolic pairs of pants of type 2.

We obtain from $T(S)$ a pants decomposition on $S$ as in the preceding
section. The cuffs of the pairs of pants decomposition will have the
hyperbolic length equal to the number assigned to the corresponding edge in
$T(S)$. The edges that do not meet a leaf always correspond to a cuff. Hence,
we have a Riemann surface structure $R^{\prime}$ exhausted by surfaces of
finite topological type $\\{R_{n}\\}_{n\in\mathbb{N}}$. We describe the
exhaustion in what follows. $R_{1}$ is the finite subsurface obtained by
gluing two pairs of pants along $E_{1}$; each of its boundary components has
length $3!$. $R_{2}$ is the union of $R_{1}$ with the pairs of pants glued
along the boundary components of $R_{1}$. We proceed by induction, and
therefore we have subsurfaces $R_{n}$ with geodesic frontier whom all
components have the same length of $(2n+1)!$. See Fig 3.1.

By Theorem 5.1 of [BŠ19] there is a choice of twist on each cuff of the pants
decomposition of $R^{\prime}$, such that the resulting surface $R$ is complete
and of the first kind.

Case 2. Genus of $S$ is positive or has a finite number of isolated ends

Note that we can always choose a core tree $T(S)$ with a finite subtree
$T^{*}$ with all of its vertices being marked and also having all possibly
finite leaves. We can choose the tree $T(S)$ such that the root vertex is part
of $T^{*}$ and deleting it will separate $T^{*}$ from the rest of $T(S)$. We
can also choose $T(S)$ such that $T(S)\setminus T^{*}$ has at most one
exterior tree (in the case of infinitely many isolated points). So we choose
$T(S)$ in this way.

Assign whatever positive numbers to the edges of $T^{*}$. Then choose an edge
$E_{1}$ of the root vertex that is not part of $T^{*}$ and then assign numbers
to the edges as in the previous case. Obtain from $T(S)$ the pants
decomposition as in section $2$ and make the twists to each cuff such that the
resulting Riemann surface $R$ is complete and of the first kind. See Figs. 3.1
and 3.1.

Figure 4(()): Core tree representing finite genus. The root is in blue.

Figure 4(()): Surface with positive genus.

The exhaustion of the $R$ will be given by: $R_{1}$ which is going to be the
subsurface with all the genus of $S$ and possibly the finite isolated points
of $Ends(S)$. The geodesic boundary of $R_{1}$ is the cuff chosen
corresponding to $E_{1}$. The subsurface $R_{2}$ is $R_{1}$ union the
periferic pair or pair of pants that $E_{1}$ bounds. The subsurface $R_{3}$ is
the union of $R_{2}$ and the perifec pair or pair of pants the geodesic
boundaries of $R_{2}$ bounds. We proceed by induction.

###  3.2 Countability

As mentioned in the introduction the idea for proving countability is to
define injective maps $\Psi_{n}:Mod(R)_{n}\rightarrow Mod(R_{n})$ for all
$n\geq N$, for some $N$. This implies that $Mod(R)$ is countable since
$Mod(R)=\bigcup_{n\in\mathbb{N}}Mod(R)_{n}$ and $Mod(R)_{m}\subset
Mod(R)_{n},n>m$. This $N$ is given by Lemma 4, which is an extension of Lemma
4 of [Mat05]. In order to prove it we need the following two Lemmas.

######  Lemma 2.

(Wolpert’s Lemma, [Wol79]) Let $c$ be a simple closed geodesic on a Riemann
surface $R$ with the length $l(c)$ and $g:R\rightarrow R$ a $K$-quasiconformal
homeomorphism. Then the geodesic length $l(g(c))$ of the free homotopy class
of $g(c)$ on $R$ satisfies

$\dfrac{1}{K}l(c)\leq l(g(c))\leq Kl(c).$

This is Lemma 8.1 of [BŠ19].

######  Lemma 3.

Let $\Sigma$ an ideal right-angled pentagon as in Fig. 5. Let $l(d)$ the
length of the geodesic orthogonal segment $d$. Then for sufficiently large
$l(a)>1$, and $l(c)>l(a)$, then

$l(d)\geq l(c)-l(a)$

.

Figure 5: Ideal Penthagon

######  Lemma 4.

Let $S$ be of infinite type and finite genus and $R$ be a Riemann surface
structure on $S$ constructed in Section 3.1. There exists $N\in\mathbb{N}$
such that for all $g:R\rightarrow R$ $K$-quasiconformal homeomorphism, $K\geq
N$, $g(R_{n})$ is freely homotopic to $R_{n}$ for every $n\geq K$. The $R_{n}$
are the exhaustion subsurfaces mentioned in Section 3.1.

###### Proof.

The goal will be to contradict Lemma 2. The curves we are going to use are the
geodesic boundaries of the subsurfaces $R_{n}$, which are closed curves of the
hyperbolic pairs of pants decomposition described in Section 3.1. Lemma 3 will
give us the $N$ such that if $R_{n},n\geq N,$ is not freely homotopic to
$g(R_{n})$, then the image of some of the boundary curves, say $c$ under $g$
is going to be large enough so that the inequality $l(g(c))\leq Kl(c)$ of
Lemma 2 cannot hold.

Case 1. $S$ is of genus $0$ and $Ends(S)$ has infinitely many isolated points.

First note that for every $n$ if $g(R_{n})$ is not freely homotopic to $R_{n}$
there is some curve $c_{n}$ of the boundary component of the $R_{n}$ such that
$g(c_{n})\cap(R\setminus\mkern
1.5mu\overline{\mkern-1.5muR_{n}\mkern-1.5mu}\mkern 1.5mu)\neq\emptyset$. This
is because the boundary components of $R_{n}$ are elements of the pairs of
pants decomposition described in Section 2 and the $R_{n}$’s are subsurfaces
with boundaries. Recall from Section 2 that we eliminated the possibility of
having infinitely many pairs of pants of type $2$, in fact at most one can
occur only in the root. Hence, the hyperbolic pairs of pants decomposition of
R consists of, at most, one hyperbolic pair of type $2$, and infinitely many
of pairs of pants of type $0$ or $1$. Suppose then we have
$g(c_{n})\cap(R\setminus\mkern
1.5mu\overline{\mkern-1.5muR_{n}\mkern-1.5mu}\mkern 1.5mu)\neq\emptyset$, for
some $c_{n}$. Since we have a direction of exhaustion and we have genus $0$,
there is some geodesic $\sigma_{n}$ homotopic to $g(c_{n})$ such that it has a
segment $s_{n}$ which cuts a hyperbolic pair of pants of type $0$ or $1$. The
segment $s_{n}$ cuts the pairs of pants in a boundary component and returns to
it in the direction of the orthogonal geodesic on the corresponding pairs of
pants that also cuts the same boundary component, see Figures 3.2 and 3.2. We
are going to give lower bounds to the length of this segment.

Figure 6(()): In red is how the segment $s_{n}$ could go in the case of pairs
of pants of type $1$. The lengths of $c_{1}$ is $(2m+1)!$ and of $c_{2}$ is
$(2m+3)!$, for some $m$.

Figure 6(()): In red is how the segment $s_{n}$ could go in the case of pairs
of pants of type $0$. The lengths of $c_{1}$ is $(2m+1)!$ and of $c_{2}$ and
$c_{3}$ are $(2m+3)!$, for some $m$.

Suppose first that the pairs of pants decomposition of $R$ have infinitely
many pairs of pants of type $1$. Then Lemma 3 gives an $N$, such that if
$s_{n}$ cuts a pair of pants of type $1$ with one geodesic boundary of length
$(2m+1)!$ and the other of length $(2m+3)!$, with $m\geq N$ then the
inequality given by Lemma 3 implies $l(s_{n})>(2m+1)!(2m+1)$. If $s_{n}$ would
cut a pairs of pants of type $0$ on the boundary component of length $(2m+1)!$
(the other two boundary components have length $(2m+3)!$), with $m\geq N$,
then the calculations for the proof of Proposition $5$ of [Mat05] also implies
$l(s_{n})>(2m+1)!(2m+1)$. Therefore in this case we can take $N=l(a)$ of Lemma
3 for the statement we want to prove. Hence if $g(R_{n})$ is not freely
homotopic to $R_{n}$ there would be a pair of pants such that $s_{n}$ cuts its
boundary component of length $(2m+1)!$ with $m\geq n\geq N$. The previous
discussion shows together with Lemma 2 that $n(2n+1)!\geq l(g(c_{n}))$, but
$l(g(c_{n}))\geq l(s_{n})>(2m+1)!(2m+1)$, contradiction. In the case where $R$
have $r\in\mathbb{N}$ pairs of pants of type $1$ we set $N=min\\{n:\text{all r
pants of type 1 are in \hskip 0.28436pt}R_{n}\\}+1$. The same argument in the
case of pairs of pants of type $0$ gives the result.

Case 2. $S$ has positive genus and possibly finite isolated ends.

We can choose $N$ as in the previous case. Note however that given our
construction of this case in section 2 the subsurface with genus or finite
isolated ends is a non wandering surface. Therefore all the arguments go
through as in the previous case.

∎

###### Proof of Theorem 1.

By Lemma 4 the restriction map $\Psi_{n}:Mod(R)_{n}\rightarrow Mod(R_{n})$ is
well defined for $n\geq N$. Recall that $Mod(R)_{n}$ are the homotopy classes
of $Map(S)$ that have a $n$-quasiconformal representative for the Riemann
surface structure we constructed. The group $Mod(R_{n})$ is the group of free
homotopy classes of quasiconformal homeomorphisms of $R_{n}$. Because of Lemma
4 every $n$-quasiconformal homeomorphism $g$, with $n\geq N$ maps any pairs of
pants (of type $0$ or $1$) $P\in R\setminus\mkern
1.5mu\overline{\mkern-1.5muR_{n}\mkern-1.5mu}\mkern 1.5mu$ to a pair of pants
of the same type. The homeomorphism $g$ also maps boundary components to
boundary components of the same length. This implies that if
$\Psi_{n}(g)=\Psi_{n}(f)$, for $m\geq n$ the difference between $g$ and $f$ on
$R_{m+1}\setminus\mkern 1.5mu\overline{\mkern-1.5muR_{m}\mkern-1.5mu}\mkern
1.5mu$ is a conformal homeomorphism freely homotopic to the identity. Hence
the difference between $f$ and $g$ on $R\setminus\mkern
1.5mu\overline{\mkern-1.5muR_{n}\mkern-1.5mu}\mkern 1.5mu$ is by possibly Dehn
twists along each cuff of $R\setminus\mkern
1.5mu\overline{\mkern-1.5muR_{n}\mkern-1.5mu}\mkern 1.5mu$, but this cannot
happen, otherwise the Dehn twists would make either $f$ or $g$
$K$-quasiconformal with $K>n$, see Theorem 3 in [Mat05]. Then $Mod(R)_{n}$ is
countable. ∎

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