$(t,0)$. This is given by $f_{t_{y}}(t,0)$. Written explicitly, it is given by
the map

$t\,\,\longrightarrow\,\,f_{t_{11}}t.$

Assuming that $f_{t_{11}}\,\neq\,0$, the order of vanishing of the above
function at $t=0$ is one. This is a valid assumption, since to compute the
intersection multiplicity, we will be intersecting with generic cycles. This
proves (8.14) on the level of cycles and hence, completes the proof of (8.3).

###  8.3. Counting one tacnodal curves

We will now prove (8.2). First of all, we note that
$[\mathsf{A}_{3}^{\mathsf{F}}]$ is a codimension $5$ class in
$\mathsf{M}^{1}_{0}$. Hence, it is of the form

$\displaystyle[\mathsf{A}_{3}^{\mathsf{F}}]$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=5,\\\ i,j\geq 0,\\\ j\leq
2\end{subarray}}C_{ij}y_{d}^{i}b_{1}^{j}.$ (8.16)

There is no term involving $y_{1}$, because it is the pullback of a class in
$\mathcal{D}_{d}\times X^{1}$. We also note that $j$ can not be greater than
$2$, since $b_{1}^{3}$ is zero. Hence, it suffices to compute the following
three numbers:

$C_{32},\qquad C_{41}\qquad\textnormal{and}\qquad C_{50}.$

Let us define $\mathsf{P}\mathsf{A}_{3}\subset\mathsf{M}^{1}_{0}$ to be the
locus $(H_{1},H_{d},p_{1})$ such that curve $H_{d}$ has a tacnode at $p_{1}$
and the line $H_{1}$ is the branch of the tacnode. Pictorially, it is denoted
by the following picture

Notice the difference between the spaces
$\overline{\mathsf{A}^{\mathsf{L}}_{3}}$ and $\mathsf{P}\mathsf{A}_{3}$; the
latter lies in the closure of the former. The space
$\mathsf{A}_{3}^{\mathsf{L}}$ is pictorially represented as follows

It will now be shown that computing intersection numbers involving the class
$[\mathsf{P}\mathsf{A}_{3}]$ will enable us to compute the numbers $C_{ij}$.
In particular, it will be shown that

$\displaystyle C_{32}$ $\displaystyle=[\mathsf{P}\mathsf{A}_{3}]\cdot
y_{d}^{\delta_{d}-3},\qquad C_{41}=[\mathsf{P}\mathsf{A}_{3}]\cdot
y_{d}^{\delta_{d}-4}b_{1}\qquad\textnormal{and}\qquad
C_{50}=[\mathsf{P}\mathsf{A}_{3}]\cdot y_{d}^{\delta_{d}-5}b_{1}^{2}.$ (8.17)

Let us justify the first term of (8.17), namely the computation of $C_{32}$.
Note that by definition,

$C_{32}=[\mathsf{A}_{3}^{\mathsf{F}}]\cdot y_{1}^{2}y_{d}^{\delta_{d}-3}.$

Hence, $C_{32}$ denotes the number of degree $d$ curves passing through
$\delta_{d}-3$ generic points and having a tacnode. We now note that
intersecting $[\mathsf{P}\mathsf{A}_{3}]$ with $y_{d}^{\delta_{d}-3}$ makes
the curve pass through $\delta_{d}-3$ points. By definition of
$\mathsf{P}\mathsf{A}_{3}$, the line is now fixed because there is a unique
line that passes through the tacnodal point and is the branch of the tacnode.
This proves the first equation of (8.17). The remaining two equations follow
similarly.

Hence, it has been shown that to prove (8.2), it suffices to compute all
intersection numbers involving the class $[\mathsf{P}\mathsf{A}_{3}]$. It will
now be explained how to compute those numbers.

Although we are enumerating curves with one singularity, the intersection
theory will be done on the two pointed space $\mathsf{M}^{2}_{0}$. Let
$(H_{1},\,H_{d},\,p_{1},\,p_{2})\,\in\,\overline{\mathsf{A}_{1}^{\mathsf{L}}\mathsf{A}^{\mathsf{L}}_{1}}$.
Now impose the condition that the two points $p_{1}$ and $p_{2}$ come
together. It will be shown shortly that when that happens, we get a curve in
$\mathsf{PA}_{3}$. The intuitive reason for that is the following picture:

Let

$\alpha\,:=\,y_{1}^{r}y_{d}^{s}b_{1}^{\varepsilon}.$

Hence, by the collision lemma it follows that

$\displaystyle[\mathsf{PA}_{3}]\cdot\alpha$
$\displaystyle\,\,=\,\,[\mathsf{A}_{1}^{\mathsf{L}}\mathsf{A}_{1}^{\mathsf{L}}]\cdot(b_{1}+b_{2}-y_{1})\cdot\alpha.$
(8.18)

Using the results of Section 8.2, we can compute all intersection numbers
involving the class
$[\mathsf{A}_{1}^{\mathsf{L}}\mathsf{A}_{1}^{\mathsf{L}}]$. Hence, using
(8.18), we can compute all intersection numbers involving the class
$[\mathsf{PA}_{3}]$.

It remains to justify (8.18). First of all recall the proof of the fact that
when a $\mathsf{T}_{1}$ point and another $\mathsf{T}_{1}$ point collide, we
get a $\mathsf{T}_{3}$ point. The proof in fact shows the following: suppose
the first derivative of $f$ vanishes at $p_{1}$ and $p_{2}$, then when the two
points coincide, the first, second and third derivatives coincide. The proof
does not in any way require the points to be smooth points of the curve.
Hence, when two nodal points lying on a line coincide, the first, second and
third derivatives along the line vanish. Furthermore, the point is a singular
point of the curve. Note that any curve in $\mathsf{PA}_{3}$ satisfies these
conditions.

We will now show that every element of $\mathsf{PA}_{3}$ lies in the closure.
We switch to affine space. Let $f$ be a curve that has a $\mathsf{PA}_{3}$
point at the origin. Hence, the Taylor expansion of $f$ is given by

$\displaystyle f(x,y)$
$\displaystyle=f_{11}xy+\frac{f_{02}}{2}y^{2}+\frac{f_{21}}{2}x^{2}y+\frac{f_{12}}{2}xy^{2}+\frac{f_{03}}{6}y^{3}$
$\displaystyle+\frac{f_{40}}{24}x^{4}+\frac{f_{31}}{6}x^{3}y+\frac{f_{22}}{4}x^{2}y^{2}+\frac{f_{13}}{6}xy^{3}+\frac{f_{04}}{24}y^{4}+\ldots$

We will now try to construct a nearby curve $f_{t}$ that has a nodal point at
$(0,\,0)$ and at $(t,\,0)$. We will also show that every nearby curve is of
the type we have constructed.

The Taylor expansion of $f_{t}$ is given by

$\displaystyle f_{t}(x,y)$
$\displaystyle=\frac{f_{t_{20}}}{2}x^{2}+f_{t_{11}}xy+\frac{f_{t_{02}}}{2}y^{2}+\frac{f_{t_{30}}}{6}x^{3}+\frac{f_{t_{21}}}{2}x^{2}y+\frac{f_{t_{12}}}{2}xy^{2}+\frac{f_{t_{03}}}{6}y^{3}$
$\displaystyle+\frac{f_{t_{40}}}{24}x^{4}+\frac{f_{t_{31}}}{6}x^{3}y+\frac{f_{t_{22}}}{4}x^{2}y^{2}+\frac{f_{13}}{6}xy^{3}+\frac{f_{t_{04}}}{24}y^{4}+\ldots$

We now impose the condition $f_{t}(t,\,0)\,=\,0$, $f_{t_{x}}(t,\,0)\,=\,0$ and
$f_{t_{y}}(t,\,0)\,=\,0$. Using the equation $f_{t_{y}}(t,\,0)\,=\,0$, we can
uniquely solve for $f_{t_{11}}$. Next, using the equation
$f_{t_{x}}(t,\,0)\,=\,0$, we can uniquely solve for $f_{t_{20}}$. Plugging
these two solutions in the equation $f_{t}(t,\,0)\,=\,0$, we can uniquely
solve for $f_{t_{30}}$.

This gives us a procedure to construct a curve $f_{t}$, close to $f$, that has
a nodal point at $(0,\,0)$ and at $(t,\,0)$. Since our solution was unique,
this implies that every nearby curve is of this type, i.e., there is only one
branch.

It remains to compute the multiplicity of the intersection. We are basically
setting the $x$-coordinate to be equal to zero. But the $x$-coordinate is $t$.
Hence, the order of vanishing is one (since there is exactly one branch). This
completes the proof of (8.18).

##  9\. Counting rational curves

###  9.1. A review of Gathmann’s approach to count curves with tangencies

In his papers [7], [8] and [9], Andreas Gathman gives a systematic approach to
solve the following question: Let $Y$ be a hypersurface inside
$\mathbb{CP}^{n}$. What are the characteristic number of rational degree $d$
curves in $\mathbb{CP}^{n}$ that are tangent to $Y$ at a given point to order
$k$? Gathmann successfully solves the above question for any ample
hypersurface and any $k$. He goes on to use this study to compute Gromov-
Witten invariants of the Quintic Threefold.

Consider a special case of Gathmann’s result when $Y$ is a line inside
$\mathbb{CP}^{2}$ and ask the following question: How many rational degree $d$
curves are there in $\mathbb{CP}^{2}$, that pass through $3d-2$ generic points
and are tangent to a given line? In this subsection we will recapitulate
Gathmann’s approach to solve this question. The next subsection describes an
alternative approach to the question based on applying Figure 1 in the setting
of stable maps (which is also discussed in [6, pp. 41] and [12, pp.
1179–1180]).

Let us now describe Gathmann’s idea. The setup is modified in order to solve a
slightly more general question. The question we will solve is as follows: How
many pairs — consisting of a line and a rational degree $d$ curve, passing
through $m$ points and $n$ points respectively — are there such that the line
is tangent to the curve and $m+n\,=\,3d+1$? The special case of $m\,=\,2$
corresponds to the line being fixed.

Let us start by describing the ambient space. Recall that
$\overline{M}_{0,0}(\mathbb{CP}^{2},\,d)$ is the compactification of the
Kontsevich moduli space of rational curves (with no marked points). Let
$\mathcal{H}$ denote the divisor that corresponds to the subspace of curves
that pass through a generic point. We note that the intersection number

$\displaystyle[\overline{M}_{0,0}(\mathbb{CP}^{2},\,d)]\cdot\mathcal{H}^{m}$

is computable via Kontsevich’s recursion formula. For dimensional reasons, the
above number is nonzero only when $m\,=\,3d-1$. On the zero pointed moduli
space, this is the only intersection number that is relevant for our purposes;
it is also called a primary Gromov-Witten invariant.

Now consider $\overline{M}_{0,1}(\mathbb{CP}^{2},\,d)$, the one marked moduli
space. As before, we have the divisor $\mathcal{H}$ which corresponds to the
subspace of curves whose image passes through a generic point. But now, there
are two other things as well. Denote the pullback (via the evaluation map) of
the hyperplane class in $\mathbb{CP}^{2}$ by $\textnormal{ev}^{*}(b_{1})$.
Finally, consider

$\mathcal{L}\longrightarrow\overline{M}_{0,1}(\mathbb{CP}^{2},\,d),$

the universal tangent bundle, whose fibre over each point is the tangent space
at that marked point. Denote the first Chern class of the dual of this bundle
by $\psi$, i.e.,

$\displaystyle\psi$ $\displaystyle\,\,:=\,\,c_{1}(\mathcal{L}^{*}).$

It is a standard fact that all the intersection numbers

$\displaystyle[\overline{M}_{0,1}(\mathbb{CP}^{2},\,d)]\cdot\mathcal{H}^{m}\cdot\textnormal{ev}^{*}(b_{1})^{n}\cdot\psi^{\theta}$
(9.1)

are computable for any choice of $m,n$ and $\theta$. This can be seen from the
paper [10, p. 311, Proposition 2.2]. When $\theta$ is greater than zero, the
above number is also called a secondary Gromov-Witten invariant.

Let us now explain the geometric idea behind Gathmann’s method to enumerate
rational curves that are tangent to a fixed line, and how to modify his method
when the line is not fixed but is free to move in a family. Denote by
$\overline{M}_{0,k}(\mathbb{CP}^{2},\,d)$ the $k$ marked moduli space. The
pullback of the hyperplane classes (via the evaluation map) are denoted by

$\displaystyle\textnormal{ev}^{*}(b_{1}^{n_{1}}),$
$\displaystyle\,\,\cdots,\,\,\textnormal{ev}^{*}(b_{k}^{n_{k}}).$

Now define $\mathsf{M}_{1}$ as

$\displaystyle\mathsf{M}_{1}$
$\displaystyle\,\,:=\,\,\mathcal{D}_{1}\times\overline{M}_{0,1}(\mathbb{CP}^{2},\,d)\times\mathbb{CP}^{2}_{1},$

where $\mathbb{CP}^{2}_{i}$ denotes a copy of $\mathbb{CP}^{2}$ and
$\mathcal{D}_{1}$ denotes the space of lines in $\mathbb{CP}^{2}$. The
corresponding hyperplane classes will be denoted by $a_{i}$ and $y_{1}$.

Note that an element of $\mathsf{M}_{1}$ comprises of a line, a one pointed
rational curve (namely an element of
$\overline{M}_{0,1}(\mathbb{CP}^{2},\,d)$), and a point of
$\mathbb{CP}^{2}_{1}$. The relevant classes that live in $\mathsf{M}_{1}$ are

$\displaystyle
y_{1},~{}~{}\mathcal{H},~{}~{}\textnormal{ev}^{*}(b_{1}),\,\psi,~{}~{}\textnormal{and}~{}~{}a_{1}.$

Since we can compute all the primary and secondary Gromov-Witten invariants
(i.e., the numbers in equation (9.1)), we can compute all the following
intersection numbers

$\displaystyle\big{[}\mathsf{M}_{1}\big{]}\cdot\mathcal{H}^{m}\cdot\textnormal{ev}^{*}(b_{1}^{n_{1}})\cdot\psi^{\theta}\cdot
y_{1}^{r}\cdot a_{1}^{s}.$ (9.2)

Let us now define $\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}$ as the
following subspace of $\mathsf{M}_{1}$:

$\displaystyle\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}$
$\displaystyle\,\,:=\,\,\\{([f_{1}],\,[u,\,y_{1}],\,x_{1})\,\in\,\mathsf{M}_{1}\,\,\big{|}\,\,u(y_{1})\,=\,x_{1},~{}~{}f_{1}(x_{1})\,=\,0\\}.$

Remark: Note the following fact: we are typically going to denote the marked
point of the domain by the letter $y_{i}$. It is not going to cause any
confusion with the other place where the letter $y_{1}$ is used, namely for
the hyperplane class of $\mathcal{D}_{1}$.

Returning to the discussion, an element of
$\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}$ can be pictorially described
as follows:

Let us now see how we can go about describing the class
$\big{[}\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\big{]}$. First of all,
we note that the condition $f_{1}(q_{1})=0$ is same as intersecting with the
class $(y_{1}+a_{1})$ (see equation (5.2)). It remains to figure out how to
express the condition $u(y_{1})=q_{1}$. Consider the map

$\displaystyle\textnormal{ev}\times\textnormal{id}_{\mathbb{CP}^{2}_{1}}:\mathsf{M}_{1}$
$\displaystyle\longrightarrow\mathbb{CP}^{2}\times\mathbb{CP}^{2}_{1},\qquad\textnormal{given
by}\qquad\textnormal{ev}\times\textnormal{id}_{\mathbb{CP}^{2}_{1}}([f_{1}],[u,y_{1}],x_{1}):=(u(y_{1}),x_{1}).$

The condition $u(y_{1})=x_{1}$ is same as intersecting with the pullback of
the diagonal, namely

$\displaystyle(\textnormal{ev}\times\textnormal{id}_{\mathbb{CP}^{2}_{1}})^{*}(\Delta_{b_{1}a_{1}}).$

Hence, we conclude that

$\displaystyle\Big{[}\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}$
$\displaystyle\,=\,\Big{(}\textnormal{ev}^{*}(b_{1}^{2})+\textnormal{ev}^{*}(b_{1})a_{1}+a_{1}^{2}\Big{)}\cdot(y_{1}+a_{1}).$
(9.3)

Since all the intersection numbers in equation (9.2) are computable, we
conclude from equation (9.3) that all the intersection numbers

$\displaystyle\Big{[}\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot\textnormal{ev}^{*}(b_{1}^{n_{1}})\cdot\psi^{\theta}\cdot
y_{1}^{r}\cdot a_{1}^{s}$ (9.4)

are computable.

Let us now define $\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}$. It is the
subspace of $\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}$, where the curve
is tangent to the line at the point $x_{1}$. It is pictorially described as
follows:

Let us now explain Gathmann’s approach to compute intersection numbers
involving the class
$\big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\big{]}$. In the next
subsection, we will give an alternative approach to compute these intersection
numbers.

Gathmann’s approach is best summarized by the following equation

$\displaystyle\Big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\Big{]}$
$\displaystyle=\Big{[}\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}\cdot\Big{(}\psi+y_{1}+a_{1}\Big{)}.$
(9.5)

Let us explain why equation (9.5) is true. We note that for a rational curve
$[u,y_{1}]$ to be tangent to the line, the differential $du|_{y_{1}}$ should
take values in the tangent space of the line. In other words, the differential
$du|_{y_{1}}$ has to vanish in the normal direction of the line. Let us
interpret this condition as the vanishing of a section of an appropriate line
bundle.

First of all, let us consider the line bundle
$\mathbb{L}\longrightarrow\big{(}\mathsf{T}_{0}\big{)}_{\textnormal{st}}$
whose fibre over each point $([f_{1}],[u,y_{1}],x_{1})$ is the tangent space
of the line $f_{1}^{-1}(0)$ at the point $x_{1}$. This is basically the same
line bundle we defined in subsection 8.1. For the convenience of the reader,
we review the definition, namely the short exact sequence into which the line
bundle fits:

$\displaystyle 0\longrightarrow$ $\displaystyle\mathbb{L}\longrightarrow
T\mathbb{CP}^{2}_{1}|_{x_{1}}\longrightarrow\gamma_{\mathcal{D}_{1}}^{*}\otimes\gamma_{\mathbb{CP}^{2}_{1}}^{*}\longrightarrow
0.$ (9.6)

The condition that $[u,y_{1}]$ is tangent to the line at $x_{1}$ (namely that
the differential $du|_{y_{1}}$ vanishes in the normal direction to the line)
can be interpreted as a section of the following line bundle

$\displaystyle\mathcal{L}^{*}\otimes\textnormal{ev}^{*}\Big{(}T\mathbb{CP}^{2}_{1}/\mathbb{L}\Big{)}.$

Using equation (9.6), we conclude that the Euler class of the above line
bundle is equal to $(\psi+y_{1}+a_{1})$ which gives us equation (9.5).

Note that using equations (9.5), (9.3), and the fact that all the intersection
numbers of equation (9.2) are computable, we conclude that all the following
intersection numbers

$\displaystyle\Big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot\textnormal{ev}^{*}(b_{1}^{n_{1}})\cdot\psi^{\theta}\cdot
y_{1}^{r}\cdot a_{1}^{s}$ (9.7)

are computable.

This idea can be pushed further to enumerate rational curves with higher order
tangency. However, starting from second order tangency, there is a non-trivial
geometric phenomenon that occurs. In the closure of curves tangent to a given
line, there will be bubble maps. These bubble maps will be in the zero locus
of the section that computes the second derivative. Hence, one has to analyse
a degenerate locus and subtract off from the Euler class. Gathmann does that
successfully in his papers [7], [8] and [9] and is able to enumerate rational
curves tangent to any order.

###  9.2. A new method to count rational curves with first order tangency

In this subsection, we will give an alternate approach to enumerate rational
curves with tangencies based on figure 1. This idea has been discussed in [6]
(see page 41) and [12, pp. 1179–1180].

Let us continue with the setup of Section 9.1. We will do intersection theory
on $\mathsf{M}_{2}$, which is defined as

$\displaystyle\mathsf{M}_{2}$
$\displaystyle\,:=\,\mathcal{D}_{1}\times\overline{M}_{0,2}(\mathbb{CP}^{2},\,d)\times\mathbb{CP}^{2}_{1}\times\mathbb{CP}^{2}_{2}.$

The relevant classes that live in $\mathsf{M}_{2}$ are

$\displaystyle
y_{1},~{}~{}\mathcal{H},~{}~{}\textnormal{ev}^{*}(b_{1}),~{}~{}\textnormal{ev}^{*}(b_{2}),~{}~{}a_{1}~{}~{}\textnormal{and}~{}~{}a_{1}.$

Define the following projection map

$\displaystyle\pi_{21}\,:\,\mathsf{M}_{2}$
$\displaystyle\,\longrightarrow\,\mathsf{M}_{1},\qquad\textnormal{given
by}\qquad\pi_{21}([f_{1}],\,[u,\,y_{1},\,y_{2}],\,x_{1},\,x_{2})\,:=\,([f_{1}],\,[u,\,y_{1}],\,x_{1}).$
(9.8)

Basically, the map forgets the second marked point on
$\overline{M}_{0,2}(\mathbb{CP}^{2},\,d)$. Furthermore, it forgets the second
factor $\mathbb{CP}^{2}_{2}$. The cycles that have been defined in the one
pointed moduli space $\mathsf{M}_{1}$ can be pulled back to the two pointed
moduli space $\mathsf{M}_{2}$ via this projection map.

Let us now define the following divisor in the two pointed moduli space
$\overline{M}_{0,2}(\mathbb{CP}^{2},\,d)$. We will denote this divisor by the
symbol

$\displaystyle\big{[}y_{1}=y_{2}\big{]}.$

The above symbol allows us to guess the divisor we will be talking about: this
divisor is represented by the space of stable maps with two marked points,
where the two marked points have coincided. It can be pictorially represented
as follows:

Usually, in the literature, it is denoted by $D(\\{\emptyset\\},{2};d,0)$ (see
[5]). Note that this space can be identified with the one marked moduli space
$\overline{M}_{0,1}(\mathbb{CP}^{2},\,d)$. Let us now explain how to intersect
with this divisor. It is a standard fact that all the primary intersection
numbers

$\displaystyle\big{[}\overline{M}_{0,2}(\mathbb{CP}^{2},\,d)\big{]}\cdot\mathcal{H}^{m}\cdot\textnormal{ev}^{*}(b_{1}^{n_{1}})\cdot\textnormal{ev}^{*}(b_{2}^{n_{2}})$
(9.9)

are computable.

Now define $\big{(}\mathsf{T}_{0}\mathsf{T}_{0}\big{)}_{\textnormal{st}}$ to
be the following subspace of $\mathsf{M}_{2}$: it comprises of a line and a
rational curve and two marked points, where the two marked points lie on the
line. It is pictorially given by

It will be shown that the following equality of cycles holds in
$\mathsf{M}_{2}$

$\displaystyle\Big{(}\textnormal{ev}^{*}(b_{1}^{2})+\textnormal{ev}^{*}(b_{1})a_{1}+a_{1}^{2}\Big{)}\cdot(y_{1}+a_{1})$
$\displaystyle\cdot\Big{(}\textnormal{ev}^{*}(b_{2}^{2})+\textnormal{ev}^{*}(b_{2})a_{2}+a_{2}^{2}\Big{)}\cdot(y_{1}+a_{2})$
$\displaystyle=\Big{[}\big{(}\mathsf{T}_{0}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}+\Big{[}\big{(}\mathsf{T}_{0}\big{)}_{\textsf{st}}\Big{]}.$
(9.10)

A slight abuse of notation is being made here. Equation (9.10) is an equality
of cycles in $\mathsf{M}_{2}$. The second term on the right hand side, namely
$\Big{[}\big{(}\mathsf{T}_{0}\big{)}_{\textsf{st}}\Big{]}$ is a cycle in
$\mathsf{M}_{1}$. However, we can identify that as a cycle in $\mathsf{M}_{2}$
via the following process: we attach a ghost bubble at the marked point and
add one more marked point on the ghost component. This is the intended meaning
of the second term of the right hand side.

The reason for why equation (9.10) is true is similar to why Theorem 5.2 is
true (for the special case $n\,=\,1$ and $k_{1}\,=\,0$). It has been shown in
subsection 9.1 that intersecting with

$\big{(}\textnormal{ev}^{*}(b_{1}^{2})+\textnormal{ev}^{*}(b_{1})a_{1}+a_{1}^{2}\big{)}\cdot(y_{1}+a_{1})$

is imposing the condition that the first marked point lies on the line.
Similarly, intersecting with

$\big{(}\textnormal{ev}^{*}(b_{2}^{2})+\textnormal{ev}^{*}(b_{2})a_{2}+a_{2}^{2}\big{)}\cdot(y_{1}+a_{2})$

imposes the condition that the second marked point lies on a line.
Intersecting with both of them gives us the condition that both the marked
points lie on the line. However, the two marked points can coincide. That
gives us the degenerate locus corresponding to the second term on the right
hand side of equation (9.10). Rewriting equation (9.10), one concludes that

$\displaystyle\Big{[}\big{(}\mathsf{T}_{0}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}=$
$\displaystyle\Big{(}\textnormal{ev}^{*}(b_{1}^{2})+\textnormal{ev}^{*}(b_{1})a_{1}+a_{1}^{2}\Big{)}\cdot(y_{1}+a_{1})$
$\displaystyle\cdot\Big{(}\textnormal{ev}^{*}(b_{2}^{2})+\textnormal{ev}^{*}(b_{2})a_{2}+a_{2}^{2}\Big{)}\cdot(y_{1}+a_{2})-\Big{[}\big{(}\mathsf{T}_{0}\big{)}_{\textsf{st}}\Big{]}.$
(9.11)

Using equation (9.11), the fact that all the intersection numbers of equation
(9.9) are computable and the fact that all the intersection numbers in
equation (9.4) are computable, one concludes that all the intersection numbers

$\displaystyle\Big{[}\big{(}\mathsf{T}_{0}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot\textnormal{ev}^{*}(b_{1}^{n_{1}})\cdot\textnormal{ev}^{*}(b_{2}^{n_{2}})\cdot
y_{1}^{r}\cdot a_{1}^{n_{1}}\cdot a_{2}^{n_{2}}$ (9.12)

are computable.

Next, let us define $\big{(}\mathsf{A}_{1}\big{)}_{\textnormal{st}}$ to be the
following subspace of the two pointed moduli space
$M_{0,2}(\mathbb{CP}^{2},\,d)$:

$\displaystyle\big{(}\mathsf{A}_{1}\big{)}_{\textnormal{st}}$
$\displaystyle:=\\{[u,y_{1},y_{2}]\in
M_{0,2}(\mathbb{CP}^{2},\,d):u(y_{1})=u(y_{2})\\}.$

Let us also define
$\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}$ to be the
following subspace of $\mathsf{M}_{2}$

$\displaystyle\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\,:=\,\\{([f_{1}],\,[u,\,y_{1},\,y_{2}],\,x_{1},\,x_{2})\,\in\,\mathsf{M}_{2}$
$\displaystyle\,:\,[u,\,y_{1},\,y_{2}]\,\in\,\big{(}\mathsf{A}_{1}\big{)}_{\textnormal{st}},$
$\displaystyle~{}u(y_{1})\,=\,x_{1},~{}u(y_{2})\,=\,x_{2},~{}f_{1}(x_{1})\,=\,0,~{}f_{1}(x_{2})\,=\,0\\}.$

Pictorially, the space
$\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}$ can be
represented as follows:

We will now explain how to compute the characteristic number of rational
curves with first order tangency. On the space
$\overline{\big{(}\mathsf{T}_{0}\mathsf{T}_{0}\big{)}}_{\textnormal{st}}$, let
us now impose the additional condition that $x_{1}\,=\,x_{2}$. By Lemma 4.1,
this is same as intersecting with $(a_{1}+a_{2}-y_{1})$. Hence,

$\displaystyle\Big{[}\big{(}\mathsf{T}_{0}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}\cdot(a_{1}+a_{2}-y_{1})$
$\displaystyle=\Big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textsf{st}}\Big{]}+2\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}.$
(9.13)

Again, we are making an abuse of notation here. Equation (9.13) is an equality
of cycles in $\mathsf{M}_{2}$. The first term on the right hand side, namely
$\Big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textsf{st}}\Big{]}$ is a cycle in
$\mathsf{M}_{1}$. However, we can identify that as a cycle in $\mathsf{M}_{2}$
via the following process: we attach a ghost bubble at the marked point and
add one more marked point on the ghost component. This is the intended meaning
of the first term of the right hand side.

Let us now see why (9.13) is true. We note that when we impose the condition
$x_{1}=x_{2}$, there are two possibilities. The first possibility is that the
corresponding points in the domain are also the same and hence it corresponds
to a point of tangency as given by the following picture:

This corresponds to the first term of the right hand side of equation (9.13).
But there is a second possibility. The points in the domain can be distinct.
This is a point of self intersection, which is a nodal point. This corresponds
to the second term in the right hand side of (9.13). The intersection occurs
with a multiplicity of $2$, because if a line intersects a nodal point of a
curve, then it contributes $2$ to the intersection. Let us denote $\mu$ as

$\displaystyle\mu$
$\displaystyle:=\mathcal{H}^{m}\cdot\textnormal{ev}^{*}(b_{1}^{n_{1}})\cdot
y_{1}^{r}\cdot a_{1}^{n_{1}}.$

Rewriting equation (9.13) and intersecting with $\mu$, we conclude that

$\displaystyle\Big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textsf{st}}\Big{]}\cdot\mu$
$\displaystyle=\Big{[}\big{(}\mathsf{T}_{0}\mathsf{T}_{0}\big{)}_{\textnormal{st}}\Big{]}\cdot(a_{1}+a_{2}-y_{1})\cdot\mu-2\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}\cdot\mu.$
(9.14)

Since all the intersection numbers in (9.12) are computable, we conclude that
the first term on the right hand side of (9.14) is computable. It will be
shown in a moment that using the result of [2], one can also compute the
second term. Hence, the right hand side of (9.14) is computable for any $\mu$.
This gives us an alternative way to compute the characteristic number of
rational curves tangent to a given line to first order. The reader can refer
to subsection 10.4 where we have tabulated explicit numbers.

This idea can actually be pursued further. Suppose we wish to enumerate curves
with second order tangency. Then as expected, we will define the space
$(\mathsf{T}_{1}\mathsf{T}_{0})_{\mathsf{st}}$. We will then impose the
condition of the two points coming together. This results in two things. First
of all we have curves with second order tangency. We will also encounter
curves with a node lying on the line, such that the line is one of the
branches of the node. Hence, we need to compute that number. In order to do
that, we will define the space
$(\mathsf{A}_{1}^{\mathsf{L}}\mathsf{T}_{0})_{\mathsf{st}}$ and require the
points to come together. The two things we get are, a nodal curve, with the
line being one of the branches of the node. The second thing we get is a curve
with a triple point. To compute the latter, we can use the result of [2]. We
have actually carried out this entire computation. The details of the
computation are available on request. However, the analysis of the degenerate
locus and its contribution to the intersection is more non-trivial. We hope to
pursue this approach in a more thorough way in future and figure out how to
enumerate rational curves with $k^{\textnormal{th}}$-order tangency.

It remains to be shown how to compute the second term on the right hand side
of (9.14). First of all, we note that intersecting
$[\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}]$ with $a_{1}$
or $\textnormal{ev}^{*}(b_{1})$ is the same thing. Hence, it suffices to give
a procedure to find intersection numbers with $\mathcal{H}^{m}\cdot
y_{1}^{r}\cdot a_{1}^{n}$. Let $\alpha:=(d,2)$ and let $N_{\alpha}(m,n)$
denote the intersection number defined in [2, pp. 5]. Here $m$ and $n$ are
non-negative integers. From the definition of $N_{\alpha}(m,n)$ and the proof
of the correspondence result, [2, pp. 13–16], we conclude that

$\displaystyle\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot
y_{1}^{r}\cdot a_{1}^{n}\,=\,$ $\displaystyle 0$ if $r\,=\,0$, (9.15a)
$\displaystyle\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot
y_{1}^{r}\cdot a_{1}^{n}\,=\,$ $N_{\alpha}(m,n)$ if $r\,=\,1$, (9.15b)
$\displaystyle\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot
y_{1}^{r}\cdot a_{1}^{n}\,=\,$ $N_{\alpha}(m,1)$ if $r\,=\,2$ and $n\,=\,0$,
(9.15c)
$\displaystyle\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot
y_{1}^{r}\cdot a_{1}^{n}\,=\,$ $N_{\alpha}(m,2)$ if $r\,=\,2$ and $n\,=\,1$,
(9.15d)
$\displaystyle\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot
y_{1}^{r}\cdot a_{1}^{n}\,=\,$ $\displaystyle 0$ if $r\,=\,2$ and
$n\,\geq\,2$, (9.15e)
$\displaystyle\Big{[}\big{(}\mathsf{A}_{1}^{\mathsf{L}}\big{)}_{\textnormal{st}}\Big{]}\cdot\mathcal{H}^{m}\cdot
y_{1}^{r}\cdot a_{1}^{n}\,=\,$ $\displaystyle 0$ if $r\,\geq\,3$. (9.15f)

Let us see why this is true. The first case, (9.15a) follows from dimensional
reasons.

Let us justify the next case, (9.15b). From the proof of the correspondence
result, [2, pp. 13–16], we conclude that $N_{\alpha}(m,n)$ denotes the number
of rational curves passing through $m$ generic points and with a choice of a
node lying at the intersection of $n$ generic lines. Once such a nodal curve
is fixed, a unique line passes through one point and the nodal point. That
precisely corresponds to the left hand side of the equation.

Next, let us justify (9.15c). We start by unwinding the left hand side.
Intersecting with $y_{1}^{2}$ corresponds to fixing a line. Now we intersect
with $\mathcal{H}^{m}$. This is equal to the number of rational curves passing
through $m$ generic points and with a choice of a node lying on a line. That
is precisely equal to $N_{\alpha}(m,1)$; again this follows from the proof of
the correspondence result.

Next, let us justify (9.15d). Again, we begin by unwinding the left hand side.
Intersecting with $y_{1}^{2}$ corresponds to fixing a line. We recall that the
nodal point will be lying on this line. Intersecting with $a_{1}$ is
restricting the nodal point to lie on another line; in other words, we are
restricting the nodal point to lie on a fixed point. Now we intersect with
$\mathcal{H}^{m}$. This is equal to the number of rational curves passing
through $m$ generic points and with a choice of a node lying on a fixed point.
That is precisely equal to $N_{\alpha}(m,2)$; again this follows from the
proof of the correspondence result. The next case, (9.15e) follows
immediately, since intersecting with $y_{1}^{2}$ restricts the nodal point to
lie on a line and intersecting with $a_{1}^{2}$ is restricting the nodal point
to lie on a point; since a line and a point do not intersect, this number is
clearly zero.

Finally, (9.15f) follows immediately since $y_{1}^{3}$ is zero.

###  9.3. Another method to count rational curves with first order tangency

We conclude this paper by illustrating that there is yet another way to
exploit Figure 1 to enumerate stable maps with tangencies. Recall that in the
previous subsection, we made the two points in the images come together. This
results in two things; curves that are tangent to the line and curves that
have a point of self intersection lying on the line. The former occurs when
the points in the domain also coincide, while the latter occurs when the
points in the domain remain distinct.

What if we could simply make the points in the domain come together? In this
final part of the paper, we will do precisely that. The idea will be
implemented by extending Kontsevich’s idea to enumerate rational curves, i.e.,
the WDVV equation. The details are as follows.

Recall that Kontsevich’s recursion formula gives an answer to the following
question: How many degree $d$ rational curves are there in $\mathbb{CP}^{2}$
that pass through $3d-1$ generic points? Denote this number by $n_{d}$. Let us
also denote $\mathsf{N}_{d}^{\mathsf{T}_{1}}$ to be the number of rational
degree $d$ curves in $\mathbb{CP}^{2}$ that pass through $3d-2$ that are
tangent to a given line. A recursive formula for
$\mathsf{N}_{d}^{\mathsf{T}_{1}}$ will be obtained by using the WDVV equation.
Note that for simplicity, the line will be kept fixed; the more general case
of keeping the line variable can be worked out with with very little extra
effort (the only issue would be notational).

Consider $\overline{M}_{0,4}(\mathbb{CP}^{2},\,d)$, the moduli space of genus
zero stable maps, with $4$ marked points. Let $\mathsf{L}$ be a fixed line
inside $\mathbb{CP}^{2}$. Define $\mathsf{X}_{d}$ to be the following subspace
of $M_{0,4}(\mathbb{CP}^{2},\,d)$: it is the subspace of rational degree $d$
curves where the image of the first two marked points lie on two distinct
points of the line $\mathsf{L}$. It is pictorially represented as follows:

We denote $\overline{\mathsf{X}}_{d}$ to be the closure of $\mathsf{X}_{d}$
inside $\overline{M}_{0,4}(\mathbb{CP}^{2},\,d)$.

Following the idea behind Kontsevich’s recursion formula, let us consider the
forgetful map

$\displaystyle\pi\,\,:\,\,\overline{M}_{0,4}(\mathbb{CP}^{2},\,d)$
$\displaystyle\,\,\longrightarrow\,\,\overline{M}_{0,4}.$

Let $[(ij|kl)]$ denote the divisor in $\overline{M}_{0,4}$ corresponding to
the wedge of two spheres and where the marked points $(y_{i},y_{j})$ lie on
the one sphere and $(y_{k},y_{l})$ lie on the other sphere. In
$\overline{M}_{0,4}(\mathbb{CP}^{2},\,d)$, define the class $\mathcal{Z}$ as

$\displaystyle\mathcal{Z}$
$\displaystyle:=\textnormal{ev}_{3}^{*}(\textnormal{pt})\cdot\textnormal{ev}_{4}^{*}(\textnormal{pt})\cdot\mathcal{H}^{3d-4}.$

Since $\overline{M}_{0,4}$ is isomorphic to $\mathbb{CP}^{1}$, any two points
determine the same divisor. Hence, $[(12|34)]$ is equal to $[(13|24)]$ as
divisors. Hence,

$\displaystyle\big{[}\overline{\mathsf{X}}_{d}\big{]}\cdot\big{[}\pi^{*}(12|34)\big{]}\cdot\mathcal{Z}$
$\displaystyle=\big{[}\overline{\mathsf{X}}_{d}\big{]}\cdot\big{[}\pi^{*}(13|24)\big{]}\cdot\mathcal{Z}.$
(9.16)

Note that the left hand side and right hand side of (9.16) denote intersection
numbers in $\overline{M}_{0,4}(\mathbb{CP}^{2},\,d)$. We will now unravel both
the sides of the equation and get a recursive formula.

We start by unwinding the left hand side of (9.16) by looking at it
geometrically. First, consider the possibility when the two points $y_{3}$ and
$y_{4}$ come together. That results in a bubble map of type $(d_{1},d_{2})$
such that

  * •

The marked points $y_{1}$ and $y_{2}$ lie on the $d_{1}$ component.

  * •

The marked points $y_{3}$ and $y_{3}$ lie on the $d_{2}$ component.

  * •

The image of the marked points $y_{1}$ and $y_{2}$ intersect the line
$\mathsf{L}$.

  * •

The image of $y_{3}$ and $y_{4}$ coincide with two generic points (this
corresponds to intersection with $\textnormal{ev}_{3}^{*}(\textnormal{pt})$
and $\textnormal{ev}_{4}^{*}(\textnormal{pt})$).

  * •

The entire configuration passes through $3d-4$ points (this corresponds to
intersecting with $\mathcal{H}^{3d-4}$).

The total number of such bubble maps is given by

$\displaystyle\sum_{d_{1}+d_{2}=d}\binom{3d-4}{3d_{1}-1}n_{d_{1}}n_{d_{2}}(d_{1}d_{2})d_{1}(d_{1}-1).$

The factor of $d_{1}d_{2}$ is to encode the number of choices for the bubble
point. The factor of $d_{1}(d_{1}-1)$ is there to encode the number of choices
where the image of the points $y_{1}$ and then $y_{2}$ can lie.

Let us now see what happens when $y_{1}$ and $y_{2}$ come together. What
happens is that we get a rational curve, tangent to the given line. The point
of tangency is given by the image of $y_{1}$ (which is also the same as the
image of $y_{2}$). Furthermore, the image of $y_{3}$ and $y_{4}$ coincide with
two generic points (which corresponds to intersection with
$\textnormal{ev}_{3}^{*}(\textnormal{pt})$ and
$\textnormal{ev}_{4}^{*}(\textnormal{pt})$) and the entire configuration
passes through $3d-4$ points (which corresponds to intersecting with
$\mathcal{H}^{3d-4}$). The total number of such objects is precisely equal to
the number of rational curves passing through $3d-2$ generic points, tangent
to a given line. That is precisely equal to $\mathsf{N}_{d}^{\mathsf{T}_{1}}$.
Hence, the left hand side of (9.16) is

$\displaystyle\mathsf{N}_{d}^{\mathsf{T}_{1}}+\sum_{d_{1}+d_{2}=d}\binom{3d-4}{3d_{1}-1}n_{d_{1}}n_{d_{2}}(d_{1}d_{2})d_{1}(d_{1}-1).$
(9.17)

Let us now unwind what is the right hand side of (9.16) by looking at it
geometrically. When the two points $y_{1}$ and $y_{3}$ come together (or
$y_{2}$ and $y_{4}$ come together), a bubble map of type $(d_{1},\,d_{2})$ is
obtained such that:

  * •

The marked points $y_{1}$ and $y_{3}$ lie on the $d_{1}$ component.

  * •

The marked points $y_{2}$ and $y_{4}$ lie on the $d_{2}$ component.

  * •

The image of the marked point $y_{1}$ intersects the line $\mathsf{L}$.

  * •

The image of the marked point $y_{2}$ intersects the line $\mathsf{L}$.

  * •

The image of $y_{3}$ coincides with a generic points (this corresponds to
intersection with $\textnormal{ev}_{3}^{*}(\textnormal{pt})$).

  * •

The image of $y_{4}$ coincides with a generic points (this corresponds to
intersection with $\textnormal{ev}_{4}^{*}(\textnormal{pt})$).

  * •

The entire configuration passes through $3d-4$ points (this corresponds to
intersecting with $\mathcal{H}^{3d-4}$).

The total number of such configurations is given by

$\displaystyle\sum_{d_{1}+d_{2}=d}\binom{3d-4}{3d_{1}-2}n_{d_{1}}n_{d_{2}}(d_{1}d_{2})(d_{1})(d_{2}).$
(9.18)

This is precisely the right hand side of (9.16). Equating (9.17) and (9.18),
we conclude that

$\displaystyle\mathsf{N}_{d}^{\mathsf{T}_{1}}$
$\displaystyle\,=\,\sum_{d_{1}+d_{2}=d}\Big{(}\binom{3d-4}{3d_{1}-2}d_{1}d_{2}-\binom{3d-4}{3d_{1}-1}d_{1}(d_{1}-1)\Big{)}n_{d_{1}}n_{d_{2}}d_{1}d_{2}.$
(9.19)

##  10\. Low degree checks

In this section we will perform various non-trivial low degree checks which
will be compared with the existing known results.

###  10.1. Counting smooth curves with tangencies: confirmation with Caporaso-
Harris

In section 5 it was shown that the following numbers

$\displaystyle\big{[}\mathsf{T}_{k_{1}}\ldots\mathsf{T}_{k_{n}}\big{]}\cdot
y_{1}^{2}y_{d}^{\delta_{d}-k},$

can be computed. We remind the reader that $k:=k_{1}+\ldots+k_{n}$. A few
values are displayed in the following table.

$d$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$
---|---|---|---|---|---|---
$\mu$ | $y_{1}^{2}y_{4}^{10}$ | $y_{1}^{2}y_{5}^{16}$ | $y_{1}^{2}y_{6}^{23}$ | $y_{1}^{2}y_{7}^{31}$ | $y_{1}^{2}y_{8}^{40}$ | $y_{1}^{2}y_{9}^{50}$
$\frac{1}{2}\big{[}\mathsf{T}_{1}\mathsf{T}_{1}\mathsf{T}_{2}\big{]}\cdot\mu$ | $0$ | $0$ | $0$ | $36$ | $144$ | $360$

Table 1.

Note that we have divided by a factor of $2$, because in the computation of
$\big{[}\mathsf{T}_{1}\mathsf{T}_{1}\mathsf{T}_{2}\big{]}\cdot\mu$ all the
tangency points are ordered. It is more natural to consider the two
$\mathsf{T}_{1}$ points as unordered.

The corresponding nonzero numbers obtained from the Caporaso-Harris by setting

$\delta:=0,\qquad\alpha:=(0)\qquad\textnormal{and}\qquad\beta:=(i,2,1),~{}~{}\forall~{}i=0,1,2.$

are as follows

$d$ | $7$ | $8$ | $9$
---|---|---|---
$N^{d,\delta}(\alpha,\beta)$ | $36$ | $144$ | $360$

Table 2.

We are assuming that the reader is familiar with the notation developed by
Caporaso-Harris in their paper [3]; we have followed that notation in Table 2.
Notice that the values in the last row of Table 1 and 2 are in agreement.

Next, the value of the following numbers

$\displaystyle\big{[}\mathsf{T}_{k_{1}}\ldots\mathsf{T}_{k_{n}}\big{]}\cdot
y_{1}^{2}y_{d}^{\delta_{d}-k-1}a_{1},$

will be displayed for certain values. This corresponds to fixing the line
(intersecting with $y_{1}^{2}$) and fixing the location of the first tangency
point (intersecting with $a_{1}$). The remaining tangency points are free. The
values obtained by using the theorems of Section 5 are

$d$ | $7$ | $8$ | $9$
---|---|---|---
$\mu$ | $y_{1}^{2}y_{7}^{30}a_{1}$ | $y_{1}^{2}y_{8}^{39}a_{1}$ | $y_{1}^{2}y_{9}^{49}a_{1}$
$\big{[}\mathsf{T}_{1}\mathsf{T}_{1}\mathsf{T}_{2}\big{]}\cdot\mu$ | $12$ | $36$ | $72$

Table 3.

Note that in this case, there is no significance of dividing by $2$, because
the two $\mathsf{T}_{1}$ points are different; the first $\mathsf{T}_{1}$
point lies on a fixed point, while the second $\mathsf{T}_{1}$ is free.

The corresponding numbers from Caporaso-Harris by setting

$\delta:=0,\qquad\alpha:=(0,1)\qquad\textnormal{and}\qquad\beta:=(i,1,1),~{}~{}\forall~{}i=0,1,2.$

are as follows:

$d$ | $7$ | $8$ | $9$
---|---|---|---
$N^{d,\delta}(\alpha,\beta)$ | $12$ | $36$ | $72$

Table 4.

The values in the last row of Table 3 and 4 are in agreement.

###  10.2. Counting one nodal curves with tangencies: confirmation with
Caporaso-Harris

Using the theorems of Section 6, the following numbers

$\displaystyle\big{[}\mathsf{A}_{1}^{\mathsf{F}}\mathsf{T}_{k_{1}}\ldots\mathsf{T}_{k_{n}}\big{]}\cdot
y_{1}^{2}y_{d}^{\delta_{d}-k-1}.$

can be computed. A few values are tabulated below.

$d$ | $7$ | $8$
---|---|---
$\mu$ | $y_{1}^{2}y_{7}^{30}$ | $y_{1}^{2}y_{8}^{39}$
$\frac{1}{2}\big{[}\mathsf{A}_{1}^{\mathsf{F}}\mathsf{T}_{1}\mathsf{T}_{1}\mathsf{T}_{2}\big{]}\cdot\mu$ | $3420$ | $19404$

Table 5.

The corresponding numbers from Caporaso-Harris by setting

$\displaystyle\begin{split}&\delta:=1,\qquad\alpha:=(0),\qquad\qquad\beta:=(0,2,1)\quad\textnormal{and}\\\
&\delta:=1,\qquad\alpha:=(0),\qquad\qquad\beta:=(1,2,1)\end{split}$

are as follows:

$d$ | $7$ | $8$
---|---|---
$N^{d,\delta}(\alpha,\beta)$ | $3420$ | $19404$

Table 6.

The values in the last row of Table 5 and 6 are in agreement.

Next, the value of the following numbers

$\displaystyle\big{[}\mathsf{A}_{1}^{\mathsf{F}}\mathsf{T}_{k_{1}}\ldots\mathsf{T}_{k_{n}}\big{]}\cdot
y_{1}^{2}y_{d}^{\delta_{d}-k-2}a_{1},$

will be displayed for certain values. This corresponds to fixing the line
(intersecting with $y_{1}^{2}$) and fixing the location of the first tangency
point (intersecting with $a_{1}$). The remaining tangency points are free. The
values obtained by using the theorems of Section 5 are

$d$ | $8$
---|---
$\mu$ | $y_{1}^{2}y_{8}^{38}a_{1}$
$\big{[}\mathsf{A}_{1}^{\mathsf{F}}\mathsf{T}_{1}\mathsf{T}_{1}\mathsf{T}_{2}\big{]}\cdot\mu$ | $4912$

Table 7.

The corresponding numbers from Caporaso-Harris by setting

$\delta:=1,\qquad\alpha:=(0,1)\qquad\textnormal{and}\qquad\beta:=(1,1,1)$

are as follows:

$d$ | $8$
---|---
$N^{d,\delta}(\alpha,\beta)$ | $4912$

Table 8.

The values in the last row of Table 7 and 8 are in agreement.

###  10.3. Counting one cuspidal cubics with tangencies: confirmation with
Ernström-Kennedy

Using the theorems of Section 7, one obtains that for $d\,=\,3$,

$\displaystyle\big{[}\mathsf{A}_{2}^{\mathsf{F}}\mathsf{T}_{1}\big{]}\cdot
y_{1}^{2}y_{3}^{6}$ $\displaystyle\,\,=\,\,60.$

This number is equal to the number of rational cuspidal cubics passing through
$6$ generic points that is tangent to a given line. This is in agreement with
the answer obtained by Ernström and Kennedy in [4].

###  10.4. Counting stable maps with tangencies: confirmation with Gathmann

Using the results of Section 9.2, the following numbers

$\displaystyle\Big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\Big{]}\cdot
y_{1}^{2}\cdot\mathcal{H}^{3d-2-n}a_{1}^{n}$

can be computed. A few numbers are tabulated in the following two tables:

$d$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---
$\mu$ | $y_{1}^{2}\mathcal{H}^{7}$ | $y_{1}^{2}\mathcal{H}^{10}$ | $y_{1}^{2}\mathcal{H}^{13}$ | $y_{1}^{2}\mathcal{H}^{16}$ | $y_{1}^{2}\mathcal{H}^{19}$ | $y_{1}^{2}\mathcal{H}^{22}$
$\big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\big{]}\cdot\mu$ | $36$ | $2184$ | $335792$ | $106976160$ | $61739450304$ | $58749399019136$

Table 9.

$d$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---
$\mu$ | $y_{1}^{2}\mathcal{H}^{6}a_{1}$ | $y_{1}^{2}\mathcal{H}^{9}a_{1}$ | $y_{1}^{2}\mathcal{H}^{12}a_{1}$ | $y_{1}^{2}\mathcal{H}^{15}a_{1}$ | $y_{1}^{2}\mathcal{H}^{18}a_{1}$ | $y_{1}^{2}\mathcal{H}^{21}a_{1}$
$\big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\big{]}\cdot\mu$ | $10$ | $428$ | $51040$ | $13300176$ | $6498076192$ | $5362556317120$

Table 10.

These are all in agreement with the numbers computed by Gathmann’s program
GROWI (that implements the formulas in [7], [8] and [9]).

Finally, Gathmann’s approach is extend in subsection 9.1 where the line can be
varied. In particular, the following intersection numbers can be computed

$\displaystyle\Big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\Big{]}\cdot
y_{1}^{r}\cdot\mathcal{H}^{3d-n-r}\cdot a_{1}^{n}.$

A few numbers are tabulated in the following two tables:

$d$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---
$\mu$ | $y_{1}\mathcal{H}^{8}$ | $y_{1}\mathcal{H}^{11}$ | $y_{1}\mathcal{H}^{14}$ | $y_{1}\mathcal{H}^{17}$ | $y_{1}\mathcal{H}^{20}$ | $y_{1}\mathcal{H}^{23}$
$\big{[}\big{(}\mathsf{T}_{1}\big{)}_{\textnormal{st}}\big{]}\cdot\mu$ | $48$ | $3720$ | $698432$ | $263129760$ | $175401698304$ | $189360514383488$

Table 11.

These values are all in agreement with the alternative approach given in
subsection 9.2 to compute these intersection numbers.

Finally, for $d=3$ one can compute (using the Theorems of Section 6) that

$\displaystyle\big{[}\mathsf{A}_{1}^{\mathsf{F}}\mathsf{T}_{1}\big{]}\cdot
y_{1}y_{3}^{8}$ $\displaystyle=48.$

This number is in agreement with the first number tabulated in Table 11.

Finally, we tabulate the values of $\mathsf{N}_{d}^{\mathsf{T}_{1}}$ using
(9.19).

$d$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---
$\mathsf{N}_{d}^{\mathsf{T}_{1}}$ | $36$ | $2184$ | $335792$ | $106976160$ | $61739450304$ | $58749399019136$

Table 12.

These are all in agreement with the numbers computed by Gathmann’s program
GROWI.

##  11\. Acknowledgement

We are grateful to Chitrabhanu Chaudhuri for several useful discussions
related to this paper. We also thank Soumya Pal for writing a Python program
to implement Caporaso-Harris formula for verification.

## References

  * [1] S. Basu and R. Mukherjee, Enumeration of curves with one singular point, J. Geom. Phys., 104 (2016), pp. 175–203.
  * [2] I. Biswas, C. Chaudhuri, A. Choudhury, R. Mukherjee, and A. Paul, Counting rational curves with an $m$-fold point, Adv Math, 431 (2023).
  * [3] L. Caporaso and J. Harris, Counting plane curves of any genus, Invent. Math., 131 (1998), pp. 345–392.
  * [4] L. Ernström and G. Kennedy, Recursive formulas for the characteristic numbers of rational plane curves, J. Algebraic Geom., 7 (1998), pp. 141–181.
  * [5] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Algebraic geometry—Santa Cruz 1995, vol. 62 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1997, pp. 45–96.
  * [6] A. Gathmann, Gromov-witten invariants of blow-ups, J. Algebraic Geom., 10 (2001), pp. 399–432.
  * [7] , Gromov-witten invariants of hypersurfaces, (2003).  Habilitationsschrift for Technischen Universität Kaiserslautern.
  * [8] , Relative Gromov-Witten invariants and the mirror formula, Math. Ann., 325 (2003), pp. 393–412.
  * [9] , The number of plane conics that are five-fold tangent to a given curve, Compos. Math., 141 (2005), pp. 487–501.
  * [10] E. Ionel, Genus one enumerative invariants in $\mathbb{P}^{2}$, Duke Math Jour, 94 (1998), pp. 279–324.
  * [11] D. McDuff and D. Salamon, $J$-holomorphic curves and symplectic topology, vol. 52 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, second ed., 2012.
  * [12] D. McDuff and K. Siegel, Counting curves with local tangency constraints, Journal of Topology, 14 (2021), pp. 1176–1242.
  * [13] A. Paul, Counting singular curves with tangencies.  https://arxiv.org/pdf/1909.03201.pdf, 2019.
  * [14] A. Paul, Enumeration of singular curves with prescribed tangencies, 2021\.  Thesis (Ph.D.)–National Institue of Science Education and Research.


possesses an approximate symmetry such that varying the dark-matter mass
$m_{\chi}$ amounts to rigidly translating the transfer function along the
$\log k$-axis. To be concrete, the translation operation can be expressed as

$\displaystyle m_{\chi}\to
m^{\prime}_{\chi}\,,~{}~{}~{}\alpha(m_{\chi})\to\alpha^{\prime}=\alpha(m^{\prime}_{\chi})\,,{}$
$\displaystyle k\to
k^{\prime}=\frac{\alpha}{\alpha^{\prime}}k\,,~{}~{}~{}T^{2}(k)\to{T^{\prime}}^{2}(k^{\prime})=T^{2}(k)\,.$
(B.1)

In this appendix, we aim to show that the underlying reason for this symmetry
is due to the fact that, the normalized momentum distributions
$g_{p}(p)/(N_{\chi})$ resulting from freeze-out (or freeze-in) are
approximately the same for different $m_{\chi}$, as we have seen in Fig.
4.666Since we shall discuss distribution functions in different spaces in this
appendix, we reserve the subscript to indicate the corresponding phase space,
such as momentum space, velocity space and wavenumber space.

We first notice that the velocity distributions $g_{v}(v)\equiv
g_{p}(p)\left|d\log p/d\log v\right|\approx g_{p}(p)$ are identical to each
other (after normalization by the area under the curve) up to some rigid
translation along the $\log v$-axis when the distributions only have non-
vanishing support for non-relativistic velocities. In particular, changing
$m_{\chi}$ to $m^{\prime}_{\chi}$ amounts to translating the velocity
distribution as

$\displaystyle v\to
v^{\prime}=v\frac{m_{\chi}}{m^{\prime}_{\chi}}\,,~{}~{}~{}g_{v}(v)\to
g^{\prime}_{v}(v^{\prime})=g_{v}(v)\,.$ (B.2)

On the other hand, the velocity distribution can be mapped into a distribution
$g_{k}(k)$ in the space of wavenumbers by mapping the velocity $v$ to the
wavenumber $k$ using the freestreaming horizon $k_{\rm hor}$ [64, 65]:

$\displaystyle k=k_{\rm hor}(v)=\xi\left[\int_{a_{\rm
prod}}^{1}\frac{da}{Ha^{2}}\frac{\gamma
v}{\sqrt{\gamma^{2}v^{2}+a^{2}}}\right]^{-1}\,,$ (B.3)

where $\xi$ is an $\mathcal{O}(1)$ constant factor. One can then define a
distribution function in the $\log k$ space

$g_{k}(k)\equiv J(v)g_{v}(v)\,,$ (B.4)

in which $J(v)\equiv\left|d\log v/d\log k\right|$ is the Jacobian. With these,
the translation in the $v$-space can be mapped to a translation in the
$k$-space which goes like

$\displaystyle k=k_{\rm hor}(v)\to k^{\prime}=k_{\rm
hor}\left(v^{\prime}\right)\,,~{}~{}~{}g_{k}(k)\to
J(v^{\prime})g^{\prime}_{v}(v^{\prime})=g_{k}^{\prime}(k^{\prime})\,.$ (B.5)

In the limit $v/a_{\rm MRE}\ll 1$, one can show numerically that since
$\log(k^{\prime}/k)$ is approximately a constant as $v$ changes, the Jacobian
$J$ is also approximately a constant. Therefore,
$g^{\prime}_{k}(k^{\prime})\approx J(v)g_{v}(v)=g_{k}(k)$, $i.e.,$ the
transformation in the $k$-space is also an approximately rigid translation.
Approximating $J$ as a true constant, we can write

$\log k^{\prime}-\log k=(\log v-\log
v^{\prime})/J=\log\left(\frac{m^{\prime}_{\chi}}{m_{\chi}}\right)^{1/J}\,,$
(B.6)

which means $k^{\prime}=\zeta k$ with
$\zeta=(m^{\prime}_{\chi}/m_{\chi})^{1/J}$.

The next step is to figure out how the squared transfer function transform. In
Ref. [64], it has been shown that the logarithmic derivative of the transfer
function at a wavenumber $k$ is correlated with the fraction of dark-matter
particles that are able to freestream a distance larger than $\sim 1/k$. In
particular, it is shown that the relation

$\frac{d\log T^{2}}{d\log k}\approx F^{2}(k)+\frac{3}{2}F(k)$ (B.7)

holds to high accuracy, where

$F(k)~{}\equiv~{}\frac{\int_{\infty}^{\log k}d\log
k^{\prime}\,g_{k}(k^{\prime})}{\int_{-\infty}^{\infty}d\log
k^{\prime}\,g_{k}(k^{\prime})}$ (B.8)

is called the _hot-fraction function_. Integrating Eq. (B.7), it is
straightforward to show that

$T^{2}(k)\approx\exp\left\\{-\int_{-\infty}^{\log k}d\log
k^{\prime}\left[F^{2}(k^{\prime})+\frac{3}{2}F(k^{\prime})\right]\right\\}\,,$
(B.9)

where we have used the fact that $\lim_{k\to 0}\log T^{2}(k)=0$. Since $F(k)$
is the integral of $g_{k}(k)$, it is easy to verify that the $k$-space
transformation implies

$\displaystyle k\to k^{\prime}\,,~{}~{}~{}F(k)\to
F^{\prime}(k^{\prime})=F(k)\,.$ (B.10)

Therefore, the squared transfer function transforms as

$\displaystyle T^{2}(k)$ $\displaystyle\to$
$\displaystyle{T^{\prime}}^{2}(k^{\prime}){}$ (B.11) $\displaystyle=$
$\displaystyle\exp\left\\{-\int_{-\infty}^{\log k^{\prime}}d\log
k^{\prime\prime}\left[{F^{\prime}}^{2}(k^{\prime\prime})+\frac{3}{2}F^{\prime}(k^{\prime\prime})\right]\right\\}{}$
$\displaystyle=$ $\displaystyle\exp\left\\{-\int_{-\infty}^{\log
k}d\log(\zeta^{-1}k^{\prime\prime})\left[F^{2}(\zeta^{-1}k^{\prime\prime})+\frac{3}{2}F(\zeta^{-1}k^{\prime\prime})\right]\right\\}{}$
$\displaystyle\approx$ $\displaystyle T^{2}(k)\,,$

where we have used $k=\zeta^{-1}k^{\prime}$ in the third line. The entire
transformation then goes like

$\displaystyle m_{\chi}\to m^{\prime}_{\chi}\,,~{}~{}~{}v\to
v^{\prime}=\frac{m_{\chi}}{m^{\prime}_{\chi}}v\,,~{}~{}~{}g_{v}(v)\to
g^{\prime}_{v}(v^{\prime})=g_{v}(v)\,,{}$ $\displaystyle k\to k^{\prime}=\zeta
k\,,~{}~{}~{}g_{k}(k)\to
g_{k}^{\prime}(k^{\prime})=g_{k}(k)\,,~{}~{}~{}T^{2}(k)\to{T^{\prime}}^{2}(k^{\prime})=T^{2}(k)\,.$
(B.12)

Noticing that $J\approx 1.1$ for $k\in[1,10^{3}]~{}h/\rm Mpc$ from numerical
result, it is therefore reasonable to identify
$\zeta=(m^{\prime}_{\chi}/m_{\chi})^{1/J}$ with
$\alpha(m_{\chi})/\alpha(m^{\prime}_{\chi})$. Thus, we have approximately
proved that the underlying reason for the translational symmetry is the
invariance of $g_{p}(p)$ (which amounts to a rigid shift of $g_{v}(v)$ in the
$\log v$ space) under the change of $m_{\chi}$ as we have seen in Fig. 4.

## References

  * [1] F. Zwicky, _Die Rotverschiebung von extragalaktischen Nebeln_ , _Helv. Phys. Acta_ 6 (1933) 110.
  * [2] F. Zwicky, _On the Masses of Nebulae and of Clusters of Nebulae_ , _Astrophys. J._ 86 (1937) 217.
  * [3] V. C. Rubin and W. K. Ford, Jr., _Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions_ , _Astrophys. J._ 159 (1970) 379.
  * [4] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [5] J. A. Tyson, R. A. Wenk and F. Valdes, _Detection of systematic gravitational lens galaxy image alignments - Mapping dark matter in galaxy clusters_ , _Astrophys. J. Lett._ 349 (1990) L1.
  * [6] D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones et al., _A direct empirical proof of the existence of dark matter_ , _Astrophys. J. Lett._ 648 (2006) L109 [astro-ph/0608407].
  * [7] P. J. E. Peebles, _PRIMEVAL ADIABATIC PERTURBATIONS: EFFECT OF MASSIVE NEUTRINOS_ , _Astrophys. J._ 258 (1982) 415.
  * [8] J. R. Bond and A. S. Szalay, _The Collisionless Damping of Density Fluctuations in an Expanding Universe_ , _Astrophys. J._ 274 (1983) 443.
  * [9] S. D. M. White, C. S. Frenk and M. Davis, _Clustering in a Neutrino Dominated Universe_ , _Astrophys. J. Lett._ 274 (1983) L1.
  * [10] L. Gao, V. Springel and S. D. M. White, _The Age dependence of halo clustering_ , _Mon. Not. Roy. Astron. Soc._ 363 (2005) L66 [astro-ph/0506510].
  * [11] V. Springel et al., _Simulating the joint evolution of quasars, galaxies and their large-scale distribution_ , _Nature_ 435 (2005) 629 [astro-ph/0504097].
  * [12] V. Springel, C. S. Frenk and S. D. M. White, _The large-scale structure of the Universe_ , _Nature_ 440 (2006) 1137 [astro-ph/0604561].
  * [13] D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de Naray and A. H. G. Peter, _Cold dark matter: controversies on small scales_ , _Proc. Nat. Acad. Sci._ 112 (2015) 12249 [1306.0913].
  * [14] A. Del Popolo and M. Le Delliou, _Small scale problems of the $\Lambda$CDM model: a short review_, _Galaxies_ 5 (2017) 17 [1606.07790].
  * [15] S. Tulin and H.-B. Yu, _Dark Matter Self-interactions and Small Scale Structure_ , _Phys. Rept._ 730 (2018) 1 [1705.02358].
  * [16] J. S. Bullock and M. Boylan-Kolchin, _Small-Scale Challenges to the $\Lambda$CDM Paradigm_, _Ann. Rev. Astron. Astrophys._ 55 (2017) 343 [1707.04256].
  * [17] L. Perivolaropoulos and F. Skara, _Challenges for $\Lambda$CDM: An update_, _New Astron. Rev._ 95 (2022) 101659 [2105.05208].
  * [18] E. D. Carlson, M. E. Machacek and L. J. Hall, _Self-interacting dark matter_ , _Astrophys. J._ 398 (1992) 43.
  * [19] D. N. Spergel and P. J. Steinhardt, _Observational evidence for selfinteracting cold dark matter_ , _Phys. Rev. Lett._ 84 (2000) 3760 [astro-ph/9909386].
  * [20] R. Dave, D. N. Spergel, P. J. Steinhardt and B. D. Wandelt, _Halo properties in cosmological simulations of selfinteracting cold dark matter_ , _Astrophys. J._ 547 (2001) 574 [astro-ph/0006218].
  * [21] M. Rocha, A. H. G. Peter, J. S. Bullock, M. Kaplinghat, S. Garrison-Kimmel, J. Onorbe et al., _Cosmological Simulations with Self-Interacting Dark Matter I: Constant Density Cores and Substructure_ , _Mon. Not. Roy. Astron. Soc._ 430 (2013) 81 [1208.3025].
  * [22] A. H. G. Peter, M. Rocha, J. S. Bullock and M. Kaplinghat, _Cosmological Simulations with Self-Interacting Dark Matter II: Halo Shapes vs. Observations_ , _Mon. Not. Roy. Astron. Soc._ 430 (2013) 105 [1208.3026].
  * [23] M. Kaplinghat, S. Tulin and H.-B. Yu, _Dark Matter Halos as Particle Colliders: Unified Solution to Small-Scale Structure Puzzles from Dwarfs to Clusters_ , _Phys. Rev. Lett._ 116 (2016) 041302 [1508.03339].
  * [24] R. Garani, M. Redi and A. Tesi, _Dark matter self-interactions in the matter power spectrum_ , _JCAP_ 07 (2022) 012 [2201.06551].
  * [25] W. Hu, R. Barkana and A. Gruzinov, _Cold and fuzzy dark matter_ , _Phys. Rev. Lett._ 85 (2000) 1158 [astro-ph/0003365].
  * [26] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, _String Axiverse_ , _Phys. Rev. D_ 81 (2010) 123530 [0905.4720].
  * [27] L. Hui, J. P. Ostriker, S. Tremaine and E. Witten, _Ultralight scalars as cosmological dark matter_ , _Phys. Rev. D_ 95 (2017) 043541 [1610.08297].
  * [28] V. Iršič, M. Viel, M. G. Haehnelt, J. S. Bolton and G. D. Becker, _First constraints on fuzzy dark matter from Lyman- $\alpha$ forest data and hydrodynamical simulations_, _Phys. Rev. Lett._ 119 (2017) 031302 [1703.04683].
  * [29] E. G. M. Ferreira, _Ultra-light dark matter_ , _Astron. Astrophys. Rev._ 29 (2021) 7 [2005.03254].
  * [30] L. Hui, _Wave Dark Matter_ , _Ann. Rev. Astron. Astrophys._ 59 (2021) 247 [2101.11735].
  * [31] R. Cen, _Decaying cold dark matter model and small-scale power_ , _Astrophys. J. Lett._ 546 (2001) L77 [astro-ph/0005206].
  * [32] M. Kaplinghat, _Dark matter from early decays_ , _Phys. Rev. D_ 72 (2005) 063510 [astro-ph/0507300].
  * [33] J. A. R. Cembranos, J. L. Feng, A. Rajaraman and F. Takayama, _SuperWIMP solutions to small scale structure problems_ , _Phys. Rev. Lett._ 95 (2005) 181301 [hep-ph/0507150].
  * [34] M. Abdelqader and F. Melia, _Decaying Dark Matter and the Deficit of Dwarf Haloes_ , _Mon. Not. Roy. Astron. Soc._ 388 (2008) 1869 [0806.0602].
  * [35] A. H. G. Peter, _Mapping the allowed parameter space for decaying dark matter models_ , _Phys. Rev. D_ 81 (2010) 083511 [1001.3870].
  * [36] A. H. G. Peter, C. E. Moody and M. Kamionkowski, _Dark-Matter Decays and Self-Gravitating Halos_ , _Phys. Rev. D_ 81 (2010) 103501 [1003.0419].
  * [37] A. H. G. Peter and A. J. Benson, _Dark-matter decays and Milky Way satellite galaxies_ , _Phys. Rev. D_ 82 (2010) 123521 [1009.1912].
  * [38] N. F. Bell, A. J. Galea and K. Petraki, _Lifetime Constraints for Late Dark Matter Decay_ , _Phys. Rev. D_ 82 (2010) 023514 [1004.1008].
  * [39] M.-Y. Wang and A. R. Zentner, _Effects of Unstable Dark Matter on Large-Scale Structure and Constraints from Future Surveys_ , _Phys. Rev. D_ 85 (2012) 043514 [1201.2426].
  * [40] M.-Y. Wang, R. A. C. Croft, A. H. G. Peter, A. R. Zentner and C. W. Purcell, _Lyman- $\alpha$ forest constraints on decaying dark matter_, _Phys. Rev. D_ 88 (2013) 123515 [1309.7354].
  * [41] M.-Y. Wang, A. H. G. Peter, L. E. Strigari, A. R. Zentner, B. Arant, S. Garrison-Kimmel et al., _Cosmological simulations of decaying dark matter: implications for small-scale structure of dark matter haloes_ , _Mon. Not. Roy. Astron. Soc._ 445 (2014) 614 [1406.0527].
  * [42] B. Audren, J. Lesgourgues, G. Mangano, P. D. Serpico and T. Tram, _Strongest model-independent bound on the lifetime of Dark Matter_ , _JCAP_ 12 (2014) 028 [1407.2418].
  * [43] G. Blackadder and S. M. Koushiappas, _Cosmological constraints to dark matter with two- and many-body decays_ , _Phys. Rev. D_ 93 (2016) 023510 [1510.06026].
  * [44] V. Poulin, P. D. Serpico and J. Lesgourgues, _A fresh look at linear cosmological constraints on a decaying dark matter component_ , _JCAP_ 08 (2016) 036 [1606.02073].
  * [45] K. K. Boddy, M. Kaplinghat, A. Kwa and A. H. G. Peter, _Hidden Sector Hydrogen as Dark Matter: Small-scale Structure Formation Predictions and the Importance of Hyperfine Interactions_ , _Phys. Rev. D_ 94 (2016) 123017 [1609.03592].
  * [46] G. Franco Abellán, R. Murgia, V. Poulin and J. Lavalle, _Implications of the $S_{8}$ tension for decaying dark matter with warm decay products_, _Phys. Rev. D_ 105 (2022) 063525 [2008.09615].
  * [47] DES collaboration, _Milky Way Satellite Census. IV. Constraints on Decaying Dark Matter from Observations of Milky Way Satellite Galaxies_ , _Astrophys. J._ 932 (2022) 128 [2201.11740].
  * [48] S. Colombi, S. Dodelson and L. M. Widrow, _Large scale structure tests of warm dark matter_ , _Astrophys. J._ 458 (1996) 1 [astro-ph/9505029].
  * [49] P. Colin, V. Avila-Reese and O. Valenzuela, _Substructure and halo density profiles in a warm dark matter cosmology_ , _Astrophys. J._ 542 (2000) 622 [astro-ph/0004115].
  * [50] P. Bode, J. P. Ostriker and N. Turok, _Halo formation in warm dark matter models_ , _Astrophys. J._ 556 (2001) 93 [astro-ph/0010389].
  * [51] V. Avila-Reese, P. Colin, O. Valenzuela, E. D’Onghia and C. Firmani, _Formation and structure of halos in a warm dark matter cosmology_ , _Astrophys. J._ 559 (2001) 516 [astro-ph/0010525].
  * [52] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese and A. Riotto, _Constraining warm dark matter candidates including sterile neutrinos and light gravitinos with WMAP and the Lyman-alpha forest_ , _Phys. Rev. D_ 71 (2005) 063534 [astro-ph/0501562].
  * [53] K. Abazajian, _Linear cosmological structure limits on warm dark matter_ , _Phys. Rev. D_ 73 (2006) 063513 [astro-ph/0512631].
  * [54] A. Boyarsky, J. Lesgourgues, O. Ruchayskiy and M. Viel, _Lyman-alpha constraints on warm and on warm-plus-cold dark matter models_ , _JCAP_ 05 (2009) 012 [0812.0010].
  * [55] M. R. Lovell, C. S. Frenk, V. R. Eke, A. Jenkins, L. Gao and T. Theuns, _The properties of warm dark matter haloes_ , _Mon. Not. Roy. Astron. Soc._ 439 (2014) 300 [1308.1399].
  * [56] A. Schneider, R. E. Smith and D. Reed, _Halo Mass Function and the Free Streaming Scale_ , _Mon. Not. Roy. Astron. Soc._ 433 (2013) 1573 [1303.0839].
  * [57] R. Kennedy, C. Frenk, S. Cole and A. Benson, _Constraining the warm dark matter particle mass with Milky Way satellites_ , _Mon. Not. Roy. Astron. Soc._ 442 (2014) 2487 [1310.7739].
  * [58] A. Schneider, _Structure formation with suppressed small-scale perturbations_ , _Mon. Not. Roy. Astron. Soc._ 451 (2015) 3117 [1412.2133].
  * [59] V. Iršič et al., _New Constraints on the free-streaming of warm dark matter from intermediate and small scale Lyman- $\alpha$ forest data_, _Phys. Rev. D_ 96 (2017) 023522 [1702.01764].
  * [60] J. Baur, N. Palanque-Delabrouille, C. Yeche, A. Boyarsky, O. Ruchayskiy, E. Armengaud et al., _Constraints from Ly- $\alpha$ forests on non-thermal dark matter including resonantly-produced sterile neutrinos_, _JCAP_ 12 (2017) 013 [1706.03118].
  * [61] J. König, A. Merle and M. Totzauer, _keV Sterile Neutrino Dark Matter from Singlet Scalar Decays: The Most General Case_ , _JCAP_ 11 (2016) 038 [1609.01289].
  * [62] R. Murgia, A. Merle, M. Viel, M. Totzauer and A. Schneider, _”Non-cold” dark matter at small scales: a general approach_ , _JCAP_ 11 (2017) 046 [1704.07838].
  * [63] F. Huang, _Cosmological Aspects of Non-Minimal Dark Sectors_ , Ph.D. thesis, Arizona U., 2019.
  * [64] K. R. Dienes, F. Huang, J. Kost, S. Su and B. Thomas, _Deciphering the archaeological record: Cosmological imprints of nonminimal dark sectors_ , _Phys. Rev. D_ 101 (2020) 123511 [2001.02193].
  * [65] K. R. Dienes, F. Huang, J. Kost, K. Manogue and B. Thomas, _Extracting dark-matter velocities from halo masses: A reconstruction conjecture_ , _Phys. Rev. D_ 106 (2022) 083506 [2101.10337].
  * [66] K. R. Dienes, F. Huang, J. Kost, B. Thomas and H.-B. Yu, _Evaluating Lyman- $\alpha$ constraints for general dark-matter velocity distributions: Multiple scales and cautionary tales_, _Phys. Rev. D_ 106 (2022) 123521 [2112.09105].
  * [67] Q. Decant, J. Heisig, D. C. Hooper and L. Lopez-Honorez, _Lyman- $\alpha$ constraints on freeze-in and superWIMPs_, _JCAP_ 03 (2022) 041 [2111.09321].
  * [68] A. Schneider, _Astrophysical constraints on resonantly produced sterile neutrino dark matter_ , _JCAP_ 04 (2016) 059 [1601.07553].
  * [69] A. Kamada and K. Yanagi, _Constraining FIMP from the structure formation of the Universe: analytic mapping from $m_{WDM}$_, _JCAP_ 11 (2019) 029 [1907.04558].
  * [70] D. C. Hooper, N. Schöneberg, R. Murgia, M. Archidiacono, J. Lesgourgues and M. Viel, _One likelihood to bind them all: Lyman- $\alpha$ constraints on non-standard dark matter_, _JCAP_ 10 (2022) 032 [2206.08188].
  * [71] R. Murgia, V. Iršič and M. Viel, _Novel constraints on noncold, nonthermal dark matter from Lyman- $\alpha$ forest data_, _Phys. Rev. D_ 98 (2018) 083540 [1806.08371].
  * [72] J. McDonald, _Thermally generated gauge singlet scalars as selfinteracting dark matter_ , _Phys. Rev. Lett._ 88 (2002) 091304 [hep-ph/0106249].
  * [73] L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, _Freeze-In Production of FIMP Dark Matter_ , _JHEP_ 03 (2010) 080 [0911.1120].
  * [74] F. Elahi, C. Kolda and J. Unwin, _UltraViolet Freeze-in_ , _JHEP_ 03 (2015) 048 [1410.6157].
  * [75] N. Bernal, M. Heikinheimo, T. Tenkanen, K. Tuominen and V. Vaskonen, _The Dawn of FIMP Dark Matter: A Review of Models and Constraints_ , _Int. J. Mod. Phys. A_ 32 (2017) 1730023 [1706.07442].
  * [76] C. Cheung, G. Elor, L. J. Hall and P. Kumar, _Origins of Hidden Sector Dark Matter I: Cosmology_ , _JHEP_ 03 (2011) 042 [1010.0022].
  * [77] C. Cheung, G. Elor, L. J. Hall and P. Kumar, _Origins of Hidden Sector Dark Matter II: Collider Physics_ , _JHEP_ 03 (2011) 085 [1010.0024].
  * [78] X. Chu, T. Hambye and M. H. G. Tytgat, _The Four Basic Ways of Creating Dark Matter Through a Portal_ , _JCAP_ 05 (2012) 034 [1112.0493].
  * [79] N. Bernal, X. Chu, C. Garcia-Cely, T. Hambye and B. Zaldivar, _Production Regimes for Self-Interacting Dark Matter_ , _JCAP_ 03 (2016) 018 [1510.08063].
  * [80] G. Krnjaic, _Freezing In, Heating Up, and Freezing Out: Predictive Nonthermal Dark Matter and Low-Mass Direct Detection_ , _JHEP_ 10 (2018) 136 [1711.11038].
  * [81] J. Berger, D. Croon, S. El Hedri, K. Jedamzik, A. Perko and D. G. E. Walker, _Dark matter amnesia in out-of-equilibrium scenarios_ , _JCAP_ 02 (2019) 051 [1812.08795].
  * [82] J. A. Evans, C. Gaidau and J. Shelton, _Leak-in Dark Matter_ , _JHEP_ 01 (2020) 032 [1909.04671].
  * [83] Y. Du, F. Huang, H.-L. Li and J.-H. Yu, _Freeze-in Dark Matter from Secret Neutrino Interactions_ , _JHEP_ 12 (2020) 207 [2005.01717].
  * [84] A. Hryczuk and M. Laletin, _Dark matter freeze-in from semi-production_ , _JHEP_ 06 (2021) 026 [2104.05684].
  * [85] P. Ghosh, P. Konar, A. K. Saha and S. Show, _Self-interacting freeze-in dark matter in a singlet doublet scenario_ , _JCAP_ 10 (2022) 017 [2112.09057].
  * [86] J. Heeck and D. Teresi, _Cold keV dark matter from decays and scatterings_ , _Phys. Rev. D_ 96 (2017) 035018 [1706.09909].
  * [87] S. Boulebnane, J. Heeck, A. Nguyen and D. Teresi, _Cold light dark matter in extended seesaw models_ , _JCAP_ 04 (2018) 006 [1709.07283].
  * [88] K. J. Bae, A. Kamada, S. P. Liew and K. Yanagi, _Light axinos from freeze-in: production processes, phase space distributions, and Ly- $\alpha$ forest constraints_, _JCAP_ 01 (2018) 054 [1707.06418].
  * [89] F. D’Eramo and A. Lenoci, _Lower mass bounds on FIMP dark matter produced via freeze-in_ , _JCAP_ 10 (2021) 045 [2012.01446].
  * [90] A. Berlin and N. Blinov, _Thermal neutrino portal to sub-MeV dark matter_ , _Phys. Rev. D_ 99 (2019) 095030 [1807.04282].
  * [91] A. Berlin, N. Blinov and S. W. Li, _Dark Sector Equilibration During Nucleosynthesis_ , _Phys. Rev. D_ 100 (2019) 015038 [1904.04256].
  * [92] J. Lesgourgues, _The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview_ , 1104.2932.
  * [93] J. Lesgourgues and T. Tram, _The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics_ , _JCAP_ 09 (2011) 032 [1104.2935].
  * [94] Y. Du, F. Huang, H.-L. Li, Y.-Z. Li and J.-H. Yu, _Revisiting dark matter freeze-in and freeze-out through phase-space distribution_ , _JCAP_ 04 (2022) 012 [2111.01267].
  * [95] K. Griest and M. Kamionkowski, _Unitarity Limits on the Mass and Radius of Dark Matter Particles_ , _Phys. Rev. Lett._ 64 (1990) 615.
  * [96] C.-P. Ma and E. Bertschinger, _Cosmological perturbation theory in the synchronous and conformal Newtonian gauges_ , _Astrophys. J._ 455 (1995) 7 [astro-ph/9506072].
  * [97] W. H. Press and P. Schechter, _Formation of galaxies and clusters of galaxies by selfsimilar gravitational condensation_ , _Astrophys. J._ 187 (1974) 425.
  * [98] A. Lapi and L. Danese, _A Stochastic Theory of the Hierarchical Clustering I. Halo Mass Function_ , _Astrophys. J._ 903 (2020) 117 [2009.07023].
  * [99] Z. J. Xu, _Dark matter halo mass functions and density profiles from mass and energy cascade_ , _Sci. Rep._ 13 (2023) 16531 [2210.01200].
  * [100] M. R. Lovell, _The halo mass function in alternative dark matter models_ , _Mon. Not. Roy. Astron. Soc._ 493 (2020) L11 [1911.11785].
  * [101] M. R. Lovell, _Towards a general parametrization of the warm dark matter halo mass function_ , 2003.01125.
  * [102] J. R. Bond, S. Cole, G. Efstathiou and N. Kaiser, _Excursion set mass functions for hierarchical Gaussian fluctuations_ , _Astrophys. J._ 379 (1991) 440.
  * [103] C. G. Lacey and S. Cole, _Merger rates in hierarchical models of galaxy formation_ , _Mon. Not. Roy. Astron. Soc._ 262 (1993) 627.
  * [104] R. K. Sheth and G. Tormen, _Large scale bias and the peak background split_ , _Mon. Not. Roy. Astron. Soc._ 308 (1999) 119 [astro-ph/9901122].
  * [105] R. K. Sheth, H. J. Mo and G. Tormen, _Ellipsoidal collapse and an improved model for the number and spatial distribution of dark matter haloes_ , _Mon. Not. Roy. Astron. Soc._ 323 (2001) 1 [astro-ph/9907024].
  * [106] M. Musso and R. K. Sheth, _One step beyond: The excursion set approach with correlated steps_ , _Mon. Not. Roy. Astron. Soc._ 423 (2012) L102 [1201.3876].
  * [107] A. Schneider, R. E. Smith, A. V. Maccio and B. Moore, _Nonlinear Evolution of Cosmological Structures in Warm Dark Matter Models_ , _Mon. Not. Roy. Astron. Soc._ 424 (2012) 684 [1112.0330].
  * [108] C. Giocoli, L. Pieri and G. Tormen, _Analytical Approach to Subhaloes Population in Dark Matter Haloes_ , _Mon. Not. Roy. Astron. Soc._ 387 (2008) 689 [0712.1476].
  * [109] E. Polisensky and M. Ricotti, _Constraints on the Dark Matter Particle Mass from the Number of Milky Way Satellites_ , _Phys. Rev. D_ 83 (2011) 043506 [1004.1459].
  * [110] A. Merle, A. Schneider and M. Totzauer, _Dodelson-Widrow Production of Sterile Neutrino Dark Matter with Non-Trivial Initial Abundance_ , _JCAP_ 04 (2016) 003 [1512.05369].
  * [111] M. Tegmark and M. Zaldarriaga, _Separating the early universe from the late universe: Cosmological parameter estimation beyond the black box_ , _Phys. Rev. D_ 66 (2002) 103508 [astro-ph/0207047].
  * [112] M. Tegmark and M. Zaldarriaga, _The Fast Fourier Transform Telescope_ , _Phys. Rev. D_ 79 (2009) 083530 [0805.4414].
  * [113] R. Hlozek et al., _The Atacama Cosmology Telescope: a measurement of the primordial power spectrum_ , _Astrophys. J._ 749 (2012) 90 [1105.4887].
  * [114] S. Chabanier, M. Millea and N. Palanque-Delabrouille, _Matter power spectrum: from Ly $\alpha$ forest to CMB scales_, _Mon. Not. Roy. Astron. Soc._ 489 (2019) 2247 [1905.08103].
  * [115] J. B. Muñoz, C. Dvorkin and F.-Y. Cyr-Racine, _Probing the Small-Scale Matter Power Spectrum with Large-Scale 21-cm Data_ , _Phys. Rev. D_ 101 (2020) 063526 [1911.11144].
  * [116] T. Castro, V. Marra and M. Quartin, _Constraining the halo mass function with observations_ , _Mon. Not. Roy. Astron. Soc._ 463 (2016) 1666 [1605.07548].
  * [117] M. M. Cueli, L. Bonavera, J. González-Nuevo and A. Lapi, _A direct and robust method to observationally constrain the halo mass function via the submillimeter magnification bias: Proof of concept_ , _Astron. Astrophys._ 645 (2021) A126 [2102.03890].
  * [118] M. Shirasaki, T. Ishiyama and S. Ando, _Virial Halo Mass Function in the Planck Cosmology_ , _Astrophys. J._ 922 (2021) 89 [2108.11038].
  * [119] S. Gu, M.-A. Dor, L. van Waerbeke, M. Asgari, A. Mead, T. Tröster et al., _On constraining cosmology and the halo mass function with weak gravitational lensing_ , _Mon. Not. Roy. Astron. Soc._ 525 (2023) 4871 [2302.00780].
  * [120] DES collaboration, _Milky Way Satellite Census. III. Constraints on Dark Matter Properties from Observations of Milky Way Satellite Galaxies_ , _Phys. Rev. Lett._ 126 (2021) 091101 [2008.00022].
  * [121] O. Newton, M. Leo, M. Cautun, A. Jenkins, C. S. Frenk, M. R. Lovell et al., _Constraints on the properties of warm dark matter using the satellite galaxies of the Milky Way_ , _JCAP_ 08 (2021) 062 [2011.08865].
  * [122] E. O. Nadler, S. Birrer, D. Gilman, R. H. Wechsler, X. Du, A. Benson et al., _Dark Matter Constraints from a Unified Analysis of Strong Gravitational Lenses and Milky Way Satellite Galaxies_ , _Astrophys. J._ 917 (2021) 7 [2101.07810].
  * [123] A. Dekker, S. Ando, C. A. Correa and K. C. Y. Ng, _Warm dark matter constraints using Milky Way satellite observations and subhalo evolution modeling_ , _Phys. Rev. D_ 106 (2022) 123026 [2111.13137].


# Large Deviations of the Estimated Cumulative Hazard Rate

Niklas Hohmann GeoZentrum Nordbayern, Fachgruppe Paläoumwelt, University of
Erlangen-Nuremberg, Loewenichstr. 28, 91054 Erlangen, Germany
<EMAIL_ADDRESS>
###### Abstract

Survivorship analysis allows to statistically analyze situations that can be
modeled as waiting times to an event. These waiting times are characterized by
the cumulative hazard rate, which can be estimated by the Nelson-Aalen
estimator or diverse confidence estimators based on asymptotic statistics. To
better understand the small sample properties of these estimators, the speed
of convergence of the estimate to the exact value is examined. This is done by
deriving large deviation principles and their rate functions for the
estimators and examining their properties. It is shown that these rate
functions are asymmetric, leading to a tendency of the estimated cumulative
hazard rate to overestimate the true cumulative hazard rate. This tendency is
strongest in the cases of (1) small sample sizes and (2) low tail
probabilities. Taking this tendency into account can improve risk assessments
of rare events and of cases where only little data is available.

Keywords: Survival Analysis; Large Deviations; Nelson-Aalen estimator

##  1 Introduction

Survival analysis is the standard framework to statistically analyze waiting
times until an event occurs (Miličič, 2008). Classical applications of
survival analysis arise in epidemiology, where the event can be the recovery
or the death of a patient, and the results of the statistical analysis can
decide on the admission or nonadmission of a new medical procedure (Sasieni
and Brentnall, 2014; Kantoff et al., 2010).
A fundamental concept in characterizing the waiting time up to an event is the
cumulative hazard rate (short CH)(Aalen, Borgan, and Gjessing, 2008). It is
estimated using the Nelson-Aalen estimator and can be complemented by
different confidence intervals and bands to compensate for uncertainties of
the results (Aalen, Borgan, and Gjessing, 2008; Aalen, 1978; Bie, Borgan, and
Liestøl, 1986; Nelson, 1969, 1972).
However these confidence area estimators are based on the asymptotic
properties of the Nelson-Aalen estimator(Bie, Borgan, and Liestøl, 1986),
making it impossible to derive analytical statements regarding their
performance for small sample sizes. As a result, no reliable assessments of
the uncertainties for small samples are available, which is problematic for
studies where larger sample sizes cannot be achieved, be it because of ethical
considerations, financial reasons, or rarity.
The aim of this paper is to derive analytical expressions regarding the small
sample properties of the estimated CH to better assess its deviations from the
true CH in these cases.
For this, the pointwise speed of convergence of the estimated CH to the true
CH is examined using the theory of large deviations.

##  2 Outline

The conventions $\ln(0):=\lim_{x\downarrow 0}\ln(x)=-\infty$ and $0\ln(0):=0$
will be used throughout the paper.
Let $X,X_{1},X_{2},\dots$ be i.i.d. positive random variables, whose values
model the waiting times for an event to occur. With the survival function
$S(t):=P(X>t)$(Kleinbaum and Klein, 2010, p. 9), the cumulative hazard rate
(short CH) can be written as(Kleinbaum and Klein, 2010, p. 294)

$H(t):=-\ln(S(t)).$ (1)

It can be estimated using the $n$-th empirical CH

$H_{n}(t):=-\ln(S_{n}(t)),$ (2)

where $S_{n}(t)=\tfrac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{(t,+\infty)}(X_{i})$ is
the $n$-th empirical survival function. The aim of this paper is to examine
the pointwise speed of convergence of $H_{n}$ towards $H$.
From the theory of large deviations, it is known that for integrable i.i.d.
$(X_{i})_{i\in\mathbb{N}}$, the relation

$P\left(\tfrac{1}{n}\sum_{i=1}^{n}X_{i}\in C\right)\approx\exp(-n\inf I(C))$
(3)

holds for a large class of sets $C$(Varadhan, 2016). The function $I$ is
called a rate function and determines the speed of convergence of the averaged
random variables towards their expectation value.
In this paper, a rate function for the $H_{n}$ is derived (section 3). The
properties of this rate function are then examined to draw conclusions about
the convergence of $H_{n}$ to $H$ (section (4).
For this, fix any $\tilde{t}\in\mathbb{R}^{+}$, and define the tail
probabilities as $p:=p(\tilde{t})=P(X>\tilde{t})$. Assume that $p\in(0,1)$,
since the cases $p=0$ and $p=1$ are trivial. Define the i.i.d. random
variables

$Y_{i}=\begin{cases}1&\text{ if }X_{i}>\tilde{t}\\\ 0&\text{ if
}X_{i}\leq\tilde{t}\end{cases}\;,$ (4)

so $Y_{i}\sim\operatorname{Ber}(p)$. The behavior of $H_{n}$ at $\tilde{t}$ is
uniquely determined by the $Y_{i}$, since

$H_{n}(\tilde{t})=-\ln\left(\tfrac{1}{n}\sum_{i=1}^{n}Y_{i}\right)\;.$ (5)

##  3 Establishing the Rate Functions

First, a large deviation principle (LDP) for the $Y_{i}$ is derived. By either
Cramér’s theorem (Cramér, 1938)(Klenke, 2008, p. 508) or by Sanov’s theorem
(Sanov, 1958)(Klenke, 2008, p. 518), the series of probability measures

$(P_{n})_{n\in\mathbb{N}}=\left(\mathcal{L}\left(\tfrac{1}{n}\sum_{i=1}^{n}Y_{i}\right)\right)_{n\in\mathbb{N}}$
(6)

satisfies a large deviation principle with rate $n$ and rate function

$\tilde{I}_{p}(x)=x\ln\left(\frac{x}{p}\right)+(1-x)\ln\left(\frac{1-x}{1-p}\right)$
(7)

for $x\in[0,1]$. This is the relative entropy of two Bernoulli distributions,
one with success probability $p$ and one with success probability $x$(Klenke,
2008, p. 515).
Applying the contraction principle (Klenke, 2008, p. 518) to this LDP and the
function $f(x)=-\ln(x)$ shows that the series
$\left(\mathcal{L}\left(-\ln\left(\tfrac{1}{n}\sum_{k=1}^{n}Y_{k}\right)\right)\right)_{n\in\mathbb{N}}$
satisfies a LDP with rate $n$ and rate function

$\displaystyle I_{p}(y)$ $\displaystyle:=\tilde{I}_{p}(f^{-1}(y))$ (8)
$\displaystyle=\exp(-y)\ln\left(\frac{\exp(-y)}{p}\right)+(1-\exp(-y))\ln\left(\frac{1-\exp(-y)}{1-p}\right)$
(9)

for $y\in[0,+\infty]$. This is the rate function of $H_{n}(\tilde{t})$, and is
displayed in fig. 1 for four different values of $p$. Next, substitute
$y=z-\ln(p)$ and define the centered rate function as

$\displaystyle J_{p}(z)$ $\displaystyle:=I_{p}(z-\ln(p))$ (10)
$\displaystyle=-zp\exp(-z)+(1-p\exp(-z))\ln\left(\frac{1-p\exp(-z)}{1-p}\right)$
(11)

for $z\in[\ln(p),+\infty]$ and $p\in(0,1)$. It is defined since the main
interest of this examination is to compare the behavior of the rate functions
$I_{p}$ for different $p$ close to the pointwise limit value $-\ln(p)$ of the
empirical cumulative hazard rate. This value is shifted to the origin in the
centered rate function and therefore allows to compare the behavior of the
empirical cumulative hazard rate at fixed distances from the respective limit
values for different $p$.

##  4 Properties of the Rate Functions

###  4.1 Monotonicity in p

In this section, it is shown that $J_{p}$, taken as a function of $p$, is
strictly increasing. This implies that the speed of convergence is decreasing
as $\tilde{t}$ increases and correspondingly the tail probabilities decrease.
Without loss of generality, it is assumed that $X$ takes on every tail
probability, so $J_{p}$ is well defined for all $p\in(0,1)$.
First, the function $J_{p}$ can not be expected to be strictly increasing in
$p$ for $z=0$, since by definition $J_{p}(0)=0$ for all $p\in(0,1)$. Therefore
the case $z=0$ will be excluded.
The first derivative of $J_{p}$ with respect to $p$ is given by

$\frac{d}{dp}J_{p}(z)=-z\exp(-z)+\frac{1-\exp(-z)}{1-p}-\exp(-z)\ln\left(\frac{1-p\exp(-z)}{1-p}\right)$
(12)

and the second derivative of $J_{p}$ with respect to $p$ by

$\frac{d^{2}}{dp^{2}}J_{p}(z)=\frac{\exp(-z)(\exp(z)-1)^{2}}{(1-p)^{2}(\exp(z)-p)}\;.$
(13)

Since the inequality $\exp(z)-p>0$ holds for all $p\in(0,1)$ and all $z$ in
the domain of $J_{p}$, termwise analysis of eq. (13) shows that the second
derivative is positive for all feasible $p$, and zero only when $z=0$, which
was excluded above. This makes the second derivative strictly positive, so
$J_{p}$ is strictly convex in $p$.
If $J_{p}$ as defined in eq. (11) is taken as a function of $p$ for $p<1$, it
is well-defined and its first derivative, evaluated at $p=0$, yields

$\frac{d}{dp}J_{0}(z)=\exp(-z)(-z-1)+1>0\text{ for }z\neq 0\;.$ (14)

So $J_{p}$ is strictly convex in $p$ and its gradient at $p=0$ is strictly
positive, therefore $J_{p}$ is strictly increasing in $p$ for $p\in(0,1)$.
This shows that the rate of convergence $J_{p}(z)$ is decreasing for all $z$
as the tail probabilities $p$ decrease.

###  4.2 Asymmetry in z

First, it is shown that $J_{p}$ is not axis symmetric with respect to the
ordinate axis. As an aid, the identity

$(1-x)\ln(1-x)=-x+\sum_{k=2}^{\infty}\frac{x^{k}}{k^{2}-k}\;\text{ for }|x|<1$
(15)

is used. Splitting the fraction in the logarithm in $J_{p}$ from equation (11)
and then applying the power series from eq. (15) with $x=p\exp(-z)$ yields

$J_{p}(z)=p\exp(-z)\left[-z+\ln\left(\tfrac{1-p}{e}\right)+\sum_{k=2}^{\infty}\frac{(p\exp(-z))^{k-1}}{k^{2}-k}\right]-\ln(1-p)$
(16)

for $p\exp(-z)<1$. Therefore $J_{p}$ is asymmetric by the asymmetry of
$\exp(-z)$.
Next, the symmetry defect of $J_{p}$, given by $|J_{p}(-z)-J_{p}(z)|$, is
examined. For this, the identities

$2\sinh(z)=\exp(z)-\exp(-z)\text{ and }2\cosh(z)=\exp(z)+\exp(-z)$ (17)

are used. With the representation of $J_{p}$ in eq. (16), directly subtracting
$J_{p}(z)$ and $J_{p}(-z)$ yields

$\displaystyle\big{|}J_{p}(-z)-J_{p}(z)\big{|}$
$\displaystyle=\Big{|}\;2zp\cosh(z)+2p\ln(\tfrac{1-p}{e})\sinh(z)$ (18)
$\displaystyle\quad\quad+\sum_{k=2}^{\infty}\frac{2p^{k}\sinh(kz)}{k^{2}-k}\;\Big{|}\;.$
(19)

Using only the second order term of the sum gives the approximation for the
symmetry defect

$\big{|}J_{p}(-z)-J_{p}(z)\big{|}\approx
2p\left(z\cosh(z)+\sinh(z)\ln(\tfrac{1-p}{e})\right)+p^{2}\sinh(2z)$ (20)

for $p\in(0,1)$ and $z\in(0,-\ln(p))$. These two statements show that the rate
of convergence is not symmetric around the true cumulative hazard rate $H$,
and will always be slower from above the cumulative hazard rate than from
below.

##  5 Example

Let $X$ be a positive random variable with distribution function $F$ and
survival function $S$. Taking the tail probabilities $p$ as functions of $t$,
meaning $p=p(t)=S(t)$, the rate function $I_{p}$ in equation (9) is a function
in the variables $t$ and $y$:

$\displaystyle I(y,t)$ $\displaystyle:=I_{p(t)}(y)$ (21)
$\displaystyle=\exp(-y)\ln\left(\frac{\exp(-y)}{S(t)}\right)+(1-\exp(-y))\ln\left(\frac{1-\exp(-y)}{F(t)}\right).$
(22)

For the case where $X$ is exponentially distributed with mean
$\tfrac{1}{\lambda}=2$, the contour plot of $I(y,t)$ is displayed in fig. 2,
alongside the cumulative hazard rate. The asymmetry of the rate functions is
clearly visible.

##  6 Discussion

The main result of this paper is that the estimate of the cumulative hazard
rate (short CH) has a tendency to overestimate the true CH, which is strongest
in the cases of (1) small sample sizes or (2) small tail probabilities. This
is a direct conclusion from the asymmetry of the rate function governing the
convergence of the estimated cumulative hazard rate in combination with the
fundamental relation of the theory of large deviations given in equation (3).
The degree of this effect is determined by the tail probabilities and
therefore unique for every distribution.
It is notable that the asymmetry observed is not generated by the distribution
of the random variables, but is rooted in the asymmetry of the rate function
of the Bernoulli distribution. It is amplified by the logarithm in the
definition of the cumulative hazard rate, and unavoidable in the sense that it
naturally arises from observing whether a random variable takes on a value
over or under a given threshold.
What remains unclear is how these results translate into the standard
framework of survival analysis, i.e. censoring and using point processes
instead of i.i.d. random variables. Especially censoring might have a strong
influence on the case with low tail probabilities, since early censoring can
render long survival times (that are commonly associated with low tail
probabilities) irrelevant. Extending the results presented in this papers to
this more general setting is relevant for applications, but requires further
work.

Figure 1: The rate function $I_{p}(y)$ (thick black line) from equation (9)
for tail probabilities $p=\exp(-0.5)$ (top left), $p=\exp(-1)$ (top right),
$p=\exp(-1.5)$ (bottom left), and $p=\exp(-2)$ (bottom right). The grey square
is located at $-\ln(p)$, which is the minimum of the rate functions and the
exact value of the cumulative hazard rate at the given tail probability. The
dotted lines indicate the values $2^{i}$ for $i=0,-1,\dots,-5$.
As the tail probabilities decrease, the rate function to the right of the grey
square becomes increasingly flat, which indicates a low rate of convergence to
the cumulative hazard rate. Although the same effect can be observed on the
left side of the grey square, it is a lot weaker. The figure was generated
using R(R version 3.2.3, 2015). Figure 2: A contour plot showing lines of
equal rate of convergence (thin black lines) to a cumulative hazard rate
(thick grey dashed line). The black contour lines are determined by $I(y,t)$
from eq. (22) for the special case of an exponential distribution with mean
$\tfrac{1}{\lambda}=2$ and are lines of equal rate of convergence for the
rates $2^{i}$, where $i=2,1,0,\dots,-8$. The grey dashed line is the
cumulative hazard rate, which is a linear function through the origin with
gradient $\tfrac{1}{2}$. The four rate functions displayed in fig. 1 are
sections through this contour plot along the dotted black lines: The grey
squares in fig. 1 are the intersection of the dotted black lines and the
dashed grey cumulative hazard rate, the thin dashed lines in fig. 1 represent
some of the contour lines from the figure shown here.
As $t$ increases and the tail probabilities decrease accordingly, the lines on
the upper left half of the picture diverge faster from the cumulative hazard
rate than on the lower right half of the picture, showing that the rate of
convergence is lower on the upper left side.
Note that since the rate functions are only determined by the tail
probabilities, and all tail probabilities occur in an exponential
distribution, the corresponding contour plot for any random variable can be
derived from this plot by a transformation of the $t$-axis. The figure was
generated using R(R version 3.2.3, 2015).

## References

  * Aalen (1978) Aalen, O. 1978.  Nonparametric inference for a family of counting processes. The Annals of Statistics 6(4):701–726.
  * Aalen, Borgan, and Gjessing (2008) Aalen, O., O. Borgan, and H. Gjessing. 2008.  Survival and event history analysis: a process point of view.  New York: Springer.
  * Bie, Borgan, and Liestøl (1986) Bie, O., Ø. Borgan, and K. Liestøl. 1986.  Confidence intervals and confidence bands for the cumulative hazard rate function and their small sample properties.  Scandinavian Journal of Statistics 9(3):221–233
  * Cramér (1938) Cramér, H.  Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Scientifiques et Industrielles 736:5–23.
  * Kantoff et al. (2010) Kantoff, P. W., T. J. Schuetz, B. A. Blumenstein, L. M. Glode, D. L. Bilhartz, M. Wyand, K. Manson, D. L. Panicali, R. Laus, J. Schlom, et al. 2010.  _Overall survival analysis of a phase ii randomized controlled trial of a poxviral-based psa-targeted immunotherapy in metastatic castration-resistant prostate cancer._ Journal of Clinical Oncology 28(7):1099.
  * Kleinbaum and Klein (2010) Kleinbaum, D. G., and M. Klein. 2010.  Survival analysis.  New York: Springer.
  * Klenke (2008) Klenke, A. 2008.  Probability Theory. A Comprehensive Course.  London: Springer.
  * Miličič (2008) Miličič, B. 2008.  Survival Analysis, p. 1367–1371.  Dodrecht, Netherlands: Springer.
  * Nelson (1969) Nelson, W. (1969)  Hazard plotting for incomplete failure data. Journal of Quality Technology 1(1):27–52.
  * Nelson (1972) Nelson, W. 1972.  Theory and applications of hazard plotting for censored failure data. Technometrics 14(4):945–966.
  * R version 3.2.3 (2015) R Core Team. 2015.  R: A Language and Environment for Statistical Computing (version 3.2.3).  Vienna, Austria: R Foundation for Statistical Computing.
  * Sanov (1958) Sanov, I. N. 1958.  On the probability of large deviations of random variables.  Technical report, North Carolina State University. Dept. of Statistics.
  * Sasieni and Brentnall (2014) Sasieni, P. D., and A. R. Brentnall. 2014.  Survival Analysis, p. 1195–1239.  New York: Springer.
  * Varadhan (2016) Varadhan, S. R. S. 2016.  Large deviations. Rhode Island: American Mathematical Society.