Constraining cosmological parameters with the cosmic microwave background Andrew Stewart Master of Science Department of Physics McGill University Mo ntr´ eal, Qu´ eb ec November 3, 2008 A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Master of Science Andrew Stewart 2008
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8/3/2019 Andrew Stewart- Constraining cosmological parameters with the cosmic microwave background
I would like to thank Robert Brandenberger for supervising all of the work pre-
sented in this thesis, for numerous enlightening conversations, and, in particular,
for his patience. Thanks to Joshua Berger and, especially, Stephen Amsel for mak-
ing their code available and answering many questions regarding the edge detection
method. A big thanks to Eric Thewalt for debugging some parts of the code and mak-
ing many helpful suggestions. I would like to thank the WMAP Science Team for the
use of the image shown in Figure 1–1 and I acknowledge the use of the Legacy Archivefor Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is pro-
vided by the NASA Office of Space Science. Thanks, also, to my family for all of their
support during my time at McGill. Last, but most certainly not least, I would like
to thank Rachel Faust, Guillaume Giroux, Martin Auger, Francois Aubin, Razvan
Gornea, John Idarraga, Amelie Bouchat, Marie-Cecile Piro, Francis-Yan Cyr-Racine,
Aaron Vincent, Nima Lashkari, Jean Lachapelle, Anke Knauf, Jamie Sully and Paul
Franche for useful distractions.
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8/3/2019 Andrew Stewart- Constraining cosmological parameters with the cosmic microwave background
2–1 Definition of the approximate gradient directions used in the edge de-tection algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2–2 Summary of the ability of the Canny algorithm to make a significant
detection of a cosmic string signal for SPT specific simulations . . . 51
2–3 Summary of the ability of the Canny algorithm to make a significantdetection of a cosmic string signal for simulations corresponding toa hypothetical CMB survey . . . . . . . . . . . . . . . . . . . . . . . 54
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2–8 Comparison of CMB maps with and without a component of instru-mental noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3–1 Magnitude of the difference between the angular power spectra of amodel with a blue tensor spectral index and a model with a standardtensor spectral index . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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Figure 1–1: The WMAP 5-year TT power spectrum along with recent results fromthe ACBAR [10] (purple), Boomerang [11] (green), and CBI [12] (red) experiments.The pink curve is the best-fit ΛCDM model to the WMAP data. [Figure from [8]]
experiment is clear and illustrates the power of the CMB to constrain cosmological
theories.
Efficient computer codes have been written for calculating the CMB anisotropy
spectra up to an arbitrary multipole moment based on a given set of cosmological
parameters and desired physical effects. The most popular of these codes are cmb-
fast [13] and camb [14], the latter of which is based on the former, with both in
wide use in the physics community. The details of how these codes are implemented
are outside the scope of this work, though we note that these programs can compute
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8/3/2019 Andrew Stewart- Constraining cosmological parameters with the cosmic microwave background
the angular power spectrum in a matter of minutes on a typical desktop computer
to very high accuracy (∼ 0.1% for camb).
In this thesis, we investigate the constraints which can by applied on two different
cosmological parameters using the CMB. In the first part of this work, we develop an
edge detection method of searching CMB temperature anisotropy maps for the effects
of cosmic strings, and we examine how it could be used to constrain the cosmic string
tension. In the second part of this work, using the alternate cosmological model string
gas cosmology as a motivation, we study how well the angular power spectrum of the
CMB can constrain the possible blue tilt of the power spectrum of tensor fluctuations,and how it compares to the constraints applied by other observations. We end with a
review and discussion of the main results emerging from each of these investigations.
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such a way that the energy density in cosmic strings scales like radiation [15]. In this
regime, the strings then contribute some fraction of the total energy. The existence
of a scaling solution is supported by numerical simulations of the evolution of the
cosmic string network [18, 19, 20].
When discussing cosmic strings it is common to work with the dimensionless
parameter Gµ, where G is Newton’s constant. The quantity Gµ is of interest be-
cause it characterizes the strength of the gravitational interaction of the strings.
The gravitational perturbations produced by the strings, and thus the density per-
turbations and the induced fluctuations in the CMB, are all of the order of Gµ [15].Until the late 1990’s, cosmic strings were studied as potential seeds for structure
formation [21], fuelled in part by the realization that the density fluctuation from
a string formed around the grand unified epoch would be of the same order as the
temperature anisotropy discovered by COBE [5]. As mentioned in the Introduction,
acoustic peaks were eventually discovered in the CMB angular power spectrum and
subsequently measured with great accuracy. This lead to cosmic strings being ruled
out as the main origin of structure in favour of the inflationary paradigm, since the
angular power spectrum predicted by cosmic strings consists of only a single broad
peak. Despite this, cosmic strings can still contribute partially to the CMB angular
power spectrum (less than 10% [22, 23]). Therefore, there still exists a great deal of
interest in cosmic strings since there are many cosmological models in which their
formation is generically predicted (see [24, 25, 26] for just a few possibilities).
The observational signatures of cosmic strings are distinct and lie within ob-
servational reach. The current bounds on the string tension come from a variety of
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Figure 2–1: The geometry of the space-time near a cosmic string. Shown here is aslice of the space-time perpendicular to the orientation of the string. The coloured
area represents a missing wedge with deficit angle φ, while the dashed lines representthe paths of photons travelling from a source to an observer, and the arrow showsthe direction of motion of the string. The photons passing on one side of the cosmicstring will appear to be Doppler shifted with respect to those passing on the otherside due to this non-trivial geometry.
motivation behind this choice is clear since the cosmic strings literally appear as
edges in the CMB temperature. Depending on the sensitivity of the edge detection
algorithm to these temperature edges, we can then place a bound on the cosmic string
tension through Equation (2.2). We expect the bounds arising from this method to
be more robust than those coming from gravitational waves since we do not need to
make assumptions about the nature of the cosmic string network. This work is a
continuation of the study presented in [36].
We are interested in the cosmic strings in the network that survive until later
times, specifically, the times relevant to the production of an edge signature in the
CMB, that is, the time of last scattering until the present time. Based on the
evolution of the network, cosmic strings are more numerous around the time of last
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scattering than later times. On today’s sky, those strings correspond to an angular
scale of approximately 1 . Therefore, an observation of the CMB with an angular
resolution substantially less than 1
is necessary in order to be able to detect the
edges related to these strings. With this in mind, we also focus on the application
of the edge detection method to high resolution surveys of the CMB, in particular
the future data from the South Pole Telescope project.
The South Pole Telescope (SPT) [37] is a 10m diameter telescope being deployed
at the South Pole research station. The telescope is designed to perform large area,
high resolution surveys of the CMB to map the anisotropies. The telescope is de-signed to provide 1′ resolution in the maps of the CMB. This makes the SPT ideal
to search for the KS-effect (even better than Planck), and we believe that with such
high resolution data our method could provide bounds on the cosmic string tension
competitive with those of pulsar timing.
The remainder of this chapter is arranged as follows: In section 2.2, we discuss
the simulated CMB maps used in our analysis with a focus on the anisotropies
coming from Gaussian fluctuations and cosmic strings. In Section 2.3, we outline
the edge detection algorithm we are using, highlighting the details of our particular
implementation. In Section 2.4, we discuss how we quantify the edge maps that are
output by the edge detection algorithm. In Section 2.5, we explain the statistical
analysis used to determine if a significant difference has been detected. In Section
2.6, we present the results of running the edge detection algorithm on simulated
CMB maps and the possible constraints on the cosmic string tension that could be
applied. Finally, in Section 2.7, we discuss our results.
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with a square grid that has a size corresponding to the angular size being simulated,
and a pixel size corresponding to the angular resolution being simulated. The pixels
in the grid are indexed by two dimensional Cartesian coordinates (x, y) and we take
the upper left corner of the grid to be the origin.
The common component in every simulated CMB map is a set of temperature
anisotropies produced by Gaussian inflationary fluctuations. The normal distribution
of these fluctuations is predicted by various cosmological models and is supported
by current observations [7]. These fluctuations must be included because they cor-
respond to an angular power spectrum like that measured in the real microwave sky[7]. In fact, we simulate the Gaussian fluctuations such that they account for all of
the observed power in the CMB. Thus, in the absence of any other effects the final
simulated map is simply equivalent to the Gaussian component and is consistent
with observations. That is, we define
T (x, y) ≡ T G(x, y) , (2.3)
where T (x, y) represents the the final temperature anisotropy map and T G(x, y) rep-
resents the Gaussian component. We signify the maps by T simply as a choice of
notation, but we note that the value of each pixel is actually that of the temperature
anisotropy δT/T .
To make a CMB map including the effects of cosmic strings, we simulate a
separate component of string induced temperature fluctuations produced via the
KS-effect. The final temperature anisotropy map is then given by a combination of
the string and Gaussian components. Denoting the string component by T S (x, y),
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where α is a scaling factor which depends on the tension of the cosmic strings in
T S (x, y). We must scale the amplitude of the Gaussian component to compensate for
the excess power we introduce by adding a component of string-induced fluctuations.
In this way the strings can contribute a fraction of the total power, while the final
map is still in agreement with current CMB survey results.
Let us comment in more detail on the nature of this scaling. We demand that theangular power of the final combined temperature map match the observed angular
power for multipole values up to the first acoustic peak, i.e. l 220. We choose
this multipole range because it is tightly constrained by current observations [7].
Then again, as mentioned above, the Gaussian component alone accounts for all
of the observed angular power in the CMB. Thus, this demand is equivalent to
requiring that the angular power of the combined map match that of a pure Gaussian
component. Working in the flat-sky approximation allows us to replace the usual
spherical harmonic analysis of the CMB fluctuations by a Fourier analysis [38]. We
can then express our condition as
|T G(k < k p)|2 = α2|T G(k < k p)|2 + |T S (k < k p)|2 , (2.5)
where k p is the wavenumber corresponding to the first acoustic peak of the angular
power spectrum of the CMB, |T S (k < k p)|2 is the average of the Fourier tempera-
ture anisotropy values from the string component for wavenumbers less than k p and
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correctly sized simulated area with the centre of the extended area, one can see that
what we essentially did when first defining the extended region was to enlarge the
actual simulation area by a Hubble volume in each direction. The reason that we
expand our simulated area in this way is because any string whose midpoint is within
a distance θH i of the actual area we want to simulate could enter into it. Thus, we
must also account for these strings which lie around the edges of the area of interest,
not only those centred within it.
Finally, when we have simulated the string network for each Hubble time step,
we sum together these fifteen sub-components pixel by pixel. This superpositionapproximates what the contribution from the entire, more complex cosmic string
network would be, and gives the final cosmic string component, T S (x, y).
In the model described above, we have fixed values for the the speed of the
strings, the length of the strings and the depth of the temperature fluctuation region
around the string. These values were obtained from particular numerical simulations
[33], however, these parameters can vary significantly for different models of the
string network (see [16] for a review) and should not be considered as established.
Figure 2–2 shows a cosmic string component simulated using the model described
above. Clearly visible are the sharp temperature discontinuities caused by individual
straight strings as well as the cumulative effect of the entire cosmic string network.
The amplitude of the fluctuations in the string component are small compared to
those appearing due to Gaussian fluctuations. The random way in which the cosmic
strings are positioned and oriented in the network is also apparent.
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Figure 2–2: Components of a simulated temperature anisotropy map. On the leftis an example of a component of Gaussian temperature fluctuations. On the right
is an example of a component of cosmic string induced temperature fluctuations.In both components, the angular size of the simulated region is 2.5 ×2.5 and theangular resolution is 1′ per pixel (22,500 pixels). In the string component the tensionof the cosmic strings was taken to be Gµ = 6 × 10−8 and the number of strings perHubble volume in the scaling solution was taken to be M = 10. The colour of a pixelrepresents the value of the temperature anisotropy at that pixel, as described by thescale below each image.
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where the maximum and minimum values of i and j are determined by the filter
length. In practise, we actually compute Gx(x, y) and Gy(x, y) by a convolution of
the temperature map with the filter using a FFT for the sake of increased speed.
With the component of the gradient in each direction known at every pixel, we can
construct a new map
G(x, y) =
G2x(x, y) + G2
y(x, y) , (2.22)
which is the map of the gradient magnitude, or edge strength, corresponding to the
original temperature anisotropy map. We can also construct a second map
θG(x, y) = arctan
Gy(x, y)
Gx(x, y)
, (2.23)
which is the map of the gradient angle, or gradient direction. In the above equation
the sign of both components is taken into account so that the angle is placed in the
correct quadrant. Therefore, the arctangent has a range of (−180 , 180 ].
In the Canny algorithm, part of the definition of a pixel that is considered to be
on an edge is that it must be a local maximum in the gradient magnitude. By local
maximum we mean that the gradient magnitude at a given pixel is larger than that of
both pixels which neighbour it along the axis defined by the gradient direction at thatpixel. Using the gradient magnitude and direction maps, it is straightforward then
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to check the local maximum condition pixel by pixel and determine which could be a
part of an edge and which could not be part of an edge. Since we are only interested
in constructing a final map of edges, if a pixel does not satisfy the local maximum
condition we immediately discard that pixel. Therefore, this process is referred to as
non-maximum suppression .
On a square grid there are only eight distinguished directions which form four
axes, namely the two directions along each coordinate axis and the two directions
along each diagonal axis. For the sake of simplicity, when referring to the eight
directions on the grid we make an analogy with the eight directions on the face of acompass (i.e the positive x-direction is equivalent to east, etc.). However, as already
mentioned, the gradient direction as calculated in Equation (2.23) can take any value
(−180 , 180 ]. Thus, in order to relate the gradient direction, or equivalently the edge
direction, to one that we can trace on the grid, we must approximate the value of
θG(x, y) at each pixel to lie along one of the eight grid directions. The definition of
the approximated gradient directions is given in Table 2–1.
For the purpose of performing the non-maximum suppression we first check the
approximated gradient direction at a given pixel to determine which of the four grid
axes corresponds to the gradient axis at that pixel. We then record the gradient
magnitude of the two pixels which neighbour the original pixel along that gradient
axis. For example, if the gradient direction is approximated as north-west then we
record the gradient magnitude of the pixel to the north-west and the pixel to the
south-east. Lastly, we compare the gradient magnitude of the original pixel to the
gradient magnitudes of the two neighbours. Only if the gradient magnitude is larger
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Table 2–1: Definition of the approximate gradient directions used in the edge detec-tion algorithm. In the left column are the different ranges of values that the gradientdirection can take. In the right column are the approximated gradient directionsmatching each of the eight directions on the grid. Depending on which range a given
pixel falls into in the left column, the gradient direction at that pixel will then bereplaced by the corresponding approximation in the right column.
Actual Gradient Direction Approximated Gradient Direction−22.5 ≤ θG(x, y) < 22.5 θG(x, y) ≃ 0 (east)
Here Gm is the mean maximum gradient magnitude computed from simulated tem-
perature maps which contain only strings. The value of Gm depends on the param-
eters of the simulation being performed, most notably the string tension, and must
be computed separately for each parameter set using a selected number of simulated
string maps. One can think of Gm as representing the strongest possible edge that
could be formed by cosmic strings alone. Therefore, with this threshold, we are sim-
ply stating that if the gradient magnitude at a given pixel is some chosen fraction of
the maximum possible, then it must be a true-edge pixel.
It is not sufficient, however, to define the edges using only one threshold becausethe gradient magnitude can fluctuate at each pixel along the length of an edge. This
variation can be caused by both instrumental noise and the random nature of the
Gaussian anisotropies. If we applied only an upper threshold, we would reject the
pixels at which the gradient magnitude fluctuates below that threshold, but should in
fact still be considered as a part of a given edge. This would lead to edges being cut
into smaller segments, making them look like dashed lines, rather than continuous
curves on the map. To avoid this, we also choose a lower gradient threshold, tl < tu,
and define a pixel which is possibly part of an edge, which we name a semi-edge
pixel , as a local maximum pixel satisfying
tlGm ≤ G(x, y) < tuGm . (2.25)
We then further assert that any semi-edge pixel which is in contact with a true-
edge pixel and has the appropriate gradient directionality is also a true-edge pixel
sharing the same edge (see the later discussion in this section for a full explanation
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in which we mark true-edge pixels as 1, semi-edge pixels as 1/2 and all rejected pixels
as 0.
We then check which semi-edge pixels are actually true-edge pixels. We begin
by searching the map for a pixel which is a 1 and has not already been examined
during the tracing of a different edge. If we find one we then check the gradient
direction at that pixel to determine the axis along which the gradient lies. Given the
gradient axis, we inspect each of the six neighbouring pixels which do not lie along
that axis for ones which are non-zero. For example, if the gradient lies along the
north-south axis then we would check the pixels to the north-west, west, south-west,south-east, east and north-east. The two directions perpendicular to the gradient
axis represent the edge axis while the other four directions represent the two axes
which are next to parallel to the edge axis. The reason that we look at the neighbours
along six directions, rather than only the two directions along the edge, is because
we are working on a grid with finite resolution. As such, a wiggle in a real string,
which occurs on a scale below the grid resolution, may manifest itself in the map
as an abrupt jump in the edge position from one pixel to the next. Even a straight
string, depending on its orientation, may appear to have one or more “steps” when
it is viewed at the resolution of the grid. Thus, we cannot expect an edge to be a
continuous chain with the next edge pixel always lying along the edge axis defined
by previous pixel. If we did not account for this, it could lead to the tracing of edges
being prematurely terminated, causing an overabundance of short edges.
If any of the six neighbouring pixels is marked as 1/2 we check the gradient
direction at that pixel. If the gradient direction is parallel or next to parallel to
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If this is the case we mark both as true-edge pixels (if necessary) and then check the
neighbours of each of those pixels separately. In this way we trace the edge along two
separate paths simultaneously, but the end result is still a single continuous curve.
The entire process described above traces a single edge in the map. When we
have finished with a particular edge, we then search for the next pixel in the map
which is a 1 and has not already been examined. If we find one, we then start from
that pixel and trace the corresponding edge until its end. When we can no longer
find a pixel which is a 1 and has not been examined, we consider all of the edges in
the map to have been traced. If there are any remaining pixels which are still markedas 1/2, we consider them not to be in contact with a true-edge pixel and we mark
them as 0. The edge detection process is then finished, and the end result is the final
map of true-edge pixels corresponding to the original temperature anisotropy map.
Figure 2–3 shows a final edge map after thresholding with hysteresis has been
performed. Once again, we show the edge map corresponding to the same cosmic
string component shown in Figure 2–2 and the same map of local maxima shown
in Figure 2–3. Many of the pixels appearing in the map of local maxima have now
been rejected, especially those with very small gradient magnitudes, and the stronger
edges are now much better defined. This is a direct result of applying the thresholds
and directionality conditions. Comparing the original temperature anisotropy map
to the final edge map, it is clear that not only is the Canny algorithm good at
locating the edges which are clearly visible, but that it is also sensitive to the faint
edges which are not easily detectable by eye.
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Figure 2–3: Maps produced by the Canny edge detection algorithm. On the left is anexample of a map of local maxima generated after non-maximum suppression. Thesize of the gradient filters used was 5 × 5 pixels. The colour of a pixel represents themagnitude of the gradient at that pixel, as described by the scale below the image.On the right is an example of a final map of edges generated after thresholding withhysteresis. The values of the thresholds used were tu = 0.25, tl = 0.10 and tc = 3.5.The value of Gm was calculated using a cosmic string tension of Gµ < 6 × 10−8. Theyellow pixels represent pixels which were determined to be on an edge. Together,these pixels show the the position, length and shape of the edges occurring in theoriginal temperature anisotropy map. In both maps, the grey pixels represent pixelswhich were discarded from the image. The above images correspond to the samecosmic string component shown in Figure 2–2.
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Figure 2–4: A histogram of edge lengths. This histogram corresponds to a simulated
CMB map without cosmic strings and without instrumental noise. The angular sizeof the map was 10 × 10 and the angular resolution was 1′ (360,000 pixels). In theedge detection algorithm the gradient filter length was 5 pixels and the thresholdswere tu = 0.25, tl = 0.10 and tc = 3.5. The value of Gm was calculated using acosmic string tension of Gµ < 6 × 10−8. The height of each bar corresponds to thetotal number of edges at that edge length. The inset plot shows a closeup of the tailof the larger plot.
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Figure 2–5: An averaged histogram of edge lengths. This averaged histogram corre-sponds to 40 simulated CMB maps without cosmic strings and without instrumental
noise. The angular size of each map was 10 × 10 and the angular resolution of each was 1′ per pixel (360,000 pixels). In the edge detection algorithm the gradientfilter length was 5 pixels and the thresholds were tu = 0.25, tl = 0.10 and tc = 3.5.The value of Gm was calculated using a cosmic string tension of Gµ < 6 × 10−8. Theheight of each bar corresponds to the mean number of edges at that edge length. Theerror bars represent a spread of 3σ from the mean value, where σ is the standarddeviation of the mean, which is calculated separately for each length. Shown hereare only the lengths in the histogram for which the mean is greater than 3σ.
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how the potential constraint on the tension changes with respect to the design of the
survey.
The SPT is capable of producing a 4,000 square degree survey of the anisotropies
in the CMB [37]. To replicate the same amount of sky coverage, we simulate 40 sep-
arate 10 × 10 maps, where the angular resolution of each of these maps is 1 ′ per
pixel, again matching that specified for the SPT. To test the edge detection method,
we simulate two separate sets of 40 maps, the first set including the effect of cosmic
strings, and the second set excluding the effect of cosmic strings. Each set of maps
gives rise to a histogram of edge lengths via the edge detection and edge lengthcounting algorithms. We then compare these two histograms using the statistical
analysis described in Section 2.5 to determine if the difference in the distributions
is significant. We repeat this process for many different values of the cosmic string
tension, until we can no longer identify a statistically significant difference in the two
histograms. Figure 2–6 shows a side by side comparison of a simulated CMB map
without a cosmic string component and a simulated CMB map which does include
a cosmic string component. The effect of the cosmic strings in the final temperature
anisotropy map is not apparent and any difference in the typical structure between
the two maps is unnoticeable by eye. Figure 2–7, on the other hand, shows a his-
togram corresponding to a set of maps without a cosmic string component and a
histogram corresponding to a set of maps with a cosmic string component. The two
histograms show that the edge detection method is in fact able to detect a difference
which is not evident by eye, with maps including strings having slightly higher mean
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values for certain lengths. Although the difference in histograms may not seem large,
this particular example would generate a significant result.
Although the angular size and resolution of the simulation are determined by
the specifications of the survey in question, the values of the other free parameters
in each step of process must also be fixed. We take the number of cosmic strings
per Hubble volume in all of the string component simulations to be M = 10 [33],
regardless of the cosmic string tension. In every run of the edge detection algorithm,
we choose the gradient filter length to be 5 pixels, the value of the upper threshold
to be tu = 0.25 and the value of the lower threshold to be tl = 0.10. These values forthe thresholds may appear small, but as one can see from the scale in Figure 2–3,
the gradient magnitude in the string component can take a large range of values.
Therefore, Gm can be quite a bit larger than the average gradient magnitude on a
string induced edge, so we must choose low values for the thresholds in order to not
throw away the entire string signal. We have not mentioned the value of the scaling
factor in the map addition, α, nor the value of the cutoff threshold, tc. The reason is,
we do not fix the value of these two parameters for all of the runs. In the case of the
scaling factor, its value must change for each given cosmic string tension, as described
by Equation (2.7). The value of the cutoff threshold, on the other hand, is chosen
deliberately based on the value of the tension, such that we get the best results from
our edge detection method. We note the value of both of these parameters when
presenting our findings.
For the SPT specific simulations, the capability of the edge detection method
to make a significant detection of the cosmic string signal for different choices of the
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Figure 2–6: Comparison of CMB maps with and without a component of cosmic
string induced fluctuations. On the left is the map without a cosmic string componentand on the right is the map with a cosmic string component. Both maps show a 2.5
× 2.5 patch of sky at 1′ resolution (22,500 pixels). The values of the free parametersin the cosmic string simulation were Gµ = 6 × 10−8 and M = 10. The scaling factorin the map component addition that produced the map on the right was α = 0.987.
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Figure 2–7: Comparison of histograms for maps with and without a componentof cosmic string induced fluctuations. Each histogram corresponds to a set of 40simulated CMB maps. The angular size of each map was 10
×10 and the angular
resolution of each was 1′ per pixel (360,000 pixels). In the maps including a cosmicstring component, the string free parameters were taken to be Gµ < 6 × 10−8 andM = 10 while the scaling factor in the map component addition was α = 0.987. Inthe edge detection algorithm the gradient filter length was 5 pixels and the thresholdswere tu = 0.25, tl = 0.10 and tc = 3.5. The value of Gm was calculated using thesame cosmic string tension given above. The height of each bar corresponds to themean number of edges at that edge length. The error bars represent a spread of 3σfrom the mean value, where σ is the standard deviation of the mean. Shown hereare only the lengths for which the mean is greater than 3σ in both histograms.
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Table 2–2: Summary of the ability of the Canny algorithm to make a significantdetection of a cosmic string signal for SPT specific simulations. Shown here arethe results corresponding to simulated CMB maps excluding instrumental noise aswell as simulated CMB maps including instrumental noise. In the first column are
different choices for the tension of the cosmic strings. In the second, third and fourthcolumns are the values of the scaling factor, cutoff threshold and p-value respectively,corresponding to each of the tensions. A p-value of less than 2.7 × 10−3 indicatesthat the simulations including cosmic strings produced significantly different resultsfrom those without cosmic strings.
Figure 2–8: Comparison of CMB maps with and without a component of instrumen-tal noise. On the left is a simulated CMB map excluding instrumental noise. Onthe right is a simulated CMB map including a component of instrumental noise withmaximum temperature fluctuation δT N,max = 10 µK. Both maps show a 2.5 × 2.5
patch of sky at 1′ resolution (22,500 pixels). Neither map includes a component of cosmic string induced temperature fluctuations.
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Table 2–3: Summary of the ability of the Canny algorithm to make a significantdetection of a cosmic string signal for simulations corresponding to a hypotheticalCMB survey. Shown here are the results corresponding to simulated CMB mapsexcluding instrumental noise and simulated CMB maps including instrumental noise.
See the caption of Table 2–2 for a description of the columns.
The calculation of the tensor-to-scalar ratio depends quite sensitively on the pa-
rameters of the cosmological model under consideration. For that reason we choose
to leave r as a free parameter in the main expressions calculated in this work. Nev-
ertheless, for the sake of examining some numerical values of the constraints derived
here, we will insert a value of r corresponding to the current upper bound into our
results. Note, however, that the bounds we derive depend only logarithmically on
r. In each case we choose to use the value of the tensor-to-scalar ratio given by
the combined three-year WMAP and lensing normalized Sloan Digital Sky Survey
(SDSS) data2 applied to the standard ΛCDM model, but including tensors [45].
3.2.1 Pulsar Timing
High precision measurements of millisecond pulsars provide a natural way to
study low frequency gravitational waves. A gravitational wave passing between the
earth and a pulsar will cause a slight change in the time of arrival of the pulse
leading to a detectable signal. In the case of gravitational waves, the fluctuating
time of arrivals will be correlated between widely spaced pulsars, producing a unique
signature. Therefore, it can be discriminated from other effects which can cause a
varying time of arrival for a single pulsar. Jenet et al. [53] have developed a technique
to make a definitive detection of a stochastic gravitational wave background which
involves cross-correlating the time derivative of the timing residuals for multiple
2 We note that this portion of the work was completed before the release of the five-year WMAPresults [7], but this new data would not have a significant effect on any part of this study.
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Plugging this value into Equation (3.3) at f = 100 Hz, we find that if Advanced
LIGO does not make a positive detection of a gravitational wave background, then
it will place a bound on the blue tilt of the tensor spectrum
nT 0.0223 ln
2.29 × 105h2
r
. (3.12)
Using r = 0.30 and h = 0.716 in this expression, Advanced LIGO would then
constrain the blue tilt of the tensor spectrum to nT 0.29.
The Laser Interferometer Space Antenna (LISA) [55] is a planned space-based
interferometer experiment operating in the mHz range. LISA will consist of threedrag-free spacecraft each at the corner of an equilateral triangle with sides of length
5 × 109 m. Each spacecraft has two optical assemblies pointed towards the other
two spacecraft forming three Michelson interferometers. This triangle formation will
orbit the sun in an Earth-like orbit separated from us by approximately fifty million
kilometres. The goal of LISA is to reach a sensitivity of [56]
h2
Ωgw(1 mHz) ≃ 1 × 10−12
. (3.13)
At the LISA sensitivity level one would expect gravitational wave signals from super-
massive black hole binaries, other binary systems and super-massive black hole for-
mation to be present. Assuming these predicted signals could somehow be removed
and LISA does not detect any primordial signal, we can plug this predicted bound
into Equation (3.3) at f = 1 mHz to obtain a limit on the blue tilt of the primordial
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Figure 3–1: Magnitude of the difference between the angular power spectra of amodel with a blue tensor spectral index and a model with a standard tensor spectralindex. Shown here are the cases nT = 0.15 (green), nT = 0.53 (yellow) and nT = 0.79(orange) respectively. The dashed line represents the cosmic variance error at eachl.
From Figure 3.3 we can clearly see that the power spectrum of the temperature
anisotropy for models with a blue tensor spectral index does not vary much from
that calculated using a standard inflationary definition of the tensor spectral index.
In fact, the difference is within the cosmic variance error at all l ≤ 1000 for each of
the three bounds calculated using the PPTA observations, LIGO observations and
the theory of BBN. Thus, we conclude that the CMB does not offer any tighter
constraints on the blue tilt of the gravitational wave spectrum than those already
calculated.
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which was the purpose of this work. The authors of [61] also discuss a constraint
on nT coming from BBN, but they take the constraint on Ωgw from BBN to be a
constant across all frequencies rather than integrating their master equation as was
done in this study. In the end they find a weaker bound on the tilt, nT 0.36, than
the one obtained here.
As mentioned in Section 3.2 , Equation (3.2) has been normalized at the scale
of cosmic microwave background observations. However, those experiments probe
scales that are approximately ten orders of magnitude larger than those probed by
the PPTA and approximately nineteen orders of magnitude larger than those probedby LIGO, with LISA probing between the two. Extrapolating between such a large
difference in scales is not straightforward and we note that in [51] the authors con-
clude from their analysis that even within the framework of the inflationary universe
paradigm the formula for the primordial gravitational wave spectrum (3.2) is too re-
strictive, and they believe it is indeed not possible to extrapolate reliably over such a
large difference in scales. Whether or not this is the case in the string gas cosmology
model should perhaps be examined more carefully in future work.
Continuing with string gas cosmology, we conclude that the current bounds on
the tilt of the gravitational wave spectrum are weak. The predicted magnitude of the
blue tilt of the gravity wave spectrum is thought to be comparable to the magnitude
of the red tilt of the spectrum of scalar metric fluctuations [50]. If the latter is taken
to agree with the current bounds, we predict a blue tilt of less than nT = 0.1 which
will not be easy to detect. There may, on the other hand, be models similar to string
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from pulsars, since less assumptions about the nature of the cosmic strings have to
be made. We also found that using the edge detection method with data from a
much larger survey observing with the same angular resolution as the SPT would
not drastically reduce the possible constraint that could be levied, reducing it by
only a factor of a few.
We have also investigated whether the angular power spectrum of the CMB
could provide a stronger constraint on the possible blue tilt of the gravitational
wave background than those imposed by other means, namely, pulsar timing and
laser interferometer observations and the theory of nucleosynthesis. As a first step,we calculated the specific constraint on the blue tilt related to each of these three
observations, which we believe represents in itself an important result. We discovered
that the tightest current bound on the blue tilt of the tensor spectrum comes from
BBN at nT 0.15, tighter than even some future gravitational wave observatories
could hope to achieve. In the end, we found that the CMB could not impose a tighter
bound than any of these three methods.
In closing, we believe that in this thesis we have shown two important results as
well as some unique ways of extracting information from the CMB. Considering the
topics focused on in this work, we have highlighted the influence that the CMB has
on a wide range of cosmological issues. Then again, by the same token, we have also
shown that alternative models are still viable, and cannot be entirely ruled out by the
precise measurements which have already been taken. On that note, looking ahead
to the future of CMB physics, we believe that there are still many more exciting
discoveries to be made.
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